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Two adjacent sides of a parallelogram ABCD are given by $\stackrel{⟶}{AB}=2\hat{i}+10\hat{j}+11\hat{k}$ and $\stackrel{⟶}{AD}=-\hat{i}+2\hat{j}+2\hat{k}.$

The side AD is rotated by an acute angle $\alpha$ in the plane of the parallelogram so that AD becomes AD'. If AD' makes a right angle with the side AB, then the cosine of the angle $\alpha$ is given by

The value of $\alpha$ lying between $[0,\pi ]$ for which the inequality $\tan \alpha >{\tan }^3\alpha$ is valid, is

The equation of the circle whose radius is 5 and which touches the circle $x^2+y^2-2x-4y-20=0$ externally at the point (5, 5), is

Given that sample space S = {1, 2, 3, 4, 6} which of the following is the sure event?

A variable plane passes through a fixed point $(3,2,1)$ and meets $x,y$ and $z$ axes at $A,B$ and $C$ respectively. A plane is drawn parallel to $yz-$plane through $A$, a second plane is drawn parallel $zx-$plane through $B$ and a third plane is drawn parallel to $xy-$plane through $C$. Then the locus of the point of intersection of these three planes, is :

If $\alpha =\underset{n\to \infty }{lim}(\frac{1}{n^3+1}+\frac{4}{n^3+1}+\frac{9}{n^3+1}+\cdots +\frac{n^2}{n^3+1})$ and $\beta =\underset{x\to 0}{lim}\frac{\sin 2x}{\sin 8x},$ then a quadratic equation whose roots are $\alpha$ and $\beta$ is

The number of five letter words containing 3 vowels and 2 consonants that can be formed using the letters of the word EQUATION so that two consonants occur together is

The number of distinct terms in the expansion of ${(x+\frac{1}{x}+x^2+\frac{1}{x^2})}^{15}$ is

(Different power of $x$ means different terms)

Which of the following is/are functions?

The range of $si{n}^{-1}x-co{s}^{-1}x$ is

Reflection of the complex number $\frac{2-i}{3+i}$ in the straight line $z(1+i)$ = $\overline{z}(i-1)$ is

At an election, a voter may vote for any number of candidates, not greater than the number to be elected.There are 10 candidates and 4 are to be elected. If the voter votes for atleast one candidate, then the number of ways in which he can vote is

The value(s) of $p$ for which the quadratic equation $2x^2+px+8=0$ has equal roots is/are

The lines 2x - 3y = 5 and 3x - 4y = 7 are the diameters of a circle of area 154 square units. The equation of the circle is

If $\underset{0}{\overset{\frac{\pi }{2}}{\int }}\frac{\cot x}{\cot x+\text{cosec}x}dx=m(\pi +n),$ then $m.n$ is equal to :

If $f\left(x\right)=ln\left(\frac{x^2+e}{x^2+1}\right),$ then range of f(x) is

If the curve $xy=R^2-16$ represents a rectangular hyperbola whose branches lie only in the quadrant in which abscissa and ordinate are opposite in sign, then

The solution set the given set of equations will be

$x+y+z=6$

$x+2y+3z=10$

$x+2y+z=1$

If $\tan 2\alpha \cdot \tan \beta =1$ and $\tan \alpha \cdot \tan \gamma =1$ then

Let $0<\theta <\frac{\pi }{2}$. If the eccentricity of the hyperbola $\frac{x^2}{{\cos }^2\theta }-\frac{y^2}{{\sin }^2\theta }=1$ is greater than $2$, then the length of its latus rectum lies in the interval:

Let $A=\left[\begin{array}{cc}\cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha \end{array}\right],(\alpha \in R)$ such that $A^{32}=\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]$. Then a value of $\alpha$ is:

The approximate value of $(1.0002{)}^{3000}$ is

If $f\left(x\right)=4x-x^2,x\in R$, then $f(a+1)-f(a-1)$ is equal to

Match the following

$x+y+z=2$

$x+y+\lambda z=1$

$x+\mu y+2y=3$

(a) $\mu =1,\lambda =1$ (i) No solution

(b) $\mu =2,\lambda =3$ (ii) Unique Solution

(c) $\mu =1,\lambda =0$ (iii) Infinite Solutions

If one of the lines of $m{y}^2+(1-m^2)xy-m{x}^2=0$ is a bisector of the angle between the lines $xy=0$, then $m$ can be

The distance between the lines $3x-4y+9=0$ and $6x-8y-15=0$ is ___ units

If one root of $a{x}^2+bx+c=0$ is reciprocal of one root of $a_1{x}^2+b_{1}x+c_1=0$, then which of the following condition is correct?

The value of

$sin\frac{\pi }{14}sin\frac{3\pi }{14}sin\frac{5\pi }{14}sin\frac{7\pi }{14}sin\frac{9\pi }{14}sin\frac{11\pi }{14}sin\frac{13\pi }{14}$ is equal to ___

For which of the following ordered pairs $(\mu ,\delta ),$ the system of linear equations

$x+2y+3z=1$

$3x+4y+5z=\mu$

$4x+4y+4z=\delta$

is inconsistent?

If $\int f\left(x\right)dx=\psi \left(x\right)\text{, then}\int x^{5}f\left(x^3\right)dx$ is equal to

From the adjoining venn diagram, find $A\cap (B\cap C)$

Let $T_n$ be the number of all possible triangles formed by joining vertices of $n$-sided regular polygon. If $T_{n+1}-T_n=10$, then the value of $n$ is :

Circles are drawn on chords of the rectanglar hyperbola $xy=4$ parallel to the line $y=x$ as diameters. All such circles pass through two fixed points whose coordinates are

If $(\frac{5}{3},3)$ is the centroid of a triangle and its two vertices are $(0,1)$ and $(2,3)$, then the coordinates of the third vertex are

The value of the definite integral $\underset{0}{\overset{\pi }{\int }}\frac{\pi \tan x}{\sec x+\tan x}dx$ is equal to

Given 3 points given by position vectors $\overline{a}=\hat{i}+\hat{j},\overline{b}=\hat{j}+\hat{k},\overline{c}=-\hat{i}-\hat{j}-\hat{k}$. Find the plane passing through these 3 points.

If $A=\{x:-2\le x<2,x\in Z\}$, then the number of proper subsets of $A$ is

If the sets $A$ and $B$ are defined as

$A=\{x:x=2n,n\in N,n<100\}B=\{x:x=3n,n\in N,n<100\}$

then, the number of elements in $AUB$ is

Find the sum of the first 15 terms of the series 3 + 5 + 7 + 9 +. . . . . . n terms.___

The sixth term in the expansion of ${[\sqrt{\left\{2^{\log (10-3^x)}\right\}}+\sqrt[5]{5\left\{2^{(x-2)\log 3}\right\}}]}^m$ is equal to $21$. If it is known that the binomial coefficient of the $2^{\text{nd}},3^{\text{rd}}$ and $4^{\text{th}}$ terms in the expansion represents respectively the first, third and fifth terms of an A.P. (the symbol log stands for logarithm to the base $10$), then sum of possible values of $x$ is

The venn diagram for the sets $A=\{4,6,8\},$ $B=\{1,2,3\}$ and $U=\{1,2,3,4,5,6,7,8\}$ can be drawn as:

Let $f:R\to R$ be defined by $f\left(x\right)=2x+|x|$. Then $f\left(2x\right)+f(-x)-f\left(x\right)$ is

Let $P\left(x\right)=a{x}^2+bx+8$ is a quadratic polynomial. If the minimum value of $P\left(x\right)$ is $6$ at $x=2,$ then

The value of $\sqrt{\frac{1+\sin A}{1-\sin A}}$ where $A\in [0,2\pi ]-\left\{\frac{\pi }{2}\right\}$ is

Find the area of the triangle formed by the sides. $x=0,x+2y=5,3x–y=1$