List of top Mathematics Questions

The angle between the straight lines, whose direction cosines are given by the equations 2l + 2m - n = 0 and mn + nl + lm = 0, is :
The equation of the plane passing through the line of intersection of the planes \( \vec{r} \cdot (\hat{i} + \hat{j} + \hat{k}) = 1 \) and \( \vec{r} \cdot (2\hat{i} + 3\hat{j} - \hat{k}) + 4 = 0 \) and parallel to the x-axis is :
The Boolean expression \((p \land q) \Rightarrow ((r \land q) \land p)\) is equivalent to :
The area of the region bounded by the parabola \((y-2)^2 = (x-1)\), the tangent to it at the point whose ordinate is 3 and the x-axis is :
If \( \lim_{x \to \infty} (\sqrt{x^2 - x + 1} - ax) = b \), then the ordered pair (a, b) is :
A box open from top is made from a rectangular sheet of dimension a \(\times\) b by cutting squares each of side x from each of the four corners and folding up the flaps. If the volume of the box is maximum, then x is equal to:
If 0 \(<\) x \(<\) 1 and y = \(\frac{1}{2}x^2 + \frac{2}{3}x^3 + \frac{3}{4}x^4 + ...\), then the value of e\(^{1+y}\) at x = \(\frac{1}{2}\) is :
The set of all values of k \(>\) -1, for which the equation
(3x\(^2\)+4x+3)\(^2\) - (k+1)(3x\(^2\)+4x+3)(3x\(^2\)+4x+2) + k(3x\(^2\)+4x+2)\(^2\) = 0 has real roots, is :
Let [\(\lambda\)] be the greatest integer less than or equal to \(\lambda\). The set of all values of \(\lambda\) for which the system of linear equations x+y+z=4, 3x+2y+5z=3, 9x + 4y + (28 + [\(\lambda\)])z = [\(\lambda\)] has a solution is :
If $\int \frac{dx}{(x^2 + x + 1)^2} = a \tan^{-1} \left( \frac{2x + 1}{\sqrt{3}} \right) + b \left( \frac{2x + 1}{x^2 + x + 1} \right) + C, x>0$ where C is the constant of integration, then the value of $9(\sqrt{3}a + b)$ is equal to _________.
If the minimum area of the triangle formed by a tangent to the ellipse $\frac{x^2}{b^2} + \frac{y^2}{4a^2} = 1$ and the co-ordinate axis is kab, then k is equal to _________.
Let the equation $x^2 + y^2 + px + (1 - p)y + 5 = 0$ represent circles of varying radius $r \in (0, 5]$. Then the number of elements in the set $S = \{q : q = p^2 \text{ and } q \text{ is an integer}\}$ is _________.
If $y^{1/4} + y^{-1/4} = 2x$, and $(x^2 - 1)\frac{d^2y}{dx^2} + \alpha x \frac{dy}{dx} + \beta y = 0$, then $|\alpha - \beta|$ is equal to _________.
If the system of linear equations
$2x + y - z = 3$
$x - y - z = \alpha$
$3x + 3y + \beta z = 3$
has infinitely many solutions, then $\alpha + \beta - \alpha\beta$ is equal to _________.
If $A = \{x \in \mathbb{R} : |x-2|>1\}$, $B = \{x \in \mathbb{R} : \sqrt{x^2-3}>1\}$, $C = \{x \in \mathbb{R} : |x-4| \geq 2\}$ and $\mathbb{Z}$ is the set of all integers, then the number of subsets of the set $(A \cap B \cap C)^c \cap \mathbb{Z}$ is _________.
$\sum_{k=0}^{20} (^{20}C_k)^2$ is equal to :
If $0<x<1$, then $\frac{3}{2}x^2 + \frac{5}{3}x^3 + \frac{7}{4}x^4 + \dots$, is equal to :
$\int_{6}^{16} \frac{\log_e x^2}{\log_e x^2 + \log_e(x^2 - 44x + 484)} \, dx$ is equal to :
Let $\frac{\sin A}{\sin B} = \frac{\sin(A - C)}{\sin(C - B)}$, where A, B, C are angles of a triangle ABC. If the lengths of the sides opposite these angles are a, b, c respectively, then :
Let $y = y(x)$ be the solution of the differential equation $\frac{dy}{dx} = 2(y + 2\sin x - 5)x - 2\cos x$ such that $y(0) = 7$. Then $y(\pi)$ is equal to :
If $U_n = \left(1 + \frac{1}{n^2}\right) \left(1 + \frac{2^2}{n^2}\right)^2 \dots \left(1 + \frac{n^2}{n^2}\right)^n$, then $\lim_{n \to \infty} (U_n)^{-\frac{4}{n^2}}$ is equal to :
If $\alpha, \beta$ are the distinct roots of $x^2 + bx + c = 0$, then $\lim_{x \to \beta} \frac{e^{2(x^2 + bx + c)} - 1 - 2(x^2 + bx + c)}{(x - \beta)^2}$ is equal to :
If for $x, y \in \mathbb{R}, x>0, y = \log_{10} x + \log_{10} x^{1/3} + \log_{10} x^{1/9} + \dots$ upto $\infty$ terms and $\frac{2 + 4 + 6 + \dots + 2y}{3 + 6 + 9 + \dots + 3y} = \frac{4}{\log_{10} x}$, then the ordered pair $(x, y)$ is equal to :
If the matrix \( A = \begin{pmatrix} 0 & 2 \\ K & -1 \end{pmatrix} \) satisfies \( A(A^3 + 3I) = 2I \), then the value of \( K \) is :
If $S = \left\{z \in \mathbb{C} : \frac{z - i}{z + 2i} \in \mathbb{R}\right\}$, then :