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 Then,

### Mark (√) against the correct answer in the following: Let . Then, ?

A.

B.

C.

D.

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 . Then, dom

A.

B.

C.

D.

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)}

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 $h$,...

### Mark (√) against the correct answer in the following: If then

A.

B.

C.

D. None of these

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 then (f of of)

A.

B.

C.

D. None of these

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 and then

A.

B.

C.

D. None of these

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...

### Mark (√) against the correct answer in the following: Let f : N → X : f(x) = 4×2 + 12x + 15. Then, (y) = ?

A.

B.

C.

D. None of these

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 : Q → Q : f(x) = (2x + 3). Then, (y) = ?

A. (2y – 3)

B.

C.

D. none of these

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 $\mathrm{f}(\mathrm{x})=\mathrm{y}$...

### Mark (√) against the correct answer in the following: Let Then, is

A. one – one and into

B. one – one and onto

C. many – one and into

D. many – one and onto

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:

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

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}...

### Mark (√) against the correct answer in the following:

f : R → R : f(x) = is

A. one – one and onto

B. one – one and into

C. many – one and onto

D. many – one and into

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...

### Mark (√) against the correct answer in the following:

f : N → N : f(x) = + x + 1 is

A. one – one and onto

B. one – one and into

C. many – one and onto

D. many – one and into

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

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 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, $a_{11}=1 . a_{12}=\frac{1}{2}, a_{13}=0, a_{13}=\frac{1}{2}$...

### Construct a matrix whose elements are

Solution: It is a $(2 \times 3)$ matrix. Therefore, it has 2 rows and 3 columns. Given that $a_{i j}=\frac{\left(t-2 \rho^{2}\right.}{2}$ Therefore, $a_{11}=\frac{1}{2}, a_{12}=\frac{9}{2},...

### Construct a matrix whose elements are

Solution: It is a $(2 \times 2)$ matrix. So, it has 2 rows and 2 columns. Given that $a_{i j}=\frac{(i+2 j)^{2}}{2}$ Therefore, $a_{11}=\frac{9}{2}, a_{12}=\frac{25}{2}$. $a_{21}=8 . a_{22}=18$...

### Construct a matrix whose elements are given by

Solution: It is $(4 \times 3)$ matrix. Therefore it has 4 rows and 3 columns Given that$a_{i j}=\frac{i}{j}$ Therefore, $a_{11}=1 . a_{12}=\frac{1}{2}, a_{13}=\frac{1}{3}$ $\begin{array}{l}...

### Construct a matrix whose elements are given by

Solution: It is given that: $a_{i j}=(2 \mid-j)$ Now, $a_{11}=(2 \times 1-1)=2-1=1$ $\begin{array}{l} a_{12}=2 \times 1-2=2-2=0 \\ a_{21}=2 \times 2-1=4-1=3 \\ a_{22}=2 \times 2-2=4-2=2 \\ a_{31}=2...

### Find all possible orders of matrices having 7 elements.

Solution: No. of entries $=($ No. of rows) $x$ (No. of columns) $=7$ If order is $(\mathrm{a} \times \mathrm{b})$ then, No. of entries = $\mathrm{a} \times \mathrm{b}$ Therefore now a $x b=7$ (in...

### If a matrix has 18 elements, what are the possible orders it can have?

Solution: No. of entries $=$ (No. of rows) $x$ (No. of columns) $=18$ If order is $(a \times b)$ then, No. of entries = $a \times b$ Therefore now $a \times b=18$ (in this case) Possible cases are...

### Write the order of each of the following matrices:

i.

ii,

Solution: i. $E=\left[\begin{array}{c}-2 \\ 3 \\ 0\end{array}\right]$ Order of matrix $=$ Number of rows $x$ Number of columns $\begin{array}{l} =(3 \times 1) \\ \text { ii, } F=[6] \end{array}$...

### Write the order of each of the following matrices:

i.

ii.

Solution: i. $C=[7-\sqrt{2} \quad 5 \quad 0]$ Order of matrix $=$ Number of rows $x$ Number of columns $=(1 \times 4)$ ii. $D=[8-3]$ Order of matrix $=$ Number of rows $x$ Number of columns $=(1...

### Write the order of each of the following matrices: i. ii.

Solution: i. $A=\left[\begin{array}{cccc}3 & 5 & 4 & -2 \\ 0 & \sqrt{3} & -1 & \frac{4}{9}\end{array}\right]$ Order of matrix $=$ Number of rows $x$ Number of columns $=(2...

### If then write

i. the order of the matrix ,

ii. the number of all entries in ,

Solution: (i) Order of matrix $=$ Number of rows $x$ Number of columns $=(3 \times 4)$ (ii) Number of entries = (Number of rows) $x$ (Number of columns) $=3 \times 4$ $=12$

### If then write

i. the number of rows in ,

ii. the number of columns in ,

Solution: (i) Number of rows $=3$ (ii) Number of columns = 4

### Let and be two functions from into , defined by and for all . Find f o and g o .

Solution: $\begin{array}{ll} \mathrm{f}(\mathrm{x})=|\mathrm{x}|+\mathrm{x} \quad & \text { (given }) \\ \mathrm{g}(\mathrm{x})=|\mathrm{x}|-\mathrm{x} \quad & \text { (given) } \end{array}$...

### Let and If , show that is one-one and onto. Hence, find

Solution: $\mathrm{f}(\mathrm{x})=\frac{x-1}{x-2} \quad \text { (as given) }$ One-One function Suppose $\mathrm{p}, \mathrm{q}$ be two arbitrary elements in $\mathrm{R}$ Therefore, $f(p)-f(q)$...

### Let be the set of all positive real numbers. show that the function is invertible. Find .

Solution: $f(x)=9 x_{2}+6 x-5 \text { (as given) }$ $\mathrm{f}(\mathrm{x})$ is invertible if $\mathrm{f}(\mathrm{x})$ is a bijection (i.e one-one onto function) One-One function Suppose p,q be two...

### Show that the function on into itself, defined by is one-one and onto. Hence, find

Solution: $\mathrm{f}(\mathrm{x})=\frac{4 x}{3 x+4} \quad$ (as given) One-One function Suppose $\mathrm{p}, \mathrm{q}$ be two arbitrary elements in $\mathrm{R}$ Therefore, $f(p)=f(q)$...

### Let

Solution: $\mathrm{f}(\mathrm{x})=\frac{1}{2}(3 \mathrm{x}+1) \quad$ (given) $\mathrm{f}(\mathrm{x})$ is invertible if $\mathrm{f}(\mathrm{x})$ is a bijection (i.e one-one onto function) One-One...

### Let f : Q → Q : f(x) = 3x —4. Show that f is invertible and find .

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

### Show that the function f : R → R : f(x) = 2x + 3 is invertible and find .

Solution: $\mathrm{f}(\mathrm{x})=2 \mathrm{x}+3$ (as given) $\mathrm{f}(\mathrm{x})$ is invertible if $\mathrm{f}(\mathrm{x})$ is a bijection (i.e one-one onto function) One-One function Suppose...

### Let A = {2, 3, 4, 5} and B = {7, 9, 11, 13}, and let f = {(2, 7), (3, 9), (4, 11), (5, 13)}. Show that f is invertible and find .

Solution: A function is invertible if it is a bijection. (i.e. One-One Onto function) One-One function $\mathrm{f}=\{(2,7),(3,9),(4,11),(5,13)\}$ It is observed that different elements of A have...

### Find the maximum and minimum values of , subject to the constraints and

The feasible region determined by $x+3 y \geq 6, x-3 y \leq 3,3 x+4 y \leq 24$ $-3 x+2 y \leq 6,5 x+y \geq 5, x \geq 0$ and $y \geq 0$ is given by The corner points of the feasible region are $A(4 /...

### 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) $\mathrm{g} \circ \mathrm{f}=\mathrm{g}(\mathrm{f}(\mathrm{x}))$...

### Let and Write down (f of).

Solution: We need to find: f of Formula used: $f o f=f(f(x))$ Given that: (i) $f=\{(1,4),(2,1)(3,3),(4,2)\}$ We have, $\text { fof }(1)=f(f(1))=f(4)=2$ $\begin{array}{l}...

### Let and . Write down o f.

Solution: We need to find: g of Formula used: $g$ o $f=g(f(x))$ Given that: (i) $f=\{(1,2),(3,5),(4,1)\}$ (ii) $g=\{(1,3),(2,3),(5,1)\}$ We have, $\begin{array}{l}...

### Let , find

Solution: We need to find: $f\{f(x)\}$ Formula used: (i) $f \circ f=f(f(x))$ Given that: (i) $f: R \rightarrow R: f(x)=3 x+2$ We have, $\begin{array}{l} f\{f(x)\}=f(f(x))=f(3 x+2) \\ \text { fo }...

### Let . Find f of.

Solution: We need to find: f o f Formula used: (i) f o $f=f(f(x))$ Given that: (i) $f: R \rightarrow R: f(x)=\left(3-x^{3}\right)^{1 / 3}$ We have, $\begin{array}{l} \text { fo }...

### 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...

### Show that is many-one and into.

Solution: We need to prove: function is many-one and into It is given that: $f: R \rightarrow R: f(x)=\left\{\begin{array}{c}1, \text { if } x \text { is rational } \\ -1, \text { if } x \text { is...

### Let Find

Solution: We need to find: $\mathrm{f}^{-1}$ It is given that: $f: R \rightarrow R: f(x)=10 x+3$ We have, $f(x)=10 x+3$ Suppose $f(x)=y$ such that $y \in R$ $\begin{array}{l} \Rightarrow y=10 x+3 \\...

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

Given $2 x-3 y<4$ $\begin{array}{l} \Rightarrow 2 x-4<3 y \\ \Rightarrow 3 y>2 x-4 \\ \Rightarrow y>\frac{2}{3} x-\frac{4}{3} \end{array}$ Consider the equation $y=\frac{2}{3}...

### Let be an invertible function. Find

Solution: We need to find: $\mathrm{f}^{-1}$ It is given that: $f: R \rightarrow R: f(x)=\frac{2 x-7}{4}$ We have, $f(x)=\frac{2 x-7}{4}$ Suppose $f(x)=y$ such that $y \in R$ $\begin{array}{l}...

### Show that the function is many-one into.

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

### Let be the set of all nonzero real numbers. Then, show that the function is oneone and onto.

Solution: We need to prove: function is one-one and onto It is given that: $f: R_{0} \rightarrow R_{0}: f(x)=\frac{1}{x}$ We have, $f(x)=\frac{1}{x}$ For, $f\left(x_{1}\right)=f\left(x_{2}\right)$...

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

Given $x+2 y>1$ $\begin{array}{l} \Rightarrow 2 y>1-x \\ \Rightarrow y>\frac{1}{2}-\frac{x}{2} \end{array}$ Consider the equation $y=\frac{1}{2}-\frac{x}{2}$ Finding points on the...

### Show that the function is neither one-one nor onto.

Solution: We need to prove: function is neither one-one nor onto It is given that: $f: R \rightarrow R: f(x)=x^{4}$ We have, $f(x)=x^{4}$ For, $f\left(x_{1}\right)=f\left(x_{2}\right)$...

### Show that the function is one-one and into.

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

### Show that the function is neither one-one nor onto.

Solution: We need to prove: function is neither one-one nor onto It is given that: $f: R \rightarrow R: f(x)=x^{2}$ Solution: We have, $f(x)=x^{2}$ For, $f\left(x_{1}\right)=f\left(x_{2}\right)$...

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

Given $x-y \leq 3$ $\Rightarrow-y \leq 3-x$ Multiplying by minus on both the sides, we'll get $\begin{array}{l} y \geq-3+x \\ y \geq x-3 \end{array}$ Consider the equation $y=x-3$. Finding points on...

### Prove that the function f: is one-one and onto.

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

### 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...

### Let and Write down the formulae for

(i) of

(ii)

Solution: (i) $g \circ f$ We need to find: $\mathrm{g}$ o $\mathrm{f}$ Formula used: $g$ o $f=g(f(x))$ It is given that: (i) $f: \mathbb{R} \rightarrow R: f(x)=\left(x^{2}+3 x+1\right)$ (ii) $g: R...

### Let and find and o and hence find (2) and

Solution: We need to find: $f$ o $g . g \circ f,(f \circ g)(2)$ and $(g \circ f)(-3)$ Formula used: (i) $f$ o $g=f(g(x))$ (ii) $g$ o $f=g(f(x))$ It is given that: (i) $f: R \rightarrow R:...

### 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 and . Show that

Solution: We need to show: $h \circ(g \circ f)=(h \circ g)$ of Formula used: (i) fo $g=f(g(x))$ (ii) $\mathrm{g} \circ \mathrm{f}=\mathrm{g}(\mathrm{f}(\mathrm{x}))$ It is given that: (i) $f:...

### Let . Find .

Solution: We need to find: $g: Z \rightarrow Z: g$ o $f=I_{Z}$ Formula used: (i) f o $\mathrm{g}=\mathrm{f}(\mathrm{g}(\mathrm{x}))$ (ii) $\mathrm{g}$ of $=g(f(x))$ It is given that: (i) $g: Z...

### Let and Show that

Solution: We need to prove: $(f \circ g)=I_{R}=(g \circ f)$. Formula used: (i) f o $g=f(g(x))$ (ii) $\mathrm{g}$ of $=\mathrm{g}(\mathrm{f}(\mathrm{x}))$ It is given that: (i) $f: R \rightarrow R:...

### Let and Find a formula for h o (g of). Show that [h o

Solution: We need to find: formula for $h \circ(g \circ f)$ We need to prove: Show that [h o $(g \circ f)] \sqrt{\frac{\pi}{4}}=0$ Formula used: f of $=f(f(x))$ It is given that: (i) $f: \mathbb{R}...

### 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 Write down the formulae for

(i)

Solution: (i) $\mathrm{g} \circ \mathrm{g}$ We need to find: $\mathrm{g} \circ \mathrm{g}$ Formula used: $\mathrm{g}$ o $\mathrm{g}=\mathrm{g}(\mathrm{g}(\mathrm{x}))$ It is given that: (i) $g: R...

### Let and Write down the formulae for.

(i) (g of) (ii) (f o g)

Solution: (i) g of We need to find: $g \circ f$. Formula used: $g$ o $f=g(f(x))$ It is given that: (i) $f: R \rightarrow R: f(x)=(2 x+1)$ (ii) $g: R \rightarrow R: g(x)=\left(x^{2}-2\right)$...

### Let and Show that (f o g).

Solution: We need to prove: $(g \circ f) \neq(f \circ g)$ Formula used: (i) $\mathrm{g}$ o $\mathrm{f}=\mathrm{g}(\mathrm{f}(\mathrm{x}))$ (ii) $f$ o $g=f(g(x))$ It is given that: (i) $f: R...

### 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...

### Let . Let and , defined by and

Find (i) f of.

Solution: (i) f of We need to find: $f$ o $f$ Formula used: $f$ f $f=f(f(x))$ It is given that: $f=\{(1,4),(2,1),(3,3),(4,2)\}$ Solution: We have, $\begin{array}{l}...

### Let . Let and , defined by and

Find (i) of (ii) fo

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: $f=\{(1,4),(2,1),(3,3),(4,2)\}$ and $g=\{(1,3),(2,1)$ $(3,2),(4,4)\}$ Solution:...

### In a culture the bacteria count is 100000 . The number is increased by in 2 hours. In how many hours will the count reach 200000 , if the rate of growth of bacteria is proportional to the number present?

Solution: Suppose $y$ be the bacteria count, therefore, we have, the rate of growth of bacteria is proportional to the no. present $\mathrm{So}_{\mathrm{s}} \frac{d y}{d t}=c y$ Where $c$ is a...

### The volume of a spherical balloon being inflated changes at a constant rate. If initially its radius is 3 units and after 3 seconds it is 6 units. Find the radius of the balloon after seconds.

Solution: It is given that: Volume $\mathrm{V}=\frac{4 \pi r^{3}}{\mathrm{a}}$ $\frac{d V}{d t}=\frac{4}{3} \pi 3 r^{2} \frac{d r}{d t}$ $\Rightarrow \frac{d V}{d t}=k$ (constant) $\begin{array}{l}...

### In a bank, principal increases at the rate of per annum. An amount of 1000 is deposited in the bank. How much will it worth after 10 years? (Given

Solution: Itis given that: rate of interest $=5 \%$ $P(\text { initial })=\operatorname{Rs} 1000$ And, $\begin{array}{l} \frac{d p}{d t}=\frac{5}{100} \times p \\ \Rightarrow \frac{d...

### In a bank, principal increases at the rate of per annum. Find the value of if ‘ 100 double itself in 10 years. (Given

Solution: It is given that: $\frac{d p}{d t}=\left(\frac{r}{100}\right) \times p$ Here, $p$ is the principal, $r$ is the rate of interest per annum and $t$ is the time in years. On solving the...

### A curve passes through the point and at any point of the curve, the slope of the tangent is twice the slope of the line segment joining the point of contact to the point . Find the equation of the curve.

Solution: $\begin{array}{l} \text { It is given that: } \frac{d y}{d x}=\frac{2(y+3)}{x+4} \\ \Rightarrow \frac{d y}{y+3}=\frac{2 d x}{x+4} \\ \Rightarrow \int \frac{d y}{y+3}=2 \int \frac{d x}{x+4}...

### A curve passes through the point and at any point of the curve, the product of the slope of its tangent and -coordinate of the point is equal to the -coordinate of the point. Find the equation of the curve.

Solution: It is given that product of slope of tangent and $y$ coordinate equals the $x$-coordinate i.e., $y \frac{d y}{d x}=x$ We have, $y d y=x d x$ $\begin{array}{l} \Rightarrow \int y d y=\int x...

### Find the equation of a curve which passes through the origin and whose differential equation is

Solution: $\begin{array}{l} \text { It is given that, } \frac{d y}{d x}=e^{x} \sin x \\ d y=e^{x} \sin x d x \\ \Rightarrow \int d y=\int e^{x} \sin x d x \end{array}$ $\text { Suppose } I=\int...

### Find the equation of the curve passing through the point whose differential equation is

Solution: We have, $\sin x \cos y d x+\cos x \sin y d y=0$ $\begin{array}{l} \Rightarrow \sin x \operatorname{cosy} d x+\cos x \operatorname{siny} d y=0 \\ \Rightarrow \tan x d x+\tan y d y=0 \\...

### Solve , given that when

Solution: We have, $\left(1+x^{2}\right) \sec 2 y d y+2 x \tan y d x=0$ It is given that, $y=\frac{\pi}{4}$ when $x=1$ $\begin{array}{l} \Rightarrow\left(1+x^{2}\right) \sec 2 y d y+2 x \tan y d x=0...

### Solve , given that when

Solution: We have $\frac{\mathrm{dy}}{\mathrm{dx}}=\mathrm{y} \cot 2 \mathrm{x}$ It is given that, $y=2$ when $x=\frac{\pi}{2}$ $\begin{array}{l} \Rightarrow \frac{d y}{y}=y \cot 2 x \\ \Rightarrow...

### Solve , given that when

Solution: We have: $\frac{\mathrm{dy}}{\mathrm{dx}}=\mathrm{y}^{2} \tan 2 \mathrm{x}$, It is given that, $y=2$ when $x=0$ $\begin{array}{l} \Rightarrow \frac{d y}{y^{2}}=\tan 2 x d x \\ \Rightarrow...

### Solve , given that when

Solution: We have, $\frac{d y}{d x}=y \tan x$ It is given that: $y=1$ when $x=0$ $\begin{array}{l} \Rightarrow \frac{d y}{d x}=y \tan x \\ \Rightarrow \frac{d y}{y}=\tan x d x \\ \Rightarrow \log...

### Solve , given that when

Solution: $\left(x^{3}+x^{2}+x+1\right) \frac{d y}{d x}=2 x^{2}+x$ For: $y=1$ when $x=0$, $\frac{d y}{d x}=\frac{x^{2}+x+x^{2}}{\left(x^{2}+1\right)(x+1)} \mathrm{dx}$ $\begin{array}{l} \int...

### Solve , given that when

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...

### Solve the differential equation , given that

Solution: $\begin{array}{l} \text { It is given that: }(x+1) \frac{d y}{d x}=2 x y \\ \Rightarrow \frac{d y}{y}=2 \frac{x}{x+1} d x \\ \Rightarrow \log y=\int 2-\frac{2}{x+1} d x \\ \Rightarrow \log...

### 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 particular solution of the differential equation , given that when

Solution: It is given that: $\frac{d y}{d x}=\frac{x(2 \log x+1)}{(\text { siny }+y \cos y)}$ $\Rightarrow \int \sin y d y+\int y \operatorname{cosy} d y=\int 2 x \log x d x+\int x d x$ Let $\int y...

### Find the particular solution of the differential equation , given that when

Solution: It is given that: $e^{x} \sqrt{1-y^{2} d x}+\frac{y}{x} d y=0$ On separating the variables we obtain, $\begin{array}{l} \Rightarrow x e^{x} d x+\frac{y}{\sqrt{1-y^{2}}} d y=0 \\...

### Solve the differential equation , given that when

Solution: $\begin{array}{l} x^{2}(1-y) d y+y^{2}\left(1+x^{2}\right) d x=0 \\ \Rightarrow \frac{(1-y)}{y^{2}} d y+\frac{\left(1+x^{2}\right)}{x^{2}} d x=0 \\ \Rightarrow \int \frac{(1-y)}{y^{2}} d...

### Find the particular solution of the differential equation , given that when .

Solution: $\begin{array}{l} \log \left(\frac{d y}{d x}\right)=3 x+4 y \\ \Rightarrow y=0 \\ \Rightarrow x=0 \\ \Rightarrow \frac{d y}{d x}=e^{3 x} e^{4 y} \\ \Rightarrow e^{-4 y} d y=e^{3 x} d x \\...

### Find the particular solution of the differential equation , given that when

Solution: $\begin{array}{l} \frac{2 x d x}{1+x^{2}}-\frac{2 y d y}{1+y^{2}}=0 \\ \Rightarrow \frac{\log \left(1+x^{2}\right)}{1+y^{2}}=0 \\ \Rightarrow\left(1+x^{2}\right)=c\left(1+y^{2}\right) \\...

### Find the particular solution of the differential equation , given that when .

Solution: It is given that: $\frac{d y}{d x}=(1+x)(1+y)$ $\Rightarrow \frac{d y}{1+y}=(1+x) d x$ $\Rightarrow \log |y+1|=\left(x+\frac{x^{2}}{2}+c\right)$ $\Rightarrow$ now, for $y=0$ and $x=1$ We...

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

Solution: Here we have, $y=\int\left(\sin ^{3} x \cos ^{2} x+x e^{x}\right) d x$ $\Rightarrow \int \cos ^{2} x\left(1-\cos ^{2} x\right) \sin x d x+\int x e^{x} d x$ On taking $\cos x$ as $t$ we...

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

Solution: $\begin{array}{l} \text { It is given that: } \frac{1}{x} \cos ^{2} y d y+\frac{1}{y} \cos ^{2} x d x=0 \\ \Rightarrow y \cos ^{2} y d y+x \cos ^{2} x d x=0 \\ \left.\Rightarrow...

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

Solution: $\frac{d y}{d x}+\sin (x+y)=\sin (x-y)$ $\begin{array}{l} \Rightarrow \frac{d y}{d x}=\sin (x-y)-\sin (x+y) \\ \Rightarrow \frac{d y}{d x}=-2 \sin y \cos x(\text { Using } \sin (A+B)-\sin...

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

Solution: By using $\sin ^{3} x=\frac{3 \sin x-\sin 3 x}{4}$ We have, $\begin{array}{l} \Rightarrow \frac{3 \sin x-\sin 3 x}{4} d x-\sin y d y=0 \\ \Rightarrow \frac{3}{4} \sin x d x-\frac{\sin 3...

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

Solution: It is given that: $\cos x(1+\cos y) d x-\sin y(1+\sin x) d y=0$ Divide the whole equation by $(1+\sin x)(1+\cos y)$, $\begin{array}{l} \Rightarrow \frac{\int \operatorname{cosxdx}}{1+\sin...

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

Solution: $\begin{array}{l} \text { It is given that: } \frac{d y}{d x}=-\frac{\text { cosxstny }}{\cos y} \\ \Rightarrow \frac{d y}{d x}=-\cos x \tan y \\ \Rightarrow \int \operatorname{coty} d...

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

Solution: $\begin{array}{l} \text { It is given that: } \frac{d y}{d x}+\frac{1+\cos 2 y}{1-\cos 2 x}=0 \\ \Rightarrow \frac{d y}{d x}=-\frac{2 \cos ^{2} y}{2 \sin ^{2} x} \\ \Rightarrow \sec ^{2} y...

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

Solution: $\text { It is given that: } \frac{d y}{d x}+\frac{\cos 2 x}{\cos x}=\frac{\cos 3 x}{\cos x}$ $\begin{array}{l} \Rightarrow \frac{d y}{d x}=\frac{\cos (x+2 x)-\cos 2 x}{\cos x} . \\...

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

Solution: $\begin{array}{l} \frac{d y}{d x}=\frac{1-\cos x}{1+\cos x} \\ d y=\frac{1-\cos x}{1+\cos x} d x \end{array}$ $\cos x$ can also be written as $\cos x=\frac{1-\tan...

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

Solution: $e^{x} \cdot x d x+\frac{y}{\sqrt{1-y^{2}}} d y=0$ On integrating, we get $\int e^{x} \cdot x d x+\int \frac{y}{\sqrt{1-y^{2}}} d y=C$ Now consider the integral $\int e^{x} \cdot x d x$...

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

Solution: $d y=x \cdot \tan ^{-1} x d x$ On integrating both the sides, we get $\begin{array}{l} \int d y=\int x \cdot \tan ^{-1} x d x \\ y=\tan ^{-1} x \int x d x-\int\left[\frac{d}{d x}\left(\tan...

### 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: $\begin{array}{l} \frac{\log y}{y} d y+\frac{x^{2}}{\csc x} d x=0 \\ \frac{\log y}{y} d y+x^{2} \cdot \sin x d x=0 \end{array}$ On integrating, we get $\int \frac{\log y}{y} d y+\int x^{2}...

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

Solution: $d y=\frac{x}{\sqrt{1-x^{4}}} d x$ Multiplying and dividing by 2 , $\begin{array}{l} d y=\frac{1}{2}, \frac{2 x}{\sqrt{1-x^{4}}} d x \\ d y=\frac{1}{2} \cdot \frac{2...

### 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: $\begin{array}{l} \frac{\cos x}{\sin x} d x+\frac{e^{y}}{e^{y}+1} d y=0 \\ \cot x d x+\frac{e^{y}}{e^{y}+1} d y=0 \end{array}$ On integrating, we get $\begin{array}{l} \int \cot x d x+\int...

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

Solution: $\begin{array}{l} \frac{d y}{d x}=e^{x} \cdot e^{y}+e^{x} \cdot e^{-y} \\ \frac{d y}{d x}=e^{x}\left(e^{y}+e^{-y}\right) \\ \frac{1}{e^{y}+e^{-y}} d y=e^{x} d x \\...

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

Solution: $\begin{array}{l} e^{y}\left(1+x^{2}\right) d y=\frac{x}{y} d x \\ e^{y} \cdot y d y=\frac{x}{1+x^{2}} d x \end{array}$ On integrating both the sides, we get $\begin{array}{l} \int e^{y}...

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

Solution: $\Rightarrow 3 . e^{x} \cdot \tan y d x=\left(e^{x}-1\right) \sec ^{2} y d y$ 3. $\frac{e^{x}}{e^{x}-1} d x=\frac{\sec ^{2} y}{\tan y} d y$ 3. $\left[\frac{1}{\frac{e^{x}-1}{e^{x}}}\right]...

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

Solution: Considering ' $\mathrm{d}$ ' as exponential 'e' $\begin{array}{l} \frac{d y}{d x}=\frac{3 e^{2 x}+3 e^{4 x}}{e^{x}+e^{-x}} \\ \frac{d y}{d x}=\frac{3 e^{2 x}+3 e^{4...

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

Solution: $\begin{array}{l} \frac{d y}{d x}=e^{x} \cdot e^{y}+x^{2} \cdot e^{y} \\ \frac{d y}{d x}=e^{y}\left(e^{x}+x^{2}\right) \\ \frac{1}{e^{y}} d y=\left(e^{x}+x^{2}\right) d x \end{array}$ On...

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

Solution: $\frac{x}{\sqrt{1+x^{2}}} d x+\frac{y}{\sqrt{1+y^{2}}} d y=0$ On integrating, we get $\begin{array}{l} \int \frac{x}{\sqrt{1+x^{2}}} d x+\int \frac{y}{\sqrt{1+y^{2}}} d y \\ =C \text {...

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

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

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

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

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

Solution: $\begin{array}{l} y(1+x) d x+x\left(1-y^{2}\right) d y=0 \\ \frac{1+x}{x} d x+\frac{1-y^{2}}{y} d y=0 \\ \frac{1}{x} d x+1 . d x+\frac{1}{y} d y-y d y=0 \end{array}$ On integrating, we get...

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

Solution: $\begin{array}{l} \frac{1-x^{2}}{x} d x=\frac{y(1+y)}{(1-y)} d y \\ {\left[\frac{1}{x}-x\right] d x=\left[\frac{y+y^{2}}{1-y}\right] d y} \\ {\left[\frac{1}{x}-x\right] d...

### 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: $\begin{array}{l} x \cdot x^{2}\left(1-y^{2}\right) d y+y \cdot y^{2}\left(1+x^{2}\right) d x=0 \\ x^{3}\left(1-y^{2}\right) d y+y^{3}\left(1+x^{2}\right) d x=0 \\ \frac{1+x^{2}}{x^{3}} d...

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

Solution: $\begin{array}{l} y \cdot \log y d x=x d x \\ \frac{1}{x} d x=\frac{1}{y \cdot \log y} d y \end{array}$ On integrating both the sides, we get $\int \frac{1}{x} d x=\int \frac{1}{y \cdot...

### 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: $\begin{array}{l} \Rightarrow x \cdot \frac{d y}{d x}+y=y^{2} \\ x \cdot \frac{d y}{d x}=y^{2}-y \\ \frac{1}{y^{2}-y} d y=\frac{1}{x} d x \\ \frac{1}{y(y-1)} d y=\frac{1}{x} d x...

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

Solution: $\begin{array}{l} \frac{d y}{d x}=-\sqrt{\frac{1-y^{2}}{1-x^{2}}} \\ \frac{1}{\sqrt{1-y^{2}}} d y=-\frac{1}{\sqrt{1-x^{2}}} d x \end{array}$ On integrating both the sides, we get...

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

Solution: $\begin{array}{l} \frac{d y}{d x}=1-y \\ \frac{1}{1-y} d y=d x \end{array}$ On integrating both the sides, we get $\begin{array}{l} \int \frac{1}{1-y} d y=\int d x \\ \Rightarrow \log...

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

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

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

Solution: $\frac{1}{1+y^{2}} d y=(1+x) d x$ On integrating both the sides, we get $\begin{array}{l} \int \frac{1}{1+y^{2}} d y=\int(1+x) d x \\ \Rightarrow \tan ^{-1} y=x+\frac{x^{2}}{2}+C...

### 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...

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

Solution: $(y+2) d y=(x-1) d x$ On integrating both the sides, we get $\begin{array}{l} \int(y+2) d y=\int(x-1) d x \\ \frac{y^{2}}{2}+2 y=\frac{x^{2}}{2}-x+c \\ y^{2}+4 y-x^{2}+2 x=c...

### The slope of the tangent to a curve at any point on it is given by , where and . If the curve passes through the point , find the equation of the curve.

Solution: It is given that: $\begin{array}{l} \Rightarrow \frac{d y}{d x}=\frac{y}{x}-\cot \frac{y}{x} \cos \frac{y}{x} \\ \Rightarrow \frac{d y}{d x}=f\left(\frac{y}{x}\right) \end{array}$...

### Find the particular solution of the differential equation , given that .

Solution: $\begin{array}{l} \Rightarrow x e^{\frac{y}{x}}-y+x \frac{d y}{d x}=0 \\ \Rightarrow x \frac{d y}{d x}=y-x e^{\frac{y}{x}} \\ \Rightarrow \frac{d y}{d x}=\left(\frac{y}{x}\right)-e^{y} \\...

### Find the particular solution of the differential equation , given that .

Solution: $\begin{array}{l} \Rightarrow x e^{\frac{y}{x}}-y+x \frac{d y}{d x}=0 \\ \Rightarrow x \frac{d y}{d x}=y-x e^{\frac{y}{x}} \\ \Rightarrow \frac{d y}{d...

### Find the particular solution of the differential equation given that when .

Solution: $\begin{array}{l} \Rightarrow \frac{d y}{d x}=\frac{y(2 y-x)}{x(2 y+x)} \\ \Rightarrow \frac{d y}{d x}=\frac{y\left(2^{y}-1\right)}{x\left(2_{x}^{\frac{y}{x}}+1\right)} \\ \Rightarrow...

### Find the particular solution of the differential equation , it being given that when

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

### Find the particular solution of the different equation. , it being given that when

Solution: $\begin{array}{l} 2 x y+y^{2}-2 x^{2} \frac{d y}{d x}=0 \\ \Rightarrow \frac{d y}{d x}=\frac{2 x y+y^{2}}{2 x^{2}}=\frac{y}{x}+\frac{y^{2}}{2 x^{2}} \\ \Rightarrow \frac{d y}{d...

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

Solution: $\begin{array}{l} \Rightarrow\left(x \cos \frac{y}{x}\right) \frac{d y}{d x}=y \cos \frac{y}{x}+x \\ \Rightarrow \frac{d y}{d x}=\frac{y \cos ^{\frac{y}{x}}+x}{x \cos \frac{y}{x}} \\...

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

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

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

Solution: $\begin{array}{l} \Rightarrow x \frac{d y}{d x}-y+x \sin \frac{y}{x}=0 \\ \Rightarrow \frac{d y}{d x}=\frac{y-x \sin \frac{y}{x}}{x}=\frac{y}{x}-\sin \frac{y}{x} \\ \Rightarrow \frac{d...

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

Solution: $\begin{array}{l} \Rightarrow \frac{d y}{d x}=\frac{y}{x}\left(\log \left(\frac{y}{x}\right)+1\right) \\ \Rightarrow \frac{d y}{d x}=f\left(\frac{y}{x}\right) \end{array}$ $\Rightarrow$...

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

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

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

Solution: $\begin{array}{l} \Rightarrow \frac{d y}{d x}=\frac{y}{x-\sqrt{x y}}=\frac{1}{\frac{x}{y}-\sqrt{\frac{x}{y}}}=\frac{1}{\left(\frac{y}{x}\right)^{-1}-\sqrt{\left(\frac{y}{x}\right)^{-1}}}...

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

Solution: $\begin{array}{l} \Rightarrow\left(x^{3}+3 x y^{2}\right) d x+\left(y^{3}+3 x^{2} y\right) d y=0 \\ \left.\left.\Rightarrow \frac{d y}{d x}=-\frac{\left(x^{3}+3 x...

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

Solution: $\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{x}+3 \mathrm{y}}{\mathrm{x}-\mathrm{y}}$ $\begin{array}{l} \Rightarrow \frac{d y}{d x}=\frac{1+3 \frac{y}{x}}{1-\frac{y}{x}} \\...

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

Solution: $\begin{array}{l} \Rightarrow \mathrm{y}^{2} \mathrm{dx}+\left(\mathrm{x}^{2}+\mathrm{xy}+\mathrm{y}^{2}\right) \mathrm{dy}=0 \\ \Rightarrow...

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

Solution: $\begin{array}{l} \Rightarrow x \frac{d y}{d x}-y=2 \sqrt{y^{2}-x^{2}} \\ \Rightarrow \frac{d y}{d x}=\frac{y+2 \sqrt{y^{2}-x^{2}}}{x} \\ \Rightarrow \frac{d y}{d x}=\frac{y}{x}+2...

### 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...

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

Solution: $\begin{array}{l} \Rightarrow x^{2} \frac{d y}{d x}=x^{2}+x y+y^{2} \\ \Rightarrow \frac{d y}{d x}=\frac{x^{2}+x y+y^{2}}{x^{2}} \\ \Rightarrow \frac{d y}{d...

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

Solution: $\begin{array}{l} \Rightarrow x^{2} \frac{d y}{d x}=2 x y+y^{2} \\ \Rightarrow \frac{d y}{d x}=\frac{2 x y+y^{2}}{x^{2}} \\ \Rightarrow \frac{d y}{d...

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

Solution: $\begin{array}{l} \Rightarrow \frac{d y}{d x}=\frac{2 x y}{x^{2}-y^{2}} \\ \Rightarrow \frac{d y}{d x}=\frac{2}{\left(\frac{y}{x}\right)^{-1}-\left(\frac{y}{x}\right)} \\ \Rightarrow...

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

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

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

Solution: $\begin{array}{l} \frac{d y}{d x}=-\frac{x^{2}-y^{2}}{3 x y} \\ \Rightarrow \frac{d y}{d x}=-\left(\frac{y}{3 x}\right)^{-1}+\left(\frac{y}{3 x}\right) \\ \Rightarrow \frac{d y}{d...

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

Solution: $\begin{array}{l} \Rightarrow \frac{d y}{d x}=-\frac{x-2 y}{2 x-y}=-\frac{1-2^{\frac{z}{x}}{x}}{2-\frac{z}{x}} \\ \Rightarrow \frac{d y}{d x}=f\left(\frac{y}{x}\right) \end{array}$...

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

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

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

Solution: $\begin{array}{l} \left(x^{2}+3 x y+y^{2}\right) d x-x^{2} d y=0 \\ \Rightarrow \frac{d y}{d x}=\frac{x^{2}+3 x y+y^{2}}{x^{2}}=1+3 \frac{y}{x}+\frac{y^{2}}{x^{2}} \\ \Rightarrow \frac{d...

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

Solution: $\begin{array}{l} (x+y) d y+(y-2 x) d x=0 \\ \Rightarrow \frac{d y}{d x}=\frac{2 x-y}{x+y}=\frac{2-\frac{y}{x}}{1+\frac{y}{x}} \\ \Rightarrow \frac{d y}{d x}=f\left(\frac{y}{x}\right)...

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

Solution: $(x-y) d y-(x+y) d x=0$ $\begin{array}{l} \Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{x}+\mathrm{y}}{\mathrm{x}-\mathrm{y}} \Rightarrow...

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

Solution: $\begin{array}{l} x^{2} d y+y(x+y) d x=0 \\ \Rightarrow \frac{d y}{d x}=-\frac{y(x+y)}{x^{2}}=-\left(\frac{y}{x}+\frac{y^{2}}{x^{2}}\right) \\ \Rightarrow \frac{d y}{d...

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

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

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

Solution: $\begin{array}{l} x d y=(x+y) d x \\ \frac{d y}{d x}=\frac{x+y}{x} \\ \Rightarrow \frac{d y}{d x}=1+\frac{y}{x} \\ \Rightarrow \frac{d y}{d x}=f\left(\frac{y}{x}\right) \end{array}$...

### For each of the following differential equations, find a particular solution satisfying the given condition : , it being given that when

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 : , given thaty when

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: , it being given that when

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: 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: 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:

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:

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:

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:

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:

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:

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:

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:

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:

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:

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:

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...

### Find the values of and so that the function is differentiable at each

Solution: Given that $f(x)$ is differentiable at each $x \in R$ For $x \leq 1$ $f(x)=x^{2}+3 x+a$ i.e. a polynomial for $x>1$ $f(x)=b x+2$, which is also a polynomial As, a polynomial function is...

### Let Show that is not derivable at .

Solution: The given function $f(x)=\left\{\begin{array}{l}(2+x), \text { if } x \geq 0 ; \\ (2-x), i f x<0\end{array}\right.$ Left Hand Derivative at $x=0$ : $\begin{array}{l}...

### Show that is neither continuous nor derivable at

Solution: L.H.L. at $x=2$ $\lim _{x \rightarrow 2^{-}} f(x)=\lim _{h \rightarrow 0} f(2-h)=\lim _{h \rightarrow 0}[2-h]=\lim _{h \rightarrow 0} 1=1$ R.H.L. at $x=2$ $\lim _{x \rightarrow 2^{+}}...

### Show that is continuous but not differentiable at

Solution: L.H.L. at $x=5$ $\lim _{x \rightarrow 5^{-}}|x-5|=\lim _{x \rightarrow 5}(5-x)=0$ R.H.L. at $\mathrm{x}=5$ $\lim _{x \rightarrow 5^{+}}|x-5|=\lim _{x \rightarrow 5}(x-5)=0$ Also...

### Show that is not differentiable at

Solution: The given function $f(x)=(x-1)^{1 / 3}$ $\mathrm{Left Hand Derivative}$ at $\mathrm{x}=1$ $\lim _{x \rightarrow 1^{-}} f(x)=\lim _{x \rightarrow 1^{-}} \frac{f(x)-f(1)}{x-1}=\lim _{h...

### Show that is continuous as well as differentiable at

Solution: It is given that: $f(x)=x^{3}$ If a function is differentiable at a point, it is necessary that it is continuous at that point. Left hand derivative at $x=3$ $\begin{array}{l} \lim _{x...

### Discuss the continuity of the function in the interval of

Solution: The given function $f(x)=|x|+|x-1|$ A function $f(x)$ is said to be continuous on a closed interval $[a, b]$ if and only if, (i) $\mathrm{f}$ is continuous on the open interval...

### Locate the point of discontinuity of the function

Solution: The given function $f(x)=\left\{\begin{array}{c}\left(x^{3}-x^{2}+2 x-2\right), \text { if } x \neq 1 \\ 4, \text { if } x=1\end{array}\right.$ L.H.L. at $\mathrm{x}=1: \lim _{\mathrm{x}...

### Show that is continuous at each point except 0 .

Solution: The given function is $f(x)=\left\{\begin{array}{l}x, \text { if } x \neq 0 \\ 1, \text { if } x=0\end{array}\right.$ L.H.L. at $\mathrm{x}=0$ $\lim _{x \rightarrow 0^{-}} f(x)=\lim _{h...

### Show that is continuous.

Solution: The given function $f(x)=\left\{\begin{array}{c}(2 x-1), \text { if } x<2 \\ \frac{3 x}{2}, \text { if } x \geq 2\end{array}\right.$ L.H.L. at $x=2$ $\lim _{x \rightarrow 2^{-}}...

### Show that function is continuous.

Solution: It is known that $\sin x$ is continuous everywhere Now considering the point $x=0$ L.H.L: $\lim _{x \rightarrow 0^{-}} f(x)=\lim _{x \rightarrow 0^{-}}\left(\frac{\sin x}{x}\right)=\lim...

### Show that sec is a continuous function.

Solution: Assume $f(x)=\sec |x|$ and a be any real number. Then, L.H.L. at $\mathrm{x}=\mathrm{a}$ $\lim _{x \rightarrow \mathrm{a}^{-}} \mathrm{f}(\mathrm{x})=\lim _{\mathrm{x} \rightarrow...

### Show that function

Solution: It is given that: $f(x)=\left\{\begin{array}{c} \frac{x^{n}-1}{x-1}, \text { when } x \neq 1 \\ n, \text { when } x=1 \end{array}\right.$ L.H.L. and $\mathrm{x}=1$ $\begin{array}{l} \lim...

### Show that function is continuous function.

Solution: It is given that: $f(x)=\left\{\begin{array}{l} (7 x+5), \text { when } x \geq 0 \\ (5-3 x), \text { when } x<0 \end{array}\right.$ Let us now calculate the limit of $f(x)$ when $x$...

### Prove that the function given is continuous but not differentiable at

Solution: $f(x)=|x-3|$ As every modulus function is continuous for all real $x, f(x)$ is continuous at $x=3$. $f(x)=f(x)=\left\{\begin{array}{l} 3-x, x<0 \\ x-3, x \geq 0 \end{array}\right.$ In...

### Find the values of a and b such that the following functions , defined as at

Solution: $: \mathrm{f}$ is continuous at $x=0$ $\lim _{x \rightarrow 0-} f(x)=\lim _{x \rightarrow 0+} f(x)$ $\lim _{x \rightarrow 0-}\left(\operatorname{asin} \frac{\pi}{2}(x+1)\right)=\lim _{x...

### Find the values of a and b such that the following functions continuous.

Solution: $f$ is continuous at $x=2$ $\begin{array}{l} \lim _{x \rightarrow 2-} f(x)=\lim _{x \rightarrow 2+} f(x)=f(2) \\ \lim _{x \rightarrow 2-}(5)=\lim _{x \rightarrow 2+}[a x+b]=5 \\...

### Show that: is continuous at

Solution: Left Hand Limit: $\lim _{x \rightarrow 2-} f(x)=\lim _{x \rightarrow 1-} x^{3}-3$ $=5$ Right Hand Limit: $\lim _{x \rightarrow 2^{-}} f(x)=\lim _{x \rightarrow 2^{-}} x^{2}+1$ $=5$...

### Show that: is continuous at

Solution: $\begin{array}{l} : \mathrm{Left Hand Limit}: \lim _{\mathrm{x} \rightarrow 1^{-}} \mathrm{f}(\mathrm{x})=\lim _{\mathrm{x} \rightarrow 1-} \mathrm{x}^{2}+1 \\ =2 \end{array}$...

### Show that function:

Solution: $\lim _{x \rightarrow 0} f(x)=\lim _{x \rightarrow 0} x^{2} \sin \frac{1}{x}$ As $\lim _{x \rightarrow 0} x^{2}=0$ and $\sin \left(\frac{1}{x}\right)$ is bounded function between $-1$ and...

### For what valve of is the following function Ans.

Solution: $f$ is continuous at $x=\frac{\pi}{2}$ $\begin{array}{l} \Rightarrow \lim _{x \rightarrow-\frac{\pi}{r}} f(x)=f\left(\frac{\pi}{2}\right) \\ \Rightarrow \lim _{x \rightarrow \frac{\pi}{r}}...

### For what value of is the following function continuous at

Solution: As, $f(x)$ is continuous at $x=2$ $\begin{array}{l} \Rightarrow \lim _{x \rightarrow 2-} 2 x+1=\lim _{x \rightarrow 2^{*}} 3 x-1=f(2) \\ \Rightarrow \lim _{x \rightarrow 2^{-}} 2 x+1=f(2)...

### Find the value of for which

; is continuous at x=-1

Solution: As, $f(x)$ is continuous at $x=0$ $\begin{array}{l} \Rightarrow \lim _{x \rightarrow-1} \frac{x^{2}-2 x-3}{x+1}=f(0) \\ \Rightarrow \lim _{x \rightarrow-1} \frac{(x-3)(x+1)}{x+1}=\lambda...