## 1. If $x$ and $y$ are two real numbers such that $x + y = 1$ and $x^3 + y^3 = 4,$ then $x^5 + y^5$ is

COMEDK 2005 Probability - Part 2

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

COMEDK 2013 Differential Equations

## 3. The value of $\Delta=\begin{bmatrix} a^2 & 2ab&b^2 \\[0.3em] b^2 & a^2&2ab \\[0.3em]2ab&b^2&a^2 \end{bmatrix}=,$

J & K CET 2013 Matrices

## 4. $\sqrt{2 + \sqrt{2+2\cos4\theta}}$

COMEDK 2008 Application of Integrals

## 5. For the curve $4x^5 = 5y^4$, the ratio of the cube of the subtangent at a point on the curve to the square of the subnormal at the same point is............................

KCET 2010 Application of Derivatives

## 6. The shortest distance between the lines $\frac{ x - 3}{3} = \frac{y +8}{-1}= \frac{z - 3}{1}$ and $\frac{ x + 3}{-3} = \frac{y +7}{2}= \frac{z - 6}{4}$ is

COMEDK 2015 Three Dimensional Geometry

## 7. In the group $(Q - \{-1\}, *\}$ ,where * is defined by $a * b = a + b + ab,$ the inverse of 3 is

COMEDK 2008 Relations and Functions - Part 2

## 8. If a class of 175 students the following data shows the number of students opting one or more subjects. Mathematics 100, Physics 70, Chemistry 40, Mathematics and Physics 30, Mathematics and Chemistry 28, Physics and Chemistry 23, Mathematics, Physics and Chemistry 18. The number of students who have opted Mathematics alone is

COMEDK 2015 Relations and Functions

## 9. $\int\limits_0^{2x} (\sin x + | \sin x |)dx =$

COMEDK 2007 Inverse Trigonometric Functions

## 10. The set of real values of x for which $f(x) = \frac {x}{Log x}$ increasing, is____________

KCET 2010 Application of Derivatives

## 11. The points $\bigg (0, \frac {8}{3} \bigg ), (1,3)$ and $(82,30)$ are vertices of

JEE Main 1986 Straight Lines

## 12. The value of the $\sin[\cot^{-1} . \{ \cos(\tan^{-1} x\} ]$ is

COMEDK 2009 Application of Integrals

## 13. The distance between the lines $\overrightarrow{r}=(4\hat{i}-7\hat{j}-\hat{k})+t(3\hat{i}-7\hat{j}+4\hat{k})\overrightarrow{r}=(7\hat{i}-14\hat{j}-5\hat{k})+s(-3\hat{i}+7\hat{j}+4\hat{k})$is equal to

KEAM 2009 Introduction to Three Dimensional Geometry

## 14. The logically equivalent proposition of $p \Leftrightarrow q$ is

KCET 2001 Mathematical Reasoning

## 15. The equation of the circle concentric with $x^2 + y^2 + 6x + 2y + 1 = 0$ and passing through the point (-2, -1) is

COMEDK 2007 Conic Sections

## 16. Coefficient of $t^{24}$ in $(1+t^2)^{12}) (1+t^{12}(1+t^{24})$ is

IIT JEE 2003 Binomial Theorem

## 17. The equation $x^{\frac{3}{4}(\log_2 x)^2+\log_2x-\frac{5}{4}=\sqrt2}$ has

IIT JEE 1989 Complex Numbers and Quadratic Equations

## 18. I f F and F are independent events such that $O < P(E) < 1$ and $O < P (F ) < 1$, then

IIT JEE 1989 Probability

## 19. Number of divisors of the form $(4n + 2), n\ge 0$ of the integer 240 is

IIT JEE 1998 Permutations and Combinations

## 20. Let $\omega=-\frac{1}{2}+i\frac{\sqrt 3}{2},$ then value of the determinant $\begin {vmatrix} 1 & 1& 1 \\ 1 & -1-\omega^2 & \omega^2 \\ 1 & \omega^2 & \omega \\ \end {vmatrix} is$

IIT JEE 2002 Complex Numbers and Quadratic Equations

## 21. The equation $\frac{x^2}{1-r}-\frac{y^2}{1+r}=1,|r|<1$ represents

IIT JEE 1981 Conic Sections

## 22. The number of solution of the equation $\tan x = \sin (x/2)$ in the interval $(- \pi , \pi )$ is

COMEDK 2005 Application of Integrals

AIEEE 2002 Sets

## 24. The function $f\left(x\right) = \left(\frac{\log_{e}\left(1+ax\right) - \log_{e}\left(1-bx\right)}{x}\right)$ is undefined at $x = 0$. The value which should be assigned to $f$ at $x = 0$ so that it is continuous at $x = 0$ is

COMEDK 2009 Statistics

## 25. Let $z_1$ and $z_2$ be nth roots of unity which subtend a right angled at the origin, then n must be of the form (where, k is an integer)

IIT JEE 2001 Complex Numbers and Quadratic Equations

## 26. The orthocentre of the triangle formed by the lines $xy = 0$ and $x+ y = 1$,is

IIT JEE 1995 Straight Lines

## 27. $\, if \, I_ n = \int^{\pi}_{ -\pi} \frac { \sin \, n \, x }{ ( 1 + \pi ^x ) \sin \, x } dx , \, n = 0 , 1 , 2 , .............., then$

IIT JEE 2009 Integrals

## 28. If $A= \begin{bmatrix}1&2\\ 0&1\end{bmatrix}$, then $A"$ is

COMEDK 2009 Matrices

## 29. Let $A$ and $B$ be two sets containing $2$ elements and $4$ elements respectively. The number of subsets of $A \times B$ having· 3 or more elements is

COMEDK 2013 Matrices

## 30. If $a^2+b^2+c^2=-2$ and $f(x)=\begin{vmatrix} 1+a^2x &(1+b^2)x & (1+c^2)x \\[0.3em] (1+a^2)x^2& 1+b^2x &(1+c^2)x \\[0.3em] (1+a^2)x & (1+b^2)x & 1+c^2x \end{vmatrix}$ then f(x) is a polynomial of degree

AIEEE 2005 Determinants