List of practice Questions

If the foot of the perpendicular drawn from the point (2,0,-3) to the plane $\pi$ is (1,-2,0) and the equation of the plane is $ax+by-3z+d=0$ then $a+b+d=$

The line $4x-3y+2 = 0$ intersects the circle $x^2+y^2-2x+6y+c=0$ at two points A, B and AB=8. If (1,k) is a point on the given circle and $k>0$, then $k =$

If $2x-3y+5=0$ and $4x-5y+7=0$ are the equations of the normals drawn to a circle and (2,5) is a point on the given circle, then the radius of the circle is

If $(\alpha,\beta)$ is the centre of the circle which passes through the point (1,-1) and cuts the circles $x^2 + y^2+2x-3y-5=0$, $x^2+y^2-3x+2y+1=0$ orthogonally, then $\alpha-5\beta =$

The centre of the circle touching the circles $x^2+y^2-4x-6y-12=0$, $x^2+y^2+6x+18y+26=0$ at their point of contact and passing through the point (1,-1) is

The number of normals that can be drawn through the point (2,0) to the parabola $y^2 = 7x$ is

If $m_1$ and $m_2$ are the slopes of the tangents drawn from the point (1,4) to the parabola $y^2 = 11x$ then $2(m_1^2 + m_2^2) =$

If a line L passing through the point A(-2,4) makes an angle of 60$^\circ$ with the positive direction of X-axis in anti-clockwise direction and B(p,q) lying in the 3$^\text{rd}$ quadrant is a point on L at the distance of 6 units from the point A, then $\sqrt{p^2+q^2-8q} =$

If the perpendicular drawn from the point (2,-3) to the straight line $4x-3y+8=0$ meets it at M(a,b) and $a^3-b^3=k^3$, then $k =$

Let Q be the image of a point P(1,2) with respect to the line $x+y+1=0$ and R be the image of Q with respect to the line $x-y-1=0$. If M and N are the midpoints of PQ and QR respectively, then MN =

If the slopes of the lines represented by the equation $6x^2+2hxy+4y^2 = 0$ are in the ratio 2:3, then the value of h such that both the lines make acute angles with the positive X-axis measured in positive direction is

If (3,-2) is the centre of the circle $S= x^2+y^2+2gx+2fy-23=0$ and A is a point on the circle S = 0 such that its distance from a point P(-1,-5) is least, then A =

Two circles which touch both the coordinate axes intersect at the points A and B. If A = (1,2), then AB =

Three companies C1, C2, C3 produce car tyres. A car manufacturing company buys 40% of its requirement from C1, 35% from C2 and 25% from C3. The company knows that 2% of the tyres supplied by C1, 3% by C2 and 4% by C3 are defective. If a tyre chosen at random from the consignment received is found defective then the probability that it was supplied by C2 is

The probability distribution of a random variable X is given below. Then, the standard deviation of X is.

If the mean and variance of a binomial distribution are $\frac{4}{3}$ and $\frac{10}{9}$ respectively, then $P(X \geq 6) =$

A straight line passing through a point (3,2) cuts X and Y-axes at the points A and B respectively. If a point P divides AB in the ratio 2:3, then the equation of the locus of point P is

By shifting the origin to the point (-1,2) through translation of axes, if $ax^2+2hxy+by^2+2gx+2fy+c=0$ is the transformed equation of $2x'^2-x'y'+y'^2-3x'+4y'-5=0$, then $2(f+g+h) =$

Let $\vec{a} = \hat{i} +2\hat{j}+3\hat{k}$, $\vec{b}=2\hat{i}-3\hat{j}+\hat{k}$ and $\vec{c}=3\hat{i}+\hat{j}-2\hat{k}$ be three vectors. If $\vec{r}$ is a vector such that $\vec{r}\cdot\vec{a} = 0$, $\vec{r}\cdot\vec{b} = -2$ and $\vec{r}\cdot\vec{c} = 6$ then $\vec{r}\cdot(3\hat{i}+\hat{j}+\hat{k})= $

Let $\vec{a}=\hat{i}-\hat{j}+\hat{k}$, $\vec{b}=\hat{i}-2\hat{j}-2\hat{k}$, $\vec{c}=6\hat{i}+3\hat{j}-2\hat{k}$ be three vectors. If $\vec{d}$ is a vector perpendicular to both $\vec{a}$, $\vec{b}$ and $|\vec{d}\times\vec{c}|=14$, then $|\vec{d}\cdot\vec{c}|=$

The mean deviation from the mean of the discrete data 2, 3, 5, 7, 11, 13, 17, 19, 22 is

Out of the given 25 consecutive positive integers, three integers are drawn. If the least integer among given 25 integers is an odd number, then the probability that the sum of the three integers drawn is an even number is

If three dice are thrown at a time, then the probability of getting the sum of the numbers on them as a prime number is

$\coth^2 x - \tanh^2 x =$

If $a=3, b=5, c=7$ are the sides of a triangle ABC, then its circumradius is