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If the latus-rectum through one focus of a hyperbola subtends a right angle at the farther vertex,then write the eccentricity of the hyperbola.

$If(1+x{)}^n=C_0+C_{1}x+C_2{x}^2+.......C_n{x}^n....(i)$
then sum of series $C_0+C_k+C_{2k}+$....can be obtained by putting all roots of equation $x^k-1=0$ in (i) & then adding vertically:
for example: sum of $C_0+C_2+C_4$.....can be obtained by putting all roots of equation $x^2=1$
i.e. $x=\pm 1$ in (i)
At $x=1C_0+C_1+C_2..........C_n=2^n$
$x=-1C_0-C_1+C_2-C_3...........=0$
Adding we get $C_0+C_2+C_4.....=2^{n-1}$

Now answer the folloiwng

If n is a multiple of 3, then $C_0+C_3+C_6+................$ equals

Find $\frac{1}{2}(A+A^{\prime })and\frac{1}{2}(A-A^{\prime }),whenA=\left[\begin{array}{ccc}0& a& b\\ -a& 0& c\\ -b& -c& 0\end{array}\right]$.

If possible, using elementary row transformations, find the inverse of the following matrices.
$$\left[\begin{array}{ccc}2& 0& -1\\ 5& 1& 0\\ 0& 1& 3\end{array}\right]$$

Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are $\hat{i}+2\hat{j}-\hat{k}$ and $-\hat{i}+\hat{j}+\hat{k}$ respectively, in the ratio 2:1
(i) Internally

If the arithmetic and geometric mean of two positive real numbers $a$ and $b(a>b)$ are $13$ and $12$ respectively, then the value of $a-b$ is

Find r if $\left(i\right){}^5{P}_r=2{}^6{P}_{r-1}\left(ii\right){}^5{P}_r{=}^6{P}_{r-1}$

The centre of the circle passing through $(0,0)$ and $(1,0)$ and touching the circle $x^2+y^2=9$ can be

The probability of India winning a test match against West Indies is $\frac{1}{2}$. Assuming independence from match to match, the probability that in a five match series India’s second win occurs at third test is

If $(1,-2)$ is a pole of the circle $x^2+y^2-10x-10y+25=0$, then the equation of the diameter which bisects the polar is

Show that the function f(x)=|x+1|+|x-1| for all x belongs to R, is not differentiable at points x= -1 and x=1.

Which among $2^{\frac{1}{2}},3^{\frac{1}{3}},4^{\frac{1}{4}},6^{\frac{1}{6}}$ and ${12}^{\frac{1}{12}}$ is the highest?

If $\overrightarrow{a},\overrightarrow{b}$ and $\overrightarrow{c}$ are non-coplanar unit vectors equally inclined to one another at an acute angle $\theta$ and $\overrightarrow{a}\times \overrightarrow{b}+\overrightarrow{b}\times \overrightarrow{c}=p\overrightarrow{a}+q\overrightarrow{b}+t\overrightarrow{c}$, then

Let for $n>1,nϵ1,{lim}_{n\to \infty }{\int }_0^{(n+1)\frac{\pi }{2}}\left(\frac{\cos x+x\sin x}{x^2+{\cos }^{2}x}\right)dx=l$ and ${lim}_{x\to \infty }\left(x^{2}ln\right(x{\cot }^{-1}x\left)\right)=m$, then

Let $\Delta PQR$ be a triangle. Let a = QR, b = RP and c = PQ. If $|a|=12,|b|=4\sqrt{3}$ and $b.c=24,$ then which of the following is/are true ?

If $△ABC$ is an acute angle triangle, then the minimum value of $\tan A+\tan B+\tan C$ is

$f\left(x\right)$ is a function defined on entire number line and is even and odd at the same time. Find the value of $f\left(10\right)\times f\left(5\right)$.

If $\alpha$ and $\beta$ are the eccentric angles of points of contract of tangents drawn from (3,2) to the ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$ then $|tan\left(\frac{\alpha -\beta }{2}\right)|=$

Let, $f\left(x\right)=x^2+6x+c,c\in R.$ If $f\left(f\right(x\left)\right)=0$ has exactly three distinct real roots, then the value of $c$ can be

If $x=sec\theta -cos\theta ,y=se{c}^{10}\theta -co{s}^{10}\theta$ and $(x^2+4){\left(\frac{dy}{dx}\right)}^2=k(y^2+4)$, then k is equal to

If $\frac{{\sin }^4\alpha }{{\sin }^2\beta }+\frac{{\cos }^4\alpha }{{\cos }^2\beta }=1$, then

Explain various methods for the treatment of goodwill on the admission of a new partner.

Two fair dice are rolled simultaneously. One of the dice shows four. The probability of other dice showing six, is equal to

A line segment joining (1, 0, 1) and the origin (0, 0, 0) is revolved about the x-axis to form a right circular cone. If (x, y, z) is any point on the cone other than the origin, then it satisfies the equation

If $|z-1|\le 2$ and $|wz-1-w^2|=a$ (where w is a cube root of unity) then complete set of values of a is

The distance of the point $(2,3)$ from the line $2x-3y+9=0$ measured along a line $x-y+1=0$ is

Question 2 (iii)

If p = -2, find the value of:
$-2p^3-3p^2+4p+7$

The coefficient of$x^7$ in $(1-x-x^2+x^3{)}^6$ is

If f(x) is differential function satisfying 6. ${\int }_0^{1}f\left(t\right)dt=2x^3-3x^2+6x+5,$ then

If the maximum and the minimum values of $\left|\begin{array}{ccc}1+{\sin }^{2}x& {\cos }^{2}x& 4\sin 2x\\ {\sin }^{2}x& 1+{\cos }^{2}x& 4\sin 2x\\ {\sin }^{2}x& {\cos }^{2}x& 1+4\sin 2x\end{array}\right|$ are $M$ and $m$ respectively, then $\frac{M}{m}$ is

From the following, find the correct relation [MP PET 1990]

The value of $co{s}^{-1}\left(cos\frac{3\pi }{2}\right)$ is

(a) $\frac{\pi }{2}$ (b) $\frac{3\pi }{2}$ (c) $\frac{5\pi }{2}$ (d) $\frac{7\pi }{2}$

If the vertex of the conic $y^2-4y=4x-4a$ always lies between the straight lines $x+y=3$ and $2x+2y-1=0$ then

If $\alpha ,\beta$ are the roots of the equation $2x^2+3x+4=0,find\frac{\alpha }{\beta }+\frac{\beta }{\alpha }$

If one root of the determinant then the other two roots are

Let $\overrightarrow{a}$ and $\overrightarrow{b}$ be two vectors of equal magnitude $5$ uints. Let $\overrightarrow{p},\overrightarrow{q}$ be vectors such that $\overrightarrow{p}=\overrightarrow{a}-\overrightarrow{b}$ and $\overrightarrow{q}=\overrightarrow{a}+\overrightarrow{b}$. If $|\overrightarrow{p}\times \overrightarrow{q}|=2\{\gamma -(\overrightarrow{a}.\overrightarrow{b}{)}^2{\}}^{1/2}$, then the value of $\gamma$ is

If $\int \frac{2dx}{\left(\right(x-5)+(x-7\left)\right)\sqrt{(x-5)(x-7)}}=f\left(g\right(x\left)\right)+c$, then

Give the journal entry for the treatment of partner's loan appearing on the assets side of the balance sheet, on dissolution of a partnership firm.

Find the equation of the perpendicular to the line segment joining (4, 3) and (-1, 1) if it cuts off an intercept -3 from y-axis.