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An equation $a_0+a_{1}x+a_2{x}^2+...+a_{99}{x}^{99}+x^{100}=0$ has root ${}^{99}{C}_0,{}^{99}{C}_1,{}^{99}{C}_2,...{}^{99}{C}_{99}$ then

The direction cosines of two lines satisfy the relations $\lambda (l+m)=n$ and $mn+nl+lm=0$.The value of $\lambda$, for which the two lines are perpendicular to each other, is

If the tangent to the curve $y=\frac{x}{x^2-3},x\in R,(x\ne \pm \sqrt{3},)$ at a point $(\alpha ,\beta )\ne (0,0)$ on it is parallel to the line $2x+6y-11=0,$ then :

If the AM and GM of two positive numbers a and b are in the ratio m:n show that
$a:b=(m+\sqrt{m^2-n^2}):(m-\sqrt{m^2-n^2}).$

For the in equation

If one root of the quadratic equation $a{x}^2$+bx+c=0 is equal to the nth power of the other root, then the value of $(a{c}^n{)}^{\frac{1}{n+1}}+(a^{n}c{)}^{\frac{1}{n+1}}=$

Point on the hyperbolawhich is nearest to the line 3x+2y+1=0 is

Find the value of $\theta$ satisfying $\left|\begin{array}{ccc}1& 1& sin3\theta \\ -4& 3& cos2\theta \\ 7& -7& -2\end{array}\right|$ = 0

The perpendicular distance of the point (2, 4, -1) from the line $\frac{x+5}{1}=\frac{y+3}{4}=\frac{6-z}{9}$ is

The set of all values of k for which lines kx + 2y + 2 = 0, 2x + ky + 3 = 0, 3x + 3y + k = 0 are concurrent is

Given that the events $A$ and $B$ are such that $P\left(A\right)=\frac{1}{2}$ , $P(A\cup B)=\frac{3}{5}$ and $P\left(B\right)=p$. The value of p if they are independent events is:

Write the length of the chord of the parabola $y^2=4ax$ which passes through the vertex and is inclined to the axis $\frac{\pi }{4}$.

Which of these equations have 3 as a solution?

$If\overline{A}=2i-3jand\overline{B}=-4i+2j,then|\overline{A}.\overline{B}|=$ ___

Total number of values of 'a' so that $x^2$ - x - a = 0 has integral roots, $aϵN$ & 5 < a < 99 is

Under what name does Wasim operate?

Angle between asymptotes of the hyperbola $3x^2-y^2=3$ is

If ${10}^n+3\times 4^{n+2}+\lambda$ is divisible by 9 for all $nepsilonN$, then the least positive integral value of $\lambda$ is

If $-4\le 8x-2\le 4$, then the minimum value of $\frac{1}{x^2}$ is

Find the number of permutations of n different things taken r at a time such that two specified things occur together ?

Let the equations of two sides of a triangle be $3x-2y+6=0$ and $4x+5y-20=0$. If the orthocentre of this triangle is at $(1,1)$, then the equation of its third side is :

The mean and standard deviation (s.d.) of $10$ observations are $20$ and $2$ respectively. Each of these $10$ observations is multiplied by $p$ and then reduced by $q$, where $p\ne 0$ and $q\ne 0$. If the new mean and standard deviation become half of their original values, then $q$ is equal to:

Let $a,x,b$ be in A.P.; $a,y,b$ be in G.P. and $a,z,b$ be in H.P. if $x=y+2$ and $a=5z$ then

Evaluate $\left|\begin{array}{cc}cos15& sin15\\ sin75& cos75\end{array}\right|$

If [.] denotes greatest integer function and f(x) = [x] $\left\{\frac{sin\frac{\pi }{[x+1]}+sin\pi [x+1]}{1+\left[x\right]}\right\},$ then

The differential coefficient of ${\log }_{10}x$ with respect to ${\log }_{x}10$ is

If $0<\alpha ,\beta <\pi ,\underset{x\to 0}{lim}{\left(\frac{(sin\alpha {)}^x+(sin\beta {)}^x}{2}\right)}^{\frac{1}{x}}=1$ then the value of $tan\left(\frac{\alpha +\beta }{3}\right)$ is

If $f\left(x\right)={\int }_0^x(1+t^3{)}^{-1/2}dt$ and g(x) is the inverse of f, then the value of $\frac{g^{\prime \prime }\left(x\right)}{g^2\left(x\right)}$ is

If $\cos \theta -\sin \theta =\frac{1}{5}$, where $0<\theta <\frac{\pi }{2}$

$\begin{array}{cccc}& \text{List I}& & \text{List II}\\ \text{(1)}& \frac{(\cos \theta +\sin \theta )}{2}& \left(p\right)& \frac{4}{5}\\ \text{(2)}& \sin 2\theta & \left(q\right)& \frac{7}{10}\\ \text{(3)}& \cos 2\theta & \left(r\right)& \frac{24}{25}\\ \text{(4)}& \cos \theta & \left(s\right)& \frac{7}{25}\end{array}$

Which of the following is the correct combination?

If $\tan A,\tan B$ are the roots the quadratic equation $\sqrt{3}{x}^2-2x-\sqrt{3}=0$, $0<A+B<\pi$, then $A+B$ is equal to

${lim}_{x\to 0}\frac{3^{2+x}-9}{x}$

If $f\left(x\right)=\{\begin{array}{cc}x,& \text{if}whenxisrational\\ 0,& \text{if}whenxisirrational\end{array}$;

$g\left(x\right)=\{\begin{array}{cc}0,& \text{if}whenxisrational\\ x,& \text{if}whenxisirrational\end{array}$;

then$(f-g$) is

${lim}_{x\to \infty }{a}^{x}sin\left(\frac{b}{a^2}\right),a,b>1$ is equal to

The edge of a metal cube is increasing at the rate of 1cm/sec. How fast its surface area increasing when its edges are 1cm each

$f\left(x\right)=$ $a{x}^2+1,x\le 1=x^2+ax+b,x>1$ is differentiable at x = 1, then (a, b) = _____

If a circle of radius $R$ passes through the origin $O$ and intersects the coordinate axes at $A$ and $B$, then the locus of the foot of perpendicular from $O$ on $AB$ is :

${lim}_{x\to 2}(\frac{x}{x-2}-\frac{4}{x^2-2x})$

Normals are drawn from a point P(h, k) with slopes $m_1,m_2,m_3$ to the parabola $C_1:y^2=4x$

If the curve C is the locus of point P with $m_1{m}_2=2$, then number of common tangents to the curve $C_1$ and curve C is

If $(1-y)(1+2x+4x^2+8x^3+16x^4+32x^5)=1-y^6,$ where $y\ne 1,x\ne \frac{1}{2}$, then the value of $\frac{y}{x}$ is

If the sum of n terms of the following series:

$\frac{2n+1}{2n-1}+3{\left(\frac{2n+1}{2n-1}\right)}^2+5{\left(\frac{2n+1}{2n-1}\right)}^3+\dots$is 36, then value of n is

Find the equivalent capacitance across A & B

In an ac dynamo, the number of turns in the armature are made four times and the angular velocity nine times. Then the peak value of induced emf will become.