List of top Mathematics Questions

Let the two variables $ x $ and $ y $ satisfy the following conditions: \[ x + y \leq 50, \quad x + 2y \leq 80, \quad 2x + y \geq 20, \quad x, y \geq 0. \] Then the maximum value of $ Z = 4x + 3y $ is:
If the pair of tangents drawn to the circle \( x^2 + y^2 = a^2 \) from the point \( (10, 4) \) are perpendicular, then \( a \) is:
The perimeter of the locus of the point \( P \) which divides the line segment \( QA \) internally in the ratio 1:2, where \( A = (4, 4) \) and \( Q \) lies on the circle \( x^2 + y^2 = 9 \), is:
The general solution of \( 4 \cos 2x - 4 \sqrt{3} \sin 2x + \cos 3x - \sqrt{3} \sin 3x + \cos x - \sqrt{3} \sin x = 0 \) is:
In \( \triangle ABC \), prove the identity: $$ a^2 \sin 2B + b^2 \sin 2A = $$ 
The \( n^{th} \) term of the series \[ 1 + (3+5+7) + (9+11+13+15+17) + \dots \] is:
In the expansion of \(\frac{2x+1}{(1+x)(1-2x)}\), the sum of the coefficients of the first 5 odd powers of \(x\) is:
If the ratio of the terms equidistant from the middle term in the expansion of \( (1 + x)^{12} \) is \( \frac{1}{256} \), then the sum of all the terms of the expansion \( (1 + x)^{12} \) is:
If \( \vec{a} = \hat{i} - \hat{j} + \hat{k} \), \( \vec{b} = \hat{i} + \hat{j} - 2\hat{k} \), \( \vec{c} = 2\hat{i} - 3\hat{j} - 3\hat{k} \), and \( \vec{d} = 2\hat{i} + \hat{j} + \hat{k} \) are four vectors, then \( (\vec{a} \times \vec{c}) \times (\vec{b} \times \vec{d}) = \):
The number of ways of arranging 2 red, 3 white, and 5 yellow roses of different sizes into a garland such that no two yellow roses come together is:
There are 2 bags each containing 3 white and 5 black balls and 4 bags each containing 6 white and 4 black balls. If a ball drawn randomly from a bag is found to be black, then the probability that this ball is from the first set of bags is:
The determinant of the matrix $$ \begin{bmatrix} a & b & c \\ a^2 & b^2 & c^2 \\ 1 & 1 & 1 \end{bmatrix} $$ is not equal to: 
If the solution of the system of simultaneous linear equations: $$ x + y - z = 6, $$ $$ 3x + 2y - z = 5, $$ $$ 2x - y - 2z + 3 = 0 $$ is \( x = \alpha, y = \beta, z = \gamma \), then \( \alpha + \beta = ?\) 
If \( A = \begin{pmatrix} 1 & 2 \\ -2 & -5 \end{pmatrix} \) and \( \alpha^2 + \beta A = 21 \) for some \( \alpha, \beta \in \mathbb{R} \), then find \( \alpha + \beta \):
If \( Z \) is a complex number such that \( |Z| \leq 3 \) and \( -\frac{\pi}{2} \leq \text{arg } Z \leq \frac{\pi}{2} \), then the area of the region formed by the locus of \( Z \) is:
The number of ways of selecting 3 numbers that are in GP from the set \( \{1, 2, 3, \dots, 100\} \) is:
It is given that in a random experiment, events A and B are such that \( P(A) = \frac{1}{4} ,P(A|B) = \frac{1}{2} \) and \( P(B|A) = \frac{2}{3} \). Then \( P(B) \) is:
When two dice are thrown, the probability of getting the sum of the values on them as 10 or 11 is:
A unit vector perpendicular to the vectors \( \bar{a} = 2\bar{i} + 3\bar{j} + 4\bar{k} \) and \( \bar{b} = 3\bar{j} + 2\bar{k} \) is:
The angle subtended by the chord \( x + y - 1 = 0 \) of the circle \( x^2 + y^2 - 2x + 4y + 4 = 0 \) at the origin is: 
The cubic equation whose roots are the squares of the roots of the equation \( 12x^3 - 20x^2 + x + 3 = 0 \) is:
There are 6 different novels and 3 different poetry books on a table. If 4 novels and 1 poetry book are to be selected and arranged in a row on a shelf such that the poetry book is always in the middle, then the number of such possible arrangements is:
The radius of the circle which cuts the circles \( x^2 + y^2 - 4x - 4y + 7 = 0 \), \( x^2 + y^2 + 4x + 6 = 0 \), and \( x^2 + y^2 + 4x + 4y + 5 = 0 \) orthogonally is:
Which of the following functions are odd? 
\[ \begin{aligned} \text{I. } & f(x) = x \left( \frac{e^x -1}{e^x +1} \right) \\[8pt] \text{II. } & f(x) = k^x + k^{-x} + \cos x \\[8pt] \text{III. } & f(x) = \log \left( x + \sqrt{x^2 +1} \right) \end{aligned} \]
If the circles \( x^2 + y^2 + 2ax + 2y - 8 = 0 \) and \( x^2 + y^2 - 2x + ay - 14 = 0 \) intersect orthogonally, then the distance between their centers is: