List of top Mathematics Questions

If \( \mathbf{b} \) and \( \mathbf{c} \) are any two non-collinear unit vectors and \( \mathbf{a} \) is any vector, then \[ (\mathbf{a} \cdot \mathbf{b})\mathbf{b} + (\mathbf{a} \cdot \mathbf{c})\mathbf{c} + \frac{(\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}))}{|\mathbf{b} \times \mathbf{c}|^2} (\mathbf{b} \times \mathbf{c}) \] is equal to:

The degree of the differential equation \( x = \frac{dy}{dx} + \frac{1}{2!} \left(\frac{dy}{dx}\right)^2 + \frac{1}{3!} \left(\frac{dy}{dx}\right)^3 + \cdots \) is:

Let \(A\) and \(B\) be two events such that \(P(A \cup B) = \frac{1}{6}\), \(P(A \cap B) = \frac{1}{4}\) and \(P(\overline{A}) = \frac{1}{4}\), where \(\overline{A}\) stands for the complement of event \(A\). Then, the events \(A\) and \(B\) are:

If \( f(x) = \int_0^1 e^{|t - x|} \, dt \) for \( 0 \leq x \leq 1 \), then the maximum value of \( f(x) \) is:

The locus of the mid-points of the chords of the circle \(x^2 + y^2 = 16\) which are tangent to the hyperbola \(9x^2 - 16y^2 = 144\) is

The number of positive integers satisfying the inequality \(^{n+1}C_{n-1} + ^{n}C_{n-1} \leq 50\) is

The intercepts on the straight line \(y = mx\) by the lines \(y = 2\) and \(y = 6\) is less than 5, then \(m\) belongs to

A line passing through origin and is perpendicular to two given lines \(2x + y + 6 = 0\) and \(4x + 2y - 9 = 0\). The ratio in which the origin divides this line, is

All the chords of the hyperbola \(3x^2 - 2y^2 - 4x + y = 0\), subtending a right angle at the origin pass through the fixed point

The vector equation of the plane through the point \(\hat{i} + \hat{j} - 2\hat{k}\) and perpendicular to the line of intersection of the planes \(\mathbf{r} \cdot (\hat{i} - \hat{j} + 3\hat{k}) = 1\) and \(\mathbf{r} \cdot (4\hat{i} - 2\hat{j} + 2\hat{k}) = 2\) is

\(\int \cos^{-3/7} x \sin^{-11/7} x \, dx\) is equal to

If \(P = \begin{bmatrix} \frac{\sqrt{3}}{2} & \frac{1}{2} -\frac{1}{2} & \frac{\sqrt{3}}{2} \end{bmatrix}\), \(A = \begin{bmatrix} 1 & 1 0 & 1 \end{bmatrix}\), and \(Q = PAP'\), then \(P'Q^{2005}P\) is

A vector perpendicular to the plane containing the points \(A(1, 1, -2)\), \(B(2, 0, -1)\), \(C(0, 2, 1)\) is

If \(Q\) is the image of the point \(P(2, 3, 4)\) under the reflection in the plane \(x - 2y + 5z = 6\), then the equation of the line \(PQ\) is

The equation of the plane passing through the line of intersection of the planes \(x + y + z = 6\) and \(2x + 3y + 4z + 5 = 0\) and perpendicular to the plane \(4x + 5y + 3z = 8\) is

\(1 + \frac{2}{2!} + \frac{3}{3!} + \frac{4}{4!} + \cdots \infty\) equals

The equation of the plane through the points \((2, 3, 1)\) and \((4, -5, 3)\) parallel to X-axis is

The number of values of \(x\), where \(f(x) = \cos x + \cos 2x\) attains its maximum is

In a \(\triangle ABC\), if \(\sin A \sin B = \frac{ab}{c^2}\), then the triangle is

If \(\theta\) is the semi-vertical angle of a cone of maximum volume and given slant height, then \(\tan \theta\) is given by

If \(x - 2y = 4\), then the minimum value of \(xy\) is

The foci of the conic section \(25x^2 + 16y^2 - 150x - 175 = 0\) are

The direction cosines \(l, m, n\) of two lines are connected by the relations \(l + m + n = 0\) and \(lm = 0\), then the angle between them is

The value of \(\int \frac{\cos x + \sin x}{\cos x + x \sin x} \, dx\) is

In a \(\triangle ABC\), if \(b = 2\), \(\angle B = 30^\circ\), then the area of the circumcircle of \(\triangle ABC\) (in sq units) is