Solution: $\begin{array}{l} \vec{a}=6 \hat{\imath}-3 \hat{\jmath}-6 \hat{k} \\ \vec{b}=\hat{\imath}+\hat{\jmath}+\hat{k} \end{array}$ Let $\vec{a}$ be written as sum of $\vec{c}+\vec{d}$, where...

### Find the angle between and , if and .

Solution: $\begin{array}{l} \text { Given: } \vec{a}=2 \hat{\imath}-\hat{\jmath}+3 \hat{k} \\ \vec{b}=3 \hat{\imath}+\hat{\jmath}+2 \hat{k} \\ \vec{a}+\vec{b}=5 \hat{\imath}+5 \hat{k} \\...

### Find a vector of magnitude , making an angle with – axis, with – axis and an acute angle with z – axis.

Solution: $\mathrm{I} \overrightarrow{\mathrm{a}} \mathrm{I}=5 \sqrt{2}$ Also it is given that $\alpha=\frac{\pi}{4}$ $\beta=\frac{\pi}{2}, \gamma=$ acute angle. So from this $\cos...

### Show that the vector is equally inclined to the coordinate axes.

Solution: $\begin{array}{l} \vec{a}=\cos \gamma=\frac{1}{\sqrt{3}} \\ \mathrm{I} \vec{a} \mathrm{I}=\sqrt{1^{2}+1^{2}+1^{2}}=\sqrt{3} \end{array}$ Since direction cosines or cosines of angle made by...

### Find the angles which the vector makes with the coordinate axes.

Solution: Direction cosines gives the angle made by the vector with $x$-axis, $y-$ axis, z-axis respectively. $\begin{array}{l} \vec{a}=3 \hat{\imath}-6 \hat{\jmath}+2 \hat{k} \\ \mathrm{I} \vec{a}...

### If is a unit vector such that , find .

Solution: Given: $\mathrm{I} \vec{a} \mathrm{I}=1$ $\begin{array}{l} (\vec{x}-\vec{a}) \cdot(\vec{x}+\vec{a})=8 \\ \vec{x} \cdot \vec{x}+\vec{x} \cdot \vec{a}-\vec{a} \cdot \vec{x}-\vec{a} \cdot...

### If and then calculate the angle between and .

Solution: $\vec{a}=\hat{\imath}+2 \hat{\jmath}-3 \hat{k}$ $\vec{b}=3 \hat{\imath}-\hat{\jmath}+2 \hat{k}$ $\begin{array}{l}2 \vec{a}+\vec{b} & =2(\hat{\imath}+2 \hat{\jmath}-3 \hat{k})+(3...

### Find the angle between the vectors and , when

i. and

ii. and

iii. and .

Solution: $\begin{array}{l} \text { i) } \vec{a}=\hat{\imath}-2 \hat{\jmath}+3 \hat{k} \\ \vec{b}=3 \hat{\imath}-2 \hat{\jmath}+\hat{k} \\ \mathrm{I} \vec{a}...

### Write the projection of vector along the vector .

Solution: $\begin{array}{l} \vec{a}=\hat{\imath}+\hat{\jmath}+\hat{k} \\ \vec{b}=\hat{\jmath} \end{array}$ $\mathrm{I} \vec{a} \mathrm{I}=\sqrt{3}=\sqrt{1^{2}+1^{2}+1^{2}}$ $\begin{array}{l}...

### Let and . Find the projection of (i) on and (ii) on .

Solution: If $\vec{a}=\mathrm{a} 1 \hat{\imath}+\mathrm{a} 2 \hat{\jmath}+\mathrm{a}_{3} \hat{k}$ and $\vec{b}=\mathrm{b}_{1} \hat{\imath}+\mathrm{b}_{2} \hat{\jmath}+\mathrm{b}_{3} \hat{k}$, then...

### Show that the vectors are mutually perpendicular unit vectors.

Solution: $\begin{array}{l} \vec{a}=\frac{1}{7}(2 \hat{\imath}+3 \hat{\jmath}+6 \hat{k}) \\ \vec{b}=\frac{1}{7}(3 \hat{\imath}-6 \hat{\jmath}+2 \hat{k}) \\ \vec{c}=\frac{1}{7}(6 \hat{\imath}+2...

### If and then find the value of so that and are orthogonal vectors.

Solution: $\begin{array}{l} \vec{a}=\hat{\imath}-1 \hat{\jmath}+7 \hat{k} \\ \vec{b}=5 \hat{\imath}-1 \hat{\jmath}+\lambda \hat{k} \\ \vec{a}+\vec{b}=6 \hat{\imath}-2 \hat{\jmath}+(7+\lambda)...

i. If and , show that is perpendicular to .

ii. If and then show that and are orthogonal.

Solution: If two vectors are perpendicular or orthogonal then their dot product is zero. $\begin{array}{l} \text { i) } \quad \vec{a}=\hat{\imath}+2 \hat{\jmath}-3 \hat{k} \\ \vec{b}=3...

### Find when

i. and

ii. and

iii. and

Solution: If $\vec{a}=\mathrm{a} 1 \hat{\imath}+\mathrm{a} 2 \hat{\jmath}+\mathrm{a} 3 \hat{k}$ and $\vec{b}=\mathrm{b}_{1} \hat{\imath}+\mathrm{b}_{2} \hat{\jmath}+\mathrm{b}_{3} \hat{k}, \text {...

### Write a unit vector in the direction of , where and are the points and respectively.

Solution: $\begin{array}{l} \overrightarrow{O P}=\hat{\imath}+3 \hat{\jmath} \\ \overrightarrow{O Q}=4 \hat{\imath}+5 \hat{\jmath}+6 \hat{k} \\ \overrightarrow{P Q}=\overrightarrow{O...

### Find the position vector of the mid-point of the vector joining the points and

Solution: $\begin{array}{l} \overrightarrow{O A}=3 \hat{\imath}+2 \hat{\jmath}+6 \hat{k} \\ \overrightarrow{O B}=\hat{\imath}+4 \hat{\jmath}-2 \hat{k} \end{array}$ Let $\mathrm{C}$ be the mid-point...

### Find the position vector of a point which divides the line joining and in the ratio (i) internally (ii) externally.

Solution: $\begin{array}{l} \overrightarrow{O A}=-2 \hat{\imath}+\hat{\jmath}+3 \hat{k} \\ \overrightarrow{O B}=3 \hat{\imath}+5 \hat{\jmath}-2 \hat{k} \end{array}$ (i) $\mathrm{R}$ divides...

### The position vectors of two points and are and respectively. Find the position vector of a point which divides externally in the ratio . Also, show that is the mid-point of the line segment .

Solution: Given: $\overrightarrow{O A}=(2 \vec{a}+\vec{b})$ $\overrightarrow{O B}=(\vec{a}-3 \vec{b})$ Position vector of $\mathrm{C}$ which divides $\mathrm{AB}$ in the ratio $1: 2$ externally is...

### Find the position vector of the point which divides the join of the points and (i) internally and (ii) externally in the ratio

Solution: Given: $\overrightarrow{O A}=(2 \vec{a}-3 \vec{b})$ $\overrightarrow{O B}=(3 \vec{a}-2 \vec{b})$ (i) Let $\mathrm{P}$ be the point that divides $\mathrm{A}, \mathrm{B}$ internally in the...

### Using vector method, show that the points and are the vertices of a rightangled triangle.

Solution: $\begin{array}{l} \overrightarrow{O A}=\hat{\imath}-\hat{\jmath} \\ \overrightarrow{O B}=4 \hat{\imath}-3 \hat{\jmath}+\hat{k} \\ \overrightarrow{O C}=2 \hat{\imath}-4 \hat{\jmath}+5...

### If the position vectors of the vertices and of a be and , respectively, prove that is equilateral.

Solution: $\begin{array}{l} \overrightarrow{O A}=\hat{\imath}+2 \hat{\jmath}+3 \hat{k} \\ \overrightarrow{O B}=2 \hat{\imath}+3 \hat{\jmath}+\hat{k} \\ \overrightarrow{O C}=3...

### Show that the points and having position vectors and respectively, are collinear.

Solution: $\begin{array}{l} \overrightarrow{O A}=\hat{\imath}+2 \hat{\jmath}+7 \hat{k} \\ \overrightarrow{O B}=2 \hat{\imath}+6 \hat{\jmath}+2 \hat{k} \\ \overrightarrow{O C}=3 \hat{\imath}+10...

### If and , find

Solution: $\begin{array}{l} \vec{a}=\hat{\imath}-2 \hat{\jmath} \\ \vec{b}=2 \hat{\imath}-3 \hat{\jmath} \\ \vec{c}=2 \hat{\imath}+3 \hat{k} \\ \vec{a}+\vec{b}+\vec{c}=(\hat{\imath}-2...

### Find a vector of magnitude 8 units in the direction of the vector .

Solution: $\vec{A}=5 \hat{\imath}-\hat{\jmath}+2 \hat{k}$ Magnitude of required vector is 8 units $\mathrm{I} \vec{A} \mathrm{I}=\sqrt{5^{2}+1^{2}+2^{2}}=\sqrt{25+1+4}=\sqrt{30}$ $\begin{array}{l}...

### Find a vector of magnitude 9 units in the direction of the vector .

Solution: $\vec{A}=-2 \hat{\imath}+\hat{\jmath}+2 \hat{k}$ $\mathrm{I} \vec{A} \mathrm{I}=\sqrt{(-2)^{2}+1^{2}+2^{2}}=\sqrt{4+1+4}=\sqrt{9}=3$ Note: Any vector $\vec{X}$ is given by $\vec{X}=$...

### If and then find a unit vector parallel to .

Solution: $\begin{array}{l} \vec{a}=\hat{\imath}+2 \hat{\jmath}-3 \hat{k} \\ \vec{b}=2 \hat{\imath}+4 \hat{\jmath}+9 \hat{k} \end{array}$ Then $\vec{a}+\vec{b}=(\hat{\imath}+2 \hat{\jmath}-3...

### If and then find a unit vector in the direction of .

Solution: $\begin{array}{l} \vec{a}=3 \hat{\imath}+\hat{\jmath}-5 \hat{k} \\ \vec{b}=\hat{\imath}+2 \hat{\jmath}-\hat{k} \end{array}$ Then $\vec{a}-\vec{b}=(3 \hat{\imath}+\hat{\jmath}-5...

### If and then find the unit vector in the direction of .

Solution: $\vec{a}=-\hat{\imath}+\hat{\jmath}-\hat{k}$ $\vec{b}=2 \hat{\imath}-\hat{\jmath}+2 \hat{k}$ Then $(\vec{a}+\vec{b})=(-\hat{\imath}+\hat{\jmath}-\hat{k})+(2 \hat{\imath}-\hat{\jmath}+2...

### Find a unit vector in the direction of the vector:

A.

B.

C.

D.

Solution: If $\vec{a}=\mathrm{a}_{1} \hat{\imath}+\mathrm{a}_{2} \hat{\jmath}+\mathrm{a}_{3} \hat{k}$, then Unit vector in the direction of $\vec{a}$ can be given by $\hat{a}=\frac{\vec{a}}{I...

### Write down the magnitude of each of the following vectors:

A.

B.

C.

D.

Solution: A. $\vec{a}=\hat{\imath}+2 \hat{\jmath}+5 \hat{k}$ If $\vec{a}=\mathrm{a}_{1} \hat{\imath}+\mathrm{a}_{2} \hat{\jmath}+\mathrm{a}_{3} \hat{k}$, then $\mathrm{I} \vec{a}...

### Mark (√) against the correct answer in the following: The general solution of the is

A.

B.

C.

D. None of these

Solution: $\frac{\mathrm{dy}}{\mathrm{dx}}+\frac{\mathrm{y}}{\mathrm{x}}=\mathrm{x}^{2}$ Comparing with $\frac{d y}{d x}+P y=Q$ Integrating factor $=\mathrm{x}$ General solution is $\mathrm{yx}=\int...

### Mark (√) against the correct answer in the following: The general solution of the DE is

A.

B.

C.

D. None of these

Solution: $\frac{d y}{d x}+y \operatorname{Cot} x=2 \operatorname{Cos} x$ Comparing with $\frac{d y}{d x}+P y=Q$ Integrating factor is $e^{\int \cot x d x}=\operatorname{Sin} x$ General solution is...

### Mark (√) against the correct answer in the following: The general solution of the

A.

B.

C.

D. None of these

Solution: $\frac{\mathrm{dy}}{\mathrm{dx}}+\mathrm{y} \tan \mathrm{x}=\mathrm{Secx}$ Comparing with $\frac{d y}{d x}+P y=Q$ Integrating factor $e^{\int \tan x d x}=\operatorname{Sec} x$ General...

### Mark (√) against the correct answer in the following: The general solution of the DE is

A.

B.

C.

D. None of these

Solution: $\frac{d y}{d x}=\frac{y}{x}+\sin \frac{y}{x}$ Let $\mathrm{y}=\mathrm{vx}$ $\begin{array}{l} \mathrm{dy} / \mathrm{dx}=\mathrm{v}+\mathrm{xdv} / \mathrm{d} \mathrm{x} \\...

### Mark (√) against the correct answer in the following: The general solution of the DE is

A.

B.

C.

D. None of these

Solution: $\begin{array}{r} (x-y) d y+(x+y) d x=0 \\ \qquad \frac{d y}{d x}=\frac{x+y}{y-x} \end{array}$ Let $\mathrm{y}=\mathrm{vx}$ $\begin{array}{l} \mathrm{dy} /...

### Mark (√) against the correct answer in the following: The general solution of the DE is

A.

B.

C.

D. None of these

Solution: $\begin{array}{r} 2 \mathrm{xydy}+\left(\mathrm{x}^{2} \quad \mathrm{y}^{2}\right) \mathrm{d} \mathrm{x}=0 \\ \frac{d y}{d x}=\frac{y^{2}-x^{2}}{2 x y} \end{array}$ Let...

### Mark (√) against the correct answer in the following: The general solution od the DE is

A.

B.

C.

D. None of these

Solution: $\mathrm{x} \frac{d y}{d x}=y+x \tan \frac{y}{x}$ Dividing both sides by $x$, we get, $\frac{d y}{d x}=\frac{y}{x}+\tan \frac{y}{x}$ Let $\mathrm{y}=\mathrm{vx}$ Differentiating both...

### Mark (√) against the correct answer in the following: The general solution of the DE is.

A.

B.

C.

D. None of these

Solution: $\begin{array}{l} \mathrm{x}^{2} \frac{d y}{d x}=x^{2}+x y+y^{2} \\ \frac{d y}{d x}=1+\frac{y}{x}+\frac{y^{2}}{x^{2}} \end{array}$ Let $\mathrm{y}=\mathrm{vx}$ $\begin{array}{l}...

### Mark (√) against the correct answer in the following: The general solution of the is

A.

B.

C.

D. None of these

Solution: $\frac{d y}{d x}=\frac{y^{2}-x^{2}}{2 x y}$ Let $\mathrm{y}=\mathrm{vx}$ $\begin{array}{l} \mathrm{dy} / \mathrm{dx}=\mathrm{v}+\mathrm{xdv} / \mathrm{dx} \\ \frac{x^{2}...

### Mark (√) against the correct answer in the following: The general solution of the

A.

B.

C.

D. None of these

Solution: $\begin{array}{r} \frac{\mathrm{dy}}{\mathrm{dx}}=\sqrt{1-\mathrm{x}^{2}} \sqrt{1-\mathrm{y}^{2}} \\ \frac{d y}{\sqrt{1-y^{2}}}=\sqrt{1-x^{2}} d x \end{array}$ $\operatorname{Let}...

### Mark (√) against the correct answer in the following: The general solution of the DE is

A.

B.

C.

D. None of these

Solution: $\log \left(\frac{d y}{d x}\right)=(a x+b y)$ $\begin{array}{l} \frac{d y}{d x}=e^{a x+b y} \\ \frac{d y}{e^{b y}}=e^{a x} d x \end{array}$ On integrating on both sides we get...

### Mark (√) against the correct answer in the following: The general solution of the is

A.

B.

C.

D. None of these

Solution: $\begin{array}{r} x \sqrt{1+y^{2}} \mathrm{dx}+\mathrm{y} \sqrt{1+x^{2}} \mathrm{dy}=0 \\ \frac{y d y}{\sqrt{1+y^{2}}}=\frac{-x d x}{\sqrt{1+x^{2}}} \end{array}$ Let $1+y^{2}=t$ and...

### Mark (√) against the correct answer in the following: the general solution of the DE is

A.

B.

C.

D. None of these

Solution: $\left(1+x^{2}\right) d y-x y d x=0$ $\frac{d y}{y}=\frac{x}{1+x^{2}} d x$ Let $1+\mathrm{x}^{2}=\mathrm{t}$ $\begin{array}{r} 2 \mathrm{x} \mathrm{dx}=\mathrm{dt} \\ \frac{d y}{y}=\frac{d...

### Mark (√) against the correct answer in the following: The solution of the DE is

A.

B.

C.

D. none of these

Solution: $\frac{d y}{d x}+y \log y \operatorname{Cot} x=0$ Let $\log \mathrm{y}=\mathrm{t}$ On differentiating we get $\begin{array}{l} \frac{1}{y} d y=d t \\ \frac{d t}{t}=-\operatorname{Cot} x d...

### Mark (√) against the correct answer in the following: the solution of the DE is

A.

B.

C.

D. none of these

Solution: $\begin{array}{l} x \operatorname{Cosydy}=\left(x e^{x} \log x+e^{x}\right) d x \\ \operatorname{Cosydy}=\frac{x \operatorname{exlogx}+e x}{x} d x \end{array}$ On integrating on both sides...

### Mark (√) against the correct answer in the following: The solution of the is

A.

B.

C.

D. none of these

Solution: $\cos x(1+\cos y) d x-\sin y(1+\sin x) d y=0$ Let $1+\cos y=t$ and $1+\sin x=u$ On differentiating both equations, we get $-\sin y d y=d t$ and $\cos x d x=d u$ Substitute this in the...

### Mark (√) against the correct answer in the following: The solution of the is

A.

B.

C.

D. None of these

Solution: $\begin{array}{l} \frac{d y}{d x}=\frac{-2 x y}{x^{2}+1} \\ \frac{d y}{y}=\frac{-2 x d x}{x^{2}+1} \end{array}$ Let $\mathrm{x}^{2}+1=\mathrm{t}$ On differentiating on both sides we get $2...

### Mark (√) against the correct answer in the following: The solution of the is

A.

B.

C.

D. None of these

Solution: $\begin{array}{l} \frac{d y}{d x}=\frac{1-\operatorname{Cos} x}{1+\operatorname{Cos} x} \\ \frac{d y}{d x}=\frac{2 \sin ^{2} \frac{x}{2}}{2 \cos ^{2} \frac{x}{2}}, \\ \frac{d y}{d x}=\tan...

### Mark (√) against the correct answer in the following: The solution of the is

A.

B.

C.

D. None of these

Solution: $\frac{\mathrm{dy}}{\mathrm{dx}}+\sqrt{\frac{1-\mathrm{y}^{2}}{1-\mathrm{x}^{2}}}=0$ $\frac{-d y}{\sqrt{1-y^{2}}}=\frac{d x}{\sqrt{1-x^{2}}}$ On integrating on both sides, we get $-\sin...

### Mark (√) against the correct answer in the following: The solution of the is

A.

B.

C.

D. None of these

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

### Mark (√) against the correct answer in the following: The solution of the DE is

A.

B.

C.

D. None of these

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

### Mark (√) against the correct answer in the following: The solution of the is.

A.

B.

C.

D. None of these

Solution: $\frac{d y}{d x}=\frac{1+y^{2}}{1+x^{2}}$ On integrating on both sides, we get $\begin{array}{l} \tan ^{-1} y=\tan ^{-1} x+c \\ \frac{y-x}{1+y x}=\mathrm{c} \\...

### Mark (√) against the correct answer in the following: The solution of the is

A.

B.

C.

D. None of these

Solution: $x \frac{d y}{d x}=\cot y$ Separating the variables, we get, $\begin{array}{c} \frac{d y}{\cot y}=\frac{d x}{x} \\ \tan y d y=\frac{d x}{x} \end{array}$ Integrating both sides, we get,...

### Mark (√) against the correct answer in the following: The solution of the is

A.

B.

C. D. None of these

Solution: $\begin{array}{l} \text { Given } x d y+y d x=0 \\ x d y=-y d x \end{array}$ $-\frac{d y}{y}=\frac{d x}{d x}$ On integrating on both sides we get, $\begin{array}{l} -\log y=\log x+c \\...

### Mark (√) against the correct answer in the following: The solution of the is

A.

B.

C.

D. None of these

Solution: $\begin{array}{l} \left(e^{x}+1\right) y d y=(y+1) e^{x} d x \\ \frac{y d y}{y+1}=\frac{e^{x} d x}{\left(e^{x}+1\right)} \end{array}$ On differentiating on both sides we get $e^{x} d x=d...

### Mark (√) against the correct answer in the following: The solution of the DE is

A.

B.

C.

D. None of these

Solution: $\begin{array}{c} \frac{d y}{d x}=e^{x+y} \\ \frac{d y}{d x}=e^{x} e^{y} \\ e^{-y} d y=e^{x} d x \end{array}$ On integrating on both sides, we get $\begin{array}{c}...

### Solve , given that when , then .

Solution: $\left(1+y^{2}\right) d x+\left(x-e^{-\tan ^{-1} y}\right) d y=0\dots (1)$ General solution for the differential equation in the form of $\frac{d y}{d x}+\mathrm{Py}=$ Qis given by, $y...

### Find the general solution for each of the following differential equations.

Solution: $(x+y+1) \frac{d y}{d x}=1-\ldots(1)$ General solution for the differential equation in the form of $\frac{d y}{d x}+\overline{\mathrm{Py}}=$ Qis given by, $y \cdot(I . F .)=\int Q \cdot(I...

### Find the general solution for each of the following differential equations.

Solution: $(\mathrm{x}+\mathrm{y}) \frac{d y}{d x}=1-\ldots(1)$ General solution for the differential equation in the form of $\frac{d y}{d x}+\mathrm{Py}=\mathrm{Q}$ is given by, $y \cdot\left(I ....

### Find the general solution for each of the following differential equations.

Solution: $y d x+\left(x-y^{2}\right) d y=0\dots (1)$ General solution for the differential equation in the form of $\frac{d y}{d x}+\mathrm{Py}=\mathrm{Q}$ General solution is given by, $y...

### A curve passes through the point and the sum of the coordinates of any point on the curve exceeds the magnitude of the slope of the tangent to the curve at that point by Find the equation of the curve.

Solution: General solution for the differential equation in the form of $\frac{d y}{d x}+\mathrm{Py}=$ Qis given by, $y \cdot(I . F .)=\int Q \cdot(I . F .) d x+c$ Where, integrating factor, $I . F...

### A curve passes through the origin and the slope of the tangent to the curve at any point is equal to the sum of the coordinates of the point. Find the equation of the curve.

Solution: General solution for the differential equation in the form of $\frac{d y}{d x}+\mathrm{Py}=$ Qis given by, $y \cdot(I . F .)=\int Q \cdot(I . F .) d x+c$ Where, integrating factor, $\text...

### Find a particular solution satisfying the given condition for each of the following differential equations. , given that when

Solution: $\frac{d y}{d x}+\operatorname{ytan} \mathrm{x}=2 \mathrm{x}+x^{2} \tan x\dots(1)$ General solution for the differential equation in the form of $\frac{d y}{d x}+\mathrm{Py}=\mathrm{Q}$...

### Find a particular solution satisfying the given condition for each of the following differential equations. , given that when

Solution: $x \frac{d y}{d x}-y=\log x-\ldots(1)$ General solution for the differential equation in the form of $\frac{d y}{d x}+\mathrm{Py}=\mathrm{Q}$ is given by, $y \cdot(I . F .)=\int Q \cdot(I...

### Find a particular solution satisfying the given condition for each of the following differential equations. , given that when

Solution: $\left(1+x^{2}\right) \frac{d y}{d x}+2 x y=4 x^{2}-\ldots(1)$ General solution for the differential equation in the form of $\frac{d y}{d x}+\mathrm{Py}=\mathrm{Q}$ is given by, $y...

### Find a particular solution satisfying the given condition for each of the following differential equations. , given that , when

Solution: $\frac{d y}{d x}+2 y=e^{-2 x} \cdot \operatorname{Sin} x\dots(1)$ General solution for the differential equation in the form of $\frac{d y}{d x}+\mathrm{Py}=\mathrm{Q}$ General solution is...

Solution: General solution for the differential equation in the form of $\frac{d y}{d x}+\mathrm{Py}=\mathrm{Q}\dots(1)$ General solution is given by, $y \cdot(I . F .)=\int Q \cdot(I . F .) d x+c$...

Solution: $\frac{\mathrm{dy}}{\mathrm{dx}}+\mathrm{y} \cdot \operatorname{Cot} \mathrm{x}=4 \mathrm{x} \operatorname{Cosec} \mathrm{x}^{--}-\mathrm{c}-\mathrm{c}-\mathrm{c}(1)\dots (1)$ General...

Solution: $\mathrm{x} \frac{d y}{d x}+y=x^{3}\dots(1)$ General solution for the differential equation in the form of $\frac{d y}{d x}+\mathrm{Py}=\mathrm{Q}$ is given by, $y \cdot(I . F .)=\int Q...

### Find the general solution for each of the following differential equations.

Solution: $\frac{d y}{d x}+2 y \tan x=\sin x\dots(1)$ To solve (1) we will use following formula $\int \tan x d x=\log _{\mid} \sec x \mid$ $\begin{array}{l} \operatorname{alog} \mathrm{b}=\log...

### Find the general solution for each of the following differential equations.

Solution: $\frac{d y}{d x}+y \cot x=\sin 2 x-\dots (1)$ General solution: For the differential equation in the form of $\frac{d y}{d x}+\mathrm{Py}=\mathrm{Q}$ General solution is given by, $y...

### Find the general solution for each of the following differential equations.

Solution: $\frac{d y}{d x}=\operatorname{ytan} x-2 \sin x-\ldots(1)$ General solution for the differential equation in the form of $\frac{d y}{d x}+\mathrm{Py}=\mathrm{Q}$ General solution is given...

### Find the general solution for each of the following differential equations.

Solution: $x \frac{d y}{d x}-y=2 x^{2} \sec x \dots (1)$ General solution for the differential equation in the form of $\frac{d y}{d x}+\mathrm{Py}=\mathrm{Q}$ General solution is given by, $y...

### Find the general solution for each of the following differential equations.

Solution: $\frac{d y}{d x}+2 \mathrm{y}(\cot x)=3 x^{2} \operatorname{cosec}^{2} x\dots (1)$ General solution for the differential equation in the form of $\frac{d y}{d x}+\mathrm{Py}=\mathrm{Q}$...

### Find the general solution for each of the following differential equations.

Solution: $\left(1+x^{2}\right) \frac{d y}{d x}+2 x y=\cot x-\ldots(1)$ General solution for the differential equation in the form of $\frac{d y}{d x}+\mathrm{Py}=\mathrm{Q}$ General solution is...

### Find the general solution for each of the following differential equations.

Solution: $\frac{d y}{d x}+y=\cos x-\sin x-\cdots \dots(1)$ General solution for the differential equation in the form of $\frac{d y}{d x}+P y=Q$ is given by, $y \cdot(I . F .)=\int Q \cdot(I . F .)...

### Find the general solution for each of the following differential equations.

Solution: $\frac{d y}{d x}+2 y=\sin x- \dots(1)$ General solution for the differential equation in the form of is given by, $y \cdot(I . F .)=\int Q \cdot(I . F .) d x+c$ Where, integrating factor,...

### Find the general solution for each of the following differential equations.

Solution: $x d y+\left(y-x^{3}\right) d x=0\dots (1)$ General solution for the differential equation in the form of is given by,$\frac{d y}{d x}+P y=Q$ $y \cdot(I . F .)=\int Q \cdot(I . F .) d x+c$...

### Find the general solution for each of the following differential equations.

Solution: $x d y-\left(y+2 x^{2}\right) d x=0\dots (1)$ General solution for the differential equation in the form of $\frac{d y}{d x}+P y=Q$ is given by $y \cdot(I . F .)=\int Q \cdot(I . F \cdot)...

### Find the general solution for each of the following differential equations.

Solution: $\left(\mathrm{y}+3 \mathrm{x}^{2}\right) \frac{d x}{d y}=\mathrm{x}-\cdots \dots(1)$ General solution for the differential equation in the form of is given by,$\frac{d y}{d x}+P y=Q$ $y...

### Find the general solution for each of the following differential equations.

Solution: $\frac{d y}{d x}+\frac{4 x}{x^{2}+1} y+\frac{1}{\left(1+x^{2}\right)^{2}}=0\dots (1)$ General solution for the differential equation in the form of $\frac{d y}{d x}+P y=Q$ is given by, $y...

### Find the general solution for each of the following differential equations.

Solution: $(1+x) \frac{d y}{d x}-y=e^{3 x}(1+x)^{2}\dots (1)$ General solution for the differential equation in the form of is given by $\frac{d y}{d x}+P y=Q$ $y \cdot(I . F .)=\int Q \cdot(I . F...

### Find the general solution for each of the following differential equations.

Solution: $\begin{array}{r} x \frac{d y}{d x}+2 y=x^{2} \log x-\ldots(1) \\ \qquad \frac{d(\log x)}{d x}=\frac{1}{x} \end{array}$ General solution for the differential equation in the form of...

### Find the general solution for each of the following differential equations.

Solution: $x \frac{d y}{d x}+y=x \log x\dots (1)$ To solve (1) we will use following formula $\begin{array}{l} \int \frac{1}{x} d x=\log x \\ a^{\log _{a} b=b} \\ \qquad \int u \cdot v d x=u . \int...

### Find the general solution for each of the following differential equations.

Solution: $\frac{d y}{d x}-y \tan x=e^{x} \sec x-\ldots(1)$ To solve (1) we will use following formula $\begin{array}{l} \quad \int \tan x d x=\log (\sec x) \\ a \log b=\log b^{a} \\ a^{\log _{a}...

### Find the general solution for each of the following differential equations.

Solution: Given Differential Equation : $\frac{\text { dy }}{\mathrm{dx}}+\frac{1}{\mathrm{x}} \cdot \mathrm{y}=\mathrm{x}^{2} \ldots \ldots \ldots \mathrm{eq}(1)$ Formula : i) $\int \frac{1}{x} d...

### Find the general solution for each of the following differential equations.

Solution: $\frac{d y}{d x}+8 y=5 e^{-3 x} \ldots(1)$ To solve (1) we will use following formula $\begin{array}{l} \int 1 d x=x \\ \int e^{k x} d x=\frac{e^{k x}}{k} \end{array}$ General solution for...

### Find the general solution for each of the following differential equations.

Solution: $\frac{d y}{d x}+3 y=e^{-2 x}\dots \dots(1)$ To solve (1) we will use following formula $\begin{array}{l} \int 1 d x=x \\ \int e^{k x} d x=\frac{e^{k x}}{k} \end{array}$ General solution...

### Find the general solution for each of the following differential equations.

Solution: $\frac{d y}{d x}+2 y=6 e^{x}$ To solve (1) we will use following formula $\begin{array}{l} \int 1 d x=x \\ \int e^{k x} d x=\frac{e^{k x}}{k} \end{array}$ General solution for the...

### Find the general solution for each of the following differential equations.

Solution: $\left(x^{2}+1\right) \frac{d y}{d x}-2 x y=\left(x^{2}+1\right)\left(x^{2}+2\right)^{-\ldots}-\ldots(1)$ To solve (1) we will use following formula $\begin{array}{l} \int...

### Find the general solution for each of the following differential equations.

Solution: $\left(1-x^{2}\right) \frac{d y}{d x}=x y=a x-\ldots(1)$ To solve (1) we will use following formula $\begin{array}{r} \int \frac{f^{\prime}(X)}{f(x)} d x=\log f(x)+\mathrm{C} \\ a \log...

### Find the general solution for each of the following differential equations.

Solution: $\left(1-x^{2}\right) \frac{d y}{d x}+x y=x \sqrt{1-x^{2}}\dots \dots (1)$ To solve (1) we will use following formula $\begin{array}{l} \int \frac{f^{\prime}(x)}{f(x)} d x=\log f(x) \\...

### Find the general solution for each of the following differential equations.

Solution: $\left(1+x^{2}\right) \frac{d y}{d x}+2 x y=\frac{1}{\left(1+x^{2}\right)}\dots \dots(1)$ To solve (1) we will use following formula $\begin{array}{l} \int \frac{f^{\prime}(x)}{f(x)} d...

### Find the general solution for each of the following differential equations.

Solution: $x \frac{d y}{d x}-y=x+1\dots \dots (1)$ To solve (1) we will use following formula $\begin{array}{l} \int \frac{1}{x} d x=\log x \\ \int x^{n} d x=\frac{x^{n+1}}{n+1}+c \\ a \log b=\log...

### Find the general solution for each of the following differential equations.

Solution: $x \frac{d y}{d x}-y+2 x^{3}\dots \dots(1)$ To solve (1) we will use following formula $\begin{array}{c} \int \frac{1}{x} d x=\log x \\ a \log b=\log b^{a} \end{array}$ $a^{\log _{a}...

### Find the general solution for each of the following differential equations.

Solution: $x \frac{d y}{d x}+y=3 x^{2}-3\dots \dots (1)$ To solve (1) we will use following formula $\begin{array}{l} \int \frac{1}{x} d x=\log x \\ a^{\log _{a} b}=\log b \end{array}$ General...

### Find the general solution for each of the following differential equations.

Solution: To solve the given equation we will use following formula $\begin{array}{l} \int \frac{1}{x} d x=\log \mathrm{x}+\mathrm{C} \\ \int x^{n} d x=\frac{x^{n+1}}{n+1}+\mathrm{c} \\ a \log...

### Find the general solution for each of the following differential equations.

Solution: $x \frac{d y}{d x}+2 y=x^{2}\dots \dots (1)$ To solve (1) we will use following formula $\begin{array}{l} \int \frac{1}{x} d x=\log x \\ \int x^{n} d x=\frac{x^{n}+1}{n+1}+c \\ a^{\log...

### Find the general solution for each of the following differential equations.

Solution: Given Differential Equation : $\frac{\text { dy }}{\mathrm{dx}}+\frac{1}{\mathrm{x}} \cdot \mathrm{y}=\mathrm{x}^{2} \ldots \ldots \ldots \mathrm{eq}(1)$ Formula : i) $\int \frac{1}{x} d...

### If the matrix A is both symmetric and skew-symmetric, show that A is a zero matrix.

Solution: The transpose of the matrix is an operation of making interchange of elements by the rule on positioned element $a_{j i}$ shifted to new position $a_{j i}$. The symmetric matrix is defined...

### If and , find .

Solution: We have $A=\left(\begin{array}{ll}2 & 3 \\ 1 & 2\end{array}\right)$. Now addition/subtraction of two matrices is possible if order of both the matrices are same and multiplication...

### If A and B are symmetric matrices of the same order, show that (AB – BA) is a skew symmetric matrix.

Solution: We have $A$ and $B$ are symmetric matrices. Therefore $A^{T}=A$ and $B^{T}=B$ The transpose of the matrix is an operation of making interchange of elements by the rule on positioned...

### If , show that .

Solution: We have $A=\left(\begin{array}{cc}\cos \alpha & \sin \alpha \\ -\sin \alpha & \cos \alpha\end{array}\right)$. The transpose of the matrix is an operation of making interchange of...

### If and , find a matrix such that

Solution: We have $A=\left(\begin{array}{ll}4 & 2 \\ 1 & 3\end{array}\right), B=\left(\begin{array}{cc}-2 & 1 \\ 3 & 2\end{array}\right)$ and $3 A-2 B+X=0$ We can have the addition...

### If and , find a matrix such that .

Solution: We have $A=\left(\begin{array}{cc}2 & -3 \\ 4 & 5\end{array}\right), B=\left(\begin{array}{cc}-1 & 2 \\ 0 & 3\end{array}\right)$ and $A+2 B+X=0$. We can have the addition...

### If , show that is symmetric

Solution: We have $\left(\begin{array}{ll}4 & 5 \\ 1 & 8\end{array}\right)$. The transpose of the matrix is an operation of making interchange of elements by the rule on positioned element...

### Find the value of and for which

Solution: We have $\left(\begin{array}{cc}x & y \\ 3 y & x\end{array}\right)\left(\begin{array}{l}1 \\ 2\end{array}\right)=\left(\begin{array}{l}3 \\ 5\end{array}\right)$. Use the...

### Find the value of and for which

Solution: We have $\left(\begin{array}{cc}2 & -3 \\ 1 & 1\end{array}\right)\left(\begin{array}{l}x \\ y\end{array}\right)=\left(\begin{array}{l}1 \\ 3\end{array}\right)$. Use the...

### If then find the least value of for which .

Solution: We have $A=\left(\begin{array}{cc}\cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha\end{array}\right)$ Use the addition rule and get $A+A^{T}=I_{2}$ as follow:...

### If and , find the matrix such that is a zero matrix

Solution: We have $A=\left(\begin{array}{cc}1 & -5 \\ -3 & 2 \\ 4 & -2\end{array}\right) ; B=\left(\begin{array}{cc}3 & 1 \\ 2 & -1 \\ -2 & 3\end{array}\right) .$ and...

### Show that

Solution: We have $\cos \theta\left(\begin{array}{cc}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta\end{array}\right)+\sin \theta\left(\begin{array}{cc}\sin \theta & -\cos...

### If A = diag (3 -2, 5) and B = diag (1 3 -4), find (A + B).

Solution: We have $A=\operatorname{diag}(3-25)=\left(\begin{array}{ccc}3 & 0 & 0 \\ 0 & -2 & 0 \\ 0 & 0 & 5\end{array}\right) ; B=$...

### If , find the values of

Solution: We have $\left(\begin{array}{cc}x & 6 \\ -1 & 2 w\end{array}\right)+\left(\begin{array}{cc}4 & x+y \\ z+w & 3\end{array}\right)=3\left(\begin{array}{cc}x & y \\ z &...

### If , find the values of .

Solution: We have $\left(\begin{array}{cc}x & 3 x-y \\ 2 x+z & 3 y-w\end{array}\right)=\left(\begin{array}{ll}3 & 2 \\ 4 & 7\end{array}\right)$. Now from the equality of matrices we...

### If , find the values of and .

Solution: We have $x\left(\begin{array}{l}2 \\ 3\end{array}\right)+y\left(\begin{array}{c}-1 \\ 1\end{array}\right)=\left(\begin{array}{c}10 \\ 5\end{array}\right)$. Use the addition rule and get...

### Find the values of and , if

Solution: We have $2\left(\begin{array}{ll}1 & 3 \\ 0 & x\end{array}\right)+\left(\begin{array}{ll}y & 0 \\ 1 & 2\end{array}\right)=\left(\begin{array}{ll}5 & 6 \\ 1 &...

### If , find the values of and

Solution: We have $\left(\begin{array}{cc}x+2 y & -y \\ 3 x & 4\end{array}\right)=\left(\begin{array}{cc}-4 & 3 \\ 6 & 4\end{array}\right)$ Now from the equality of matrices we can...

### Construct a matrix whose elements are given by

Solution: We have $a_{i j}=\frac{1}{2}|-3 i+j|^{2}$ Now $\begin{array}{l} a_{11}=\frac{|-3(1)+1|}{2}=1, a_{12}=\frac{|-3(1)+2|}{2}=\frac{9}{2}, a_{13}=\frac{|-3(1)+3|}{2}=\frac{9}{2} \\...

### Construct a matrix whose elements are given by

Solution: We have $a_{i j}=\frac{1}{2}(i-2 j)^{2}$ Now $\begin{array}{l} a_{11}=\frac{(1-2(1))^{2}}{2}=\frac{1}{2}, a_{12}=\frac{(1-2(2))^{2}}{2}=\frac{9}{2} \\ a_{21}=\frac{(2-2(1))^{2}}{2}=0...

### Using elementary row transformations, find the inverse of each of the following matrices:

Solution: We have $A=\left(\begin{array}{ccc}-1 & 1 & 2 \\ 1 & 2 & 3 \\ 3 & 1 & 1\end{array}\right)$ To get the inverse we will proceed by augmented matrix with elementary...

### Using elementary row transformations, find the inverse of each of the following matrices:

Solution: We have $A=\left(\begin{array}{ccc}1 & 2 & 3 \\ 2 & 5 & 7 \\ -2 & -4 & -5\end{array}\right)$. To get the inverse we will proceed by augmented matrix with elementary...

### Using elementary row transformations, find the inverse of each of the following matrices:

Solution: We have $A=\left(\begin{array}{ccc}3 & -1 & -2 \\ 2 & 0 & -1 \\ 3 & -5 & 0\end{array}\right)$ To get the inverse we will proceed by augmented matrix with elementary...

### Using elementary row transformations, find the inverse of each of the following matrices:

Solution: We have $A=\left(\begin{array}{lll}3 & 0 & 2 \\ 1 & 5 & 9 \\ 6 & 4 & 7\end{array}\right)$ To get the inverse we will proceed by augmented matrix with elementary row...

### Using elementary row transformations, find the inverse of each of the following matrices:

Solution: We have $A=\left(\begin{array}{lll}0 & 1 & 2 \\ 1 & 2 & 3 \\ 3 & 1 & 1\end{array}\right)$ To get the inverse we will proceed by augmented matrix with elementary row...

### Using elementary row transformations, find the inverse of each of the following matrices:

Solution: We have $A=\left(\begin{array}{ll}4 & 0 \\ 2 & 5\end{array}\right)$. To get the inverse we will proceed by augmented matrix with elementary row transformation process is as follow:...

### Using elementary row transformations, find the inverse of each of the following matrices:

Solution: We have $A=\left(\begin{array}{cc}2 & -3 \\ 4 & 5\end{array}\right)$ To get the inverse we will proceed by augmented matrix with elementary row transformation process is as follow:...

### Using elementary row transformations, find the inverse of each of the following matrices:

Solution: We have $A=\left(\begin{array}{cc}2 & 5 \\ -3 & 1\end{array}\right)$ To get the inverse we will proceed by augmented matrix with elementary row transformation process is as follow:...

### Using elementary row transformations, find the inverse of each of the following matrices:

Solution: We have $A=\left(\begin{array}{cc}1 & 2 \\ 2 & -1\end{array}\right)$ To get the inverse we will proceed by augmented matrix with elementary row transformation process is as follow:...

### Using elementary row transformations, find the inverse of each of the following matrices:

Solution: We have $A=\left(\begin{array}{ll}1 & 2 \\ 3 & 7\end{array}\right)$ To get the inverse we will proceed by augmented matrix with elementary row transformation process is as follow:...

### If , show that .

Solution: Given that $A=\left[\begin{array}{cc}\cos \alpha & \sin \alpha \\ -\sin \theta & \cos \alpha\end{array}\right]$. We wil find $A$ $A^{\prime}=\left[\begin{array}{cc}\cos \alpha...

### For each of the following pairs of matrices and , verify that : and

Solution: Take $\mathrm{C}=\mathrm{AB}$ $\begin{array}{l} C=\left[\begin{array}{ccc} -1 & 2 & -3 \\ 4 & -5 & 6 \end{array}\right]\left[\begin{array}{cc} 3 & -4 \\ 2 & 1 \\ -1...

### For each of the following pairs of matrices and , verify that :

Solution: Take $C=A B$ $\begin{array}{l} C=\left[\begin{array}{c} -1 \\ 2 \\ 3 \end{array}\right]\left[\begin{array}{lll} -2 & -1 & -4 \end{array}\right] \\...

### For each of the following pairs of matrices and , verify that :

Solution: Take $C=A B$ $\begin{array}{l} C=\left[\begin{array}{rr} 3 & -1 \\ 2 & -2 \end{array}\right]\left[\begin{array}{ll} 1 & -3 \\ 2 & -1 \end{array}\right] \\...

### For each of the following pairs of matrices and , verify that :

Solution: Take $\mathrm{C}=\mathrm{A} 8$ $\begin{array}{l} C=\left[\begin{array}{ll} 1 & 3 \\ 2 & 4 \end{array}\right]\left[\begin{array}{ll} 1 & 4 \\ 2 & 5 \end{array}\right] \\...

### Express the matrix as sum af two matrices such that and is symmetric and the other is skew-symmetric.

Solution: Given that $A=\left[\begin{array}{lll}3 & 2 & 5 \\ 4 & 1 & 3 \\ 0 & 6 & 7\end{array}\right]$, to express as sum of symmetric matrix $P$ and skew symmetric matrix...

### Express the matrix as the sum of a symmetric matrix and a skew-symmetric matrix.

Solution: Given that $\mathrm{A}=\left[\begin{array}{rr}3 & -4 \\ 1 & -1\end{array}\right]$,to express as the sum of symmetric matrix $\mathrm{P}$ and skew symmetric matrix Q. $A=P+Q$ Where...

### Express the matrix as the sum of a symmetric matrix and a skew-symmetric matrix.

Solution: Given that $A=\left[\begin{array}{cc}2 & 3 \\ -1 & 4\end{array}\right]$, As for a symmetric matrix $A^{\prime}=A$ hence $\begin{array}{l} A+A^{\prime}=2 A \\...

### Show that the matrix is skew-symmetric.

Solution: We have $A=\left(\begin{array}{ccc}0 & a & b \\ -a & 0 & c \\ -b & -c & 0\end{array}\right)$. The transpose of the matrix is an operation of making interchange of...

### If , show that is skew-symmetric.

Solution: We have $A=\left(\begin{array}{ll}3 & -4 \\ 1 & -1\end{array}\right)$ The transpose of the matrix is an operation of making interchange of elements by the rule on positioned...

### If , show that is symmetric.

Solution: We have $A=\left(\begin{array}{ll}4 & 1 \\ 5 & 8\end{array}\right)$. The transpose of the matrix is an operation of making interchange of elements by the rule on positioned element...

### If and , verify that

Solution: We have $P=\left(\begin{array}{cc}3 & 4 \\ 2 & -1 \\ 0 & 5\end{array}\right)$ and $Q=\left(\begin{array}{cc}7 & -5 \\ -4 & 0 \\ 2 & 6\end{array}\right)$. The...

### If and , verify that .

Solution: We have $A=\left(\begin{array}{ccc}3 & 2 & -1 \\ -5 & 0 & -6\end{array}\right)$ and $B=\left(\begin{array}{ccc}-4 & -5 & -2 \\ 3 & 1 & 8\end{array}\right)$....

### If , verify that .

Solution: We have $A=\left(\begin{array}{ccc}2 & -3 & 5 \\ 0 & 7 & -4\end{array}\right)$. The transpose of the matrix is an operation of making interchange of elements by the rule on...

### If , find the value of

Solution: We have $A=\left(\begin{array}{ll}2 & 3 \\ 5 & 7\end{array}\right), B=\left(\begin{array}{cc}1 & -3 \\ -2 & 4\end{array}\right)$ and $C=\left(\begin{array}{cc}-4 & 6 \\...

### If and , find .

Solution: We have $A=\left(\begin{array}{cc}1 & 0 \\ -1 & 7\end{array}\right), B=\left(\begin{array}{cc}0 & 4 \\ -1 & 7\end{array}\right)$. (i) Let's compute first $A^{2}$ $A^{2}=A...

### Give an example of three matrices A, B, C such that AB = AC but B ≠ C.

Solution: We have $\boldsymbol{A B}=\boldsymbol{A} \boldsymbol{C}$ but $\boldsymbol{B} \neq \boldsymbol{C}$. WE need to find: $\boldsymbol{A}, \boldsymbol{B}$. Let's take...

### Given an example of two matrices A and B such that A ≠ O, B ≠ O, AB = O and BA ≠ O.

Solution: We have $\boldsymbol{A} \neq \boldsymbol{O}, \boldsymbol{B} \neq \boldsymbol{O}, \boldsymbol{A B}=\boldsymbol{O}$ and $\boldsymbol{B A} \neq \boldsymbol{O}$. We need to find:...

### If , find , where .

Solution: We have $A=\left(\begin{array}{cc}3 & 4 \\ -4 & -3\end{array}\right)$ and equation $f(x)=x^{2}-5 x+7$. (i) Let us compute first $A^{2}$ $A^{2}=A A=\left(\begin{array}{cc} 3 & 4...

### Find the values of and for which

Solution: We have $\boldsymbol{A}=\left(\begin{array}{cc}\boldsymbol{a} & \boldsymbol{b} \\ -\boldsymbol{a} & \boldsymbol{2} \boldsymbol{b}\end{array}\right),...

### If , find

Solution: We have $A=\left(\begin{array}{lll}1 & 2 & 3 \\ 4 & 5 & 6 \\ 3 & 2 & 5\end{array}\right), B=\left(\begin{array}{lll}1 & x & 1\end{array}\right)$ and...

### If , show that

Solution: We have $A=\left(\begin{array}{cc}\cos \alpha & \sin \alpha \\ -\sin \alpha & \cos \alpha\end{array}\right)$ and to show $A^{2}=$ $\left(\begin{array}{cc}\cos ^2 \alpha & \sin...

### If , show that

Solution: We have $\boldsymbol{F}(\boldsymbol{X})=\left(\begin{array}{ccc}\cos x & -\sin x & 0 \\ \sin x & \cos x & 0 \\ 0 & 0 & 1\end{array}\right)$ and to show...

### Find the matrix A such that A. .

Solution: We have $\boldsymbol{B}=\left(\begin{array}{ll}\mathbf{2} & \mathbf{3} \\ \mathbf{4} & \mathbf{5}\end{array}\right)$ and $\boldsymbol{C}=\left(\begin{array}{cc}\mathbf{0} &...

### If , find the value of a and such that

Solution: We have $\boldsymbol{A}=\left(\begin{array}{ll}\mathbf{3} & \mathbf{2} \\ \mathbf{1} & \mathbf{1}\end{array}\right)$. To find $\boldsymbol{a}, \boldsymbol{b}$ such that...

### If , find and such that .

Solution: We have $\boldsymbol{A}=\left(\begin{array}{ll}\mathbf{3} & \mathbf{1} \\ \mathbf{7} & \mathbf{5}\end{array}\right)$. To find $\boldsymbol{x}, \boldsymbol{y}$ such that...

### Find the values of and , when

Solution: We have $A=\left(\begin{array}{cc}2 & -3 \\ 1 & 1\end{array}\right), B=\left(\begin{array}{l}1 \\ 3\end{array}\right)$ and $X=\left(\begin{array}{l}x \\ y\end{array}\right)$. To...

### If , find , where

Solution: We have $\boldsymbol{A}=\left(\begin{array}{cc}-1 & 2 \\ 3 & 1\end{array}\right)$. Now addition/subtraction of two matrices is possible if order of both the matrices are same and...

### If , find so that .

Solution: We have $\boldsymbol{A}=\left(\begin{array}{ll}\mathbf{3} & -\mathbf{2} \\ \mathbf{4} & -\mathbf{2}\end{array}\right)$. Now addition/subtraction of two matrices is possible if...

### Show that the matrix satisfies the equation .

Solution: We have $\boldsymbol{A}=\left(\begin{array}{ll}\mathbf{2} & \mathbf{3} \\ \mathbf{1} & \mathbf{2}\end{array}\right)$. Now addition of two matrices is possible if order of both the...

### If , show that .

Solution: We have $\boldsymbol{A}=\left(\begin{array}{cc}\mathbf{3} & \mathbf{1} \\ -\mathbf{1} & \mathbf{2}\end{array}\right)$. Now addition of two matrices is possible if order of both the...

### If then find

Solution: We have $\boldsymbol{A}=\left(\begin{array}{cc}\mathbf{2} & \mathbf{- 2} \\ -\mathbf{3} & \mathbf{4}\end{array}\right)$. Now addition of two matrices is possible if order of both...

### If and , find .

Solution: We have $\boldsymbol{A}=\left(\begin{array}{cc}\mathbf{2} & \mathbf{- 1} \\ \mathbf{3} & \mathbf{2}\end{array}\right)$ and $\boldsymbol{B}=\left(\begin{array}{cc}\mathbf{0} &...

### If , show that

Solution: We have $\boldsymbol{A}=\left(\begin{array}{ccc}\mathbf{4} & -\mathbf{1} & -\mathbf{4} \\ \mathbf{3} & \mathbf{0} & -\mathbf{4} \\ \mathbf{3} & -\mathbf{1} &...

### If , show that

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

### Verify that , when

Solution: We have $\boldsymbol{A}=\left(\begin{array}{ll}\mathbf{1} & \mathbf{2} \\ \mathbf{3} & \mathbf{4}\end{array}\right), \boldsymbol{B}=\left(\begin{array}{cc}\mathbf{2} &...

### For the following matrices, verify that :

Solution: We have $\boldsymbol{A}=\left(\begin{array}{ccc}\mathbf{2} & \mathbf{3} & \mathbf{- 1} \\ \mathbf{3} & \mathbf{0} & \mathbf{2}\end{array}\right),...

### For the following matrices, verify that :

Solution: We have $A=\left(\begin{array}{lll}1 & 2 & 5 \\ 0 & 1 & 3\end{array}\right), B=\left(\begin{array}{ccc}2 & 3 & 0 \\ 1 & 0 & 4 \\ 1 & -1 &...

### If and . show that is a zero matrix.

Solution: We have $\boldsymbol{A}=\left(\begin{array}{ccc}\mathbf{0} & \boldsymbol{c} & -\boldsymbol{b} \\ -\boldsymbol{c} & \mathbf{0} & \boldsymbol{a} \\ \boldsymbol{b} &...

### Show that in each of the following cases:

Solution: We have $\boldsymbol{A}=\left(\begin{array}{ccc}\mathbf{1} & \mathbf{3} & -\mathbf{1} \\ \mathbf{2} & \mathbf{2} & -\mathbf{1} \\ \mathbf{3} & \mathbf{0} &...

### Show that in each of the following cases:

Solution: We have $A=\left(\begin{array}{lll}1 & 2 & 1 \\ 3 & 4 & 2 \\ 1 & 3 & 2\end{array}\right)$ and $B=\left(\begin{array}{ccc}10 & -4 & -1 \\ -11 & 5 & 0...

### Show that in each of the following cases:

Solution: We have $A=\left(\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta\end{array}\right)$ and $B=\left(\begin{array}{cc}\cos \phi & -\sin \phi \\ \sin \phi...

### Show that in each of the following cases:

Solution: We have $\boldsymbol{A}=\left(\begin{array}{lll}\mathbf{1} & \mathbf{2} & \mathbf{3} \\ \mathbf{0} & \mathbf{1} & \mathbf{0} \\ \mathbf{1} & \mathbf{1} &...

### Show that in each of the following cases :

Solution: We have $\boldsymbol{A}=\left(\begin{array}{cc}\mathbf{5} & -\mathbf{1} \\ \mathbf{6} & \mathbf{7}\end{array}\right)$ and $\boldsymbol{B}=\left(\begin{array}{ll}\mathbf{2} &...

### Compute and , which ever exists when

Solution: We have $\boldsymbol{B}=\left(\begin{array}{ccc}\mathbf{1} & \mathbf{0} & \mathbf{1} \\ -\mathbf{1} & \mathbf{2} & \mathbf{1}\end{array}\right)$ and...

### Compute and BA, which ever exists when

Solution: We have $\boldsymbol{A}=\left(\begin{array}{llll}\mathbf{1} & 2 & 3 & 4\end{array}\right)$ and $\boldsymbol{B}=\left(\begin{array}{l}\mathbf{1} \\ \mathbf{2} \\ \mathbf{3} \\...

### Compute AB and BA, which ever exists when

Solution: We have $\boldsymbol{A}=\left(\begin{array}{ccc}\mathbf{0} & \mathbf{1} & -\mathbf{5} \\ \mathbf{2} & \mathbf{4} & \mathbf{0}\end{array}\right)$ and...

### Compute AB and BA, which ever exists when

Solution: We have $\boldsymbol{A}=\left(\begin{array}{cc}-\mathbf{1} & \mathbf{1} \\ -\mathbf{2} & \mathbf{2} \\ -\mathbf{3} & \mathbf{3}\end{array}\right)$ and...

### Compute AB and BA, which ever exists when

Solution: We have $\boldsymbol{A}=\left(\begin{array}{cc}\mathbf{2} & \mathbf{- 1} \\ \mathbf{3} & \mathbf{0} \\ \mathbf{- 1} & \mathbf{4}\end{array}\right)$ and...

### If then write the value of

Solution: $\text { If }\left[\begin{array}{ll} a & b \\ c & d \end{array}\right]=\left[\begin{array}{ll} e & f \\ g & h \end{array}\right]$ Therefore $a=e, b=f, c=g, d=h$ It is given...

### Find the value of from the following equation :

Solution: It is given that $\begin{array}{l} 2\left[\begin{array}{ll} 1 & 3 \\ 0 & x \end{array}\right]+\left[\begin{array}{ll} y & 0 \\ 1 & 2...

### Find the value of and , when

i.

Solution: (i) Given $2\left(\begin{array}{lc} x & 5 \\ 7 y-3 \end{array}\right)+\left(\begin{array}{c} 3-4 \\ 12 \end{array}\right)=\left(\begin{array}{cc} 7 & 6 \\ 1514 \end{array}\right)$...

### Find the value of and , when

i.

ii.

Solution: (i) Given $\left(\begin{array}{l}\boldsymbol{x}+\boldsymbol{y} \\ \boldsymbol{x}-\boldsymbol{y}\end{array}\right)=\left(\begin{array}{l}8 \\ \mathbf{4}\end{array}\right)$. By equality of...

### If and , find:

(i)

(ii)

Solution: If $Z=\operatorname{diag}[a, b, c]$, then it can be written as $Z=\left[\begin{array}{lll} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c \end{array}\right]$ Therefore, $A+2...

### Find the matrix such that where and

Solution: It is given that $2 A-B+X=0$ $\begin{array}{l} 2\left(\left[\begin{array}{ll} 3 & 1 \\ 0 & 2 \end{array}\right]\right)-\left[\begin{array}{cc} -2 & 1 \\ 0 & 3...

### If and , find a matrix such that

Solution: It is given that $A+B-C=0$ $\begin{array}{c} {\left[\begin{array}{cc} -2 & 3 \\ 4 & 5 \\ 1 & -6 \end{array}\right]+\left[\begin{array}{cc} 5 & 2 \\ -7 & 3 \\ 6 & 4...

### Find matrix , if

Solution: It is given that $\left[\begin{array}{ccc}3 & 5 & -9 \\ -1 & 4 & -7\end{array}\right]+x=\left[\begin{array}{lll}6 & 2 & 3 \\ 4 & 8 & 6\end{array}\right]$...

### Find matrices and , if and

Solution: $\operatorname{Add} 2(2 A-B)$ and $(2 B+A)$ $\begin{array}{l} 2(2 A-B)+(2 B+A)=2\left(\left[\begin{array}{ccc} 6 & -6 & 0 \\ -4 & 2 & 1...

### Find matrices and , if and

Solution: Add $(A+B)$ and $(A-B)$ We obtain $(A+B)+(A-B)=\left[\begin{array}{ccc}1 & 0 & 2 \\ 5 & 4 & -6 \\ 7 & 3 & 8\end{array}\right]+\left[\begin{array}{ccc}-5 & -4...

### If , find

Solution: $5 A=\left[\begin{array}{ccc}5 & 10 & -15 \\ 2 & 3 & 4 \\ 1 & 0 & -5\end{array}\right]$ $A=\left[\begin{array}{ccc}\frac{5}{5} & \frac{10}{5} &...

### Let and Compute

Solution: $\begin{array}{l} \left.5 A-3 B+4 C=5\left(\left[\begin{array}{ccc} 0 & 1 & -2 \\ 5 & -1 & -4 \end{array}\right]\right)-3\left(\begin{array}{ccc} 1 & -3 & -1 \\ 0...

### Let and Find:

i.

Solution: $\begin{array}{l} \text { i. } A-2 B+3 C \\ \text { A- } 2 B+3 C=\left[\begin{array}{ll} 2 & 4 \\ 3 & 2 \end{array}\right]-2\left(\left[\begin{array}{cc} 1 & 3 \\ -2 & 5...

### Let and Find:

i.

ii. B –

Solution: i. $\begin{array}{l} A+2 B=\left[\begin{array}{ll} 2 & 4 \\ 3 & 2 \end{array}\right]+2\left(\left[\begin{array}{cc} 1 & 3 \\ -2 & 5 \end{array}\right]\right) \\...

### If and , find

Solution: $\begin{array}{l} 2 A=2\left(\left[\begin{array}{ccc} 3 & 1 & 2 \\ 1 & 2 & -3 \end{array}\right]\right) \\ =\left[\begin{array}{ccc} 6 & 2 & 4 \\ 2 & 4 & -6...

### If and , verify that

Solution: $\begin{array}{l} (A+B)+C=\left(\left[\begin{array}{cc} 3 & 5 \\ -2 & 0 \\ 6 & -1 \end{array}\right]+\left[\begin{array}{cc} -1 & -3 \\ 4 & 2 \\ -2 & 3...

### If and , verify that

Solution: $\begin{array}{l} A+B=\left[\begin{array}{ccc} 2 & -3 & 5 \\ -1 & 0 & 3 \end{array}\right]+\left[\begin{array}{ccc} 3 & 2 & -2 \\ 4 & -3 & 1...

### Mark (√) against the correct answer in the following: The range of is

A.

B.

C.

D. none of these

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 . Then, range

A.

B.

C.

D. none of these

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 . Then, dom (f) and range (f) are respectively

A. and

B. and

C. R and R +

D. and

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 . Then,

A.

B.

C.

D. None of these

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 . Then, ?

A.

B.

C.

D. None of these

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...