Explore topic-wise InterviewSolutions in .

If $co{s}^{2}A+co{s}^{2}C=si{n}^{2}B,$ then $△$ ABC is

[MP PET 1991]

The circle $x^2+y^2=1$ cuts the x-axis at P and Q. Another circle with centre at Q and variable radius intersects to first circle at R above the X-axis and the line segment PQ at S. The maximum area of the triangle QSR is

An ellipse has OB as a semi-minor axis. F and F' are its foci and the angle FBF' is a right angle. Then, the eccentricity of the ellipse is ___

The value of ${\tan }^{-1}\sqrt{\frac{a(a+b+c)}{bc}}+{\tan }^{-1}\sqrt{\frac{b(a+b+c)}{ca}}+{\tan }^{-1}\sqrt{\frac{c(a+b+c)}{ab}},$ where $a,b,c>0,$ is

If the $5^{th}$ term of a G.P. is $\frac{1}{3}$ and $9^{th}$ term is $\frac{16}{243}$, then the $4^{th}$ term will be

If p is the length of perpendicular from the origin to the line whose intercepts on the axes are a and b, then show that $\frac{1}{p^2}=\frac{1}{a^2}+\frac{1}{b^2}$

$\underset{x\to \infty }{lim}\frac{{\cos }^{2}x}{x}=$

The equation of circle which passes through focus of parabola $x^2=4y$ and touches it at $(6,9)$ is

If $\sin B=\frac{1}{5}\sin (2A+B),$ then $\frac{tan(A+B)}{tanA}$ is equal to

$$\begin{array}{cc}TrigonometricEquations& GeneralSolutions\\ 1.tanx=2& P.2n\pi \pm \frac{2\pi }{3},n\in I\\ 2.sinx=\frac{\sqrt{3}}{2}& Q.n\pi -\frac{\pi }{4},n\in I\\ 3.cosx=\frac{-1}{2}& R.n\pi +(-1{)}^n\frac{\pi }{3},n\in I\\ 4.cotx=-1& S.n\pi +ta{n}^{-1}\left(2\right),n\in I\end{array}$$

If (1+i)(1+2i)(1+3i)......(1+ni) = a+ib, then 2.5.10.....(1+$n^2$) is equal to

The orthocentre of the triangle formed by the vertices $(5,0),(0,0)$ and $(\frac{5}{2},\frac{5\sqrt{3}}{2})$ is

Two parabolas $x^2=4y$ and $y^2=4x$ intersect at two distinct points out of which one of them is origin then the other point will be

If ${}^{\prime }abc{d}^{\prime }$ is a $4$ digit number where $a>b\ge c>d$, then the number of such numbers are

There are 10 points in a plane no three of which are in the same straight line, excepting 4 points, which are collinear.

Find the probability that

(i) number of straight lines obtained from the pairs of these points.

(ii) number of triangles that can be formed with the vertices as these points.

If sin B= $\frac{1}{5}sin(2A+B),then\frac{tan(A+B)}{tanA}$ is equal to

$\underset{x\to \frac{\pi }{2}}{lim}\frac{(1-tan\left(\frac{x}{2}\right))(1-sinx)}{(1+tan\left(\frac{x}{2}\right))(\pi -2x{)}^3}is$

If truth value for $p\to (q\vee r)$ is false, then the truth values of $p,q,r$ are respectively :

Arun wants to predict the runs India will score in the next ball. The present score is 100 runs in 25 overs and he wants to predict the score when it is 25 overs and 1 ball. What is the sample set? (India is batting first)

The sum of co-efficients of all even degree terms in $x$ in the expansion of $(x+\sqrt{x^3-1}{)}^6+(x-\sqrt{x^3-1}{)}^6,(x>1)$ is equal to:

If ${}^n{C}_3{+}^n{C}_4$>${}^{n+1}{C}_3,$then,

If $\sin \theta =n\sin (\theta +2\alpha )$, then the value of $\tan (\theta +\alpha )$ is

(where $n$ is a constant)

For all natural numbers n, ${10}^{2n-1}+1$ is divisible by

If cos (x-y) cos(z-t) = cos(x+y) cos (z+t), then tanx tany + tanz tant is equal to

If $\alpha ,\beta$ are zeroes of a polynomial $6x^2+x–2$, then the polynomial whose zeroes are $2\alpha +3\beta$ and $3\alpha +2\beta$ , is .

The value of the expression ${\tan }^{-1}\left(\frac{\sqrt{2}}{2}\right)+{\sin }^{-1}\left(\frac{\sqrt{5}}{5}\right)-{\cos }^{-1}\left(\frac{\sqrt{10}}{10}\right),$ is

If a, b, c, d are in GP, prove that

$(a^2+b^2+c^2)(b^2+c^2+d^2)=(ab+bc+cd{)}^2$

The rational term(s) in the expansion of ${(\sqrt{2}+(3{)}^{\frac{1}{5}})}^{10}$ is/are

$\underset{h\to 0}{lim}\{\frac{1}{h\sqrt[3]{38+h}}-\frac{1}{2h}\}\text{is equal to}$

In $△ABC,$ if $A,B$ and $C$ represent the angles of a triangle, then the maximum value of $\cos A+\cos B+\cos C$ is

The side length of the square whose area is equal to the $7^{th}$ term of the sequence $2,\sqrt{8},4,\dots ,$ is

If the arithmetic mean and harmonic mean between two numbers are $27$ and $12$ respectively, then the geometric mean of those two numbers is

If $\int \frac{\cos x}{{\sin }^{3}x(1+{\sin }^{6}x{)}^{2/3}}dx=f\left(x\right)(1+{\sin }^{6}x{)}^{1/\lambda }+c$, where $c$ is a constant of integration, then $\lambda f\left(\frac{\pi }{3}\right)$ is equal to :

Which of the following statements is/are true about an even function f(x)?

Find the general solution of each equation:

$\left(i\right)\sqrt{3}cotx+1=0$

$\left(ii\right)cosecx+\sqrt{2}=0$

Find the value of ${\sum }_{i=1}^{\infty }$ ${\sum }_{j=1}^{\infty }$ ${\sum }_{k=1}^{\infty }$ $\frac{1}{2^i{2}^j{2}^k}$.

Given 3 points with position vectors $\overline{p_1},\overline{p_2}and\overline{p_3}$ which form the vertices of a triangle with side lengths $a=|\overline{p_2}-\overline{p_1}|,b=|\overline{p_3}-\overline{p_2}|,c=|\overline{p_1}-\overline{p_3}|$. Then the in-centre is given by

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

From the top of a building $21\text{m}$ high, the angle of elevation and depression of the top and the foot of another buillding are ${30}^{\circ }$ and ${45}^{\circ }$ respectively. The height of the second building is

If $(2,1,3),(3,2,5),(1,2,4)$ are the mid points of the sides $BC,CA,AB$ of $△ABC$ respectively, then vertex $A$ is:

If $f\text{'}\left(x\right)=|x|-\left\{x\right\}$ where $x$ denotes the fractional part of $x$, then $f\left(x\right)$ is decreasing in

Let $(a,b)$ be a point on a circle which passes through $(-3,1)$ and touches the line $x+y=2$ at the point $(1,1).$ If maximum possible value of $a$ is $\alpha ,$ then a quadratic equation with rational coefficients whose one root is $\alpha ,$ is

The set of values of $m$ for which $f\left(x\right)=x^2-(m-3)x+m$ intersects the positive direction of $x-$axis atleast once, is