List of top Mathematics Questions

In the expansion of $(1 + x)^n$, $\frac{{}^nC_1}{{}^nC_0} + 2 \cdot \frac{{}^nC_2}{{}^nC_1} + 3 \cdot \frac{{}^nC_3}{{}^nC_2} + \dots + n \cdot \frac{{}^nC_n}{{}^nC_{n-1}}$ is equal to:

The length of a rectangle is five times the breadth. If the minimum perimeter of the rectangle is $180$ cm, then:

The real value of $\alpha$ for which $\frac{1 - i \sin \alpha}{1 + 2i \sin \alpha}$ is purely real is:

If $\triangle ABC$ is right-angled at $C$, then the value of $\tan A + \tan B$ is:

If in two circles, arcs of the same length subtend angles $30^\circ$ and $78^\circ$ at the centre, then the ratio of their radii is:

If $[x]^2 - 5[x] + 6 = 0$, where $[x]$ denotes the greatest integer function, then:

Two finite sets have m and a elements respectively. The total number of subsets of the first set is 56 more than the total number of subsets of the second set. The values of m and a respectively are:

If \( \cos^2 84^\circ + \sin^2 126^\circ - \sin 84^\circ \cos 126^\circ = K \) and \( \cot A + \tan A = 2K \), then the possible values of \( \tan A \) are:

 Let \( R \) be a relation on \( \mathbb{Z} \times \mathbb{Z} \) defined by \((a, b) R (c, d)\) if and only if \(ad - bc\) is divisible by 5.
Then \( R \) is:

Let \( e_1 \) be the eccentricity of the hyperbola $$ \frac{x^2}{16} - \frac{y^2}{9} = 1 $$ and \( e_2 \) be the eccentricity of the ellipse $$ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, \quad a > b, $$ which passes through the foci of the hyperbola. If \( e_1 e_2 = 1 \), then the length of the chord of the ellipse parallel to the x-axis and passing through (0, 2) is: 

The sum of the coefficients of \( x^{2/3} \) and \( x^{-2/5} \) in the binomial expansion of $$ \left( x^{2/3} + \frac{1}{2} x^{-2/5} \right)^9 $$ is: 

The area (in square units) of the region enclosed by the ellipse $$ x^2 + 3y^2 = 18 $$ in the first quadrant below the line \( y = x \) is: 

Consider the relations $R_1$ and $R_2$ defined as \[a R_1 b \iff a^2 + b^2 = 1 \quad \text{for all } a, b \in \mathbb{R},\]and \[(a, b) R_2 (c, d) \iff a + d = b + c \quad \text{for all } (a, b), (c, d) \in \mathbb{N} \times \mathbb{N}.\]Then:

Let a relation \( R \) on \( \mathbb{N} \times \mathbb{N} \) be defined as: $$(x_1, y_1) \, R \, (x_2, y_2) \text{ if and only if } x_1 \leq x_2 \text{ or } y_1 \leq y_2.$$ 
Consider the two statements:
[(I)] \( R \) is reflexive but not symmetric. 
[(II)] \( R \) is transitive.
Then which one of the following is true:

The sum of all rational terms in the expansion of $$ \left( \frac{1}{2^5} + \frac{1}{5^3} \right)^{15} $$ is equal to: 

Evaluate: $$ \lim_{n \to \infty} \frac{1}{n^{k+1}} \left[ 2^k + 4^k + 6^k + \dots + (2n)^k \right]. $$ 

Find the general solution of the differential equation: $$ (xy + y^2)dx - (x^2 - 2xy)dy = 0. $$ is 

Let \( f : \mathbb{R} \rightarrow \mathbb{R} \) be a function defined by
$$ f(x) = \frac{x}{(1 + x^4)^{1/4}} $$
and \( g(x) = f(f(f(x))) \). Then
$$ 18 \int_{\sqrt[3]{\frac{8}{3}}}^{\sqrt[3]{\frac{4}{3}}} x^3 g(x) \, dx $$
equals: 

If a real-valued function $$ f(x) = \begin{cases} \frac{2x^2 + k(2x) + 9}{3x^2 - 7x - 6}, & \text{for } x \neq 3, \\ l, & \text{for } x = 3 \end{cases} $$ is continuous at \( x = 3 \) and \( l \) is a finite value, then \( l - k = \): 

If the differential equation $$ x \, dy + (y + y^2 x) \, dx = 0 $$ with condition \( y = 1 \) at \( x = 1 \), then the solution is: