List of top Mathematics Questions

Let \( \vec{b} = 3i - 2j + k \) and \( \vec{c} = i - j - k \) be two vectors. If \( \vec{a} \) is a vector such that \( \vec{a} + \vec{b} + \vec{c} = 0 \), then \( | \vec{a} \times \vec{b} + \vec{b} \times \vec{c} + \vec{c} \times \vec{a} | \) is:
If \( 2i\hat{i} - j\hat{j} - 3j\hat{k} \) and \( -3i\hat{i} + 4j\hat{j} - 4k\hat{k} \) are the position vectors of three points A, B, and C respectively, then \( \Delta ABC \) is:
Let \( \pi_1 \) be the plane determined by the vectors \( \overline{\mathbf{i}} + 2\overline{\mathbf{j}} - 2\overline{\mathbf{k}} \) and \( \overline{\mathbf{i}} + 3\overline{\mathbf{j}} - 2\overline{\mathbf{k}} \). Let \( \pi_2 \) be the plane determined by the vectors \( \overline{\mathbf{j}} + 2\overline{\mathbf{k}} \) and \( 3\overline{\mathbf{k}} - 2\overline{\mathbf{i}} \). If \( \theta \) is the angle between \( \pi_1 \) and \( \pi_2 \), then \( \cos \theta = \)
If $\vec{a} = 2\hat{i} - t\hat{j} - 2\hat{k}$ and $\vec{b} = 6\hat{i} + 2\hat{j} - 3\hat{k}$ are two vectors such that $\vec{a} \cdot \vec{b}$ is minimum, then $t=$
If $\vec{a} = 2\hat{i} - \hat{j} + 3\hat{k}$, $\vec{b} = -3\hat{i} + 5\hat{j} - 4\hat{k}$, $\vec{c} = 6\hat{i} - 4\hat{j} + 5\hat{k}$, then $(\vec{a} \times \vec{b}) \cdot (\vec{b} \times \vec{c}) = $
Planes $\pi_1$ and $\pi_2$ are defined by vectors. If $|\vec{a}| = \sqrt{14}$ and $\vec{a}$ is parallel to their intersection, then $|\vec{a} \cdot (\hat{i} + \hat{j} + \hat{k})| = $
The mean deviation from the mean for the data $ 6, 7, 10, 12, 13, 4, 12, 16 $ is:
If the sum of squares of the deviations from the mean of the data \( x_i \) (\( i = 1, 2, \dots, n \)) is \( n\overline{x}^2 \), where \( \overline{x} \) is the mean of \( x_i \)'s, then the sum of squares of \( x_i \)'s is
Students of two sections A and B of a class show the following results in a test conducted for 100 marks. Then, the comparison of variability between Section A and Section B is:
Students of two sections A and B of
If \( X \) is a random variable such that \[ P(X = -2) = P(X = -1) = P(X = 2) = P(X = 1) = \frac{1}{6}, \quad P(X = 0) = \frac{1}{3}, \] then the mean of \( X \) is:
An analysis of monthly wages paid to the workers of two jute mills A and B gives the following data:
\begin{tabular}{|c|c|c|} \hline & Mill - A & Mill - B
\hline No. of workers & 500 & 600
Average daily wage (in rupees) & 186 & 175
Variance of distribution of wages & 81 & 100
\hline \end{tabular} Then:
If \( f(x) = \begin{cases} \sqrt{\pi - \cos^{-1} x}, & x = -1 \\ \frac{\sqrt{2(1 + x)}}{\pi + \cos^{-1} x}, & x \neq -1 \end{cases} \) is right continuous at \( x = -1 \), then \( \lambda = \)
Let \( S_n = 1 + 3x + 9x^2 + 27x^3 + \ldots + n \text{ terms} -\frac{1}{3}<x<\frac{1}{3} \). If \( f(x) = S_n \), then \( f(x) \) is discontinuous at the point \( x = \):

If \( f(x) = \begin{cases} 1 + 6x - 3x^2, & x \leq 1 \\ x + \log_2(b^2 + 7), & x > 1 \end{cases} \) is continuous at all real \( x \), then \( b \) is:

Let \( f(x) \) be a real valued function. If \( f'(x) \) is a constant for all \( x \in \mathbb{R}, f(0) = 1 \) and \( f'(0) = 2 \), then
If \( f : \mathbb{R} \to \mathbb{R} \) is defined by: \[ f(x) = \begin{cases} \sin x - \sin \left( \frac{x}{2} \right), & x<0 \\ \frac{x}{\sqrt{x^2 + \sqrt{x^2}}}, & x>0 \end{cases} \] If \( f \) is continuous on \( \mathbb{R} \), then \( f(0) = \)
Let \(A(1,15),\ B(3,-12),\ C(6,12)\) be three consecutive turning points of a continuous curve \(y = f(x)\). If \(f(x) = 0\) only for \(x = \alpha\) and \(x = \beta\), then \[ |\beta - \alpha| < ? \]
The area enclosed between the curve \( y = \log_e(x + e) \) and the coordinate axes is:

The ratio in which the YZ-plane divides the line segment formed by joining the points (-2, 4, 7) and  (3, -5, 8)  is  2 : m.  
The value of m is:

If the coordinates of the points A and B are (3, 3)  and  (7, 6),  then the length of the portion of the line AB intercepted between the axes is:

The interval in which the function \( f(x) = 4x^2 + x \) is decreasing is:
If \( \int \frac{e^x (1 + \sin x)}{1 + \cos x} \, dx = e^x f(x) + C \), then \( f(x) \) is equal to:
Simplify \( i^{57} + \frac{1}{i^{25}} \) and find its value:
Evaluate the integral: \[ I = \int \frac{x + 3}{(x + 4)^2} e^x \,dx \]
The derivative of \( e^{x^3} \) with respect to \( \log x \) is: