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Basic Mathematics

127 Questions Start Allen Test
All Exams Allen Objective Errorless
Q51 Allen ALGEBRA OF VECTORS MCQ
Which of the following sets of concurrent forces may be in equilibrium?
A.
$F_{1} = 3N$ , $F_{2} = 5N$ , $F_{3} = 1N$ , $F_{4} = 10N$
B.
$F_{1} = 3N$ , $F_{2} = 5N$ , $F_{3} = 9N$ , $F_{4} = 4N$
C.
$F_{1} = 3N$ , $F_{2} = 5N$ , $F_{3} = 6N$ , $F_{4} = 15N$
D.
$F_{1} = 3N$ , $F_{2} = 5N$ , $F_{3} = 15N$ , $F_{4} = 5N$
Q52 Allen ALGEBRA OF VECTORS MCQ
Five forces 2N, $\sqrt{3}$ N, 5N, $\sqrt{3}$ N and 2N respectively act at a particle P as shown in the figure. The resultant force on the particle P is.
A.
10 N making angle $60^{\circ}$ with x-axis.
B.
10 N making angle $60^{\circ}$ with y-axis.
C.
20 N along y-axis
D.
None of these
Q53 Allen ALGEBRA OF VECTORS MCQ
If $\vec{a}+\vec{b}+\vec{c}=0$ . The angle between $\vec{a}$ and $\vec{b}$ , $\vec{b}$ and $\vec{c}$ are $150^{\circ}$ and $120^{\circ}$ , respectively. Then, the magnitude of vectors $\vec{a}$ , $\vec{b}$ and $\vec{c}$ are in ratio of:-
A.
1:2:3
B.
$1:2:\sqrt{3}$
C.
$\sqrt{3}:2:1$
D.
$2:\sqrt{3}:1$
Q54 Allen ALGEBRA OF VECTORS MCQ
The magnitudes of vectors $\vec{A},\vec{B}$ and $\vec{C}$ are respectively 7, 24 and 25 units and $\vec{A}+\vec{B}=\vec{C}$ , then the angle between $\vec{A}$ and $\vec{B}$ is:
A.
0
B.
$\pi/3$
C.
$\pi/2$
D.
$\pi/4$
Q55 Allen ALGEBRA OF VECTORS MCQ
If vectors $\vec{P}, \vec{Q}$ and $\vec{R}$ have magnitudes 5, 12 and 13 units and $\vec{P} + \vec{Q} = \vec{R}$ , the angle between $\vec{P}$ and $\vec{R}$ is:
A.
$\cos^{-1}\left(\frac{5}{12}\right)$
B.
$\cos^{-1}\left(\frac{5}{13}\right)$
C.
$\cos^{-1}\left(\frac{12}{13}\right)$
D.
$\cos^{-1}\left(\frac{2}{13}\right)$
Q56 Allen ALGEBRA OF VECTORS MCQ
If the vectors $\left(\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}\right)$ and $3\hat{i}$ form two sides of a triangle, then $3^{rd}$ side of the triangle can be:
A.
$\sqrt{18}$ unit
B.
$\sqrt{6}$ unit
C.
$\frac{3}{\sqrt{2}}$ unit
D.
both (1) & (2)
Q57 Allen ALGEBRA OF VECTORS MCQ
The minimum number of vectors of equal magnitude required to produce a zero resultant is:
A.
2
B.
3
C.
4
D.
more than 4
Q58 Allen ALGEBRA OF VECTORS MCQ
How many minimum number of coplanar vectors having different magnitudes can be added to give zero resultant?
A.
2
B.
3
C.
4
D.
5
Q59 Allen ALGEBRA OF VECTORS MCQ
How many minimum number of vectors in different planes can be added to give zero resultant?
A.
2
B.
3
C.
4
D.
5
Q60 Allen ALGEBRA OF VECTORS MCQ
What happens, when we multiply a vector by 2?
A.
direction reverses and unit changes
B.
direction remains same and magnitude is doubled
C.
direction remains unchanged and unit changes
D.
none of these
Q61 Allen RESOLUTION OF VECTOR MCQ
What is the maximum number of components into which a vector can be split?
A.
2
B.
3
C.
4
D.
Infinite
Q62 Allen RESOLUTION OF VECTOR MCQ
What is the maximum number of rectangular components into which a vector can be split in its own plane?
A.
2
B.
3
C.
4
D.
Infinite
Q63 Allen RESOLUTION OF VECTOR MCQ
What is the maximum number of rectangular components into which a vector can be split in space?
A.
2
B.
3
C.
4
D.
āˆž
Q64 Allen RESOLUTION OF VECTOR MCQ
The unit vector along $\hat{\mathrm{i}} - 2\hat{\mathrm{j}}$ is:
A.
$\frac{\hat{\mathrm{i}} - 2\hat{\mathrm{j}}}{\sqrt{5}}$
B.
$\hat{\mathrm{i}} +\hat{\mathrm{j}}$
C.
$\frac{\hat{\mathrm{i}} + \hat{\mathrm{j}}}{\sqrt{2}}$
D.
$\frac{\hat{\mathrm{i}} - \hat{\mathrm{j}}}{\sqrt{5}}$
Q65 Allen RESOLUTION OF VECTOR MCQ
If $\vec{\mathrm{A}} +\vec{\mathrm{B}}$ is a unit vector along y-axis and $\vec{\mathrm{A}} = \hat{\mathrm{i}} -\hat{\mathrm{j}} +\hat{\mathrm{k}}$ , then what is $\vec{\mathrm{B}}?$
A.
$\hat{\mathrm{j}} +\hat{\mathrm{k}}$
B.
$\hat{\mathrm{j}} -\hat{\mathrm{k}}$
C.
$\hat{\mathrm{i}} +\hat{\mathrm{j}} +\hat{\mathrm{k}}$
D.
$2\hat{\mathrm{j}} -\hat{\mathrm{i}} -\hat{\mathrm{k}}$
Q66 Allen RESOLUTION OF VECTOR MCQ
If a unit vector is represented by $0.3\hat{i}-0.4\hat{j}+ck$ , then the value of 'c' is:
A.
$\sqrt{0.75}$
B.
$\sqrt{0.25}$
C.
$\sqrt{0.01}$
D.
$\sqrt{0.39}$
Q67 Allen RESOLUTION OF VECTOR MCQ
$\hat{\mathbf{e}}_{\mathrm{r}}$ is unit vector along radius of a circle shown in figure. $\hat{\mathbf{e}}_{\mathrm{r}}$ can be represented as :-
A.
$\cos \theta \hat{\mathrm{i}} +\sin \theta \hat{\mathrm{j}}$
B.
$\sin \theta \hat{\mathrm{i}} +\cos \theta \hat{\mathrm{j}}$
C.
$\cos \theta \hat{\mathrm{i}} -\sin \theta \hat{\mathrm{j}}$
D.
$-\cos \theta \hat{\mathrm{i}} +\sin \theta \hat{\mathrm{j}}$
Q68 Allen RESOLUTION OF VECTOR MCQ
Forces 7N, 24N, 25N act at a point in mutually perpendicular directions. The magnitude of the resultant force is:
A.
19 N
B.
13 N
C.
26 N
D.
$25\sqrt{2}$ N
Q69 Allen RESOLUTION OF VECTOR MCQ
The angle that the vector $\vec{\mathrm{A}} = 2\hat{\mathrm{i}} +3\hat{\mathrm{j}}$ makes with x-axis is:
A.
$\tan^{-1}(3 / 2)$
B.
$\tan^{-1}(2 / 3)$
C.
$\sin^{-1}(2 / 3)$
D.
$\cos^{-1}(3 / 2)$
Q70 Allen RESOLUTION OF VECTOR MCQ
What vector must be added to the other vectors $\hat{\mathrm{i}} - 2\hat{\mathrm{j}} + 2\hat{\mathrm{k}}$ and $2\hat{\mathrm{i}} + \hat{\mathrm{j}} - \hat{\mathrm{k}}$ , so that the resultant may be a unit vector along y-axis?
A.
$2\hat{\mathrm{i}} + \hat{\mathrm{j}} - \hat{\mathrm{k}}$
B.
$-3\hat{\mathrm{i}} + 2\hat{\mathrm{j}} - \hat{\mathrm{k}}$
C.
$-2\hat{\mathrm{i}} - \hat{\mathrm{j}} - \hat{\mathrm{k}}$
D.
$3\hat{\mathrm{i}} - 2\hat{\mathrm{j}} - \hat{\mathrm{k}}$
Q71 Allen RESOLUTION OF VECTOR MCQ
The unit vector parallel to the resultant of the vectors $\vec{A} = 4\hat{i} + 3\hat{j} + 6\hat{k}$ and $\vec{B} = 2\hat{i} - 3\hat{j} + 2\hat{k}$ is:
A.
$\frac{1}{10}\left[8\hat{i} + 6\hat{k}\right]$
B.
$\frac{1}{10}\left[6\hat{i} + 8\hat{k}\right]$
C.
$\frac{1}{10}\left[6\hat{i} + 6\hat{k} + 6\hat{j}\right]$
D.
$\frac{1}{10}\left[6\hat{j} + 8\hat{k}\right]$
Q72 Allen RESOLUTION OF VECTOR MCQ
If $\vec{a}=2\hat{i}+2\hat{j}-\hat{k}$ and $\vec{b}=\hat{i}+\hat{j}+\hat{k}$ . Find a vector $\vec{c}$ which is parallel to $\vec{a}$ but has magnitude three times that of $\vec{b}$ .
A.
$\vec{c}=3\hat{i}+3\hat{j}+3\hat{k}$
B.
$\vec{c}=2\sqrt{3}\hat{i}+2\sqrt{3}\hat{j}-\sqrt{3}\hat{k}$
C.
$\vec{c}=6\hat{i}+6\hat{j}-3\hat{k}$
D.
$\vec{c}=3\hat{i}+6\hat{j}-6\hat{k}$
Q73 Allen RESOLUTION OF VECTOR MCQ
The velocity of a particle is $\vec{v} = (\hat{i} + \hat{j} - \hat{k})m/s$ . A force of $10\sqrt{3}N$ parallel to velocity in vector form is:
A.
$(10\hat{i} + 10\hat{j} - 10\hat{k})N$
B.
$(\hat{i} + \hat{j} - \hat{k})10\sqrt{3}N$
C.
$(\hat{i} + \hat{j} - \hat{k})N$
D.
None of these
Q74 Allen RESOLUTION OF VECTOR MCQ
The direction cosines of a vector $\sqrt{2}\hat{\mathrm{i}} +\sqrt{2}\hat{\mathrm{j}} +\hat{\mathrm{k}}$ are:-
A.
$\frac{\sqrt{2}}{\sqrt{5}},\frac{\sqrt{2}}{\sqrt{5}},\frac{1}{\sqrt{5}}$
B.
$\frac{1}{\sqrt{5}},\frac{1}{\sqrt{5}},\frac{1}{5}$
C.
$\frac{1}{5},\frac{1}{5},\frac{1}{\sqrt{5}}$
D.
$\frac{1}{\sqrt{5}},\frac{1}{\sqrt{5}},\frac{1}{\sqrt{5}}$
Q75 Allen RESOLUTION OF VECTOR MCQ
Vector $\vec{P}$ makes angles $\alpha, \beta \& \gamma$ with the X, Y and Z axes respectively, then $\cos^{2}\alpha + \cos^{2}\beta + \cos^{2}\gamma =$
A.
0
B.
1
C.
2
D.
3