Cross Product

Q.

If the length of a vector is $21$ and direction ratios be $2,-3,6$ then its direction cosines are.

1. $\frac{2}{21},-\frac{1}{7},\frac{2}{7}$

2. $\frac{2}{7},-\frac{3}{7},\frac{6}{7}$

3. $\frac{2}{7},\frac{3}{7},\frac{6}{7}$

4. None of these.

Q.

Let the vectors a and b be such that $|a|=3$ and $|b|=\frac{\sqrt{2}}{3}$, then $a\times b$ is a unit vector, if the angle between a and b is

a) $\frac{\pi }{6}$

b) $\frac{\pi }{4}$

c) $\frac{\pi }{3}$

d) $\frac{\pi }{2}$

Q. Find the principal value of ${\cot }^{-1}\left(\sqrt{3}\right).$

Q. Column Matching:

$$\begin{array}{cc}\text{Column (I)}& \text{Column (II)}\\ \begin{array}{cc}\text{(A)}& \text{In}{R}^2,\text{if the magnitude of the projection}\\ & \text{vector of the vector}\alpha \hat{i}+\beta \hat{j}\text{on}\sqrt{3}\hat{i}+\hat{j}\\ & \sqrt{3}\text{and if}\alpha =2+\sqrt{3}\beta ,\text{then possible}\\ & \text{value(s) of}|\alpha |\text{is (are)}\end{array}& \text{(P) 1}\\ & \\ \begin{array}{cc}\text{(B)}& \text{Let}a\text{and}b\text{be real numbers such that}\\ & \text{the function}\\ & f\left(x\right)=\{\begin{array}{c}-3a{x}^2-2,x<1\\ bx+a^2,x\ge 1\end{array}\\ & \text{is differentiable for all}x\in R.\text{Then}\\ & \text{possible value(s) of}a\text{is (are)}\end{array}& \text{(Q) 2}\\ & \\ \begin{array}{cc}\text{(C)}& \text{Let}\omega \ne 1\text{be a complex cube root of}\\ & \text{unity. If}(3-3\omega +2{\omega }^2{)}^{4n+3}+\\ & (2+3\omega -3{\omega }^2{)}^{4n+3}+(-3+2\omega +3{\omega }^2{)}^{4n+3}=0,\\ & \text{then possible value(s) of}n\text{is (are)}\end{array}& \text{(R) 3}\\ & \\ \begin{array}{cc}\text{(D)}& \text{Let the harmonic mean of two positive real}\\ & \text{numbers}a\text{and}b\text{be}4.\text{If}q\text{is a positive real}\\ & \text{number such that}a,5,q,b\text{is an arithmetic}\\ & \text{progression, then the value(s) of}|q-a|\text{is (are)}\end{array}& \text{(S) 4}\\ & \\ & \text{(T) 5}\\ & \end{array}$$
Option $\text{(D)}$ matches with which of the elements of right hand column?

1. $P$
2. $Q$
3. $R$
4. $S$
5. $T$

Q.

Let O be the origin and OX, OY, OZ be three unit vectors in the directions of the sides QR, RP, PQ respectively, of a triangle PQR.
If the triangle PQR varies, then the minimum value of $\cos (P+Q)+\cos (Q+R)+\cos (R+P)$ is

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

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

3. $\frac{5}{3}$

4. $-\frac{5}{3}$

Q.

If $\overrightarrow{a}$ and $\overrightarrow{b}$ are vectors such that $|\overrightarrow{a}+\overrightarrow{b}|=\sqrt{29}$ and $\overrightarrow{a}\times (2\hat{i}+3\hat{j}+4\hat{k})=(2\hat{i}+3\hat{j}+4\hat{k})\times \overrightarrow{b},$ then a possible value of $(\overrightarrow{a}+\overrightarrow{b}).(-7\hat{i}+2\hat{j}+3\hat{k})$ is

1. 0

2. 3

3. 4

4. 8

Q. If $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c},\overrightarrow{a}+\overrightarrow{b},\overrightarrow{b}+\overrightarrow{c}$ and $\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}$ are vectors of same magnitude, none of which is null vector, then which of the following is false

1. Angle between $\overrightarrow{b}$ and $\overrightarrow{c}$ will always be obtuse
2. $\overrightarrow{a}-\overrightarrow{c}$ and $\overrightarrow{b}$ are mutually perpendicular vectors
3. $\overrightarrow{a}+\overrightarrow{c}$ will be perpendicular to $\overrightarrow{b}$
4. $\overrightarrow{a}$ and $\overrightarrow{c}$ are mutually perpendicular vectors

Q. Let $\overrightarrow{a}$ and $\overrightarrow{b}$ be two unit vectors, then $\overrightarrow{a}\times \overrightarrow{b}$ will also be a unit vector if:

1. $\overrightarrow{a}=\overrightarrow{b}$
2. $\overrightarrow{a}\perp \overrightarrow{b}$
3. $\overrightarrow{a}||\overrightarrow{b}$
4. $\overrightarrow{a}=m\overrightarrow{b},m>1$

Q.

Let $a=2\hat{i}+\hat{j}-2\hat{k},b=\hat{i}+\hat{j}$ and $c$ be a vector such that $|c-a|=3,|(a\times b)\times c|=3$ and the angle between $c$ and $a\times b$ is ${30}^{\circ }$. Then, $a\cdot c$ is equal to

1. $\frac{25}{8}$

2. 2

3. 5

4. $\frac{1}{8}$

Q. Column Matching:

$$\begin{array}{cc}\text{Column (I)}& \text{Column (II)}\\ \begin{array}{cc}\text{(A)}& \text{In}{R}^2,\text{if the magnitude of the projection}\\ & \text{vector of the vector}\alpha \hat{i}+\beta \hat{j}\text{on}\sqrt{3}\hat{i}+\hat{j}\\ & \text{is}\sqrt{3}\text{and if}\alpha =2+\sqrt{3}\beta ,\text{then possible}\\ & \text{value(s) of}|\alpha |\text{is (are)}\end{array}& \text{(P) 1}\\ & \\ \begin{array}{cc}\text{(B)}& \text{Let}a\text{and}b\text{be real numbers such that}\\ & \text{the function}\\ & f\left(x\right)=\{\begin{array}{c}-3a{x}^2-2,x<1\\ bx+a^2,x\ge 1\end{array}\\ & \text{is differentiable for all}x\in R.\text{Then}\\ & \text{possible value(s) of}a\text{is (are)}\end{array}& \text{(Q) 2}\\ & \\ \begin{array}{cc}\text{(C)}& \text{Let}\omega \ne 1\text{be a complex cube root of}\\ & \text{unity. If}(3-3\omega +2{\omega }^2{)}^{4n+3}+\\ & (2+3\omega -3{\omega }^2{)}^{4n+3}+(-3+2\omega +3{\omega }^2{)}^{4n+3}=0,\\ & \text{then possible value(s) of}n\text{is (are)}\end{array}& \text{(R) 3}\\ & \\ \begin{array}{cc}\text{(D)}& \text{Let the harmonic mean of two positive real}\\ & \text{numbers}a\text{and}b\text{be}4.\text{If}q\text{is a positive real}\\ & \text{number such that}a,5,q,b\text{is an arithmetic}\\ & \text{progression, then the value(s) of}|q-a|\text{is (are)}\end{array}& \text{(S) 4}\\ & \\ & \text{(T) 5}\\ & \end{array}$$
Option $\text{(D)}$ matches with which of the elements of right hand column?

1. $P$
2. $Q$
3. $R$
4. $S$
5. $T$

Q.

If $\overrightarrow{a}$ and $\overrightarrow{b}$ are vectors such that $|\overrightarrow{a}+\overrightarrow{b}|=\sqrt{29}$ and $\overrightarrow{a}\times (2\hat{i}+3\hat{j}+4\hat{k})=(2\hat{i}+3\hat{j}+4\hat{k})\times \overrightarrow{b},$ then a possible value of $(\overrightarrow{a}+\overrightarrow{b}).(-7\hat{i}+2\hat{j}+3\hat{k})$ is

1. 3

2. 4

3. 0

4. 8

Q. Let $\overrightarrow{a}=2\hat{i}+\hat{j}-2\hat{k},\overrightarrow{b}=\hat{i}+\hat{j}$. If $\overrightarrow{c}$ is a vector such that $\overrightarrow{a}\cdot \overrightarrow{c}=|\overrightarrow{c}|,|\overrightarrow{a}-\overrightarrow{c}|=2\sqrt{2}$ and the angle between $\overrightarrow{a}\times \overrightarrow{b}$ and $\overrightarrow{c}$ is $\frac{\pi }{6}$, then

1. $|(\overrightarrow{a}\times \overrightarrow{b})\times \overrightarrow{c}|=\frac{3}{2}$
2. $|\overrightarrow{c}|=1$
3. $|(\overrightarrow{a}\times \overrightarrow{b})\times \overrightarrow{c}|=\frac{2}{3}$
4. $|\overrightarrow{c}|=4$

Q. Let $A_1{A}_2{A}_3....A_9$ be a nine-sided regular polygon with side length $2$ units. The difference between the lengths of the diagonals $A_1{A}_5$ and $A_2{A}_4$ equals

1. $\sqrt{12}-2$
2. $6$
3. $2$
4. $2+\sqrt{12}$

Q. Area of parallelogram whose adjacent sides are $\overrightarrow{a}=\hat{i}-\hat{j}+3\hat{k}$ and$\overrightarrow{b}=2\hat{i}-7\hat{j}+\hat{k}$ is

1. $15\sqrt{2}$ sq. units
2. $20$ sq. units
3. $25\sqrt{14}$ sq. units
4. $30$ sq. units

Q. Column Matching:

$$\begin{array}{cc}\text{Column (I)}& \text{Column (II)}\\ \begin{array}{cc}\text{(A)}& \text{In}{R}^2,\text{if the magnitude of the projection}\\ & \text{vector of the vector}\alpha \hat{i}+\beta \hat{j}\text{on}\sqrt{3}\hat{i}+\hat{j}\\ & \sqrt{3}\text{and if}\alpha =2+\sqrt{3}\beta ,\text{then possible}\\ & \text{value(s) of}|\alpha |\text{is (are)}\end{array}& \text{(P) 1}\\ & \\ \begin{array}{cc}\text{(B)}& \text{Let}a\text{and}b\text{be real numbers such that}\\ & \text{the function}\\ & f\left(x\right)=\{\begin{array}{c}-3a{x}^2-2,x<1\\ bx+a^2,x\ge 1\end{array}\\ & \text{is differentiable for all}x\in R.\text{Then}\\ & \text{possible value(s) of}a\text{is (are)}\end{array}& \text{(Q) 2}\\ & \\ \begin{array}{cc}\text{(C)}& \text{Let}\omega \ne 1\text{be a complex cube root of}\\ & \text{unity. If}(3-3\omega +2{\omega }^2{)}^{4n+3}+\\ & (2+3\omega -3{\omega }^2{)}^{4n+3}+(-3+2\omega +3{\omega }^2{)}^{4n+3}=0,\\ & \text{then possible value(s) of}n\text{is (are)}\end{array}& \text{(R) 3}\\ & \\ \begin{array}{cc}\text{(D)}& \text{Let the harmonic mean of two positive real}\\ & \text{numbers}a\text{and}b\text{be}4.\text{If}q\text{is a positive real}\\ & \text{number such that}a,5,q,b\text{is an arithmetic}\\ & \text{progression, then the value(s) of}|q-a|\text{is (are)}\end{array}& \text{(S) 4}\\ & \\ & \text{(T) 5}\\ & \end{array}$$
Option $\text{(D)}$ matches with which of the elements of right hand column?

1. $P$
2. $Q$
3. $R$
4. $S$
5. $T$

Q. Let the unit vector $a$ and $b$ be perpendicular to each other and the unit vector $c$ be inclined at an angle $\theta$ to both $a$ and $b$. If $c=xa+yb+z(a\times b)$, then

1. $x=\sin \theta ,y=\cos \theta ,z=-\cos 2\theta$
2. $x=y=\cos \theta ,z^2=\cos 2\theta$
3. $x=\cos \theta ,y=\sin \theta ,z=\cos 2\theta$
4. $x=y=\cos \theta ,z^2=-\cos 2\theta$

Q. If $A,B,C,D$ are any four points in space, then $|\overrightarrow{AB}\times \overrightarrow{CD}+\overrightarrow{BC}\times \overrightarrow{AD}+\overrightarrow{CA}\times \overrightarrow{BD}|=$$m\times$Area of $△ABC$. Find m.

Q. If $2\hat{i}+3\hat{j}+4\hat{k}$ and $\hat{i}-\hat{j}+\hat{k}$ are two adjacent sides of a parallelogram, then the area of the parallelogram will be

1. 
2. 
3. Cannot be calculated from the given data
4. none of the above

Q. If the vectors $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ from the sides $BC,CA,AB$ respectively of triangle $ABC$, then

1. $\overrightarrow{a}.\overrightarrow{b}=\overrightarrow{b}.\overrightarrow{c}=\overrightarrow{c}.\overrightarrow{a}$
2. $\overrightarrow{a}\times \overrightarrow{b}+\overrightarrow{b}\times \overrightarrow{c}+\overrightarrow{c}\times \overrightarrow{a}=0$
3. $\overrightarrow{a}\times \overrightarrow{b}=\overrightarrow{b}\times \overrightarrow{c}=\overrightarrow{c}\times \overrightarrow{a}$
4. $\overrightarrow{a}.\overrightarrow{b}+\overrightarrow{b}.\overrightarrow{c}+\overrightarrow{c}.\overrightarrow{a}=0$

Q. Suppose the area of the $\Delta ABC$ is $10\sqrt{3}$. Length of segments $AC$ and $AB$ be $5$ and $8$ respectively. Then the angle $A$ is (are) :

1. ${45}^{\circ }$ or ${135}^{\circ }$
2. ${30}^{\circ }$ or ${150}^{\circ }$
3. ${90}^{\circ }$
4. ${60}^{\circ }$ or ${120}^{\circ }$

Q. The area common to the circle $x^2+y^2=16a^2$ and the parabola $y^2=6ax$ is

1. $\frac{4a^2}{3}(4\pi -\sqrt{3})$
2. $\frac{4a^2}{3}(8\pi -3)$
3. None of these
4. $\frac{4a^2}{3}(4\pi +\sqrt{3})$

Q.

Let $\Delta PQR$ be a triangle. Let a = QR, b = RP and c = PQ. If $|a|=12,|b|=4\sqrt{3}$ and $b.c=24,$ then which of the following is/are true ?

1. $\frac{|c{|}^2}{2}-|a|=12$

2. $\frac{|c{|}^2}{2}+|a|=30$

3. $|a\times b+c\times a|=48\sqrt{3}$

4. $a.b=-72$

Q.

Let O be the origin and OX, OY, OZ be three unit vectors in the directions of the sides QR, RP, PQ respectively, of a triangle PQR.
$|OX\times OY|=$

1. sin (Q + R)

2. sin (P+Q)

3. sin (P + R)

4. sin 2R

Q.

Let O be the origin and OX, OY, OZ be three unit vectors in the directions of the sides QR, RP, PQ respectively, of a triangle PQR.
If the triangle PQR varies, then the minimum value of $\cos (P+Q)+\cos (Q+R)+\cos (R+P)$ is

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

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

3. $\frac{5}{3}$

4. $-\frac{5}{3}$

Q. lf a line makes angles $\frac{\pi }{12},\frac{5\pi }{12}$ with $OY,OZ$ respectively where $O=(0,0,0)$, then the angle made by that line with $OX$ is

1. ${45}^o$
2. ${90}^o$
3. ${60}^o$
4. ${30}^o$

Q. Find the angles between the following pairs of vectors: $i-j+k,-i+j+2k,$
and $i-j+k,-i+j+2k$

1. $\frac{\pi }{6}$
2. $\frac{\pi }{2}$
3. $\frac{\pi }{3}$
4. $\frac{\pi }{4}$

Q. Column Matching:

$$\begin{array}{cc}\text{Column (I)}& \text{Column (II)}\\ \begin{array}{cc}\text{(A)}& \text{In}{R}^2,\text{if the magnitude of the projection}\\ & \text{vector of the vector}\alpha \hat{i}+\beta \hat{j}\text{on}\sqrt{3}\hat{i}+\hat{j}\\ & \text{is}\sqrt{3}\text{and if}\alpha =2+\sqrt{3}\beta ,\text{then possible}\\ & \text{value(s) of}|\alpha |\text{is (are)}\end{array}& \text{(P) 1}\\ & \\ \begin{array}{cc}\text{(B)}& \text{Let}a\text{and}b\text{be real numbers such that}\\ & \text{the function}\\ & f\left(x\right)=\{\begin{array}{c}-3a{x}^2-2,x<1\\ bx+a^2,x\ge 1\end{array}\\ & \text{is differentiable for all}x\in R.\text{Then}\\ & \text{possible value(s) of}a\text{is (are)}\end{array}& \text{(Q) 2}\\ & \\ \begin{array}{cc}\text{(C)}& \text{Let}\omega \ne 1\text{be a complex cube root of}\\ & \text{unity. If}(3-3\omega +2{\omega }^2{)}^{4n+3}+\\ & (2+3\omega -3{\omega }^2{)}^{4n+3}+(-3+2\omega +3{\omega }^2{)}^{4n+3}=0,\\ & \text{then possible value(s) of}n\text{is (are)}\end{array}& \text{(R) 3}\\ & \\ \begin{array}{cc}\text{(D)}& \text{Let the harmonic mean of two positive real}\\ & \text{numbers}a\text{and}b\text{be}4.\text{If}q\text{is a positive real}\\ & \text{number such that}a,5,q,b\text{is an arithmetic}\\ & \text{progression, then the value(s) of}|q-a|\text{is (are)}\end{array}& \text{(S) 4}\\ & \\ & \text{(T) 5}\\ & \end{array}$$
Option $\text{(D)}$ matches with which of the elements of right hand column?

1. $P$
2. $Q$
3. $R$
4. $S$
5. $T$

Q. If arg$\left(\frac{z_1-\frac{z}{\left|z\right|}}{\frac{z}{\left|z\right|}}\right)=\frac{\pi }{2}$ and $\left|\frac{z}{\left|z\right|}-z_1\right|=3$ then $\left|z_1\right|$ equals to

1. $\sqrt{3}$
2. $2\sqrt{2}$
3. $\sqrt{26}$
4. $\sqrt{10}$

Q. If $△$ ABC is an acute angled triangle and BD is perpendicular to AC, then the value of $\frac{\sin A}{\sin C}$

1. $\frac{AB}{BC}$
2. $\frac{BC}{AB}$
3. $\frac{BC}{CD}$
4. $\frac{CD}{BC}$

Q. The vectors $\overrightarrow{a}=\hat{i}+2\hat{j}+3\hat{k};\overline{b}=2\hat{i}-\hat{j}+\hat{k}$ & $\overrightarrow{c}=3\hat{i}+\hat{j}+4\hat{k}$ are so placed that the end point of one vector is the starting point of the next vector.Then the vectors are

1. not coplanar
2. coplanar but cannot form a triangle
3. coplanar but can form a triangle
4. coplanar & can form a right angled triangle