Algebra Questions - JEE

When |\mathbf{c} - \mathbf{a}| attains least value, then the value of |\mathbf{c}| is

Let $x_1$ and $x_2$ ($x_1 > x_2$) are the roots of the equation $9^{\log_9(x^2-4x+5)} = x-1$, then the value of $\tan(x_1)\pi + \sec(x_2)\pi$ is:

Let $\vec{a}=\hat{i}-3\hat{j}+7\hat{k}$, $\vec{b}=2\hat{i}-\hat{j}+\hat{k}$ and $\vec{c}$ be a vector such that $(\vec{a}+2\vec{b})\times\vec{c}=3(\vec{c}\times\vec{a})$. If $\vec{a}\cdot\vec{c}=130$, then $\vec{b}\cdot\vec{c}$ is equal to _____

If a = i + 2j − 2k, b = 2i − j + k and c = i + 3j − k, then a × (b × c) is equal to

If the coefficients of $x^r$ and $x^{r+1}$ in the expansion of $(3+7x)^{29}$ are equal, then $r=$

66. Let $\vec{a} = \hat{i} + \hat{j} + \sqrt{2}\hat{k}$, $\vec{b} = b_1\hat{i} + b_2\hat{j} + \sqrt{2}\hat{k}$ and $\vec{c} = 5\hat{i} + \hat{j} + \sqrt{2}\hat{k}$ be three vectors such that projection vector of $\vec{b}$ on $\vec{a}$ is $\vec{a}$. If $\vec{a} + \vec{b}$ is perpendicular to $\vec{c}$, then $|\vec{b}|$ is equal to ________.

Let $\vec{a}=4\hat{i}+3\hat{j}$ and $\vec{b}=3\hat{i}-4\hat{j}+5\hat{k}$. If $(\vec{a}+\vec{b})\perp(\lambda\vec{a}-\vec{b})$, find $\lambda$.

If $\vec{c} \cdot \vec{d} = 0$, $5|a|^2 + 6\vec{a}\cdot\vec{b} - 8|b|^2 = 0$, find $\hat{a}\cdot\hat{b}$.

If $\vec{a}, \vec{b}, \vec{c}$ are non-coplanar vectors and $\lambda$ is a real number then $[\lambda(\vec{a}+\vec{b})\ \lambda^2\vec{b}\ \lambda\vec{c}] = [\vec{a}\ \vec{b}+\vec{c}\ \vec{b}]$ for

Let $x_1, x_2, \ldots, x_{10}$ be the roots of the polynomial equation $x^{10} + x^9 + \cdots + x + 1 = 0$. Then the value of $\displaystyle\sum_{n=1}^{10}\left(\dfrac{1}{1-x_n}\right)$:

Let $\vec{a}=\hat{i}+\hat{j}+\hat{k}$, $\vec{c}=\hat{j}-\hat{k}$ and a vector $\vec{b}$ be such that $\vec{a}\times\vec{b}=\vec{c}$ and $\vec{a}\cdot\vec{b}=3$. Then $|\vec{b}|$ equals

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 a = $2\mathbf{i} + \mathbf{j} + \mathbf{k}$, b = $\mathbf{i} + 2\mathbf{j} - \mathbf{k}$ and c is a unit vector coplanar to them. If c is perpendicular to a, then c is equal to

Let v = $2\mathbf{i} + \mathbf{j} - \mathbf{k}$ and w = $\mathbf{i} + 3\mathbf{k}$. If u is a unit vector and the maximum value of $[\mathbf{u}, \mathbf{v}, \mathbf{w}] = \lambda$, then the value of $\lambda - 51$ is

A unit vector a makes an angle $\frac{\pi}{4}$ with the Z-axis. If a + i + j is a unit vector, then a is equal to

Let $\vec{a} = \vec{i} + \vec{j} + \vec{k}$, $\vec{b} = \vec{i} - \vec{j} + \vec{k}$ and $\vec{c} = \vec{i} - \vec{j} - \vec{k}$ be three vectors. A vector $\vec{v}$ in the plane of $\vec{a}$ and $\vec{b}$, whose projection on $\vec{c}$ is $\frac{1}{3}$, is given by

Ex. 97 Let a = 2i + 3j - 6k, b = 2i - 3j + 6k and c = -2i + 3j + 6k. Let a₁ be the projection of a on b and a₂ be the projection of a₁ on c. Then a₂ is equal to

If $a_1$ and $a_2$ are two values of $a$ for which the unit vector $\vec{a} = a\hat{i} + 2a\hat{j} - \frac{1}{2}\hat{k}$ is linearly dependent with $\hat{i} + b\hat{j} - 2\hat{k}$, then $\frac{1}{a_1} + \frac{1}{a_2}$ is equal to

If internal and external bisectors of ∠A of △ABC meet the base BC at D and E respectively (D and E lie on same side of B), then which relation holds?

If $a^2 + b^2 + c^2 = 1$ where $a, b, c \in \mathbb{R}$, then the maximum value of $(4a - 3b)^2 + (5b - 4c)^2 + (3c - 5a)^2$ is

Vectors b = (tan α, −1, 2 sin α/2) and c = (tan α, tan α, −3 sin α/2) are orthogonal and vectors a = (1, 3, sin 2α) makes an obtuse angle with the Z-axis, then the value of α is

The vectors $u = (al + a_1l_1)\mathbf{i} + (am + a_1m_1)\mathbf{j} + (an + a_1n_1)\mathbf{k}$, $v = (bl + b_1l_1)\mathbf{i} + (bm + b_1m_1)\mathbf{j} + (bn + b_1n_1)\mathbf{k}$ and $w = (cl + c_1l_1)\mathbf{i} + (cm + c_1m_1)\mathbf{j} + (cn + c_1n_1)\mathbf{k}$

The vectors $\vec{a}$ and $\vec{b}$ are not perpendicular and $\vec{c}$ and $\vec{d}$ are two vectors satisfying: $\vec{b} \times \vec{c} = \vec{b} \times \vec{d}$ and $\vec{a} \cdot \vec{d} = 0$. Then the vector $\vec{d}$ is equal to

The value of a so that the volume of the parallelepiped formed by i + aj + k, j + ak, and ai + k becomes minimum is

Given a = 3î + 2ĵ + xk̂ and b = î − ĵ + k̂. Then the minimum value of |a × b| is

If $|\vec{c}|^2 = 60$ and $\vec{c} \times (\hat{i} + 2\hat{j} + 5\hat{k}) = \vec{0}$, then a value of $\vec{c} \cdot (-7\hat{i} + 2\hat{j} + 3\hat{k})$ is

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 $\dfrac{1}{x+1}+\dfrac{1}{y+1}+\dfrac{1}{z+1}=$

Algebra Questions (626)