Algebra Questions - JEE

If $\hat{a}$ and $\hat{b}$ are unit vectors and $|\hat{a}+\hat{b}|=\sqrt{3}$, find $(2\hat{a}-\hat{b})\cdot(3\hat{a}+2\hat{b})$.

Let $\vec{a},\vec{b}$ be two non-zero non-collinear vectors. Then for any scalar $\lambda\neq0$, $\vec{a}=\lambda\vec{b}$ if and only if

A line passes through point $A$ with position vector $\vec{v}=\hat{j}-\hat{k}$ and is parallel to the vector $\hat{i}+\hat{j}$. For any point $P$ on this line, which of the following is/are true?

[JEE Main 2019] Let $\vec{a}=\hat{i}-\hat{j}$ and $\vec{b}=-\hat{i}+\hat{j}+\hat{k}$ be two given vectors. Let $\vec{c}=\vec{a}\times\vec{b}$. Then which of the following is NOT true?

The altitude of a parallelepiped whose three coterminous edges are $\vec{A}=\hat{i}+\hat{j}+\hat{k}$, $\vec{B}=2\hat{i}+4\hat{j}-\hat{k}$, $\vec{C}=\hat{i}+\hat{j}+3\hat{k}$, with $\vec{A}$ and $\vec{B}$ as the base, is

[JEE Main 2021] If a unit vector $\hat{a}$ makes angles $\dfrac\pi3$ with $\hat{i}$, $\dfrac\pi4$ with $\hat{j}$ and an acute angle $\theta$ with $\hat{k}$, then $\theta$ equals

If the coefficient of $x^{10}$ in $(ax^2+\frac{1}{bx})^{11}$ is equal to the coefficient of $x^{-10}$ in $(ax-\frac{1}{bx^2})^{11}$, then $a\cdot b$ is

If $p,q,r$ are in AP and $p^2,q^2,r^2$ are in HP, then $p,q,r$ are

$\vec{a}, \vec{b}, \vec{c}$ are 3 vectors, such that $\vec{a} + \vec{b} + \vec{c} = \vec{0}$, $|\vec{a}| = 1$, $|\vec{b}| = 2$, $|\vec{c}| = 3$, then $\vec{a}\cdot\vec{b} + \vec{b}\cdot\vec{c} + \vec{c}\cdot\vec{a}$ is equal to

The number of 5-letter words that can be formed from letters of CONSIDER using each letter at most once and beginning with C and ending with R is

A velocity 1/4 m/s is resolved into two components along $OA$ and $OB$ making angles 30° and 45°, respectively, with the given velocity. Then the component along $OB$ is

Let $(1+x+x^2)^{30} = \displaystyle\sum_{r=0}^{60} a_r x^r$. If $\alpha a_{21} = \beta a_{20} + \gamma a_{19}$, $(\alpha,\beta,\gamma\in\mathbb{N})$, then $\alpha+\beta+\gamma$ can be

In a group of 65 people, 40 like cricket, 10 like both cricket and tennis. How many like tennis only?

If $\alpha,\beta,\gamma$ are the roots of $x^3+ax^2+bx+c=0$ and $\alpha+\beta=0$, then

The resultant $R$ of two forces acting on a particle is at right angles to one of them and its magnitude is one third of the other force. The ratio of larger force to smaller one is

The number of 3-digit numbers divisible by 7 is

Number of real numbers $x$ satisfying both $x^{2026}+x^{2025}+x^{2024}+\ldots+x+1>0$ and $x^{2027}<1$ is

[JEE Main 2019] Let $\vec{a}$ and $\vec{b}$ be unit vectors and $\alpha$ be the angle between them. Then $\vec{a}+\vec{b}$ is a unit vector if

$D=\begin{vmatrix}x+1&x+2&x+a\\x+2&x+3&x+b\\x+3&x+4&x+c\end{vmatrix}=0$ if $a,b,c$ are in

If $\log_{0.3}(x-1)<\log_{0.09}(x-1)$, then $x$ lies in

Which of the following is always true?

[JEE Main 2022] Let $\vec{a}=\hat{i}+2\hat{j}+3\hat{k}$, $\vec{b}=\hat{i}-\hat{j}+2\hat{k}$, $\vec{c}=5\hat{i}+3\hat{j}-\hat{k}$. If $\alpha$ is the projection of $(\vec{a}+\vec{b})$ on $\vec{c}$, and $\beta$ is the projection of $\vec{c}$ on $(\vec{a}+\vec{b})$, find $6(\alpha+\beta)$.

Number of correct statements is $k$. Then $2k$ is: I) $A=\{1,2,3,4,5,6,7\}$, $B=\{T\subseteq A:\text{either }1\notin T\text{ or }2\in T\}$, $C=\{T\subseteq A:\text{sum of elements is prime}\}$. Number of elements in $B\cup C$ is 107. II) If sum of $n$ terms of any sequence is quadratic in $n$, then the sequence is AP. III) $x=(8\sqrt{3}+13)^{13}$ and $y=(7\sqrt{2}+9)^9$: $[x]+[y]$ is even. IV) $(x+(x^3-1)^{1/2})^5+(x-(x^3-1)^{1/2})^5$ is polynomial of degree 7.

The non-zero vectors $\vec{a}$, $\vec{b}$ and $\vec{c}$ are related by $\vec{a} = 8\vec{b}$ and $\vec{c} = -7\vec{b}$. Then the angle between $\vec{a}$ and $\vec{c}$ is

If $(1+x)^{50}=a_0+a_1x+\cdots+a_{50}x^{50}$, then $\dfrac{a_1+a_2+\cdots+a_{25}}{a_{26}+a_{27}+\cdots+a_{50}}$ equals

The number of integral values of $a$ for which the equation $x^4 - (a+2)x^3 + 2ax^2 + 4(a-2)x - 16 = 0$ has at least two positive roots; $a \in [-10, 10]$ is/are

Algebra Questions (626)