List of top Mathematics Questions

Let the mean and variance of 8 numbers -10, -7, -1, x, y, 9, 2, 16 be \( 2 \) and \( \frac{293}{4} \), respectively. Then the mean of 4 numbers x, y, x+y+1, |x-y| is:

A rectangle is formed by the lines \( x = 0 \), \( y = 0 \), \( x = 3 \) and \( y = 4 \). Let the line \( L \) be perpendicular to \( 3x + y + 6 = 0 \) and divide the area of the rectangle into two equal parts. Then the distance of the point \( \left(\frac{1}{2}, -5\right) \) from the line \( L \) is equal to :

Let \( y = y(x) \) be the solution of the differential equation \( x^2 dy + (4x^2 y + 2\sin x)dx = 0 \), \( x>0 \), \( y\left(\frac{\pi}{2}\right) = 0 \). Then \( \pi^4 y\left(\frac{\pi}{3}\right) \) is equal to :

A building construction work can be completed by two masons A and B together in 22.5 days. Mason A alone can complete the construction work in 24 days less than mason B alone. Then mason A alone will complete the construction work in :

Let \( f(x) = \int \frac{(2 - x^2) \cdot e^x{(\sqrt{1 + x})(1 - x)^{3/2}} dx \). If \( f(0) = 0 \), then \( f\left(\frac{1}{2}\right) \) is equal to :}

Let the domain of \( f(x) = \log_3 \log_3 \log_7 (9x - x^2 - 13) \) be (m, n). Let the hyperbola \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) have eccentricity \( \frac{n}{3} \) and latus rectum \( \frac{8m}{3} \). Then \( b^2 - a^2 \) is equal to :

The value of the integral \( \int_{\frac{\pi}{24}}^{\frac{5\pi}{24}} \frac{dx}{1 + \sqrt[3]{\tan 2x}} \) is :

Among the statements :
I: If the given determinants are equal, then \( \cos^2\alpha + \cos^2\beta + \cos^2\gamma = \frac{3}{2} \), and
II: If the polynomial determinant equals \( px + q \), then \( p^2 = 196q^2 \), identify the truth value.

The vertices B and C of a triangle ABC lie on the line \( \frac{x}{1} = \frac{1-y}{2} = \frac{z-2}{3} \). The coordinates of A and B are (1, 6, 3) and (4, 9, 6) respectively and C is at a distance of 10 units from B. The area (in sq. units) of \( \triangle ABC \) is:

If \( \alpha \) and \( \beta \) (\( \alpha<\beta \)) are the roots of the equation \( (-2 + \sqrt{3})(\sqrt{x} - 3) + (x - 6\sqrt{x}) + (9 - 2\sqrt{3}) = 0 \), \( x \ge 0 \), then \( \sqrt{\frac{\beta}{\alpha}} + \sqrt{\alpha\beta} \) is equal to:

Let \( f(x) = \begin{cases} \frac{ax^2 + 2ax + 3}{4x^2 + 4x - 3}, & x \neq -\frac{3}{2}, \frac{1}{2} \\ b, & x = -\frac{3}{2}, \frac{1}{2} \end{cases} \) be continuous at \( x = -\frac{3}{2} \). If \( f(x) = \frac{7}{5} \), then \( x \) is equal to :

Let \( \alpha \) and \( \beta \) respectively be the maximum and the minimum values of the function \( f(\theta) = 4\left(\sin^4\left(\frac{7\pi}{2} - \theta\right) + \sin^4(11\pi + \theta)\right) - 2\left(\sin^6\left(\frac{3\pi}{2} - \theta\right) + \sin^6(9\pi - \theta)\right) \). Then \( \alpha + 2\beta \) is equal to :

Let the direction cosines of two lines satisfy the equations : \( 4l + m - n = 0 \) and \( 2mn + 5nl + 3lm = 0 \). Then the cosine of the acute angle between these lines is :

Number of solutions of \( \sqrt{3} \cos 2\theta + 8 \cos \theta + 3\sqrt{3} = 0, \theta \in [-3\pi, 2\pi] \) is:

Let the line \( y - x = 1 \) intersect the ellipse \( \frac{x^2}{2} + \frac{y^2}{1} = 1 \) at the points A and B. Then the angle made by the line segment AB at the center of the ellipse is:

The value of \[ \sum_{r=1}^{20}\sqrt{\left|\pi\left(\int_0^r x|\sin \pi x|\,dx\right)\right|} \] is:

For some \( \theta\in\left(0,\frac{\pi}{2}\right) \), let the eccentricity and the length of the latus rectum of the hyperbola \[ x^2-y^2\sec^2\theta=8 \] be \( e_1 \) and \( l_1 \), respectively, and let the eccentricity and the length of the latus rectum of the ellipse \[ x^2\sec^2\theta+y^2=6 \] be \( e_2 \) and \( l_2 \), respectively. If \[ e_1^2=\frac{2}{e_2^2}\left(\sec^2\theta+1\right), \] then \[ \left(\frac{l_1l_2}{e_1^2e_2^2}\right)\tan^2\theta \] is equal to:

If \[ k=\tan\!\left(\frac{\pi}{4}+\frac{1}{2}\cos^{-1}\!\left(\frac{2}{3}\right)\right) +\tan\!\left(\frac{1}{2}\sin^{-1}\!\left(\frac{2}{3}\right)\right), \] then the number of solutions of the equation \[ \sin^{-1}(kx-1)=\sin^{-1}x-\cos^{-1}x \] is:

In a G.P., if the product of the first three terms is \(27\) and the set of all possible values for the sum of its first three terms is \( \mathbb{R} - (a,b) \), then \( a^2+b^2 \) is equal to:

Let \( f \) be a polynomial function such that \[ f(x^2+1)=x^4+5x^2+2,\quad \text{for all } x\in\mathbb{R}. \] Then \[ \int_0^3 f(x)\,dx \] is equal to:

If \[ \int \frac{1-5\cos^2 x}{\sin^5 x\cos^2 x}\,dx=f(x)+C, \] where \( C \) is the constant of integration, then \[ f\!\left(\frac{\pi}{6}\right)-f\!\left(\frac{\pi}{4}\right) \] is equal to:

Let \( ABC \) be an equilateral triangle with orthocenter at the origin and the side \( BC \) lying on the line \( x+2\sqrt{2}\,y=4 \). If the coordinates of the vertex \( A \) are \( (\alpha,\beta) \), then the greatest integer less than or equal to \( |\alpha+\sqrt{2}\beta| \) is:

Let \( A, B, C \) be three \( 2\times2 \) matrices with real entries such that \[ B=(I+A)^{-1} \quad \text{and} \quad A+C=I. \] If \[ BC=\begin{bmatrix}1 & -5 \\-1 & 2\end{bmatrix} \quad \text{and} \quad B\begin{bmatrix}x_1\\x_2\end{bmatrix} =\begin{bmatrix}12\\-6\end{bmatrix}, \] then \( x_1+x_2 \) is:

Let \( S=\{1,2,3,4,5,6,7,8,9\} \). Let \( x \) be the number of 9-digit numbers formed using the digits of the set \( S \) such that only one digit is repeated and it is repeated exactly twice. Let \( y \) be the number of 9-digit numbers formed using the digits of the set \( S \) such that only two digits are repeated and each of these is repeated exactly twice. Then:

If \( \alpha,\beta \) where \( \alpha<\beta \), are the roots of the equation \[ \lambda x^2-(\lambda+3)x+3=0 \] such that \[ \frac{1}{\alpha}-\frac{1}{\beta}=\frac{1}{3}, \] then the sum of all possible values of \( \lambda \) is: