Explore topic-wise InterviewSolutions in .

1. -52

If $\int \frac{\sin 2x}{a^2{\cos }^{2}x+b^2{\sin }^{2}x}dx=k\cdot \ln |a^2{\cos }^{2}x+b^2{\sin }^{2}x|+C,$ where $a\ne b$, then $k$ is equal to

(where $C$ is integration constant)

Let $f_p\left(\alpha \right)=e^{\frac{i\alpha }{p^2}}.e^{\frac{2i\alpha }{p^2}}.e^{\frac{3i\alpha }{p^2}}.e^{\frac{4i\alpha }{p^2}}\dots e^{\frac{i\alpha }{p}}$, (where $i=\sqrt{-1}$ and $p\in N$) then ${lim}_{n\to \infty }{f}_n\left(\pi \right)$ is

The number of common tangents to the circles $x^2+y^2+6x+6y+14=0$ and $x^2+y^2-2x-4y-4=0$ is

sin x sin 2x
sin 3x

The value of $co{s}^2(\frac{\pi }{6}+\theta )-si{n}^2(\frac{\pi }{6}-\theta )$ is

Evaluate the determinants.

$\left|\begin{array}{cc}{x}^2-x{+}_1& x-1\\ x+1& x+1\end{array}\right|$

Which of the following is/are true?

(1) $(1+i{)}^n$ = $2^{\frac{n}{2}}$$(sin\frac{n\pi }{4}+icos\frac{n\pi }{4})$

If $(1+i{)}^n$ = ${}^n{C}_0$ + ${}^n{C}_1$i - ${}^n{C}_2$ - ${}^n{C}_3$i + ${}^n{C}_4$...............

(2) ${}^n{C}_0$ - ${}^n{C}_2$ + ${}^n{C}_4$ - ${}^n{C}_6$................ = $2^{\frac{n}{2}}$ $sin\frac{n\pi }{4}$

(3) ${}^n{C}_1$ - ${}^n{C}_3$ + ${}^n{C}_5$ - ${}^n{C}_7$....................= $2^{\frac{n}{2}}cos\frac{n\pi }{4}$

Find the product of the roots of the equation $\left|x^2\right|-4\left|x\right|-21=0$

Semi latus rectum of the parabola $y^2=4ax$, is the _____ mean between segments of any focal chord of the parabola.

Let $f\left(x\right)=e^{ax}+e^{bx},$ where $a\ne b$ and $f^{\prime \prime }\left(x\right)-2f^{\prime }\left(x\right)-15f\left(x\right)=0$ for all $x\in R$. Then the product $ab$ is equal to

The equation has a positive slope and a negative

y-intercept.

The normal
to the curve x² = 4y passing (1, 2) is

(A) x
+ y = 3 (B) x − y = 3

(C) x
+ y = 1 (D) x − y = 1

Find the value of
${\cos }^{-1}\frac{3}{5}+{\cos }^{-1}\frac{5}{13}.$

If $\cos \alpha +\cos \beta +\cos \gamma =\sin \alpha +\sin \beta +\sin \gamma =0,$ then $\cos (2\alpha -\beta -\gamma )+\cos (2\beta -\gamma -\alpha )+\cos (2\gamma -\alpha -\beta )=$

Let $\overrightarrow{a}=\hat{i}+\hat{j}$ and $\overrightarrow{b}=2\hat{i}-\hat{k}$ then the point of intersection of the line $\overrightarrow{r}\times \overrightarrow{a}=\overrightarrow{b}\times \overrightarrow{a}$ and $\overrightarrow{r}\times \overrightarrow{b}=\overrightarrow{a}\times \overrightarrow{b}$

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

$y=si{n}^{-1}\left(2x\sqrt{1-x^2}\right),-\frac{1}{\sqrt{2}}<x<\frac{1}{\sqrt{2}}$

Let $c_1\&c_2$ be the centers and $r_1\&r_2$ be the radius of two circles. Then

Cases Conditions

p. 1.$|r_1-r_2|<c_1{c}_2<r_1+r_2$

q. 2. $|r_1-r_2|=c_1{c}_2$

r. 3.$c_1{c}_2<|r_1+r_2|$

s.

4. $r_1+r_2<c_1{r}_2$

t. 5. $r_1+r_2=c_1{r}_2$

The line $x-2y=0$ will be a bisector of the angle between the lines represented by the equation $x^2-2hxy-2y^2=0$, if $h=$

What is the number of ways of choosing 6 cards from a pack of 52 playing cards?

If a chord which is normal to the parabola $y^2=4ax$ at one end subtends a right angle at the vertex, then its slope can be

Show that:
(i) $sin{50}^{\circ }cos{85}^{\circ }=\frac{1-\sqrt{2}sin{35}^{\circ }}{2\sqrt{2}}$
(ii) $sin{25}^{\circ }cos{115}^{\circ }=\frac{1}{2}(sin{140}^{\circ }-1)$

If $f\left(x\right)=\prod _{n=1}^{10}(x+n{)}^{(20+n)},$ then $\frac{f\left(20\right)}{f^{\prime }\left(20\right)}=$

The equation of the normal to the curve $y={(1+x)}^{2y}+{\cos }^2\left({\sin }^{-1}x\right)$ at $x=0$ is

The domain of the function $f\left(x\right)={\sin }^{-1}\left(\frac{e^x+e^{-x}}{2}\right)$ is

If $\tan \left(\frac{\pi }{9}\right),$ $x,$ $\tan \left(\frac{7\pi }{18}\right)$ are in arithmetic progression and $\tan \left(\frac{\pi }{9}\right),$ $y,$ $\tan \left(\frac{5\pi }{18}\right)$ are also in arithmetic progression, then $|x-2y|$ is equal to

If $\alpha ,\beta ,\gamma$ are the roots of $p{x}^3+q{x}^2+r=0,$ then the value of the determinant

$\left|\begin{array}{ccc}\alpha \beta & \beta \gamma & \gamma \alpha \\ \beta \gamma & \gamma \alpha & \alpha \beta \\ \gamma \alpha & \alpha \beta & \beta \gamma \end{array}\right|$ is

Which of the following points lies on the locus of the foot of perpendicular drawn upon any tangent to the ellipse, $\frac{x^2}{4}+\frac{y^2}{2}=1$ from any of its foci?

Solve the following system of equations in R.
$1\le |x-2|\le 3$

A tangent PT is drawn to the circle $x^2+y^2=4$ at the point $P(\sqrt{3},1)$. A straight line L, perpendicular to PT is a tangent to the circle $(x-3{)}^2+y^2=1$
A common tangent of the two circles is

For $a>0$, let the curves $C_1:y^2=ax$ and $C_2:x^2=ay$ intersect at origin $O$ and a point $P$. Let the line $x=b(0<b<a)$ intersects the chord $OP$ and the $x$-axis at points $Q$ and $R$, respectively. If the line $x=b$ bisects the area bounded by the curves, $C_1$ and $C_2$, and the area of $\Delta OQR=\frac{1}{2}$, then '$a$' satisfies the equation: