RS Aggarwal

### Mark (√) against the correct answer in the following: Let Then, A. B. C. D. none of these

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: 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)}

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 and then A. 0 B. 1 C. D.

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 $h$,...

### A medicine company has factories at two places, and Y. From these places, supply is made to each of its three agencies situated at and . the monthly requirement of the agencies are respectively 40 packets, 40 packets and 50 packets of medicine, while the production capacity of the factories at and are 60 packets and 70 packets respectively. The transportation costs per packet from the factories to the agencies are given as follows.

How many packets from each factory should be transported to each agency so that the cost of transportation is minimum? Also, find the minimum cost. Solution: Let $x$ packets of medicines be...

### Construct a matrix whose elements are given by

Solution: It is a (3 $x 4)$ matrix. Therefore, it has 3 rows and 4 columns. Given that $a_{i j}=\frac{|-a \|+| l}{2}$ Therefore, $a_{11}=1 . a_{12}=\frac{1}{2}, a_{13}=0, a_{13}=\frac{1}{2}$...

### Let and let be a function from to State whether is one-one.

Solution: We need to state: Whether $\mathrm{f}$ is one-one Given that: $f=\{(1,4),(2,5),(3,6)\}$ Here the function is defined from $A \rightarrow B$ For a function to be one-one if the images of...

### Let and Find o and o .

Solution: We need to find: $g$ o f and f o $\mathrm{g}$ Formula used: (i) f o $\mathrm{g}=\mathrm{f}(\mathrm{g}(\mathrm{x}))$ (ii) $\mathrm{g} \circ \mathrm{f}=\mathrm{g}(\mathrm{f}(\mathrm{x}))$...

### Graph the solution sets of the following inequations:

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

### Graph the solution sets of the following inequations:

Given $x+y \geq 4$ $\Rightarrow y \geq 4-x$ Consider the equation $y=4-x$. Finding points on the coordinate axes: If $x=0$, the $y$ value is 4 i.e, $y=4$ $\Rightarrow$ the point on the $Y$ axis is...

### If be a greatest integer function and be an absolute value function, find the value of

Solution: We need to find: $(f \circ g)\left(\frac{-3}{2}\right)+(g \circ f)\left(\frac{4}{3}\right)$ Formula used: (i) $f$ o $g=f(g(x))$ (ii) $g \circ f=g(f(x))$ It is given that: (i) $f$ is a...

### Let , prove that o

Solution: We need to prove: fo $f=f$ Formula used: $f$ o $f=f(f(x))$ It is given that: (i) f: $\mathbf{R} \rightarrow \mathrm{R}: \mathrm{f}(x)=|\mathrm{x}|$ Solution: We have, fo...

### Let and be defined as and Find (i) (ii) (f o g).

Solution: (i) $\mathrm{g} \circ \mathrm{f}$ We need to find: $g$ of Formula used: $g$ o $f=g(f(x))$ It is given that: $f=\{(3,1),(9,3),(12,4)\}$ and $g=\{(1,3),(3,3),(4,9),(5,9)\}$ Solution: We...

### Solve the differential equation , given that

Solution: We have, $\begin{array}{l} \frac{d y}{d x}=y \sin 2 x \\ \Rightarrow \frac{d y}{y}=\sin 2 x d x \\ \Rightarrow \log y=-\frac{\cos 2 x}{2}+c \end{array}$ For $y=1, x=0$, we have,...

### Find the general solution of each of the following differential equations:

Solution: $\frac{1}{\tan ^{-1} x \cdot\left(1+x^{2}\right)} d x+\frac{1}{y} d y=0$ On integrating, we get $\int \frac{1}{\tan ^{-1} x \cdot\left(1+x^{2}\right)} d x+\int \frac{1}{y} d y=c$ Now...

### Find the general solution of each of the following differential equations:

Solution: $\begin{array}{l} \frac{d y}{d x}+\frac{y(1+x)}{x(1+y)}=0 \\ \frac{1+y}{y} d y+\frac{1+x}{x} d x=0 \\ \frac{1}{y} d y+1 . d y+\frac{1}{x} d x+1 . d x=0 \end{array}$ On integrating, we get...

### Find the general solution of each of the following differential equations:

Solution: $\begin{array}{l} \left(1-x^{2}\right) d y=-x y(1-y) d x \\ \left(1-x^{2}\right) d y=x y(y-1) d x \\ \frac{1}{y(y-1)} d y=\frac{x}{1-x^{2}} d x \\ \end{array}$ On integrating both the...

### Find the general solution of each of the following differential equations:

Solution: $\frac{y}{1+y^{2}} d y=\frac{x}{1-x^{2}} d x$ Multiplying 2 in both Left Hand Side and Right Hand Side, $\frac{2 y}{1+y^{2}} d y=\frac{2 x}{1-x^{2}} d x$ On integrating both sides, we get...

### Find the general solution of each of the following differential equations:

Solution: $\begin{array}{l} x^{2}(y+1) d x+y^{2}(x-1) d y=0 \\ x^{2}(y+1) d x=-y^{2}(x-1) d y \\ x^{2}(y+1) d x=y^{2}(1-x) d y \\ \frac{x^{2}}{(1-x)} d x=\frac{y^{2}}{y+1} d y \end{array}$ Adding...

### Find the general solution of each of the following differential equations:

Solution: $d y=\frac{x}{x^{2}+1} d x$ Multiplying and dividing by 2 in numerator and denominator of Right Hand Side, $y=\frac{1}{2} \cdot\left(\frac{2 x}{x^{2}+1} d x\right)$ On integrating both the...

### In each of the following differential equation show that it is homogeneous and solve it.

Solution: $\begin{array}{l} \frac{d x}{d y}=\frac{x y-x^{2}}{y^{2}}=\frac{x}{y}-\left(\frac{x}{y}\right)^{2} \\ \Rightarrow \frac{d x}{d y}=f\left(\frac{x}{y}\right) \end{array}$ $\Rightarrow$ the...

### For each of the following differential equations, find a particular solution satisfying the given condition: where and when

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:

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...