Assertion (A): | The graph between \(P\) and \(Q\) is a straight line when \(\frac{P}{Q}\) is constant. |
Reason (R): | The straight-line graph means that \(P\) is proportional to \(Q\) or \(P\) is equal to a constant multiplied by \(Q\). |
1. | Both (A) and (R) are True and (R) is the correct explanation of (A). |
2. | Both (A) and (R) are True but (R) is not the correct explanation of (A). |
3. | (A) is True but (R) is False. |
4. | Both (A) and (R) are False |
Two forces are such that the sum of their magnitudes is \(18~\text{N}\) and their resultant is perpendicular to the smaller force and the magnitude of the resultant is \(12~\text{N}\). Then the magnitudes of the forces will be:
1. \(12~\text{N}, 6~\text{N}\)
2. \(13~\text{N}, 5~\text{N}\)
3. \(10~\text{N}, 8~\text{N}\)
4. \(16~\text{N}, 2~\text{N}\)
The projection of a vector \(\overrightarrow r = 3\hat i + \hat j + 2\hat k\) on the \(XY\)-plane has a magnitude of:
1. \(3\)
2. \(4\)
3. \(\sqrt{14}\)
4. \(\sqrt{10}\)
The angle which the vector \(\overrightarrow{A} = 2 \hat{i} + 3 \hat{j}\) makes with the \(y\text-\)axis, where \(\hat i\) and \(\hat j\) are unit vectors along \(x\text-\) and \(y\text-\)axis, respectively, is:
1. \(\cos^{-1}\left(\frac{3}{5}\right)\)
2. \(\cos^{-1}\left(\frac{2}{3}\right)\)
3. \(\tan^{-1}\left(\frac{2}{3}\right)\)
4. \(\sin^{-1}\left(\frac{2}{3}\right)\)
The components of a vector along the \(x\) and \(y\) directions are \((n+1)\) and \(1\), respectively. If the coordinate system is rotated by an angle \(\theta\), then the components change to \(n\) and \(3\). The value of \(n\) will be:
1. \(2\)
2. \(\cos60^{\circ}\)
3. \(\sin 60^{\circ}\)
4. \(3.5\)
Given that \(\overrightarrow {C}= \overrightarrow {A}+\overrightarrow {B}\)\(\overrightarrow {C}\) makes an angle \(\alpha\)
1. \(\alpha \) cannot be less than \(\beta\)
2. \(\alpha <\beta, ~\text{if}~A<B\)
3. \(\alpha <\beta, ~\text{if}~A>B\)
4. \(\alpha <\beta, ~\text{if}~A=B\)
Two forces, \(1\) N and \(2\) N, act along with the lines \(x=0\) and \(y=0\). The equation of the line along which the resultant lies is given by:
1. \(y-2x =0\)
2. \(2y-x =0\)
3. \(y+x =0\)
4. \(y-x =0\)
If the magnitude of the sum of two vectors is equal to the magnitude of the difference between the two vectors, the angle between these vectors is:
1. \(90^{\circ}\)
2. \(45^{\circ}\)
3. \(180^{\circ}\)
4. \(0^{\circ}\)
Six vectors \(\overrightarrow a ~\text{through}~\overrightarrow f\) 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. \(\overrightarrow b + \overrightarrow e = \overrightarrow f\)
If a vector \(2\hat{i}+3\hat{j}+8\hat{k}\) is perpendicular to the vector \(-4\hat{i}+4\hat{j}+\alpha \hat{k},\) then the value of \(\alpha\) will be:
1. \(-1\)
2. \(\frac{-1}{2}\)
3. \(\frac{1}{2}\)
4. \(1\)