List of top Mathematics Questions

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 __________.
Let \(A = \{n \in N : n \text{ is a 3-digit number}\}\), \(B = \{9k + 2 : k \in N\}\) and \(C = \{9k + l : k \in N\}\) for some \(l\) (0<l<9) If the sum of all the elements of the set \(A \cap (B \cup C)\) is \(274 \times 400\), then \(l\) is equal to __________
The statement among the following that is a tautology is :
Two vertical poles are 150 m apart and the height of one is three times that of the other. If from the middle point of the line joining their feet, an observer finds the angles of elevation of their tops to be complementary, then the height of the shorter pole (in meters) is :
The tangent to the curve $y = x^3$ at the point $P(t, t^3)$ meets the curve again at Q, then the ordinate of the point which divides PQ internally in the ratio 1:2 is :