Question 13 Marks
If $\vec{a}, \vec{b}, \vec{c}$ are three vectors equal in magnitude and $\vec{a}+\vec{b}+\vec{c}=0$, then find the value of $\vec{a} \cdot \vec{b}+\vec{b} \cdot \vec{c}+\vec{c} \cdot \vec{a}$.
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Find a unit vector along perpendicular to vectors $(\vec{a}+\vec{b})$ and $(\vec{a}-\vec{b})$ where $\vec{a}=2 \hat{i}+\hat{j}-2 \hat{k}$ and $\vec{b}=\hat{i}-2 \hat{j}+2 \hat{k}$.
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Find the value of $\int \frac{1}{1-\tan x} d x$.
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Find the value of $\int \frac{x e^x}{(x+1)^2} d x$.
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If $x^y=y^x$, then find $\frac{d y}{d x}$.
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If $x ^{ y }= e ^{ x - y }$, then prove that $\frac{d y}{d x}=\frac{\log x}{(1+\log x)^2}$.
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Prove that $\cos ^{-1} \frac{63}{65}+2 \tan ^{-1} \frac{1}{5}=\sin ^{-1} \frac{3}{5}$.
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Solve : $\tan ^{-1} 2 x+\tan ^{-1} 3 x=\frac{\pi}{4}$.
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If $\vec{a}+\vec{b}, \vec{b}+\vec{c}$ and $\vec{c}+\vec{a}$ are coplanar, then show that $\vec{a}, \vec{b}$ and $\vec{c}$ will be coplanar.
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If $\theta$ is the angle between two unit vectors $\hat{a}$ and $\hat{b}$ then show that $\cos \frac{\theta}{2}=\frac{1}{2}|\hat{a}+\hat{b}|$.
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Find the value of $\int_0^1 \sin ^{-1}\left(\frac{2 x}{1+x^2}\right) d x$.
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Find the value of $\int \frac{1}{\sqrt{\sin ^3 x \sin (x+\alpha)}} d x$.
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If $y=\log \left[x+\sqrt{a^2+x^2}\right]$, then prove that $\left(a^2+x^2\right) \frac{d^2 y}{d x^2}+x \frac{d y}{d x}=0$.
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If $y=\left(\sin ^{-1} x\right)^2$ then prove that $\left(1-x^2\right) \frac{d^2 y}{d x^2}-x \frac{d y}{d x}-2=0$.
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Solve the equation $\tan ^{-1}\left(\frac{1-x}{1+x}\right)=\frac{1}{2} \tan ^{-1} x, x>0$.
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Prove that: $\tan ^{-1} 1+\tan ^{-1} \frac{1}{2}+\tan ^{-1} \frac{1}{3}=\frac{\pi}{2}$.
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Prove that $\left[\begin{array}{lll}\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}} & \overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}} & \overrightarrow{\mathrm{c}}+\overrightarrow{\mathrm{a}}\end{array}\right]=2\left[\begin{array}{ll}\overrightarrow{\mathrm{a}} & \overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{c}}\end{array}\right]$.
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Prove that $\vec{a}, \vec{b}, \vec{c}$ are coplanar, if and only if $\vec{a} \times \vec{b}, \vec{b} \times \vec{c}, \vec{c} \times \vec{a}$ are coplanar.
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Find the value of $\int \frac{\sin x}{\sin (x-a)} d x$.
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Find the value of $\int \frac{\mathrm{dx}}{\mathrm{e}^{\mathrm{x}}-1}$.
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If $x=\sqrt{a^{\sin ^{-1} t}}$ and $y=\sqrt{a^{\cos ^{-1} t}}$ then prove that $\frac{d y}{d x}=-\frac{y}{x}$.
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If $(\cos x)^y=(\sin y)^x$ then find the value of $\frac{d y}{d x}$.
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Prove that: $\tan ^{-1}\left[\frac{\sqrt{1+x}+\sqrt{1-x}}{\sqrt{1+x}-\sqrt{1-x}}\right]=\frac{\pi}{4}+\frac{1}{2} \cos ^{-1} x$.
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Prove that : $\tan ^{-1} x+\cot ^{-1}(x+1)=\tan ^{-1}\left(x^2+x+1\right)$
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$D , E$ and $F$ are mid points of the sides of triangle $ABC$. If ' $O$ ' be any point, then prove that $\overrightarrow{O A}+\overrightarrow{O B}+\overrightarrow{O C}=\overrightarrow{O D}+\overrightarrow{O E}+\overrightarrow{O F}$
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Find the vector along vector $5 \hat{i}-\hat{j}+2 \hat{k}$ whose magnitude is 8 units.
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Prove that $\int_0^{\pi / 2} \frac{\sqrt{\sin x}}{\sqrt{\sin x}+\sqrt{\cos x}} d x=\frac{\pi}{4}$
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Find the integral : $\int \frac{d x}{1+\cos x+\sin x}$
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Find $\frac{d y}{d x}$, if $y =\sqrt{\frac{(x-1)(x-2)}{(x-3)(x-4)(x-5)}}$.
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If function $F ( x )=\left\{\begin{array}{l}k x+1, x \leq \pi \\ \cos x, x>\pi\end{array}\right.$ is continuous at $x=\pi$ then find $k$.
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Prove that- $\tan ^{-1}\left(\frac{63}{16}\right)=\sin ^{-1}\left(\frac{5}{13}\right)+\cos ^{-1}\left(\frac{3}{5}\right)$.
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Solve the following: $\tan ^{-1}(1)+\cos ^{-1}\left(-\frac{1}{2}\right)+\sin ^{-1}\left(-\frac{1}{2}\right)$.
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If $\vec{a}, \vec{b}, \vec{c}$ are perpendicular to each other and of equal magnitudes, then prove that the vector $\vec{a}+\vec{b}+\vec{c}$ makes equal angle with $\vec{a}, \vec{b}$ and $\vec{c}$
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For any vector $\vec{a}$ prove that $|\vec{a} \times \hat{i}|^2+|\vec{a} \times \hat{j}|^2+|\vec{a} \times \hat{k}|^2=2|a|^2$
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Find the value of $\int \frac{d x}{\sqrt{7-6 x-x^2}}$
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Find the value of integration $\int_1^4\{|x-1|+|x-2|+|x-3|\} d x$
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If $x^2+y^2=t-\frac{1}{t}$ and $x^4+y^4=t^2+\frac{1}{t^2}$, then prove that $x \frac{d^2 y}{d x^2}+2 \frac{d y}{d x}=0$
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If $y=\sqrt{\log x+\sqrt{\log x+\sqrt{\log x+\ldots \ldots \ldots \infty}}}$, then find $\frac{d y}{d x}$.
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Prove that: $\cos ^{-1} \frac{12}{13}+\sin ^{-1} \frac{4}{5}=\tan ^{-1} \frac{63}{16}$
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If $\tan ^{-1} x+\tan ^{-1} y+\tan ^{-1} z=\frac{\pi}{2}$, then prove that $x y+y z+z x=1$
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