List of top Mathematics Questions

Let A be a 3 × 3 real matrix such that det(A) = 6 and
\(adj\ A=\begin{pmatrix} 1 & -1 & 2 \\ 5 & 7 & 1 \\ -1 & 1 & 1 \end{pmatrix}\)
where adj A denotes the adjoint of A.
Then the trace of A equals __________ (round off to 2 decimal places)

Let f ∶ \(\R^2 → \R\) be the function defined by f(x, y) = x² - 12y. If M and m be the maximum value and the minimum value, respectively, of the function f on the circle x² + y² = 49, then |M| + |m| equals __________

The value of
\(\int^2_0 \int_0^{2-x}(x+y)^2e^{\frac{2y}{x+y}}\ dy\ dx\)
equals __________ (round off to 2 decimal places)

Let \(A=\begin{pmatrix} 1 & -1 & 2 \\ -1 & 0 & 1 \\ 2 & 1 & 1 \end{pmatrix}\) and let \(\begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix}\)be an eigenvector corresponding to the smallest eigenvalue of A, satisfying \(x^2_1+x^2_2+x^2_3=1\). Then the value of |x₁| + |x₂| + |x₃| equals __________ (round off to 2 decimal places)

The equation of the hyperbola with vertices at

\[ (0, \pm 6) \quad \text{and} \quad e = \frac{5}{3} \]

is:

The value of the integral \[ \int_0^{\frac{\pi}{2}} \frac{\sqrt{\sin x}}{\sqrt{\sin x} + \sqrt{\cos x}} \, dx \] is:

For what value of \( k \), does the equation \[ 9x^2 + y^2 = k(x^2 - y^2 - 2x) \] represent the equation of a circle?

The area enclosed between the graph of \( y = x^3 \) { and the lines} \[ x = 0, \, y = 1, \, y = 8 { is:} \]

Consider the ellipse
\(\frac{x^2}{4} + \frac{y^2}{3} = 1\).
Let \(H (α, 0), 0 < α < 2\) , be a point. A straight line drawn through H parallel to y-axis crosses the ellipse and its auxiliary circle at points E and F respectively, in the first quadrant. The tangents to the ellipse at the point E intersects the positive x-axis at a point G. Suppose the straight line joining F and the origin makes an angle \(\phi\) with the positive x-axis.

List-I		List-II
(I)	If \(\Phi = \frac{\pi}{4}\), then the area of the triangle FGH is	P	\(\frac{(\sqrt{3}-1)^4}{8}\)
(II)	If \(\Phi = \frac{\pi}{3}\), then the area of the triangle FGH is	Q	1
(III)	If \(\Phi = \frac{\pi}{6}\), then the area of the triangle FGH is	R	\(\frac{3}{4}\)
(IV)	If \(\Phi = \frac{\pi}{12}\), then the area of the triangle FGH is	S	\(\frac{1}{2\sqrt{3}}\)
		T	\(\frac{3\sqrt{3}}{2}\)

The radius of the sphere is increased by 100% then the volume of the resultant sphere is increased by

For positive integer $n$, define $f(n)=n+\frac{16+5 n-3 n^2}{4 n+3 n^2}+\frac{32+n-3 n^2}{8 n+3 n^2}+\frac{48-3 n-3 n^2}{12 n+3 n^2}+\ldots+\frac{25 n-7 n^2}{7 n^2}$.
Then, the value of $\displaystyle\lim _{n \rightarrow \infty} f(n)$ is equal to

If A = {1,2,3........,10} then number of subsets of A containing only odd number is

If \( A = \{3, 5, 7\} \) and \( B = \{1, 2, 3, 5\} \), then \( A \times B \cap B \times A \) is equal to:

The value of \( 3^{3-\log_3 5} \) is

If \( \alpha, \beta \) are the roots of \( x^2 - x - 1 = 0 \), and \( A_n = \alpha^n + \beta^n \), then Arithmetic mean of \( A_{n-1} \) and \( A_n \) is

If \( (\vec{a} \times \vec{b}) \times \vec{c} = \vec{a} \times (\vec{b} \times \vec{c}) \), then

Suppose that the temperature at a point (x, y) on a metal plate is \( T(x, y) = 4x^2 - 4xy + y^2 \). An ant, walking on the plate, traverses a circle of radius 5 centered at the origin. What is the highest temperature encountered by the ant?

If \( x^m y^n = (x+y)^{m+n} \), then \( \frac{dy}{dx} \) is

If \( \csc\theta - \cot\theta = 2 \), then the value of \( \csc\theta \) is

The greatest number which on dividing 1657 and 2037 leaves remainders 6 and 5 respectively is

The first three moments of a distribution about 2 are 1, 16, -40 respectively. The mean and variance of the distribution are

The 10th and 50th percentiles of the observation 32, 49, 23, 29, 118 respectively are

Let a, b, c be distinct non-negative numbers. If the vectors \( a\hat{i} + a\hat{j} + c\hat{k} \), \( \hat{i} + \hat{k} \) and \( c\hat{i} + c\hat{j} + b\hat{k} \) lie in a plane, then c is

The domain of the function \( f(x) = \frac{\cos^{-1}x}{[x]} \) is

If the roots of the quadratic equation \( x^2 + px + q = 0 \) are tan 30\(^{\circ}\) and tan 15\(^{\circ}\) respectively, then the value of \( 2 + p - q \) is