## 1. The angle between two asymptotes of the hyperbola $\frac{x^{2}}{25} - \frac{y^{6}}{16} = 1$ is

COMEDK 2012 Conic Sections

## 2. $^{\frac{\pi}{2}}\int_{0}\frac{sin 2x}{1+2 cos^{2}x}dx$ is equal to

KEAM 2013 Integrals

## 3. If $\lambda\left(3\hat{i}+2\hat{j}-6\hat{k}\right)$ is a unit vector, then the values of $\lambda$ are

KEAM 2014 Vector Algebra

## 4. If $x = 2y + 3$ is a focal chord of the ellipse with eccentricity 3/4, then the lengths of the major and minor axes are

COMEDK 2011 Conic Sections

## 5. If $x^r$ occurs in the expansion of $\left( x + \frac{1}{x} \right)^n$, then its coefficient is

COMEDK 2009 Binomial Theorem

## 6. The expression$\frac{tan\, A}{1-cot\, A}+\frac{cot\, A}{1-tan \, A}$ can be written as

JEE Main 2013 Trigonometric Functions

## 7. The product of the perpendiculars from the foci on any tangent to the ellipse $5x^2 + 8y^2 = 40$ is

COMEDK 2011 Conic Sections

## 8. If $z$ is a non-real complex number, then the minimum value of $\frac{Im\,z^{5}}{\left(Im\,z\right)^{5}}$ is :

JEE Main 2015 Complex Numbers and Quadratic Equations

## 9. Let g(x) = 1 + x - [x] and f(x) = $\begin{cases} - 1 , & x < 0 \\ 0 , & x = 0 \\ 1, & x > 0 \end{cases}$ . Then for all x, f(g(x)) is equal to

JEE Advanced 2001 Relations and Functions

## 10. The value of $\sin \left[2 \cos^{-1} \frac {\sqrt {5}}{3} \right]$is

KCET 2007 Inverse Trigonometric Functions

## 11. $\int\limits_{-\pi 2}^{\pi 2}\log\left(\frac{2-\sin x}{2+\sin x}\right)dx =$

COMEDK 2008 Inverse Trigonometric Functions

## 12. The maximum value of $xe^{ -x}$ is

KCET 2012 Application of Derivatives

## 13. If $\omega (\ne 1)$ be a cube root of unity and $(1 + \omega^2)^n = (1 + \omega^4)^n,$ then the least positive value of n is

IIT JEE 2004 Complex Numbers and Quadratic Equations

## 14. A certain item is manufactured by 3 factory $F_1, F_2$ and $F_3$ with $30\%$ of item made in $F_1, 20\%$ in $F_2$ and $50\%$ in $F_3$. It is found that $2\%$ of the items produced by $F_1, 3\%$ of the items produced by $F_2$ and $4\%$ of the items produced by $F_3$ are defective. Suppose that an items selected at random from the stock is found defective. What is the probability that the item came from $F_1$?

COMEDK 2014 Probability - Part 2

## 15. If $\int^x_1 \frac{dt}{|t\, \sqrt{t^2 - 1}} = \frac{\pi}{6}$ , then $x$ can be equal to

VITEEE 2010 Integrals

## 16. The number of real values of $\lambda$ for which the system of linear equations $2x + 4y - \lambda z = 0$ $4x + \lambda y + 2z = 0$ $\lambda x + 2y + 2z = 0$ has infinitely many solutions, is :

JEE Main 2017 Determinants

## 17. $\tan\left[\frac{1}{2} \sin^{-1} \left(\frac{2x}{1+x^{2}}\right) + \frac{1}{2} \cos^{-1} \left(\frac{1-x^{2}}{1+x^{2}}\right)\right] =$

COMEDK 2007 Inverse Trigonometric Functions

## 18. If the equations $x^2+2x+3=0$ and $ax^2+bx+c=0, a, b, c \in R$ have a common root, then $a : b : c$ is

JEE Main 2013 Complex Numbers and Quadratic Equations

## 19. If $A$ is any square matrix of order $3 \times 3$ then $|3A|$ is equal to

KCET 2016 Determinants

## 20. The point on the parabola $y^2 = 64x$ which is nearest to the line $4x + 3y + 35 = 0$ has coordinates

WBJEE 2014 Application of Derivatives

## 21. Suppose X is one end of the major axis of the ellipse $\frac{x^2}{9} + \frac{y^2}{b^2} = 1$ (b < 3) and F is that focus which is farther from X. Let Y denote one end of the minor axis. If $FYX$ is a right angled triangle, then the length of the latus rectum of the ellipse is

COMEDK 2005 Conic Sections

## 22. If $\overrightarrow {a}$ and $\overrightarrow {b}$ are two unit vectors inclined at an angle $\frac {\pi}{3}$, then the value of $| \overrightarrow {a} + \overrightarrow {b} |$ is

KCET 2014 Vector Algebra

## 23. The rate of change of the surface area of a sphere of radius $r$, when the radius is increasing at the rate of $2\, cm/s^{-1}$ is proportional to

VITEEE 2013 Application of Derivatives

## 24. The simplified form of $\tan^{-1}$ $\left(\frac{x}{y}\right)$ $- \tan^{-1}$ $\left(\frac{x-y}{x+y}\right)$ is equal to

KCET 2016 Inverse Trigonometric Functions

## 25. For every integer $n$, let $a_n \ and \ b_n$ be real numbers. Let function $f : R$ $\rightarrow$ R be given by f (x) = $\Bigg \{ \begin{array} \ a_n + sin \ \pi x, \\ b_n + cos \ \pi x, \\ \end{array} \begin{array} \ for \,x \in [2n, 2n + 1 ] \\ for \, x \in (2n - 1, 2n) \\ \end{array}$ for all integers $n$. If $f$ is continuous, then which of the following hold(s) for all $n$ ?

IIT JEE 2012 Continuity and Differentiability

## 26. If $ax^2 - y^2 + 4x - y = 0$ represents a pair of lines, then $a$ =

COMEDK 2011 Straight Lines

## 27. If $n$ is a positive integer which is relatively prime to $6,$ and if $n$ has $8$ divisors, then $12n$ has

COMEDK 2005 Probability - Part 2

## 28. The value of tan $\left(1^{\circ}\right)+$ tan $\left(89^{\circ}\right)$ is equal to

KEAM 2014 Trigonometric Functions

## 29. Two dice are thrown simultaneously. The probability of obtaining a total score of 5 is

KCET 2014 Probability

## 30. The equation $\frac{x^2}{2-r}+\frac{y^2}{r-5}+1 \,=\,0$ represents an ellipse iff

KCET 2000 Conic Sections