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Let $x^2=4ky,k>0$, be a parabola with vertex $A$. Let $BC$ be its latus rectum. An ellipse with center on $BC$ touches the parabola at $A$, and cuts $BC$ at points $D$ and $E$ such that $BD=DE=EC$ ($B,D,E,C$ in that order). The eccentricity of the ellipse is

If ${1690}^{2608}+{2608}^{1690}$ is divided by $7$, then the remainder is

The length of perpendicular drawn from the point (5,4, -1) to the line $\overrightarrow{r}=\hat{i}+\lambda (2\hat{i}+9\hat{j}+5\hat{k})$ is

If $(1+x)(1+x^2)(1+x^4)\dots (1+x^{128})=\sum _{r=0}^n{x}^r,$ then $n$ is equal to

The coefficient of $x^5$ in the expansion of ${(2-x+3x^2)}^6$ is

The set of value(s) of $x$ for which $f\left(x\right)=e^{\ln (-x^2+5x-6)}$ and $g\left(x\right)=-x^2+5x-6$ are identical functions is

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 :

Column Matching:



$$\begin{array}{cc}\text{Column (I)}& \text{Column (II)}\\ \begin{array}{cc}\text{(A)}& \text{In a triangle}△XYZ,\text{let}a,b\text{and}c\text{be the}\\ & \text{lengths of the sides opposite to the angles}\\ & X,Y\text{and}Z,\text{respectively. If}2(a^2-b^2=c^2\\ & \text{and}\lambda =\frac{\sin (X-Y)}{\sin Z},\text{then possible values}\\ & \text{of}n\text{for which}\cos \left(n\pi \lambda \right)=0\text{is (are)}\end{array}& \text{(P) 1}\\ & \\ \begin{array}{cc}\text{(B)}& \text{In a triangle}△XYZ,\text{let}a,b\text{and}c\text{be}\\ & \text{the lengths of the sides opposite to the}\\ & \text{angles}X,Y\text{and}Z,\text{respectively. If}\\ & 1+\cos 2X-2\cos 2Y=2\sin X\sin Y,\text{then}\\ & \text{possible value(s) of}\frac{a}{b}\text{is (are)}\end{array}& \text{(Q) 2}\\ & \\ \begin{array}{cc}\text{(C)}& \text{In}{R}^2,\text{let}\sqrt{3}\hat{i}+\hat{j},\hat{i}+\sqrt{3}\hat{j}\text{and}\beta \hat{i}+(1-\beta )\hat{j}\\ & \text{be the position vectors of}X,Y\text{and}Z\text{with}\\ & \text{respect to the origin}O,\text{respectively. If the}\\ & \text{distance of}Z\text{from the bisector of the acute}\\ & \text{angle of}\overrightarrow{OX}\text{with}\overrightarrow{OY}\text{is}\frac{3}{\sqrt{2}},\text{then possible}\\ & \text{value(s) of}|\beta |\text{is (are)}\end{array}& \text{(R) 3}\\ & \\ \begin{array}{cc}\text{(D)}& \text{Suppose that}F\left(\alpha \right)\text{denotes the area of the}\\ & \text{region bounded by}x=0,x=2,y^2=4x\\ & \text{and}y=|\alpha x-1|+|\alpha x-2|+\alpha x,\text{where}\\ & \alpha \in \{0,1\}.\text{Then the value(s) of}F\left(\alpha \right)+\frac{8}{3}\sqrt{2}\text{,}\\ & \text{when}\alpha =0\text{and}\alpha =1,\text{is (are)}\end{array}& \text{(S) 5}\\ & \\ & \text{(T) 6}\\ & \end{array}$$

Option $\text{(D)}$ matches with which of the elements of right hand column?

The value of ${}^{30}{C}_0-\frac{{}^{30}{C}_1}{2}+\frac{{}^{30}{C}_2}{3}-\cdots \cdots +\frac{{}^{30}{C}_{30}}{31}$ is

The mean weight of $150$ students in a certain class is $60$kg. The mean weight of boys in the class is $70$kg and that of girls if $55$kg, then number of boys and girls respectively are

Given the sight distance as 120 m, the height of the driver's eye as 1.5 m, the height of the obstacle as 15 cm and the grade difference of the intersecting gradients as 0.09, the required length of the summit parabola curve is

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If a, b, c are $p^{th}$, $q^{th}$, and $r^{th}$ terms of a G.P., then $\left(\frac{c}{b}{)}^p\right(\frac{b}{a}{)}^r(\frac{a}{c}{)}^q$ is equal to

Let $2si{n}^{2}x+3sinx-2>0$ and $x^2-x-2<0$ (x is measured in radians). Then x lies in the interval

Consider the function $f\left(x\right)=x^2+bx+c,$ where $D=b^2-4c>0.$ If $b<0,c>0$ then the number of points of non-differentiability of $g\left(x\right)=|f\left(|x|\right)|$ is

Let $\alpha ,\beta ,\gamma$ be the roots of equations, $x^3+a{x}^2+bx+c=0,(a,b,c\in R$ and $a,b\ne 0).$ If the system of the equations (in $u,v,w)$ given by $\alpha u+\beta v+\gamma w=0;$ $\beta u+\gamma v+\alpha w=0;$ $\gamma u+\alpha v+\beta w=0$ has non-trivial solutions, then the value of $\frac{a^2}{b}$ is

If $\frac{cos(A-B)}{cos(A+B)}+\frac{cos(C+D)}{cos(C-D)}=0$, prove that $tanAtanBtanCtanD=-1$

Let $y\left(x\right)$ be the solution of the differential equation $2x^{2}dy+(e^y-2x)dx=0,x>0.$ If $y\left(e\right)=1$, then $y\left(1\right)$ is equal to

Let $f\left(x\right)$ be a twice differentiable function and $f^{\prime \prime }\left(0\right)=5$, then $\underset{x\to 0}{lim}\frac{3f\left(x\right)-4f\left(3x\right)+f\left(9x\right)}{x^2}$ is equal to:

The plane containing the line $\frac{x-3}{2}=\frac{y+2}{-1}=\frac{z-1}{3}$ and also containing its projection on the plane $2x+3y-z=5$, contains which one of the following points ?

Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse

6)The radius of hydrogen atom in the ground state is 0.53Å. The radius of ₃Li²⁺ in the similar state is

Let the area of the triangle formed by $(1,2),(-3,0)$ and any point on the line $x-2y+k=0$ is $5$ sq. units. If possible values of $k$ are $k_1,k_2$ then locus of foot of the perpendicular drawn from $(k_1,0)$ to a variable line passing through $(0,k_2)$ is

Let $f:R\to R,g:R\to R$ and $h:R\to R$ be differentiable functions such that $f\left(x\right)=x^3+3x+2,g\left(f\right(x\left)\right)=x$ and $h\left(g\right(g\left(x\right)\left)\right)=x$ for all $x\in R.$ Then

The general solution of the differential equation $(x+y+1)dy=dx$ is (where $C$ is a constant of integration)

The number of real solutions of $(x-1)(x+1)(2x+1)(2x-3)=15$ is

All face cards from pack of $52$ playing cards are removed. From remaining $40$ cards, two are drawn randomly, without replacement, then probability of drawing a pair (same denominations) is

If the sum of n terms of an A.P. is
and
its m^th term is 164, find the value of m.

The range of $f\left(x\right)=\frac{1}{5\sin x-6}$ is

Integrate the following functions.
$\int \frac{1}{x+xlogx}dx.$

The equation of straight line passing through point $A(-1,3,-2)$ and equally inclined to positive co-ordinate axes is

Find the area of the region in the
first quadrant enclosed by x-axis, line
and
the circle

If H is the harmonic mean between p and q, then the value of $\frac{H}{p}+\frac{H}{q}$ is

Find the number of combinations of n different objects taken r at a time.

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