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List of top Mathematics Questions

If $ \alpha, \beta $ are the roots of the equation $ x^2 - 6x - 2 = 0 $, $ \alpha > \beta $, and $ a_n = \alpha^n - \beta^n, n \geq 1 $, then the value of $ \frac{a_{10} - 2 a_8}{2 a_9} $ is:

S is the sample space and A, B are two events of a random experiment. Match the items of List A with the items of List B.

Then the correct match is:

Evaluate the integral \[ \int \frac{x^4 + 1}{x^6 + 1} dx. \]

If $ X \sim B(5, p) $ is a binomial variate such that $ p(X = 3) = p(X = 4) $, then $ P(|X - 3|<2) = \dots $

If two numbers $x$ and $y$ are chosen one after the other at random with replacement from the set of numbers $ \{1, 2, 3, \ldots, 10\} $, then the probability that $ |x^2 - y^2| $ is divisible by 6 is:

The area (in square units) of the smaller region lying above the X-axis and bounded between the circle \[ x^2 + y^2 = 2ax \] and the parabola \[ y^2 = ax \]

A person is known to speak false once out of 4 times. If that person picks a card at random from a pack of 52 cards and reports that it is a king, then the probability that it is actually a king is

Evaluate $ \int (\log x)^m x^n dx $.

Evaluate the integral: \[ I = \int_{-5\pi}^{5\pi} \left(1 - \cos 2x \right)^{\frac{5}{2}} dx. \]

If the ratio of the terms equidistant from the middle term in the expansion of $(1 + x)^{12}$ is $\frac{1}{256}$, then the sum of all the terms of the expansion $(1 + x)^{12}$ is:

The descending order of magnitude of the eccentricities of the following hyperbolas is: A. A hyperbola whose distance between foci is three times the distance between its directrices. B. Hyperbola in which the transverse axis is twice the conjugate axis. C. Hyperbola with asymptotes $ x + y + 1 = 0, x - y + 3 = 0 $.

A test containing 3 objective type of questions is conducted in a class. Each question has 4 options and only one option is the correct answer. No two students of the class have answered identically and no student has written all correct answers. If every student has attempted all the questions, then the maximum possible number of students who have written the test is:

If $ \omega $ is a complex cube root of unity and if $ Z $ is a complex number satisfying $ |Z - 1| \leq 2 $ and \[ |\omega^2 Z - 1 - \omega| = a, \] then the set of possible values of $ a $ is:

The solution of the differential equation \[ x dy - y dx = \sqrt{x^2 + y^2} dx \] when $ y(\sqrt{3}) = 1 $ is:

If \[ \cos x \frac{dy}{dx} - y \sin x = 6x, \quad (0 < x < \frac{\pi}{2}) \quad \text{and} \quad y(\frac{\pi}{3}) = 0, \quad \text{then} \quad y(\frac{\pi}{6}) = \]

A line $ L $ passes through $ (1,2,-3) $ and $ (3,3,-1) $, and a plane $ \pi $ passes through $ (2,1,-2), (-2,-3,6), (0,2,-1) $. If $ \theta $ is the angle between $ L $ and $ \pi $, then $ 27 \cos^2 \theta = $ ?

In a class consisting of 40 boys and 30 girls, 30% of the boys and 40% of the girls are good at Mathematics. If a student selected at random from that class is found to be a girl, then the probability that she is not good at Mathematics is:

The algebraic equation of degree 4 whose roots are the translates of the roots of the equation $ x^4 + 5x^3 + 6x^2 + 7x + 9 = 0 $ by $ -1 $ is:

Evaluate the integral \[ \int \frac{x^4 + 1}{x^6 + 1} dx. \]

If $$ \lim\limits_{x \to \infty} \frac{\left(\sqrt{2x+1} + \sqrt{2x-1}\right) + \left(\sqrt{2x+1} - \sqrt{2x-1}\right) P x^4 - 16} {(x+\sqrt{x^2 - 2}) + (x - \sqrt{x^2 - 2})} = 1, $$ then P = ?

The angle between the planes $ \vec{r} \cdot (12\hat{i} + 4\hat{j} - 3\hat{k}) = 5 $ and $ \vec{r} \cdot (5\hat{i} + 3\hat{j} + 4\hat{k}) = 7 $ is:

If $ n \geq 2 $ is a natural number and $ 0<\theta<\frac{\pi}{2} $, then \[ \int \frac{(\cos^n \theta - \cos \theta)^{1/n}}{\cos^{n+1} \theta} \sin \theta \, d\theta = \]