List of top Mathematics Questions

If the shortest distance between the lines
\(\vec{r} = (-\hat{i} + 3\hat{k}) + \lambda(\hat{i} - a\hat{j})\) and \(\vec{r} = (-\hat{j} + 2\hat{k}) + \mu(\hat{i} - \hat{j} + \hat{k})\)
is \(\sqrt{\frac23}\), then the integral value of a is equal to ____________.

Let \(A\left(\frac{3}{\sqrt{a}}, \sqrt{a}\right)\), \(a \gt 0\), be a fixed point in the xy-plane. The image of A in y-axis be B and the image of B in x-axis be C. If \(D(3\cos\theta, a\sin\theta)\) is a point in the fourth quadrant such that the maximum area of \(\Delta ACD\) is 12 square units, then a is equal to ____________.

Let a line having direction ratios, 1, \(-4\), 2 intersect the lines \(\frac{x - 7}{3} = \frac{y - 1}{-1} = \frac{z + 2}{1}\) and \(\frac{x}{2} = \frac{y - 7}{3} = \frac{z}{1}\) at the points A and B. Then \((AB)^2\) is equal to ____________.

In an examination, there are 5 multiple choice questions with 3 choices, out of which exactly one is correct. There are 3 marks for each correct answer, \(-2\) marks for each wrong answer and 0 mark if the question is not attempted. Then, the number of ways a student appearing in the examination gets 5 marks is ____________.

Let
\(\underset{0 \le x \le 2}{\mathrm{Max}}\left\{\frac{9 - x^2}{5 - x}\right\} = \alpha\) and \(\underset{0 \le x \le 2}{\mathrm{Min}}\left\{\frac{9 - x^2}{5 - x}\right\} = \beta\).
If \(\int_{\beta - \frac83}^{2\alpha - 1} \mathrm{Max}\left\{\frac{9 - x^2}{5 - x},\ x\right\}dx = \alpha_1 + \alpha_2\log_e\left(\frac{8}{15}\right)\), then \(\alpha_1 + \alpha_2\) is equal to ____________.

Let S be the region bounded by the curves \(y = x^3\) and \(y^2 = x\). The curve \(y = 2|x|\) divides S into two regions of areas \(R_1, R_2\). If \(\max\{R_1, R_2\} = R_2\), then \(\frac{R_2}{R_1}\) is equal to ____________.

The number of points where the function
\(f(x) = \begin{cases} |2x^2 - 3x - 7| & \text{if } x \le -1 \\ [4x^2 - 1] & \text{if } -1 \lt x \lt 1 \\ |x + 1| + |x - 2| & \text{if } x \ge 1 \end{cases}\)
\([t]\) denotes the greatest integer \(\le t\), is discontinuous is ____________.

Let \(f(\theta) = \sin\theta + \int_{-\pi/2}^{\pi/2} (\sin\theta + t\cos\theta)f(t)\,dt\). Then the value of \(\left|\int_0^{\pi/2} f(\theta)\,d\theta\right|\) is ____________.

Let \(S = \{\sqrt{n} : 1 \le n \le 50 \text{ and } n \text{ is odd}\}\). Let \(a \in S\) and
\(A = \begin{bmatrix} 1 & 0 & a \\ -1 & 1 & 0 \\ -a & 0 & 1 \end{bmatrix}\).
If \(\sum_{a \in S} \det(\operatorname{adj} A) = 100\lambda\), then \(\lambda\) is equal to :

Let \(S = \left\{\theta \in [-\pi, \pi] - \left\{\pm\frac{\pi}{2}\right\} : \sin\theta\tan\theta + \tan\theta = \sin 2\theta\right\}\). If \(T = \sum_{\theta \in S} \cos 2\theta\), then \(T + n(S)\) is equal to :

Let \(x^2 + y^2 + Ax + By + C = 0\) be a circle passing through (0, 6) and touching the parabola \(y = x^2\) at (2, 4). Then A + C is equal to __________.

If the tangent at the point \((x_1, y_1)\) on the curve \(y = x^3 + 3x^2 + 5\) passes through the origin, then \((x_1, y_1)\) does NOT lie on the curve :

Let \(\lambda x - 2y = \mu\) be a tangent to the hyperbola \(a^2x^2 - y^2 = b^2\). Then \(\left(\frac{\lambda}{a}\right)^2 - \left(\frac{\mu}{b}\right)^2\) is equal to :

The domain of the function
\(f(x) = \frac{\cos^{-1}\left(\frac{x^2-5x+6}{x^2-9}\right)}{\log_e(x^2-3x+2)}\)
is :

If
\(\{a_i\}_{i=1}^{n}\)
, where n is an even integer, is an arithmetic progression with common difference 1, and
\(\sum_{i=1}^{n} a_i = 192,\ \sum_{i=1}^{n/2} a_{2i} = 120\)
, then n is equal to :

The sum of absolute maximum and absolute minimum values of the function
\(f(x) = |2x^2 + 3x - 2| + \sin x \cos x\)
in the interval [0, 1] is :