If $\mathrm{f}(x)=3^x;\mathrm{g}(x)=4^x$, then $\frac{{\mathrm{f}}^{\prime }(0)-{\mathrm{g}}^{\prime }(0)}{1+{\mathrm{f}}^{\prime }(0){\mathrm{g}}^{\prime }(0)}$ is

If $x=\frac{5}{1-2\mathrm{i}},\mathrm{i}=\sqrt{-1}$, then the value of $x^3+x^2-x+22$ is

Two cards are drawn successively with replacement from a well-shuffled pack of 52 cards. Then mean of number of tens is

$\int x\sqrt{\frac{2\sin (x^2+1)-\sin 2(x^2+1)}{2\sin (x^2+1)+\sin 2(x^2+1)}}dx=$

$\text{ For all real }x\text{, the minimum value of }\frac{1-x+x^2}{1+x+x^2}\text{ is }$

If $\overline{a}=\hat{i}+4\hat{j}+2\hat{k},\overline{b}=3\hat{i}-2\hat{j}+7\hat{k},\overline{c}=2\hat{i}-\hat{j}+4\hat{k}$, then a vector $\bar{\mathrm{d}}$ which is parallel to vector $\bar{\mathrm{a}}\times \bar{\mathrm{b}}$ and which $\bar{\mathrm{c}}\cdot \bar{\mathrm{d}}=15$, is

If the vertices of a triangle are $(-2,3),(6,-1)$ and $(4,3)$, then the co-ordinates of the circumcentre of the triangle are

The solution of $\frac{\mathrm{d}x}{\mathrm{d}y}+\frac{x}{y}=x^2$ is

If $\int \frac{\cos 8x+1}{\cot 2x-\tan 2x}\mathrm{d}x=\mathrm{A}\cos 8x+\mathrm{c}$, where $\mathrm{c}$ is an arbitrary constant, then the value of $\mathrm{A}$ is

The set of all points, where the derivative of the functions $\mathrm{f}(x)=\frac{x}{1+|x|}$ exists, is

In triangle $\mathrm{ABC}$ with usual notations $\mathrm{b}=\sqrt{3},\mathrm{c}=1,\mathrm{m}\mathrm{\angle }\mathrm{A}={30}^{\circ }$, then the largest angle of the triangle is

A fair die is tossed twice in succession. If $\mathrm{X}$ denotes the number of fours in two tosses, then the probability distribution of $\mathrm{X}$ is given by

If $\mathrm{f}(x)=\frac{3x+4}{5x-7}$ and $\mathrm{g}(x)=\frac{7x+4}{5x-3}$, then $\mathrm{f}(\mathrm{g}(x))=$

If the function $f$ is given by $f(x)=x^3-3(a-2)x^2+3ax+7$, for some $\mathrm{a}\in \mathbb{R}$, is increasing in $(0,1]$ and decreasing in $[1,5)$, then a root of the equation $\frac{\mathrm{f}(x)-14}{(x-1{)}^2}=0(x\ne 1)$ is

The unit vector perpendicular to each of the vectors $\overline{a}+\overline{b}$ and $\overline{a}-\overline{b}$, where $\overline{a}=\hat{i}+\hat{j}+\hat{k}$ and $\bar{\mathrm{b}}=3\hat{\mathrm{i}}-2\hat{\mathrm{j}}+5\hat{\mathrm{k}}$ is

The logical statement $(\sim (\sim \mathrm{p}\vee \mathrm{q})\vee (\mathrm{p}\wedge \mathrm{r}))\wedge (\sim \mathrm{q}\wedge \mathrm{r})$ is equivalent to

Let $\overline{a}=2\hat{i}+\hat{j}-2\hat{k},\overline{b}=\hat{i}+\hat{j}$ and $\overline{c}$ be a vector such that $|\overline{c}-\overline{a}|=4,|(\overline{a}\times \overline{b})\times \overline{c}|=3$ and the angle between $\bar{\mathrm{c}}$ and $\bar{\mathrm{a}}\times \bar{\mathrm{b}}$ is $\frac{\pi }{6}$, then $\bar{\mathrm{a}}\cdot \bar{\mathrm{c}}$ is equal to

If $a$ and $b$ are positive number such that $a>b$, then the minimum value of $a\sec \theta -b\tan \theta (0<\theta <\frac{\pi }{2})$ is

$\text{ If }l=\underset{x\to 0}{lim}\frac{x}{|x|+x^2}\text{, then the value of }l\text{ is }$

If $y=[(x+1)(2x+1)(3x+1)\dots (\mathrm{n}x+1){]}^{\frac{3}{2}}$, then $\frac{\mathrm{d}y}{\mathrm{d}x}$ at $x=0$ is

If $\cos x+\cos y-\cos (x+y)=\frac{3}{2}$, then

The joint equation of the lines pair of lines passing through the point $(3,-2)$ and perpendicular to the lines $5x^2+2xy-3y^2=0$ is

If the line $\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-2}{4}$ meets the plane $x+2y+3z=15$ at the point $P$, then the distance of $\mathrm{P}$ from the origin is

If $\mathrm{A}$ and $\mathrm{B}$ are two events such that $\mathrm{P}(\mathrm{A})=\frac{1}{3},\mathrm{P}(\mathrm{B})=\frac{1}{5},\mathrm{P}(\mathrm{A}\cup \mathrm{B})=\frac{1}{3}$, then the value of $\mathrm{P}\left({\mathrm{A}}^{\prime }/{\mathrm{B}}^{\prime }\right)+\mathrm{P}\left({\mathrm{B}}^{\prime }/{\mathrm{A}}^{\prime }\right)$ is

If the general solution of the equation $\frac{\tan 3x-1}{\tan 3x+1}=\sqrt{3}$ is $x=\frac{\mathrm{n}\pi }{\mathrm{p}}+\frac{7\pi }{\mathrm{q}},\mathrm{n},\mathrm{p},\mathrm{q},\in \mathrm{Z}$, then $\frac{p}{q}$ is

If the area of the parallelogram with $\overline{a}$ and $\overline{b}$ as two adjacent sides is $16\mathrm{sq}$. units, then the area of the parallelogram having $3\bar{\mathrm{a}}+2\bar{\mathrm{b}}$ and $\bar{\mathrm{a}}+3\bar{\mathrm{b}}$ as two adjacent sides (in sq. units) is

The value of $\int (1-\cos x)\cdot {\mathrm{cosec}}^{2}xdx$ is

If ${\cos }^{-1}\sqrt{\mathrm{p}}+{\cos }^{-1}\sqrt{1-\mathrm{p}}+{\cos }^{-1}\sqrt{1-\mathrm{q}}=\frac{3\pi }{4}$, then $\mathrm{q}$ is

The slope of the tangent to a curve $y=\mathrm{f}(x)$ at $(x,\mathrm{f}(x))$ is $2x+1$. If the curve passes through the point $(1,2)$, then the area (in sq. units), bounded by the curve, the $\mathrm{X}$-axis and the line $x=1$, is

$A$ rod $AB,13$ feet long moves with its ends $A$ and $B$ on two perpendicular lines $OX$ and $OY$ respectively. When $A$ is 5 feet from $O$, it is moving away at the rate of $3\mathrm{feet}/\sec$. At this instant, $\mathrm{B}$ is moving at the rate

If $\mathrm{f}(x)=\{\begin{array}{cc}\frac{x-3}{|x-3|}+\mathrm{a}& ,x<3\\ \mathrm{a}+\mathrm{b}& ,x=3\\ \frac{|x-3|}{x-3}+\mathrm{b},& x>3\end{array}$

Is continuous at $x=3$, then the value of $\mathrm{a}-\mathrm{b}$ is

The equation of the tangent to the curve $y=\sqrt{9-2x^2}$, at the point where the ordinate and abscissa are equal, is

If $\bar{\mathrm{a}}=2\hat{\mathrm{i}}+3\hat{\mathrm{j}}+2\hat{\mathrm{k}},\bar{\mathrm{b}}=2\hat{\mathrm{i}}+\hat{\mathrm{j}}-\hat{\mathrm{k}}$ and $\bar{\mathrm{c}}=3\hat{\mathrm{i}}-\hat{\mathrm{j}}$ are such that $\overline{a}+\lambda \overline{b}$ is perpendicular to $\overline{c}$, then the value of $\lambda$ is

If a circle passes through points $(4,0)$ and $(0,2)$ and its centre lies on $\mathrm{Y}$-axis. If the radius of the circle is $r$, then the value of $r^2-r+1$ is

If $\mathrm{A}=\left[\begin{array}{ll}\mathrm{i}& 1\\ 1& 0\end{array}\right]$ where $\mathrm{i}=\sqrt{-1}$ and $\mathrm{B}={\mathrm{A}}^{2029}$, then ${\mathrm{B}}^{-1}=$

The solution of the differential equation $\frac{\mathrm{d}y}{\mathrm{d}x}+\frac{y}{x}=\sin x$ is

Let $f:[-1,2]\to [0,\mathrm{\infty })$ be a continuous function such that $\mathrm{f}(x)=\mathrm{f}(1-x),\mathrm{\forall }x\in [-1,2]$

Let ${\mathrm{R}}_1={\int }_{-1}^{2}x\mathrm{f}(x)\mathrm{d}x$ and ${\mathrm{R}}_2$ be the area of the region bounded by $y=\mathrm{f}(x),x=-1,x=2$ and the $\mathrm{X}$-axis, then ${\mathrm{R}}_2$ is

The equation of line passing through the point $(1,2,3)$ and perpendicular to the lines $\frac{x-2}{3}=\frac{y-1}{2}=\frac{z+1}{-2}$ and $\frac{x}{2}=\frac{y}{-3}=\frac{z}{1}$ is

Let a random variable $\mathrm{X}$ have a Binomial distribution with mean 8 and variance 4. If $\mathrm{P}(\mathrm{X}\le 2)=\frac{\mathrm{K}}{2^{16}}$, then $\mathrm{K}$ is

$\pi +({\sin }^{-1}\frac{4}{5}+{\sin }^{-1}\frac{5}{13}+{\sin }^{-1}\frac{16}{65})$ is equal to

If the angles of a triangle are in the ratio $4:1:1$, then the ratio of the longest side to its perimeter is

If truth value of logical statement $(p\leftrightarrow \sim q)\to (\sim p\wedge q)$ is false, then the truth values of $p$ and $q$ are respectively

The teacher wants to arrange 5 students on the platform such that the boy $B_1$ occupies second position and the girls $G_1$ and $G_2$ are always adjacent to each other, then the number of such arrangements is

If $\bar{\mathrm{a}}=\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}},\bar{\mathrm{b}}=4\hat{\mathrm{i}}+3\hat{\mathrm{j}}+4\hat{\mathrm{k}}$ and $\bar{\mathrm{c}}=\hat{\mathrm{i}}+\alpha \hat{\mathrm{j}}+\beta \hat{\mathrm{k}}$ are linearly dependent vectors and $|\overline{c}|=\sqrt{3}$, then the values of $\alpha$ and $\beta$ are respectively.

At present a firm is manufacturing 1000 items. It is estimated that the rate of change of production $\mathrm{P}$ w.r.t. additional number of worker $x$ is given by $\frac{\mathrm{dp}}{\mathrm{d}x}=100-12\sqrt{x}$. If the firm employees 9 more workers, then the new level of production of items is

The maximum value of $z=3x+5y$ subject to the constraints $3x+2y\le 18,x\le 4,y\le 6,x,y\ge 0$, is

If $\mathrm{I}=\int \sin (\log (x))\mathrm{d}x$, then $\mathrm{I}$ is given by

If both mean and variance of 50 observations $x_1,x_2,\dots \dots ,x_{50}$ are equal to 16 and 256 respectively, then mean of ${(x_1-5)}^2,{(x_2-5)}^2,\dots \dots {(x_{50}-5)}^2$ is

The angle between the line $\frac{x+1}{2}=\frac{y-2}{1}=\frac{z-3}{-2}$ and plane $x-2y-\lambda z=3$ is ${\cos }^{-1}\left(\frac{2\sqrt{2}}{3}\right)$, then value of $\lambda$ is