## 1. If the distance of the point $P (1, - 2,1)$ from the plane $x + 2y - 2z = a$, where $a > 0$, is 5, then the foot of the perpendicular form $P$ to the plane is

IIT JEE 2010 Introduction to Three Dimensional Geometry

## 2. The general solution of the differential equation $(e^x + 1)y dy = (y + 1)e^x \,dx$ is

COMEDK 2006 Differential Equations

## 3. A line $I$ passing through the origin is perpendicular to the lines $l_1 : (3+t)\hat{i} + (-1 + 2t)\hat{j} + (4 + 2t)\hat{k}, - \infty < t< \infty$ $l_2 : (3 + 2s)\hat{i} + (3 + 2s)\hat{j} + (2 + s) \hat{k} , - \infty < s < \infty$ Then, the coordinate(s) of the point(s) on $l_2$ at a distance of $\sqrt{17}$ from the point of intersection of $l$ and $l_1$ is (are)

JEE Advanced 2013 Three Dimensional Geometry

## 4. Negation of the statement $p$ : for every real number, either $x \geq$ or $x < 1$ is

AIEEE 2010 Mathematical Reasoning

## 5. The value of $\int \frac{10^{x 2}}{\sqrt{10^{-x} - 10^{x}}}dx$ is

COMEDK 2009 Inverse Trigonometric Functions

## 6. For $x \in R, \lim_{x \to\infty} \left(\frac{x-3}{x+2}\right)^{x} =$

IIT JEE 2000 Limits and Derivatives

## 7. $\cos^2 \frac{\pi}{12} + \cos^2 \frac{\pi}{4} \cos^2 \frac{\pi}{15} =$

COMEDK 2009 Application of Integrals

## 8. A straight line meets the coordinates axes at A and B, so that the centroid of the triangle OAB is (1, 2). Then the equation of the line AB is

COMEDK 2015 Straight Lines

## 10. 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

## 11. If $\log_2 \: \sin x - \log_2 \cos x -\log_2(1 - \tan^2x) = - 1$, then

COMEDK 2008 Trigonometric Functions

## 12. A tetrahedron has vertices at $O (0, 0, 0), A(l, 2, 1) B(2, 1, 3 )$ and $C(-1, 1, 2)$. Then the angle between the faces OAB and ABC will be

AIEEE 2003 Introduction to Three Dimensional Geometry

## 13. Let $T_n$ be the number of all possible triangles formed by joining vertices of an n-sided regular polygon. If $T_{n+1} - T_n = 10$, then the · value of 11 is

COMEDK 2013 Binomial Theorem

## 14. If $C$ is the centre of the ellipse $\frac{x^{2}}{16} + \frac{y^{2}}{9} = 1$ and S isone of the foci, then the ratio of CS to semi-minor axis of the ellipse is

COMEDK 2008 Conic Sections

## 15. Two lines $L_1 : x =5 , \frac{y}{3 - \alpha} = \frac{z}{-2}$ and $L_2 = x = \alpha , \frac{y}{-1} = \frac{z}{2 - \alpha}$ are coplanar. Then, $\alpha$ can take value (s)

JEE Advanced 2013 Three Dimensional Geometry

## 16. If the system of equations $x + y + z = 6$ $x + 2y + 3z = 10$ $x + 2y + lambda z = 0$ has a unique solution, then $\lambda$ is not equal to

AIEEE 2012 Determinants

## 17. If $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ and $\overrightarrow{d}$ $(\overrightarrow{a}\times\overrightarrow{b}).(\overrightarrow{c}\times\overrightarrow{d})=1$ and $\overrightarrow{a}. \overrightarrow{c}=\frac{1}{2},$ then

IIT JEE 2009 Vector Algebra

## 18. Let $a > 0, b > 0$ and $c>0$. Then, both the roots of the equation $ax^2+bx+c=0$

IIT JEE 1979 Complex Numbers and Quadratic Equations

## 19. The sum $\displaystyle \sum_{i-0}^{m} \binom{10}{i} \binom{20}{m-i},$ where $\binom{p}{q}=0 \, if \, p>q,$ is maximum when m is equal to

IIT JEE 2002 Binomial Theorem

## 20. If $|\vec{a} \times \vec{b}| = 5$ and $|\vec{a} .\vec{b}| = 3$ , then $|\vec{a}|^2 |\vec{b}|^2$ is equal to

COMEDK 2012 Vector Algebra

## 21. The value of the integral $\int^1_0 \sqrt \frac{1 + x }{ 1 + x } \, dx \$ is

IIT JEE 2004 Integrals

## 22. If B is a non-singular matrix and A is a square matrix such that $B^{-1}\,AB$ exists, then $det\,(B^{-1}\,AB)$ is equal to

KEAM 2009 Matrices

## 23. If the line $2x + 3y + k = 0$ is at tangent to the circle $x^2 + y^2 - 6x - 8y = 0$, then the value of k is

COMEDK 2006 Conic Sections

EAMCET 2009 Sets

## 25. If the system of equations $x - k y - z = 0, kx - y - z = 0, x + y - z = 0$ has a non-zero solution, then possible values of k are

IIT JEE 2000 Determinants

## 26. The number of integers greater than 6000 that can be formed using the digits $3, 5, 6, 7$ and $8$ without repetition is

JEE Main 2015 Permutations and Combinations

## 27. $\frac{\sin x - \sin 3x}{\sin^{2} x -\cos^{2} x}$ is equal to

COMEDK 2015 Trigonometric Functions

## 28. If $A = \begin{bmatrix}1&0\\ 1&1\end{bmatrix}$ and $A^8 = aA +bI,$ then $(a , b) =$

COMEDK 2015 Matrices

## 29. The domain of the derivative of the function $f(x) = \begin{cases} \tan^{-1} x & \quad \text{if } | x | \le 1 \\ \frac{1}{2} (|x | - 1) & \quad \text{if } |x | > 1 \end{cases}$ is

IIT JEE 2002 Continuity and Differentiability

## 30. $\lim_{x \to0} \frac{\sqrt{1- \cos2x}}{\sqrt{2 } x }$

IIT JEE 2002 Limits and Derivatives