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If $\overrightarrow{a},\overrightarrow{b},$ and $\overrightarrow{c}$ are unit vectors such that $\overrightarrow{a}+2\overrightarrow{b}+2\overrightarrow{c}=\overrightarrow{0},$ then $|\overrightarrow{a}\times \overrightarrow{c}|$ is equal to :

In a triangle $\Delta ABC$, the sum of the lengths of two sides is represented by $p$ and the product of the lengths of the same two sides is $q$. Let $c$ be the third side. If $p^2=c^2+2q$, then the area of the circumcircle of $\Delta ABC$ is

If $a,c,b$ are three terms of a geometric progression, then the line $ax+by+c=0$

If the coefficient of the middle term in the expansion of $(1+x{)}^{2n+2}$ is $\alpha$ and the coefficients of middle terms in the expansion of $(1+x{)}^{2n+1}$ are $\beta$ and $\gamma$, then relation between $\alpha ,\beta$ and $\gamma$ is-

The sum of the distinct real values of $\mu ,$ for which the vectors, $\mu \hat{i}+\hat{j}+\hat{k},\hat{i}+\mu \hat{j}+\hat{k},\hat{i}+\hat{j}+\mu \hat{k}$ are co-planar, is:

The smallest positive value of $x$ which satisfies the equation ${\log }_{\cos x}\sin x+{\log }_{\sin x}\cos x=2$ is

Let $\frac{a_1}{a_2},\frac{b_1}{b_2},\frac{c_1}{c_2}$ be the consecutive terms of an arithmetic progression. If $a_1{x}^2+2b_{1}x+c_1=0\text{and}{a}_2{x}^2+2b_{2}x+c_2=0$ have a common root, then $a_2,b_2,c_2$ are in

The variance of first $10$multiples of $3$ is

The value of $\sin \frac{\pi }{n}+\sin \frac{3\pi }{n}+\sin \frac{5\pi }{n}+\cdots \text{to}n\text{terms}$ is

If $\frac{(2x+3)(4-3x{)}^3(x-4)}{x^5(x+2{)}^2}\le 0$, then $x\in$

All the integral values of $a$ for which the quadratic equation $(x-a)(x-10)+1=0$ has integral roots, are

Mean and variance of $20$ observations are $10$ and $4$, respectively. It was found, that in place of $11,9$ was taken by mistake, then correct variance is

Suppose that water is emptied from a spherical tank of radius 10 cm. If the depth of the water in the tank is 4 cm and is decreasing at the rate of 2 cm/sec, them the radius of the top surface of water is decreasing at the rate of

Given diagrams are examples of:

For an A.P., $a_1,a_2,a_3,\dots \dots$, if $a_3+a_5+a_8=11$ and $a_4+a_2=-2$, then the value of $a_1+a_6+a_7$ is

If $P=\left[\begin{array}{ccc}1& \alpha & 3\\ 1& 3& 3\\ 2& 4& 4\end{array}\right]$ is the adjoint of a $3\times 3$ matrix $A$ and $|A|=4$, then $\alpha$ is equal to :

For three sets $A,B\&C,$ if $A\subset B,A\subset C$ then $A\cap (B\cup C)=$

If $\alpha ,\beta$ be the roots of the equation $a{x}^2+bx+c=0$. Let $S_n={\alpha }^n+{\beta }^n,$ for $n\ge 1$

If $\Delta =\left|\begin{array}{ccc}3& 1+S_1& 1+S_2\\ 1+S_1& 1+S_2& 1+S_3\\ 1+S_2& 1+S_3& 1+S_4\end{array}\right|$, then $\Delta$ is equal to

Equation of the line passing through the mid-point of intercepts made by the circle $x^2+y^2-17x-16y+60=0$ is

The value(s) of $\lambda$ for which the line $y=x+\lambda$ touches the ellipse

$9x^2+16y^2=144$ is/are:

For positive integers n₁,n₂ the value of the expression $(1+i{)}^{n_1}$+$(1+i^3{)}^{n_1}$+$(1+i^5{)}^{n_2}$ + $(1+i^7{)}^{n_2}$ where i=$\sqrt[2]{2-1}$ is a real number if and only if

If $\int x^5{e}^{-x^2}dx=g\left(x\right)e^{-x^2}+c,$ where $c$ is a constant of integration, then $g(-1)$ is equal to :

Three critics review a book. Odds in favour of th ebook are $5:2,4:3$ and $3:4$, respectively, for the three critics. The probability that majority are in favor off the book is

The smallest positive term of the sequence $25,22\frac{3}{4},20\frac{1}{2},18\frac{1}{4},\cdots$ is

If $x^2+x+1=0$ and $x^2+ax+b=0$ have a common root, then the minimum value of $(x-a{)}^2+2b$ is

The number of real roots of the equation $(x+1)(x+2)(x+3)(x+4)=120$ is

If $\tan A=\frac{x\sin B}{1-x\cos B}$ and $\tan B=\frac{y\sin A}{1-y\cos A}$ then $\frac{\sin A}{\sin B}=$

Find the image of the point $P(2,-3,1)$ about the line

$\frac{x+1}{2}=\frac{y-3}{3}=\frac{z+2}{-1}$

The set of all natural numbers divisible by $5$ and less than $35$ can be written in set roster form as:

A series, whose $n^{th}$ term is $\left(\frac{n}{x}\right)+y$, then sum of $r$ terms will be :

If f and g are continuous on [0, a] and satisfy f(x) = f(a - x) and g(x) + g(x - a) = 2, then ${\int }_0^{a}f\left(x\right)g\left(x\right)\mathrm{dx}$is equal to

Solution of $|x||x+1|=2$ is

If $x-y=\frac{1}{3}$ and ${\cos }^2\left(\pi x\right)-{\sin }^2\left(\pi y\right)=\frac{1}{2}$, then $(x,y)$ can be

If the locus of mid point of the chords of the parabola $y^2=4ax$ which passes through a fixed point $(h,k)$ is also a parabola, then length of its latus rectum (in units) is

An ellipse, with foci at $(0,2)$ and $(0,–2)$ and minor axis of length $4$ units, passes through which of the following points?

In an increasing geometric progression, the sum of the first term and the last term is $66$, the product of the second terms from the beginning and the end is $128$ and sum of all terms is $126$. Then the number of terms in the progression is

Image of a point P(2, -3, 1 ) with respect to line L is I. Find the coordinates of I, if foot of the perpendicular of P with respect to L is $(\frac{-22}{7},\frac{-3}{14},\frac{-13}{14})$

If $\alpha$ is a repeated root of $a{x}^2+bx+c=0$ then $\underset{x\to \alpha }{lim}\frac{\sin (a{x}^2+bx+c)}{(x-\alpha {)}^2}$ is

Locus of the point which divided the double ordinates of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ in the ratio $1:2$ internally is

Equation of the parabola whose axis is $y=x$, distance from origin to vertex is $\sqrt{2}$ and distance from origin to focus is $2\sqrt{2},$ is (Focus and vertex lie in 1st quadrant) :

Three boys and three girls are to be seated around a circular table. Among them, the boy $X$ does not want any girl neighbour and the girl $Y$ does not want any boy neighbour. Then the number of possible arrangements is

Let $A,B,C,D$ be four concyclic points in order in which $AD:AB=CD:CB$. If $A,B,C$ are represented by complex numbers $a,b,c$, then vertex $D$ can be represented as

Two sets are given as $X=\{10,11,12,13\}$ and $Y=\{2,3,4,6,12,18,36\},$ then $X\cup Y=$

The solution of the inequality $\frac{4x+4}{2}>-x$ is

The polynomial $(x+y{)}^9$ is expanded in decreasing powers of $x.$ The second and third terms have equal values when evaluated at $x=p$ and $y=q,$ where $p$ and $q$ are positive numbers whose sum is one. The value of $p$ is

Let ${\log }_{a}3=2$ and ${\log }_{b}8=3.$ If $\alpha =\left[{\log }_{a}b\right]+1,$ where $[.]$ denotes the greatest integer function and $\beta$ is the integral part of ${\log }_{\sqrt{2}}\left(\sqrt{\alpha +\sqrt{\alpha +\sqrt{\alpha +\sqrt{\alpha +\cdots \text{upto}\infty }}}}\right),$ then