List of practice Questions

If \( (x^2-3x+2)e^{\frac{y}{x-1}} = x+2 \) then \( \left(\frac{dy}{dx}\right)_{x=0} = \)

If \( y = f(\cosh x) \) and \( f'(x) = \log(x + \sqrt{x^2-1}) \) then \( \frac{d^2 y}{dx^2} = \)

If \( f(x)=\sqrt{\cos ^{-1} \sqrt{1-x^{2}}} \), then \( f^{\prime}\left(\frac{1}{2}\right)= \)

If the direction cosines of two lines satisfy the equations \( 2l+m-n=0 \), \( l^2-2m^2+n^2=0 \) and \( \theta \) is the angle between the lines then \( \cos\theta = \)

If A(0,3,4), B(1,5,6), C(-2,0,-2) are the vertices of a triangle ABC and the bisector of angle A meets the side BC at D, then AD =

If the product of the perpendicular distances from any point on the hyperbola \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) to its asymptotes is \( \frac{36}{13} \) and its eccentricity is \( \frac{\sqrt{13}}{3} \), then \( a - b = \)

If S and S' are the foci of an ellipse \( \frac{x^2}{169} + \frac{y^2}{144} = 1 \) and the point B lying on positive Y-axis is one end of its minor axis, then the incentre of the triangle SBS' is

The focal distance of a point (5,5) on the parabola \( x^2-2x-4y+5=0 \) is

If the normal drawn at P(8,16) to the parabola \( y^2=32x \) meets the parabola again at Q, then the equation of the tangent drawn at Q to the parabola is

A circle C passing through the point (1,1) bisects the circumference of the circle \( x^2+y^2-2x=0 \). If C is orthogonal to the circle \( x^2+y^2+2y-3=0 \) then the centre of the circle C is

If A, B are the points of contact of the tangents drawn from the point (-3,1) to the circle \( x^2+y^2-4x+2y-4=0 \), then the equation of the circumcircle of the triangle PAB is

If \( x-2y=0 \) is a tangent drawn at a point P on the circle \( x^2+y^2-6x+2y+c=0 \), then the distance of the point (6,3) from P is

If the equation of the circle passing through the points (-1,0), (-1,1), (1,1) is \( ax^2+ay^2+2gx+2fy-2=0 \) then \( a = \)

If \( 4x^2+12xy+9y^2+2gx+2fy-1=0 \) represent a pair of parallel lines then

If a line L passing through a point A(2,3) intersects another line \( 4x-3y-19=0 \) at the point B such that \( AB=4 \), then the angle made by the line L with positive X-axis in anti-clockwise direction is

If \( (h,k) \) is the new origin to be chosen to eliminate first degree terms from the equation \( S = 2x^2 - xy - y^2 - 3x + 3y = 0 \) by translation and if \( \theta \) is the angle with which the axes are to be rotated about the origin in anticlockwise direction to eliminate xy-term from \( S = 0 \), then \( \tan 2\theta = \)

If the points A(2,3), B(3,2) form a triangle with a variable point \( p(t, t^2) \), where t is a parameter, then the equation of the locus of the centroid of triangle ABC is

If three dice are thrown, then the mean of the sum of the numbers appearing on them is

Urn A contains 6 white and 2 black balls; urn B contains 5 white and 3 black balls and urn C contains 4 white and 4 black balls. If an urn is chosen at random and a ball is drawn at random from it, then the probability that the ball drawn is white is

If three cards are drawn randomly from a pack of 52 playing cards then the probability of getting exactly one spade card, exactly one king and exactly one card having a prime number is

If three smallest squares are chosen at random on a chess board then the probability of getting them in such a way that they are all together in a row or in a column is

If \( \bar{a} \) and \( \bar{b} \) are two vectors such that \( |\bar{a}|=5 \), \( |\bar{b}|=12 \) and \( |\bar{a}-\bar{b}|=13 \) then \( |2\bar{a}+\bar{b}| = \)

The point of intersection of the line joining the points \( \bar{i} + 2\bar{j} + \bar{k} \), \( 2\bar{i} - \bar{j} - \bar{k} \) and the plane passing through the points \( \bar{i}, 2\bar{j}, 3\bar{k} \) is

The position vectors of two points A and B are \( \bar{i} + 2\bar{j} + 3\bar{k} \) and \( 7\bar{i} - \bar{k} \) respectively. The point P with position vector \( -2\bar{i} + 3\bar{j} + 5\bar{k} \) is on the line AB. If the point Q is the harmonic conjugate of P, then the sum of the scalar components of the position vector of Q is

Let the angles A, B, C of a triangle ABC be in arithmetic progression. If the exradii \( r_1, r_2, r_3 \) of triangle ABC satisfy the condition \( r_3^2 = r_1 r_2 + r_2 r_3 + r_3 r_1 \), then \( b = \)