Vector Addition

Q.

The area of the region bounded by curves $y=x\log x$ and $y=2x-2x^2$ is

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

2. $\frac{3}{12}$ sq units

3. $\frac{7}{12}$sq units

4. None of these

Q. Find the magnitude of resultant force.

1. $\sqrt{2}F$
2. $\sqrt{3}F$
3. $\sqrt{7}F$
4. 10 $F$

Q.

A particle has displacement of 12 m towards east and 5 m towards north then 6 m vertically upward. The sum of these displacements is

1. 12

2. 10.04 m

3. 14.31 m

4. None of these

Q.

If each side of a triangle is doubled, then the percentage increase in its area is $200\%$.

1. True
2. False

Q. When two vectors, $\overrightarrow{a}$ and $\overrightarrow{b}$ are added, the magnitude of the resultant vector is always

1. Greater than $(|\overrightarrow{a}|+|\overrightarrow{b}|)$
2. less than or equal to $(|\overrightarrow{a}|+|\overrightarrow{b}|)$
3. less than $(|\overrightarrow{a}|+|\overrightarrow{b}|)$
4. equal to $(|\overrightarrow{a}|+|\overrightarrow{b}|)$

Q.

Magnitude of the resultant of two vectors, $A$ and $B$ is given by square root of _____________.

Q. Six vectors $\overrightarrow{a}$to $\overrightarrow{f}$ have the directions as indicated in the figure. Which of the following statements may be true?

1. $\overrightarrow{b}+\overrightarrow{c}=-\overrightarrow{f}$
2. $\overrightarrow{d}+\overrightarrow{c}=\overrightarrow{f}$
3. $\overrightarrow{d}+\overrightarrow{e}=\overrightarrow{f}$
4. $\overrightarrow{b}+\overrightarrow{e}=\overrightarrow{f}$

Q. Two forces of $3\text{N}$ and $4\text{N}$ are acting at a point such that the angle between them is $60\text{degrees}$. Find the resultant force.

1. Mangnitude of $R=8\text{N}$ and angle $q={\tan }^{-1}0.472$
2. Mangnitude of $R=6\text{N}$ and angle $q={\tan }^{-1}0.572$
3. Mangnitude of $R=6.08\text{N}$ and angle $q={\tan }^{-1}0.472$
4. Mangnitude of $R=6.08\text{N}$ and angle $q={\tan }^{-1}0.872$

Q. If $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$are three mutually perpendicular unit vectors, then $|\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}|$ is equal to

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

Q. Two forces each numerically equal to $25\text{N}$ are acting as shown in the following figure. What is the resultant vector?

1. $|\overrightarrow{R}|=25\text{N}$ at an angle of ${60}^{\circ }$ clockwise from $\overrightarrow{A}$
2. $|\overrightarrow{R}|=25\text{N}$ at an angle of ${120}^{\circ }$ clockwise from $\overrightarrow{A}$
3. $|\overrightarrow{R}|=25\text{N}$ at an angle of ${30}^{\circ }$ clockwise from $\overrightarrow{A}$
4. $|\overrightarrow{R}|=30\text{N}$ at an angle of ${75}^{\circ }$ clockwise from $\overrightarrow{A}$

Q.

For the vectors $\overrightarrow{A}$and $\overrightarrow{B}$ making an angle $\theta$. Which one of the following relations is correct?

1. $\overrightarrow{A}\times \overrightarrow{B}=\overrightarrow{B}\times \overrightarrow{A}$

2. $\overrightarrow{A}\times \overrightarrow{B}=AB\sin \left(\theta \right)$

3. $\overrightarrow{A}.\overrightarrow{B}=AB\cos \left(\theta \right)$

4. $\overrightarrow{A}\times \overrightarrow{B}=-\overrightarrow{B}\times \overrightarrow{A}$

Q. Let $\overrightarrow{a}$ and $\overrightarrow{b}$ be two non -null vectors such that $|\overrightarrow{a}+\overrightarrow{b}|=|\overrightarrow{a}-2\overrightarrow{b}|$. Then the value of $\frac{|\overrightarrow{a}|}{|\overrightarrow{b}|}$ may be

1. $\frac{1}{4}$
2. $\frac{1}{8}$
3. 1
4. 2

Q. Maximum and minimum magnitudes of the resultant of two vectors of magnitudes $P$ and $Q$ are found to be in the ratio of $3:1$. Then

1. $P=Q$
2. $P=2Q$
3. $P=4Q$
4. $P=Q/3$

Q. If $F_1=10\text{N}$ and $F_2=20\text{N}$ as shown in the figure, the resultant $R$ will be [Hint: $\cos {65}^{\circ }=0.423$]

1. $15.87\text{N}$
2. $35.87\text{N}$
3. $25.87\text{N}$
4. $20.87\text{N}$

Q.

The area of the region bounded by the curves ${\mathrm{y}}^2=\mathrm{x}$and $\mathrm{y}=\left|\mathrm{x}\right|$is:

1. $\frac{1}{3}$$squnits$

2. $\frac{1}{6}$ $squnits$

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

4. $1$$squnits$

Q. Two vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ of equal magnitude $2\text{unit}$ are at an angle of ${60}^{\circ }$ to each other. Find the ratio of |$\overrightarrow{a}+\overrightarrow{b}$| and |$\overrightarrow{a}-\overrightarrow{b}$|

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

Q.

Can you use tan on non right triangles?

Q.

The greatest and least magnitude of the resultant of two forces of constant magnitude is$F$and $G$. When the forces act at an angle$2\alpha$, the resultant in magnitudes is equal to

1. $\sqrt{F^2{\cos }^2\alpha +G^2{\sin }^2\alpha }$

2. $\sqrt{F^2{\sin }^2\alpha +G^2{\cos }^2\alpha }$

3. $\sqrt{F^2+G^2}$

4. $\sqrt{F^2-G^2}$

Q. Six forces, $9.81N$ each, acting at a point are coplanar. If the angles between neighbouring forces are equal, then the resultant is

1. $0N$
2. $9.81N$
3. $2\times 9.81N$
4. $3\times 9.81N$

Q. The resultant of two vectors $\overrightarrow{P}$ and $\overrightarrow{Q}$ is $\overrightarrow{R}$. If magnitude of $\overrightarrow{Q}$ is doubled, the new resultant is perpendicular to $\overrightarrow{P}.$ Then $|\overrightarrow{R}|$ equals

1. $|\overrightarrow{P}|$
2. $|(\overrightarrow{P}+\overrightarrow{Q})|$
3. $|\overrightarrow{Q}|$
4. $|\overrightarrow{P}-\overrightarrow{Q}|$

Q.

Area Of The Quadrilateral Formed With The Foci Of The Hyperbola $\left(\frac{X_2}{A_2}\right)-\left(\frac{Y_2}{B_2}\right)=1$ And $\left(\frac{X_2}{A_2}\right)-\left(\frac{Y_2}{B_2}\right)=-1$ Is

Q. For a given speed, which two angles will produce the same range?

1. 35 and 65 degrees
2. 0 and 45 degrees
3. 30 and 60 degrees
4. 55 and 15 degrees

Q. The minimum number of vectors of unequal magnitudes required to produce a zero resultant is

1. 2
2. 3
3. 4
4. More than 4

Q. Two vectors $\overrightarrow{A}$ and $\overrightarrow{B}$ lie in a plane, another vector $\overrightarrow{C}$ lies outside this plane, then the resultant of these three vectors i.e.$\overrightarrow{A}+\overrightarrow{B}+\overrightarrow{C}$

1. Can be zero.
2. Cannot be zero.
3. Lies in the plane containing $\overrightarrow{A}+\overrightarrow{B}$
4. Lies in the plane containing $\overrightarrow{C}$

Q.

The area of the region bounded by the straight lines $\mathrm{x}=0$ and $\mathrm{x}=2$ and the curves $\mathrm{y}=2^{\mathrm{x}}$ and $\mathrm{y}=2\mathrm{x}-{\mathrm{x}}^2$ is equal to

1. $\left[\left(\frac{2}{\log \left(2\right)}\right)-\left(\frac{4}{3}\right)\right]$

2. $\left[\left(\frac{3}{\log \left(2\right)}\right)-\left(\frac{4}{3}\right)\right]$

3. $\left[\left(\frac{1}{\log \left(2\right)}\right)-\left(\frac{4}{3}\right)\right]$

4. $\left[\left(\frac{4}{\log \left(2\right)}\right)-\left(\frac{3}{2}\right)\right]$

Q. If $\overrightarrow{C}=\overrightarrow{A}+\overrightarrow{B}$, then the magnitude of $\overrightarrow{B}$ is

1. $6\sqrt{5}\text{N}$
2. $3\sqrt{5}\text{N}$
3. $5\sqrt{5}\text{N}$
4. $4\sqrt{5}\text{N}$

Q. Two forces $\overrightarrow{F_1}=2\hat{i}+2\hat{j}N$ and $\overrightarrow{F_2}=3\hat{j}+4\hat{k}N$ are acting on a particle. The resultant force acting on the particle is

1. $2\hat{i}+5\hat{j}+4\hat{k}$
2. $2\hat{i}-5\hat{j}-4\hat{k}$
3. $\hat{i}-3\hat{j}-2\hat{k}$
4. $\hat{i}-\hat{j}-\hat{k}$

Q.

Two vectors having equal magnitudes of A make an angle θ with each other. Find the magnitude B of the resultant vector and the angle it makes with any one of the vectors?

1. B = A $cos\frac{\theta }{2}$ , $\alpha =0$

2. B = 2 A cos $\frac{\theta }{2}$ , $\alpha =\frac{\theta }{2}$

3. B = A cos $\frac{\theta }{2}$ , $\alpha =\frac{\theta }{2}$

4. B = 2 A cos $\frac{\theta }{2}$ , $\alpha =\theta$

Q. Two vectors have magnitudes $3$ units and $6$ units respectively. What should be the angle between these two vectors if the resultant vector is perpendicular to the smaller one?

1. ${60}^{\circ }$
2. ${120}^{\circ }$
3. ${150}^{\circ }$
4. ${180}^{\circ }$

Q. For the resultant of two vectors to be maximum, the angle between them should be

1. 0 degree
2. 45 degrees
3. 90 degrees
4. 180 degrees