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If the line $y=mx+7\sqrt{3}$ is normal to the hyperbola $\frac{x^2}{24}-\frac{y^2}{18}=1$, then a value of $m$ is

If P = (1, 0), Q = (-1, 0) and R = (2, 0) are three given points, then the locus of a point S satisfying the relation

$S{Q}^2+S{R}^2=2S{P}^2$ is

Let the median of $21$ observations is $40.$ If the observations greater than the median are increased by $6$, then the median of new data will be

Let $P$ be a square matrix satisfying $P^2=I-P,$ where $I$ is identity matrix. If $P^n=5I-8P,$ then the value of $n$ is

A variable circle passes through the fixed point $A\left(p,q\right)$ and touches $X-axis$. The locus of the other end of the diameter through $A$ is

Let $f:R\to R$ be a function defined by $f\left(x\right)=x^3+x^2+x-1.$ If $g$ is the inverse of $f,$ then $g^{\prime }\left(2\right)$ is

If set A has p element and set B has q elements, then the number of element in set (A $\times$ B) is _____.

The number of terms in the expansion of $(x^3+9x^2+27x+27{)}^{25}$ is

If $\sin \theta +\sin \varphi =a$ and $\cos \theta +\cos \varphi =b$, $(a\ne b,a\ne 0,b\ne 0)$ then

The coefficient of $\frac{1}{x}$ in the expansion of $(1+x{)}^n{(1+\frac{1}{x})}^n$ is

If A is a square matrix of order 4 and the value of |A| is equal to 2. Then the value of |Adj(A)| is,

If the number of terms in the expansion of $(x+y+z{)}^n$ is $231$, then the value of $n$ is

Two sets $A=\{a,b,c,d\}$ and $B=\{b,c,d,x\}$ are equal sets, iff $x=$

In the circle given below, let $OA=1$ unit, $OB=13$ unit and $PQ\perp OB.$ Then, the area of the triangle $PQB$ (in square units) is :

For $x\in R-\left\{b\right\},$ if $y=\frac{(x-a)(x-c)}{x-b}$ will assume all real values, then

$li{m}_{x\to 0}$ $\frac{\left(x^{3}cotx\right)}{1-cosx}$ =





[AI CBSE ; DSSE 1988]

The sum of the series $1-3x+5x^2-7x^3+\dots \infty$, when $|x|<1$ is

The distance between the points $(a\sin {30}^{\circ },0)$ and $(0,a\cos {60}^{\circ })$ is

The planes x-cy-bz=0, cx-y+az=0 and bx+ay-z=0 pass through a straight line, where a,b,c are non-zero constants. Then the value of $a^2+b^2+c^2+2abc$ is

The harmonic conjugate of the point $R(2,4)$ with respect to the points $P(2,2)$ and $Q(2,5)$ is

Let $E_1$ and $E_2$ be two ellipse whose centres are at the origin. The major axes of $E_1$ and $E_2$ lie along x-axis and y-axis respectively. Let S be the circle $x^2+(y-1{)}^2=2$ the straight line $x+y=3$ touches the curves S, $E_1$ and $E_2$ at P, q and R, respectively.

Suppose that $PQ=PR=\frac{2\sqrt{2}}{3}.$ If $e_1$ and $e_2$ are the eccentricities of $E_1$ and $E_2$ respectively, then the correct expression(s) is/are

If $z$ be a complex number satisfying $|Re\left(z\right)|+|Im\left(z\right)|=4$, then $|z|$ cannot be:

If coefficient of $(2r+3{)}^{th}$ and $(r-1{)}^{th}$ terms in the expansion of $(1+x{)}^{15}$ are equal, then value of r is

Solve $|x-2|\ge 5$

Value of $C_0+2C_1+3C_2+4C_3+\dots +(n+1)C_n$ is

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( where $C_r={}^n{C}_r$)

Out of the $200$ residents of the colony, $120$ can speak Hindi, $60$ can speak English and $50$ can speak Tamil. $10$ residents can speak both Hindi and English, $15$ residents can speak English and Tamil, while $15$ residents can speak both Hindi and Tamil. $10$ residents can speak all three languages. find the number of residents who can speak Tamil but not English.

Number of points of intersection of diagonals of polygon with $2009$ sides which are situated inside the polygon.

The value of $x$ if ${\log }_{10}{x}^2-7x-6=1-{\log }_{10}5$

The equation of the ellipse with foci at $(\pm 5,0)$ and $x=\frac{36}{5}$ as one directrix, is

The normal at a point $P$ on the ellipse $x^2+4y^2=16$ intersects the $x$-axis at $Q$. If $M$ is the midpoint of the line segment $PQ$, then the locus of $M$ intersects the latus rectum of the given ellipse at points

If $\tan \theta =\frac{\sin \alpha -\cos \alpha }{\sin \alpha +\cos \alpha };(0<\theta <\pi )\text{then}\sin \alpha +\cos \alpha and\sin \alpha -\cos \alpha$

must be equal to

Let n be a positive integer such that $sin\frac{\pi }{2^n}+cos\frac{\pi }{2^n}=\frac{\sqrt{n}}{2}$. Then

Consider a point given by position vector $\overline{a}.$ Distance of this point from the plane given $\overline{r}.\overline{n}=d$ will be.

If $3x+4y=12\sqrt{2}$ is a tangent to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{9}=1$ for some $a\in R$, then the distance between the foci of the ellipse is :

The coefficient of $x^7$ in the expression $(1+x{)}^{10}+x(1+x{)}^9+x^2(1+x{)}^8+\dots +x^{10}$ is :

If $(n+1)!=12(n-1)!$, then the value of $n$ is

Conjugate of z is the mirror image of z along _______ on the Argand plane.

The sum of $\frac{3}{1\cdot 2}\left(\frac{1}{2}\right)+\frac{4}{2\cdot 3}{\left(\frac{1}{2}\right)}^2+\frac{5}{3\cdot 4}{\left(\frac{1}{2}\right)}^3+\cdots$ upto $n$ terms is

Number of ordered pairs (x,y) satisfying |y|=cosx, y=$si{n}^{-1}\left(sinx\right),x\in [-2\pi ,3\pi ]$ is

If $B=\left[\begin{array}{ccc}5& 2\alpha & 1\\ 0& 2& 1\\ \alpha & 3& -1\end{array}\right]$ is the inverse of a $3\times 3$ matrix $A,$ then the sum of all values of $\alpha$ for which $det\left(A\right)+I=0,$ is

Let A and B be two sets such that $n\left(A\right)=20,n(A\cup B)=42\text{and}$

$n(A\cap B)=5,$ then $n(B-A)=$

Q(h, k) is the foot of the perpendicular of P(3, 6) on the line x - 2y + 4 =0.

If the slope of PQ is m, find $m^2$.