Vectors

Q. Two vectors $A$ and $B$ have equal magnitude. If the magnitude of $(A+B)$ is ${}^{\prime }{n}^{\prime }$ times the magnitude of $(A-B)$, then angle between vectors $A$ and $B$ is

1. ${\sin }^{-1}\left(\frac{n^2-1}{n^2+1}\right)$
2. ${\sin }^{-1}\left(\frac{n-1}{n+1}\right)$
3. ${\cos }^{-1}\left(\frac{n^2-1}{n^2+1}\right)$
4. ${\cos }^{-1}\left(\frac{n-1}{n+1}\right)$

Q. Three particles $P,Q$ and $R$ are moving along the vectors $\overrightarrow{A}=\hat{i}+\hat{j},\overrightarrow{B}=\hat{j}+\hat{k}$ and $\overrightarrow{C}=-\hat{i}+\hat{j}$ respectively. They strike on a point and start to move in different directions. Now particle $P$ is moving normal to the plane which contains vector $\overrightarrow{A}$ and $\overrightarrow{B}$ . Similarly, particle $Q$ is moving normal to the plane which contains vector $\overrightarrow{A}$ and $\overrightarrow{C}$ . The angle between the direction of motion of $P$ and $Q$ is ${\cos }^{-1}\left(\frac{1}{\sqrt{x}}\right)$. Then the value of $x$ is

Q.

A particle of mass 2 kg moves on a circular path with constant speed 10mm/s find change in speed and magnitude of change in velocity when particle completes half revolution

Q. Let the angle between two non-zero vectors $\overrightarrow{A}$ and $\overrightarrow{B}$ be ${120}^{{}^0}$ and resultant be $\overrightarrow{C}$, then

1. $\left|\overrightarrow{C}\right|$ must be equal to $|\overrightarrow{A}-\overrightarrow{B}|$
   
2. $\left|\overrightarrow{C}\right|$ must be less than $|\overrightarrow{A}-\overrightarrow{B}|$
3. $\left|\overrightarrow{C}\right|$ must be greater than $|\overrightarrow{A}-\overrightarrow{B}|$
4. $\left|\overrightarrow{C}\right|$ may be equal to $|\overrightarrow{A}-\overrightarrow{B}|$
   

Q. Two vectors $\overrightarrow{X}$ and $\overrightarrow{Y}$ have equal magnitude. The magnitude of $(\overrightarrow{X}-\overrightarrow{Y})$ is $n$ times the magnitude of $(\overrightarrow{X}+\overrightarrow{Y})$. The angle between $\overrightarrow{X}$ and $\overrightarrow{Y}$ is:

1. ${\cos }^{-1}\left(\frac{n^2+1}{n^2-1}\right)$
2. ${\cos }^{-1}\left(\frac{n^2+1}{-n^2-1}\right)$
3. ${\cos }^{-1}\left(\frac{n^2-1}{-n^2-1}\right)$
4. ${\cos }^{-1}\left(\frac{-n^2-1}{n^2-1}\right)$

Q.

An aeroplane is moving in a circular path with a speed 250 km/h what is change in velocity in half revolution

Q.

A body is moving with a uniform velocity of 10 m/s on a circular path of diameter 2.0m. calculate

1) the difference between the magnitude of the displacement of the body and the distance covered in half a round and

2) the magnitude of the changes in velocity of the body in half a round.

Q. The magnitudes of four pairs of displacement vectors are given. Which pair of displacement vectors cannot be added to give a resultant vector of magnitude 4 cm?

1. 1 cm, 1 cm
2. 1 cm, 3 cm
3. 1 cm, 5 cm
4. 1 cm, 7 cm

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

1. $\hat{i}-3\hat{j}+3\hat{k}$
2. $-3\hat{i}-3\hat{j}-\hat{k}$
3. $3\hat{i}-\hat{j}+3\hat{k}$
4. $\hat{i}+3\hat{j}-3\hat{k}$

Q. A point traversed half a circle of radius R=160cm during time interval t=10sec. Calculate the following quantities averaged over that time (a) the mean velocity (b) the modulus of mean velocity vector || (c) the modulus of the mean vector of the total acceleration ||

Q. If the sum of two unit vectors is a unit vector, then magnitude of difference is

1. √2
2. √3
3. 1/√2
4. √5

Q. The magniude of vectors \(\overrightarrow {OA}, \overrightarrow {OB}\) and \(\overrightarrow {OC}\) In the given figure are equal.\(\overrightarrow {OA}+\overrightarrow {OB}-\overrightarrow {OC}\) with x-axis will be

Q. The sum of the magnitudes of two forces is $18\text{N}$ and the magnitude of their resultant is $12\text{N}$. If the resultant makes an angle of ${90}^{\circ }$ with the smaller force, then find the magnitude of the forces.

1. $5\text{N},13\text{N}$
2. $6\text{N},12\text{N}$
3. $\text{None of these}$
4. $10\text{N},8\text{N}$

Q. The sum of the magnitudes of two forces is $18\text{N}$ and the magnitude of their resultant is $12\text{N}$. If the resultant makes an angle of ${90}^{\circ }$ with the smaller force, then find the magnitude of the forces.

1. $5\text{N},13\text{N}$
2. $6\text{N},12\text{N}$
3. $10\text{N},8\text{N}$
4. $\text{None of these}$

Q. Magnitude of resultant of $3$ vectors $\overrightarrow{OA},\overrightarrow{OB}$, and $\overrightarrow{OC}$ as shown below is

1. $r$
2. $2r$
3. $r(1+\sqrt{2})$
4. $r(\sqrt{2}-1)$

Q. Part of commentary by a commentator of a race is given. What can you infer about it? "The racing car takes a circular turn at a constant velocity of 84 km per hour. "

1. Commentary is right, car can turn at constant velocity.
2. Commentary is wrong, velocity does not remain constant while turning.
3. Such high speed is not possible.
4. None of the above.

Q. Magnitude of resultant of $3$ vectors $\overrightarrow{OA},\overrightarrow{OB}$, and $\overrightarrow{OC}$ as shown below is

1. $r$
2. $2r$
3. $r(1+\sqrt{2})$
4. $r(\sqrt{2}-1)$

Q. A radioactive element $X$ converts into another stable element $Y$. Half-life of $X$ is $2h$. Initially, only $X$ is present. After time $t$, the ratio of atoms of $X$ and $Y$ is found to be $1:4$. Then $t$ in hours is

1. $2$
2. $4$
3. between $4$ and $6$
4. $6$

Q. A radioactive element $X$ converts into another stable element $Y$. Half life of $X$ is $2hrs$. Initially only $X$ is present. After time $t$, the ratio of atoms of $X$ and $Y$ is found to be $1:4$, then $t$ in hours:

1. $4$
2. $2$
3. $6$
4. Between $4$ and $6$

Q. If the sum of two unit vectors is a unit vector, then magnitude of their difference is

1. $\sqrt{2}$
2. $\sqrt{3}$
3. $\frac{1}{\sqrt{2}}$
4. $\sqrt{5}$

Q. Two forces each of 10N act at an angle ${60}^0$ with each other. The magnitude and direction of the resultant with respect to one of the vectors is:

1. $\sqrt{10}N,{30}^0$
2. $10\sqrt{3}N,{30}^0$
3. $10\sqrt{2}N,{120}^0$
4. $20N,{120}^0$

Q. The resultant of two vectors A and B subtends an angle of ${45}^{\circ }$ with either of them. The magnitude of the resultant is

1. zero
2. $\sqrt{2}A$
3. A
4. 2A

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

1. $\hat{i}-3\hat{j}+3\hat{k}$
2. $-3\hat{i}-3\hat{j}-\hat{k}$
3. $3\hat{i}-\hat{j}+3\hat{k}$
4. $\hat{i}+3\hat{j}-3\hat{k}$

Q. The sum of the magnitude of two forces at a point is 16 N. If their resultant is normal to the smaller force and has a magnitude of 8 N. Then two forces are

1. 6 N, 10 N
2. 8 N, 8 N
3. 4 N, 12 N
4. 2 N, 14 N

Q. If $S_A$ and $S_B$ represents the displacement of block and plank w.r.t. ground respectively when block seperates, then

1. $S_A$=$4V\sqrt{\frac{l}{3g}}-\frac{2}{3}l$
2. $S_A$=$4V\sqrt{\frac{l}{g}}-\frac{2}{3}l$
3. $S_B$=$4V\sqrt{\frac{l}{3g}}-\frac{5}{3}l$
4. $S_B$=$4V\sqrt{\frac{l}{g}}-\frac{1}{3}l$

Q. Part of commentary by a commentator of a race is given. What can you infer about it? "The racing car takes a circular turn at a constant velocity of 84 km per hour. "

1. Commentary is right, car can turn at constant velocity.
2. Commentary is wrong, velocity does not remain constant while turning.
3. Such high speed is not possible.
4. None of the above.