If x cos θ = y cos
Question: If $x \cos \theta=y \cos \left(\theta+\frac{2 \pi}{3}\right)=z \cos \left(\theta+\frac{4 \pi}{3}\right)$, then write the value of $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}$ Solution: Given: $x \cos \theta=y\left(\cos \theta \cos \frac{2 \pi}{3}-\sin \theta \sin \frac{2 \pi}{3}\right)=z\left(\cos \theta \cos \frac{4 \pi}{3}-\sin \theta \sin \frac{4 \pi}{3}\right)$ $\Rightarrow x \cos \theta=y\left(-\frac{1}{2} \cos \theta-\frac{\sqrt{3}}{2} \sin \theta\right)=z\left(-\frac{1}{2} \cos \theta+...
Read More →If α + β − γ = π
Question: If + = and sin2 +sin2 sin2 = sin sin cos , then write the value of . Solution: Given: $\gamma=-[\pi-(\alpha+\beta)]$ Also, $\lambda=\frac{\sin ^{2} \alpha+\sin ^{2} \beta-\sin ^{2}[-(\pi-(\alpha+\beta)]}{\sin \alpha \sin \beta \cos (-(\pi-(\alpha+\beta))}$ $=\frac{\sin ^{2} \alpha+\sin ^{2} \beta-(\sin (\alpha+\beta))^{2}}{-(\sin \alpha \sin \beta \cos (\alpha+\beta))} \quad[\sin (\pi-\theta)=\sin \theta$ and $\cos (\pi-\theta)=-\cos \theta]$ $=\frac{\sin ^{2} \alpha+\sin ^{2} \beta-\s...
Read More →The value of cos
Question: The value of $\cos \frac{\pi}{12}-\sin \frac{\pi}{12}$ is __________________ Solution: $\cos \frac{\pi}{12}-\sin \frac{\pi}{12}$ multiply and divide above equation by $\sqrt{2}$ i. e. $\sqrt{2}\left[\frac{1}{\sqrt{2}} \cos \frac{\pi}{12}-\frac{1}{\sqrt{2}} \sin \frac{\pi}{12}\right]$ $=\sqrt{2}\left[\frac{\cos \pi}{4} \frac{\cos \pi}{12}-\frac{\sin \pi}{4} \frac{\sin \pi}{12}\right]$ $=\sqrt{2}\left[\cos \left(\frac{\pi}{4}+\frac{\pi}{12}\right)\right]$ [using identify $\cos (a+b)=\cos...
Read More →The value of tan 5x tan 3x tan 2x – tan 5x + tan 3x + tan 2x is
Question: The value of tan 5xtan 3xtan 2x tan 5x+ tan 3x+ tan 2xis ____________. Solution: Consider, $\tan 5 x \tan 3 x \tan 2 x-\tan 5 x+\tan 3 x+\tan 2 x$ ..(1) Since $5 x=3 x+2 x$ i. e. $\tan 5 x=\tan (3 x+2 x)$ $\tan 5 x=\frac{\tan 3 x+\tan 2 x}{1-\tan 3 x \tan 2 x}$ i.e. $\tan 5 x(1-\tan 3 x \tan 2 x)=\tan 3 x+\tan 2 x$ i.e. $\tan 5 x-\tan 5 x \tan 3 x \tan 2 x=\tan 3 x+\tan 2 x$ i.e. $\tan 5 x \tan 3 x \tan 2 x-\tan 5 x+\tan 3 x+\tan 2 x=0$ Hence value of $\tan 5 x \tan 3 x \tan 2 x-\tan 5...
Read More →If tan x
Question: If $\tan x=\frac{1}{2}$ and $\tan y=\frac{1}{3}$, then the value of $x+y$ is ________________ Solution: Given: $\tan x=\frac{1}{2}$ $\tan y=\frac{1}{3}$ Since $\tan (x+y)=\frac{\tan x+\tan y}{1-\tan x \tan y}$ $=\frac{\frac{1}{2}+\frac{1}{3}}{1-\frac{1}{2} \times \frac{1}{3}}$ $=\frac{\frac{3+2}{6}}{\frac{6-1}{6}}$ $=\frac{5}{6} \times \frac{6}{5}$ $\tan (x+y)=1$ i. e. $x+y=\tan ^{-1}$ hence, value of $x+y$ is $\frac{\pi}{4}$...
Read More →If cos
Question: If $\cos ^{2}\left(\frac{\pi}{6}+x\right)-\sin ^{2}\left(\frac{\pi}{6}-x\right)=k \cos 2 x$ then $k=$ __________________ Solution: Given: $\cos ^{2}\left(\frac{\pi}{6}+x\right)-\sin ^{2}\left(\frac{\pi}{6}-x\right)=K \cos 2 x$ L.H.S. is $\left(\cos \left(\frac{\pi}{6}+x\right)\right)^{2}-\left(\sin \left(\frac{\pi}{6}-x\right)\right)^{2}$ $=\left(\cos \frac{\pi}{6} \cos x-\sin \frac{\pi}{6} \sin x\right)^{2}-\left(\sin \frac{\pi}{6} \cos x-\cos \frac{\pi}{6} \sin x\right)^{2}$ using id...
Read More →If sin x cos y
Question: If $\sin x \cos y=\frac{1}{4}$ and $3 \tan x=4 \tan y$, then $\sin (x-y)$ is equal to ____________________. Solution: Given $\sin x \cos y=\frac{1}{4}$ and $3 \tan x=4 \tan y$ i. e. $\tan x=\frac{4}{3} \tan y$ i. e. $\frac{\tan x}{\tan y}=\frac{4}{3}$ i. e. $\frac{\sin x}{\cos x} \frac{\cos y}{\sin y}=\frac{4}{3}$ $\Rightarrow \frac{1 / 4}{\cos x \sin y}=\frac{4}{3}$ $\Rightarrow \cos x \sin y=\frac{3}{16}$ $\therefore \sin (x-y)=\sin x \cos y-\cos x \sin y$ $=\frac{1}{4}-\frac{3}{16}$...
Read More →The value of cot
Question: The value of $\cot \left(\frac{\pi}{4}+x\right) \cot \left(\frac{\pi}{4}-x\right)$ is ______________ Solution: $\cot \left(\frac{\pi}{4}+x\right) \cot \left(\frac{\pi}{4}-x\right)$ $\frac{1}{\tan (\pi / 4+x)} \times \frac{1}{\tan (\pi / 4-x)}$ using identities : $\tan (a+b)=\frac{\tan a+\tan b}{1-\tan a \tan b}$ and $\tan (a-b)=\frac{\tan a-\tan b}{1+\tan a \tan b}$ $=\frac{1}{\frac{\tan \pi / 4+\tan x}{1-\tan \pi / 4 \tan x}} \times \frac{1}{\frac{\tan \pi / 4-\tan x}{1+\tan \frac{\pi...
Read More →If
Question: If $\frac{x}{\cos \theta}=\frac{y}{\cos \left(\theta-\frac{2 \pi}{3}\right)}=\frac{z}{\cos \left(\theta+\frac{2 \pi}{3}\right)}$, then $x+y+z=$ ____________ . Solution: $\frac{x}{\cos \theta}=\frac{y}{\cos \left(\theta-\frac{2 \pi}{3}\right)}=\frac{z}{\cos \left(\theta+\frac{2 \pi}{3}\right)}$ say $=K$ i. e. $x=K \cos \theta, y=K \cos \left(\theta-\frac{2 \pi}{3}\right), z=K \cos \left(\theta+\frac{2 \pi}{3}\right)$ $\therefore x+y+z=K\left[\cos \theta+\cos \left(\theta-\frac{2 \pi}{3}...
Read More →If cos (A−B)=
Question: If $\cos (A-B)=\frac{3}{5}$ and $\tan A \tan B=2$, then $\sin A \sin B=$ ______________ . Solution: $\cos (A-B)=\frac{3}{5}$ Given and $\tan A \tan B=2$ Since $\cos (A-B)=3 / 5$ i. e. $\cos A \cos B+\sin A \sin B=3 / 5$ i. e. $\cos A \cos B\left(1+\frac{\sin A \sin B}{\cos A \cos B}\right)=3 / 5$ i. e. $\cos A \cos B(1+\tan A \tan B)=3 / 5$ i.e. $\cos A \cos B(1+2)=3 / 5$ $\cos A \cos B=\frac{3}{5} \times \frac{1}{3}$ $\cos A \cos B=\frac{1}{5}$ $\therefore \sin A \sin B=\frac{3}{5}-\f...
Read More →If cos (A−B)=
Question: If $\cos \|(A-B)=\frac{3}{5}$ and $\tan A \tan B=2$, then $\sin A \sin B=$ ___________ . Solution: $\cos (A-B)=\frac{3}{5}$ Given and $\tan A \tan B=2$ Since $\cos (A-B)=3 / 5$ i. e. $\cos A \cos B+\sin A \sin B=3 / 5$ i. e. $\cos A \cos B\left(1+\frac{\sin A \sin B}{\cos A \cos B}\right)=3 / 5$ i. e. $\cos A \cos B(1+\tan A \tan B)=3 / 5$ i.e. $\cos A \cos B(1+2)=3 / 5$ $\cos A \cos B=\frac{3}{5} \times \frac{1}{3}$ $\cos A \cos B=\frac{1}{5}$ $\therefore \sin A \sin B=\frac{3}{5}-\fr...
Read More →If A+B=
Question: If $A+B=\frac{\pi}{4}$, then $(1+\tan A)(1+\tan B)=$ Solution: Since $A+B=\frac{\pi}{4}$ $(1+\tan A)(1+\tan B)=?$ $\tan (A+B)=\tan \frac{\pi}{4}$ i. e. $\frac{\tan A+\tan B}{1-\tan A \tan B}=1$ i. e. $\tan A+\tan B=1-\tan A \tan B$ $\tan A+\tan B+\tan A \tan B=1$ i. e. $\tan A+1+\tan B(1+\tan A)=1+1$ i. e. $(1+\tan A)(1+\tan B)=2$ Hence, value of $(1+\tan A)(1+\tan B)=2$....
Read More →If A−B=
Question: If $A-B=\frac{\pi}{4}$, then $(1+\tan A)(1-\tan B)=$ Solution: If $A-B=\frac{\pi}{4}$ then $(1+\tan A)(1-\tan B)=?$ Since $A-B=\frac{\pi}{4}$ i. e. $\tan (A-B)=\tan \frac{\pi}{4}$ i. e. $\frac{\tan A-\tan B}{1+\tan A+\tan B}=1$ i. e. $\tan A-\tan B=1+\tan A \tan B$ i. e. $\tan A-\tan B-\tan A \tan B=1$ i.e. $\tan A(1-\tan B)-\tan B+1=1+1 \quad$ (adding 1 both sides) i. e. $(\tan A+1)(1-\tan B)=2$ i. e. value of $(1+\tan A)(1-\tan B)=2$...
Read More →If sinθ + cosθ = 1,
Question: If sin+ cos= 1, then the value of sin 2is ___________. Solution: Given $\sin \theta+\cos \theta=1$ By squaring both sides, we get $(\sin \theta+\cos \theta)^{2}=(1)^{2}$ i. e. $\sin 2 \theta+\cos ^{2} \theta+2 \sin \theta \cos \theta=1$ i. e. $1+2 \sin \theta \cos \theta=1$ i.e. $2 \sin \theta \cos \theta=0$ i.e. $\sin 2 \theta=0$ Hence, value of $\sin 2 \theta$ is 0 ....
Read More →The minimum value of 4 cos x – 3 sin x + 7 is
Question: The minimum value of 4 cosx 3 sinx+ 7 is _________. Solution: $4 \cos x-3 \sin x+7$ Express $4 \cos x-3 \sin x$ as $a \cos (x+A)$ i. e. $4 \cos x-3 \sin x=a[\cos x \cos A-\sin x \sin A]$ on compairing coefficients of $\sin x$ and $\cos x$, we get, $4=a \cos A$ and $-3=-a \sin A$ i.e. $a \cos A=4$ and $a \sin A=3$ i. e. $\frac{a \sin A}{a \cos A}=\frac{3}{4}$ i.e. $\tan A=\frac{3}{4}$ i.e. $A=\tan ^{-1}\left(\frac{3}{4}\right)$ and $(a \sin A)^{2}+(a \cos A)^{2}=9+16$ $a^{2}\left(\sin ^...
Read More →The maximum value of 3 cos x + 4 sin x + 5 is
Question: The maximum value of 3 cosx+ 4 sinx+ 5 is _________. Solution: $3 \cos x+4 \sin x+5$ first express $3 \cos x+4 \sin x$ as $a \cos (x+A)$ i. e. $3 \cos x+4 \sin x=a[\cos x \cos A-\sin x \sin A]$ Now, equate the coefficients of $\sin x$ and $\cos x$ We get, $a \cos A=3$ $-a \sin A=4$ i. e $\frac{a \sin A}{a \cos A}=\frac{-4}{3}$ i. e $\tan A=-\frac{4}{3}$ i. e $A=\tan ^{-1}\left(\frac{-4}{3}\right)$ also $(a \cos A)^{2}+(a \sin A)^{2}=9+16=25$ $a^{2}=25$ $a=5$ $\therefore 3 \cos x+4 \sin...
Read More →The minimum value of 3 cos x + 4 sin x + 8 is
Question: The minimum value of 3 cosx+ 4 sinx+ 8 is (a) 5 (b) 9 (c) 7 (d) 3 Solution: $3 \cos x+4 \sin x+8$ first express $3 \cos x+4 \sin x$ as a $\cos (x+A)$ $3 \cos x+4 \sin x=a[\cos x \cos A-\sin x \sin A]$ i. e $3 \cos x+4 \sin x=a \cos x \cos A-a \sin x \sin A$ Now, equate the coefficients of $\sin x$ and $\cos x$ we get, $a \cos A=3$ and $-a \sin A=4$ i. e $\frac{a \sin A}{a \cos A}=\frac{-4}{3}$ i. e $\tan A=\frac{-4}{3}$ i. e $A=\tan ^{-1}\left(\frac{-4}{3}\right)$ also, $(a \cos A)^{2}...
Read More →The value of sin
Question: The value of $\sin \left(\frac{\pi}{4}+\theta\right)-\cos \left(\frac{\pi}{4}-\theta\right)$ is (a) 2 cos (b) 2 sin (c) 1 (d) 0 Solution: $\sin \left(\frac{\pi}{4}+\theta\right)-\cos \left(\frac{\pi}{4}-\theta\right)$ $=\sin \frac{\pi}{4} \cos \theta+\cos \frac{\pi}{4} \sin \theta-\left[\cos \frac{\pi}{4} \cos \theta+\sin \frac{\pi}{4} \sin \theta\right]$ Using identities : $\sin (a+b)=\sin a \cos b+\cos a \sin b$ $\cos (a-b)=\cos a \cos b+\sin a \sin b$ $=\frac{1}{\sqrt{2}} \cos \thet...
Read More →If
Question: If $\alpha+\beta=\frac{\pi}{4}$, then the value of $(1+\tan \alpha)(1+\tan \beta)$ is (a) 1 (b) 2 (c) 2 (d) not defined Solution: Given $\alpha+\beta=\frac{\pi}{4}$ $(1+\tan \alpha)(1+\tan \beta)$ $1+\tan \alpha+\tan \beta+\tan \alpha \tan \beta$ $=1+\tan (\alpha+\beta)(1-\tan \alpha \tan \beta)+\tan \alpha \tan \beta$ using identity : $\tan (a+b)=\frac{\tan a+\tan b}{1-\tan a \tan b}$ $=1+\tan (\pi / 4)(1-\tan \alpha \tan \beta)+\tan \alpha \tan \beta(\because \alpha+\beta=\pi / 4$ gi...
Read More →Find a particular solution of the differential equation
Question: Find a particular solution of the differential equation $\frac{d y}{d x}+y \cot x=4 x \operatorname{cosec} x(x \neq 0)$, given that $y=0$ when $x=\frac{\pi}{2}$ Solution: The given differential equation is: $\frac{d y}{d x}+y \cot x=4 x \operatorname{cosec} x$ This equation is a linear differential equation of the form $\frac{d y}{d x}+p y=Q$, where $p=\cot x$ and $Q=4 x \operatorname{cosec} x$. Now, I.F $=e^{\int \rho d x}=e^{\int \cot x d x}=e^{\log |\sin x|}=\sin x$ The general solu...
Read More →If tan 69° + tan 66° − tan 69° tan 66° = 2k,
Question: If tan 69 + tan 66 tan 69 tan 66 = 2k, thenk= (a) $-1$ (b) $\frac{1}{2}$ (c) $-\frac{1}{2}$ (d) None of these Solution: (c) $\frac{-1}{2}$ $\tan 135^{\circ}=\tan \left(90^{\circ}+45^{\circ}\right)$ $=-\tan 45^{\circ}$ $=-1$ Or, $\tan \left(69^{\circ}+66^{\circ}\right)=\frac{\tan 69^{\circ}+\tan 66^{\circ}}{1-\tan 69^{\circ} \tan 66^{\circ}}$ $\Rightarrow-1=\frac{\tan 69^{\circ} \tan 66^{\circ}}{1-\tan 69^{\circ} \tan 66^{\circ}}$ $\Rightarrow \tan 69^{\circ}+\tan 66^{\circ}-\tan 69^{\c...
Read More →If cos (A − B) =
Question: If $\cos (A-B)=\frac{3}{5}$ and $\tan A \tan B=2$, then (a) $\cos A \cos B=\frac{1}{5}$ (b) $\cos A \cos B=-\frac{1}{5}$ (c) $\sin A \sin B=-\frac{1}{5}$ (d) $\sin A \sin B=-\frac{1}{5}$ Solution: (a) $\frac{1}{5}$ $\tan A \tan B=\frac{\sin A \sin B}{\cos A \cos B}=2 \quad$ (Given) $\quad \ldots$ (1) Also, $\cos (A-B)=\frac{3}{5}$ $\Rightarrow \cos A \cos B+\sin A \sin B=\frac{3}{5}$ $\therefore \sin A \sin B=\frac{3}{5}-\cos A \cos B \quad \ldots(2)$ Substituting eq (2) in eq (1), we ...
Read More →Solve the differential equation
Question: Solve the differential equation $\left[\frac{e^{-2 \sqrt{x}}}{\sqrt{x}}-\frac{y}{\sqrt{x}}\right] \frac{d x}{d y}=1(x \neq 0)$ Solution: $\left[\frac{e^{-2 \sqrt{x}}}{\sqrt{x}}-\frac{y}{\sqrt{x}}\right] \frac{d x}{d y}=1$ $\Rightarrow \frac{d y}{d x}=\frac{e^{-2 \sqrt{x}}}{\sqrt{x}}-\frac{y}{\sqrt{x}}$ $\Rightarrow \frac{d y}{d x}+\frac{y}{\sqrt{x}}=\frac{e^{-2 \sqrt{x}}}{\sqrt{x}}$ This equation is a linear differential equation of the form $\frac{d y}{d x}+P y=Q$, where $P=\frac{1}{\...
Read More →The maximum value of sin
Question: The maximum value of $\sin ^{2}\left(\frac{2 \pi}{3}+x\right)+\sin ^{2}\left(\frac{2 \pi}{3}-x\right)$ is (a) 1/2 (b) 3/2 (c) 1/4 (d) 3/4 Solution: (b) 3/2 $\frac{2 \pi}{3}=120^{\circ}$ Let $f(x)=\sin ^{2}(90+30+x)+\sin ^{2}(90+30-x)$ $=[\cos (30+x)]^{2}+[\cos (30-x)]^{2} \quad[$ Using $\sin (90+A)=\cos A]$ $=\left[\frac{\sqrt{3}}{2} \cos x-\frac{1}{2} \sin x\right]^{2}+\left[\frac{\sqrt{3}}{2} \cos x+\frac{1}{2} \sin x\right]^{2}$ $=\frac{3}{4} \cos ^{2} x+\frac{1}{4} \sin ^{2} x-\fra...
Read More →Find a particular solution of the differential equation
Question: Find a particular solution of the differential equation $(x-y)(d x+d y)=d x-d y$, given that $y=-1$, when $x=0$ (Hint: put $x-y=t$ ) Solution: $(x-y)(d x+d y)=d x-d y$ $\Rightarrow(x-y+1) d y=(1-x+y) d x$ $\Rightarrow \frac{d y}{d x}=\frac{1-x+y}{x-y+1}$ $\Rightarrow \frac{d y}{d x}=\frac{1-(x-y)}{1+(x-y)}$ ...(1) Let $x-y=t$. $\Rightarrow \frac{d}{d x}(x-y)=\frac{d t}{d x}$ $\Rightarrow 1-\frac{d y}{d x}=\frac{d t}{d x}$ $\Rightarrow 1-\frac{d t}{d x}=\frac{d y}{d x}$ Substituting the...
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