List of top Mathematics Questions

If \(\alpha\) satisfies the equation \(x^2 + x + 1 = 0\) and \((1 + \alpha)^7 = A + B \alpha + C \alpha^2\), \(A, B, C \geq 0\), then \(5(3A - 2B - C)\) is equal to ______.

Let \(k\in R\). If \(\)\(\)\(\)\(\lim\limits_{x→0+}(\sin(\sin kx)+\cos x+x)^{\frac{2}{x}}=e^6\), then the value of k is

Let the set of all values of \( p \), for which \[ f(x) = (p^2 - 6p + 8)(\sin^2 2x - \cos^2 2x) + 2(2 - p)x + 7 \] does not have any critical point, be the interval \( (a, b) \). Then \( 16ab \) is equal to _____ .

Let a line passing through the point \((-1, 2, 3)\) intersect the lines\( L_1 : \frac{x - 1}{3} = \frac{y - 2}{2} = \frac{z + 1}{-2} \) at \( M(\alpha, \beta, \gamma) \) and\( L_2 : \frac{x + 2}{-3} = \frac{y - 2}{-2} = \frac{z - 1}{4} \) at \( N(a, b, c) \).Then the value of \( \frac{(\alpha + \beta + \gamma)^2}{(a + b + c)^2} \) equals ____.

Let \(a=3\sqrt2\) and \(b=\frac{1}{5^{1/6}\sqrt5}\). If x, y ∈ R are such that
\(3x+2y=\log_a(18)^{\frac{5}{4}}\) and
\(2x-y=\log_b(\sqrt{1080})\), 
then 4x + 5y is equal to _______.

The square root of \( 7 + 24i \) is:

Let $f(x) = \frac{1}{7 - \sin 5x}$ be a function defined on $\mathbb{R}$. Then the range of the function $f(x)$ is equal to:

If \( \mathbf{a}, \mathbf{b}, \mathbf{c} \) are the position vectors of the points \( A, B, C \) respectively and \[ 5\mathbf{a} - 3\mathbf{b} - 2\mathbf{c} = \mathbf{0}, \] then find the ratio in which the point \( C \) divides the line segment \( BA \) externally.

Let the coefficient of \( x^r \) in the expansion of \((x + 3)^{n-1} + (x + 3)^{n-2} (x + 2) + (x + 3)^{n-3} (x + 2)^2 + \ldots + (x + 2)^{n-1}\)
be \( \alpha_r \). If \( \sum_{r=0}^n \alpha_r = \beta^n - \gamma^n \), \( \beta, \gamma \in \mathbb{N} \), then the value of \( \beta^2 + \gamma^2 \) equals \(\_\_\_\_\_\).

A software company sets up m number of computer systems to finish an assignment in 17 days. If 4 computer systems crashed on the start of the second day, 4 more computer systems crashed on the start of the third day and so on, then it took 8 more days to finish the assignment. The value of m is equal to :

If the coefficient of \(x^{30}\) in the expansion of \(\left(1 + \frac{1}{x}\right)^6 (1 + x^2)^7 (1 - x^3)^8, \, x \neq 0\) is \(\alpha\), then \(|\alpha|\) equals ____.

If the mean of the following probability distribution of a random variable \( X \): \[ \begin{array}{|c|c|c|c|c|c|} \hline X & 0 & 2 & 4 & 6 & 8 \\ \hline P(X) & a & 2a & a+b & 2b & 3b \\ \hline \end{array} \] is \( \frac{46}{9} \), then the variance of the distribution is:

Let A be a square matrix such that \(AA^T=I\).Then \(\frac{1}{2}A[(A+A^T)^2+(A-A^T)^2]\) is equal to

The number of ways in which 21 identical apples can be distributed among three children such that each child gets at least 2 apples, is

If a sin 45°= b cosec 30°, then the value of \(\frac {a^4}{b^4}\) is

If \(f(x) =ln(\frac {1-x^2}{1+x^2})\) then value of \(225(f'(x) – f''(x))\) at \(x=\frac 12\)

Consider a line \( L \) passing through the points \( P(1, 2, 1) \) and \( Q(2, 1, -1) \). If the mirror image of the point \( A(2, 2, 2) \) in the line \( L \) is \( (\alpha, \beta, \gamma) \), then \( \alpha + \beta + 6\gamma \) is equal to .

Let a die rolled till 2 is obtained. The probability that 2 obtained on even numbered toss is equal to: