List of top Mathematics Questions

The shortest distance between the lines \( \bar{r} = (3\bar{i} - 5\bar{j} + 2\bar{k}) + t(4\bar{i} + 3\bar{j} - \bar{k}) \) and \( \bar{r} = (\bar{i} + 2\bar{j} - 4\bar{k}) + s(6\bar{i} + 3\bar{j} - 2\bar{k}) \) is

If \( \bar{a} = \bar{i} - 2\bar{j} - 2\bar{k} \) and \( \bar{b} = 2\bar{i} + \bar{j} + 2\bar{k} \) are two vectors then \( (\bar{a} + 2\bar{b}) \times (3\bar{a} - \bar{b}) = \)

The general solution of the differential equation \( 2dx + dy = (6xy + 4x - 3y)dx \) is

If \( \cos x \frac{dy}{dx} = y \sin x - 1, x \neq (2n+1)\frac{\pi}{2}, n \in \mathbb{Z} \) is the differential equation corresponding to the curve \( y = f(x) \) and \( f(0)=1 \) then \( f(x)= \)

\( \int_{-2 \pi}^{2 \pi} \sin ^{4} x \cos ^{6} x d x= \)

\( \int_{0}^{\pi / 4} x^{2} \sin 2 x d x= \)

\( \int_{0}^{2} \sqrt{(x+3)(2-x)} d x= \)

If \( \int(x+2) \sqrt{x^{2}-x+2} d x=\frac{1}{3} f(x)+\frac{5}{8} g(x)+\frac{35}{16} h(x)+c \) then \( f(-1)+g(-1)+h\left(\frac{1}{2}\right)= \)

\( \int \left( \frac{1-\log x}{1+(\log x)^2} \right)^2 dx = \)

\( \int e^{4x}(\sin 3x - \cos 3x) dx = \)

\( \int \frac{2\sin x - 3\cos x}{4\cos x - 3\sin x} dx = \)

If \( 1^\circ = 0.0175 \) radians, then the approximate value of \( \sec 58^\circ \) is

If the rate of change of the slope of the tangent drawn to the curve \( y = x^3 - 2x^2 + 3x - 2 \) at the point (2,4) is k times the rate of change of its abscissa, then k =

The acute angle between the curves \( y = 3x^2 - 2x - 1 \) and \( y = x^3 - 1 \) at their point of intersection which lies in the first quadrant is

If \( x = \frac{t^2}{1+t^5} \), \( y = \frac{2t^3}{1+t^5} \) and \( t \neq -1 \) is a parameter then \( \frac{dy}{dx} = \)

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