List of top Mathematics Questions

Two friends were born in the year 2000. The probability that they have the same birthday is :

A box contains cards numbered 6 to 50. A card is drawn at random from the box. The probability that the drawn card has a number which is a perfect square, is :

If probability of winning a game is \(p\), then probability of losing the game is:

Two dice are rolled together. The probability of getting a doublet is:

Two dice are rolled together. The probability of getting a doublet is:

A card is drawn from a well-shuffled deck of 52 playing cards. The probability that drawn card is a red queen, is:

A card is drawn from a well-shuffled deck of 52 playing cards. The probability that drawn card is a red queen, is:

Two friends were born in the year 2000. The probability that they have the same birthday is :

Let \( \alpha = \sum_{k=0}^{n} \left( \frac{\binom{n}{k}}{k+1} \right)^2 \) and \( \beta = \sum_{k=0}^{n-1} \left( \frac{\binom{n}{k} \binom{n}{k+1}}{k+2} \right) \).
If \( 5\alpha = 6\beta \), then \( n \) equals __________.

\(^{n-1}C_r = (k^2 - 8)  ^{n}C_{r+1}\) if and only if:

If \( A \) denotes the sum of all the coefficients in the expansion of \( (1 - 3x + 10x^2)^n \) and \( B \) denotes the sum of all the coefficients in the expansion of \( (1 + x^2)^n \), then:

If the coefficient of \(x^{30}\) in the expansion of \(\left(1 + \frac{1}{x}\right)^6 (1 + x^2)^7 (1 - x^3)^8, \, x \neq 0\) is \(\alpha\), then \(|\alpha|\) equals ____.

Let \( a \) be the sum of all coefficients in the expansion of \( (1 - 2x + 2x^2)^{2023} (3 - 4x^2 + 2x^3)^{2024} \). and \( b = \lim_{x \to 0} \frac{\int_0^x \frac{\log(1 + t)}{t^{2024} + 1} \, dt}{x^2} \).If the equations \( cx^2 + dx + e = 0 \) and \( 2bx^2 + ax + 4 = 0 \) have a common root, where \( c, d, e \in \mathbb{R} \), then \( d : c : e \) equals

In the expansion of \[ (1 + x)(1 - x^2) \left( 1 + \frac{3}{x} + \frac{3}{x^2} + \frac{1}{x^3} \right)^5, \quad x \neq 0, \]the sum of the coefficients of \( x^3 \) and \( x^{-13} \) is equal to ____

Let the coefficient of \( x^r \) in the expansion of \((x + 3)^{n-1} + (x + 3)^{n-2} (x + 2) + (x + 3)^{n-3} (x + 2)^2 + \ldots + (x + 2)^{n-1}\)
be \( \alpha_r \). If \( \sum_{r=0}^n \alpha_r = \beta^n - \gamma^n \), \( \beta, \gamma \in \mathbb{N} \), then the value of \( \beta^2 + \gamma^2 \) equals \(\_\_\_\_\_\).

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

Let \[ a = 1 + \frac{{^2C_2}}{3!} + \frac{{^3C_2}}{4!} + \frac{{^4C_2}}{5!} + \dots,\]
\[ b = 1 + \frac{{^1C_0 + ^1C_1}}{1!} + \frac{{^2C_0 + ^2C_1 + ^2C_2}}{2!} + \frac{{^3C_0 + ^3C_1 + ^3C_2 + ^3C_3}}{3!} + \dots \]Then \( \frac{2b}{a^2} \) is equal to _____

If the coefficients of \( x^4 \), \( x^5 \), and \( x^6 \) in the expansion of \( (1 + x)^n \) are in arithmetic progression, then the maximum value of \( n \) is:

If the constant term in the expansion of $\left(\frac{\sqrt[5]{3}}{x}+\frac{2x}{\sqrt[3]{5}}\right)^{12}$, $x \neq 0$, is $\alpha \times 2^8 \times \sqrt[5]{3}$, then $25\alpha$ is

Let $\alpha = \sum_{r=0}^n (4r^2 + 2r + 1) \binom{n}{r}$ and $\beta = \left( \sum_{r=0}^n \frac{\binom{n}{r}}{r+1} \right) + \frac{1}{n+1}$. If $140 < \frac{2\alpha}{\beta} < 281$, then the value of $n$ is _____.

If the term independent of \(x\) in the expansion of \[ \left( \sqrt{ax^2} + \frac{1}{2x^3} \right)^{10} \] is 105, then \(a^2\) is equal to:

Let \(\vec a =3î + ĵ -2k̂\), \(\vec b =4 î + ĵ +7 k̂\) and \(\vec c =î -3 ĵ +4 k̂\) be 3 vectors. If a vector \(\vec p\)  satisfies \(\vec p_x \vec b=\vec c_x\vec b\) and \(\vec p_x\vec a = 0\) then \(\vec p.( î - ĵ - k̂ )\) is equal to

Let the position vectors of the vertices \( A, B \) and \( C \) of a triangle be \[ 2\mathbf{i} + 2\mathbf{j} + \mathbf{k}, \quad \mathbf{i} + 2\mathbf{j} + 2\mathbf{k} \quad \text{and} \quad 2\mathbf{i} + \mathbf{j} + 2\mathbf{k} \] respectively. Let \( l_1, l_2 \) and \( l_3 \) be the lengths of the perpendiculars drawn from the ortho center of the triangle on the sides \( AB, BC \) and \( CA \) respectively. Then \( l_1^2 + l_2^2 + l_3^2 \) equals:

Let O be the origin and the position vector of A and B be \(2\hat{i}+2\hat{j}+\hat{k}\) and \(2\hat{i}+4\hat{j}+4\hat{k}\) respectively. If the internal bisector of\(\angle AOB\) meets the line AB at C, then the length of OC is