List of top Mathematics Questions

A body cools according to Newton’s law from \(100^\circ\text{C}\) to \(60^\circ\text{C}\) in 20 minutes. The temperature of the surroundings being \(20^\circ\text{C}\), the temperature of the body after one hour is

If \(A(3,-2,2)\), \(B(2,\lambda+1,5)\) are the end points of the diameter of a circle and if the point \((5,6,-1)\) lies on the circle, then \(\lambda =\)

The domain of the real valued function \[ f(x) = \frac{x + 2}{9 - x^2} \] is

If \(A, B, C\) are the angles of \(\triangle ABC\), then with usual notations, \[ \frac{c^2 - a^2 + b^2}{a^2 - b^2 + c^2} = \]

The rate of growth of bacteria is proportional to the bacteria present. If it is found that the number doubles in 3 hours, then the number of times the bacteria are increased in 6 hours is

The joint equation of two lines through the origin, each of which makes an angle of \(30^\circ\) with the line \(x + y = 0\), is

If \(\displaystyle \sin\!\left(\frac{x+y}{x-y}\right) = \tan \frac{\pi}{5}\), then find \(\displaystyle \frac{dy}{dx}\).

If \[ A = \begin{bmatrix} 2 & 3 \\ 1 & 2 \end{bmatrix}, \quad B = \begin{bmatrix} 1 & 0 \\ 3 & 1 \end{bmatrix}, \] then find \((AB)^{-1}\).

An urn contains 4 red and 5 white balls. Two balls are drawn one after the other without replacement. Find the probability that both the balls are red.

The probability mass function of a random variable \(X\) is given by \[ P(X = x) = \frac{5cx}{2^5}, \quad x = 0,1,2,3,4,5 \] \[ P(X = x) = 0, \quad \text{otherwise} \] Then find \(P(X \le 2)\).

Find the value of the following expression:
\[ 5^2 + 6^2 + 7^2 + \cdots + 20^2 = \]

With usual notations in \( \triangle ABC \), \(a = 3\), \(c = 2\) and \(\sin C = \dfrac{2}{3}\), then \(\angle A =\)

The acute angle between the line \[ \vec{r} = (i + 2j + k) + \lambda (i + j + k) \] and the plane \[ \vec{r} \cdot (2i - j + k) = 5 \] is

The probability density function of a random variable \(X\) is given by \[ f(x) = \frac{k}{\sqrt{x}}, \quad 0 \le x \le 4 \] \[ f(x) = 0, \quad \text{otherwise} \] Find \(P(1<X<4)\).

The line through the points \( (1, 4), (-5, 1) \) intersects the line \( 4x + 3y - 5 = 0 \) in the point

Which of the following matrix is invertible?
\[ A_1 = \begin{pmatrix} 4 & 2 \\ 2 & 1 \end{pmatrix} \] \[ A_2 = \begin{pmatrix} -1 & -2 & 3 \\ 4 & 5 & 7 \\ 2 & 4 & -6 \end{pmatrix} \] \[ A_3 = \begin{pmatrix} 1 & 0 & 0 \\ 5 & 2 & 1 \\ 7 & 2 & 1 \end{pmatrix} \] \[ A_4 = \begin{pmatrix} 1 & 0 & 1 \\ 0 & 2 & 3 \\ 1 & 2 & 1 \end{pmatrix} \]

If the Cartesian equation of the line is \[ x - 1 = 2y + 3 = 3 - z, \] then its vector equation is

The parametric equations of the line passing through \( A(3, 4, -7) \), \( B(1, -1, 6) \) are

The area of the region bounded by the parabola \[ y^2 = 8x \quad \text{and its latus rectum is} \]

If \[ |3x - 2| \leq \frac{1}{2}, \] then \( x \in \)

Evaluate the integral: \[ \int e^{\cos^{-1}x} \left[ x - \sqrt{1 - x^2} \right] \frac{dx}{\sqrt{1 - x^2}}. \]

With usual notations, in \( \triangle ABC \), if \( a = 2 \), \( b = 3 \), \( c = 5 \) and \[ \frac{\cos A}{a} + \frac{\cos B}{b} + \frac{\cos C}{c} = \frac{k}{7} + \frac{30}{30}, \] then \( k = \)

Evaluate the integral: \[ \int \frac{dx}{\cos x \sqrt{\cos 2x}}. \]

The rate of growth of bacteria is proportional to the number present. If initially there were 1000 bacteria and the number doubles in 1 hour, then the number of bacteria after \( 2\frac{1}{2} \) hours are (Given \( \sqrt{2} = 1.414 \))

If the equation \[ x^2 - 3xy + y^2 + 3x - 5y + 2 = 0 \] represents a pair of lines, where \( \theta \) is the angle between them, then the value of \( \csc^2 \theta \) is