Algebra Questions - JEE

Number of values of $x$ satisfying the system $x+y=1$, $(x+y)^2+\left(\dfrac{x^2}{y}+\dfrac{y^2}{x}\right)^2=0$

The value of $\begin{vmatrix}1+a^2-b^2&2ab&-2b\\2ab&1-a^2+b^2&2a\\2b&-2a&1-a^2-b^2\end{vmatrix}$ is

Let $\vec{a}=\hat{i}+2\hat{j}+3\hat{k}$. A vector $\vec{b}$ satisfies $\vec{a}\cdot\vec{b}=|\vec{b}|^2$ and $|\vec{a}-\vec{b}|^2=7$. Find $|\vec{b}\times\vec{a}|^2$.

If two points $P$ and $Q$ are on the curve $y = 2^{x+1}$, such that $\overrightarrow{OP} \cdot \vec{i} = -1$ and $\overrightarrow{OQ} \cdot \vec{i} = 2$, where $\vec{i}$ is a unit vector along the $x$-axis, then $|\overrightarrow{OQ} - \overrightarrow{OP}|$ is equal to

If $a,b$ are roots of $x^2-5x+6=0$ and $c,d$ are roots of $x^2-5x+6=0$... Actually: $a,b$ roots of $x^2-px+q=0$ and $c,d$ roots of $x^2-px+r=0$. Then $(a-c)(a-d)(b-c)(b-d)=$

Let $f(x)=x^2+bx+c$, minimum value of $f(x)$ is $-5$, then absolute value of the difference of the roots of $f(x)$ is

The value of $\left(1+\cos\dfrac{\pi}{9}\right)\left(1+\cos\dfrac{3\pi}{9}\right)\left(1+\cos\dfrac{5\pi}{9}\right)\left(1+\cos\dfrac{7\pi}{9}\right)=$

Number of real solutions of $2^x+3^x-4^x+6^x-9^x=1$

The number of ways to arrange letters of BANANA so that no two N's appear together is

Letters of COCHIN are permuted and all permutations are arranged in alphabetical order. The number of words before COCHIN is

If the minimum value of $\dfrac{x^2y^2-2xy^2+2y^2+4xy-4y+4}{xy^2+2y}$ is '$k$' $\forall\,x,y\in\mathbb{R}^+$, then $[10k]$ equals (where $[\cdot]$ is GIF)

The number of real solutions of $\sin\left(e^x\right)=5^x+5^{-x}$ is

Here, m = cos(π/4) = 1/√2 and n = cos(π/2) = 0. If l² + m² + n² = 1, find the value of l.

For any complex number $z$, $|z|^2+|z-1|^2+|z-2|^2+|z-i|^2$ is minimum when $z=$

If coefficient of $x^2y^3z^4$ in $(x+y+z)^n$ is $A$ (where $A\ne 0$, $n\in\mathbb{N}$), then coefficient of $x^4y^4z$ is

Coefficient of $x^{99}$ in $x^{100}+2x^{99}(1+x)+3(1+x)^2x^{98}+\cdots+101(1+x)^{100}$ is

The number of integers satisfying $\log_{1/2}|x-3| \ge -1$ is

Let $q$ be the maximum integral value of $p$ in $[0,10]$ for which the roots of the equation $x^2+px+\frac{5p}{4}=0$ are rational. Then the area of region $\{(x,y):0\le y\le(x-q)^2, 0\le x\le q\}$ is (in square units)

Let $f(x)=\int_x^{x^2}\dfrac{dt}{\ln t}$. Then $\lim_{x\to1}\dfrac{f(x)}{x-1}$ is

Let the height of a triangle be l, where a triangle has a base of 5 units. Points are given as A(1, -1, 2), B(-2, 1, 0) (direction cosine of the line), and C(3, 0, 4). Find the area of the triangle (in square units, rounded to 3 decimal places).

Let $\vec{a}=2\hat{i}-\hat{j}+2\hat{k}$ and $\vec{b}=\hat{i}+2\hat{j}-\hat{k}$. A vector $\vec{c}$ satisfies $\vec{a}\times\vec{c}=\vec{b}$ and $\vec{a}\cdot\vec{c}=3$. Find $|\vec{c}|^2$.

$A=\{x:x^{18}=1\}$, $B=\{x:x^{12}=1\}$. If $A\cap B=\{x:x^n=1\}$, then $n=$

With two forces acting at a point, the maximum effect is obtained when their resultant is 4 N. If they act at right angles, their resultant is 3 N. Then the factors are

Number of ways of selecting a committee of 3 women and 4 men from 8 women and 6 men, if a particular woman refuses to serve if a particular man is in the committee is

If the sum of the squares of the reciprocals of the roots $\alpha$ and $\beta$ of the equation $3x^2+\lambda x-1=0$ is 15, then $6(\alpha^3+\beta^3)^2=$

The projections of a vector on the three coordinate axes are 6, $-3$, 2, respectively. The direction cosines of the vector are

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:

Algebra Questions (626)