List of top Mathematics Questions

If \((1,-2,2)\) and \((2,6,-3)\) are the direction ratios of two straight lines, then the direction cosines of the line bisecting an angle between these two lines are

Let \[ f(x)=\frac{([x]+|x|-x)x}{\sin|x|} \] be a real valued function. If \[ \alpha=\lim_{x\to0^-}f(x),\qquad \beta=\lim_{x\to0^+}f(x) \] then

Let D be harmonic conjugate of point C with respect to points \[ A(1,-3,5),\qquad B(5,-3,1) \] If C divides AB in ratio \(3:5\), then point dividing CD in ratio \(1:2\) is

If both foci of a hyperbola having eccentricity \(\sqrt3\) lie on x-axis and x coordinates of foci are roots of equation \[ x^2-4x+1=0 \] then length of chord passing through focus and perpendicular to transverse axis is

If \(\theta\) is acute angle between tangents drawn from point \((3,4)\) to ellipse \[ \frac{x^2}{25}+\frac{y^2}{9}=1 \] then \(\theta=\)

Let \(S,S'\) be foci and \(B\) one end of minor axis of an ellipse. If \[ \angle SBS'=120^\circ \] and area of triangle \(SBS'\) is \(\sqrt3\), then length of latus rectum 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 a straight line \[ y=mx+c \] touches the circle \[ x^2+y^2=4 \] and parabola \[ y^2=4x \] then \(2m^2=\)

\(T_1,T_2\) 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

Let \(L_{1}\equiv3x+4y-1=0,\; L_{2}\equiv8x-6y+1=0,\; L_{3}\equiv12x+9y-1=0\) be three tangents drawn to the circle \[ x^2+y^2+2gx+2fy+c=0 \] and \(L_1>0,\;L_2>0,\;L_3>0\) at the centre \((-g,-f)\). Then \(g+2f=\)

The external centre of similitude for circles \[ x^2+y^2+10x-16y-11=0 \] and \[ x^2+y^2-2x+4y-4=0 \] 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 circle \(S\) passing through origin cuts another circle \[ x^2+y^2-6x+8y+16=0 \] orthogonally and makes a chord of maximum length on line \[ x-y-2=0 \] then one diameter of circle \(S\) is

If \[ l_1x+m_1y+n_1=0 \] and \[ l_2x+m_2y+n_2=0 \] are tangents drawn from point \((2,-1)\) to circle \[ x^2+y^2=4 \] then \(n_1+n_2=\)

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

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 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

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 \(AB=\)

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

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

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

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

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