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