List of practice Questions

If $n \geq 2$ is a positive integer, then the sum of the series ${ }^{n+1} C_{2}+2\left({ }^{2} C_{2}+{ }^{3} C_{2}+{ }^{4} C_{2}+\ldots .+{ }^{n} C_{2}\right)$ is:

Let $\binom{n}{k}$ denote $^nC_k$.
If $A_k = \sum_{i=0}^{9} \binom{9}{i} \binom{12}{12-k+i} + \sum_{i=0}^{8} \binom{8}{i} \binom{13}{13-k+i}$ and $A_4 - A_3 = 190 p$, then p is equal to ___________.

If ${}^{20}C_r$ is the co-efficient of $x^r$ in the expansion of $(1+x)^{20$, then the value of $\sum_{r=0}^{20} r^2 \cdot {}^{20}C_r$ is equal to :}

$3 \times 7^{22} + 2 \times 10^{22} - 44$ when divided by 18 leaves the remainder _________.

If $\left(\frac{3^6}{4^4}\right)k$ is the term independent of $x$ in the binomial expansion of $\left(\frac{x}{4} - \frac{12}{x^2}\right)^{12}$, then $k$ is equal to ___________

If the coefficient of $ a^7b^8 $ in the expansion of $ (a + 2b + 4ab)^{10} $ is $ K \cdot 2^{16} $, then K is equal to _________.

If the coefficients of $x^7$ in $\left(x^2 + \frac{1}{bx}\right)^{11}$ and $x^{-7}$ in $\left(x - \frac{1}{bx^2}\right)^{11}$, $b \neq 0$, are equal, then the value of b is equal to :

The value of $ 2^{15}C_1 + 2^{15}C_2 - 3^{15}C_3 + \cdots - 15^{15}C_{15} + \binom{14}{1} + \binom{14}{3} + \binom{14}{5} + \cdots + \binom{14}{11} $ is :

The term independent of x in the expansion of [ (x + 1)/(x^{2/3} - x^{1/3} + 1) - (x - 1)/(x - x^{1/2}) ]¹⁰, x ≠ 1, is equal to ________.

Let n C_r denote the binomial coefficient of x^r in the expansion of (1 + x)^n. If ∑_{k=0}^{10} (2² + 3k) n C_k = α 3^{10} + β 2^{10}, α, β ∈ R, then α + β is equal to ________.

A possible value of 'x', for which the ninth term in the expansion of
$\left\{ 3^{\log_3\sqrt{25^{x-1}+7}} + 3^{(-\frac{1}{8})\log_3(5^{x-1}+1)} \right\}^{10}$ in the increasing powers of $3^{(-\frac{1}{8})\log_3(5^{x-1}+1)}$ is equal to 180, is :

If $b$ is very small as compared to the value of $a$, so that the cube and other higher powers of $\frac{b}{a}$ can be neglected in the identity $\frac{1}{a-b} + \frac{1}{a-2b} + \frac{1}{a-3b} + ....... + \frac{1}{a-nb} = \alpha n + \beta n^2 + \gamma n^3$, then the value of $\gamma$ is :

The term independent of '$x$' in the expansion of $\left( \frac{x+1}{x^{2/3}-x^{1/3}+1} - \frac{x-1}{x-x^{1/2}} \right)^{10}$, where $x \neq 0, 1$ is equal to __________.

The ratio of the coefficient of the middle term in the expansion of $(1+x)^{20}$ and the sum of the coefficients of two middle terms in expansion of $(1+x)^{19}$ is __________.

Let (1 + x + 2x²)²⁰ = a₀ + a₁x + a₂x² + ⋯ + a₄₀x⁴⁰. Then, a₁ + a₃ + a₅ + ⋯ + a₃₇ is equal to :

If n ≥ 2 is a positive integer, then the sum of the series $^{n+1}C_2 + 2( ^2C_2 + ^3C_2 + ^4C_2 + ....... + ^nC_2 )$ is :

The maximum value of k for which the sum $\sum_{i=0}^k \binom{10}{i} \binom{15}{k-i} + \sum_{i=0}^{k+1} \binom{12}{i} \binom{13}{k+1-i}$ exists, is equal to __________.

If the projection of the vector $\hat{i} + 2\hat{j} + \hat{k}$ on the sum of the two vectors $2\hat{i} + 4\hat{j} - 5\hat{k}$ and $-\lambda\hat{i} + 2\hat{j} + 3\hat{k}$ is 1, then $\lambda$ is equal to ___________.

Let $\vec{a} = \hat{i} + \hat{j} + \hat{k}$ and $\vec{b} = \hat{j} - \hat{k}$. If $\vec{c}$ is a vector such that $\vec{a} \times \vec{c} = \vec{b}$ and $\vec{a} \cdot \vec{c} = 3$, then $\vec{a} \cdot (\vec{b} \times \vec{c})$ is equal to :

Let $\vec{a} = \hat{i} + 5\hat{j} + \alpha\hat{k}$, $\vec{b} = \hat{i} + 3\hat{j} + \beta\hat{k}$ and $\vec{c} = -\hat{i} + 2\hat{j} - 3\hat{k}$ be three vectors such that, $|\vec{b} \times \vec{c}| = 5\sqrt{3}$ and $\vec{a}$ is perpendicular to $\vec{b}$. Then the greatest amongst the values of $|\vec{a}|^2$ is _________.

Let $\vec{a}$ and $\vec{b}$ be two vectors such that $|2\vec{a} + 3\vec{b}| = |3\vec{a} + \vec{b}|$ and the angle between $\vec{a}$ and $\vec{b}$ is $60^\circ$. If $\frac{1}{8} \vec{a}$ is a unit vector, then $|\vec{b}|$ is equal to :

Let $\vec{a}, \vec{b}, \vec{c}$ be three vectors mutually perpendicular to each other and have same magnitude. If a vector $\vec{r}$ satisfies $\vec{a} \times \{(\vec{r} - \vec{b}) \times \vec{a}\} + \vec{b} \times \{(\vec{r} - \vec{c}) \times \vec{b}\} + \vec{c} \times \{(\vec{r} - \vec{a}) \times \vec{c}\} = \vec{0}$, then $\vec{r}$ is equal to :

If $\vec{P} \times \vec{Q} = \vec{Q} \times \vec{P}$, the angle between $\vec{P}$ and $\vec{Q}$ is $\theta$ (0°<$\theta$<360°). The value of '$\theta$' will be ________°.

Let $\vec{a} = \hat{i} + \alpha\hat{j} + 3\hat{k}$ and $\vec{b} = 3\hat{i} - \alpha\hat{j} + \hat{k}$. If the area of the parallelogram whose adjacent sides are represented by the vectors $\vec{a}$ and $\vec{b}$ is 8$\sqrt{3}$ square units, then $\vec{a} \cdot \vec{b}$ is equal to ________ .

Let $\vec{a} = \hat{i}+\hat{j}+2\hat{k}$ and $\vec{b} = -\hat{i}+2\hat{j}+3\hat{k}$. Then the vector product $(\vec{a}+\vec{b}) \times ((\vec{a} \times ((\vec{a}-\vec{b}) \times \vec{b})) \times \vec{b})$ is equal to :