Algebra Questions - JEE

For $x\in\mathbb{R}$, $x\ne -1$, $(1+x)^{2016}+x(1+x)^{2015}+x^2(1+x)^{2014}+\cdots+x^{2016}=\sum_{i=0}^{2016}a_i x^i$. Then $a_{17}=$

The number of solutions of $2\sin^2 x+3\sin x-2=0$ in $[0,2\pi]$ is

The number of ways of distributing 10 different books among 4 students (S1 to S4) such that S1 gets 2 books, S2 gets 3 books, S3 gets 2 books and S4 gets 3 books is

Vector coplanar with $\vec{a} = \hat{i} - \hat{j}$ and $\vec{b} = \hat{i} + 2\hat{j}$ is given by

If the vectors $\vec{a} = x\hat{i} + y\hat{j} + z\hat{k}$ and such that $\vec{a}$, $\vec{c}$ and $\vec{b}$ form a right handed system, then $\vec{c}$ is

Given that $\vec{a}$, $\vec{b}$ and $\vec{c}$ are unit vectors. We know that $|\vec{a}+\vec{b}+\vec{c}|^2 = |\vec{a}|^2 + |\vec{b}|^2 + |\vec{c}|^2 + 2(\vec{a}\cdot\vec{b}+\vec{b}\cdot\vec{c}+\vec{c}\cdot\vec{a}) \geq 0$. Find the minimum value of $|\vec{a}-\vec{b}|^2+|\vec{b}-\vec{c}|^2+|\vec{c}-\vec{a}|^2$.

If a unit vector $\vec{a}$ makes angles $\pi/3$ with $\hat{i}$, $\pi/4$ with $\hat{j}$ and $\theta \in (0, \pi)$ with $\hat{k}$, then a value of $\theta$ is ______ (in degrees).

The vectors $\overrightarrow{AB} = 3\hat{i} + 4\hat{k}$ and $\overrightarrow{AC} = 5\hat{i} - 2\hat{j} + 4\hat{k}$ are the sides of a triangle $ABC$. The length of the median through $A$ is

In a right angle $\triangle ABC$, $\angle A = 90°$ and sides $a, b, c$ are, respectively, 5 cm, 4 cm and 3 cm. If a force $\vec{F}$ has moments 0, 9 and 16 in N cm units, respectively, about vertices $A$, $B$ and $C$, then magnitude of $\vec{F}$ is

Let $\vec{a}=2\hat{i}-\hat{j}+\hat{k}$ and $\vec{b}=\hat{i}+2\hat{j}-\hat{k}$. If $\vec{c}=\alpha\vec{a}+\beta\vec{b}$ satisfies $\vec{c}\times\vec{a}=\vec{b}$, find $\alpha+\beta$.

Let 61. Let $\vec{a} = 2\hat{i} + \lambda_1\hat{j} + 3\hat{k}$, $\vec{b} = 4\hat{i} + (3-\lambda_2)\hat{j} + 6\hat{k}$ and $\vec{c} = 3\hat{i} + 6\hat{j} + (\lambda_3 - 1)\hat{k}$ be three vectors such that $\vec{b} = 2\vec{a}$ and $\vec{a}$ is perpendicular to $\vec{c}$. Then a possible value of $(\lambda_1, \lambda_2, \lambda_3)$ is:

A tangent is drawn to the curve $y=x^2$ at a point $A(x_1,y_1)$. The scalar product $\overrightarrow{AR}\cdot\overrightarrow{AP}$ at point $P(a,a^2)$ equals (where $A$ is any point on the curve)

Let $\vec{a}=q_1\hat{i}+q_2\hat{j}+q_3\hat{k}$ make equal angles with OX, OY, OZ and $|\vec{a}|=\sqrt{3}$. If the projection of $\vec{a}$ on $\hat{i}+\hat{j}-\hat{k}$ is 1, find $q_1+q_2+q_3$.

The distance of the point having position vector 2i + 6j + 3k from the straight line passing through the point (2, 3, 4) and parallel to the vector i + 4j - 6k is

The projection of the vector $\vec{i} - \vec{j}$ on the vector $\vec{i} + \vec{j}$ is

Let $\vec{a}$, $\vec{b}$ and $\vec{c}$ be non-zero vectors such that $(\vec{a}\times\vec{b})\times\vec{c} = \dfrac{1}{3}|\vec{b}||\vec{c}|\vec{a}$. If $\theta$ is the acute angle between the vectors $\vec{b}$ and $\vec{c}$, then $\sin\theta$ equals

For a triangle ABC, let $\vec{p}=\overrightarrow{BC}$, $\vec{q}=\overrightarrow{CA}$ and $\vec{r}=\overrightarrow{BA}$. If $|\vec{p}|=2\sqrt{3}$, $|\vec{q}|=2$ and $\cos\theta=\dfrac{1}{\sqrt{3}}$, where $\theta$ is the angle between $\vec{p}$ and $\vec{q}$, then $|\vec{p}\times(\vec{q}-3\vec{r})|^2+3|\vec{r}|^2$ is equal to:

Let $PQR$ be a triangle such that $\overrightarrow{PQ}=-2\hat{i}-\hat{j}+2\hat{k}$ and $\overrightarrow{PR}=a\hat{i}+b\hat{j}-4\hat{k}$, $a,b\in\mathbb{Z}$. Let $S$ be the point on QR which is equidistant from the lines PQ and PR. If $|\overrightarrow{PR}|=9$ and $\overrightarrow{PS}=\hat{i}-7\hat{j}+2\hat{k}$, then the value of $3a-4b$ is ___.

Direction vectors of two lines $L_1$ and $L_2$ are $\hat{i}-\hat{j}+\hat{k}$ and $2\hat{i}+\hat{j}-\hat{k}$ respectively. The angle between the lines is

A, B, C, D are four points in space and satisfy $|\vec{AB}| = 3, |\vec{BC}| = 7, |\vec{CD}| = 11$ and $|\vec{DA}| = 9$. Then find the value of $\vec{AC} \times \vec{BD}$.

The scalar product of the vector $\hat{i} + \hat{j} + \hat{k}$ with a unit vector along the sum of vectors $2\hat{i} + 4\hat{j} - 5\hat{k}$ and $\lambda\hat{i} + 2\hat{j} + 3\hat{k}$ is equal to one. The value of $\lambda$ is

Find $\frac{\Delta_2}{\Delta}$

Let $\vec{a}=-5\hat{i}+\hat{j}-3\hat{k}$ and $\vec{b}=\hat{i}+2\hat{j}-4\hat{k}$. Let $\vec{c}=\left(\left(\left(\vec{a}\times\vec{b}\right)\times\hat{i}\right)\times\hat{i}\right)\times\hat{i}$. Then $\vec{c}\cdot(-\hat{i}+\hat{j}+\hat{k})$ is equal to:

Let $\vec{a}$, $\vec{b}$ and $\vec{c}$ be three non-zero vectors such that $\vec{b}$ and $\vec{c}$ are non-collinear. If $\vec{a}+5\vec{b}$ is collinear with $\vec{c}$, $\vec{b}+6\vec{c}$ is collinear with $\vec{a}$, and $\vec{a}+\alpha\vec{b}+\beta\vec{c}=\vec{0}$, then $\alpha+\beta$ is equal to:

Let $\vec{a}=\hat{i}+\alpha\hat{j}+\beta\hat{k}$, $\alpha,\beta\in\mathbb{R}$. Let $\vec{b}$ be a vector such that the angle between $\vec{a}$ and $\vec{b}$ is $\dfrac{\pi}{4}$ and $|\vec{b}|^2=6$. If $\vec{a}\cdot\vec{b}=3\sqrt{2}$, then the value of $(\alpha^2+\beta^2)|\vec{a}\times\vec{b}|^2$ is equal to:

If O is the origin and the position vector of A is 4i + 5j, then a unit vector parallel to OA is

Find the volume of a parallelepiped with coterminous edges of lengths 8, 6, and 2.

Let a, b, c be three non-coplanar vectors and d be a non-zero vector, which is perpendicular to a + b + c. If d = $\sin x$(\a × b) + $\cos y$(b × c) + 2(c × a), then the minimum value of $x^2 + y^2$ is

If x, y are two non-zero and non-collinear vectors satisfying \[[(a-2)\alpha^2 + (b-3)\alpha + c]\mathbf{x} + [(a-2)\beta^2 + (b-3)\beta + c]\mathbf{y} + [(a-2)\gamma^2 + (b-3)\gamma + c](\mathbf{x} \times \mathbf{y}) = 0\] where $\alpha, \beta, \gamma$ are three distinct real numbers, then find the value of $a^2 + b^2 + c^2 - 4$.

Let a = $\alpha \mathbf{i} + 2\mathbf{j} - 3\mathbf{k}$, b = $\mathbf{i} + 2\alpha \mathbf{j} - 2\mathbf{k}$ and c = $2\mathbf{i} - \alpha \mathbf{j} + \mathbf{k}$. Then the value of $\alpha$ such that $\{(\mathbf{a} \times \mathbf{b}) \times (\mathbf{b} \times \mathbf{c})\} \times (\mathbf{c} \times \mathbf{a}) = 0$, is

A vector a has components 2p and 1 with respect to a rectangular Cartesian system. This system is rotated through a certain angle about the origin in the counter-clockwise sense. If with respect to the new system, a has components (p+1) and 1, then find p.

Max \{|PQ|\} is

If a = 2i − 3j + k, b = −i + k, c = 2j − k, then the area (in sq units) of parallelogram with diagonals a + b and b + c will be

If D, E and F are respectively the mid-points of AB, AC and BC in $\triangle ABC$, then BE + AF is equal to

If a × (b × c) = 0, then

Let $ABCD$ be a tetrahedron in which position vectors of $A, B, C$ and $D$ are $\vec{i} + \vec{j} + \vec{k}, 2\vec{i} + 2\vec{j} + 2\vec{k}, 3\vec{i} + 2\vec{j} + \vec{k}$ and $2\vec{i} + 3\vec{j} + 2\vec{k}$. If $ABC$ be the base of tetrahedron then height of tetrahedron is:

The position vector of a point C with respect to B is $\mathbf{i} + \mathbf{j}$ and that of B with respect to A is $\mathbf{i} - \mathbf{j}$. Find the position vector of C with respect to A.

The value of $\frac{q}{p} + 2\cos\theta$ is

Let a = 2i + j − 2k, b = i + j and c be a vector such that |c − a| = 3, |(a × b) × c| = 3 and the angle between c and a × b is 30°. Then a · c is equal to

If $\vec{a}, \vec{b}, \vec{c}$ and $\vec{d}$ are four non-coplanar unit vectors, $\vec{a}, \vec{b}, \vec{c}$ are mutually perpendicular, such that $\vec{d}$ makes equal angles with all the three vectors $\vec{a}, \vec{b}, \vec{c}$, then:

If $\vec{a}, \vec{b}, \vec{c}$ are unit vectors such that $\vec{a} \cdot \vec{b} = 0 = \vec{a} \cdot \vec{c}$ and the angle between $\vec{b}$ and $\vec{c}$ is $\frac{\pi}{3}$. Then the value of $|\vec{a} \times \vec{b} - \vec{a} \times \vec{c}|$ is ______.

Let the vectors $\vec{a}, \vec{b}, \vec{c}$ and $\vec{d}$ be such that $\vec{c} - 3\vec{a} = -\vec{b} - \vec{a}$. Then the points with position vectors as $\vec{a}, \vec{b}, \vec{c}$ and $\vec{d}$ are:

253. Let $\vec{a}, \vec{b}, \vec{c}$ be three non-zero vectors satisfying $\vec{a} = \vec{b} \times \vec{c} + 2\vec{b}$ where $|\vec{b}| = |\vec{c}| = 2$ and $|\vec{a}| \leq 4$. The sum of possible value(s) of $|2\vec{a} + \vec{b} + \vec{c}|$ is:

If the equation $x^3+bx^2+cx+1=0$ ($b<c$) has only one real root $\alpha$. Then the value of $2\tan^{-1}(\text{cosec}\,\alpha)+\tan^{-1}(2\sin\alpha\sec^2\alpha)$ is

Line $L_1$ is parallel to a vector $\vec{a} = 3\hat{i} + 2\hat{j} + 4\hat{k}$ and passes through a point $A(7,6,2)$ and the line $L_2$ is parallel to a vector $\vec{b} = 2\hat{i} + \hat{j} + 3\hat{k}$ and passes through a point $B(5, 3, 4)$. Now a line $L_3$ parallel to a vector $\vec{c} = 2\hat{i} + 2\hat{j} + \hat{k}$ intersects the lines $L_1$ and $L_2$ at points $C$ and $D$ respectively then $|\overrightarrow{CD}|$ is_______.

Let $\vec{a} = 2\hat{i} + \hat{j} - 2\hat{k}$ and $\vec{b} = \hat{i} + \hat{j}$. If $\vec{c}$ is a vector such that $\vec{a}\vec{c} = |\vec{c}||\vec{a} - \vec{a}| = 2\sqrt{2}$ and the angle between $(\vec{a} \times \vec{b})$ and $\vec{c}$ is $30°$, then the value of $|(\vec{a} \times \vec{b}) \times 2\vec{c}|$ is _______.

If $\vec{a} \times (\vec{b} \times \vec{c}) + (\vec{a}\vec{b})\vec{b} = (4 - 2\beta - \sin \alpha)\vec{b} + (\beta^2 - 1)\vec{c}$ and $\vec{c} \cdot \vec{c} = \vec{a} \cdot \vec{c}$ where $\vec{b}$ and $\vec{c}$ are non-collinear and $\alpha, \beta$ are scalars then $\beta = _______.

Let $\vec{a} = \hat{i} + \hat{j} + \hat{k}$, $\vec{b} = x_1\hat{i} + x_2\hat{j} + x_3\hat{k}$, where $x_1, x_2, x_3 \in \{-3, -2, -1, 0, 1, 2\}$. Number of possible vectors $\vec{b}$ such that $\vec{a}$ and $\vec{b}$ are mutually perpendicular, is $p$ then $\frac{p}{5} = _______.

If in a triangle $ABC$, $\overrightarrow{BC} = \frac{\vec{u}}{|\vec{u}|\vec{v}|}$ and $\overrightarrow{AC} = \frac{2\vec{u}}{|\vec{u}|}$ where $|\vec{u}||\vec{v}|$, then $1 + \cos 2A + \cos 2B + \cos 2C = _______.

Algebra Questions (626)