Definition of Vector

Q.

If $z=x+iy$ is a complex number satisfying $\left|z-i\mathrm{Re}\left(z\right)\right|=\left|z-\mathrm{Im}\left(z\right)\right|$ then $z$ lies on

1. $y=x$

2. $y=-x$

3. $y=x+1$

4. $y=-x+1$

Q. Find a(vector) . b(vector) and a(vector)*b(vector) if a= i+2j+k and magnitude of b=3 acting along c(vector) = i+j+k.

Q.

Let $S=\{x\in R:x\ge 0$ and $2|\sqrt{x}-3|+\sqrt{x}(\sqrt{x}-6)+6=0\}$. Then $S$:

1. contains exactly two elements.

2. contains exactly four elements.

3. is an empty set.

4. contains exactly one element.

Q.

Let $z$ and $w$ be two complex numbers such that $w=z\overline{z}-2z+2$, $\left|\frac{(z+i)}{(z–3i)}\right|=1$ and $Re\left(w\right)$ has a minimum value. Then, the minimum value of $n\in N,$ for which $w^n$ is real, is equal to ____

Q.

A certain orbital has $\mathrm{n}=4$ and ${\mathrm{m}}_l=-3$.The number of radial nodes in the orbital is__________. (Round off to the nearest integer)

Q. If $|\overrightarrow{a}|=16,$ $|\overrightarrow{b}|=4,$ then, $\sqrt{|\overrightarrow{a}\times \overrightarrow{b}{|}^2+|\overrightarrow{a}.\overrightarrow{b}{|}^2}=$

1. 4
2. 64
3. 16
4. 8

Q.

If $\overrightarrow{x}$and $\overrightarrow{y}$be two non-zero vectors such that $\left|\overrightarrow{x}+\overrightarrow{y}\right|=\left|\overrightarrow{x}\right|$ and $2\overrightarrow{x}+\lambda \overrightarrow{y}$is perpendicular to $\overrightarrow{y}$ , then the value of $\lambda$ is

Q.

The shortest distances of the point $(1,2,3)$ from the axis $X,Y$and $Z$ respectively are

1. $1,2,3$

2. $\sqrt{5},\sqrt{13},\sqrt{10}$

3. $\sqrt{10},\sqrt{13},\sqrt{5}$

4. $\sqrt{13},\sqrt{10},\sqrt{5}$

Q.

What are the $4$ operations of sets ?

Q.

If $f\left(x\right)=2x^3+m{x}^2-13x+n$ and $2,3$ are the roots of the equation $f\left(x\right)=0$, then the values of $m$ and $n$ are:

1. $-5,-30$

2. $-5,30$

3. $5,30$

4. None of these

Q.

Solve the equation $4x^2-5x-3=0$ and give your answer to correct two decimals.

Q.

The solution of equation $\left|z\right|-z=1+2i$ is

1. $\frac{3}{2}+2i$

2. $\frac{3}{2}-2i$

3. $3-2i$

4. none of these

Q.

Vector of magnitude 1 is called.

1. Constant vector

2. Zero vector

3. Unit vector

4. None of these

Q.

he vector$\overrightarrow{a}=\alpha \hat{i}+2\hat{j}+\beta \hat{k}$ lies in the plane of the vectors $\overrightarrow{b}=\hat{i}+\hat{j}$ and $\overrightarrow{c}=\hat{j}+\hat{k}$ and bisects the angle between $\overrightarrow{b}$ and $\overrightarrow{c}$. Then, which of the following gives possible values of $\alpha$ and $\beta$?

1. $\alpha =1,\beta =1$

2. $\alpha =2,\beta =2$

3. $\alpha =1,\beta =2$

4. $\alpha =2,\beta =1$

Q.

The solution of the equation ${(3+2\sqrt{2})}^{{\mathrm{x}}^2-8}+{(3+2\sqrt{2})}^{8-{\mathrm{x}}^2}=6$ are

1. $3\pm 2\sqrt{2}$

2. $\pm 1$

3. $\pm 3\sqrt{3},\pm 2\sqrt{2}$

4. $\pm 7,\pm \sqrt{3}$

5. $\pm 3,\pm \sqrt{7}$

Q. If the position vectors of the vertices $A,B$ and $C$ of a $△ABC$ are respectively $4\hat{i}+7\hat{j}+8\hat{k},2\hat{i}+3\hat{j}+4\hat{k}$ and $2\hat{i}+5\hat{j}+7\hat{k}$, then the position vector of the point, where the bisector of $\angle A$ meets $BC$ is :

1. $\frac{1}{2}(4\hat{i}+8\hat{j}+11\hat{k})$
2. $\frac{1}{4}(8\hat{i}+14\hat{j}+19\hat{k})$
3. $\frac{1}{3}(6\hat{i}+13\hat{j}+18\hat{k})$
4. $\frac{1}{3}(6\hat{i}+11\hat{j}+15\hat{k})$

Q. Let $\overrightarrow{a}=4\hat{i}+5\hat{j}-\hat{k},\overrightarrow{b}=\hat{i}-4\hat{j}+5\hat{k},\text{and}\overrightarrow{c}=3\hat{i}-\hat{j}-\hat{k}$. Find a vector $\overrightarrow{d}$ which is perpendicular to both $\overrightarrow{c}$ and $\overrightarrow{b}$ and $\overrightarrow{d}$. $\overrightarrow{a}=21$.

Q.

Find a vector of magnitude 6 , which is perpendicular to both the vectors $2\hat{i}-\hat{j}+2\hat{k}\text{and}4\hat{i}-\hat{j}+3\hat{k}.$

Q.

For the curve $\sqrt{\mathrm{x}}+\sqrt{\mathrm{y}}=1,$find $\frac{\mathrm{dy}}{dx}$at $\left(\frac{1}{4},\frac{1}{4}\right)$ is

1. $1$

2. $\frac{1}{2}$

3. $-1$

4. $2$

Q.

A vector has components $2p$ and $1$ with respect to a rectangular Cartesian system. This system is rotated clockwise. If this respect to the new system has components $p+1$ and $1$, then?

1. $p=0$

2. $p=1\mathrm{or}p=-\frac{1}{3}$

3. $p=-1$

4. $p=1\mathrm{or}p\mathit{=}-1$

Q.

If the vector $a=2i+3j+6k$ and $b$ are collinear and $\left|b\right|=21$, then $b$ is equal to:

1. $\pm \left(2i+3j+6k\right)$

2. $\pm 3\left(2i+3j+6k\right)$

3. $\left(i+j+k\right)$

4. $\pm 21\left(2i+3j+6k\right)$

Q. If $\overrightarrow{a}$ and $\overrightarrow{b}$ are any two unit vectors, then the greatest positive integer in the range of $\frac{3|\overrightarrow{a}+\overrightarrow{b}|}{2}+2|\overrightarrow{a}-\overrightarrow{b}|$ is

Q.

Find the roots of $x^2-3x-10=0$

Q.

Let $a,b,c$ be three non-coplanar vectors such that $r_1=a-b+c$, $r_2=b+c-a$, $r_3=c+a+b$, $r=2a-3b+4c$. If $r={\lambda }_1{r}_1+{\lambda }_2{r}_2+{\lambda }_3{r}_3$, then

1. ${\lambda }_1=7$

2. ${\lambda }_1+{\lambda }_3=6$

3. ${\lambda }_1+{\lambda }_2+{\lambda }_3=4$

4. ${\lambda }_2+{\lambda }_3=2$

Q. If a variable line drawn through the point of intersection of straight lines $\frac{x}{\alpha }+\frac{y}{\beta }=1$ and $\frac{x}{\beta }+\frac{y}{\alpha }=1$ meets the coordinate axes in $A$ and $B$, then the locus of the mid-point of $AB$ is:

1. $\alpha \beta (x+y)=xy(\alpha +\beta )$
2. $\alpha \beta (x+y)=2(\alpha +\beta )xy$
3. $(\alpha +\beta )(x+y)=2\alpha \beta xy$
4. None of these

Q.

The ends of the latus rectum of the conic $x^2+10x-16y+25=0$ are

1. $\left(3,-4\right),\left(13,4\right)$

2. $\left(-3,-4\right),\left(13,-4\right)$

3. $\left(3,-4\right),\left(-13,4\right)$

4. $\left(5,-8\right),\left(-5,8\right)$

Q. 8. If a, b , c and d are non-coplanar vectors , then d. (a(b(cd))) is equal to _______.

Q.

The inequality $\left|z-4\right|$ less than $\left|z-2\right|$ represents the region given by

1. $\text{Re(z)}$ greater than $0$

2. $\text{Re(z)}$ less than $0$

3. $\text{Re(z)}$ greater than $2$

4. $\text{Re(z)}$ greater than $3$

Q.

If $b$and$c$ are any two non - collinear unit vectors and $a$ is any vector,

then $\frac{(a.b)b+(a.c)c+a.(bxc)}{|bxc|.(bxc)}$ is equal to

1. $0$

2. $a$

3. $b$

4. $c$

Q. Let $f:R\to R$ be a function defined by $f(x+1)=\frac{f\left(x\right)-5}{f\left(x\right)-3}\forall x\in R$ then which of the following statements are true?

1. $f\left(2021\right)=f\left(2025\right)$
2. $f\left(2017\right)=f\left(2020\right)$
3. $f\left(2018\right)=f\left(2026\right)$
4. $f\left(7\right)=f\left(10\right)$