List of top Mathematics Questions

Let S1 : x² + y² = 9 and S2 : (x - 2)² + y² = 1. Then the locus of center of a variable circle S which touches S1 internally and S2 externally always passes through the points :
Let a complex number be w = 1 - √3 i. Let another complex number z be such that |z w| = 1 and arg(z) - arg(w) = π/2. Then the area of the triangle with vertices origin, z and w is equal to :
Define a relation R over a class of n × n real matrices A and B as "ARB iff there exists a non-singular matrix P such that P A P⁻¹ = B". Then which of the following is true ?
Let a and b be two non-zero vectors perpendicular to each other and |a| = |b|. If |a × b| = |a|, then the angle between the vectors (a + b + (a × b)) and a is equal to :
The area bounded by the curve 4y² = x²(4 - x)(x - 2) is equal to :
Let g(x) = ∫₀ˣ f(t) dt, where f is continuous function in [0, 3] such that 1/3 ≤ f(t) ≤ 1 for all t ∈ [0, 1] and 0 ≤ f(t) ≤ 1/2 for all t ∈ (1, 3]. The largest possible interval in which g(3) lies is :
Let S1 be the sum of first 2n terms of an arithmetic progression. Let S2 be the sum of first 4n terms of the same arithmetic progression. If (S2 - S1) is 1000, then the sum of the first 6n terms of the arithmetic progression is equal to :
A pole stands vertically inside a triangular park ABC. Let the angle of elevation of the top of the pole from each corner of the park be π/3. If the radius of the circumcircle of ΔABC is 2, then the height of the pole is equal to :
Let the centroid of an equilateral triangle ABC be at the origin. Let one of the sides of the equilateral triangle be along the straight line x+y=3. If R and r be the radius of circumcircle and incircle respectively of ΔABC, then (R + r) is equal to :
In a triangle ABC, if |BC| = 8, |CA| = 7, |AB| = 10, then the projection of the vector AB on AC is equal to :
If 15 sin⁴ α + 10 cos⁴ α = 6, for some α ∈ R, then the value of 27 sec⁶ α + 8 cosec⁶ α is equal to :
If P and Q are two statements, then which of the following compound statement is a tautology ?
Let in a series of 2n observations, half of them are equal to a and remaining half are equal to -a. Also by adding a constant b in each of these observations, the mean and standard deviation of new set become 5 and 20, respectively. Then the value of a² + b² is equal to :
Let \( P = \begin{bmatrix} 3 & -1 & -2 \\ 2 & 0 & \alpha \\ 3 & -5 & 0 \end{bmatrix} \), where \( \alpha \in \mathbb{R} \). Suppose \( Q = [q_{ij}] \) is a matrix satisfying \( PQ = k I_3 \) for some non-zero \( k \in \mathbb{R} \). If \( q_{23} = -\dfrac{k}{8} \) and \( |Q| = \dfrac{k^3}{2} \), then \( \alpha^2 + k^2 \) is equal to __________.
The equation of the plane passing through the point (1, 2, -3) and perpendicular to the planes $3x + y - 2z = 5$ and $2x - 5y - z = 7$, is :
The system of linear equations 3x - 2y - kz = 10, 2x - 4y - 2z = 6, x + 2y - z = 5m is inconsistent if :
If \( f(x) = [x - 1]\cos\!\left( \frac{2x - 1}{2}\pi \right) \), then \( f \) is :
The function \( f(x) = \dfrac{4x^3 - 3x^2}{6} - 2 \sin x + (2x - 1)\cos x \) :
\( \displaystyle \lim_{x \to 0} \frac{\int_{0}^{x^2} \sin(\sqrt{t}) \, dt}{x^3} \) is equal to :
Let \( p \) and \( q \) be two positive numbers such that \( p + q = 2 \) and \( p^4 + q^4 = 272 \). Then \( p \) and \( q \) are roots of the equation :
If \(\int_{-a}^{a} (|x| + |x-2|) dx = 22, (a>2)\) and \([x]\) denotes the greatest integer \(\le x\), then \(\int_{a}^{-a} (x + [x]) dx\) is equal to ______
\(\lim_{n \to \infty} \tan \left\{ \sum_{r=1}^{n} \tan^{-1} \left( \frac{1}{1 + r + r^2} \right) \right\}\) is equal to __________.
If the least and the largest real values of \(\alpha\), for which the equation \(|z| + \alpha(z - 1) + 2i = 0\) (\(z \in C\) and \(i = \sqrt{-1}\)) has a solution, are \(p\) and \(q\) respectively; then \(4(p^2 + q^2)\) is equal to _________
The minimum value of \(\alpha\) for which the equation \(\frac{4}{\sin x} + \frac{1}{1 - \sin x} = \alpha\) has at least one solution in \((0, \frac{\pi}{2})\) is ____________
Let \(M\) be any \(3\times3\) matrix with entries from the set {0, 1, 2}. The maximum number of such matrices, for which the sum of diagonal elements of \(M^T M\) is seven, is __________.