## 1. The ellipse $\frac {x^2} {25}+ \frac {y^2}{16}=1$ and the hyperbola $\frac {x^2} {25}- \frac {y^2}{16}=1$ have in common

KCET 2006 Conic Sections

## 2. If $y =e^{\log_e[1+x+x^2+.........]}$, then $\frac {dy}{dx}$=

KCET 2012 Continuity and Differentiability

## 3. Let $P (x) = a_0 + a_1x^2 + a_2x^4 +..... a_nx^{2n}$ be a polynomial in a real variable x with $0 < a_0 < a_1...< a_n$. The function P (x) has :

JEE Advanced 1986 Application of Derivatives

## 4. The domain of definition of $f(x) = \frac{\log_2(x + 3) }{x^2 + 3x +2}$ is

JEE Advanced 2001 Relations and Functions

## 5. A rectangular sheet of fixed perimeter with sides having their lengths in the ratio 8 : 15 is converted into an open rectangular box by folding after removing squares of equal area from all four corners. If the total area of removed squares is 100, the resulting box has maximum volume. Then the lengths of the sides of the rectangular sheet are

JEE Advanced 2013 Application of Derivatives

## 6. The amplitude of $\frac {1+i\sqrt{3}} {\sqrt {3}+i}$ is .................

KCET 2005 Complex Numbers and Quadratic Equations

## 7. The area of the region enclosed between the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ and its auxiliary circle is

COMEDK 2007 Conic Sections

## 8. If the positive numbers a,b,c,d are in AP. Then, $abc, abd, acd, bcd$ are

IIT JEE 2001 Sequences and Series

## 9. The maximum value of $f(x) = \frac{\log x}{x} , 0 < x < \infty$ is

COMEDK 2007 Statistics

## 10. The angle of elevation of the top of a tower from the top and bottom of a building of height $'a'$ are $30°$ and $45°$ respectively. If the tower and building stand at the same level, the height of the tower is

COMEDK 2008 Probability - Part 2

## 11. Two circles centered at $(2, 3)$ and $(5, 6)$ intersect each other. If the radii are equal, the equation of the common chord is__________

KCET 2010 Conic Sections

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

JEE Advanced 2013 Introduction to Three Dimensional Geometry

## 13. The equation $x^{\frac{3}{4}(\log_2 x)^2+\log_2x-\frac{5}{4}=\sqrt 2}$ has

IIT JEE 1989 Complex Numbers and Quadratic Equations

## 14. $\int \frac{Sec \, x}{Sec \,x + \tan \, x} dx =$

KCET 2008 Integrals

## 15. If $2x^2 + 2y^2 + 4x + 5y +1 = 0$ and $3x^2 + 3y^2 + 6x - 7y + 3k = 0$ are orthogonal, then value of k is.......

KCET 2011 Conic Sections

## 16. If Z= $\frac {(\sqrt {3}+ i)^3 (3i+4)^2}{{(8+6i)^2}}$ then |Z| is equal to

KCET 2015 Complex Numbers and Quadratic Equations

## 17. For three vectors $\overrightarrow{u},\overrightarrow{v},\overrightarrow{w}$which of the following expressions is not equal to any of the remaining three?

IIT JEE 1998 Vector Algebra

## 18. The solution of the differential equation $e ^{-x} dy (y +1) dy + (Cos^2 x - Sin 2x)y (dx) = 0$ subjected to the condition that $y = 1$ when $x = 0$ is

KCET 2006 Differential Equations

## 19. The relation R on the set $A = \{ x | x | < 3, x \in Z\}$ is defined by $R = \{ (x, y) | x | , x \neq -1\}$ . Then the number of elements in the power set of $R$ is

COMEDK 2008 Relations and Functions - Part 2

## 20. The locus of the midpoint of the intercept of the line $x \cos \alpha + y \sin \alpha = p$ between the coordinate axes is

COMEDK 2009 Straight Lines

## 21. The domain of definition of the function f(x) given by the equation $2^x + 2^y = 2$ is

JEE Advanced 2000 Relations and Functions

## 22. If $a_1, a_2, a_3,......, a_n$,.... are in G.P., then the value of the determinant $\begin{vmatrix}\log a_{n}& \log a_{n+1}&\log a_{n+2}\\ \log a_{n+3}& \log a_{n+4}&\log a_{n+5}\\ \log a_{n+6} &\log a_{n+7}& \log a_{n+8}\end{vmatrix}$, is

AIEEE 2004 Determinants

## 23. A bag contains 17 tickets numbered from 1 to 17. A ticket is draw at random, then another ticket is drawn at random, then another ticket is drawn without replacing the first one . The probability that both the tickets may show even numbers is

KCET 2018 Probability

## 24. Let $f (x) = ax^2 + bx + c, a \ne 0$ and $\Delta =b^2 -4ac.$ If $a+ \beta,$ $a^2+ \beta^2$ and $a^3+ \beta^3$ are in GP, then

IIT JEE 2005 Sequences and Series

## 25. The distance of the point P(a, b, c) from the x-axis is

KCET 2014 Three Dimensional Geometry

## 26. Let $f\left(x\right) = \frac{\log\left(1+ex\right)-\log\left(1-x\right)}{x} , x\ne0$ . Then $f$ is continuous at $x = 0$ if $f(0)$ =

COMEDK 2008 Statistics

## 27. Slope of Normal to the curve $y=x^2 - \frac {1}{x^2}$ at (-1,0) is

KCET 2015 Application of Derivatives

## 28. If $n = 2020!$ then $\frac {1}{\log_2n}+\frac {1}{\log_3n}+\frac {1}{\log_4n}+............+\frac {1}{\log_{2020} n}$

KCET 2009 Sequences and Series

## 29. If $\alpha$ and $\beta$ are the roots of $x^2 - ax + b^2 = 0$, then $\alpha^2 + \beta^2$ is equal to

KCET 2015 Complex Numbers and Quadratic Equations

## 30. One hundred identical coins, each with probability p, of showing up heads are tossed once. If $0 < p < 1$ and the probability of heads showing on 50 coins is equal to that of heads showing on 51 coins, then the value of p is

IIT JEE 1988 Probability