List of top Mathematics Questions asked in JEE Main

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 __________.

Let $m$ and $n$ be the coefficients of the seventh and thirteenth terms respectively in the expansion of \[\left( \frac{1}{3}x^{\frac{1}{3}} + \frac{1}{2x^{\frac{2}{3}}} \right)^{18}.\]Then \[\left(\frac{n}{m}\right)^{\frac{1}{3}}\]is:

The number of elements in the set S = {(x, y, z) : x, y, z ∈ Z, x + 2y + 3z = 42, x, y, z ≥ 0} equals ____

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

Suppose \( 2 - p \), \( p \), \( 2 - \alpha \), \( \alpha \) are the coefficients of four consecutive terms in the expansion of \( (1 + x)^n \). Then the value of \( p^2 - \alpha^2 + 6\alpha + 2p \) equals

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:

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

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 ____

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:

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 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: 

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:

The coefficient of \( x^{70} \) in \( x^2 (1+x)^{98} + x^3 (1+x)^{97} + x^4 (1+x)^{96} + \ldots + x^{54} (1+x)^{46} \) is \( ^{99}C_p - ^{46}C_q \).
Then a possible value to \( p + q \) is:

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

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

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 \( \vec{a} \) and \( \vec{b} \) be two vectors such that \( |\vec{b}| = 1 \) and \( |\vec{b} \times \vec{a}| = 2 \). Then \( \left| (\vec{b} \times \vec{a}) - \vec{b} \right|^2 \) is equal to

Let \[ \vec{a} = \hat{i} + \hat{j} + \hat{k}, \quad \vec{b} = 2\hat{i} + 4\hat{j} - 5\hat{k}, \quad \text{and} \quad \vec{c} = x\hat{i} + 2\hat{j} + 3\hat{k}, \, x \in \mathbb{R}. \] If \( \vec{d} \) is the unit vector in the direction of \( \vec{b} + \vec{c} \) such that \( \vec{a} \cdot \vec{d} = 1 \), then \( (\vec{a} \times \vec{b}) \cdot \vec{c} \) is equal to:

Consider three vectors $\vec{a}, \vec{b}, \vec{c}$. Let $|\vec{a}| = 2, |\vec{b}| = 3$ and $\vec{a} = \vec{b} \times \vec{c}$. If $\alpha \in [0, \frac{\pi}{3}]$ is the angle between the vectors $\vec{b}$ and $\vec{c}$, then the minimum value of $27|\vec{c}| - |\vec{a}|^2$ is equal to:

\[\text{If } \lambda>0, \text{ let } \theta \text{ be the angle between the vectors }\vec{a} = \hat{i} + \lambda \hat{j} - 3 \hat{k} \text{ and } \vec{b} = 3 \hat{i} - \hat{j} + 2 \hat{k}.\text{ If the vectors } \vec{a} + \vec{b} \text{ and } \vec{a} - \vec{b} \text{ are mutually perpendicular, then the value of } (14 \cos \theta)^2 \text{ is equal to.}\]

Let \(\vec{a}\) = 2$\hat{i}$ + 5$\hat{j}$ - $\hat{k}$, $\vec{b}$ = 2$\hat{i}$ - 2$\hat{j}$ + 2$\hat{k}$
and $\vec{c}$ be three vectors such that
($\vec{c}$ + $\hat{i}$) $\times$ ($\vec{a}$ + $\vec{b}$ + $\hat{i}$) = $\vec{a}$ $\times$ ($\vec{c}$ + $\hat{i})$ . $\vec{a}$.$\vec{c}$ = -29,)
then $\vec{c}$.(-2$\hat{i}$ + $\hat{j}$ + $\hat{k}$) is equal to :

Let \(\vec{a} = \hat{i} - 3\hat{j} + 7\hat{k}, \quad \vec{b} = 2\hat{i} - \hat{j} + \hat{k}, \quad \text{and} \quad \vec{c} \text{ be a vector such that}\) \((\vec{a} + 2\vec{b}) \times \vec{c} = 3(\vec{c} \times \vec{a}).\)
If \(\vec{a} \cdot \vec{c} = 130\), then \(\vec{b} \cdot \vec{c}\) is equal to \(\_\_\_\_\_\_\_\_ .\)

Let a unit vector which makes an angle of \(60^\circ\) with \( 2\hat{i} + 2\hat{j} - \hat{k} \) and an angle of \(45^\circ\) with \( \hat{i} - \hat{k} \) be \( \vec{C} \). Then \( \vec{C} + \left( -\frac{1}{2} \hat{i} + \frac{1}{3\sqrt{2}} \hat{j} - \frac{\sqrt{2}}{3} \hat{k} \right) \) is: