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Basic Mathematics

127 Questions Start Allen Test
All Exams Allen Objective Errorless
Q76 Allen 7. RESOLUTION OF VECTOR MCQ
Find the direction cosines of vector $(\vec{a}-\vec{b})$ , if $\vec{a}=2\hat{i}+3\hat{j}+\hat{k}$ and $\vec{b}=2\hat{i}+2\hat{j}+3\hat{k}$
A.
$0,\frac{1}{\sqrt{5}},\frac{-2}{\sqrt{5}}$
B.
0, 0, 0
C.
$\frac{1}{\sqrt{5}},\frac{-2}{\sqrt{5}},\frac{2}{\sqrt{5}}$
D.
1, 1, -1
Q77 Allen 7. RESOLUTION OF VECTOR MCQ
Three forces are acting on a particles as shown in the figure. To have the resultant forces only along the y-direction, the magnitude of the minimum additional force needed is :-
A.
0.866 N
B.
1.732 N
C.
0.5 N
D.
4 N
Q78 Allen 7. RESOLUTION OF VECTOR MCQ
The force $\vec{\mathrm{F}} = (\hat{\mathrm{i}} +\mathrm{b}\hat{\mathrm{j}} +3\hat{\mathrm{k}})\mathrm{N}$ is rotated through an angle $\alpha$ , then it becomes $\{2\hat{\mathrm{i}} +(2\mathrm{b} - 1)\hat{\mathrm{j}} +\hat{\mathrm{k}}\} \mathrm{N}$ The value of b is :-
A.
2
B.
$-\frac{2}{3}$
C.
$\frac{2}{3}$
D.
Both (1) and (2)
Q79 Allen 8. DOT PRODUCT MCQ
If $\vec{P}\cdot\vec{Q}=-PQ$ , then angle between $\vec{P}$ and $\vec{Q}$ is:
A.
$0^{\circ}$
B.
$180^{\circ}$
C.
$45^{\circ}$
D.
$60^{\circ}$
Q80 Allen 8. DOT PRODUCT MCQ
A force $\vec{\mathrm{F}} = (2\hat{\mathrm{i}} + 3\hat{\mathrm{j}})\mathrm{N}$ acts on a body and displaces it by $\vec{S} = (3\hat{\mathrm{i}} + 4\hat{\mathrm{j}})\mathrm{m}$ . The work done $(W = \overline{F} \cdot \overline{S})$ by the force is:
A.
10J
B.
12J
C.
18J
D.
25J
Q81 Allen 8. DOT PRODUCT MCQ
A force $(2\hat{i}+2\hat{j})$ N displaces an object through a distance $(2\hat{i}-3\hat{j})$ m. The work $(W=\vec{F}\cdot\vec{S})$ done is:
A.
-2J
B.
12J
C.
5J
D.
13J
Q82 Allen 8. DOT PRODUCT MCQ
A force of 14 N acts on a particle along the vector $(3\hat{i}+2\hat{j}-6\hat{k})$ . If the particle displaces from $(0,0,0)$ to $(2,4,-2)$ , the work $(W=\vec{F}\cdot\vec{S})$ done by force on the particle is :-
A.
10J
B.
52J
C.
-48J
D.
14 J
Q83 Allen 8. DOT PRODUCT MCQ
If a vector $(2\hat{i}+3\hat{j}+8\hat{k})$ is perpendicular to the vector $(4\hat{i}-4\alpha\hat{j}+\alpha\hat{k})$ , then the value of $\alpha$ is:
A.
-1
B.
1/2
C.
-1/2
D.
2
Q84 Allen 8. DOT PRODUCT MCQ
The vector $\vec{\mathrm{P}} = a\hat{\mathrm{i}} + a\hat{\mathrm{j}} + 3\hat{\mathrm{k}}$ and $\vec{\mathrm{Q}} = a\hat{\mathrm{i}} - 2\hat{\mathrm{j}} - \hat{\mathrm{k}}$ are perpendicular to each other. The positive value of a is:
A.
3
B.
2
C.
1
D.
zero
Q85 Allen 8. DOT PRODUCT MCQ
A vector perpendicular to $(4\hat{i}+3\hat{j})$ may be:
A.
$4\hat{i}+3\vec{j}$
B.
$7\hat{k}$
C.
$6\hat{i}$
D.
$3\hat{i}+4\hat{j}$
Q86 Allen 8. DOT PRODUCT MCQ
Let $\vec{A} = \hat{i} A\cos \theta -\hat{j} A\sin \theta$ , be any vector. Another vector $\vec{B}$ which is normal to $\vec{A}$ is :
A.
$\hat{\mathrm{i}} B\cos \theta +\hat{\mathrm{j}} B\sin \theta$
B.
$\hat{\mathrm{i}} B\sin \theta +\hat{\mathrm{j}} B\cos \theta$
C.
$\hat{\mathrm{i}} B\sin \theta -\hat{\mathrm{j}} B\cos \theta$
D.
$\hat{\mathrm{i}} A\cos \theta -\hat{\mathrm{j}} A\sin \theta$
Q87 Allen 8. DOT PRODUCT MCQ
If $\hat{\mathbf{n}} = a\hat{\mathbf{i}} + b\hat{\mathbf{j}}$ is perpendicular to the vector $(\hat{\mathbf{i}} - \hat{\mathbf{j}})$ , then the value of $a$ and $b$ may be:
A.
1, 0
B.
-2, 0
C.
3, 0
D.
$\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}$
Q88 Allen 8. DOT PRODUCT MCQ
The vector sum of two forces is perpendicular to their vector difference. In that case, the force:
A.
Are equal to each other.
B.
Are equal to each other in magnitude.
C.
Are not equal to each other in magnitude.
D.
Cannot be predicted.
Q89 Allen 8. DOT PRODUCT MCQ
Given that A = B and $\vec{A} \perp \vec{B}$ . What is the angle between $(\vec{A} + \vec{B})$ and $(\vec{A} - \vec{B})$ ?
A.
$30^{0}$
B.
$60^{0}$
C.
$90^{0}$
D.
$180^{0}$
Q90 Allen 8. DOT PRODUCT MCQ
A parallelogram is formed with $\vec{a}$ and $\vec{b}$ as the sides. Let $\vec{d}_{1}$ and $\vec{d}_{2}$ be the diagonals of the parallelogram. The value of $a^{2} + b^{2}$ is :-
A.
$d_{1}^{2} + d_{2}^{2}$
B.
$\frac{d_{1}^{2} + d_{2}^{2}}{2}$
C.
$(d_{1} + d_{2})^{2}$
D.
$d_{1}^{2} + d_{2}^{2} + d_{1}d_{2}$
Q91 Allen 8. DOT PRODUCT MCQ
The angle between the two vectors $\vec{A} = 3\hat{i} +4\hat{j} +5\hat{k}$ and $\vec{B} = 6\hat{i} +8\hat{j} +10\hat{k}$ will be:
A.
$0^{\circ}$
B.
$180^{\circ}$
C.
$90^{\circ}$
D.
$45^{\circ}$
Q92 Allen 8. DOT PRODUCT MCQ
What is the angle between $\vec{A}$ and the resultant of $(\vec{A} + 2\hat{B})$ and $(2\vec{A} - 2\hat{B})$ ?
A.
$0^{\circ}$
B.
$\tan^{-1}\left(\frac{A}{B}\right)$
C.
$\tan^{-1}\left(\frac{B}{A}\right)$
D.
$\tan^{-1}\left(\frac{A - B}{A + B}\right)$
Q93 Allen 8. DOT PRODUCT MCQ
The angle between vectors $(\hat{\mathrm{i}}+\hat{\mathrm{j}})$ and $(\hat{\mathrm{i}}+\hat{\mathrm{k}})$ is:
A.
$90^{\circ}$
B.
$180^{\circ}$
C.
$0^{\circ}$
D.
$60^{\circ}$
Q94 Allen 8. DOT PRODUCT MCQ
The angle between $2\hat{i}+\hat{j}+2\hat{k}$ and $\hat{i}-\hat{j}+\hat{k}$ is :-
A.
$30^{\circ}$
B.
$60^{\circ}$
C.
$\cos^{-1}\left(\frac{1}{\sqrt{3}}\right)$
D.
$\cos^{-1}\left(\frac{2}{3}\right)$
Q95 Allen 8. DOT PRODUCT MCQ
What is the projection of $3\hat{i} + 4\hat{k}$ on the z-axis?
A.
3
B.
4
C.
5
D.
zero
Q96 Allen 8. DOT PRODUCT MCQ
What is the projection of $\vec{B}$ on $\vec{A}$ ?
A.
$\vec{A}.\vec{B}$
B.
$\vec{A}.\hat{B}$
C.
$\vec{B}.\hat{A}$
D.
$\hat{A}.\hat{B}$
Q97 Allen 9. CROSS PRODUCT MCQ
If $\hat{\mathrm{i}},\hat{\mathrm{j}}$ and $\hat{\mathrm{k}}$ are unit vectors along X, Y & Z axis respectively, then tick the wrong statement:
A.
$\hat{\mathrm{i}}\times \hat{\mathrm{i}} = \vec{0}$
B.
$\hat{\mathrm{i}}\times \hat{\mathrm{j}} = \hat{\mathrm{k}}$
C.
$\hat{\mathrm{i}}.\hat{\mathrm{j}} = 1$
D.
$\hat{\mathrm{i}}\times \hat{\mathrm{k}} = -\hat{\mathrm{j}}$
Q98 Allen 9. CROSS PRODUCT MCQ
The magnitude of the vector product of two vectors $\vec{A}$ and $\vec{B}$ may be:
A.
Equal to 0
B.
Less than AB
C.
Equal to AB
D.
All of above
Q99 Allen 9. CROSS PRODUCT MCQ
If $\vec{\mathrm{P}}\times \vec{\mathrm{Q}} = \vec{\mathrm{R}}$ , then which of the following statements is true?
A.
$\vec{\mathrm{R}}\bot \vec{\mathrm{P}}$
B.
$\vec{\mathrm{R}}\bot \vec{\mathrm{Q}}$
C.
$\vec{\mathrm{R}}\bot (\vec{\mathrm{P}} +\vec{\mathrm{Q}})$
D.
All of above
Q100 Allen 9. CROSS PRODUCT MCQ
Two vectors $\vec{P}$ and $\vec{Q}$ are inclined to each other at angle $\theta$ . Which of the following is the unit vector perpendicular to $\vec{P}$ and $\vec{Q}$ ?
A.
$\frac{\vec{P}\times\vec{Q}}{P.Q}$
B.
$\frac{\hat{P}\times\hat{Q}}{\sin\theta}$
C.
$\frac{\vec{P}\times\vec{Q}}{|\vec{P}\times\vec{Q}|}$
D.
Both 2 and 3