List of practice Questions

\(T_1,T_2\) are points of contact of a transverse common tangent drawn to circles \[ x^2+y^2+4x-10y+4=0 \] and \[ x^2+y^2-6x+8y+9=0 \] If \(T_1T_2\) is horizontal line, midpoint of segment \(T_1T_2\) is

If \(P(t_1)\) and \(Q(t_2)\) are two points on the parabola \[ y^2=7x \] If \(t_1=2\) and \(t_2=-4\), then the length of chord \(PQ\) is

If the inclination of a straight line \[ x-y+1=0 \] with another straight line \(L\) is \(30^\circ\) and \(m\) is the slope of line \(L\), then \[ m^2+1= \]

If \(A(2,1)\), \(B(4,k)\), \(C(3,4)\) are vertices of triangle right angled at B and \(k\) is not an odd number, then equation of line joining orthocentre and circumcentre is

Let \(ABC\) be a triangle and \(A=(-2,3)\). If \[ 7x-y+2=0 \] and \[ 4x-7y+44=0 \] are medians drawn through vertices B and C respectively, then \(AB=\)

Orthocentre of triangle formed by pair of lines \[ 2x^2-xy-3y^2=0 \] and line \[ x-y+4=0 \] is

If circle \[ x^2+y^2+2x+4y+k=0 \] lies totally inside third quadrant and point \[ \left(-\frac12,-\frac12\right) \] lies outside circle, then set of all real values of \(k\) is

A bag A contains 3 red, 2 white and 2 black balls and another bag B contains 1 red, 2 white and 4 black balls. A die is thrown to select a bag. If an odd prime number appears on the die, a ball is drawn from bag A; otherwise from bag B. If the ball drawn is black, then the probability that it is drawn from bag B is

If a random variable \(X\) has the following probability distribution, then the mean of \(X\) is

In a Poisson distribution with parameter \(\lambda\), if \[ 5P(X=3)=P(X=5) \] then \[ P(X=2)= \]

If a straight line passing through the point \((2,3)\) intersects X-axis at A and Y-axis at B, then the locus of a point dividing AB in the ratio \(2:3\) is

When origin is shifted to point \[ \left(-\frac47,\frac67\right) \] and transformed equation of \[ 2x^2+5xy+4y^2-2x-4y+2=0 \] is \[ ax^2+35xy+by^2+2gx+2fy+c=0 \] then

If a 4-digit number is chosen from all possible 4-digit numbers, probability of getting exactly three odd digits and one even digit is

Let \[ S=\{2,3,5,7,11,13\} \] Consider all onto functions from \(S\) to \(S\). If function \(f\) is chosen randomly, probability that \[ f(3)& gt;3f(2) \] is

The standard deviation of the data \[ 2,3,4,5,6,7,10,11,13,19 \] is

Let \[ \vec a=\vec i+2\vec j+\vec k \] and \[ \vec b=2\vec i-\vec j+\vec k \] be two vectors. If vector \[ \vec r=x\vec i+y\vec j+2\vec k \] is along the bisector of angle between \(\vec a\) and \(\vec b\), then \[ |\vec r|= \]

If the points with position vectors \[ x\vec i+2\vec j+y\vec k \] \[ \vec i-2\vec j+2x\vec k \] and \[ 2\vec i+3\vec j-\vec k \] are collinear, then \[ 10x-25y= \]

Let \[ \vec a=2\vec i-\vec j-3\vec k,\qquad \vec b=\vec i+3\vec j-2\vec k,\qquad \vec c=3\vec i-2\vec j+\vec k \] If magnitude of projection of \[ \vec a+\lambda\vec b \] on \(\vec c\) is \[ \frac{10}{\sqrt{14}} \] then sum of squares of magnitudes of all such vectors is

Let \[ \vec a=\vec i-2\vec j+2\vec k \] and \[ \vec b=2\vec i+3\vec j-6\vec k \] If \[ \alpha\vec i+\beta\vec j+\gamma\vec k \] is perpendicular to plane of \[ 2\vec a+\vec b \] and \[ \vec b-\vec a \] such that \[ \alpha+\beta+\gamma=46 \] then \[ \alpha-2\beta+3\gamma= \]

Let \[ \vec a=3\vec i-2\vec j+5\vec k,\qquad \vec b=\vec i+3\vec j-2\vec k \] If \(\vec c\) is a vector such that \[ \vec b\times\vec c=\vec a \] and \[ \vec b\cdot\vec c=5 \] then \[ 14\vec c\times\vec a= \]

If \(\alpha,\beta\in\left(-\frac{\pi}{2},\frac{\pi}{2}\right)\), \[ \cos^4\alpha=\frac1{16},\qquad \sin^4\beta=\frac1{16} \] then \[ \cos\alpha+\cos\beta= \]

Number of solutions of equation \[ 3^{2\sin^2x}+3^{2\cos^2x}=6 \] lying in interval \[ [-\pi,\pi] \] is

If \[ (2\sin^{-1}x)^3=\pi^3-(2\cos^{-1}x)^3 \] then one value of \[ \cos(2\sin^{-1}x-3\cos^{-1}x) \] is

Evaluate \[ e^{Sinh^{-1}(2\sqrt2)}+e^{Cosh^{-1}(3)} \]

In triangle ABC, if \[ \frac{a}{b+c}+\frac{c}{a+b}=1 \] and \[ s=r+a \] then \[ \sin A+\sin B+\sin C= \]