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Let the locus of foot of the perpendicular drawn from $(1,-2)$ to the family of lines $(3+\lambda )x+(3\lambda -2)y=8-\lambda$ is $s=0$. If the line's joining origin to the point of intersections of $s=0$ with the line $x+by=1$ makes an angle of ${60}^{\circ }$ then sum of all possible values of $b$ is

The number of ways $10$ boys can be divided into $2$ groups of $5$, such that two particular boys are in the different groups, is

Solution of the differential equation
$cosxdy=y(sinx-y)dx,0<x<\frac{\pi }{2}$ is

Find the equation of the hyperbola satisfying the give conditions: Foci (0, ±13), the conjugate axis is of length 24.

Questions) If both roots of quadratic equation (2-x)(x+1)=p are distinct and positive , then find interval in which P lie

$\frac{1}{tan3A-tanA}$ -

$\frac{1}{cot3A-cotA}$ =

If A + 2I = 0, find A if the order of I is 2 $\times$ 2.

If A = $\left[\begin{array}{cc}0& 2\\ 3& 5\end{array}\right]$, B = $\left[\begin{array}{ccc}3& 4& 5\\ 2& 3& 1\\ 1& 3& 7\end{array}\right]$ then A + B =

The 10th and 18th terms of an A.P. are 41 and 73 respectively. Find 26th term.

Find the equation of the circle with centreand radius

differentiate x^x with respect to x log x

Let $S$ be the circle in the $xy$-plane defined by the equation $x^2+y^2=4$.



Let $E_1{E}_2$ and $F_1{F}_2$ be the chords of $S$ passing through the point $P_0(1,1)$ and parallel to the $x$-axis and the $y$-axis, respectively. Let $G_1{G}_2$ be the chord of $S$ passing through $P_0$ and having slope $-1$. Let the tangents to $S$ at $E_1$ and $E_2$ meet at $E_3$, the tangents to $S$ at $F_1$ and $F_2$ meet at $F_3$, and the tangents to $S$ at $G_1$ and $G_2$ meet at $G_3$. Then, the points $E_3,F_3$, and $G_3$ lie on the curve

The number of non negative integral solutions of the equation $x+y+3z=33$ 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

Differentiate the
function with respect to x.

Evaluate the definite integrals.
${\int }_0^{\frac{\pi }{4}}tanxdx.$

If the line x-y+k=0 is a normal to $y^2=4ax$ then the value of k is

Which of the following can be treated as an Elementary Row transformation.

If $0<a<b<1,$ then which of the following is true

Let $X$ follows binomial distribution with parameters $n,p$. If $P(X=3)=P(X=5)$ and $p>\frac{1}{2}$, then

Using binomial theorem evaluate each of the following:

$\left(i\right)\left(96{)}^3\right(ii\left)\right(102{)}^5\left(iii\right)\left(101{)}^4\right(iv\left)\right(98{)}^5$

The largest interval for which $x^{22}-x^{19}+x^{14}-x^5+1>0$ is

Solveis
equal to

(A)

(B).

(C)

(D)

Two trains running in opposite directions cross a man standing on the platform in $27$ seconds and $17$ seconds respectively and they cross each other in $23$ seconds. The ratio of their speeds is

Prove that sin2x+ sin2(x+π/3)+sin2(x-π/3)=3/2