Explore topic-wise InterviewSolutions in .

If the function $f\left(x\right)=2x^3-9a{x}^2+12a^{2}x+1,$ where $a>0,$ attains its local maximum and local minimum at $p$ and $q$ respectively such that $p^2=q,$ then $a$ is equal to

If the fractional part of the number $\frac{2^{403}}{15}$ is $\frac{k}{15}$, then $k$ is equal to :

Find the term independent of x in the expasion of the following expressions:

$\left(i\right){(\frac{3}{2}{x}^2-\frac{1}{3x})}^9\left(ii\right){(2x+\frac{1}{3x^2})}^9\left(iii\right){(2x^2-\frac{3}{3x^3})}^{25}\left(iv\right){(3x-\frac{2}{2x^2})}^{15}\left(v\right)(\sqrt{\frac{x}{3}}+{\frac{3}{2x^2}}^{10})\left(vi\right){(x-\frac{1}{x^2})}^{3n}\left(vii\right){(\frac{1}{2}{x}^{1/3}+x^{-1/5})}^8\left(viii\right)(1+x+2x^3){(\frac{3}{2}{x}^2-\frac{1}{3x})}^9\left(ix\right){(\sqrt[3]{3x}+\frac{1}{2\sqrt[3]{3x}})}^{18},x>2\left(x\right){(\frac{3}{2}{x}^2-\frac{1}{3x})}^6$

The integrating factor of the differential equation $\frac{dy}{dx}(1+x)-xy=1-x$ is

Six boys and six girls sit in a row alternatively in x ways and at a round table (again alternatively) in y ways. Then

Two sides of a parallelogram are along the lines $4x+5y=0$ and $7x+2y=0$. If the equation of one of the diagonals of the parallelogram is $11x+7y=9$, then other diagonal passes through the point

If $\theta =178°$, then the value of $\sin \theta \sqrt{1+{\cot }^2\theta }+\cos \theta \sqrt{1+{\tan }^2\theta }$ is

If abscissae and ordinates of the points $A(x_1,y_1)$ and $B(x_2,y_2)$ are the roots of the quadratic equation $x^2-x-1=0$ and $y^2-2y=0$ respectively, then the distance $AB$(in units) is

If $cos(x-y),cosx,cos(x+y)$ are three distinct numbers which are harmonic progression and $cosx\ne cosy$ then $1+cosy=$

If the standard deviation of the numbers 2, 3, a and 11 is 3.5, then which of the following is true?

Let $\overrightarrow{a}=\hat{i}+\hat{j};\overrightarrow{b}=2\hat{i}-\hat{k}$. Then, vector $\overrightarrow{r}$ satisfying the equations $\overrightarrow{r}\times \overrightarrow{a}=\overrightarrow{b}\times \overrightarrow{a}$ and $\overrightarrow{r}\times \overrightarrow{b}=\overrightarrow{a}\times \overrightarrow{b}$ is

solve the inequality $cosx\le -\frac{1}{2}$

If $A=\left[\begin{array}{ccc}i& 0& 0\\ 0& i& 0\\ 0& 0& i\end{array}\right]$,where $i=\sqrt{-1},$ then $A^{4n+1}$ is:

($n\in N$)

If p + q + r = a + b + c = 0, then the determinant
$\Delta =\left|\begin{array}{ccc}pa& qb& rc\\ qc& ra& pb\\ rb& pc& qa\end{array}\right|$ equals

Find $\frac{dy}{dx}$in the following questions:

$y=co{s}^{-1}\left(\frac{1-x^2}{1+x^2}\right),0<x<1.$

If $\frac{2\pi }{3}<\alpha <\pi$, then the distance between the points $(\sin \alpha ,0)$ and $(0,\cos \alpha )$ is

Let $\overrightarrow{a}$ and $\overrightarrow{b}$ be two vectors such that $|\overrightarrow{a}|=1,$ $|\overrightarrow{b}|=4$ and $\overrightarrow{a}\cdot \overrightarrow{b}=2.$ If $\overrightarrow{c}=(2\overrightarrow{a}\times \overrightarrow{b})-3\overrightarrow{b},$ then the angle between $\overrightarrow{b}$ and $\overrightarrow{c}$ is

Total number of values in $(-2\pi ,2\pi )$ and satisfying

$lo{g}_{|cosx|}|sinx|+lo{g}_{|sinx|}|cosx|=2$ is

The second derivative of the function $f\left(e^x\right)$ with respect to $x$ is

The locus of the midpoints of all chords of the parabola $y^2=4ax$ through its vertex is another parabola with directrix

Art the foot of a mountain, the angle of elevation of its summit is ${45}^{\circ }$. After ascending 1 km towards the mountain up an incline of ${30}^{\circ }$, the elevation changes to ${60}^{\circ }$ (as shown in the given figure). Fin dthe height of the mountain. [Given : $\sqrt{3}=\mathrm{1.73.}]$

The true angle of dip at a given place on earth is equal to ${53}^{\circ }$. If the plane of dip circle is at an angle of ${60}^{\circ }$ with the magnetic meridian, then the apparent angle of dip will be (Take $\tan {53}^{\circ }=\frac{4}{3})$



$[$0.77 Mark$]$

Let $f\left(x\right)=\sqrt{x+4}$. Then the point $c$ that satisfies the mean value theorem for the function on the interval $[0,5]$, is

Let $f$ be a real valued function satisfying $f(x+y)=f\left(x\right)+f\left(y\right)$ for all $x,y.$ If $f\left(1\right)=\frac{1}{2}$, then the value of $f\left(16\right)$ is

Show that function f: R →
{x ∈ R: −1
< x < 1} defined by f(x) =,
x ∈R is
one-one and onto function.

If $\alpha ,\beta \in I^{+}$ are the roots of the equation $x^2+ax+b=0,$ where $a=-\frac{i}{\pi }\times \ln i$ and $b\in R,$ then the number of possible pairs of $(\alpha ,\beta )$ is/are

Ashu studies at Byju's classes and her probability of selection in IIT-JEE is $\frac{4}{5}$. Ridhima took coaching at FIIT-JEE and the probability of her selection is $\frac{2}{3}$. What is the probability that only 1 of them cracks the Exam?

The equation of the plane which contains the line $\frac{x}{4}=\frac{y}{2}=\frac{z}{1}$ and is perpendicular to the plane containing the lines $\frac{x-4}{1}=\frac{y-5}{4}=\frac{z-9}{2}$ and $\frac{x+8}{4}=\frac{y+6}{2}=\frac{z-2}{1}$