List of practice Questions

Let $\vec{a}, \vec{b}, \vec{c}$ be three vectors such that $|\vec{a}|=\sqrt{31}, 4|\vec{b}|=|\vec{c}|=2$ and $2(\vec{a} \times \vec{b})=3(\vec{c} \times \vec{a})$ If the angle between $\vec{b}$ and $\vec{c}$ is $\frac{2 \pi}{3}$, then $\left(\frac{\vec{a} \times \vec{c}}{\vec{a} \cdot \vec{b}}\right)^2$ is equal to _____

If the foot of the perpendicular drawn from $(1,9,7)$ to the line passing through the point $(3,2,1)$ and parallel to the planes $x+2 y+z=0$ and $3 y-z=3$ is $(\alpha, \beta, \gamma)$, then $\alpha+\beta+\gamma$ is equal to
The foot of perpendicular of the point $(2,0,5)$ on the line $\frac{x+1}{2}=\frac{y-1}{5}-\frac{z+1}{-1}$ is $(a, \beta, \gamma)$. Then which of the following is NOT correct?
Let $\vec{a}=\hat{i}-\hat{j}+2 \hat{k}$ and $\vec{b}$ be a vector such that $\vec{a} \times \vec{b}=2 \hat{i}-\hat{k}$ and $\vec{a} \cdot \vec{b}=3$ Then the projection of $\vec{b}$ on the vector $\vec{a}-\vec{b}$ is :-
Let \(\vec{a}=-\hat{i}-\hat{j}+\hat{k}, \vec{a} \cdot \vec{b}=1\) and \(\vec{a} \times \vec{b}=\hat{i}-\hat{j}\). Then \(\vec{a}-6 \vec{b}\) is equal to
Let $P Q R$ be a triangle. The points $A, B$ and $C$ are on the sides $Q R, R P$ and $P Q$ respectively such that $\frac{Q A}{A R}=\frac{R B}{B P}=\frac{P C}{C Q}=\frac{1}{2}$. Then $\frac{A \text { Area }(\triangle P Q R)}{\text { Area }(\triangle A B C)}$ is equal to

Let $\vec{u}=\hat{i}-\hat{j}-2 \hat{k}, \vec{v}=2 \hat{i}+\hat{j}-\hat{k}, \vec{v} \cdot \vec{w}=2$ and $\vec{v} \times \vec{w}=\vec{u}+\lambda \vec{v}$.  Then $\vec{u} \cdot \vec{w}$ is equal to

Let\(\overrightarrow{ a }=2 \hat{i}-7 \hat{j}+5 \hat{k}, \overrightarrow{ b }=\hat{i}+\hat{k} and \overrightarrow{ c }=\hat{i}+2 \hat{j}-3 \hat{k}\) be three given vectors If \(\overrightarrow{ r }\) is a vector such that\( \vec{r} \times \vec{a}=\vec{c} \times \vec{a}  \ and \  \vec{r} \cdot \vec{b}=0,\) then \(|\vec{r}|\) is equal to :

Let \(\vec{a}\) = 2\(\widehat{i}\)+3\(\widehat{j}\)+4\(\widehat{k}\), \(\vec{b}\) = \(\widehat{i}\)-2\(\widehat{j}\)-2\(\widehat{k}\), \(\vec{c}\) = -\(\widehat{i}\)+4\(\widehat{j}\)+3\(\widehat{k}\) and \(\vec{d}\) is a vector perpendicular to \(\vec{b}\) and \(\vec{c}\),  \(\vec{a}\).\(\vec{d}\) = 18, then find |\(\vec{a}\)x\(\vec{d}\)|2

The sum of all values of \( \alpha \), for which the points whose position vectors are:

\[ \mathbf{r_1} = \hat{i} - 2\hat{j} + 3\hat{k}, \quad \mathbf{r_2} = 2\hat{i} - 3\hat{j} + 4\hat{k}, \quad \mathbf{r_3} = (\alpha+1)\hat{i} + 2\hat{k}, \quad \mathbf{r_4} = \hat{j} + 2\hat{k} \]

are coplanar, is equal to:

If \(\vec{a}\), \(\vec{b}\), \(\vec{c}\) are three non-zero vectors and \(\hat{n}\) is a unit vector perpendicular to \(\vec{c}\) such that \(\vec{a}\)=\(\alpha \vec{b}\)-\(\hat{n}\), \(\alpha \neq 0\) and \(\vec{b}\) .\( \vec{c}\)=12, then |\(\vec{c} \times(\vec{a} \times \vec{b})\)| is equal to:
If \(\mathbf{a}, \mathbf{b}, \mathbf{c}\) are three non-zero vectors and \(\hat{\mathbf{n}}\) is a unit vector perpendicular to \(\mathbf{c}\) such that: \(\mathbf{a} = \alpha \mathbf{b} - \hat{\mathbf{n}}, \, (\alpha \neq 0)\) and \( \mathbf{b} \cdot \mathbf{c} = 12 \), then \( \mathbf{c} \times (\mathbf{a} \times \mathbf{b}) \) is equal to:
If \(\overrightarrow a =\hat i+2\hat k,\overrightarrow b =\hat i+\hat j+\hat k,\overrightarrow c =7\hat i−3\hat j+4\hat k,r ×\hat b +\hat b ×\overrightarrow c =\overrightarrow0\) and \(\overrightarrow r .\overrightarrow a =0\). Then \(\overrightarrow r. \overrightarrow c\)  is equal to 
Let \(\overrightarrow 𝑎 =4\hat 𝑖+3\hat 𝑗ˆ\) and \(\overrightarrow 𝑏 =3\hat 𝑖−4\hat 𝑗\)+\(5\hat 𝑘\). If \(\overrightarrow 𝑐\)  is a vector such that \(\overrightarrow 𝑐 ⋅(\overrightarrow 𝑎 ×\overrightarrow𝑏⃗ )+25=0\), \(\overrightarrow𝑐 ⋅(\hat𝑖+\hat𝑗+ \hat𝑘)=4\), and projection of \(\overrightarrow𝑐\)  on \(\overrightarrow𝑎\)  is \(1\) , then the projection of \(\overrightarrow 𝑐\)  on \(\overrightarrow 𝑏\) equals 
If four distinct points with position vectors \(\overrightarrow a,\overrightarrow b,\overrightarrow c\)  and \(\overrightarrow d\)  are coplanar, then \([\overrightarrow a \overrightarrow b \overrightarrow c]\) is equal to
Let for a triangle ABC,
\(\vec{AB}=-2\hat{i}+\hat{j}+3\hat{k} \)
\(\vec{CB}=α\hat{i}+β\hat{j}+γ\hat{k} \)
\(\vec{CA}=4\hat{i}+3\hat{j}+δ\hat{k} \)
If \(δ>0\) and the area of the triangle ABC is \(5\sqrt6\), then \(\vec{CB}.\vec{CA}\) is equal to
Let \( \mathbf{a} \) be a non-zero vector parallel to the line of intersection of the two planes described by \( i + j + k \) and \( -i - j - k \). If \( \theta \) is the angle between the vector \( \mathbf{a} \) and the vector \( \mathbf{b} = -2i - 2j + 2k \), and \( \left| \mathbf{a} \right| = 6 \), then ordered pair \( (\mathbf{a} \cdot \mathbf{b}) \) is equal to:
Let \( f(x) = 2x^n + \lambda \), where \( \lambda \in \mathbb{R} \) and \( n \in \mathbb{N} \) . Given that \( f(4) = 133 \) and \( f(5) = 255 \), What is the sum of all the positive integer divisors of \( f(3) - f(2) \)?

The total number of six digit numbers, formed using the digits 4,5,9 only and divisible by 6 , is __

If ${ }^{2 n+1} P _{n-1}:{ }^{2 n-1} P _n=11: 21$, then $n^2+n+15$ is equal to :

If ${ }^{2 n+1} P _{n-1}:{ }^{2 n-1} P _n=11: 21$, then $n^2+n+15$ is equal to :

The number of ways to distribute 20 chocolates among three students such that each student gets atleast one chocolate is

If 2nC3 : nC3 = 10, then \(\frac{n^{2}+3n}{n^{2}-3n+4}\) is equal to

Find all the four letter words with two vowels and 2 consonants from the word UNIVERSE?
Rank of the word PUBLIC is?