Maximise

Question: Maximise Z = 3x+ 4y Subject to the constraints: $x+y \leq 4, x \geq 0, y \geq 0$. Solution: The feasible region determined by the constraints, $x+y \leq 4, x \geq 0, y \geq 0$, is as follows. The corner points of the feasible region are O (0, 0), A (4, 0), and B (0, 4). The values of Z at these points are as follows. Therefore, the maximum value of Z is 16 at the point B (0, 4)....

In a rectangle ABCD, AB = 20 cm,

Question: In a rectangle $A B C D, A B=20 \mathrm{~cm}, \angle B A C=60^{\circ}$, calculate side $B C$ and diagonals $A C$ and $B D$. Solution: We have drawn the following figure SinceABCDis a rectangle Therefore, $\angle A B C=\angle B C D=90^{\circ}$ Now, consider $\triangle A B C$ We know that sum of all the angles of any triangle is Therefore, $\angle B A C+\angle A B C+\angle A C B=180^{\circ}$.....(1) Now by substituting the values of known angles $\angle B A C$ and $\angle A B C$ in equat...

The value of sin

Question: The value of $\sin \frac{\pi}{10} \sin \frac{13 \pi}{10}$ is (a) $\frac{1}{2}$ (b) $-\frac{1}{2}$ (c) $-\frac{1}{4}$ (d) 1 Solution: $\sin \frac{\pi}{10} \sin \frac{13 \pi}{10}$ $=\sin \frac{\pi}{10} \sin \left(\pi+\frac{3 \pi}{10}\right)$ $=\sin \frac{\pi}{10}\left(-\sin \frac{3 \pi}{10}\right) \quad(\because \sin (\pi+\theta)=-\sin \theta)$ $\therefore \sin \frac{\pi}{10} \sin \frac{13 \pi}{10}=-\sin \frac{\pi}{10} \sin \frac{3 \pi}{10} \quad \ldots$ (1) Let $\theta=\frac{\pi}{10} \q...

If sin θ + cos θ = 1,

Question: If sin+ cos= 1, then the value of sin 2is equal to (a) 1 (b) $\frac{1}{2}$ (c) 0 (d) 1 Solution: Given, $\sin \theta+\cos \theta=1$ By squaring both sides, we get, $(\sin \theta+\cos \theta)^{2}=(1)^{2}=1$ i.e. $\sin ^{2} \theta+\cos ^{2} \theta+2 \sin \theta \cos \theta=1$ i. e. $1+2 \sin \theta \cos \theta=1$ Since $\sin ^{2} \theta+\cos ^{2} \theta=1$ (using identity) 1.e. $2 \sin \theta \cos \theta=0$ $\Rightarrow \sin 2 \theta=0$ Hence, the correct answer is option C....

If tan A

Question: If $\tan A=\frac{1}{2}, \tan B=\frac{1}{3}$, then $\tan (2 A+B)$ is equal (a) 1 (b) 2 (c) 3 (d) 4 Solution: $\tan A=\frac{1}{2}, \quad \tan B=\frac{1}{3}$ then $\tan (2 A+B)=\frac{\tan 2 A+\tan B}{1-\tan 2 A \tan B}$ Using Identities : $\left(\tan (a+b)=\frac{\tan a+\tan b}{1-\tan a \tan b}\right)$ $=\frac{\left(\frac{2 \tan A}{1-\tan ^{2} A}\right)+\tan B}{1-\left(\frac{2 \tan A}{1-\tan ^{2} A}\right) \tan B} \quad$ (using identity $\left.\tan 2 a=\frac{2 \tan a}{1-\tan ^{2} a}\right)...

If ∆ABC is a right triangle such that ∠C = 90°,

Question: If $\triangle A B C$ is a right triangle such that $\angle C=90^{\circ}, \angle A=45^{\circ}$ and $B C=7$ units. Find $\angle B, A B$ and $A C$. Solution: We are given the following information in the form of the triangle It is required to find $\angle B$ and length of sides $A B$ and $A C$ In $\triangle A B C \angle C=90^{\circ}$ Now we know that sum of all the angles of any triangle is $180^{\circ}$ Therefore, $\angle A+\angle B+\angle C=180^{\circ}$.....(1) Now by substituting the v...

cos 2θ cos 2ϕ + sin

Question: cos 2 cos 2ϕ + sin2( ϕ) sin2(+ ϕ) is equal to (a) sin 2 (+ ϕ) (b) cos 2 (+ ϕ) (c) sin 2 ( ϕ) (d) cos 2 ( ϕ) Solution: $\cos 2 \theta \cos 2 \phi+\sin ^{2}(\theta-\phi)-\sin ^{2}(\theta+\phi)$ $=\cos 2 \theta \cos 2 \phi+\sin (\theta-\phi+\theta+\phi) \sin (\theta-\phi-\theta-\phi)$ (using identity $\left.\sin ^{2} A-\sin ^{2} B=\sin (A+B) \sin (A-B)\right)$ $=\cos 2 \theta \cos 2 \phi+\sin (2 \theta) \sin (-2 \phi)$ $=\cos 2 \theta \cos 2 \phi-\sin 2 \theta \sin 2 \phi \quad(\because \...

The value of tan 75° – cot75° is

Question: The value of tan 75 cot75 is (a) $2 \sqrt{3}$ (b) $2+\sqrt{3}$ (c) $2-\sqrt{3}$ (d) 1 Solution: $\tan 75^{\circ}-\cot 75^{\circ}$ $=\tan 75^{\circ}-\frac{1}{\tan 75^{\circ}}$ $=\frac{\tan ^{2} 75^{\circ}-1}{\tan 75^{\circ}}$ $=-\frac{\left(1-\tan ^{2} 75^{\circ}\right)}{\tan 75^{\circ}} \times \frac{2}{2} \quad$ (multiply and divide by 2 ) $=-2\left[\frac{1-\tan ^{2} 75^{\circ}}{2 \tan 75^{\circ}}\right]$ $=-2\left[\frac{1}{\tan 150^{\circ}}\right] \quad$ (Using identity $\left.\tan 2 ...

The value of tan 75° – cot75° is

Question: The value of tan 75 cot75 is (a) $2 \sqrt{3}$ (b) $2+\sqrt{3}$ (c) $2-\sqrt{3}$ (d) 1 Solution: $\tan 75^{\circ}-\cot 75^{\circ}$ $=\tan 75^{\circ}-\frac{1}{\tan 75^{\circ}}$ $=\frac{\tan ^{2} 75^{\circ}-1}{\tan 75^{\circ}}$ $=-\frac{\left(1-\tan ^{2} 75^{\circ}\right)}{\tan 75^{\circ}} \times \frac{2}{2} \quad$ (multiply and divide by 2 ) $=-2\left[\frac{1-\tan ^{2} 75^{\circ}}{2 \tan 75^{\circ}}\right]$ $=-2\left[\frac{1}{\tan 150^{\circ}}\right] \quad$ (Using identity $\left.\tan 2 ...

The value of

Question: The value of $\frac{1-\tan ^{2} 15^{\circ}}{2}$ is (a) 1 (b) $\sqrt{3}$ (c) $\frac{\sqrt{3}}{2}$ (d) 2 Solution: $\frac{1-\tan ^{2} 15^{\circ}}{1+\tan ^{2} 15^{\circ}}$ $\left[\right.$ Using identity : $\left.\frac{1-\tan ^{2} A}{1+\tan ^{2} A}=\cos 2 A\right]$ We have, $\frac{1-\tan ^{2} 15^{\circ}}{1+\tan ^{2} 15^{\circ}}=\cos 2\left(15^{\circ}\right)$ $=\cos 30^{\circ}$ $=\frac{\sqrt{3}}{2}$ Hence, the correct answer is option C....

In a right triangle ABC, right angled at C,

Question: In a right triangle $A B C$, right angled at $C$, if $\angle B=60^{\circ}$ and $A B=15$ units. Find the remaining angles and sides. Solution: We are given the following triangle with related information It is required to find $\angle A, \angle C$ and length of sides $A C$ and $B C$ $\triangle A B C$ is right angled at $C$ Therefore, $\angle C=90^{\circ}$ Now we know that sum of all the angles of any triangle is $180^{\circ}$ Therefore, $\angle A+\angle B+\angle C=180^{\circ}$.....(1) N...

The value of cos

Question: The value of $\cos ^{2} 48^{\circ}-\sin ^{2} 12^{\circ}$ is (a) $\frac{\sqrt{5}+1}{8}$ (b) $\frac{\sqrt{5}-1}{8}$ (c) $\frac{\sqrt{5}+1}{5}$ (d) $\frac{\sqrt{5}+1}{2 \sqrt{2}}$ Solution: $\cos ^{2} 48^{\circ}-\sin ^{2} 12^{\circ}$ $=\cos \left(48^{\circ}+12^{\circ}\right) \cos \left(48^{\circ}-12^{\circ}\right) \quad\left[\cos (A+B) \cos (A-B)=\cos ^{2} A-\sin ^{2} B\right]$ $=\cos 60^{\circ} \cos 36^{\circ}$ $=\frac{1}{2} \times\left(\frac{\sqrt{5}+1}{4}\right)$ $=\frac{\sqrt{5}+1}{8}...

If tanα=

Question: If $\tan \alpha=\frac{1}{7}, \tan \beta=\frac{1}{3}$, then $\cos 2 \alpha$ is equal to (a) $\sin 2 \beta$ (b) $\sin 4 \beta$ (c) $\sin 3 \beta$ (d) $\cos 2 \beta$ Solution: It is given that $\tan \alpha=\frac{1}{7}$ and $\tan \beta=\frac{1}{3}$. Now, $\tan 2 \beta=\frac{2 \tan \beta}{1-\tan ^{2} \beta}$ $=\frac{2 \times \frac{1}{3}}{1-\frac{1}{9}}$ $=\frac{\frac{2}{3}}{\frac{8}{9}}$ $=\frac{3}{4}$ $\therefore \tan (\alpha+2 \beta)=\frac{\tan \alpha+\tan 2 \beta}{1-\tan \alpha \tan 2 \b...

If tanα=

Question: If $\tan \alpha=\frac{1}{7}, \tan \beta=\frac{1}{3}$, then $\cos 2 \alpha$ is equal to (a) $\sin 2 \beta$ (b) $\sin 4 \beta$ (c) $\sin 3 \beta$ (d) $\cos 2 \beta$ Solution: It is given that $\tan \alpha=\frac{1}{7}$ and $\tan \beta=\frac{1}{3}$. Now, $\tan 2 \beta=\frac{2 \tan \beta}{1-\tan ^{2} \beta}$ $=\frac{2 \times \frac{1}{3}}{1-\frac{1}{9}}$ $=\frac{\frac{2}{3}}{\frac{8}{9}}$ $=\frac{3}{4}$ $\therefore \tan (\alpha+2 \beta)=\frac{\tan \alpha+\tan 2 \beta}{1-\tan \alpha \tan 2 \b...

The planes:

Question: The planes: $2 x-y+4 z=5$ and $5 x-2.5 y+10 z=6$ are (A) Perpendicular (B) Parallel (C) intersect $y$-axis (C) passes through $\left(0,0, \frac{5}{4}\right)$ Solution: The equations of the planes are $2 x-y+4 z=5$ (1) $5 x-2.5 y+10 z=6$ (2) It can be seen that, $\frac{a_{1}}{a_{2}}=\frac{2}{5}$ $\frac{b_{1}}{b_{2}}=\frac{-1}{-2.5}=\frac{2}{5}$ $\frac{c_{1}}{c_{2}}=\frac{4}{10}=\frac{2}{5}$ $\therefore \frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}$ Therefore, the given pla...

If tanx=

Question: If $\tan x=\frac{a}{b}$, then $b \cos 2 x+a \sin 2 x$ is equal to (a) $a$ (b) $b$ (c) $\frac{a}{b}$ (d) $\frac{b}{a}$ Solution: Given: $\tan x=\frac{a}{b}$ Now, $b \cos 2 x+a \sin 2 x$ $=b\left(\frac{1-\tan ^{2} x}{1+\tan ^{2} x}\right)+a\left(\frac{2 \tan x}{1+\tan ^{2} x}\right)$ $=b\left(\frac{1-\frac{a^{2}}{b^{2}}}{1+\frac{a^{2}}{b^{2}}}\right)+a\left(\frac{2 \times \frac{a}{b}}{1+\frac{a^{2}}{b^{2}}}\right)$ $=\frac{b\left(b^{2}-a^{2}\right)}{a^{2}+b^{2}}+\frac{2 a^{2} b}{a^{2}+b^...

If sin (A − B) = sin A cos B − cos A sin B and cos (A − B) = cos A cos B + sin A sin B,

Question: If $\sin (A-B)=\sin A \cos B-\cos A \sin B$ and $\cos (A-B)=\cos A \cos B+\sin A \sin B$, find the values of $\sin 15^{\circ}$ and $\cos 15^{\circ}$. Solution: Given: $\sin (A-B)=\sin A \cos B-\cos A \sin B \ldots \ldots(1)$ $\cos (A-B)=\cos A \cos B+\sin A \sin B$....(2) To find: The values of $\sin 15^{\circ}$ and $\cos 15^{\circ}$ In this problem we need to find $\sin 15^{\circ}$ and $\cos 15^{\circ}$ Hence to get $15^{\circ}$ angle we need to choose the value of $A$ and $B$ such th...

Distance between the two planes:

Question: Distance between the two planes: $2 x+3 y+4 z=4$ and $4 x+6 y+8 z=12$ is (A)2 units (B)4 units (C)8 units (D) $\frac{2}{\sqrt{29}}$ units Solution: The equations of the planes are $2 x+3 y+4 z=4$ ...(1) $4 x+6 y+8 z=12$ $\Rightarrow 2 x+3 y+4 z=6$ ...(2) It can be seen that the given planes are parallel. It is known that the distance between two parallel planes, $a x+b y+c z=d_{1}$ and $a x+b y+c z=d_{2}$, is given by, $D=\left|\frac{d_{2}-d_{1}}{\sqrt{a^{2}+b^{2}+c^{2}}}\right|$ $\Rig...

If n=1, 2, 3

Question: If $n=1,2,3, \ldots$, then $\cos \alpha \cos 2 \alpha \cos 4 \alpha \ldots \cos 2^{n-1} \alpha$ is equal to (a) $\frac{\sin 2 n \alpha}{2 n \sin \alpha}$ (b) $\frac{\sin 2^{n} \alpha}{2^{n} \sin 2^{n-1} \alpha}$ (c) $\frac{\sin 4^{n-1} \alpha}{4^{n-1} \sin \alpha}$ (d) $\frac{\sin 2^{n} \alpha}{2^{n} \sin \alpha}$ (e) None of these Solution: (d) $\frac{\sin 2^{n} \alpha}{2^{n} \sin \alpha}$ $\because \cos \alpha \cos 2 \alpha \cos 4 \alpha \ldots \cos 2^{n-1} \alpha=\frac{\sin 2^{n} \a...

Question: $\frac{\sin 5 x}{\sin x}$ is equal to (a) $16 \cos ^{4} x-12 \cos ^{2} x+1$ (b) $16 \cos ^{4} x+12 \cos ^{2} x+1$ (c) $16 \cos ^{4} x-12 \cos ^{2} x-1$ (d) $16 \cos ^{4} x+12 \cos ^{2} x-1$ Solution: (a) $16 \cos ^{4} x-12 \cos ^{2} x+1$ To find : $\frac{\sin 5 x}{\sin x}$ Now, $\sin 5 x=\sin (3 x+2 x)$ $=\sin 3 x \cos 2 x+\cos 3 x \sin 2 x$ $=\left(3 \sin x-4 \sin ^{3} x\right)\left(1-2 \sin ^{2} x\right)+\left(4 \cos ^{3} x-3 \cos x\right)(2 \sin x \cos x)$ $=\left(3 \sin x-6 \sin ^{...

Prove that if a plane has the intercepts a, b, c and is at a distance of P units from the origin, then

Question: Prove that if a plane has the intercepts $a, b, c$ and is at a distance of $P$ units from the origin, then $\frac{1}{a^{2}}+\frac{1}{b^{2}}+\frac{1}{c^{2}}=\frac{1}{p^{2}}$ Solution: The equation of a plane having interceptsa,b,cwithx,y, andzaxes respectively is given by, $\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1$ $\ldots(1)$ The distance (p) of the plane from the origin is given by, $p=\left|\frac{\frac{0}{a}+\frac{0}{b}+\frac{0}{c}-1}{\sqrt{\left(\frac{1}{a}\right)^{2}+\left(\frac{1}{b}...

If A = 30° and B = 60°, verify that

Question: If $A=30^{\circ}$ and $B=60^{\circ}$, verify that (i) $\sin (A+B)=\sin A \cos B+\cos A \sin B$ (ii) $\cos (A+B)=\cos A \cos B-\sin A \sin B$ Solution: (i) Given $A=30^{\circ}$ and $B=60^{\circ} \ldots \ldots$ (1) To verify: $\sin (A+B)=\sin A \cos B+\cos A \sin B$.....(2) Now consider LHS of the expression to be verified in equation (2) Therefore, $\sin (A+B)=\sin (30+60)$ $=\sin 90$ $=1$ Now consider RHS of the expression to be verified in equation (2) Therefore; $\sin A \cos B+\cos A...

The value of

Question: The value of $\frac{\sin 5 \alpha-\sin 3 \alpha}{\cos 5 \alpha+2 \cos 4 \alpha+\cos 3 \alpha}=$ (a) $\cot \alpha / 2$ (b) $\cot \alpha$ (c) $\tan \alpha / 2$ (d) None of these Solution: (c) $\tan \alpha / 2$ $\frac{\sin 5 \alpha-\sin 3 \alpha}{\cos 5 \alpha+2 \cos 4 \alpha+\cos 3 \alpha}=\frac{\sin 5 \alpha-\sin 3 \alpha}{\cos 5 \alpha+\cos 3 \alpha+2 \cos 4 \alpha}$ $=\frac{2 \sin \alpha \cos 4 \alpha}{2 \cos 4 \alpha \cos \alpha+2 \cos 4 \alpha}$ $=\frac{2 \sin \alpha \cos 4 \alpha}{...

Find the vector equation of the line passing through the point

Question: Find the vector equation of the line passing through the point $(1,2,-4)$ and perpendicular to the two lines: $\frac{x-8}{3}=\frac{y+19}{-16}=\frac{z-10}{7}$ and $\frac{x-15}{3}=\frac{y-29}{8}=\frac{z-5}{-5}$ Solution: Let the required line be parallel to the vector $\vec{b}$ given by, $\vec{b}=b_{1} \hat{i}+b_{2} \hat{j}+b_{3} \hat{k}$ The position vector of the point $(1,2,-4)$ is $\vec{a}=\hat{i}+2 \hat{j}-4 \hat{k}$ The equation of the line passing through $(1,2,-4)$ and parallel t...

The value of tan x+tan

Question: The value of $\tan x+\tan \left(\frac{\pi}{3}+x\right)+\tan \left(\frac{2 \pi}{3}+x\right)$ is (a) 3 tan 3x (b) tan 3x (c) 3 cot 3x (d) cot 3x Solution: (a) 3 tan 3x $\frac{\pi}{3}=60^{\circ}, \frac{2 \pi}{3}=120^{\circ}$ $\tan x+\tan \left(60^{\circ}+x\right)+\tan \left(120^{\circ}+x\right)=\tan x+\frac{\tan 60^{\circ}+\tan x}{1-\tan 60^{\circ} \tan x}+\frac{\tan 120^{\circ}+\tan x}{1-\tan 120^{\circ} \tan x}$ $=\tan x+\frac{\sqrt{3}+\tan x}{1-\sqrt{3} \tan x}+\frac{(-\sqrt{3}+\tan x)...