List of top Mathematics Questions

If \(u = \sin\left(\frac{x}{y}\right),\ x = e^t,\ y = t^2\), then \[ t^6 \left(\frac{du}{dt}\right)^2 \div e^{2t}(t - 2)^2 = \]
Let the circle \( S \) be concentric with the circle \( x^2 + y^2 - 2x + ky + 4 = 0 \). If one of the diameters of \( S \) lies along the line \( 3x - 2y + 4 = 0 \) and the length of the diameter is 6, then the radius of the circle \( S \) is
If \( f'(x) = a \cos x + b \sin x \), \( f'(0) = 4 \), \( f(0) = 3 \), and \( f\left( \frac{\pi}{2} \right) = 5 \), then \( f(x) = \)
If \( f(x) \) is a function such that \( f'(x) = \sqrt{f^2(x) - 1} \) and \( f(0) = 1 \), then \( f(1) = \):
If the mean and varian of a binomial variable $X$ are 2 and 1 respectively, then $P(X>1) = $
The area (in square units) bounded by \[ x = 4, \quad y = -4, \quad \text{and} \quad y = x \quad \text{is:} \]
The line \( x + y = k \) meets the curve \( x^2 + y^2 - 2x - 4y + 2 = 0 \) at two points \( A \) and \( B \). If \( O \) is the origin and \( \angle AOB = 90^\circ \), then the value of \( k \) (\( k>1 \)) is
Let \( A = \begin{pmatrix} 0 & 3 & 5 & -7 \\ 8 & 0 & -1 & 0 \\ 6 & -1 & 0 & 0 \end{pmatrix} \) and \( X = \begin{pmatrix} x \\ y \\ z \end{pmatrix} \). If \( D = [\alpha \, \beta \, \gamma]^T \) is the solution of \( X^T B^T = A^T X \), then \( D^T A = \)
If the equation \( x^4 + ax^3 + bx^2 + cx + d = 0 \) has three equal roots, then the root is:
If \[ \frac{d}{dx}\left( \frac{x^2}{(x+2)(2x+3)} \right) = \frac{-A}{(x+2)^2} + \frac{B}{(2x+3)^2}, \] then the value of \( A + B = \):
A pair of dice is thrown. Then the probability that either of the dice shows 2 when their sum is 6 is:
Tangent \( L_1 = 3x - 4y - 8 = 0 \) and the chord \( L_2 = x + y - 1 = 0 \) are at a distance of 2 and \( \sqrt{2} \) units respectively from the centre of a circle \( S \). \((h, k)\) is the centre of \( S \) such that \( h^2 + k^2 = 13 \). If the midpoint of the chord \( L_2 = 0 \) is \( (\alpha, \beta) \) and the radius of the circle is \( r \), then \( \alpha + \beta + r = \):
If \( 2i\hat{i} - j\hat{j} - 3j\hat{k} \) and \( -3i\hat{i} + 4j\hat{j} - 4k\hat{k} \) are the position vectors of three points A, B, and C respectively, then \( \Delta ABC \) is:
If $\vec{a} = 2\hat{i} - t\hat{j} - 2\hat{k}$ and $\vec{b} = 6\hat{i} + 2\hat{j} - 3\hat{k}$ are two vectors such that $\vec{a} \cdot \vec{b}$ is minimum, then $t=$
If a circle \(S\) passing through the points \(A(1, 2)\) and \(B(2, 1)\) has its centre \(C\) located in the third quadrant at a distance of \(\frac{7}{\sqrt{2}}\) units from \(AB\), then the point \(P(1, -2)\)
If a set \( A \) has \( n \) elements, then the number of functions defined from \( A \) to \( A \) that are not one-one is:
If \( f(x) = \begin{vmatrix} 1 & 6 + x & 36 + x^2 \\ 0 & x - 3 & 3x^2 - 27 \\ 0 & 2x - 4 & 8x^2 - 32 \end{vmatrix} \), then \( \lim\limits_{x \to 1} \frac{f(x)}{f(-x)} \) is:
If \( \alpha \in \mathbb{R} \setminus \{-1\} \) and \[ f(x) = |x| + \alpha |x|(|x| - 1), \] then the number of points at which \( f(x) \) is not differentiable is:
If $\log y$ is an integrating factor of $\frac{dx}{dy}+P(y)x=Q(y)$, then $P(y)=$

Match the items of List - A with those of the entries of List - B.

If \( \frac{6x^3 + 7x^2 - 14x + 11}{6x^3 + x^2 - 10x + 3} = \frac{a}{x + p} + \frac{b}{qx + 3} + \frac{c}{3x + p} \), then \( a + b \) is:
If a straight line \( L \) perpendicular to the line \( 3x - 4y = 6 \) forms a triangle of area 6 square units with coordinate axes, then the minimum perpendicular distance from the point \( (1,1) \) to the line \( L \) is:
If $\int_0^{2\pi} |x \sin x| dx = k\pi$, then $k=$
If \( 1 + \sqrt{1 + a} = (1 + \sqrt{1 - a}) \cot \alpha \) and \( 0<a<1 \), then \( \sin 4\alpha = \)
If p and q are the x and y intercepts respectively of the line passing through the points \( (a \cos \alpha, b \sin \alpha) \) and \( (a \cos \beta, b \sin \beta) \), then: