List of practice Questions

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 :
Let $\vec{a}=\hat{i}+\hat{j}+\hat{k}$, $\vec{b}$ and $\vec{c}=\hat{j}-\hat{k}$ be three vectors such that $\vec{a} \times \vec{b} = \vec{c}$ and $\vec{a} \cdot \vec{b} = 1$. If the length of projection vector of the vector $\vec{b}$ on the vector $\vec{a} \times \vec{c}$ is $l$, then the value of $3l^2$ is equal to _________.
Let $\vec{a} = \hat{i} + 2\hat{j} - \hat{k}$, $\vec{b} = \hat{i} - \hat{j}$ and $\vec{c} = \hat{i} - \hat{j} - \hat{k}$. If $\vec{r}$ is a vector such that $\vec{r} \times \vec{a} = \vec{c} \times \vec{a}$ and $\vec{r} \cdot \vec{b} = 0$, then $\vec{r} \cdot \vec{a}$ is equal to __________.
Let three vectors \(\vec{a}, \vec{b}\) and \(\vec{c}\) be such that \(\vec{c}\) is coplanar with \(\vec{a}\) and \(\vec{b}, \vec{a} \cdot \vec{c} = 7\) and \(\vec{b}\) is perpendicular to \(\vec{c}\), where \(\vec{a} = -\hat{i} + \hat{j} + \hat{k}\) and \(\vec{b} = 2\hat{i} + \hat{k}\), then the value of \(|2\vec{a} + \vec{b} + \vec{c}|^2\) is __________
Let $\vec{a}$, $\vec{b}$ and $\vec{c}$ be three vectors such that $\vec{a} = \vec{b} \times (\vec{b} \times \vec{c})$. If magnitudes of the vectors $\vec{a}$, $\vec{b}$ and $\vec{c}$ are $\sqrt{2}, 1$ and 2 respectively and the angle between $\vec{b}$ and $\vec{c}$ is $\theta$ ($0<\theta<\frac{\pi}{2}$), then the value of $1+\tan\theta$ is equal to :
Let $\vec{a}=\hat{i}-\alpha\hat{j}+\beta\hat{k}$, $\vec{b}=3\hat{i}+\beta\hat{j}-\alpha\hat{k}$ and $\vec{c}=-\alpha\hat{i}-2\hat{j}+\hat{k}$, where $\alpha$ and $\beta$ are integers. If $\vec{a} \cdot \vec{b} = -1$ and $\vec{b} \cdot \vec{c} = 10$, then $(\vec{a} \times \vec{b}) \cdot \vec{c}$ is equal to ________.
Let the vectors $(2 + a + b)\hat{i} + (a + 2b + c)\hat{j} - (b + c)\hat{k}$, $(1 + b)\hat{i} + 2\hat{j} - b\hat{k}$ and $(2 + b)\hat{i} + 2\hat{j} + (1 - b)\hat{k}$, $a, b, c \in \mathbb{R}$ be co-planar. Then which of the following is true ?
Rotation of axes: Vector ā (3p, 1) becomes (p+1, √10) in new system. Find p.
The sum of all 3-digit numbers less than or equal to 500, that are formed without using the digit "1" and they all are multiple of 11, is ___________.
The number of three-digit even numbers, formed by the digits 0, 1, 3, 4, 6, 7 if the repetition of digits is not allowed, is _________.
If ${}^1P_1 + 2 \cdot {}^2P_2 + 3 \cdot {}^3P_3 + \dots + 15 \cdot {}^{15}P_{15} = {}^qP_r - s$, where $0 \leq s \leq 1$, then $q + s + {}^qC_{r - s}$ is equal to _________.
A number is called a palindrome if it reads the same backward as well as forward. For example 285582 is a six digit palindrome. The number of six digit palindromes, which are divisible by 55, is _________.