List of top Mathematics Questions

Let \( a, b, c, d \) be non-zero numbers. If the point of intersection of the lines \[ 4ax + 2ay + c = 0 \text{and} 5bx + 2by + d = 0 \] lies in the fourth quadrant and is equidistant from the two, then the relation between \( a, b, c, d \) is
Which of the following numbers is the coefficient of \( x^{100} \) in the expansion of \[ \log_e \left( \frac{1 + x}{1 + x^2} \right), \text{where} |x| < 1? \]
The locus of the mid-point of all chords of the parabola \( y^2 = 4x \) which are drawn through its vertex is:
If \[ \prod_{i=1}^{n} \tan (\alpha_i) = 1 \forall \alpha_i \in \left[0, \frac{\pi}{2}\right] \text{where} i = 1, 2, 3, \dots, n. \] Then the maximum value of \[ \prod_{i=1}^{n} \sin (\alpha_i) \] is
For a group of 100 candidates, the mean and standard deviation of scores were found to be 40 and 15 respectively. Later on, it was found that the scores 25 and 35 were misread as 52 and 53 respectively. Then the corrected mean and standard deviation corresponding to the corrected figures are

The negation of \( \sim S \vee ( \sim R \wedge S) \) \(\text{ is equivalent to}\)

The range of values of \( \theta \) in the interval \( \left( 0, \pi \right) \) such that the points \( (3, 2) \) and \( (\cos \theta, \sin \theta) \) lie on the same sides of the line \( x + y - 1 = 0 \), is:
Number of points at which \( f(x) \) is not differentiable, where \[ f(x) = |\cos x| + 3 \text{in} [-\pi, \pi] \]

Let A and B be sets. \[A \cap X = B \cap X = \varnothing \quad \text{and} \quad A \cup X = B \cup X \quad \text{for some set } X,\ \text{find the relation between } A \text{ and } B.\]
 

The value of \[ \int_{-\frac{\pi}{3}}^{\frac{\pi}{3}} \frac{x \sin x}{\cos^2 x} \, dx \text{ is:} \]

If \( f(x) = \lim_{x \to 0} \frac{6^x - 3^x - 2^x + 1}{\log_e 9 (1 - \cos x)} \) \(\text{ is a real number, then }\) \( \lim_{x \to 0} f(x) = \)

If \( \mathbf{a} \) and \( \mathbf{b} \) are vectors in space, given by \( \mathbf{a} = \frac{\hat{i} - 2\hat{j}}{\sqrt{5}} \) and \( \mathbf{b} = \frac{2\hat{i} + \hat{j} + 3\hat{k}}{\sqrt{14}} \), then the value of \[ (2\mathbf{a} + \mathbf{b}) \cdot \left[ (\mathbf{a} \times \mathbf{b}) \times ( \mathbf{a} - 2\mathbf{b} ) \right] \] is:
In an examination of nine papers, a candidate has to pass in more papers than the number of papers in which he fails in order to be successful. The number of ways in which he can be unsuccessful is:
If a vector having magnitude of 5 units, makes equal angle with each of the three mutually perpendicular axes, then the sum of the magnitude of the projections on each of the axis is
A point \( P \) in the first quadrant, lies on \( y^2 = 4ax \), \( a > 0 \), and keeps a distance of \( 5a \) units from its focus. Which of the following points lies on the locus of \( P \)?

Consider the following frequency distribution table. 
\[ \begin{array}{|c|c|} \hline \textbf{Class Interval} & \textbf{Frequency} \\ \hline 10-20 & 180 \\ \hline 20-30 & f_1 \\ \hline 30-40 & 34 \\ \hline 40-50 & 180 \\ \hline 50-60 & 136 \\ \hline 60-70 & 50 \\ \hline 70-80 & f_2 \\ \hline \end{array} \] If the total frequency is 685 and the median is 42.6, then the values of \( f_1 \) and \( f_2 \) are 
 

A bag contains different kinds of balls in which there are 5 yellow, 4 black, and 3 green balls. If 3 balls are drawn at random, then find the probability that no black ball is chosen.
Find foci of the equation \( x^2 + 2x - 4y^2 + 8y - 7 = 0 \)
Largest value of \( \cos^2 \theta - 6 \sin \theta \cos \theta + 3 \sin^2 \theta + 2 \)
The graph of the function \( f(x) = \log_e \left( x^3 + \sqrt{x^6 + 1} \right) \) is symmetric about:

Let \[ f(x) = \frac{x^2 - 1}{|x| - 1}. \] \(\text{Then the value of}\) \[ \lim_{x \to 1} f(x) \text{ is:} \]

If \( |x - 6| = |x^2 - 4x| - |x^2 - 5x + 6| \), \(\text{ where \( x \) is a real variable.}\)

Given two events \( A \) and \( B \) such that the odds in favour of \( A \) are 2:1 and the odds in favour of \( A \cup B \) are 3:1. Consistent with this information, the smallest and largest value for the probability of event \( B \) are given by
If \( x_k = \cos \left( \frac{2\pi k}{n} \right) + i \sin \left( \frac{2\pi k}{n} \right) \), then \[ \sum_{k=1}^{n} x_k = ? \]

If \[ \mathbf{a} = \hat{i} - \hat{k}, \mathbf{b} = x\hat{i} + \hat{j} + (1 - x)\hat{k}, \mathbf{c} = y\hat{i} + x\hat{j} + (1 + x - y)\hat{k}, \] \(\text{then }\) [\(\mathbf{a}\)  \(\mathbf{b}\)  \(\mathbf{c}\)] \(\text{ depends on:}\)