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In a binomial distribution mean and variance are $\frac{11}{4}$ and $\frac{15}{16}$ respectively, then the probability of success is

If the $2^{\text{nd}},3^{\text{rd}}$ and $4^{\text{th}}$ terms in the expansion of $(x+a{)}^n$ are $240,720$ and $1080$ respectively, then the sum of odd numbered terms in the expansion is

The equation of the circle passing through the point (2,-1) and having two diameters along the pair of lines $2x^2+6y^2-x+y-7xy-1=0is$

Evaluate $\underset{0}{\overset{\pi /2}{\int }}\frac{3\cos x-5\sin x}{3\sin x+5\cos x}dx$

The sides of a rhombus $ABCD$ are parallel to the lines, $x-y+2=0$ and $7x-y+3=0$. If the diagonals of the rhombus intersect at $P(1,2)$ and the vertex $A$ (different from the origin) is on the $y-$axis, then the ordinate of $A$ is

If $1+(1-2^2\cdot 1)+(1-4^2\cdot 3)+(1-6^2\cdot 5)+\cdots +(1-{20}^2\cdot 19)=$$\alpha -220\beta$, then an ordered pair $(\alpha ,\beta )$ is equal to

If one of the roots of the equation $a{x}^3-b{x}^2+cx+d=0\forall a,b,c,d\in R^{+}$ is positive, then the number of negative roots is and the number of imaginary roots is

If the $D.R^{\prime }s$ of a line are $(1+\lambda ,1-\lambda ,2)$ and it makes an angle of ${60}^{\circ }$ with $Y$ axis, then the value of $\lambda$ can be

Let $[.]$ denote the greatest integer function and $f\left(x\right)=\{\begin{array}{cc}\left[x\right]\left(\frac{e^{1/x}-1}{e^{1/x}+1}\right),& x<0\\ b,& x=0\\ \left[x\right]\left(\frac{e^{1/x}-1}{e^{1/x}+1}\right)+a,& x>0\end{array}$. If $f\left(x\right)$ is continuous at $x=0$, then the value of $a+b$ is

Given the matrices $A$ and $B$ as $A=\left[\begin{array}{cc}1& -1\\ 4& -1\end{array}\right]$ and $B=\left[\begin{array}{cc}1& -1\\ 2& -2\end{array}\right]$. The two matrices $X$ and $Y$ are such that $XA=B$ and $AY=B$. Then $3(X+Y)$ is equal to

If the $2^{nd}\text{and}{5}^{th}$ terms of G.P. are 24 and 3 respectively, then the sum of $1^{st}$ six terms is

A circle whose centre is the point of intersection of the lines 2x -3y + 4 = 0 and 3x + 4y - 5 = 0 passes through the origin. Find its equation.

If a hyperbola has length of its conjugate axis equal to $5$ unit and the distance between its foci is $13$ unit, then the eccentricity of the hyperbola is

If $\sum _{k=1}^n\sum _{r=1}^{k}r=a{n}^3+b{n}^2+cn+d$, then the value of $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}$ is

Sum of first n terms in the following series

${cot}^{-1}$3 + ${cot}^{-1}$7 + ${cot}^{-1}$13 + ${cot}^{-1}$21 + .........n terms.is given by

The solution set of $\cos 5\theta =-\frac{1}{2}$ is

In the following cases,
find the coordinates of the foot of the perpendicular drawn from the
origin.

(a) (b)

(c) (d)

If the distance between the directrices be thrice the distance between the foci, then eccentricity of ellipse is

Number of solutions for the system of equations $3x+4y+2z=1,3x+2y+4z=4$ and $6x+8y+4z=3$

The area of the region bounded by the curve $y=(x^2+2{)}^2+2x$ between the ordinates x = 0, x = 2 is

If $I_n=\int \frac{\sin nx}{\cos x}dx$, then $I_n+I_{n-2}$ is:

(where $n\in N,n\ge 2$ and $c$ is constant of integration)

If $a$ and $b$ are real numbers such that $(2+\alpha {)}^4=a+b\alpha$, where $\alpha =\frac{-1+i\sqrt{3}}{2},$ then $a+b$ is equal to

Which of the following binary operation is not commutative?

Prove the following :

${\tan }^{-1}\left(\frac{2}{11}\right)+{\tan }^{-1}\left(\frac{7}{24}\right)={\tan }^{-1}\left(\frac{1}{2}\right)$

If $\underset{x\to 0}{lim}\frac{\underset{0}{\overset{x}{\int }}\frac{t^2}{\sqrt{a+t}}dt}{bx-\sin x}=1,$ then

What is the period of f(x)= 2tanx +3 ?

Let $E_1$ and $E_2$ be two ellipses whose centers are in origin. The major axes of $E_1$ and $E_2$ lies along the $x-$axis and $y-$axis, respectively. Let $S$ be the circle $x^2+(y-1{)}^2=2.$. The straight lin $x+y=3$ touches the curves $S$, $E_1$ and $E_2$ at $P,Q$ and $R,$ respectively. Supoose the $PQ=PR=\frac{2\sqrt{2}}{3}.$If $e_1$ and $e_2$ are the eccentricities of $E_1$ and $E_2$, respectively, then the correct expression(s) is(are)

Find out the wrong number in the series given below :

$5,6,10,19,35,69$

If both mean and the standard deviation of $50$ observation $x_1,x_2,...,x_{50}$ are equal to $16$, then the mean of $(x_1-4{)}^2,(x_2-4{)}^2,...,(x_{50}-4{)}^2$ is :

Distance between the
two planes:
and

is

(A)2
units (B)4 units (C)8 units

(D)

If A + B = $\frac{\pi }{4}$ where A, B $\in R^{+}$, then the minimum value of (1+tanA) (1+tanB) is always equal to

The value of ${\sin }^{-1}\left(\sin 3\right)+{\cos }^{-1}\left(\cos 4\right)+{\tan }^{-1}\left(\tan 6\right)+{\cot }^{-1}\left(\cot 5\right)$ is equal to

Find the largest natural number 'a' for which the maximum value of f(x)=a-1 + 2x - x^{​​​​​​2} is smaller than the minimum value of g(x)= x^{​​​​​​2} -2ax +10-2a.

If the radius of a circle is at least $7\text{cm}$, then the minimum area of the circle is : (in $\text{sq. cm}$) $(\text{use}\pi =\frac{22}{7})$