Algebra Questions - JEE

Three forces P, Q, R act along the bisectors of the angles of a triangle ABC. By Lami's theorem, which of the following is correct?

If the unit vectors \mathbf{e}_1 and \mathbf{e}_2 are inclined at an angle 2\theta and |\mathbf{e}_1 - \mathbf{e}_2| , then for \theta \in [0, \pi], \theta may lie in the interval

If the vectors a = i − j + 2k, b = 2i + 4j + k and c = λi + j + μk are mutually orthogonal, then (λ, μ) is equal to

Let $\vec{m}$ and $\vec{n}$ are unit vectors and $\vec{w}$ is vector such that $\vec{m} \times \vec{n} + \vec{m} = \vec{w}$ and $\vec{w} \times \vec{m} = \vec{n}$. Then find the value of $[\vec{m}, \vec{n}, \vec{w}]$.

[JEE Main 2020] Let $\vec{a},\vec{b},\vec{c}$ be three vectors such that $\vec{a}\neq\vec{0},\;|\vec{b}|=4,\;|\vec{c}|=2$. Given $\vec{a}=\vec{b}\times(2\vec{a}+\lambda\vec{c})$, $\lambda>0$. If angle between $\vec{b}$ and $\vec{c}$ is $\pi/3$ and $(\vec{a}\times\vec{b})\cdot\vec{c}=|\vec{a}|$, then $\lambda$ equals

Let a vector $\hat{i}+\sqrt{2}\,\hat{j}+\sqrt{2}\,\hat{k}$ be obtained by rotating the vector $\sqrt{3}\,\hat{j}$ by an angle $45°$ about the origin in the clockwise direction to the first quadrant. Then the area of the triangle formed by the vector $(\hat{i}+\sqrt{2}\,\hat{j}+\sqrt{2}\,\hat{k})$ with the coordinate axes is equal to:

[JEE Main 2019] Let $\vec{a}=\hat{i}+2\hat{j}-\sqrt2\hat{k}$ and $\vec{b}=\sqrt2\hat{i}-\hat{j}+\sqrt2\hat{k}$. If $\vec{c}=\vec{a}\times(\vec{a}\times\vec{b})$, then $|\vec{c}|$ equals

Let $\hat{a},\hat{b},\hat{c}$ be three mutually perpendicular unit vectors and $\vec{d}=\lambda(\hat{a}+\hat{b}+\hat{c})$. If $|\vec{d}-\hat{a}|^2+|\vec{d}-\hat{b}|^2+|\vec{d}-\hat{c}|^2=8$, find $\lambda$.

[JEE Main 2020] Let $\vec{a},\vec{b},\vec{c}$ be three vectors such that $\vec{a}\neq\vec{0},\;|\vec{b}|=4,\;|\vec{c}|=2$. Given $\vec{a}=\vec{b}\times(2\vec{a}+\lambda\vec{c})$, $\lambda>0$. If angle between $\vec{b}$ and $\vec{c}$ is $\pi/3$ and $(\vec{a}\times\vec{b})\cdot\vec{c}=|\vec{a}|$, then $\lambda$ 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

Four points $A(1,-1,1),\;B(3,1,1),\;C(6,3,1)$ and $D(6,-1,-1)$ taken in order are the vertices of

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)

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

Given that $\vec{u} = \hat{i} + \hat{j}$, $\vec{v} = \hat{i} - \hat{j}$, $\vec{\omega} = \hat{i} + 2\hat{j} + 3\hat{k}$. Let $\vec{n} = a\hat{i} + b\hat{j} + c\hat{k}$ be a unit vector such that $\vec{u} \cdot \vec{n} = 0$ and $\vec{v} \cdot \vec{n} = 0$. Find $|\vec{\omega} \cdot \vec{n}|$.

70. If $\vec{a}$, $\vec{b}$ and $\vec{c}$ are unit vectors, then $|\vec{a} - \vec{b}|^2 + |\vec{b} - \vec{c}|^2 + |\vec{c} - \vec{a}|^2$ does not exceed ________.

$f: (0,\infty)\to(0,\infty)$ is differentiable. $\int_0^{f(x)}t^2\,dt=\int_0^x t^2 f(t)\,dt$, $f(1)=3$. Then $8f(2)$ is

Given $\vec{a} = \alpha\hat{i} + \hat{j} + 3\hat{k}$, $\vec{b} = 2\hat{i} + \hat{j} - \alpha\hat{k}$ and $\vec{c} = \alpha\hat{i} - 2\hat{j} + 3\hat{k}$ are coplanar. If $S$ is the set of all values of $\alpha$, then find $S$.

If $\begin{vmatrix}a&b&c\\x&y&z\\p&q&r\end{vmatrix}=k$, then $\begin{vmatrix}6x&2y&2z\\-3p&-q&-r\\3a&b&c\end{vmatrix}=$

If $t_1$ and $t_2$ are the times of flight of two particles having the same initial velocity $u$ and range $R$ on the horizontal, then $t_1^2 + t_2^2$ is equal to

If $\omega\ne 1$ is a cube root of unity, then $(1+\omega)(1+\omega^2)(1+\omega^4)(1+\omega^8)\cdots$ to $2n$ factors is

Let $S=\{z\in\mathbb{C}:|z-1|^2+|z+1|^2=4\}$. If $|z+\bar{z}+|z||_{\max}=p+q\sqrt{r}$ (simplified), then $p+q+r=$

If the roots of $x^2+bx+c=0$ are real and unequal, those of $x^2+bx+(c+1)=0$ are

If $a,b$ are roots of $x^2-px+q=0$ with $0<a<1<b$, which is NOT necessarily true?

If $x=\log_a(bc)$, $y=\log_b(ca)$, $z=\log_c(ab)$, then $\dfrac{1}{x+1}+\dfrac{1}{y+1}+\dfrac{1}{z+1}=$

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

Algebra Questions (626)