## 1. If three distinct numbers are chosen randomly from the first 100 natural numbers, then the probability that all three of them are divisible by both 2 and 3, is

IIT JEE 2004 Probability

## 2. If $p \to (\sim q \vee r)$ is false then the truth values of $p,q,r.$ are

COMEDK 2012 Mathematical Reasoning

## 3. The area in square units of the region bounded by $y^2 = 9x$ and $y = 3x$ is

COMEDK 2015 Linear Programming

## 4. If the events A and B are independent if $P(A') =\frac {2}{3}$ and $P(B')= \frac {2}{7}$ then $P(A \cap B)$ is equal to

KCET 2014 Probability

## 6. $f(x) = \begin{cases} 2a -x & \quad \text{when} n \text{ -a < x < a}\\ 3x-2a & \quad \text{when } n \text{ a < x}\\ \end{cases}$ Then which of the following is true ?

KCET 2013 Limits and Derivatives

## 7. If $(\omega\,\neq\,1)$ is a cube root of unity , then $\begin{vmatrix} 1 &1+i+\omega^2 &\omega^2 \\[0.3em] 1-i&-1 & \omega^2-1 \\[0.3em] -i & -1+\omega-i& -1 \end{vmatrix}=$

AIEEE 2002 Determinants

## 8. Consider an infinite geometric series with first term a and common ratio r. If its sum is 4 and the second term is 3/4 then

IIT JEE 2000 Sequences and Series

## 9. If the product of the matrix $B = \begin{bmatrix}2&6&4\\ 1&0&1\\ -1&1&-1\end{bmatrix}$ with a matrix $A$ has the inverse $C = \begin{bmatrix}-1&0&1\\ 1&1&3\\ 2&0&2\end{bmatrix}$ then $A^{-1}$ equals

COMEDK 2014 Matrices

## 10. $\int\limits \frac{x^3 -1}{x^3 + x} dx =$

COMEDK 2015 Integrals

## 11. $\tan^{-1}\frac{1}{3}+\tan^{-1}\frac{1}{7}+\tan^{-1}\frac{1}{18}+.........+\tan^{-1}\left(\frac{1}{n^2+n+1}\right)+....to\infty$ is equal to

KCET 2000 Inverse Trigonometric Functions

## 12. If $A = \begin {bmatrix} 1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & -2 & 4 \end {bmatrix} ,6A^{-1}=A^2+cA+dI,$ then $(c,d)$ is

IIT JEE 2005 Matrices

## 13. If the circles $x^2 + y^2 + 2gx + 2fy = 0$ and $x^2 + y^2 + 2g'x + 2f'y = 0$ touch each other, then

COMEDK 2009 Conic Sections

## 14. The circle passing through (1, - 2) and touching the axis of x at (3; 0) also passes through the point

COMEDK 2013 Conic Sections

## 15. The number of common tangents to the circles $x^2+y^2-4x-6y-12=0$ and $x^2+y^2+6x+18y+26=0$

JEE Main 2015 Conic Sections

## 16. The solution of the differential equation, $\frac{dy}{dx} = \frac{y}{x}+\frac{g\left(y /x\right)}{g'\left(y / x\right)}$, where g is a differentiable function is

COMEDK 2005 Differential Equations

## 17. The distance between the line $\overrightarrow{r}=(2\hat{i}+2\hat{j}\hat{k})+\lambda(2\hat{i}+\hat{j}-2\hat{k})$ and the plane $\overrightarrow{r}.(\hat{i}+2\hat{j}-\hat{k})$ =10 is equal to

KEAM 2010 Introduction to Three Dimensional Geometry

## 18. The area (in sq units) of the quadrilateral formed by the tangents at the end points of the latusrectum recta to the ellipse $\frac{x^2}{9} +\frac{y^2}{ 5}=1 is$

IIT JEE 2015 Conic Sections

## 19. If $P = (1,0), Q = (-1,0)$ and $R= (2,0)$ are three given points, then locus of the points satisfying the relation $SQ^2 + SR^2 = 2SP^2,$ is

IIT JEE 1988 Straight Lines

KCET 2010 Sets

## 21. If the lines $\frac{1+x}{3}=\frac{y-2}{2\alpha}=\frac{z-3}{2}$ and $\frac{x-1}{3\alpha}=y-1=\frac{6-z}{5}$ are Perpendicular then the value of $\alpha$ is

KEAM 2009 Introduction to Three Dimensional Geometry

## 22. The continued product of the four values of $\left(\cos \frac{\pi}{3} + i \sin \frac{\pi}{3} \right)^{3/ 4 }$ is

COMEDK 2011 Complex Numbers and Quadratic Equations

## 23. At present, a firm is manufacturing 2000 items. It is estimated that the rate of change of production $P$ w.r.t. additional number of workers x is given by $\frac{dP}{dx} = 100 - 12 \sqrt{x}$ .If the firm employs 25 more workers, then the new level of production of items is

COMEDK 2013 Differential Equations

## 24. If $x_{r} = \cos \frac{\pi}{2^{r}} + i \sin \frac{\pi}{2^{r}}$, then $x_{1} .x_{2} . x_{3} ....$ to $\infty =$

COMEDK 2011 Complex Numbers and Quadratic Equations

## 25. If a, b, c, are non zero complex numbers satisfying $a^2 + b^2 + c^2 = 0$ and $\begin{vmatrix}b^{2} + c^{2}&ab&ac\\ ab&c^{2} +a^{2}&bc\\ ac&bc&a^{2}+b^{2}\end{vmatrix} = ka^{2}b^{2}c^{2}$ , then $k$ is equal to

AIEEE 2012 Determinants

## 26. Let A be a $2 \times 2$ matrix with non-zero entries and let $A^2$ = I, where I is $2 \times 2$ identity matrix. Define Tr (A) = Sum of diagonal elements of A and |A | = determinant of matrix A. Statement-1. Tr (A) = 0 Statement-2. | A | = 1

AIEEE 2008 Determinants

## 27. If $f(x) = \begin{cases} \frac{e^{3x} - 1}{4x} & \quad \text{for} x \neq 0 \\ \frac{k + x}{4} & \quad \text{for } x= 0 \end{cases}$ is continuous at $x = 0$, then $k =$

COMEDK 2012 Statistics

## 28. If $\vec{a}, \vec{b}, \vec{c}$ are three non-zero vectors such that each one of them are perpendicular to the sum of the other two vectors, then the value of $| \vec{a}, \vec{b}, \vec{c}|^2$ is

COMEDK 2015 Vector Algebra

## 29. If the lines $\frac{x-2}{1}=\frac{y-3}{1}=\frac{z-4}{-k}$ and $\frac{x-1}{k}=\frac{y-4}{2}=\frac{z-5}{1}$ are coplanar, then k can have

JEE Main 2012 Introduction to Three Dimensional Geometry

## 30. If $a_n=\sqrt{7+\sqrt{7+\sqrt{7+...}}}$ having n radical singns ,then which is true

AIEEE 2002 Principle of Mathematical Induction