List of top Mathematics Questions

Consider the following statements:
Assertion (A): When \( x, y, z \) are positive numbers, then \[ \tan^{-1}\left( \sqrt{\frac{x(x+y+z)}{yz}} \right) + \tan^{-1}\left( \sqrt{\frac{y(x+y+z)}{xz}} \right) + \tan^{-1}\left( \sqrt{\frac{z(x+y+z)}{xy}} \right) = \pi \] Reason (R): \( \tan^{-1}a + \tan^{-1}b = \tan^{-1}\left( \frac{a+b}{1-ab} \right) \) if \( a>0 \) and \( b>0 \).

If \( 3\sin(\alpha-\beta) = 5\cos(\alpha+\beta) \) and \( \alpha+\beta \neq \frac{\pi}{2} \), then \( \frac{\tan(\frac{\pi}{4}-\alpha)}{\tan(\frac{\pi}{4}-\beta)} = \)

\( \frac{\sin 1^\circ + \sin 2^\circ + \dots + \sin 89^\circ}{2(\cos 1^\circ + \cos 2^\circ + \dots + \cos 44^\circ) + 1} = \)

If \( 5\sin\theta + 3\cos\left(\theta + \frac{\pi}{3}\right) + 3 \) lies between \( \alpha \) and \( \beta \) (including \( \alpha, \beta \) also), then \( (\alpha-\beta)(\alpha+\beta-6) = \)

If \( \frac{x^2-3}{(x+2)(x^2+1)} = \frac{A}{x+2} + \frac{Bx+C}{x^2+1} \), then \( 3A+2B-C = \)

When \( |x|>3 \), the coefficient of \( \frac{1}{x^n} \) in the expansion of \( x^{3/2} (3+x)^{1/2} \) is

The constant term in the expansion of \( \left(1+\frac{1}{x}\right)^{20} \left(30x(1+x)^{29} + (1+x)^{30}\right) \) is

If all the letters of the word MOST are permuted and the words (with or without meaning) thus obtained are arranged in the dictionary order then the rank of the word STOM when counted from the rank of the word MOST, is

The number of integers lying between 1000 and 10000 such that the sum of all the digits in each of those numbers becomes 30 is

The number of non negative integral solutions of the equation \( x+y+z+t=10 \) when \( x \ge 2, z \ge 5 \) is

If \( \alpha, \beta, \gamma \) are the roots of the equation \( x^3 - Px^2 + Qx - R = 0 \) and \( (\alpha-2)^2, (\beta-2)^2, (\gamma-2)^2 \) are the roots of the equation \( x^3 - 5x^2 + 4x = 0 \), then the possible least value of \( P+Q+R \) is

The number of all common roots of the equation \( x^4 - 10x^3 + 37x^2 - 60x + 36 = 0 \) and the transformed equation of it obtained by increasing any two distinct roots of it by 1, keeping the other two roots fixed, is

If the equation \( x^2 - 3ax + a^2 - 2a - K = 0 \) has different real roots for every rational number \( a \), then \( K \) lies in the interval

If \( l \) is the maximum value of \( -3x^2+4x+1 \) and \( m \) is the minimum value of \( 3x^2+4x+1 \), then the equation of the hyperbola having foci at \( (l,0), (7m,0) \) and eccentricity as 2 is

In Argand plane, no value of \( \sqrt[3]{1-i\sqrt{3}} \) lie in

If \( n, K \in \mathbb{N} \) such that \( n \neq 3K \), then \( (\sqrt{3}+i)^{2n} + (\sqrt{3}-i)^{2n} = \)

If \( |Z|=2 \), \( Z_1 = \frac{Z}{2}e^{i\alpha} \) and \( \theta \) is the amp(Z), then \( \frac{Z_1^n - Z_1^{-n}}{Z_1^n + Z_1^{-n}} = \)

\( \omega \) is a complex cube root of unity and \( Z \) is a complex number satisfying \( |Z-1| \le 2 \). The possible values of \( r \) such that \( |Z-1| \le 2 \) and \( |\omega Z - 1 - \omega^2| = r \) have no common solution are

A and B are two non-square matrices. If \( P = A + B \), \( Q = A^TB \), \( R = AB^T \), then the matrices whose order is equal to the order of A are

A, C are \( 3 \times 3 \) matrices. B, D are \( 3 \times 1 \) matrices. If \( AX=B \) has a unique solution and \( CX=D \) has an infinite number of solutions, then

If \( A+2B = \begin{bmatrix} 1 & 2 & 0 \\ 6 & -3 & 3 \\ -5 & 3 & 1 \end{bmatrix} \) and \( 2A-B = \begin{bmatrix} 2 & -1 & 5 \\ 2 & -1 & 6 \\0 & 1 & 2 \end{bmatrix} \), then \( \text{tr}(A) - \text{tr}(B) = \)

If \( \left| \begin{matrix} 1 & 2 & 3-\lambda \\ 0 & -1-\lambda & 2 \\ 1-\lambda & 1 & 3 \end{matrix} \right| = A\lambda^3 + B\lambda^2 + C\lambda + D \), then \( D+A = \)

The value of the greatest positive integer \( k \), such that \( 49^k + 1 \) is a factor of \( 48(49^{125} + 49^{124} + \dots + 49^2 + 49 + 1) \) is

The inverse of the function \( y = \frac{10^x - 10^{-x}}{10^x + 10^{-x}} + 1 \) is \( x = \)