Question 12 Marks
Find the general solution of differential equation $\frac{d x}{d y}+(\tan y) x=\sec ^2 y$.
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With the help of integration find the complete area of circle $x^2+y^2=1$.
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If $\vec{a}=2 \hat{i}+2 \hat{j}+3 \hat{k}, \vec{b}=-\hat{i}+2 \hat{j}+\hat{k}$ and $\vec{c}=3 \hat{i}+\hat{j}$ are such that $\vec{a}+\lambda \vec{b}$ is perpendicular on vector $\vec{c}$, then find the value of $\lambda$.
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Find the value $k$, so that the given function is continuous at $x=5$
$
f(x)=\left\{\begin{array}{ccc}
k x+1 & \text { if } & x \leq 5 \\
3 x-5 & \text { if } & x>5
\end{array}\right.
$
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If $A$ and $B$ are independent events show that the probability of happening at least one of $A$ and $B$ will be $1-P\left(A^{\prime}\right) P\left(B^{\prime}\right)$
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Find that minimum value of a for which the function $f(x)=x^2+a x+1$ is increasing in interval $(1,2)$.
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If $A=\left[\begin{array}{ll}2 & 5 \\ 3 & 8\end{array}\right]$ then find $A ^{-1}$.
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Prove the points $(a, b+c),(b, c+a),(c, a+b)$ are collinear.
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If $y=x^x+x^a+a^x+a^a$, then find $\frac{d y}{d x}$.
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Find the area bounded by parabola $x^2=4 y$ and line $y=3$ (draw figure).
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Find the value of $\int \frac{d x}{x\left(x^5+3\right)}$.
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Find the maximum and minimum value of function $f(x)=x+\frac{1}{x}$.
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If $A=\left[\begin{array}{lll}1 & 2 & 3\end{array}\right], B=\left[\begin{array}{l}3 \\ 2 \\ 1\end{array}\right]$ then find $AB$ and $BA$.
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Find a vector of magnitude 8 along with the vector $i+2 j+2 k$.
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The probability distribution of a random variable $\mathrm{X}$ is following, where $\mathrm{k}$ is any number $\mathrm{P}(X)=\left\{\begin{array}{l}k \text { if } X=0 \\ 2 k \text { if } X=1 \\ 3 k \text { if } X=2 \\ 0 \quad \text { otherwise }\end{array}\right.$
(a) Find the value of $\mathrm{k}$
(b) Find the value of $\mathrm{P}(\mathrm{X}<2), \mathrm{P}(\mathrm{X} \leq 2), \mathrm{P}(\mathrm{X} \geq 2)$
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If the area of a triangle is 18 sq units and vertices are $(x, 7) ;(2,2)$ and $(10,8)$, then find the value of $x$.
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If $\mathrm{A}=\left[\begin{array}{cc}\cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha\end{array}\right]$ and $\mathrm{A}+\mathrm{A}^{\prime}=\mathrm{I}$, then find the value of $\alpha$
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If $f, g: R \rightarrow R$ is defined such that $f(x)=x^2+1, g(x)=2 x-3$, then find the value of $fog(x)$, gof $(x)$ and $gog (3)$.
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Find the value of $\int \frac{x^2-1}{x^2+1} d x$.
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Find that point on curve $y=x^2-2 x+3$, the tangent drawn on that is parallel to line $2 \mathrm{x}-\mathrm{y}+9=0$.
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Find the area bounded by parabola $x^2=4 x$ and line $y=x$
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Find the value of $\int \frac{1}{x-\sqrt{x}} d x$.
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Prove that the function $y=\sin ^p \theta \cdot \cos ^q \theta$ has maxima at $\theta=\tan ^{-1}(\sqrt{p / q})$.
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Find the equation of tangent to the curve $y=x^2-2 x+3$, which is parallel to the line $2 x + y +9=0$.
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If the function $f(x)=\left\{\begin{array}{cc}\frac{k \cos x}{\pi-2 x} & , x \neq \frac{\pi}{2} \\ 5 & , x=\frac{\pi}{2}\end{array}, \quad\right.$ is continuous at $x=\frac{\pi}{2}$, then find the value of $k$.
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If $\left|\begin{array}{ll}2 & 3 \\ y & x\end{array}\right|=3,\left|\begin{array}{ll}x & y \\ 4 & 2\end{array}\right|=5$, then find the value of $x$ and $y$.
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Prove that $\left|\begin{array}{ccc}x+4 & 2 x & 2 x \\ 2 x & x+4 & 2 x \\ 2 x & 2 x & x+4\end{array}\right|=(5 x+4)(x-4)^2$
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If $A=\left[\begin{array}{c}-2 \\ 4 \\ 5\end{array}\right], B=\left[\begin{array}{lll}1 & 3 & 6\end{array}\right]$, then show that $(A B)^T=B^T \cdot A^T$
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Consider the function $f(x)=2 x+3, f: R \rightarrow R$, prove that $F$ is invertible. Also find the inverse function of $F$.
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If vector $\vec{a}=2 \hat{i}-\hat{j}+\hat{k}, \vec{b}=3 \hat{i}+\hat{j}-2 \hat{k}$ then find a unit vector in the direction of $\vec{a} \times \vec{b}$.
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Find the projection of vector $\hat{i}-\hat{j}$ on vector $\hat{i}+\hat{j}$.
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If $y=\tan ^{-1}\left(\frac{2^{x+1}}{1-4^x}\right)$ then find $\frac{d y}{d x}$.
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Find a unit vector perpendicular to vectors $(\vec{a}+\vec{b})$ and $(\vec{a}-\vec{b})$, when $\vec{a}=2 \hat{i}+\hat{j}-2 \hat{k}, \vec{b}=\hat{i}-2 \hat{j}+2 \hat{k}$
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If $\left[\begin{array}{lll}x & 4 & 1\end{array}\right]\left[\begin{array}{ccc}2 & 0 & 1 \\ 1 & 0 & 2 \\ 0 & 2 & -4\end{array}\right]\left[\begin{array}{c}-2 \\ 4 \\ 1\end{array}\right]=0$ then find the value of $x$.
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If a relation $R$ is defined in a set $A$, then show that if $R$ is symmetric then $R=R^{-1}$
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Show that the points $2 \hat{i},-\hat{i}-4 \hat{j},-i+4 \hat{j}$ represent an isosceles triangle.
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If A = {1, 2, 3}, B {2, 1), then find the number of non empty relations from A to B.
Question 382 Marks
Find the interval in which the function $f(x)=x^2-6 x+5$ is strictly increasing.
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If $y=\log \left[x+\sqrt{a^2+x^2}\right]$ then find $\frac{d y}{d x}$.
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If $\mathrm{A}=\left[\begin{array}{lll}2 & -4 & 3\end{array}\right]$ and $\mathrm{B}=\left[\begin{array}{r}2 \\ -4 \\ 8\end{array}\right]$ then find $(\mathrm{AB})^{\mathrm{T}}$.
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Show that the function $F(x)=\frac{1}{(x-a)}$ is not continuous at point $x=a$.
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If $|\vec{a}|=10,|\vec{b}|=2, \vec{a} \cdot \vec{b}=12$, then find the value of $\vec{a} \times \vec{b}$.
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Find the general solution of differential equation $\frac{d y}{d x}+\sqrt{\frac{1-y^2}{1-x^2}}=0$.
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Find the area bounded by parabola $x^2=y$ and a straight line $y=x+2$ and $x$-axis.
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Evaluate $\int \frac{e^{2 x}-1}{e^{2 x}+1} d x$
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Find the maximum and minimum value of function $f(x)=x^2-4 x+8$ in the interval $[1,5]$.
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Prove that the function $f(x)=\log (\sin x)$ is increasing in the interval $\left(0, \frac{\pi}{2}\right)$.
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Examine the continuity of function $f(x)=\left\{\begin{array}{cc}\frac{x e^{\frac{1}{x}}}{1+e^{\frac{1}{x}}}, & x \neq 0 \\ 0 & , x=0\end{array}\right.$ at $x =0$.
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If $A=\left[\begin{array}{cc}2 & 3 \\ 1 & -4\end{array}\right]$, then find $A ^{-1}$.
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If the matrix $A=\left[\begin{array}{ccc}2 & 0 & 1 \\ 2 & 1 & 3 \\ -1 & -1 & 0\end{array}\right]$, then find the value of $A^2-5 A+6 I$
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