List of practice Questions

A circle C cuts the circles \(x^{2}+y^{2}-4x+6y+4=0\) and \(x^{2}+y^{2}+6x-4y+9=0\) orthogonally. If origin lies on this circle C, then the radius of the circle C is

Let \(x+y=0\) be the equation of the latus rectum of a parabola. Let the axis of this parabola pass through the point (1, 1). If \(x+y-2\sqrt{2}=0\) is the equation of the directrix of the parabola, then its vertex is

If the tangent drawn from the point \(P(5,3)\) to the parabola \(y^{2}=x\) is at a distance of \(\frac{1}{\sqrt{5}}\) units from the vertex of the parabola and touches the parabola at the point Q, then \(PQ=\)

X axis is the major axis and origin is the centre of an ellipse. If the distance between its directrices is \(\frac{18}{\sqrt{5}}\) and the ratio between the distances from the centre of this ellipse to its focus and its corresponding directrices is \(5:9\), then the length of its latus rectum is

Points \((h,k)\), \((1,2)\), \((-3,4)\) lie on \(L_1\). Line \(L_2\) through \((h,k)\) and \((4,3)\) is perpendicular to \(L_1\). Find \(\frac{k}{h}\).

If the sum of the slopes of the lines given by \(x^2-2cxy-7y^2=0\) is four times their product, find the value of \(c\).

If one of the lines \(my^2 + (1-m^2)xy - mx^2 = 0\) is a bisector of the angle between the lines \(xy = 0\), find \(m\).

If a circle \(S\) passes through \((a, b)\) and cuts the circle \(x^2 + y^2 = 4\) orthogonally, then find the locus of the center of \(S\).

Find the interval of \(\lambda\) for which exactly two common tangents can be drawn to \(x^2+y^2-4x-4y+6=0\) and \(x^2+y^2-10x-10y+\lambda=0\).

The sum of the squares of the lengths of the chords intercepted on \(x^2+y^2=16\) by the lines \(x+y=n, n \in \mathbb{N}\) is:

Box I contains 30 cards numbered 1 to 30 and Box II contains 20 cards numbered 31 to 50. A box is selected at random and a card is drawn. If the number is non-prime, the probability that it came from Box I is:

A random variable X takes values 0, 1, 2, 3 and its mean is 1.3. If $P(X=3)=2P(X=1)$ and $P(X=2)=0.3$, then find $P(X=0)$.

In a binomial distribution with 5 trials, the probabilities of 1 success and 2 successes are 0.4096 and 0.2048 respectively. Find the variance of the distribution.

A variable circle passes through the fixed point \(A(p,q)\) and touches the X-axis. The locus of the other end of the diameter through A is

The origin is shifted to (2, 3) and axes are rotated through angle \(\theta\). If \(3x^2+2xy+3y^2-18x-22y+50=0\) transforms to \(4x^2+2y^2-1=0\), then \(\theta =\)

If the image of \(P(2,3)\) in a line \(L\) is \(Q(4,5)\), find the image of \(R(0,0)\) in the same line \(L\).

$\overline{V}=2\overline{i}+\overline{j}-\overline{k}$ and $\overline{W}=\overline{i}+3\overline{k}$. If $\overline{U}$ is a unit vector, then the maximum value of the scalar triple product $[\overline{U}\overline{V}\overline{W}]$ is

$\overline{b}$ and $\overline{c}$ are non-collinear vectors and $\overline{a}$ is a vector such that $(\overline{c}\cdot\overline{c})\overline{a}=\overline{c}$. If $\overline{a}\times(\overline{b}\times\overline{c})+(\overline{a}\cdot\overline{b})\overline{b}=(4-2\beta-sin~\alpha)\overline{b}+(\beta^{2}-1)\overline{c}$, then the values of the scalars $\alpha$ and $\beta$ are

The mean of 5 observations is 5. If three of the observations are 1, 2, 6 and the other two observations are such that each is greater than 5, then the mean deviation from the mean of the observations is

A hunter is firing at a target. He has only 10% chance of hitting it in one round. The number of rounds he must fire in order to have at least 50% chance of hitting the target at least once, is

Two symmetric cubical dice are rolled once. Match the items of Column-I with the items of Column-II. center tabular|l|l|l|l| 2|c|Column-I & 2c|Column-II
A & Probability that the numbers appearing are equal & I & 1/12
B & Probability that the numbers are all distinct & II & 5/36
C & Probability that the sum of numbers is 10 & III & 1/6
D & Probability that the sum of numbers is 6 & IV & 4/36
tabular center

For a biased die, the probabilities for different faces are given by P(1)=0.1, P(2)=0.32, P(3)=0.21, P(4)=0.15, P(5)=0.05, P(6)=0.17. The die is tossed and it is known that either face 1 or 2 turned up. The probability that it is face 1 is:

In a \(\Delta ABC\) if \( a:b:c = 5:6:7 \), then the ratio of the radius of the circumcircle to that of the incircle is:

In a \(\triangle ABC\), let \(r\) be the inradius and \(r_1,r_2,r_3\) be the exradii opposite to vertices \(A,B,C\) respectively. Match the items of List-I with List-II. \[ \begin{array}{|c|c|} \hline \text{List-I} & \text{List-II}\\ \hline \text{A. } rr_1=r_2r_3 & \text{I. } \Delta^2\\ \text{B. } r_1+r_2=r_3-r & \text{II. } \angle A=90^\circ\\ \text{C. } \dfrac1{r_1}+\dfrac1{r_2}+\dfrac1{r_3} & \text{III. } \angle C=90^\circ\\ \text{D. } rr_1r_2r_3 & \text{IV. } s^2\\ & \text{V. } \dfrac1r\\ \hline \end{array} \]

If the median AD of \(\Delta ABC\) is bisected at the point E and BE is produced to meet the side AC at F. Then the vector \( \overline{BF} = \)