List of practice Questions

If a tangent to the circle $x^2+y^2+2x+2y+1=0$ is radical axis of the circles $x^2+y^2+2gx+2fy+c=0$ and $2x^2+2y^2+3x+8y+2c=0$, then

If the angle between the tangents drawn to the parabola $y^2=4x$ from the points on the line $4x-y=0$ is $\frac{\pi}{3}$, then the sum of the abscissae of all such points is

When the coordinate axes are rotated about the origin through an angle $\frac{\pi}{4}$ in the positive direction, the equation $ax^2+2hxy+by^2=c$ is transformed to $25x'^2+9y'^2=225$, then $(a+2h+b-\sqrt{c})^2=$

$y-x=0$ is the equation of a side of a triangle ABC. The orthocentre and circumcentre of the triangle ABC are respectively (5,8) and (2,3). The reflection of orthocentre with respect to any side of the triangle lies on its circumcircle. Then the radius of the circumcircle of the triangle is

Two families of lines are given by $ax+by+c=0$ and $4a^2+9b^2-c^2-12ab=0$. Then the line common to both the families is

Two non parallel sides of a rhombus are parallel to the lines $x+y-1=0$ and $7x-y-5=0$. If (1,3) is the centre of the rhombus and one of its vertices $A(\alpha, \beta)$ lies on $15x-5y=6$, then one of the possible values of $(\alpha+\beta)$ is

If the equations $3x^2+2hxy-3y^2=0$ and $3x^2+2hxy-3y^2+2x-4y+c=0$ represent the four sides of a square, then $\frac{h}{c}= $

The radius of the circle having three chords along y-axis, the line $y=x$ and the line $2x+3y=10$ is

A and B are two events of a random experiment such that $P(B)=0.4$, $P(A \cap \bar{B}) = 0.5$, $P(A \cup B) + P(A|B) = 1.15$, then $P(A)=$

There are two boxes each containing 10 balls. In each box, few of them are black balls and rest are white. A ball is drawn at random from one of the boxes and found that it is black. If the probability that the black ball drawn is from the second box is $\frac{1}{5}$, then number of black balls in the first box is

In a shelf there are three mathematics and two physics books. A student takes a book randomly. If he randomly takes, successively for three times by replacing the book already taken every time, then the mean of the number of mathematics books which is treated as random variable is

A(2,0), B(0,2), C(-2,0) are three points. Let a, b, c be the perpendicular distances from a variable point P on to the lines AB, BC and CA respectively. If a, b, c are in arithmetic progression, then the locus of P is

In a quadrilateral ABCD, $\angle A = \frac{2\pi}{3}$ and AC is the bisector of angle A. If $15|AC| = 5|AD| = 3|AB|$, then the angle between $\vec{AB}$ and $\vec{BC}$ is

$\vec{a}, \vec{b}, \vec{c}$ are three non-coplanar and mutually perpendicular vectors of same magnitude K. $\vec{r}$ is any vector satisfying $\vec{a}\times((\vec{r}-\vec{b})\times\vec{a}) + \vec{b}\times((\vec{r}-\vec{c})\times\vec{b}) + \vec{c}\times((\vec{r}-\vec{a})\times\vec{c}) = \vec{0}$, then $\vec{r} =$

Consider the following
Assertion (A): The two lines $\vec{r} = \vec{a}+t(\vec{b})$ and $\vec{r}=\vec{b}+s(\vec{a})$ intersect each other.
Reason (R): The shortest distance between the lines $\vec{r}=\vec{p}+t(\vec{q})$ and $\vec{r}=\vec{c}+s(\vec{d})$ is equal to the length of projection of the vector $(\vec{p}-\vec{c})$ on $(\vec{q}\times\vec{d})$.
The correct answer is

The mean deviation about median of the numbers $3x, 6x, 9x, ..., 81x$ is 91, then $|x|=$

Functions are formed from the set $A = \{a_1, a_2, a_3\}$ to another set $B = \{b_1, b_2, b_3, b_4, b_5\}$. If a function is selected at random, the probability that it is a one-one function is

The number of real solutions of $\tan^{-1}x + \tan^{-1}(2x) = \frac{\pi}{4}$ is

Consider the following statements
Statement-I: $\cosh^{-1}x = \tanh^{-1}x$ has no solution
Statement-II: $\cosh^{-1}x = \coth^{-1}x$ has only one solution
The correct answer is

If the angular bisector of the angle A of the triangle ABC meets its circumcircle at E and the opposite side BC at D, then $DE\cos\frac{A}{2} =$

In a triangle ABC, $a=5, b=4$ and $\tan\frac{C}{2} = \sqrt{\frac{7}{9}}$, then its inradius r =

Two adjacent sides of a triangle are represented by the vectors $2\vec{i}+\vec{j}-2\vec{k}$ and $2\sqrt{3}\vec{i}-2\sqrt{3}\vec{j}+\sqrt{3}\vec{k}$. Then the least angle of the triangle and perimeter of the triangle are respectively

A plane $\pi_1$ contains the vectors $\vec{i}+\vec{j}$ and $\vec{i}+2\vec{j}$. Another plane $\pi_2$ contains the vectors $2\vec{i}-\vec{j}$ and $3\vec{i}+2\vec{k}$. $\vec{a}$ is a vector parallel to the line of intersection of $\pi_1$ and $\pi_2$. If the angle $\theta$ between $\vec{a}$ and $\vec{i}-2\vec{j}+2\vec{k}$ is acute, then $\theta=$

If $\frac{3x+1}{(x-1)^2(x^2+1)} = \frac{A}{x-1} + \frac{B}{(x-1)^2} + \frac{Cx+D}{x^2+1}$, then $2(A-C+B+D)=$

If $\tan(\frac{\pi}{4}+\frac{\alpha}{2}) = \tan^3(\frac{\pi}{4}+\frac{\beta}{2})$, then $\frac{3+\sin^2\beta}{1+3\sin^2\beta}=$