The area of the region bounded by x-axis and the curves defined by $y=tanx$, $-\frac{\pi}{3}\leq x\leq \frac{\pi}{3}$ and $y=cotx$, $-\frac{\pi}{6}\leq x\leq \frac{3\pi}{2}$ is

If $\log (1-x+x^2)={{a}}_1x+{{a}}_{2{}^{{}^{}}}{x}^2+{{{}{{a}}_3{x}^3+.\ldots.}}^{}$  then ${{a}}_3+{{a}}_6+{{a}}_9+.\ldots.$ is equal to

$\lim_{x\to \infty} (\frac{x+7}{x+2})^{x+5}$ equal to

The number of common tangents to the circle  $x^2+y^2=4$ and $x^2+y^2-6x-8y=24$ is

In a ΔABC, if $\tan ^2\frac{A}{2}+\tan ^2\frac{B}{2}+\tan ^2\frac{C}{2}=k$ , then k is always

Let $\vec{a}=2\widehat{i}\, +\widehat{j}\, +2\widehat{k}$ , $\vec{b}=\widehat{i}-\widehat{j}+2\widehat{k}$ and $\vec{c}=\widehat{i}+\widehat{j}-2\widehat{k}$ are are three vectors. Then, a vector in the plane of $\vec{a}$ and $\vec{c}$ whose projection on $\vec{b}$ is of magnitude $\frac{1}{\sqrt{6}}$ is

For what value of p, the polynomial  $x^4-3x^3+2px^2-6$ is exactly divisible by $(x-1)$

If $F(\theta)=\begin{bmatrix}{\cos \theta} & {-\sin \theta} & {0} \\ {\sin \theta} & {\cos \theta} & {0} \\ {0} & {0} & {1}\end{bmatrix}$ , then $F(\theta)F(\alpha)$ is equal to

If $\frac{n!}{2!(n-2)!}$ and $\frac{n!}{4!(n-4)!}$ are in the ratio 2:1, then the value of n is

The locus of the point of intersection of tangents to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ which meet right angles is

If the position vector of A and B relative to O be $\widehat{i}\, -4\widehat{j}+3\widehat{k}$ and $-\widehat{i}\, +2\widehat{j}-\widehat{k}$ respectively, then the median through O of ΔABC is:

The general value of $\theta$, satisfying the equation $\sin \theta=\frac{-1}{2},\, \tan \theta=\frac{1}{\sqrt[]{3}}$

The area of the triangle formed by the vertices whose position vectors are $3\widehat{i}+\widehat{j}$ , $5\widehat{i}+2\widehat{j}+\widehat{k}$ , $\widehat{i}-2\widehat{j}+3\widehat{k}$ is

The standard deviation of 20 numbers is 30. If each of the numbers is increased by 4, then the new standard deviation will be

The function $f(x)=\frac{x}{1+x\tan x}$ , $0\leq x\leq\frac{\pi}{2}$ is maximum when

If $f\colon R\rightarrow R$ is defined by $f(x)=\begin{cases}{\frac{x+2}{{x}^2+3x+2}} & {,\, if\, x\, \in R-\{-1,-2\}} \\ {-1} & {,if\, x=-2} \\ {0} & {,if\, x=-1}\end{cases}$ , then f(x) is continuous on the set

A polygon has 44 diagonals, the number of sides are

The probability of occurrence of two events E and F are 0.25 and 0.50, respectively. the probability of their simultaneous occurrence is 0.14. the probability that neither E nor F occur is

If $y={\tan }^{-1}\lgroup{\frac{3x-{x}^3}{1-3{x}^2}}\rgroup\, ,\, \frac{-1}{\sqrt[]{3}}{\lt}x{\lt}\frac{1}{\sqrt[]{3}}$ then $\frac{dy}{dx}$ is

The value of $\tan 9{^{\circ}}-\tan 27{^{\circ}}-\tan 63{^{\circ}}+\tan 81{^{\circ}}$ is equal to

In a triangle, if the sum of two sides is x and their product is y such that (x+z)(x-z)=y, where z is the third side of the triangle , then triangle is

If $H_1,H_2,\ldots,H_n$ are n harmonic means between a and b $(b\ne a)$;,then $\frac{{{H}}_n+a}{{{H}}_n-a}+\frac{{{H}}_n+b}{{{H}}_n-b}$

If $y=\sin ^{-1}(\frac{{x}^2+1}{\sqrt[]{1+3{x}^2+{x}^4}}),\, (x>0),$ then  $\frac{dy}{dx}$=

If $32\, \tan ^8\theta=2\cos ^2\alpha-3\cos \alpha$ and $3\, \cos \, 2\theta=1$, then the general value of $\alpha$ =

In a triangle ABC, $a\cos ^2\frac{C}{2}+\, c\, \, {\cos }^2\frac{A}{2}=\frac{3b}{2}$ then the sides of the triangle are in

If |k|=5 and 0° ≤ θ ≤ 360°, then the number of distinct solutions of 3cos⁡θ + 4sin⁡θ = k is

If θ is acute angle between the pair of lines $x^2-7xy+12y^2=0$, then $\frac{2\cos \theta+3\sin \theta}{4\sin \theta+5\cos \theta}=$

The four geometric means between 2 and 64 are

The lines $px+qy=1$ and $qx+py=1$ are respectively the sides AB, AC of the triangle ABC and the base BC is bisected at $(p,q)$. Equation of the median of the triangle through the vertex A is

If the vectors $a\hat{i}+\hat{j}+\hat{k},\hat{i}+b\hat{j}+\hat{k},\hat{i}+\hat{j}+c\hat{k}$ , $(a,b,c\ne1)$ are coplanar, then $\frac{1}{1-a}+\frac{1}{1-b}+\frac{1}{1-c}=$

Let $\vec{a}=\hat{i}+\hat{j}$ and  $\vec{b}=2\hat{i}-\hat{k}$, the point of intersection of the lines $\vec{r}\times\vec{a}=\vec{b}\times\vec{a}$  and  $\vec{r}\times\vec{b}=\vec{a}\times\vec{b}$  is

If the system of equations $3x-y+4z=3$ ,  $x+2y-3z=-2$ , $6x+5y+λz=-3 $   has atleast one solution, then $λ=$

If a variable takes values 0, 1, 2,…, 50 with frequencies $1,\, {{50}}_{{{C}}_1},{{50}}_{{{C}}_2},\ldots..,{{50}}_{{{C}}_{50}}$, then the AM is

If A={1,2,3,4} and B={3,4,5}, then the number of elements in (A∪B)×(A∩B)×(AΔB)

If n is an integer between 0 to 21, then find a value of n for which the value of $n!(21-n)!$ is  minimum

Suppose $A_1,A_2,\ldots,A_{30}$ are 30 sets each with five elements and $B_1,B_2,B_3,\ldots,B_n$ are n sets (each with three elements) such that  $\bigcup ^{30}_{i=1}{{A}}_i={{\bigcup }}^n_{j=1}{{B}}_i=S\, $ and each element of S belongs to exactly ten of the $A_i$'s and exactly 9 of the $B^{\prime}_j$'s. Then $n=$

The eccentric angle of the extremities of latus-rectum of the ellipse $\frac{{x}^2}{{a}^2}^{}+\frac{{y}^2}{{b}^2}^{}=1$ are given by

If α≠β and $\alpha^2=5\alpha-3,\beta^2=5\beta-3$, then the equation whose roots are $\frac{\alpha}{\beta}$ and $\frac{\beta}{\alpha}$ is

The probability that a man who is x years old will die in a year is p. Then, amongst n persons $A_1,A_2,\ldots A_n$ each x year old now, the probability that ${{A}}_1$ will die in one year and (be the first to die ) is

Angle between $\vec{a}$ and  $\vec{b}$ is $120{^{\circ}}$. If $|\vec{b}|=2|\vec{a}|$ and the vectors , $\vec{a}+x\vec{b}$ ,   $\vec{a}-\vec{b}$ are at right angle, then $x=$

If a number x is selected at random from natural numbers 1,2,…,100, then the probability for $x+\frac{100}{x}{\gt}29$ is

If X and Y are two sets, then X∩Y ' ∩ (X∪Y) ' is

$\int {e}^x(\sinh x+\cosh x)dx$

If $\vec{e_1}=(1,1,1)$ and $\vec{e_2}=(1,1,-1)$ and $\vec{a}$ and $\vec{b}$  and two vectors such that $\vec{e_2}=\vec{a}+2\vec{b}$ , then angle between $\vec{a}$ and $\vec{b}$

If P(1,2), Q(4,6), R(5,7) and S(a,b) are the vertices of a parallelogram PQRS, then

If $a\, \cos \theta+b\, \sin \, \theta=2$ and $a\, \sin \, \theta-b\, \cos \, \theta=3$ , then ${a}^{2^{}}+{b}^2=$

If three thrown of three dice, the probability of throwing triplets not more than twice is

$\int {3}^{{3}^{{3}^x}}.{3}^{{3}^x}.{3}^xdx$ is equal to

There are 50 questions in a paper. Find the number of ways in which a student can attempt one or more questions :

Consider the following frequency distribution table.

Choose the correct option for the remainder when X = 1! + 2! + 3! + ...........+ 100! is divided by 24