List of top Mathematics Questions

Evaluate the integral: \[ \int \frac{x+1}{x^3 - 1}dx \]

P and Q are the ends of a diameter of the circle \( x^2+y^2=a^2(a>\frac{1}{\sqrt{2}}) \). \( s \) and \( t \) are the lengths of the perpendiculars drawn from P and Q onto the line \( x+y=1 \) respectively. When the product \( st \) is maximum, the greater value among \( s, t \) is:

Let \( P(x) = x^4 + ax^3 + bx^2 + cx + d \) be such that \( x = 0 \) is the only real root of \( P'(x) = 0 \). If \( P(-1)<P(1) \), then in the interval \( [-1,1] \):

If \( x = \sqrt{2}e^t(\sin t - \cos t) \) and \( y = \sqrt{2}e^t(\sin t + \cos t) \), then \( \left[ \frac{d^2y}{dx^2} \right]_{t=\frac{\pi}{4}} \) is:

If the volume of a sphere is increasing at the rate of 12 \( \text{cm}^3/\text{sec} \), then the rate (in \( \text{cm}^2/\text{sec} \)) at which its surface area is increasing when the diameter of the sphere is 12 cm is:

If \( y = \operatorname{Sin}^{-1} \frac{\sqrt{1+\sin x}+\sqrt{1-\sin x}}{\sqrt{1+\sin x}-\sqrt{1-\sin x}} \) and \( -\frac{3\pi}{2}<x<-\frac{\pi}{2} \), then \( \frac{dy}{dx} \) is:

If a function \( f(x) \) is given as: \[ f(x) = \begin{cases} \frac{\sqrt{1+ax^2+bx^3}-\sqrt{1-ax^2-bx^3}}{x^2}, & x<0
5, & x = 0
\frac{\tan3x-\sin3x}{bx^3}, & x>0 \end{cases} \] and is continuous at \( x = 0 \), then the geometric mean of \( a \) and \( b \) is:

If \( y = \log(\sec(\tan^{-1}x)) \) for \( x>0 \), then \( \frac{dy}{dx} \) at \( x = 1 \) is:

Evaluate the limit: \[ \lim_{x \to \infty} \frac{3x+4\cos^2x}{\sqrt{x^2-5\sin^2x}} \]

If \( \theta \) is the acute angle between the tangents drawn from the point \( (1,1) \) to the hyperbola \( 4x^2-5y^2-20=0 \), then \( \tan\theta \) is:

A plane \( \pi \) given by \( ax+by+11z+d = 0 \) is perpendicular to the planes \( 2x-3y+z=4 \), \( 3x+y-z=5 \), and the perpendicular distance from the origin to the plane \( \pi \) is \( \sqrt{6} \) units. If all the intercepts made by the plane \( \pi \) on the coordinate axes are positive, then \( d = \):

The quadratic equation whose roots are \[ l = \lim_{\theta \to 0} \left( \frac{3\sin\theta - 4\sin^3\theta}{\theta} \right) \] \[ m = \lim_{\theta \to 0} \left( \frac{2\tan\theta}{\theta(1-\tan^2\theta)} \right) \] is:

A line segment \( PQ \) has the length 63 and direction ratios \( (3, -2, 6) \). If this line makes an obtuse angle with the X-axis, then the components of the vector \( \vec{PQ} \) are:

If \( A(2,-1,1) \), \( B(2,5,1) \) and \( C(0,-2,3) \) are the vertices of a triangle, and \( D \) is the point of intersection of the side \( BC \) and the internal angular bisector of angle \( A \), then \( AD = \):

The equations of the asymptotes of a hyperbola are \( x+y+3=0 \), \( 2x-y+1=0 \). If \( (1,-2) \) is a point on this hyperbola, then the equation of its conjugate hyperbola is:

If \( \theta \) is the angle between the tangents drawn from the point \( (-1, -1) \) to the circle \( x^2+y^2-4x-6y+c=0 \) and \( \cos\theta = -\frac{7}{25} \), then the radius of the circle is:

If the equation of the polar of the point \( (\alpha, -1) \) with respect to the circle \( x^2+y^2-4x-6y-12=0 \) is \( y = \beta \), then \( 4(\alpha+\beta) = \):

The lengths of the two focal chords of the parabola \( y^2 = 16x \) is 25 units each. If these two chords cut the parabola at \( A, B, C, D \), then the area (in sq. units) of the quadrilateral formed by \( A, B, C, D \) is:

If the tangents drawn from a point \( P \) to the ellipse \( 4x^2+9y^2-16x+54y+61=0 \) are perpendicular, then the locus of \( P \) is:

The radius of the circle passing through the points of intersection of the circles \( x^2+y^2+2x+4y+1=0 \), \( x^2+y^2-2x-4y-4=0 \), and intersecting the circle \( x^2+y^2=6 \) orthogonally is:

If the power of the point \( (1,6) \) with respect to the circle \( x^2+y^2+4x-6y-a=0 \) is \( -16 \), then \( a \) is:

From a point \( P(-4, 0) \), two tangents are drawn to the circle \( x^2+y^2-4x-6y-12=0 \) touching the circle at \( A \) and \( B \). If the equation of the circle passing through \( P, A, B \) is \( x^2+y^2+2gx+2fy+c=0 \), then \( (g,f) = \):

Let \( A(5,4) \) and \( B(5,-4) \) be two points. If \( P \) is a point in the coordinate plane such that \( |APB| = \frac{\pi}{4} \), then the point \( P \) lies on the curve:

When the axes are rotated through an angle \( \theta \) about the origin in the anticlockwise direction and then translated to the new origin (2, -2), if the transformed equation of \( x^2+y^2=4 \) is \( X^2+Y^2+aX+bY+c=0 \), then \( a+b+c= \):

If \( \left(\frac{2}{3},0\right) \) is the centroid of the triangle formed by the lines \( 4x^2 - y^2 = 0 \) and \( lx + my + n = 0 \), then \( l+m+n= \):