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The sum of the first n terms is $\frac{1}{2}$ + $\frac{3}{4}$ + $\frac{7}{8}$ + $\frac{15}{16}$ + .......... is

If $\alpha ,\beta ,\gamma$ are in A.P, then $\frac{{\sin }^2\alpha -{\sin }^2\gamma }{\sin \alpha \cos \alpha -\sin \gamma \cos \gamma }$ is equal to

A sphere $S$ which passes through origin and the image of it's center in the plane $x+y+z=3$ is $(0,0,0)$. If $a$ be the area of the cross section made by the plane then

The area (in sq units) of the quadrilateral formed by the tangents at the endpoints of the latus recta to the ellipse $\frac{x^2}{9}+\frac{y^2}{5}=1$ is___ .

The angle between the normals to the parabola $y^2=24x$ at points $(6,12)$ and $(6,-12)$ is :

The equation of an ellipse, centred at origin and passing through the points $(4,3)$ and $(-1,4),$ is

If a unit vector $\overrightarrow{a}$ makes angles $\frac{\pi }{3}$ with $\hat{i}$, $\frac{\pi }{4}$ with $\hat{j}$ and $\theta \in (0,\pi )$ with $\hat{k}$, then a value of $\theta$ is :

If the maximum and minimum values of the determinant

$\left|\begin{array}{ccc}1+{\cos }^{2}x& {\sin }^{2}x& \cos 2x\\ {\cos }^{2}x& 1+{\sin }^{2}x& \cos 2x\\ {\cos }^{2}x& {\sin }^{2}x& 1+\cos 2x\end{array}\right|$

are $\alpha$ and $\beta$ respectively, then which of the following is correct?

The range of the function $f\left(x\right)=\frac{x+3}{|x+3|},x\ne -3$ is

The least value of $\sec A+\sec B+\sec C$ in an acute angle triangle is

The value of $P$ for which the equation $(P^3-3P^2+2P)x^2+(P^3-P)x+P^3+3P^2+2P=0$ has exactly one root at infinity is

Third term in the expression $(x+a{)}^7$ is

The middle point of the line segment joining (3, -1)and (1, 1) is shifted by two units (in the sense of increasing y) Perpendicular to the line segment. Then, the coordinates of the point in the new position are

The number of non-negtive integer(s) which lie in between the maximum and minimum value of $x$ if $-5\le 2x-4\le -1$ is

Which of the following equation has exactly one root as $0$?
$(a,b,c>0)$

The solution of the differential equation, $\frac{dy}{dx}=(x-y{)}^2$, when $y\left(1\right)=1$, is :

Exhaustive values of $x$ satisfying the equation $|x^4-x^2-12|=|x^4-9|-|x^2+3|$ is -

If $\theta \in (0,2\pi )$ and $2\cos \theta =\sqrt{3}\cos {10}^{\circ }-\sin {10}^{\circ }$, then $\theta$ can be

If $\overrightarrow{a}\mathrm{and}\overrightarrow{b}$are two unit vectors such that

$\overrightarrow{a}=\hat{i}$ and $\overrightarrow{b}=\hat{j}$

then the angle between

Let $S$ be the standard deviation of $n$ observations. Each of the $n$ observations is multiplied by a constant $C$. Then the standard deviation of the resulting number is

Find the orthogonal trajectory of $x^2+y^2=c$

Which of the following can best represent the graph of $f\left(x\right)=e^{\left\{x\right\}}$?

[Note: $\left\{x\right\}$ denotes the fractional part of $x$]

If a, b, c are the roots of x³ - x² - 2004 = 0. Then the value of is equal to

If $X=\{1,2,3,4,5\},Y=\{1,3,5,7,9\},$ then which among the following is not a relation from $X$ to $Y$

$\int e^x\left[\frac{x^3+x+1}{(1+x^2{)}^{3/2}}\right]dx$ is equal to

A box contains 15 transistors, 5 of which are defective. An inspector takes out one transistor at random, examines it for defects, and replaces it. After it has been replaced another inspector does the same things, and then so does a third inspector. The probability that atleast one of the inspectors finds a defective transistor, is equal to

Cofactor of 4 in the determinant $\left|\begin{array}{ccc}1& 2& -3\\ 4& 5& 0\\ 2& 0& 1\end{array}\right|$ is equal to

If $\frac{sin(x+y)}{sin(x-y)}=\frac{a+b}{a-b}\text{, then}\frac{tanx}{tany}=$

Let A be a square matrix of order n and B be its adjoint, then for a scalar K, $|AB+K{I}_n|$ is ___

The value of $\sum _{n=0}^{100}{i}^{n!}$ equals (where $i=\sqrt{-1})$

The integral $\int \frac{e^{3{\log }_{e}2x}+5e^{2{\log }_{e}2x}}{e^{4{\log }_{e}x}+5e^{3{\log }_{e}x}-7e^{2{\log }_{e}x}}dx,x>0,$ is equal to :

(where c is a constant of integration)

General solutions of $x$ for which $2\sin x+1=0$ is :

Let $P$ and $Q$ be distinct points on the parabola $y^2=2x$ such that a circle with $PQ$ as diameter passes through the vertex $O$ of the parabola. If $P$ lies in the first quadrant and the area of the triangle $△OPQ$ is $3\sqrt{2}\text{sq. units}$, then which of the following is/are the coordinates of $P$?

If $I_1=\underset{0}{\overset{1}{\int }}{e}^{-x}{\cos }^{2}xdx$,

$I_2=\underset{0}{\overset{1}{\int }}{e}^{-x^2}{\cos }^{2}xdx$ and

$I_3=\underset{0}{\overset{1}{\int }}{e}^{-x^3}dx$ ; then :

Let $A$ and $B$ be two sets containing $4$ and $7$ elements respectively. If the minimum and maximum number of elements in $A\cup B$ are $m$ and $n$ respectively, then $m+n$ is

If ${\log }_{a}x>y$ and $a>1$. Then

Equation of the hyperbola with length of the latusrectum $\frac{9}{2}$ and e = $\frac{5}{4}$ is

Two sets given as$A=\{2,6,5,7,8,3\}$and $B=\{6,3,5,8,7,2,1\},$ then tap the bubbles having elements in $A\cap B.\$$

If 3X + 2Y = I and 2X - Y = O, where I and O are unit and null matrices of order 3 respectively, then

The weighted mean of the first n natural numbers whose weights are equal to the corresponding numbers is

If ${\sin }^{2}x$ + ${\cos }^{2}y$ = 2 ${\sec }^{2}z$, find the value of ${\cos }^{2}x$ + $si{n}^{2}y+2si{n}^{2}z$.

If $\sec x+\tan x=p$, then which of the following is/are correct?