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The sum of $\sum _{r=0}^n(-1{)}^r\frac{{}^n{C}_r}{{}^{r+2}{C}_r}$ is

Let $a_1,a_2,a_3,\dots$ be an A.P. such that $a_3+a_5+a_8=11$ and $a_4+a_2=-2$. Then the value of $a_1+a_6+a_7$ is

Which of the following relation in x and y is general solution of the differential equation $\frac{dy}{dx}=y$

If $A^3=0$, then $I+A+A^2$ equals

If $f\left(x\right)={\log }_e\left(\frac{1-x}{1+x}\right),|x|<1$, then $f\left(\frac{2x}{1+x^2}\right)$ is equal to:

The sum of two positive numbers is $6$ times their geometric mean. Then the ratio of these two numbers is

6 women and 5 men are to be seated in a row so that no 2 men can sit together. Number of ways they can be seated is

Let $\int \frac{dx}{x^{2008}+x}=\frac{1}{p}ln\left(\frac{x^q}{1+x^r}\right)+C$ where $p,q,rϵN$ and need not be distinct, then the value of (p+q+r) equals

The power set of $C=\left\{\varphi \right\}$ is

Let $g\left(x\right)={\int }_0^{x}f\left(t\right)dt$ where $\frac{1}{2}\le f\left(t\right)\le 1,tϵ[0,1]$ and $0\le f\left(t\right)\le \frac{1}{2}$ for $tϵ(1,2)$, then [IIT Screening 2000]

The ratio in which the line joining points (1,-2,3) and (4,2,-1) is divided by XOY plane is

If $f\left(x\right)=2\left[x\right]+\cos ([-\pi ]x)$, where $[.]$ represents greatest integer function, then

Let $f\left(x\right)=\{\begin{array}{cc}x\left[\frac{1}{x}\right]+x\left[x\right]& \text{if}x\ne 0\\ 0& \text{if}x=0\end{array}$ where $[\cdot ]$ denotes the greatest integer function, then which of the following is true?

If $f\left(x\right)=ax+b$, where $a$ and $b$ are integers, $f(–1)=–5$ and $f\left(3\right)=3$, then $a$ and $b$ are equal to , respectively.

A series of concentric ellipses $E_1,E_2,\dots ,E_n$ are drawn such that $E_n$ touches the extremities of the major axis of $E_{n-1}$ and the foci of $E_n$ coincide with the extemities of minor axis of $E_{n-1}.$ If the eccentricites of the ellipse are independent of $n,$ then the value of the eccentricity, is

If a set $Y$ is a singleton set, then $n\left(Y\right)=$

Sum of solutions of $x^2-\sqrt{x^2}=|x-1|+5$ is

Let $\alpha ,\beta$ be the roots of $a{x}^2+bx+c=0,a\ne 0$ and ${\alpha }_1,-\beta$ be the roots of $a_1{x}^2+b_{1}x+c_1=0,a_1\ne 0$. Then the quadratic equation whose roots are $\alpha ,{\alpha }_1$ is

Which of the following function is a monotonically increasing function?

A normal to the hyperbola, $4x^2-9y^2=36$ meets the co-ordinate axes $x$ and $y$ at $A$ and $B$, respectively. If the parallelogram $OAPB$ ($O$ being the origin) is formed, then the locus of $P$ is :

The number of all the possible triplets $(a_1,a_2,a_3)$ such that $a_1+a_{2}cos\left(2x\right)+a_{3}si{n}^2\left(x\right)=0$ for all x is

If the sum of $n$ terms of the series $5+7+13+31+85+\dots$ is $\frac{1}{2}(a^n+bn+c)$, then

$\underset{n\to \infty }{lim}\sum _{r=1}^{n}co{t}^{-1}(r^2+\frac{3}{4})$ is

A variable plane which remains at a constant distance p from the origin cuts the coordinate axes in A, B, C. The locus of the centroid of the tetrahedron OABC is $y^2{z}^2+z^2{x}^2+x^2{y}^2=k{x}^2{y}^2{z}^2$, where k is equal to

If the curve $y=|x-3|$ touches the parabola $y^2=\lambda (x-4),\lambda >0,$ then length of latus rectum of the parabola(in units) is

The area of the triangle formed by the tangent and the normal to the parabola $y^2=4ax$, both drawn at the same end of the latus rectum, and the axis of the parabola is

The number of common tangents to the circles $x^2+y^2+2x+8y-23=0and{x}^2+y^2-4x-10y+19=0$ is

If $y={\left(\cos x\right)}^{x^2}$, then $\frac{dy}{dx}$ is equal to



यदि $y={\left(\cos x\right)}^{x^2}$, तब $\frac{dy}{dx}$ का मान है

In an examination, out of $100$ students, $75$ passed in English, $60$ passed in Mathematics and $45$ passed in both English and Mathematics. The number of students who passed in none of the two subjects, is

(Assume all students gave both the exams)

The length of major axis of the ellipse $(5x-10{)}^2+(5y+15{)}^2=\frac{(3x-4y+7{)}^2}{4}$ is

The value of $2c_0+\frac{2^2}{2}{C}_1+\frac{2^3}{3}{C}_2+\frac{2^4}{4}{C}_3+....+\frac{2^{11}}{11}{C}_{10}is$

If the line

$\frac{x-x_1}{a_1}=\frac{y-y_1}{b_1}=\frac{z-z_1}{c_1}$

makes an angle $\theta$ with the plane $a_{2}x+b_{2}y+c_{2}z=d$, then which of the following is correct -

If in a frequency distribution, the mean and median are 21 and 22 respectively, then its mode is approximately

A man walks a distance of 3 units, from the origin towards the north - east $\left(N{45}^{\circ }E\right)$ direction. From there, he walks a distance of 4 units towards the north - west $\left(N{45}^{\circ }E\right)$ direction to reach a point P. Then the position of P in the Argand plane is:

The eccentricity of the hyperbola whose length of the latus rectum is equal to 8 and the length of its conjugate axis is equal to half of the distance between its foci, is___ .

If from a point $P,$ tangents $PQ$ and $PR$ are drawn to the ellipse $\frac{x^2}{2}+y^2=1,$ such that equation of $QR$ is $x+3y=1,$ then the coordinates of $P$ is

The lines joining the origin to the point of intersection of $3x^2+mxy-4x+1=0$ and $2x+y-1=0$ are at right angles. Then all possible values of $m$ lie in the interval

The straight lines $3x+4y=5$ and $4x-3y=15$ intersect at the point $A$. On these lines points $B$ and $C$ are chosen so that $AB=AC$.Possible equation(s) of the line $BC$ passing through $(1,2)$ is/are

The value of

$(\frac{30}{0})(\frac{30}{10})-(\frac{30}{1})(\frac{30}{11})+(\frac{30}{2})(\frac{30}{12})...+(\frac{30}{20})(\frac{30}{30})$ is; where $(\frac{n}{r}){=}^n{C}_r$

If $x_1,x_2,x_3,....,x_n$ are in A.P. whose common difference is $\alpha$, then the value of $sin\alpha (sec{x}_{1}sec{x}_2+sec{x}_{2}sec{x}_3+......+sec{x}_{n-1}sec{x}_n)=$

The coordinates of the vertex of a parabola represented by $y=a{x}^2+bx+c$ is, . Take D as discriminant;

Vertex of the parabola formed by taking reflection of $y=4x^2-4x+3$ along the line $y=x$ will be

Let $PQ$ be a chord of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ which subtends right angle at the centre $(0,0).$ Then its distance from the centre is equal to

If $xϵR$ and $m=\frac{x^2}{(x^4-2x^2+4)}$, then m lies in the interval

Let,

$A=\left[\begin{array}{ccc}4& 6& -1\\ 3& 0& 2\\ 1& -2& 5\end{array}\right],B=\left[\begin{array}{cc}2& 4\\ 0& 1\\ -1& 2\end{array}\right]$

$\text{and}C=\left[\begin{array}{ccc}1& 2& 3\end{array}\right]$

The expression which is not defined is:

If $\alpha ,\beta$ are the solutions of $\cot x=-\sqrt{3}$ in $[0,2\pi ]$ and $\alpha ,\gamma$ are the solutions of $\text{cosec}x=-2$ in $[0,2\pi ],$ then