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Find out the wrong number in the series given below :

$242,462,572,427,671,264$

For some $\theta \in (0,\frac{\pi }{2})$, if the eccentricity of the hyperbola, $x^2-y^2{\sec }^2\theta =10$ is $\sqrt{5}$ times the eccentricity of the ellipse, $x^2{\sec }^2\theta +y^2=5$, then the length of the latus rectum of the ellipse, is:

X and Y are two continuous random variables with probability density functions $f_x\left(x\right)$ and $f_y\left(y\right)$ respectively. If E[•] indicates the expectation operator, then select the correct one of the following relations:

The value of when m is equal to

Integrate the following functions.
$\int \frac{x+3}{x^2-2x-5}dx.$

Let $2{(1+x^3)}^{100}=\sum _{i=0}^{100}\{a_i{x}^i-\cos \left(\frac{\pi }{2}\right(x+i\left)\right)\}.$ If $\sum _{i=0}^{50}{a}_{2i}=2^k,$ then the value of $k$ is

If one root of the quadratic equation $2x^2+ax-6=0$ is 2, find the value of a.

Let $P,Q$ be the end points of the chord of contact of the point $R(2,5)$ with respect to $y^2=8x.$ The length of the intercept made by the circle with $PQ$ as a diameter, on the $y-$axis is equal to

If the sum of $n$ terms of an A.P. is given by $S_n=3n^2-4n,$ then its ${50}^{th}$ term is

If $x=3\sin t,y=3\cos t,$ then $\frac{dy}{dx}$ at $t=\frac{\pi }{3}$ is equal to

Difference between the maximum and the minimum value of the function $f\left(x\right)=({\sin }^{-1}x{)}^2+({\cos }^{-1}x{)}^2$ is :

The smallest positive integral value of $n$ for which $(1+i{)}^{2n}=(1-i{)}^{2n}$ is

The sum of residues of $f\left(z\right)=\frac{2z}{(z-1{)}^2(z-2)}$ at its singular point is

If D, G and R denote respectively the number of degrees, grades and radians in an angle, then

If $f\left(x\right)=\left|\begin{array}{ccc}sinx& sina& sinb\\ cosx& cosa& cosb\\ tanx& tana& tanb\end{array}\right|,$

where $0<a<b<\frac{\pi }{2}$
then the equation
$f^{\prime }\left(x\right)=0$ has in the interval $(a,b)$

Refere to question 7 above. Find the maximum value of Z.

Let $Z$ be a complex number and $\overline{Z}$ denotes the conjugate of $Z$. If $2Z-3\overline{Z}=\frac{-27+23i}{1+i}$, then which of the following is/are correct?

Solve $\sqrt{x-2}\ge -1$

The circles $z\overline{z}+z\overline{a_1}+a_1\overline{z}+b_1=0,b_1\in R$ and $z\overline{z}+z\overline{a_2}+\overline{z_2}{a}_2+b_2=0,b_2\in R$ will intersect orthogonally if

Let $y=y\left(x\right)$ be the solution of the differential equation $xdy=(y+x^3\cos x)dx$ with $y\left(\pi \right)=0$, then $y\left(\frac{\pi }{2}\right)$ is equal to

The maximum area bounded by the curves $y^2=4ax,y=ax$ and $y=\frac{x}{a}(1<a\le 2)$

If A is a skew-symmetric matrix of order 3, then the matrix $A^4$ is

The derivative of $e^{\ln \left(\sin x\right)}$ with respect to $x$ where $x\in (0,\frac{\pi }{2}),$ is



[1 mark]

Find the missing number in the series.

11, 14, 27, ?, 119, 216

If ${\tan }^{2}x=1-{\alpha }^2$, then the possible values of $\alpha$ satisfying the equation $\sec x+{\tan }^{3}x\text{cosec}x=(2-{\alpha }^2{)}^{3/2}$ is

The sum of all possible values of $\theta$ where $\theta \in (0,\frac{\pi }{2}),$ satisfying the equation $\left|\begin{array}{ccc}1+{\sin }^2\theta & {\cos }^2\theta & 4\sin 4\theta \\ {\sin }^2\theta & 1+{\cos }^2\theta & 4\sin 4\theta \\ {\sin }^2\theta & {\cos }^2\theta & 1+4\sin 4\theta \end{array}\right|=0,$ is

The outcome of each of $30$ items was observed; $10$ items gave an outcome $\frac{1}{2}-d$ each, $10$ items gave outcome $\frac{1}{2}$ each and the remaining $10$ items gave outcome $\frac{1}{2}+d$ each. If the variance of this outcome data is $\frac{4}{3},$ then $|d|$ equals

If $a$ and $b$ are positive integers, $f$ is a function defined for positive numbers and attains only positive values such that $f\left(yf\right(x\left)\right)=x^a{y}^b$, then

If $f\left(x\right)+2f(1-x)=6x\forall x\in R,$ then $\frac{f(-x)-f\left(x\right)}{2}$ equals