Class 11 Archives - Noon Academy

Class 11

$\text { If } A=\left[\begin{array}{ccc} 2 & -3 & -5 \\ -1 & 4 & 5 \\ 1 & -3 & -4 \end{array}\right] \text { and } B=\left[\begin{array}{ccc} 2 & -2 & -4 \\ -1 & 3 & 4 \\ 1 & -2 & -3 \end{array}\right] \text {, shown that } A B=A \text { and } B A=B \text {. }$

CBSE, Class 12, Exercise 5C, Maths, Matrices, RS Aggarwal

Solution: We have $\boldsymbol{A}=\left(\begin{array}{ccc}\mathbf{2} & -\mathbf{3} & -\mathbf{5} \\ -\mathbf{1} & \mathbf{4} & \mathbf{5} \\ \mathbf{1} & -\mathbf{3} &...

If $A=\left[\begin{array}{rrr}1 & -3 & 2 \\ 2 & 0 & 2\end{array}\right]$ and $B=\left[\begin{array}{ccc}2 & -1 & -1 \\ 1 & 0 & -1\end{array}\right]$ , find a matrix $C$ such that $(A+B+C)$ is a zero matrix.

CBSE, Class 12, Exercise 5B, Maths, Matrices, RS Aggarwal

Solution: It is given that $A+B+C$ is zero matrix i.e $A+B+C=0$ $\begin{array}{l} {\left[\begin{array}{ccc} 1 & -3 & 2 \\ 2 & 0 & 2 \end{array}\right]+\left[\begin{array}{ccc} 2...

A ray of light incident at an angle θ on a refracting face of a prism emerges from the other face normally. If the angle of the prism is 5o and the prism is made of a material of refractive index 1.5, the angle of incidence is
a) 7.5o
b) 5o
c) 15o
d) 2.5o

CBSE, Class 12, NCERT Exemplar, Physics, Physics, Ray Optics and Optical Instruments

Answer: a) 7.5o The distance between the refracting surfaces is negligible with thin prisms, thus the prism angle (A) is very small. Because A = r1 + r2, if A is tiny, both r1 and r2 will be little...

Mark (√) against the correct answer in the following: The range of $f(x)=a^{x}$ , where $a>0$ is
A. $[-\infty, 0]$
B. $[-\infty, 0)$
C. $[0, \infty)$
D. $(0, \infty)$

CBSE, Class 12, Functions, Maths, RS Aggarwal

Solution: Option(D) is correct. $f(x)=a x$, where $a>0$ Case 1 : When $x<0$, then ax lies between $(0,1)$ Case 2 : When $x \geq 0$, then $a x \geq 1$ Union of above two cases, gives us the...

Mark (√) against the correct answer in the following: The range of $f(x)=x+\frac{1}{x}$ is
A. $[-2,2]$
B. $[2, \infty)$
C. $(-\infty,-2]$
D. none of these

CBSE, Class 12, Functions, Maths, RS Aggarwal

Solution: Option(D) is correct. $f(x)=x+\frac{1}{x}$ The range of the function can be given by putting values of $\mathrm{x}$ and find $\mathrm{y}$. $$\begin{tabular}{|l|l|} \hline $\mathrm{X}$ &...

Mark (√) against the correct answer in the following: Let $f(x)=\frac{x^{2}}{\left(1+x^{2}\right)}$ . Then, range $(f)=?$
A. $[1, \infty)$
B. $[0,1)$
C. $[-1,1]$
D. $(0,1]$

CBSE, Class 12, Functions, Maths, RS Aggarwal

Solution: Option (B) is correct. $\mathrm{f}(\mathrm{x})=\frac{x^{2}}{\left(1+x^{2}\right)}$ The range of $f(x)$ can be found out by putting $f(x)=y$ $\begin{array}{l}...

Mark (√) against the correct answer in the following: Let $\mathrm{f}(\mathrm{x})=\frac{1}{\left(1-\mathrm{x}^{2}\right)}$ . Then, range $(\mathrm{f})=?$
A. $(-\infty, 1]$
B. $[1, \infty)$
C. $[-1,1]$
D. none of these

CBSE, Class 12, Functions, Maths, RS Aggarwal

Solution: Option(B) is correct. $f(x)=\frac{1}{\left(1-x^{2}\right)}$ The range of $f(x)$ can be found out by putting $f(x)=y$ $\begin{array}{l} \mathrm{y}=\frac{1}{\left(1-x^{2}\right)} \\...

Mark (√) against the correct answer in the following: Let $f(x)=\log (1-x)+\sqrt{x}^{2}-1$ . Then, $\operatorname{dom}(f)=?$
A. $(1, \infty)$
B. $(-\infty,-1]$
C. $[-1,1)$
D. $(0,1)$

CBSE, Class 12, Functions, Maths, RS Aggarwal

Solution: Option (B) is correct. $f(x)=\log (1-x)+\sqrt{x^{2}-1}$ Solving inequality, $\log (1-x) \geq 0$ $\Rightarrow 1-x \geq \mathrm{e}^{0} \quad \begin{array}{c}\text { (Log taken to the...

Mark (√) against the correct answer in the following: Let $f(x)=x^{3}$ . Then, dom (f) and range (f) are respectively
A. $R$ and $R$
B. $R^{+}$ and $R^{+}$
C. R and R +
D. $R^{+}$ and $R$

CBSE, Class 12, Functions, Maths, RS Aggarwal

Solution: Option(A) is correct. $f(x)=x^{3}$ $f(x)$ can assume any value, therefore domain of $f(x)$ is $R$ Range of the function can be positive or negative Real numbers, as the cube of any number...

Mark (√) against the correct answer in the following: Let $f(x)=x^{2}$ . Then, dom (f) and range (f) are respectively.
A. $R$ and $R$
B. $R^{+}$ and $R^{+}$
C. $R$ and $R^{+}$
D. $R$ and $R-\{0\}$

CBSE, Class 12, Functions, Maths, RS Aggarwal

Solution: Option(C) is correct. $f(x)=x_{2}$ $f(x)$ can assume any value, therefore domain of $f(x)$ is $R$ Range of the function can only be positive Real numbers, as the square of any number is...

Mark (√) against the correct answer in the following: Let $f(x)=\sqrt{\log }\left(2 x-x^{2}\right)$ . Then, $\operatorname{dom}(f)=?$
A. $(0,2)$
B. $[1,2]$
C. $(-\infty, 1]$
D. None of these

CBSE, Class 12, Functions, Maths, RS Aggarwal

Solution: Option(C) is correct. $f(x)=\sqrt{\log \left(2 x-x^{2}\right)}$ For $f(x)$ to be defined $2 x-x^2$ should be positive. Solving inequality, (Log taken to the opposite side of the equation...

Mark (√) against the correct answer in the following: Let $f(x)=\sqrt{\cos } x$ . Then, $\operatorname{dom}(f)=?$
A. $\left[0, \frac{\pi}{2}\right]$
B. $\left[\frac{3 \pi}{2}, 2 \pi\right]$
C. $\left[0, \frac{\pi}{2}\right] \cup\left[\frac{3 \pi}{2}, 2 \pi\right]$
D. none of these

CBSE, Class 12, Functions, Maths, RS Aggarwal

Solution: Option(C) is correct. $\mathrm{f}(\mathrm{x})=\sqrt{\cos x}$ As per graph of $\sqrt{\cos x}$ the domain is $\left[0, \frac{\pi}{2}\right] \cup\left[\frac{3 \pi}{2}, 2 \pi\right]$

Mark (√) against the correct answer in the following: Let $f(x)=\cos ^{-1}(3 x-1)$ . Then, $\operatorname{dom}(f)=$ ?
A. $\left(0, \frac{2}{3}\right)$
B. $\left[0, \frac{2}{3}\right]$
C. $\left[\frac{-2}{3}, \frac{2}{3}\right]$
D. None of these

CBSE, Class 12, Functions, Maths, RS Aggarwal

Solution: Option(B) is correct. $\begin{array}{l} \mathrm{f}(\mathrm{x})=\cos ^{-1}(3 \mathrm{x}-1) \end{array}$ Domain for function $\cos ^{-1} \mathrm{x}$ is $[-1,1]$ and range is $[0, \pi]$ When...

Mark (√) against the correct answer in the following: Let $f(x)=\cos ^{-1} 2 x$ . Then, $\operatorname{dom}(f)=?$
A. $[-1,1]$
B. $\left[\frac{-1}{2}, \frac{1}{2}\right]$
C. $\left[\frac{-\pi}{2}, \frac{\pi}{2}\right]$
D. $\left[\frac{-\pi}{4}, \frac{\pi}{4}\right]$

CBSE, Class 12, Functions, Maths, RS Aggarwal

Solution: Option(B) is correct. $f(x)=\cos ^{-1}2 x$ Domain for function $\cos ^1 \mathrm{x}$ is $[-1,1]$ and range is $[0, \pi]$ When a function is multiplied by an integer, the domain of the...

Mark (√) against the correct answer in the following: Let $f(x)=\frac{\sin ^{-1} x}{x} \cdot$ Then, $\operatorname{dom}(f)=?$
A. $(-1,1)$
B. $[-1,1]$
C. $[-1,1]-\{0\}$
D. none of these

CBSE, Class 12, Functions, Maths, RS Aggarwal

Solution: Option(B) is correct. $f(x)=\frac{\sin ^{-1} x}{x}$ The domain of the function is defined for $\mathrm{x} \neq 0$ domain of $\sin ^{-1} x$ is $[-1,1]$ So, domain of...

Mark (√) against the correct answer in the following: Let $\mathrm{f}(\mathrm{x})=\frac{\mathrm{x}}{\left(\mathrm{x}^{2}-1\right)}$ . Then, $\operatorname{dom}(\mathrm{f})=$ ?
A. $R$
B. $R-\{1\}$
C. $R-\{-1\}$
D. $R-\{-1,1\}$

CBSE, Class 12, Functions, Maths, RS Aggarwal

Solution: Option(D) is correct. $\mathrm{f}(\mathrm{x})=\frac{x}{\left(x^{2}-1\right)}$ The domain of the function is defined for $\begin{array}{l} \mathrm{x}^{2}-1 \neq 0 \\ \Rightarrow \mathrm{x}...

Mark (√) against the correct answer in the following: Let $f(x)=e^{\sqrt{x^{2}-1}} \cdot \log (x-1)$ . Then, $\operatorname{dom}(f)=$ ?
A. $(-\infty, 1]$
B. $[-1, \infty)$
C. $(1, \infty)$
D. $(-\infty,-1] \cup(1, \infty)$

CBSE, Class 12, Functions, Maths, RS Aggarwal

Solution: Option(C) is correct. $\mathrm{f}(\mathrm{x})=e^{\sqrt{x^{2}-1}} \cdot \log (x-1)$ The domain of the function is defined for $\begin{array}{l} \mathrm{x}-1>0 \quad \text { and }...

Mark (√) against the correct answer in the following: Let $f(x) \sqrt{\frac{x-1}{x+4}}$ . Then, dom (f) – ?
A. $[1,4)$
B. $[1,4]$
C. $(-\infty, 4]$
D. $(-\infty, 1] \cup(4, \infty)$

CBSE, Class 12, Functions, Maths, RS Aggarwal

Solution: Option(D) is correct. $\mathrm{f}(\mathrm{x})=\sqrt{\frac{x-1}{x+4}}$ The domain of the function can be defined for $\sqrt{\frac{x-1}{x+4}} \geq 0$ $\begin{array}{l} \Rightarrow...

Mark (√) against the correct answer in the following: Let $f(x)=\sqrt{9-x^{2}}$ . Then, dom $(f)=?$
A. $[-3,3]$
B. $[-\infty,-3]$
C. $[3, \infty)$
D. $(-\infty,-3] \cup(4, \infty)$

CBSE, Class 12, Functions, Maths, RS Aggarwal

Solution: Option(A) is correct. $\mathrm{f}(\mathrm{x})=\sqrt{9-x^{2}}$ The domain of the function can be defined for $\sqrt{9-x^{2}} \geq 0$ $\begin{array}{l} \Rightarrow \sqrt{9-x^{2}} \geq 0 \\...

Mark (√) against the correct answer in the following: If f = {(1, 2), (3, 5), (4, 1)} and g = {(2, 3), (5, 1), (1, 3)} then (g o f) = ?
A. {(3, 1), (1, 3), (3, 4)}
B. {(1, 3), (3, 1), (4, 3)}
C. {(3, 4), (4, 3), (1, 3)}
D. {(2, 5), (5, 2), (1, 5)}

CBSE, Class 12, Functions, Maths, RS Aggarwal

Solution: Option(B) is correct. $\begin{array}{l} \mathrm{f}=\{(1,2),(3,5),(4,1)\} \\ \mathrm{g}=\{(2,3),(5,1),(1,3)\} \end{array}$ According to the combination of $\mathrm{f}$ and $\mathrm{g}$,...

Mark (√) against the correct answer in the following: If $f(x)=x^{2}, g(x)=\tan x$ and $h(x)=\log x$ then $\{h \circ(g \circ f)\}\left(\sqrt{\frac{\pi}{4}}\right)=?$
A. 0
B. 1
C. $\frac{1}{x}$
D. $\frac{1}{2} \log \frac{\pi}{4}$

CBSE, Class 12, Functions, Maths, RS Aggarwal

Solution: Option(A) is correct. $\begin{array}{l} f(x)=x ^2 \\ g(x)=\tan x \\ h(x)=\log x \end{array}$ According to the combination of $f, g$ and $h$,...

Mark (√) against the correct answer in the following: If $f(x)=8 x^{3}$ and $g(x)=x^{1 / 3}$ then $(g \circ f)(x)=?$
A. $x$
B. $2 \mathrm{x}$
C. $\frac{x}{2}$
D. $3 x^{2}$

CBSE, Class 12, Functions, Maths, RS Aggarwal

Solution: Option(B) is correct. $f(x)=8 x^{3}$ $g(x)=x^{1 / 3}$ According to the combination of $\mathrm{f}$ and $\mathrm{g}$, $\operatorname{gof}(\mathrm{x})=\mathrm{f}(\mathrm{f}(\mathrm{x}))$...

Mark (√) against the correct answer in the following: If $f(x)=x^{2}-3 x+2$ then $(f \circ f)(x)=?$
A. $x^{4}$
B. $x^{4}-6 x^{3}$
C. $x^{4}-6 x^{3}+10 x^{2}$
D. None of these

CBSE, Class 12, Functions, Maths, RS Aggarwal

Solution: Option(D) is correct. $f(x)=x^2-3 x+2$ According to the combination of $\mathrm{f}$ and $\mathrm{f}$, $\operatorname{fof}(x)=f(f(x))$ Therefore, fof $(x)=f(f(x))$ $\begin{array}{l}...

Mark (√) against the correct answer in the following: If $f(x)=\sqrt[3]{3-x^{3}}$ then (f of) $(x)=?$
A. $\mathrm{x}^{1 / 3}$
B. $x$
C. $\left(1-\mathrm{x}^{1 / 3}\right)$
D. None of these

CBSE, Class 12, Functions, Maths, RS Aggarwal

Solution: Option(B) is correct. $f(x) \sqrt[3]{3-x^{3}}$ According to the combination of $f$ and $f$, $\text { fof }(x)=f(f(x))$ Therefore, fof $(x)=f(f(x))$...

Mark (√) against the correct answer in the following: If $f(x)=\frac{1}{(1-x)}$ then (f of of) $(x)=?$
A. $\frac{1}{(1-3 x)}$
B. $\frac{x}{(1+3 x)}$
C. $\mathrm{x}$
D. None of these

CBSE, Class 12, Functions, Maths, RS Aggarwal

Solution: Option(C) is correct. $\mathrm{f}(\mathrm{x})=\frac{1}{(1-x)}$ According to the combination of $\mathrm{f}$ and $\mathrm{f}$, fofof $(x)=f(f(f(x)))$ Therefore, fof $(x)=f(f(f(x))$...

Mark (√) against the correct answer in the following: If $f\left(x+\frac{1}{x}\right)=x^{2}+\frac{1}{x^{2}}$ then $f(x)=?$
A. $x^{2}$
B. $\left(x^{2}-1\right)$
C. $\left(x^{2}-2\right)$
D. None of these

CBSE, Class 12, Functions, Maths, RS Aggarwal

Solution: Option(C) is correct. $\begin{array}{l} \mathrm{f}\left(\mathrm{x}+\frac{1}{x}\right)=\left(\mathrm{x} ^2+\frac{1}{x^{2}}\right) \\ \Rightarrow...

Mark (√) against the correct answer in the following: If $f(x)=\left(x^{2}-1\right)$ and $g(x)=(2 x+3)$ then $(g \circ f)(x)=?$
A. $\left(2 x^{2}+3\right)$
B. $\left(3 x^{2}+2\right)$
C. $\left(2 x^{2}+1\right)$
D. None of these

CBSE, Class 12, Functions, Maths, RS Aggarwal

Solution: Option(C) is correct. $\begin{array}{l} \mathrm{f}(\mathrm{x})=(\mathrm{x}^2-1) \\ \mathrm{g}(\mathrm{x})=(2 \mathrm{x}+3) \end{array}$ According to the combination of $\mathrm{f}$ and...

A manufacturing company makes two types of teaching aids $A$ and $B$ of mathematics for class XII. Each type of A requires 9 labor hours of fabricating and 1 labor hour for finishing. Each type of B requires 12 labors hour for fabricating and 3 labor hour for finishing. For fabricating and finishing, the maximum labor hours available per week are 180 and 30 respectively. The company makes a profit of $\pm 80$ on each piece of type A and Rs.120 on each piece of type B. how many pieces of type A and type B. How many pieces of type $A$ and type B should be manufactured per week to get a maximum profit? Make it as an LLP and solve graphically. What is the maximum profit per week?

CBSE, Class 12, Exercise 33B, Linear Programming, Maths, RS Aggarwal

Let the company make $x$ no of $1^{\text {st }}$ type of teaching aid and y no of $2^{\text {nd }}$ type of teaching aid. $\therefore$ According to the question, $9 x+12 y \leq 180, x+3 y \leq 30, x...

One kind of cake requires $200 \mathrm{~g}$ of flour and $25 \mathrm{~g}$ of fat, another kind of cake requires $100 \mathrm{~g}$ of flour and $50 \mathrm{~g}$ , of fat. Find the maximum number of cakes which can be made from $5 \mathrm{~kg}$ of flour and $1 \mathrm{~kg}$ of fat, assuming. that there is no shortage of the other ingredients used in making the cakes. Make it an LPP and solve it graphically.

CBSE, Class 12, Exercise 33B, Linear Programming, Maths, RS Aggarwal

Let the company make $x$ no of $1^{\text {st }}$ kind and $y$ no of $2^{\text {nd }}$ cakes. $\therefore$ According to the question, $200 x+100 y \leq 5000,25 x+50 y \leq 1000, x \geq 0, y \geq 0$...

A company manufacture two types of toys A and B. type A requires 5 minutes each for cutting and 10 minutes for each assembling. Type B requires 8 minutes each for cutting and 8 minutes each for assembling. There are 3 hours available for cutting and 4 hours available for assembling in a day. He earns a profit of $\pm 50$ each on type $A$ and $\pm 60$ each on type B. How many toys of each type should the company manufacture in a day to maximize the profit?

CBSE, Class 12, Exercise 33B, Linear Programming, Maths, RS Aggarwal

Let the company manufacture $x$ and y numbers of toys $A$ and $B$. $\therefore$ According to the question, $5 x+8 y \leq 180,10 x+8 y \leq 240, x \geq 0, y \geq 0$ Maximize $Z=50 x+60 y$ The...

Mark (√) against the correct answer in the following: If $\mathrm{f}(\mathrm{x})=\frac{(4 \mathrm{x}+3)}{(6 \mathrm{x}-4)}, \mathrm{x} \neq \frac{2}{3}$ then $(\mathrm{f} \circ \mathrm{f})(\mathrm{x})=?$
A. $x$
B. $(2 x-3)$
C. $\frac{4 x-6}{3 x+4}$
D. None of these

CBSE, Class 12, Functions, Maths, RS Aggarwal

Solution: Option(A) is correct. $\mathrm{f}(\mathrm{x})=\frac{(4 x+3)}{(6 x-4)}, \mathrm{x} \neq \frac{2}{3}$ According to the combination of $\mathrm{f}$ and $\mathrm{f}$,...

Mark (√) against the correct answer in the following: Let f : N → X : f(x) = 4×2 + 12x + 15. Then, $f^{- 1}$ (y) = ?
A. $\frac{1}{2}(\sqrt{y-4}+3)$
B. $\frac{1}{2}(\sqrt{y-6}-3)$
C. $\frac{1}{2}(\sqrt{y-4}+5)$
D. None of these

CBSE, Class 12, Functions, Maths, RS Aggarwal

Solution: Option(B) is correct. $\mathrm{f}: \mathrm{N} \rightarrow \mathrm{X}: \mathrm{f}(\mathrm{x})=4 \mathrm{x} 2+12 \mathrm{x}+15$ We need to find $\mathrm{f}-1$, Suppose...

Mark (√) against the correct answer in the following:
Let $f: R-\left\{\frac{-4}{3}\right\} \rightarrow-\left\{\frac{4}{3}\right\}: f(x)=\frac{4 x}{(3 x+4)}$ . Then $f^{-1}(y)=?$
A. $\frac{4 y}{(4-3 y)}$
B. $\frac{4 y}{(4 y+3)}$
C. $\frac{4 y}{(3 y-4)}$
D. None of these

CBSE, Class 12, Functions, Maths, RS Aggarwal

Solution: Option(A) is correct. $\text { f: } R-\left\{-\frac{4}{3}\right\} \rightarrow-\left\{\frac{4}{3}\right\}: f(x)=\frac{4 x}{(3 x+4)}$ We need to find $\mathrm{f}-1$ Suppose $f(x)=y$...

Mark (√) against the correct answer in the following: Let f : Q → Q : f(x) = (2x + 3). Then, $f^{ - 1}$ (y) = ?
A. (2y – 3)
B. $\frac{1}{(2 y-3)}$
C. $\frac{1}{2}(y-3)$
D. none of these

CBSE, Class 12, Functions, Maths, RS Aggarwal

Solution: Option(C) is correct. $\mathrm{f}: \mathrm{Q} \rightarrow \mathrm{Q}: \mathrm{f}(\mathrm{x})=(2 \mathrm{x}+3)$ We need to find $\mathrm{f}-1$ Suppose $\mathrm{f}(\mathrm{x})=\mathrm{y}$...

Maximize $=60 x+15 y$ , subject to the constraints $x+y \leq 50,3 x+y \leq 90, x, y \geq 0$

CBSE, Class 12, Exercise 33B, Linear Programming, Maths, RS Aggarwal

The feasible region determined by the constraints $x+y \leq 50,3 x+y \leq 90, x, y \geq 0 .$ is given by The corner points of feasible region are $A(0,0), B(0,50), C(20,30), D(30,0)$. The values of...

Mark (√) against the correct answer in the following:
Let A and B be two non – empty sets and let
f : (A × B) → (B × A) : f(a, b) = (b, a). Then, f is
A. one – one and into
B. one – one and onto
C. many – one and into
D. many – one and onto

CBSE, Class 12, Functions, Maths, RS Aggarwal

Solution: Option(B) is correct. One-One Function Suppose $\mathrm{p}_{1}, \mathrm{p}_{2}, \mathrm{q}_{1}, \mathrm{q}_{2}$ be two arbitrary elements in $\mathrm{R}$ + Therefore, $f\left(p_{1},...

Anil wants to invest at the most Rs. $12000$ in bonds A and B .According to rules, he has to invest at least Rs.2000 in bond $A$ and at least Rs.4000 in bond $B$ . if the rate of interest of bond $A$ is $8 \%$ per annum and on bond $B$ , it is $10 \%$ per annum, how should he invest his money for maximum interest? Solve the problem graphically.

CBSE, Class 12, Exercise 33B, Linear Programming, Maths, RS Aggarwal

Let the invested money in bond $A$ be $x$ and in bond $B$ be $y$. $\therefore$ According to the question, $\mathrm{X}+\mathrm{y} \leq 12000, \mathrm{x} \geq 2000, \mathrm{y} \geq 4000$ Maximize...

Mark (√) against the correct answer in the following: Let $f: N \rightarrow N: f(x)=\left\{\begin{array}{l}\frac{1}{2}(n+1) \text {, when } n \text { is odd } \\ \frac{n}{2}, \text { when } n \text { is even. }\end{array}\right.$ Then, $\mathrm{f}$ is
A. one – one and into
B. one – one and onto
C. many – one and into
D. many – one and onto

CBSE, Class 12, Functions, Maths, RS Aggarwal

Solution: Option(D) is correct. $f:\mathrm{N} \rightarrow \mathrm{N}: \mathrm{f}(\mathrm{x})=$ $f: N \rightarrow N: f(x)=\left\{\begin{array}{l}\frac{1}{2}(n+1) \text {, when } n \text { is odd } \\...

Mark (√) against the correct answer in the following:
Let $A=R-\{3\}$ and $B=R-\{1\}$ . Then $f: A \rightarrow A: f(x)=\frac{(x-2)}{(x-3)}$ is
A. one – one and into
B. one – one and onto
C. many – one and into
D. many – one and onto

CBSE, Class 12, Functions, Maths, RS Aggarwal

Solution: Option(B) is correct. f: $\mathrm{A} \rightarrow \mathrm{B}: \mathrm{f}(\mathrm{x})=\frac{(x-2)}{(x-3)}$ Where, $\mathrm{A}=\mathrm{R}-\{3\}$ and $\mathrm{B}=\mathrm{R}-\{1\}$ One-One...

Mark (√) against the correct answer in the following:
f : C → R : f(z) = |z| is
A. one – one and into
B. one – one and onto
C. many – one and into
D. many – one and onto

CBSE, Class 12, Functions, Maths, RS Aggarwal

Solution: Option() is correct. One-One function Suppose $\mathrm{p}, \mathrm{q}$ be two arbitrary elements in $\mathrm{R}+$ Therefore, $f(p)=f(q)$ $\begin{array}{l}...

A dealer wishes to purchase a number of fans and sewing machines. He has only Rs.5760 to invest and has space for at most 20 items. A fan costs him Rs.360 and a sewing machine $\pm240$ . He expects to sell a fan at a profit of $\approx 22$ and a sewing machine at a profit of $\pm 18 .$ Assuming that he can sell all the items that he buys, how should he invest his money to maximize the profit? Solve the graphically and find the maximum profit.

CBSE, Class 12, Exercise 33B, Linear Programming, Maths, RS Aggarwal

Let the number of fans bought be $x$ and sewing machines bought be $y$. $\therefore$ According to the question, $360 x+240 y \leq 5760, x+y \leq 20, x \geq 0, y \geq 0$ Maximize $Z=22 x+18 y$ The...

Mark (√) against the correct answer in the following:
f : R → R : f(x) = cos x is
A. one – one and into
B. one – one and onto
C. many – one and into
D. many – one and onto

CBSE, Class 12, Functions, Maths, RS Aggarwal

Solution: Option (C) is correct. One-one function $\cos x$ graph cuts y axis repeatedly, hence it is many-one. Onto function Range of $f(x)$ is $[-1,1]$ Co-domain is $\mathrm{R}$ So here, Range of...

Kellogg is a new cereal formed of a mixture of bran and rice, that contains at least 88 grams of protein and at least 36 milligrams of iron. Knowing that bran contains 80 grams of protein and 40 milligrams of iron per kilograms, and that rice contains 100 grams of protein and 30 milligrams of iron per kilogram, find the minimum cost producing this new cereal if bran costs $\pm 5$ per kilogram and rice costs $\pm4$ per kilogram.

CBSE, Class 12, Exercise 33B, Linear Programming, Maths, RS Aggarwal

Let $x$ and $y$ be number of kilograms of bran and rice. $\therefore$ According to the question, $80 x+100 y \geq 88,40 x+30 y \geq 36, x \geq 0, y \geq 0$ Minimize $Z=5 x+4 y$ The feasible region...

Mark (√) against the correct answer in the following: $\mathrm{f}:\left[\frac{-\pi}{2}, \frac{\pi}{2}\right] \rightarrow[-1,1]: \mathrm{f}(\mathrm{x})=\sin \mathrm{x}$ is
A. one – one and into
B. one – one and onto
C. many – one and into
D. many – one and onto

CBSE, Class 12, Functions, Maths, RS Aggarwal

Solution: Graph Option (B) is correct. One-one Function According to the graph for $\sin (\mathrm{x})$, for given range of $\left[\frac{-\pi}{2}, \frac{\pi}{2}\right], \mathrm{f}(\mathrm{x})$ is not...

Mark (√) against the correct answer in the following:
f : R + → R + : f(x) = $e^x$ is
A. many – one and into
B. many – one and onto
C. one – one and into
D. one – one and onto

CBSE, Class 12, Functions, Maths, RS Aggarwal

Solution: Option(D) is correct. One-One function Suppose $\mathrm{p}, \mathrm{q}$ be two arbitrary elements in $\mathrm{R}+$ Therefore, $f(p)=f(q)$ $\begin{array}{l} \Rightarrow...

Mark (√) against the correct answer in the following:
f : R → R : f(x) = $x^3$ is
A. one – one and onto
B. one – one and into
C. many – one and onto
D. many – one and into

CBSE, Class 12, Functions, Maths, RS Aggarwal

Solution: Option(B) is correct. One-One function Suppose $\mathrm{p}, \mathrm{q}$ be two arbitrary elements in $\mathrm{R}$ Therefore, $f(p)=f(q)$ $\begin{array}{l} \Rightarrow...

A company manufactures two types of toys, A and B. Type A requires 5 minutes each for cutting and 10 minutes each for assembling. Type B required 8 minutes each for cutting and 8 minutes each for assembling. There are 3 hours available for cutting and 4 hours available for assembling in a day. The profit is $\pm 50$ each. on type A and $\pm 60$ each on type B. how many toys of each types should the company manufactures in a day to maximize the profit?

CBSE, Class 12, Exercise 33B, Linear Programming, Maths, RS Aggarwal

Let the company manufacture $x$ and y numbers of toys $A$ and $B$. $\therefore$ According to the question, $5 x+8 y \leq 180,10 x+8 y \leq 240, x \geq 0, y \geq 0$ Maximize $Z=50 x+60 y$ The...

A manufacture produces two types of steel trunks. He has two machines, A and B. The first type of trunk requires 3 hours on machine $A$ and 3 hours on machine $B$ . The second type required 3 hours on machine $A$ , and 2 hours on Machine A and 2 hours on machine B. Machine A and B can work at most for 18 hours and 15 hours per day respectively. He earns a profit of $\pm 30$ and Rs. 25 per trunk of the first type and second type respectively. How may trunks of each type must he make each day to make the maximum profit?

CBSE, Class 12, Exercise 33B, Linear Programming, Maths, RS Aggarwal

Let the manufacturer manufacture $\mathrm{x}$ and y numbers of type 1 and type 2 trunks. $\therefore$ According to the question, $3 x+3 y \leq 18,3 x+2 y \leq 15, x \geq 0, y \geq 0$ Maximize $Z=30...

A small firm manufactures items $A$ and $B$ . The total number of items that it can manufacture in a day is at most 24 . Item A takes one hour to make while item B take only half an hour. The maximum time available per day is 16 hours. If the profit on one unit item $A$ be $\pm 300$ and that on one unit of item $B$ be $\pm 160$ , how many of each type of item should be produced to maximize the profit? Solve the problem graphically.

CBSE, Class 12, Exercise 33B, Linear Programming, Maths, RS Aggarwal

Let the firm manufacture $x$ number of A and y number of B products. $\therefore$ According to the question, $x+y \leq 24, x+0.5 y \leq 16, x \geq 0, y \geq 0$ Maximize $Z=300 x+160 y$ The feasible...

Mark (√) against the correct answer in the following:
f : R → R : f(x) = $x^2$ is
A. one – one and onto
B. one – one and into
C. many – one and onto
D. many – one and into

CBSE, Class 12, Functions, Maths, RS Aggarwal

Solution: Option(D) is correct. One-One function Suppose $\mathrm{p}, \mathrm{q}$ be two arbitrary elements in $\mathrm{R}$ Therefore, $f(p)=f(q)$ $\begin{array}{l} \Rightarrow...

A firm manufactures two types of product, A and B, and sells them at a profit of $\pm 5$ per unit of type $A$ and $\pm 3$ per unit of type B. Each product is processed on two machines, $M_{1}$ and $M_{2} .$ one unit of type A requires one minute of processing time on $M_{1}$ and two minutes of processing time on $M_{2} ;$ whereas one unit of type $B$ requires one minute of processing time on $M_{1}$ and one minute on $M_{2} .$ Machines $M_{1}$ and $M_{2}$ are respectively available for at most 5 hours and 6 hours in a day. Find out how many units of each type of product the firm should produce a day in order to maximize the profit. Solve the problem graphically.

CBSE, Class 12, Maths, RS Aggarwal

Let the firm manufacture $x$ number of Aand y number of $B$ products. $\therefore$ According to the question, $X+y \leq 300,2 x+y \leq 360, x \geq 0, y \geq 0$ Maximize $Z=5 x+3 y$ The feasible...

Mark (√) against the correct answer in the following:
f : N → N : f(x) = $x^2$ + x + 1 is
A. one – one and onto
B. one – one and into
C. many – one and onto
D. many – one and into

CBSE, Class 12, Functions, Maths, RS Aggarwal

Solution: Option (B) is correct. One-One function Suppose $\mathrm{p}, \mathrm{q}$ be two arbitrary elements in $\mathrm{N}$ Therefore, $f(p)=f(q)$ $\begin{array}{l} \Rightarrow...

Mark (√) against the correct answer in the following:
f : N → N : f(x) = 2x is
A. one – one and onto
B. one – one and into
C. many – one and onto
D. many – one and into

CBSE, Class 12, Functions, Maths, RS Aggarwal

Solution: Option (B) is correct. One-One function Suppose $\mathrm{p}, \mathrm{q}$ be two arbitrary elements in $\mathrm{N}$ Therefore, $f(p)=f(q)$ $\begin{array}{l} \Rightarrow 2 \mathrm{p}=2...

A housewife wishes to mix together two kinds of food, $X$ and $Y$ , in such a way that the mixture contains at least 10 units of vitamin $A, 12$ units of vitamin $B$ and 8 units of vitamin $C .$ The vitamin contents of $1 \mathrm{~kg}$ of each food are given below.

CBSE, Class 12, Exercise 33B, Linear Programming, Maths, RS Aggarwal

If $1 \mathrm{~kg}$ of food $\mathrm{X}$ cost $\pm 6$ and $1 \mathrm{~kg}$ of food $\mathrm{Y}$ costs $\pm 10$, find the minimum cost of the mixture which will produce the diet. Solution: Let $x$...

A diet for a sick person must contain at least 4000 units of vitamins, 50 units of mineral and 1400 calories. Two food, $A$ and $B$ , are available at a cost of $\pm 4$ and $\pm 3$ per unit respectively. If one unit of A contains 200 units of vitamins, 1 unit of mineral and 40 calories, and 1 unit of B contains 100 units of vitamins, 2 units of mineral and 40 calories, find what combination of foods should be used to have the least cost.

CBSE, Class 12, Exercise 33B, Linear Programming, Maths, RS Aggarwal

Let $x$ and $y$ be number of units of food $A$ and $B$. $\therefore$ According to the question, $200 x+100 y \geq 4000, x+2 y \geq 50,40 x+40 y \geq 1400, x \geq 0, y \geq 0$ Minimize $Z=4 x+3 y$...

A dietician wishes to mix two types of food, $X$ and $Y$ , in such a way that the vitamin contents of the mixture contains at least 8 units of vitamin $A$ and 10 units of vitamin $C$ . Food $X$ contains 2 units/kg of vitamin $A$ and 1 unit $/ \mathrm{kg}$ of vitamin $C$ , while food $Y$ contains 1 unit/kg of vitamin $A$ and 2 units/kg of vitamin $C$ . It costs $\pm 5$ per $\mathrm{kg}$ to purchase the food $\mathrm{X}$ and $\pm7$ per kg to purchase the food Y. Determine the minimum cost of such a mixture.

CBSE, Class 12, Exercise 33B, Linear Programming, Maths, RS Aggarwal

Let $x$ and $y$ be number of units of $X$ and $Y$. $\therefore$ According to the question, $2 x+y \geq 8, x+2 y \geq 10, x \geq 0, y \geq 0$ Minimize $Z=5 x+7 y$ The feasible region determined $2...

A firm is engaged in breeding pigs. The pigs are fed on various products grown on the farm. They need certain nutrients, named as $X, Y, Z$ . the pigs are fed on two products, A and B. One unit of product A contain 36 unit of $X, 3$ units of $Y$ and 20 units of $Z$ , while one unit of product $B$ contain 6 units of $X, 12$ units of $Y$ and 10 units of $Z$ . the minimum requirement of $X, Y, Z$ are 108 units, 36 units and 100 units respectively. Product A costs $\pm 20$ per unit and product B costs $\pm40$ per unit. How many units of each product must be taken to minimize the cost? Also, find the minimum cost.

CBSE, Class 12, Exercise 33B, Linear Programming, Maths, RS Aggarwal

Let $x$ and $y$ be number of units of products of $A$ and $B$. $\therefore$ According to the question, $36 x+6 y \geq 108,3 x+12 y \geq 36,20 x+10 y \geq 100, x \geq 0, y \geq 0$ Minimize $Z=20 x+40...

An oil company has two depots, $A$ and $B$ , with capacities of $7000 \mathrm{~L}$ and $4000 \mathrm{~L}$ respectively. The company is to supply oil to three pumps, $D, E, F$ , whose requirements are $4500 \mathrm{~L}, 3000 \mathrm{~L}$ , and $3500 \mathrm{~L}$ respectively. The distances (in $\mathrm{km}$ ) between the depots and the petrol pumps are given in the following table:
$\begin{tabular}{|c|c|c|} \hline & \multicolumn{2}{|c|}{ Distance $($ in $\mathrm{km})$} \\ \hline To & $A$ & $B$ \\ \hline From & $A$ & $B$ \\ \hline$D$ & 7 & 3 \\ \hline$E$ & 6 & 4 \\ \hline$F$ & 3 & 2 \\ \hline \end{tabular}$
Assuming that the transportation cost per $\mathrm{km}$ is re 1 per litre, how should the delivery be scheduled in order that the transportation cost is minimum?

CBSE, Class 12, Exercise 33B, Linear Programming, Maths, RS Aggarwal

Let $x$ liters of petrol be transported from $A$ to $D$ and y liters of petrol be transported from $A$ to $E$. Therefore, $7000-(x+y)$ will be transported to $F$. Also, ( $4500-x$ ) liters of...

Arrange the bonds in order of increasing ionic character in the molecules: LiF, $K _{2} O , N _{2}, SO _{2}$ , and $CIF _{3}$ .

CBSE, Chemical Bonding and Molecular Structure, Chemistry, Class 12, NCERT Exemplar

Solution: The difference in electronegativity between constituent atoms determines the ionic character of a molecule. As a result, the greater the difference, the greater the ionic character of a...

Explain with the help of suitable example polar covalent bond.

CBSE, Chemical Bonding and Molecular Structure, Chemistry, Class 12, NCERT Exemplar

Solution: The bond pair of electrons are not shared equally when two unique atoms with different electronegativities join to form a covalent bond. The bond pair is attracted to the nucleus of an...

Define electronegativity. How does it differ from electron gain enthalpy?

CBSE, Chemical Bonding and Molecular Structure, Chemistry, Class 12, NCERT Exemplar

Solution: "Electronegativity refers to an atom's ability to attract a bond pair of electrons towards itself in a chemical compound." Sr. No Electronegativity Electron affinity 1 A tendency to...

Write the significance/applications of dipole moment.

CBSE, Chemical Bonding and Molecular Structure, Chemistry, Class 12, NCERT Exemplar

Solution: There is a difference in electro-negativities of constituents of the atom in a heteronuclear molecule, which causes polarisation. As a result, one end gains a positive charge, while the...

Although both $CO _{2}$ and $H _{2} O$ are triatomic molecules, the shape of the $H _{2} O$ molecule is bent while that of $CO _{2}$ is linear. Explain this on the basis of dipole moment.

CBSE, Chemical Bonding and Molecular Structure, Chemistry, Chemistry, Class 12, NCERT Exemplar

Solution: $CO_2$ has a dipole moment of 0 according to experimental results. And it's only possible if the molecule's shape is linear, because the dipole moments of the C-O bond are equal and...

Use Lewis symbols to show electron transfer between the following atoms to form cations and anions :(iii) Al and N.

CBSE, Chemical Bonding and Molecular Structure, Chemistry, Class 12, NCERT Exemplar

Solution: Below is a list of Lewis symbols. To form a cation, a metal atom loses one or more electrons, while a nonmetal atom gains one or more electrons. Ionic bonds are formed between cations and...

Use Lewis symbols to show electron transfer between the following atoms to form cations and anions : (i) K and S (ii) Ca and O

CBSE, Chemical Bonding and Molecular Structure, Chemistry, Class 12, NCERT Exemplar

Temperature dependence of resistivity ρ(T) of semiconductors, insulators, and metals is significantly based on the following factors:
a) number of charge carriers can change with temperature T
b) time interval between two successive collisions can depend on T
c) length of material can be a function of T
d) mass of carriers is a function of T

CBSE, Class 12, Current Electricity, NCERT Exemplar, Periodic Classification of Elements, Physics, Some Special Series

The correct answer is a) number of charge carriers can change with temperature T b) time interval between two successive collisions can depend on T

Write the resonance structures for $SO _{3}, NO _{2}$ , and $NO _{3}{ }^{-}$

CBSE, Chemical Bonding and Molecular Structure, Chemistry, Class 12, NCERT Exemplar

Solution: Resonance is the phenomenon that allows a molecule to be expressed in multiple ways, none of which fully explain the molecule's properties. The molecule's structure is called a resonance...

$H _{3} PO _{3}$ can be represented by structures 1 and 2 shown below. Can these two structures be taken as the canonical forms of the resonance hybrid representing $H _{3} PO _{3}$ ? If not, give reasons for the same.

CBSE, Chemical Bonding and Molecular Structure, Chemistry, Class 12, NCERT Exemplar

Solution: The positions of the atoms remain constant in canonical forms, but the positions of the electrons change. The positions of atoms change in the given canonical forms. As a result, they...

Explain the important aspects of resonance with reference to the $CO _{3}{ }^{2-}$ ion.

CBSE, Chemical Bonding and Molecular Structure, Chemistry, Class 12, NCERT Exemplar

Solution: However, while the carbonate ion cannot be represented by a single structure, the properties of the ion can be described by two or more different resonance structures. The actual structure...

Define Bond length.

CBSE, Chemical Bonding and Molecular Structure, Chemistry, Class 12, NCERT Exemplar

Solution: Bond length is defined as the equilibrium distance between the nuclei of two bonded atoms in a molecule.

How do you express the bond strength in terms of bond order?

CBSE, Chemical Bonding and Molecular Structure, Chemistry, Class 12, NCERT Exemplar

Solution: During the formation of a molecule, the extent of bonding that occurs between two atoms is represented by the bond strength of the molecule. As the bond strength increases, the bond...

Although geometries of $NH _{3}$ and $H _{2} O$ molecules are distorted tetrahedral, bond angle in water is less than that of Ammonia. Discuss.

CBSE, Chemical Bonding and Molecular Structure, Chemistry, Class 12, NCERT Exemplar

Solution: Ammonia's central atom (N) has one lone pair and three bond pairs. In water, the central atom (O) has two lone pairs and two bond pairs. As a result, the two bond pairs repel the two lone...

Discuss the shape of the following molecules using the VSEPR model: $BeCl _{2}, BCl _{3}, S i C l _{4}, AsF _{5}, H _{2} S , PH _{3}$

CBSE, Chemical Bonding and Molecular Structure, Chemistry, Class 12, NCERT Exemplar

Solution: $BeCl_2$ The central atom does not have a lone pair, but it does have two bond pairs. As a result, its shape is AB2, or linear. $BCl_3$ The central atom has three bond pairs but no lone...

Write the favourable factors for the formation of an ionic bond.

CBSE, Chemical Bonding and Molecular Structure, Chemistry, Class 12, NCERT Exemplar

Solution: Ionic bonds are formed when one or more electrons are transferred from one atom to another. As a result, the ability of neutral atoms to lose or gain electrons is required for the...

Equipotential surfaces a) are closer in regions of large electric fields compared to regions of lower electric fields b) will be more crowded near sharp edges of a conductor c) will be more crowded near regions of large charge densities d) will always be equally spaced

CBSE, Class 11, Electrostatic Potential and Capacitance, NCERT Exemplar, Physics

The correct answer is a) are closer in regions of large electric fields compared to regions of lower electric fields b) will be more crowded near sharp edges of a conductor c) will be more crowded...

Define the octet rule. Write its significance and limitations

CBSE, Chemical Bonding and Molecular Structure, Chemistry, Class 12, NCERT Exemplar

Solution: “Atoms can combine either by transferring valence electrons from one atom to another or by sharing their valence electrons in order to achieve the closest inert gas configuration by having...

Draw the Lewis structures for the following molecules and ions : $H _{2} S , SiCl _{4}, BeF _{2}, CO _{3}^{2-}, HCOOH$

CBSE, Chemical Bonding and Molecular Structure, Chemistry, Class 12, NCERT Exemplar

Solution: The lewis dot structures are:

Write Lewis symbols for the following atoms and ions: Sand $S ^{2}-; Al$ and $Al ^{3+} ; H$ and $H ^{-}$

CBSE, Chemical Bonding and Molecular Structure, Chemistry, Class 12, NCERT Exemplar

Solution: For S and S2- A sulphur atom has only 6 valence electrons, which is a very small number. As a result, the Lewis dot symbol for the letter S is The presence of a...

Consider a uniform electric field in the z direction. The potential is a constant a) in all space b) for any x for a given z c) for any y for a given z d) on the x-y plane for a given z

CBSE, Class 11, Electrostatic Potential and Capacitance, NCERT Exemplar, Physics

The correct answer is b) for any x for a given z c) for any y for a given z d) on the x-y plane for a given z

Write Lewis dot symbols for atoms of the following elements :e) N f) Br

CBSE, Chemical Bonding and Molecular Structure, Chemistry, Class 12, NCERT Exemplar

Solution: Nitrogen atoms have only five valence electrons in total. As a result, the Lewis dot symbol for N is Bromine, because the atom has only seven valence electrons. As a result,...

A parallel plate capacitor is made of two dielectric blocks in series. One of the blocks has thickness d1 and dielectric constant k1 and the other has thickness d2 and dielectric constant k2 as shown in the figure. This arrangement can be thought of as a dielectric slab of thickness d = d1 + d2 and effective dielectric constant k. The k is a) k1d1 + k2d2/d1+d2 b) k1d1 + k2d2/k1 + k2 c) k1k2 (d1 + d2)/(k1d1 + k2d2) d) 2k1k2/k1 + k2

CBSE, Class 11, Electrostatic Potential and Capacitance, NCERT Exemplar, Physics

The correct answer is c) k1k2 (d1 + d2)/(k1d1 + k2d2)

Write Lewis dot symbols for atoms of the following elements :c) B d) O

CBSE, Chemical Bonding and Molecular Structure, Chemistry, Class 12, NCERT Exemplar

Solution: Boron atoms have only three valence electrons, which is a very small number. As a result, the Lewis dot symbols for B are as follows: The oxygen atom has only six valence...

Equipotential at a great distance from a collection of charges whose total sum is not zero are approximately a) spheres b) planes c) paraboloids d) ellipsoids

CBSE, Class 11, Electrostatic Potential and Capacitance, NCERT Exemplar, Physics

The correct answer is a) spheres

The electrostatic potential on the surface of a charged conducting sphere is 100V. Two statements are made in this regard: S1: At any point inside the sphere, the electric intensity is zero S2: At any point inside the sphere, the electrostatic potential is 100V Which of the following is a correct statement? a) S1 is true but S2 is false b) Both S1 and S2 are false c) S1 is true, S2 is also true, and S1 is the cause of S2 d) S1 is true, S2 is also true but the statements are independent

CBSE, Class 11, Electrostatic Potential and Capacitance, NCERT Exemplar, Physics

The correct answer is c) S1 is true, S2 is also true, and S1 is the cause of S2

Figure shows some equipotential lines distributed in space. A charged object is moved from point A to point B. a) the work done in fig (i) is the greatest b) the work done in fig (ii) is least c) the work done is the same in fig (i), fig (ii), and fig (iii) d) the work done in fig (iii) is greater than fig (ii) but equal to that in fig (i)

CBSE, Class 11, Electrostatic Potential and Capacitance, NCERT Exemplar, Physics

The correct answer is c) the work done is the same in fig (i), fig (ii), and fig (iii)

A capacitor of 4μF is connected as shown in the circuit. The internal resistance of the battery is 0.5Ω. The amount of charge on the capacitor plates will be a) 0 b) 4μC c) 16μC d) 8μC

CBSE, Class 11, Electrostatic Potential and Capacitance, NCERT Exemplar, Physics

The correct answer is d) 8μC

If $A=\left[\begin{array}{cccc}5 & -2 & 6 & 1 \\ 7 & 0 & 8 & -3 \\ \sqrt{2} & \frac{3}{5} & 4 & 3\end{array}\right]$ then write
i. the elements $a_{23}, a_{31}, a_{14}, a_{33}, a_{22}$ of $A$ .

CBSE, Class 12, Exercise 5A, Maths, Matrices, RS Aggarwal

Solution: (i) $a_{i j}=$ element of $t^{\text {th }}$ row and $j^{\text {th }}$ column $\begin{array}{l} a_{23}=8 \\ a_{31}=\sqrt{2} \\ a_{14}=1 \\ a_{33}=4 \\ a_{22}=0 \end{array}$

Let $f: \mathbb{N} \rightarrow R: f(x)=4 x^{2}+12 x+15 .$ Show that f: $N \rightarrow$ range $(f)$ is invertible. Find $f^{-1}$

CBSE, Class 12, Exercise 2D, Functions, Maths, RS Aggarwal

Solution: $\mathrm{f}(\mathrm{x})=4 \mathrm{x} 2+12 \mathrm{x}+15 \quad \text { (as given) }$ $\mathrm{f}(\mathrm{x})$ is invertible if $\mathrm{f}(\mathrm{x})$ is a bijection (i.e one-one onto...

Show that the function $f$ on $A=R-\left\{\frac{2}{3}\right\}$ , defined as $f(x)=\frac{4 x+3}{6 x-4}$ is one-one and onto. Hence, find $f^{-1}$ ,

CBSE, Class 12, Exercise 2D, Functions, Maths, RS Aggarwal

Solution: $\mathrm{f}(\mathrm{x})=\frac{4 x+3}{6 x-4} \quad \text { (given) }$ $\mathrm{f}(\mathrm{x})$ is invertible if $\mathrm{f}(\mathrm{x})$ is a bijection (i.e one-one onto function) One-One...

Write Lewis dot symbols for atoms of the following elements :
a) Mg
b) Na

CBSE, Chemical Bonding and Molecular Structure, Chemistry, Class 12, NCERT Exemplar

Solution: Only two valence electrons exist in the magnesium atom. As a result, the Lewis dot symbols for Mg are as follows: Only one valence electron exists in the sodium atom. As a...

Explain the formation of a chemical bond.

CBSE, Chemical Bonding and Molecular Structure, Chemistry, Class 12, NCERT Exemplar

Answer: "A chemical bond is an attractive force that holds a chemical species' constituents together." For chemical bond formation, many theories have been proposed, including valence shell electron...

The velocity-displacement graph of a particle is shown in the figure. a) Write the relation between v and x. b) Obtain the relation between acceleration and displacement and plot it.

CBSE, Class 11, Motion in a straight line, NCERT Exemplar, Physics

a) Consider the point P(x,v) at any time t on the graph such that angle ABO is θ such that tan θ = AQ/QP = (v0-v)/x = v0/x0 When the velocity decreases from v0 to zero during the displacement, the...

A man runs across the roof-top of a tall building and jumps horizontally with the hope of landing on the roof of the next building which is of a lower height than the first. If his speed is 9 m/s, the distance between the two buildings is 10 m and the height difference is 9 m, will he be able to land on the next building?

Class 11, Motion in a straight line, NCERT Exemplar, Physics

For a free fall at 9m, the horizontal distance covered by the man should be at least 10 m. u = 0 a = 10 m/s2 s = 9 m t = t s = ut + 1/2 at2 Substituting the values, we get t = √9/3 = 3/√5 sec The...

A ball is dropped and its displacement vs time graph is as shown in the figure where displacement x is from the ground and all quantities are positive upwards. a) Plot qualitatively velocity vs time graph b) Plot qualitatively acceleration vs time graph

CBSE, Class 11, Motion in a straight line, NCERT Exemplar, Physics, Physics

a) At t=0 and v=0 , v-t graph is: b) At x = 0, a-t graph is:

A positively charged particle is released from rest in a uniform electric field. The electric potential energy of the charge a) remains a constant because the electric field is uniform b) increases because the charge moves along the electric field c) decreases because the charge moves along the electric field d) decreases because the charge moves opposite to the electric field

CBSE, Class 11, Electrostatic Potential and Capacitance, NCERT Exemplar, Physics

The correct answer is c) decreases because the charge moves along the electric field

A ball is dropped from a building of height 45 m. Simultaneously another ball is thrown up with a speed 40 m/s. Calculate the relative speed of the balls as a function of time.

CBSE, Class 11, Motion in a straight line, NCERT Exemplar, Physics

V = v1 = ? U = 0 h = 45 m a = g t = t V = u + at v1 = 0 + gt v1 = gt Therefore, when the ball is thrown upward, v1 = -gt V = v2 u = 40 m/s a = g t = t V = u + at v2 = 40 – gt The relative velocity...

It is a common observation that rain clouds can be at about a kilometre altitude above the ground. a) If a rain drop falls from such a height freely under gravity, what will be its speed? Also, calculate in km/h b) A typical rain drop is about 4 mm diameter. Momentum is mass x speed in magnitude. Estimate its momentum when it hits ground. c) Estimate the time required to flatten the drop. d) Rate of change of momentum is force. Estimate how much force such a drop would exert on you. e) Estimate the order of magnitude force on umbrella. Typical lateral separation between two rain drops is 5 cm.

CBSE, Class 11, Motion in a straight line, NCERT Exemplar, Physics

a) Velocity attained by the rain drop which is falling freely through the height h is: v2 = u2 – 2g(-h) As u = 0 v = √2gh = 100√2 m/s = 510 km/h b) Diameter of the drop, d = 2r = 4 mm Radius of the...

A motor car moving at a speed of 72 km/h cannot come to a stop in less than 3 s while for a truck this time interval is 5 s. On a highway the car is behind the truck both moving at 72 km/h. The truck gives a signal that it is going to stop at emergency. At what distance the car should be from the truck so that it does not bump onto the truck. Human response time is 0.5 s.

CBSE, Class 11, Motion in a straight line, NCERT Exemplar, Physics

For truck, u = 20 m/s v = 0 a = ? t = 5s v = u + at a = 4 m/s2 For car, t = 3 s u = 20 m/s v = 0 a = ac v = u + at ac = -20/3 m/s2 Let s be the distance between the car and the truck when the truck...

A monkey climbs up a slippery pole for 3 seconds and subsequently slips for 3 seconds. Its velocity at time t is given by v(t) = 2t (3 – t); 0

CBSE, Class 11, Motion in a straight line, NCERT Exemplar, Physics

a) For maximum velocity v(t) dv(t)/dt = 0 Substituting the value for v, we get t = 1.5 seconds b) For average velocity = total distance/time taken Average velocity = 3 m And the average velocity is...

A man is standing on top of a building 100 m high. He throws two balls vertically, one at t = 0 and other after a time interval. The later ball is thrown at a velocity of half the first. The vertical gap between first and second ball is +15m at t = 2s. The gap is found to remain constant. Calculate the velocity with which the balls were thrown and the exact time interval between their throw.

CBSE, Class 11, Motion in a straight line, NCERT Exemplar, Physics

Let the speed of ball 1 = u1 = 2u m/s Then the speed of ball 2 = u2 = u m/s The height covered by ball 1 before coming to rest = h1 The height covered by ball 2 before coming to rest = h2 We know...

A bird is tossing between two cars moving towards each other on a straight road. One car has a speed of 18 m/h while the other has the speed of 27 km/h. The bird starts moving from first car towards the other and is moving with the speed of 36 km/h and when the two cars were separated by 36 km. What is the total distance covered by the bird? What is the total displacement of the bird?

CBSE, Class 11, Motion in a straight line, NCERT Exemplar, Physics

The relative speed of the cars = 27 + 18 = 45 km/h When the two cars meet together, time t is given as t = distance between cars/relative speed of cars = 36/(27+18) t = 4/5 h Therefore, distance...

A particle executes the motion described by x(t) = x0 (1 – e-γt) where t ≥ 0, x0 > 0 a) Where does the particles start and with what velocity? b) Find maximum and minimum values of x(t), v(t), a(t). Show that x(t) and a(t) increase with time and v(t) decreases with time.

CBSE, Class 11, Motion in a straight line, NCERT Exemplar, Physics

a) x(t) = x0 (1 – e-γt) v(t) = dx(t)/dt = +x0 γ e-γt a(t) = dv/dt = x0 γ2 e-γt v(0) = x0 γ b) x(t) is minimum at t = 0 since t = 0 and [x(t)]min = 0 x(t) is maximum at t = ∞ since t = ∞ and...

An object falling through a fluid is observed to have acceleration given by a = g – bv where g = gravitational acceleration and b is constant. After a long time of release, it is observed to fall with constant speed. What must be the value of constant speed?

CBSE, Class 11, Motion in a straight line, NCERT Exemplar, Physics

The concept used in this question will be based on the behaviour of a spherical object when it is dropped through a viscous fluid. When a spherical body of radius r is dropped, it is first...

Give example of a motion where x>0, v<0, a>0 at a particular instant.

CBSE, Class 11, Motion in a straight line, NCERT Exemplar, Physics

Let the motion be represented as: x(t) = A + Be– γ t Let A>B and γ >0 Velocity is x(t) = dx/dt = -Be– γ t Acceleration is a(t) = dx/dt = B γ 2e– γ t Therefore, it can be said that x(t) > 0,...

Give examples of a one-dimensional motion where a) the particle moving along positive x-direction comes to rest periodically and moves forward b) the particle moving along positive x-direction comes to rest periodically and moves backwardπ

CBSE, Class 11, Motion in a straight line, NCERT Exemplar, Physics

When an equation has sine and cosine functions, the nature is periodic. a) When the particle is moving in positive x-direction, it is given as t > sin t When the displacement is as a function of...

A uniformly moving cricket ball is turned back by hitting it with a bat for a very short time interval. Show the variation of its acceleration with taking acceleration in the backward direction as positive.

CBSE, Class 11, Motion in a straight line, NCERT Exemplar, Physics

The force which is generated by the bat is known as impulsive force. When the effect of gravity is ignored, it can be said that the ball moves with a uniform speed horizontally and returns back to...

Refer to the graphs below and match the following:

CBSE, Motion in a straight line, NCERT Exemplar, NCERT Exemplar, Physics

Graph Characteristics a) i) has v > 0 and a < 0 throughout b) ii) has x > 0 throughout and has a point with v = 0 and a point with a = 0 c) iii) has a point with zero displacement for t...

A ball is bouncing elastically with a speed 1 m/s between walls of a railway compartment of size 10 m in a direction perpendicular to walls. The train is moving at a constant velocity of 10 m/s parallel to the direction of motion of the ball. As seen from the ground, a) the direction of motion of the ball changes every 10 seconds b) speed of ball changes every 10 seconds c) average speed of ball over any 20 seconds intervals is fixed d) the acceleration of ball is the same as from the train

CBSE, Class 11, Motion in a straight line, NCERT Exemplar, Physics

The correct option is b) speed of ball changes every 10 seconds, c) average speed of ball over any 20 seconds intervals is fixed, and d) the acceleration of the ball is the same as from the train

A spring with one end attached to a mass and the other to a rigid support is stretched and released. a) magnitude of acceleration, when just released is maximum b) magnitude of acceleration, when at equilibrium position is maximum c) speed is maximum when mass is at equilibrium position d) magnitude of displacement is always maximum whenever speed is minimum

CBSE, Class 11, Motion in a straight line, NCERT Exemplar, Physics

The correct answer is a) magnitude of acceleration, when just released is maximum and c) speed is maximum when mass is at equilibrium position

For the one-dimensional motion, describe by x = t – sint a) x(t)>0 for all t>0 b) v(t)>0 for all t>0 c) a(t)>0 for all t>0 d) v(t) lies between 0 and 2

CBSE, Class 11, Motion in a straight line, NCERT Exemplar, Physics

The correct answer is a) x(t)>0 for all t>0 and d) v(t) lies between 0 and 2

A graph of x versus t is shown in the figure. Choose correct alternatives from below. a) the particle was released from rest at t=0 b) at B, the acceleration a>0 c) at C, the velocity and the acceleration vanish d) average velocity for the motion A and D is positive e) the speed at D exceeds that at E

CBSE, Class 11, Motion in a straight line, NCERT Exemplar, Physics

The correct answer is a) the particle was released from rest at t=0, c) at C, the velocity and the acceleration vanish and e) the speed at D exceeds that at E

The variation of quantity A with quantity B, plotted in figure describes the motion of a particle in a straight line. a) quantity B may represent time b) quantity A is velocity if motion is uniform c) quantity A is displacement if motion is uniform d) quantity A is velocity if motion is uniformly accelerated

CBSE, Class 11, Motion in a straight line, NCERT Exemplar, Physics

The correct answer is a) quantity B may represent time, c) quantity A is displacement if motion is uniform, and d) quantity A is velocity if motion is uniformly accelerated

At a metro station, a girl walks up a stationary escalator in time t1. If she remains stationary on the escalator, then the escalator take her up in time t2. The time taken by her to walk up on the moving escalator will be a) (t1 + t2)/2 b) t1t2/(t2 – t1) c) t1t2/(t2 + t1) d) t1 – t2

CBSE, Class 11, Motion in a straight line, NCERT Exemplar, Physics

The correct answer is c) t1t2/(t2 + t1)

The displacement of a particle is given by x = (t-2)2 where x is in metres and t is seconds. The distance covered by the particle in first 4 seconds is a) 4 m b) 8 m c) 12 m d) 16 m

CBSE, Class 11, Motion in a straight line, NCERT Exemplar, Physics

The correct answer is b) 8 m

A vehicle travels half the distance L with speed V1 and the other half with speed V2, then its average speed is a) (V1+V2)/2 b) (2V1+V2)/(V1+V2) c) (2V1V2)/(V1+V2) d) L(V1+V2)/V1V2

CBSE, Class 11, Guides, Motion in a straight line, NCERT Exemplar, Physics

The correct answer is c) (2V1V2)/(V1+V2)

A lift is coming from 8th floor and is just about to reach 4th floor. Taking ground floor as origin and positive direction upwards for all quantities, which one of the following is correct? a) x<0, v<0, a>0 b) x>0, v<0, a<0 c) x>0, v<0, a>0 d) x>0, v>0, a<0

CBSE, Class 11, Guides, Motion in a straight line, NCERT Exemplar, Physics

The correct answer is a) x<0, v<0, a<0 The value of x and v becomes negative as the lift is moving from the 8th floor to the 4th floor whereas acceleration is acting upwards and stays...

Among the four graphs, there is only one graph for which average velocity over the time interval (0,T) can vanish for a suitably chosen T. Which one is it?

CBSE, Class 11, Motion in a straight line, NCERT Exemplar, Physics

The correct answer is (b)

A brick manufacture has two depots, $P$ and $Q$ , with stocks of 30000 and 20000 bricks respectively. He receives order from three building $A, B, C$ , for 15000,20000 and 15000 bricks respectively. The costs of transporting 1000 bricks to the building from the depots are given below. How should the manufacture fulfill the orders so as to keep the cost of transportation minimum?

CBSE, Class 12, Exercise 33B, Linear Programming, Maths, RS Aggarwal

Let $x$ bricks be transported from $P$ to $A$ and y bricks be transported from $P$ to $B$. Therefore, $30000-(x+y)$ will be transported to $C$. Also, (15000-x) bricks, ( $20000-y)$ bricks and...

Two godowns, A and B, have a grain storage capacity of 100 quintals and 50 quintals respectively. Their supply goes to three ration shops, D, E and $F$ , whose requirements are 60,50 and 40 quintals respectively. The costs of transportation per quintal from the godowns to the shops are given in the following table.

CBSE, Class 12, Exercise 33B, Linear Programming, Maths, RS Aggarwal

$$ \begin{tabular}{|c|c|c|} \hline & \multicolumn{2}{|c|}{ Cost of transportation (in 2 perquintal) } \\ \hline To & From & B & B \\ \hline$D$ & $6.00$ & 400 \\ \hline$E$ & $3.00$ & $2.00$ \\...

A gardener has a supply of fertilizers of the type 1 which consist of $10 \%$ nitrogen and $6 \%$ phosphoric acid, and of the type II which consist of $5 \%$ nitrogen and $10 \%$ phosphoric acid. After testing the soil condition, he finds that he needs at least $14 \mathrm{~kg}$ of nitrogen and $14 \mathrm{~kg}$ of phosphoric acid for his crop. If the type – I fertilizer costs 60 paise per kg and the type – II fertilizer costs 40 paise per kg, determine how many kilograms of each type of fertilizer should be used so that the nutrient requirement are met at a minimum cost. What is the minimum cost?

CBSE, Class 12, Exercise 33B, Linear Programming, Maths, RS Aggarwal

Let $x$ and $y$ be number of kilograms of fertilizer I and II, $\therefore$ According to the question, $0.10 x+0.05 y \geq 14,0.06 x+0.10 y \geq 14, x \geq 0, y \geq 0$ Minimize $Z=0.60 x+0.40 y$...

A publisher sells a hardcover edition of a book for $\pm 72$ and a paperback edition of the same for $\pm 40 .$ Costs to minutes of printing time although the hardcover edition requires 10 minutes of binding time and the paperback edition requires only 2 minutes. Both the printing and binding operations have 4800 minutes available each week. How many of each type of books should be produced in order to maximize the profit? Also, find the maximum profit per week.

CBSE, Class 12, Exercise 33B, Linear Programming, Maths, RS Aggarwal

Let $x$ and $y$ be number of hardcover and paperback edition of the book. $\therefore$ According to the question, $5 x+5 y \leq 4800,10 x+2 y \leq 4800, x \geq 0, y \geq 0$ Maximize $Z=(72 x+40...

A man owns a field area $1000 \mathrm{~m}^{2}$ . He wants to plant fruit trees in it. He has a sum of $\pm 1400$ to purchase young trees. He has the choice of two types of trees. Type A requires $10 \mathrm{~m}^{2}$ of ground per trees and costs $\pm 20$ per tree, and type B requires $20 \mathrm{~m}^{2}$ of ground per tree and costs $\pm 25$ per tree. When full grown, a type A tree produces an average of $20 \mathrm{~kg}$ of fruit which can be sold at a profit $\pm 2$ per $\mathrm{kg}$ and type $-\mathrm{B}$ tree produces an average of $40 \mathrm{~kg}$ of fruit which can be sold at a profit of $\pm 1.50$ per $\mathrm{kg}$ . How many of each type should be planted to achieve maximum profit when tree are full grown? What is the maximum profit?

CBSE, Class 12, Exercise 33B, Linear Programming, Maths, RS Aggarwal

Let $x$ and $y$ be number of $A$ and B trees. $\therefore$ According to the question, $20 x+25 y \leq 1400,10 x+20 y \leq 1000, x \geq 0, y \geq 0$ Maximize $Z=40 x+60 y$ The feasible region...

A manufacture makes two product, A and B. product A sells at $\mp 200$ each and takes $\frac{1}{2}$ hour to make. Product B sells at $\{300$ each and takes 1 hour to make. There is a permanent order for 14 of product $A$ and 16 of product B. A working week consist of 40 hours of production and the weekly turnover must not be less than \mp10000. If the profit on each of the product $A$ is $\mp 20$ and on product $B$ , it is $\mp 30$ then how many of each should be produced so that the profit is maximum? Also, find the maximum profit.

CBSE, Class 12, Exercise 33B, Linear Programming, Maths, RS Aggarwal

Let $x$ and $y$ be number of $A$ and $B$ products. $\therefore$ According to the question. $0.5 x+y \leq 40,200 x+300 y \geq 10000, x \geq 14, y \geq 16$ Maximize $Z=20 x+30 y$ The feasible region...

A manufacture makes two types, A and B, of teapots. Three machines are needed for the manufacture and the time required for each teapot on the machines is given below. Each machine is available for a maximum of 6 hours per day. If the profit on each teapot of type $A$ is 75 paise and that on each teapot of type $B$ is 50 paise, show that 15 teapots of type $A$ and 30 of type B should be manufactured in a day to get the maximum profit.

CBSE, Class 12, Exercise 33B, Linear Programming, Maths, RS Aggarwal

$$ \begin{tabular}{|l|l|l|l|} \hline Machine & \multicolumn{2}{|l|}{ Time (in minutes) } \\ \hline Type & I & II & III \\ \hline A & 12 & 18 & 6 \\ \hline B & 6 & 0 & 9 \\ \hline \end{tabular} $$...

A small firm manufactures gold rings and chains. The combined number of rings and chains manufactured per day is at most 24. It takes 1 hour to make a ring and half an hour for a chain. The maximum number of hour to available per day is 16 . If the profit on a ring is 300 and that on a chain is 190, how many of each should be manufactured daily so as to maximize the profit?

CBSE, Class 12, Exercise 33B, Linear Programming, Maths, RS Aggarwal

Let $x$ and $y$ be number of gold rings and chains. $\therefore$ According to the question, $x+y \leq 24, x+0.5 y \leq 16, x \geq 0, y \geq 0$ Maximize $Z=300 x+190 y$ The feasible region determined...

A company producing soft drinks has a contrast which requires a minimum of 80 units of chemical $A$ and 60 , units of chemical $B$ to go in each bottle of the drink. The chemical are available in a prepared mix from two different suppliers. Supplier $X$ has a mix of 4 units of $A$ and 2 units of $B$ that costs Rs.10, and the supplier $Y$ has a mix of 1 unit of $A$ and 1 unit of $B$ that costs \mp4. How many mixes from $X$ and $Y$ should the company purchase to honor the contract requirement and yet minimize the cost?

CBSE, Class 12, Exercise 33B, Linear Programming, Maths, RS Aggarwal

Let $x$ and $y$ be number of mixes from suppliers $X$ and $Y$. $\therefore$ According to the question, $4 x+y \geq 80,2 x+y \geq 60, x \geq 0, y \geq 0$ Minimize $Z=10 x+4 y$ The feasible region...

A small manufacture has employed 5 skilled men and 10 semiskilled men and makes an article in two qualities, a deluxe model and an ordinary model. The making of a deluxe model requires 2 hours work by a skilled man and 2 hours work by a semiskilled man. The ordinary model requires 1 hour by a skilled man and 3 hours by a semiskilled man. By union rules, no man can work more than 8 hours per day. The manufacture gains \mp 15 on the deluxe model and \mp 10 on the ordinary model. How many of each type should be made in order to maximize his total daily profit? Also, find the maximum daily profit.

CBSE, Class 12, Exercise 33B, Linear Programming, Maths, RS Aggarwal

Let $x$ and $y$ be number of deluxe article manufactured and ordinary article manufactured. $\therefore$ According to the question, $2 x+y \leq 40,2 x+3 y \leq 80, x \geq 0, y \geq 0$ Maximize $Z=15...

A toy company manufactures two types of dolls, A and B. Each doll of type B take twice as long to produce as one of type A, and the company would have time to make a maximum of 2000 per day, if it produces only type A. the supply of plastic is sufficient to produce 1500 dolls per day (both A and B combined). Type B requires a fancy dress of which there are only 600 per day available. If the company makes profit of Rs.3 and $\pm5$ per dolls respectively on dolls A and B, how many of each should be produced per day in order to maximize the profit? Also, find the maximum profit.

CBSE, Class 12, Exercise 33B, Linear Programming, Maths, RS Aggarwal

Let $x$ and $y$ be number of doll A manufactured and doll B manufactured. $\therefore$ According to the question, $x+y \leq 1500, x+2 y \leq 2000, y \leq 600, x \geq 0, y \geq 0$ Maximize $Z=3 x+5...

A manufacture of a line of patent medicines is preparing a production plan on medicines $A$ and $B$ . There are sufficient ingredients available to make 20000 bottles of $A$ and 40000 bottles of B but there are only 45000 bottles into which either of the medicines can be put. Furthermore, it takes 3 hours to prepare enough material to fill 1000 bottles of A and it takes 1 hour to prepare enough material to fill 1000 bottles of $B$ , and there are 66 hours available for this operation. The profit is $\mp 8$ per bottle for $A$ and $\mp 7$ per bottle for $B$ . How should the manufacture schedule the production in order to maximize his profit? Also, find the maximum profit.

CBSE, Class 12, Exercise 33B, Linear Programming, Maths, RS Aggarwal

Let $x$ and $y$ be number of bottles of medicines $A$ and $B$ be prepared. $\therefore$ According to the question, $x+y \leq 45000,3 x+y \leq 66000, x \leq 20000, y \leq 40000 \cdot x \geq 0, y \geq...

A manufactures produces two types of soap bars using two machines, A and B. A is operated for 2 minutes and $B$ for 3 minutes to manufacture the first type, while it takes 3 minutes on machine $A$ and 5 minutes on machine B to manufacture the second type. Each machine can be operated at the most for 8 hours per day. The two types of soap bars are sold at a profit of $0.25$ and $\approx 0.50$ each. Assuming that the manufacture can sell all the soap bars he can manufacture, how many bars of soap of each type should be manufactured per day so as to maximize his profit?

CBSE, Class 12, Exercise 33B, Linear Programming, Maths, RS Aggarwal

Let $x$ and $y$ be number of soaps be manufactured of $1^{\text {st }}$ and $2^{\text {nd }}$ type. $\therefore$ According to the question, $2 x+3 y \leq 480,3 x+5 y \leq 480, x \geq 0, y \geq 0$...

A firm manufactures two types of products, $A$ and $B$ , and sells them at a profit of $\approx 2$ on type $A$ and B. Each product is processed on two machines, $M_{1}$ and $M_{2}$ . Type A requires one minute of processing time on $M_{1}$ and two minutes on $M_{2} .$ Type $B$ requires one minute on $M_{1}$ and one minute on $M_{2}$ is available for not more than 6 hours 40 minutes while $M_{2}$ is available for at most 10 hours a day. Find how many products of each type the firm should produce each day in order to get maximum profit.

CBSE, Class 12, Exercise 33B, Linear Programming, Maths, RS Aggarwal

Let the firm manufacture $x$ number of Aand y number of $B$ products. $\therefore$ According to the question, $X+y \leq 400,2 x+y \leq 600, x \geq 0, y \geq 0$ Maximize $Z=2 x+2 y$ The feasible...

A dealer wishes to purchase a number of fans and sewing machines. He has only $\pm 5760$ to invest and space and $\pm 18$ on a sewing machine. Assuming that he can sell all the items he can buy, how should he invest the money in order to maximize the profit?

CBSE, Class 12, Exercise 33B, Linear Programming, Maths, RS Aggarwal

Let the number of fans bought be $x$ and sewing machines bought be $y$. $\therefore$ According to the question, $360 x+240 y \leq 5760, x+y \leq 20, x \geq 0, y \geq 0$ Maximize $Z=22 x+18 y$ The...

Question while B can stitch 10 shirts and 4 pairs of trousers per day. How many days should each of them work if it is desired to produce at least 60 shirts and 32 pairs of trousers at a minimum labor cost?

CBSE, Class 12, Exercise 33B, Linear Programming, Maths, RS Aggarwal

Let the total number of days tailor $A$ work be $x$ and tailor $B$ be $y$. $\therefore$ According to the question, $6 x+10 y \geq 60,4 x+4 y \geq 32, x \geq 0, y \geq 0$ Minimize $Z=300 x+400 y$ The...

A manufacture produces nuts and bolts for industrial machinery. It takes 1 hour of work on machine $A$ and 3 hours on machine B to produces a packet of nuts while it takes 3 hours on machine $A$ and 1 hours on machine B to produce a packet of bolts. He earns a profit \mp17.50 per packet on nuts and \mp7 per packet on bolts. How many packets of each should be produced each day so as to maximize his profit if he operates, each machine for at the most 12 hours a day? Also find the maximum profit.

CBSE, Class 12, Exercise 33B, Linear Programming, Maths, RS Aggarwal

Let the number of packets of nuts and bolts be $x$ and y respectively. $\therefore$ According to the question, $x+3 y \leq 12,3 x+y \leq 12, x \geq 0, y \geq 0$ Maximize $Z=17.50 x+7 y$ The feasible...

The most reactive amine towards dilute hydrochloric acid is ___________.

Chemistry, Class 11, NCERT Exemplar

Solution: Option (ii) is the answer. Reason: The reactivity of amines is proportional to their basicity. If the R group is, the order of basicity is secondary amine ...

A man has $\pm1500$ to purchase rice and wheat. A bag of rice and a bag of wheat cost \mp 180 and 120 respectively. He has storage capacity of 10 bags only. He earns a profit of $\mp 11$ and 78 per bag of rice and wheat respectively. How many bags of each must he buy to make maximum profit?

CBSE, Class 12, Exercise 33B, Linear Programming, Maths, RS Aggarwal

Let the number of wheat and rice bags be $x$ and $y$. $\therefore$ According to the question, $120 x+180 y \leq 1500, x+y \leq 10, x \geq 0, y \geq 0$ Maximize $Z=8 x+11 y$ The feasible region...

A small firm manufactures necklace and bracelets. The total number of necklace and bracelet that it can handle per day is at most 24. It takes 1 hour to make a bracelet and half an hour to make a necklace. The maximum number of hours available per day is 16 . If the profit on a necklace is $\pm 100$ and that on a bracelet is $\pm 300$ , how many of each should be produced daily to maximize the profit? It is being given that at least one of each must be produced.

CBSE, Class 12, Exercise 33B, Linear Programming, Maths, RS Aggarwal

Let the firm manufacture $x$ number of necklaces and y number of bracelets a day. $\therefore$ According to the question, $x+y \leq 24,0.5 x+y \leq 16 x \geq 1, y \geq 1$ Maximize $Z=100 x+300 y$...

Mr.Dass wants to invest Rs 12000 in public provident fund (PPF) and in national bonds. He has to invest at $\mathrm{~ l e a s t ~ Rs. 1 0 0 0 ~ i n ~ P P F ~ a n d ~ a t ~ l e a s t ~ }$ bonds is $15 \%$ per annum, how should he invest the money to eam maximum annual income? Also find the maximum annual income.

CBSE, Class 12, Exercise 33B, Linear Programming, Maths, RS Aggarwal

Let the invested money in PPF be $x$ and in national bonds be $y$. $\therefore$ According to the question, $\begin{array}{l} \mathrm{X}+\mathrm{y} \leq 12000 \\ \mathrm{x} \geq 1000, \mathrm{y} \geq...

Maximize $Z=3 x+5 y$ , subject to the constraints $x+2 y \leq 2000, x+y \leq 1500, y \leq 00, y \leq 600, x \geq 0$ and $y \geq 0$

CBSE, Class 12, Exercise 33B, Linear Programming, Maths, RS Aggarwal

The feasible region determined by the $X+2 y \leq 2000, x+y \leq 1500, y \leq 600, x \geq 0$ and $y \geq 0$ is given by The corner points of the feasible region are $A(0,0), B(0,600), C(800,600),...

Minimize $Z=2 x+3 y$ , subject to the constraints $x \geq 0, y \geq 0, x+2 y \geq 1$ and $x+2 y \leq 10$

CBSE, Class 12, Exercise 33B, Linear Programming, Maths, RS Aggarwal

The feasible region determined by the $x \geq 0, y \geq 0, x+2 y \geq 1$ and $x+2 y \leq 10$ is given by The corner points of the feasible region is $A\left(0 \frac{1}{2}\right), B(0,5), C(10,0),...

Find the minimum value of $Z=3 x+5 y$ , subject to the constraints $-2 x+y \leq 4, x+y \geq 3, x-2 y \leq 2, x \geq 0$ and $y \geq 0$

CBSE, Class 12, Exercise 33B, Linear Programming, Maths, RS Aggarwal

The feasible region determined by the $-2 x+y \leq 4, x+y \geq 3, x-2 y \leq 2, x \geq 0$ and $y_{z} 0$ is given by Here the feasible region is unbounded. The vertices of the region are $A(0,4),...

Maximize $Z=4 x+9 y$ , subject to the constraints $x \geq 0, y \geq 0, x+5 y \leq 200,2 x+3 y \leq 134$

CBSE, Class 12, Exercise 33B, Linear Programming, Maths, RS Aggarwal

The feasible region determined by the constraints $x \geq 0, y \geq 0, x+5 y \leq 200,2 x+3 y \leq 134$ is given by The corner points of feasible region are $A(10,38), B(0,40), C(0,0), D(67,0)$. The...

Find the maximum value of $Z=7 \mathrm{X}+7 \mathrm{Y}$ , subject to the constraints. $x \geq 0, y \geq 0, x+y \geq 2$ and $2 x+3 y \leq 6$

CBSE, Class 12, Exercise 33B, Linear Programming, Maths, RS Aggarwal

The feasible region determined by the constraints $x \geq 0, y \geq 0$, $x+y \geq 2,2 x+3 y \leq 6$ is given by The corner points of the feasible region is $A(0,2), B(2,0), C(3,0)$. The values of...

Find the linear constraints for which the shaded area in the figure given is the solution set.

CBSE, Class 12, Exercise 33A, Linear Programming, Maths, RS Aggarwal

Solution: Consider A: Given line $x-y=1$ $\Rightarrow y=x-1$ As the region given in the figure is above the $y$ - intercept's coordinates $(0,-1)$, $\begin{array}{l} \Rightarrow y \geq x-1 \\...

Show that the solution set of the following linear constraints is empty: $x-2 y \geq 0,2 x-y \leq-2, x \geq 0$ and $y \geq 0$

CBSE, Class 12, Exercise 33A, Linear Programming, Maths, RS Aggarwal

Consider the inequation $x-2 y \geq 0$ $\begin{array}{l} \Rightarrow x \geq 2 y \\ \Rightarrow y \leq \frac{x}{2} \end{array}$ consider the equation $y=\frac{x}{2}$. This equation's graph is a...

Solve each of the following systems of simultaneous inequations: $3 x+4 y \geq 12, x \geq 0, y \geq 1$ and $4 x+7 y \leq 28$

CBSE, Class 12, Exercise 33A, Linear Programming, Maths, RS Aggarwal

Consider the inequation $3 x+4 y \geq 12$ $\Rightarrow 4 y \geq 12-3 x$ $\Rightarrow y \geq 3-\frac{3}{4} x$ Consider the equation $y=3-\frac{3}{4} x$ Finding points on the coordinate axes: If...

Solve each of the following systems of simultaneous inequations: $x-2 y \geq 0,2 x-y \leq-2$

CBSE, Class 12, Exercise 33A, Linear Programming, Maths, RS Aggarwal

Consider the inequation $x-2 y \geq 0$ : $\begin{array}{l} \Rightarrow x \geq 2 y \\ \Rightarrow y \leq \frac{x}{2} \end{array}$ consider the equation $y=\frac{x}{2}$. This equation's graph is a...

Let $f: R \rightarrow R: f(x)=10 x+7 .$ Find the function $g: R \rightarrow R: g \circ f=f \circ g=I_{g^{-}}$

CBSE, Class 12, Exercise 2C, Functions, Maths, RS Aggarwal

Solution: We need to find: the function $g: R \rightarrow R: g \circ f=f$ o $g=I_{g}$ Formula used: (i) $g$ o $f=g(f(x))$ (ii) f o $g=f(g(x))$ Given that: $f: \mathbb{R} \rightarrow \mathbb{R}:...

Solve each of the following systems of simultaneous inequations: $2 x+y>1$ and $2 x-y \geq-3$

CBSE, Class 12, Exercise 33A, Linear Programming, Maths, RS Aggarwal

Consider the inequation $2 x+y>1:$ $ \Rightarrow y>1-2 x$ Consider the equation $y=1-2 x$ Finding points on the coordinate axes: If $x=0$, the y value is 1 i.e, $y=1$ $\Rightarrow$ the point...

Graph the solution sets of the following inequations: $y-2 \leq 3 x$

CBSE, Class 12, Exercise 33A, Linear Programming, Maths, RS Aggarwal

Given $y-2 \leq 3 x$ $\Rightarrow y \leq 3 x+2$ Consider the equation $y=3 x+2$ Finding points on the coordinate axes: If $x=0$, the $y$ value is 2 i.e, $y=2$ $\Rightarrow$ the point on $Y$ axis is...

Let $\mathrm{f}: \mathrm{R} \rightarrow \mathrm{R}$ and $\mathrm{g}: \mathrm{R} \rightarrow \mathrm{R}$ defined by $\mathrm{f}(\mathrm{x})=\mathrm{x}^{2}$ and $g(\mathrm{x})=(\mathrm{x}+1)$ . Show that $\mathrm{g}$ o $\mathrm{f} \neq \mathrm{f} \circ \mathrm{g}$ .

CBSE, Class 12, Exercise 2C, Functions, Maths, RS Aggarwal

Solution: We need to prove: $g$ o $f \neq f$ o $g$ Formula used: (i) f o $\mathrm{g}=\mathrm{f}(\mathrm{g}(\mathrm{x}))$ (ii) $g$ of $=g(f(x))$ Given that: (i) $f: R \rightarrow R: f(x)=x^{2}$ (ii)...

Prove that the function f: $N \rightarrow N: f(x)=3 x$ is one-one and into.

CBSE, Class 12, Exercise 2C, Functions, Maths, RS Aggarwal

Solution: We need to prove: function is one-one and into It is given that: $f: N \rightarrow N: f(x)=3 x$ We have, $f(x)=3 x$ For, $f\left(x_{1}\right)=f\left(x_{2}\right)$ $\begin{array}{l}...

Let $f: R \rightarrow R: f(x)=(2 x+1)$ and $g: R \rightarrow R: g(x)=\left(x^{2}-2\right)$ Write down the formulae for.
(i) (f of) (ii) $(\mathrm{g} \circ \mathrm{g})$

CBSE, Class 12, Exercise 2B, Functions, Maths, RS Aggarwal

Solution: (i) f o f We need to find: $f$ o $f$ Formula used: $f$ o $f=f(f(x))$ It is given that: (i) $f: R \rightarrow R: f(x)=(2 x+1)$ Solution: We have, $\begin{array}{l} \text { fof }=f(f(x))=f(2...

Solve $\frac{\mathrm{dy}}{\mathrm{dx}}=\mathrm{x}(2 \log \mathrm{x}+1)$ , given that $\mathrm{y}=0$ when $\mathrm{x}=2$

CBSE, Class 11, Differential Equations with Variable Separable, Exercise 19B, Maths, RS Aggarwal

Solution: Given that, $\frac{d y}{d x}=2 x \log x+x$, On integrating we get, $\begin{array}{l} y=\int(2 x \log x+x) d x \\ y=\int 2 x \log x d x+x d x \\ y=\left(\int 2 x d x\right) \log...

For each of the following differential equations, find a particular solution satisfying the given condition : $\frac{\mathrm{dy}}{\mathrm{dx}}=\mathrm{y} \tan \mathrm{x}$ , it being given that $y=1$ when $x=0$

CBSE, Class 12, Differential Equations with Variable Separable, Excercise 19A, Maths, RS Aggarwal

Solution: On rearranging the terms we obtain: $\begin{array}{l} \frac{d y}{y}=\tan x d x \\ \Rightarrow \int \frac{d y}{y}=\int \tan x d x+c \\ \Rightarrow \log |y|=\log |\sec x|+\log c \end{array}$...

For each of the following differential equations, find a particular solution satisfying the given condition : $x d y=\left(2 x^{2}+1\right) d x(x \neq 0)$ , given thaty $=1$ when $x=1$

CBSE, Class 12, Differential Equations with Variable Separable, Excercise 19A, Maths, RS Aggarwal

Solution: On rearranging the terms we obtain: $\begin{array}{l} d y=\frac{2 x^{2}+1}{x} d x \\ \Rightarrow d y=2 x d x+\frac{1}{x} d x \end{array}$ On integrating both sides we obtain:...

For each of the following differential equations, find a particular solution satisfying the given condition: $\frac{\mathrm{dy}}{\mathrm{dx}}=-4 \mathrm{xy}^{2}$ , it being given that $\mathrm{y}=1$ when $\mathrm{x}=0$

CBSE, Class 12, Differential Equations with Variable Separable, Excercise 19A, Maths, RS Aggarwal

Solution: On rearranging the terms we obtain: $\frac{d y}{y^{2}}=-4 x d x$ On integrating both sides we obtain: $\Rightarrow \int \frac{d y}{y^{2}}=-\int 4 x d x+c$ $\Rightarrow...

For each of the following differential equations, find a particular solution satisfying the given condition: $\cos \left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)=\mathrm{a}, \cdot$ where $\mathrm{a} \in \mathrm{R}$ and $\mathrm{y}=2$ when $\mathrm{x}=0$

CBSE, Class 12, Differential Equations with Variable Separable, Excercise 19A, Maths, RS Aggarwal

Solution: $\begin{array}{l} \cos \left(\frac{d y}{d x}\right)=a \\ \Rightarrow \frac{d y}{d x}=\cos ^{-1} a \\ \Rightarrow d y=\cos ^{-1} a d x \end{array}$ On integrating both sides we obtain:...

Find the general solution of each of the following differential equations: $\cos x(1+\cos y) d x-\sin y(1+\sin x) d y=0$

CBSE, Class 12, Differential Equations with Variable Separable, Excercise 19A, Maths, RS Aggarwal

Solution: On rearranging the terms we obtain: $\frac{\cos x d x}{(1+\sin x)}=\frac{\sin y d y}{(1+\operatorname{cosy})}$ On integrating both the sides we obtain: $\Rightarrow \int \frac{\cos x d...

Find the general solution of each of the following differential equations: $\sec ^{2} x \tan y d x+\sec ^{2} y \tan x d y=0$

CBSE, Class 12, Differential Equations with Variable Separable, Excercise 19A, Maths, RS Aggarwal

Solution: On rearranging the terms we obtain: $\frac{\sec ^{2} x d x}{\tan x}=-\frac{\sec ^{2} y d y}{\tan y}$ On integrating both sides we obtain: $\begin{array}{l} \Rightarrow \int \frac{\sec ^{2}...

Find the general solution of each of the following differential equations: $e^{x} \tan y d x+\left(1-e^{x}\right) \sec ^{2} y d y=0$

CBSE, Class 12, Differential Equations with Variable Separable, Excercise 19A, Maths, RS Aggarwal

Solution: On rearranging all the terms we obtain: $\frac{e^{x} d x}{1-e^{x}}=-\frac{\sec ^{2} y d y}{\operatorname{tany}}$ On integrating both sides we obtain: $\begin{array}{l} \Rightarrow \int...

Find the general solution of each of the following differential equations: $e^{2 x-3 y} d x+e^{2 y-3 x} d y=0$

CBSE, Class 12, Differential Equations with Variable Separable, Excercise 19A, Maths, RS Aggarwal

Solution: $e^{2 x} e^{-3 y} d x+e^{2 y} e^{-3 x} d y=0$ On rearringing the terms we obtain: $\begin{array}{l} \Rightarrow \frac{e^{2 x} d x}{e^{-3 x}}=-\frac{e^{2 y} d y}{e^{-3 y}} \\ \Rightarrow...

Find the general solution of each of the following differential equations: $\frac{d y}{d x}=e^{x} e^{-y}+x^{2} e^{-y}$

CBSE, Class 12, Differential Equations with Variable Separable, Excercise 19A, Maths, RS Aggarwal

Solution: $\begin{array}{l} \text { It is given that: } \frac{d y}{d x}=e^{x} e^{-y}+x^{2} e^{-y} \\ \Rightarrow \frac{d y}{d x}=e^{-y}\left(e^{x}+x^{2}\right) \\ \Rightarrow \frac{d...

Find the general solution of each of the following differential equations: $\left(e^{x}+e^{-x}\right) d y-\left(e^{x}-e^{-x}\right) d x=0$

CBSE, Class 12, Differential Equations with Variable Separable, Excercise 19A, Maths, RS Aggarwal

Solution: $\begin{array}{l} \left(e^{x}+ e^{-x}\right) d y-\left(e^{x}-e^{-x}\right) d x=0 \\ \Rightarrow d y=\frac{e^{x}-e^{-x}}{e^{x}+e^{-x}} d x \end{array}$ On integrating both sides we obtain,...

Find the general solution of each of the following differential equations: $\frac{d y}{d x}=e^{x+y}$

CBSE, Class 12, Differential Equations with Variable Separable, Excercise 19A, Maths, RS Aggarwal

Solution: $\frac{d y}{d x}=e^{x} e^{y}$ On rearringing the terms we obtain: $\Rightarrow \frac{d y}{e^{y}}=e^{x} d x$ On integrating both sides we obtain, $\begin{array}{l} \Rightarrow \int \frac{d...

Find the general solution of each of the following differential equations: $(x-1) \frac{d y}{d x}=2 x^{3} y$

CBSE, Class 12, Differential Equations with Variable Separable, Excercise 19A, Maths, RS Aggarwal

Solution: $(x-1) \frac{d y}{d x}=2 x^{3} y$ On separating the variables we obtain: $\begin{array}{l} \Rightarrow \frac{d y}{y}=2 x^{3} \frac{d x}{(x-1)} \\ \Rightarrow \frac{d...

Find the general solution of each of the following differential equations: $\frac{\mathrm{dy}}{\mathrm{dx}}=1-\mathrm{x}+\mathrm{y}-\mathrm{xy}$

CBSE, Class 12, Differential Equations with Variable Separable, Excercise 19A, Maths, RS Aggarwal

Solution: $\begin{array}{l} \Rightarrow \frac{d y}{d x}=1-x+y-x y=1+y-x(1+y) \\ \Rightarrow \frac{d y}{d x}=(1+y)(1-x) \end{array}$ On rearranging the terms we obtain: $\Rightarrow \frac{d...

Find the general solution of each of the following differential equations: $\frac{\mathrm{dy}}{\mathrm{dx}}=1+\mathrm{x}+\mathrm{y}+\mathrm{xy}$

CBSE, Class 12, Differential Equations with Variable Separable, Excercise 19A, Maths, RS Aggarwal

Solution: $\begin{array}{l} \frac{d y}{d x}=1+x+y+x y=1+y+x(1+y) \\ \Rightarrow \frac{d y}{d x}=(1+y)(1+x) \end{array}$ On rearranging the terms we obtain: $\Rightarrow \frac{d y}{1+y}=(1+x) d x$ On...

Find the general solution of each of the following differential equations: $x^{4} \frac{d y}{d x}=-y^{4}$

CBSE, Class 12, Differential Equations with Variable Separable, Excercise 19A, Maths, RS Aggarwal

Solution: $\begin{array}{l} x^{4} \frac{d y}{d x}=-y^{4} \\ \Rightarrow \frac{d y}{-y^{4}}=\frac{d x}{x^{4}} \end{array}$ On integrating both sides we obtain, $\begin{array}{l} \Rightarrow \int...

Find the general solution of each of the following differential equations: $\frac{\mathrm{dy}}{\mathrm{dx}}=\left(1+\mathrm{x}^{2}\right)\left(1+\mathrm{y}^{2}\right)$

CBSE, Class 12, Differential Equations with Variable Separable, Excercise 19A, Maths, RS Aggarwal

Solution: $\frac{d y}{d x}=\left(1+x^{2}\right)\left(1+y^{2}\right)$ On rearranging the terms,we obtain: $\Rightarrow \frac{d y}{1+y^{2}}=\left(1+x^{2}\right) d x$ On integrating both sides we...

Let $f(x)=\left\{\begin{array}{c}(2-x), \text { when } x \geq 1 \\ x, \text { when } 0 \leq x \leq 1\end{array}\right.$ Show that $f(x)$ is continuous but not differentiable at $x=1$

CBSE, Class 12, Continuity and Differentiability, Exercise 9C, Maths, RS Aggarwal

Solution: L.H.L. at $x=1$ $\operatorname{Lim}_{x \rightarrow 1^{-}} f(x)=\lim _{x \rightarrow 1} x=1$ $f(x)=x$ is polynomial function and a polynomial function is continuous everywhere R.H.L. at...

Show that constant function is always differentiable.

CBSE, Class 12, Continuity and Differentiability, Exercise 9C, Maths, RS Aggarwal

Solution: Asumme that $a$ is any constant number. Therefore, $f(x)=a$ $\mathrm{f}^{\prime}(\mathrm{x})=\lim _{\mathrm{h} \rightarrow 0}...

Discuss the continuity of $f(x)=[x]$

CBSE, Class 12, Continuity and Differentiability, Exercise 9B, Maths, RS Aggarwal

Solution: Assume that $n$ be any integer $[\mathrm{x}]=$ Greatest integer less than or equal to $x$ Some values of $[x]$ for specific values of $x$ $\begin{array}{l} {[3]=3} \\ {[4.4]=4} \\...

Show that sec $x$ is a continuous function.

CBSE, Class 12, Continuity and Differentiability, Exercise 9B, Maths, RS Aggarwal

Solution: Assume $f(x)=\sec x$ So, $f(x)=\frac{1}{\cos x}$ $f(x)$ is not defined when $\cos x=0$ And $\cos x=0$ when, $x=\frac{\pi}{2}$ and odd multiples of $\frac{\pi}{2}$ like $-\frac{\pi}{2}$...

Show that function $f(x)=\left\{\begin{array}{cl}\sin x, & \text { if } x<0 \\ x, & \text { if } x \geq 0\end{array}\right.$ is continuous.

CBSE, Class 12, Continuity and Differentiability, Excercise 9B, Maths, RS Aggarwal

Solution: It is given that: $f(x)=\left\{\begin{array}{c} \sin x, \text { if } x<0 \\ x, \text { if } x \geq 0 \end{array}\right.$ L.H.L. at $\mathrm{x}=0$ $\lim _{x \rightarrow 0^{-}}...

For what value of $k$ is the following function $f(x)=\left\{\begin{array}{c}\frac{x^{2}-9}{x-3}, \text { when } x \neq 3 \\ k, \text { when } x=3\end{array}\right.$ is continuous at $x=3$ Ans. $k=6$

CBSE, Class 12, Continuity and Differentiability, Exercise 9A, Maths, RS Aggarwal

Solution: As, $f(x)$ is continuous at $x=3$ $\begin{array}{l} \Rightarrow \lim _{x \rightarrow 3} \frac{x^{2}-9}{x-3}=f(3) \\ \Rightarrow \lim _{x \rightarrow 3} \frac{(x-3)(x+3)}{x-3}=f(3) \\...

Prove that $f(x)=\left\{\begin{array}{c} \frac{\sin 3 x}{x}, \text { when } x \neq 0 \\ 1, \text { when } x=0 \end{array} \text { is discontinuous at } x=0\right.$

CBSE, Class 12, Continuity and Differentiability, Exercise 9A, Maths, RS Aggarwal

Solution: Left Hand Limit: $\lim _{\mathrm{x} \rightarrow 0-} \mathrm{f}(\mathrm{x})=\lim _{\mathrm{x} \rightarrow 0-} \frac{\sin 3 \mathrm{x}}{\mathrm{x}}$ $=3$ $\left[\lim _{x \rightarrow a}...

Prove that $f(x)=\left\{\begin{array}{c} \frac{x^{2}-x-6}{x-3}, \text { when } x \neq 3 \text { is continuous at } x=3 \\ 5, \text { when } x=3 \end{array}\right.$

CBSE, Class 12, Continuity and Differentiability, Excercise 9A, Maths, RS Aggarwal

Solution: Left Hand Limit: $\lim _{x \rightarrow 3-} f(x)=\lim _{x \rightarrow 3-} \frac{x^{2}-x-6}{x-3}$ $=\lim _{\mathrm{x} \rightarrow 3-} \frac{(\mathrm{x}+2)(\mathrm{x}-3)}{\mathrm{x}-3}$ [By...

Two schools A and B want to award their selected students on the values of sincerity, truthfulness and helpfulness. The school A wants to award ₹ x each, ₹ y each and ₹ z each for the three respective values to 3, 2 and 1 students respectively with total award money of ₹ 1,600. School B wants to spend ₹ 2,300 to award its 4, 1 and 3 students on the respective values (by giving the same award money to the three values as before). If the total amount of award for one prize on each value is ₹ 900, using matrices, find the award money for each value. Apart from these three values, suggest one more value which should be considered for award. HINT: By the given data, we have
$3 x+2 y+z=1600$
$4x+y+3 z=2300$
$x+y+z=900$

CBSE, Class 12, Excercise 8A, Maths, RS Aggarwal, System of Linear Equations

Solution: Assume the amount considered for sincerity, truthfulness and helpfulness are $x, y$ and $z$ respectively. As per the questions, $3 x+2 y+z=1600$ $\begin{array}{l} 4 x+y+3 z=2300 \\...

An amount of ₹ 5000 is put into three investments at 6%, 7% and 8% per annum respectively. The total annual income from these investments is ₹358. If the total annual income from first two investments is ₹70 more than the income from the third, find the amount of each investment by the matrix method. HINT: Let these investments be ₹x, ₹y and ₹z, respectively. Then, $x+y+z=5000, \ldots$ (i) $\begin{array}{l} \frac{6 x}{100}+\frac{7 y}{100}+\frac{8 z}{100}=358 \Rightarrow \\ 6 x+7 y+8 z=35800 \ldots (ii) \end{array}$ And, $\frac{6 x}{100}+\frac{7 y}{100}=\frac{8 z}{100}+70$ $\Rightarrow 6 x+7 y-8 z=7000 . \ldots \text { (iii) }$

CBSE, Class 12, Excercise 8A, Maths, RS Aggarwal, System of Linear Equations

Solution: Suppose the investments are $\mathrm{x} \mathrm{x}$, Fy and $\mathrm{F} \mathrm{z}$, respectively. Therefore, $x+y+z=5000$ $\begin{array}{l} \frac{6 x}{100}+\frac{7 y}{100}+\frac{8...

The cost of 4 kg potato, 3 kg wheat and 2 kg of rice is ₹ 60. The cost of 1 kg potato, 2 kg wheat and 3 kg of rice is ₹45. The cost of 6 kg potato, 2 kg wheat and 3 kg of rice is ₹70. Find the cost of each item per kg by matrix method.

CBSE, Class 12, Excercise 8A, Maths, RS Aggarwal, System of Linear Equations

Solution: Suppose the price of 1kg potato, wheat and rice is $x$, $y$ and $z$ respectively. As per the question, $4x + 3y + 2z = 60$ $x+ 2y + 3z = 45$ $6x + 2y + 3z = 70$ Now converting the...

The sum of three numbers is 2. If twice the second number is added to the sum of first and third, we get 1. On adding the sum of second and third numbers to five times the first, we get 6. Find the three numbers by using matrices.

CBSE, Class 12, Excercise 8A, Maths, RS Aggarwal, System of Linear Equations

Solution: Assume the numbers are $\mathrm{x}, \mathrm{y}$ and $\mathrm{z}$. As per the question, $\begin{array}{l} x+y+z=2 \\ x+2 y+z=1 \\ 5 x+y+z=6 \end{array}$ Now converting the following...

$\begin{array}{l} \frac{1}{x}-\frac{1}{y}+\frac{1}{z}=4 ; \frac{2}{x}+\frac{1}{y}-\frac{3}{z}=0 \\ \frac{1}{x}+\frac{1}{y}+\frac{1}{z}=2 \cdot(x, y, z \neq 0) \end{array}$

CBSE, Class 12, Excercise 8A, Maths, RS Aggarwal, System of Linear Equations

Solution: We need to find: $-x, y, z$, The given set of lines are : $\begin{array}{l} \frac{1}{x}-\frac{1}{y}+\frac{1}{z}=4 \\ \frac{2}{x}+\frac{1}{y}-\frac{3}{z}=0 \\...

$\begin{array}{l} \frac{2}{x}-\frac{3}{y}+\frac{3}{z}=10, \frac{1}{x}+\frac{1}{y}+\frac{1}{z}=10 \\ \frac{3}{x}-\frac{1}{y}+\frac{2}{z}=13 \end{array}$

CBSE, Class 12, Excercise 8A, Maths, RS Aggarwal, System of Linear Equations

Solution: We need to find: $-x, y, z$ The given set of lines are : - $\begin{array}{l} \frac{2}{x}-\frac{3}{y}+\frac{3}{z}=10 \\ \frac{1}{x}+\frac{1}{y}+\frac{1}{z}=10 \\...

If $A=\left(\begin{array}{ccc}1 & -2 & 0 \\ 2 & 1 & 3 \\ 0 & -2 & 1\end{array}\right)$ and $B=\left(\begin{array}{ccc}7 & 2 & -6 \\ -2 & 1 & -3 \\ -4 & 2 & 5\end{array}\right)$ , find $A B$ Hence, solve the system of equations:
$x-2 y=10$
$2 x+y+3 z=8$ and
$-2 y+z=7$ HINT: $A B=(11) /=A\left(\frac{1}{11} B\right)=/$ $A^{-1}=\left(\frac{1}{11}\right) B$

CBSE, Class 12, Excercise 8A, Maths, RS Aggarwal, System of Linear Equations

Solution: It is given, $\begin{array}{l} A=\left[\begin{array}{ccc} 1 & -2 & 0 \\ 2 & 1 & 3 \\ 0 & -2 & 1 \end{array}\right], B=\left[\begin{array}{ccc} 7 & 2 & -6 \\...

If $A=\left(\begin{array}{ccc}2 & 1 & 1 \\ 1 & -2 & -1 \\ 0 & 3 & -5\end{array}\right)$ , find $A^{-1}$ Using $A^{-1}$ , solve the following system of linear equations:
$2 x+y+z=1$
$x-2 y-z=\frac{3}{2}$
$3 y-5 z=9$ HINT: Here $A=\left(\begin{array}{ccc}2 & 1 & 1 \\ 1 & -2 & -1 \\ 0 & 3 & -5\end{array}\right)$ $\mathrm{X}=\left(\begin{array}{l} x \\ y \\ z \end{array}\right) \text { and } \mathrm{B}=\left(\begin{array}{l} 1 \\ 3 / 2 \\ 9 \end{array}\right)$

CBSE, Class 12, Excercise 8A, Maths, RS Aggarwal, System of Linear Equations

Solution: It is given, $\begin{array}{l} A=\left[\begin{array}{ccc} 2 & 1 & 1 \\ 1 & -2 & -1 \\ 0 & 3 & -5 \end{array}\right] \\ A^{-1}=\frac{1}{|A|} \operatorname{adj}(A)...

If $A=\left(\begin{array}{ccc}2 & -3 & 5 \\ 3 & 2 & -4 \\ 1 & 1 & -2\end{array}\right)$ , find $A^{-1}$ . Using $A^{-1}$ , solve the following system of equations:
$2 x-3 y+5 z=11$
$3 x+2 y-4 z=-5$
$x+y-2 z=-3$

CBSE, Class 12, Excercise 8A, Maths, RS Aggarwal, System of Linear Equations

Solution: It is given, $A=\left[\begin{array}{ccc} 2 & -3 & 5 \\ 3 & 2 & -4 \\ 1 & 1 & -2 \end{array}\right]$ $\mathrm{A}^{-1}=\frac{1}{|A|} \operatorname{adj}(A)$...

Solve each of the following systems of equations using matrix method.
$4x + 3y + 2z = 60$ ;
$x + 2y + 3z = 45$ ;
$6x + 2y + 3z = 70$ .

CBSE, Class 12, Excercise 8A, Maths, RS Aggarwal, System of Linear Equations

Solution: We need to find: - $x , y , z$ The given set of lines are : $\begin{array}{l} 4 x+3 y+2 z=60 \\ x+2 y+3 z=45 \\ 6 x+2 y+3 z=70 \end{array}$ Now converting the following equations in matrix...

Solve each of the following systems of equations using matrix method.
$x - y = 3$ ;
$2x + 3y + 4z = 17$ ;
$y + 2z = 7$ .

CBSE, Class 12, Excercise 8A, Maths, RS Aggarwal, System of Linear Equations

Solution: We need to find: $-\mathrm{x}, \mathrm{y}, \mathrm{z}$ The given set of lines are : - $\begin{array}{l} x-y=3 \\ 2 x+3 y+4 z=17 \\ y+2 z=7 \end{array}$ Now, converting the following...

Solve each of the following systems of equations using matrix method.
$x - 2y + z = 0$ ;
$y - z = 2$ ;
$2x - 3z = 10$ .

CBSE, Class 12, Excercise 8A, Maths, RS Aggarwal, System of Linear Equations

Solution: We need to find: $-\mathrm{x}, \mathrm{y}, \mathrm{z}$ The given set of lines are : - $\begin{array}{l} x-2 y+z=0 \\ y-z=2 \\ 2 x-3 z=10 \end{array}$ Now converting the following equations...

Solve each of the following systems of equations using matrix method.
$5x - y = - 7$ ;
$2x + 3z = 1$ ;
$3y - z = 5$ .

CBSE, Class 12, Excercise 8A, Maths, RS Aggarwal, System of Linear Equations

Solution: We need to find: $-\mathrm{x}, \mathrm{y}, \mathrm{z}$ The given set of lines are : - $\begin{array}{l} 5 x-y=-7 \\ 2 x+3 z=1 \\ 3 y-z=5 \end{array}$ Now, converting the following...

Solve each of the following systems of equations using matrix method.
$x - y - 2z = 3$ ;
$x + y = 1$ ;
$x + z = - 6$ .

CBSE, Class 12, Excercise 8A, Maths, RS Aggarwal, System of Linear Equations

Solution: We need to find: $-\mathrm{x}, \mathrm{y}, \mathrm{z}$ The given set of lines are : - $\begin{array}{l} x-y-2 z=3 \\ x+y=1 \\ x+z=-6 \end{array}$ Now converting the following equations in...

Solve each of the following systems of equations using matrix method.
$x + 2y + z = 4$ ;
$- x + y + z = 0$ ;
$x - 3y + z = 4$ .

CBSE, Class 12, Excercise 8A, Maths, RS Aggarwal, System of Linear Equations

Solution: We need to find: $-x, y, z$ The given set of lines are : - $\begin{array}{l} x+2 y+z=4 \\ -x+y+z=0 \\ x-3 y+z=4 \end{array}$ Now converting the following equations in matrix form,...

Solve each of the following systems of equations using matrix method.
$2x + y - z = 1$ ;
$x - y + z = 2$ ;
$3x + y - 2z = - 1$ .

CBSE, Class 12, Excercise 8A, Maths, RS Aggarwal, System of Linear Equations

Solution: We need to find: $-\mathrm{x}, \mathrm{y}, \mathrm{z}$ The given set of lines are : - $\begin{array}{l} 2 x+y-z=1 \\ x-y+z=2 \\ 3 x+y-2 z=-1 \end{array}$ Now, converting the following...

Solve each of the following systems of equations using matrix method.
$x + y - z = 1$ ;
$3x + y - 2z = 3$ ;
$x - y - z = - 1$ .

CBSE, Class 12, Excercise 8A, Maths, RS Aggarwal, System of Linear Equations

Solution: We need to find: - $x , y , z$ The given set of lines are : - $x + y - z = 1$ $\begin{array}{l} 3 x+y-2 z=3 \\ x-y-z=-1 \end{array}$ Now, converting the following equations in matrix form,...

Solve each of the following systems of equations using matrix method.
$3x - 4y + 2z = - 1$ ;
$2x + 3y + 5z = 7$ ;
$x+ z = 2$ .

CBSE, Class 12, Excercise 8A, Maths, RS Aggarwal, System of Linear Equations

Solution: We need to find: - $x , y , z$ The given set of lines are : - $\begin{array}{l} 3 x-4 y+2 z=-1 \\ 2 x+3 y+5 z=7 \\ x+z=2 \end{array}$ Now, converting the following equations in matrix...

Solve each of the following systems of equations using matrix method.
$6x - 9y - 20z = - 4$ ;
$4x - 15y + 10z = - 1$ ;
$2x - 3y - 5z = - 1$ .

CBSE, Class 12, Excercise 8A, Maths, RS Aggarwal, System of Linear Equations

Solution: We need to find: $-\mathrm{x}, \mathrm{y}, z$ The given set of lines are : - $\begin{array}{l} 6 x-9 y-20 z=-4 \\ 4 x-15 y+10 z=-1 \\ 2 x-3 y-5 z=-1 \end{array}$ Now, converting the...

Solve each of the following systems of equations using matrix method.
$x - y + 2z = 7$ ;
$3x + 4y - 5z = - 5$ :
$2x - y + 3z = 12$ .

CBSE, Class 12, Excercise 8A, Maths, RS Aggarwal, System of Linear Equations

Solution: We need to find: $-\mathrm{x}, \mathrm{y}, \mathrm{z}$ The given set of lines are : - $\begin{array}{l} x-y+2 z=7 \\ 3 x+4 y-5 z=-5 \\ 2 x-y+3 z=12 \end{array}$ Now, converting the...

Solve each of the following systems of equations using matrix method.
$4x - 5y - 11z = 12$ ;
$x - 3y + z = 1$ ;
$2x + 3y - 7z = 2$ .

CBSE, Class 12, Excercise 8A, Maths, RS Aggarwal, System of Linear Equations

Solution: We need to find: $-\mathrm{x}, \mathrm{y}, \mathrm{z}$ The given set of lines are: $\begin{array}{l} 4 x-5 y-11 z=12 \\ x-3 y+z=1 \\ 2 x+3 y-7 z=2 \end{array}$ Now, converting the...

Solve each of the following systems of equations using matrix method.
$2x + 3y + 3z = 5$ ;
$x - 2y + z = - 4$ ;
$3x - y - 2z = 3$ .

CBSE, Class 12, Excercise 8A, Maths, RS Aggarwal, System of Linear Equations

Solution: We need to find: - $x, y, z$ The given set of lines are : - $\begin{array}{l} 2 x+3 y+3 z=5 \\ x-2 y+z=-4 \\ 3 x-y-2 z=3 \end{array}$ Now, converting the following equations in matrix...