List of top Mathematics Questions

Find the value of the integral \( \int_0^{\frac{\pi}{2}} \sin^2(x) \, dx \).
If \[ \frac{x^4}{(x^2+1)(x-2)} = f(x) + \frac{Ax+B}{x^2+1} + \frac{C}{x-2} \] then \( f(14) + 2A - B = \) ?
Let \( z \neq 1 \) be a complex number and let \( \omega = x + iy, y \neq 0 \). If \[ \frac{\omega -\overline{\omega}z}{1 -z} \] is purely real, then \( |z| \) is equal to
Let \[ f(x) = \frac{|5 - x|(x + 5)}{\tan(x - 5)} \quad {for} \quad x \neq 5. \] Then \[ \lim_{x \to 5} f(x) { is equal to:} \]

If the real-valued function

\[ f(x) = \sin^{-1}(x^2 - 1) - 3\log_3(3^x - 2) \]

is not defined for all \( x \in (-\infty, a] \cup (b, \infty) \), then what is \( 3^a + b^2 \)?

The equation of the common tangent to the parabola \(y^2 = 8x\) and the circle \(x^2 + y^2 = 2\) is \(ax + by + 2 = 0\). If \(-\frac{a}{b}>0\), then \(3a^2 + 2b + 1 =\)
The ratio of the sum and product of the roots of the quadratic equation \(7x^2-12x+18=0\) is

Let \( I = \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \frac{\tan^2 x}{1+5^x} \, dx \). Then:

If \( f(x) = \ln \left( \frac{x^2 + e}{x^2 + 1} \right) \), then the range of \( f(x) \) is:
If \( f(x) = \cos^{-1 } \left( \frac{\sqrt{2x^2 + 1}}{x^2 + 1} \right) \), then the range of \( f(x) \) is:
Let
\[ I(R) = \int_0^R e^{-R \sin x} \, dx, \quad R > 0. \]
Which of the following is correct?
A circle passes through the points \( (2, 0) \) and \( (1, 2) \). If the power of the point \( (0, 2) \) with respect to this circle is 4, then the radius of the circle is
The acceleration \( f \) (in ft/sec\(^2\)) of a particle after a time \( t \) seconds starting from rest is given by:
\[ f = 6 - \sqrt{1.2t}. \]
Then the maximum velocity \( v \) and the time \( T \) to attain this velocity are:
Evaluate the limit: \[ \lim_{\theta \to \frac{\pi}{2}} \frac{8\tan^4\theta + 4\tan^2\theta + 5}{(3 - 2\tan\theta)^4} \]
If the volume of a cylinder is 500 m3 and the area of its base is 25 m2, then its height (in m) is
The coefficient of \(a^{10}b^7c^3\) in the expansion of \((bc + ca + ab)^{10}\) is:
The value of \(log_2 32\) is

{If \(f(x)\) is a quadratic function such that \(f\left(\frac{1}{x}\right) = f(x) + f\left(\frac{1}{1-x}\right)\), then \(\sqrt{f\left(\frac{2}{3}\right) + f\left(\frac{3}{2}\right)} =\)}

Let \( A = \{1, 3, 7, 9, 11\} \) and \( B = \{2, 4, 5, 7, 8, 10, 12\} \). Then the total number of one-one maps \( f: A \to B \), such that \( f(1) + f(3) = 14 \), is:
For the function $f(x) = 2x^3 - 9x^2 + 12x - 5, x \in [0, 3]$, match List-I with List-II:
List-IList-II
(A) Absolute maximum value(I) 3
(B) Absolute minimum value(II) 0
(C) Point of maxima(III) -5
(D) Point of minima(IV) 4
Let A, B, C denote the set of students in a college who play football, basketball, and cricket respectively. If \( n(A) = 60 \), \( n(B) = 55 \), \( n(C) = 70 \), \( n(A \cup B \cup C) = 100 \) and \( n(A \cap B \cap C) = 20 \), then the number of students who play exactly two of these sports is
If $f(x) = x^2 + bx + 1$ is increasing in the interval $[1, 2]$, then the least value of $b$ is:
The radius of the base of a cone is increasing at the rate of 3 cm/minute and the altitude is decreasing at the rate of 4 cm/minute. The rate of change of lateral surface when the radius is 7 cm and altitude is 24 cm is:
If the area of a rectangle is \(112\ m^2\) and its length is \(6\ m\) more than the breadth, then the breadth of the rectangle is
Let the vectors \( \overrightarrow{AB} = 2\hat{i} + 2\hat{j} + \hat{k} \) and \( \overrightarrow{AC} = 2\hat{i} + 4\hat{j} + 4\hat{k} \) be two sides of a triangle ABC. If \( G \) is the centroid of \( \triangle ABC \), then \( \frac{22}{7} |\overrightarrow{AG}|^2 + 5 = \): (a) 25