List of practice Questions

Let $R$ be the relation on the set $R$ of all real numbers, defined by $aRb$ If $|a - b| \le 1$. Then, $R$ is

$S$ and $T$ are the foci of an ellipse and $B$ is an end of the minor axis. If $STB$ is an equilateral triangle, then the eccentricity of the ellipse is

A box contains two white balls, three black balls and four red balls. In how many ways can three balls be drawn from the box if at least one black ball is to be included in the draw?

If $4a^2 + b^2 + 2c^2 + 4ab - 6ac - 3bc = 0$, the family of lines $ax + by + c = 0$ is concurrent at one or the other of the two points-

If a tangent having slope of $-\frac{4}{3}$ to the ellipse $\frac{x^{2}}{18}+\frac{y^{2}}{34}=1$ intersects the major and minor axes in points $A$ and $B$ respectively, then the area of $\Delta OAB$ is equal to (O is the centre of the ellipse)

The sum of the series $ \displaystyle\sum_{r = 0}^{n}\left(-1\right)^{r}\, ^{n}C_{r}\left(\frac{1}{2^{r}}+\frac{3^{r}}{2^{2r}}+\frac{7^{r}}{2^{3r}}+\frac{15^{r}}{2^{4r}}+...m \text{terms}\right)$ is

The angle of intersection of the circles $x^{2}+y^{2}-x+y-8=0$ and $x^{2}+y^{2}+2 x+2 y-11=0$ is

The vector $b = 3j + 4k$ is to be written as the sum of a vector $b_1$ parallel to $a = i +j$ and a vector $b_2$ perpendicular to $a$. Then $b_1$ is equal to

If the points ($x_1$, $y_1$), ($x_2$, $y_2$) and ($x_3$, $y_3$) are collinear, then the rank of the matrix $\begin{bmatrix}x_{_1}&y_{_1}&1\\ x_{_2}&y_{_2}&1\\ x_{_3}&y_{_3}&1\end{bmatrix}$will always be less than

The value of the determinant $\begin{vmatrix}1&cos \left(\alpha-\beta\right)&cos \alpha\\ cos \left(\alpha -\beta \right)&1&cos \beta\\ cos \alpha &cos \beta &1\end{vmatrix}$ is

The number of integral values of $K$, for which the equation $7\, \cos \,x + 5\, \sin\, x = 2K + 1$ has a solution, is

The line joining two points $A(2,0), B(3,1)$ is rotated about $A$ in anti-clockwise direction through an angle of $15^{\circ}$. The equation of the line in the now position,is

The line $2x+\sqrt{6y}=2$ is a tangent to the curve $x^{2}-2y^{2}=4$. The point of contact is

The value of $2 \,tan^{-1} (cosec \,tan^{-1} x - tan \,cot^{-1} x))$ is

The value of the integral $\int\limits ^{1/2}_{-1/2}\left[\left(\frac{x+1}{x-1}\right)^{^2}+\left(\frac{x+1}{x-1}\right)^{^2}-2\right]^{^{1/2}}\:\:dx$ is

If $\alpha$ and $\beta$ are the roots of $x^{2}-ax+b=0$ and if $\alpha^{n}+\beta^{n}=V_{_n},$ then

If | $x^2 - x - 6 | = x + 2$, then the values of $x$ are

The solution of the differential equation $\frac{dy}{dx}=\frac{yf '\left(x\right)-y^{2}}{f \left(x\right)}$ is

A fluid is in streamline flow across a horizontal pipe of variable area of cross section. For this which of the following statements is correct?

If the three lines $x - 3y = p, \,ax + 2y = q$ and $ax + y = r$ form a right-angled triangle then :

On the sides $AB, BC, CA$ of a $\Delta ABC, 3, 4, 5$ distinct points (excluding vertices $A, B, C$) are respectively chosen. The number of triangles that can be constructed using these chosen points as vertices a re :

A light ray emerging from the point source placed at $P( 1,3)$ is reflected at a point $Q$ in the axis of $x$. If the reflected ray passes through the point $R (6, 7)$, then the abscissa of $Q$ is:

All the students of a class performed poorly in Mathematics. The teacher decided to give grace marks of $10$ to each of the students. Which of the following statistical measures will not change even after the grace marks were given?

Let $R=\left\{ \left(3,3\right) \left(5,5\right), \left(9,9\right), \left(12,12\right), \left(5,12\right), \left(3,9\right), \left(3,12\right), \left(3,5\right)\right\}$ be a relation on the set $A=\left\{3,5,9,12\right\}$. Then, $R$ is:

The term independent of $x$ in expansion of $\bigg(\frac{x+1}{x^{2/3}-x^{1/3}+1}-\frac{x-1}{x-x^{1/2}}\bigg)^{10}$ is