List of top Mathematics Questions

Evaluate the integral: \[ I = \int_{0}^{32\pi} \sqrt{1 - \cos 4x} \,dx \]
The end of a latus rectum of the ellipse \(3x^2 + 4y^2 = 12\) is lying in the third quadrant. If the normal drawn at \(L_1\) to this ellipse intersects the ellipse again at the point \(P(a,b)\), then find the value of \(a\):
The equation of the normal drawn to the curve \( y^3 = 4x^5 \) at the point \( (4,16) \) is:

Let \( A = [a_{ij}] \) be a \( 3 \times 3 \) matrix with positive integers as its elements. The elements of \( A \) are such that the sum of all the elements of each row is equal to 6, and \( a_{22} = 2 \). 
 

\(\lim_{x \to 0}\) \(\frac{3^{\sin x} - 2^{\tan x}}{\sin x}\) =

The numerically greatest term in the expansion of (3x - 16y)15 when x = 2⁄3 and y = 3⁄2 is ?

If \( p \) and \( q \) are the real numbers such that the 7th term in the expansion of \( \left( \frac{5}{p^3} \cdot \frac{3q}{7} \right)^8 \) is 700, then \( 49p^2 = \):
If \(\tan A<0\) and \(\tan 2A = -\frac{4}{3}\), then \(\cos 6A\) is:
If \[ \int \sqrt{\csc x + 1} \, dx = k \tan^{-1} (f(x)) + c, \] then \[ \frac{1}{k} f\left(\frac{\pi}{6}\right) = ? \]
The equation \( 16x^4 + 16x^3 - 4x - 1 = 0 \) has a multiple root. If \( \alpha, \beta, \gamma, \delta \) are the roots of this equation, then \( \frac{1}{\alpha^4} + \frac{1}{\beta^4} + \frac{1}{\gamma^4} + \frac{1}{\delta^4} = \)
Among the 4-digit numbers that can be formed using the digits \(\{1,2,3,4,5,6\}\) without repeating any digit, the number of such numbers which are divisible by 6 is \;?

With respect to the roots of the equation \( 3x^3 + bx^2 + bx + 3 = 0 \), match the items of List-I with those of List-II.

A man has 7 relatives, 4 of them are ladies and 3 gents; his wife has 7 other relatives, 3 of them are ladies and 4 gents. The number of ways they can invite them to a party of 3 ladies and 3 gents so that there are 3 of man's relatives and 3 of wife's relatives, is 

\(\frac{1}{3.6} + \frac{1}{6.9} + \frac{1}{9.12} + \dots\) up to 9 terms \(=\)
The point \((p, q)\) is the point of intersection of a latus rectum and an asymptote of the hyperbola \(9x^2 - 16y^2 = 144\). If \(p>0\) and \(q>0\), then \(q = \ldots\)

Given below are two statements, one is labelled as Assertion (A) and the other one labelled as Reason (R).
Assertion (A): \[ 1 + \frac{2.1}{3.2} + \frac{2.5.1}{3.6.4} + \frac{2.5.8.1}{3.6.9.8} + \dots \infty = \sqrt{4} \] Reason (R): \[ |x| <1, \quad (1 - x)^{-1} = 1 + nx + \frac{n(n+1)}{1.2} x^2 + \frac{n(n+1)(n+2)}{1.2.3} x^3 + \dots \] 

If \[ f(x) = 3x^{15} - 5x^{10} + 7x^5 + 50\cos(x - 1), \] then \[ \lim_{h \to 0} \frac{f(1 - h) - f(1)}{h^2 + 3h} \] is:

The variance of the following continuous frequency distribution is:

class intervalFrequency
0 -- 42
4 -- 83
8 -- 122
12 -- 161

 

The area of the region enclosed by the curves \( y^2 = 4(x+1) \) and \( y^2 = 5(x-4) \) is:
The equations of the directrices of the ellipse \(9x^2 + 4y^2 - 18x - 16y - 11 = 0\) are:
If \( 3^{2n+2} - 8n - 9 \) is divisible by \( 2^p \) \(\forall n \in \mathbb{N}\), then the maximum value of \( p \) is

\[ \textbf{If } | \text{Adj} \ A | = x \text{ and } | \text{Adj} \ B | = y, \text{ then } \left( | \text{Adj}(AB) | \right)^{-1} \text{ is } \]

\[ D = \begin{vmatrix} -\frac{bc}{a^2} & \frac{c}{a} & \frac{b}{a} \\ \frac{c}{b} & -\frac{ac}{b^2} & \frac{a}{b} \\ \frac{b}{c} & \frac{a}{c} & -\frac{ab}{c^2} \end{vmatrix} \]

(a,b) is the point of concurrency of the lines \(x-3y+3=0\), \(Kx+y+k=0\) and \(2x+y-8=0\). If the perpendicular distance from the origin to the line \(L: ax-by+2k=0\) is \(p\), then the perpendicular distance from the point \((2,3)\) to \(L=0\) is:
A rhombus is inscribed in the region common to the two circles \[ x^2 + y^2 - 4x - 12 = 0 \] and \[ x^2 + y^2 + 4x - 12 = 0. \] If the line joining the centres of these circles and the common chord of them are the diagonals of this rhombus, then the area (in Sq. units) of the rhombus is: