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Basic Mathematics

127 Questions Start Allen Test
All Exams Allen Objective Errorless
Q26 Allen DEFINITION & TYPES OF VECTOR MCQ
A physical quantity which has a direction and obeys triangle law of vector addition:
A.
must be a vector
B.
may be a vector
C.
must be a scalar
D.
none of the above
Q27 Allen DEFINITION & TYPES OF VECTOR MCQ
The forces, which meet at one point and their lines of action lie in one plane, are called:
A.
non-concurrent and non-coplanar forces
B.
non-concurrent and coplanar forces
C.
concurrent and non-coplanar forces
D.
concurrent and coplanar forces
Q28 Allen DEFINITION & TYPES OF VECTOR MCQ
Which of the following physical quantities is not an axial vector?
A.
angular velocity
B.
angular momentum
C.
velocity
D.
torque
Q29 Allen DEFINITION & TYPES OF VECTOR MCQ
The direction of the angular velocity vector is along:
A.
Along the tangent of circular path
B.
Along the direction of radius vector
C.
Opposite to the direction of radius vector
D.
Along the axis of rotation
Q30 Allen DEFINITION & TYPES OF VECTOR MCQ
If $\hat{\mathbf{n}}$ is a unit vector in the direction of the vector $\vec{\mathbf{A}}$ , then:
A.
$\hat{\mathbf{n}} = \frac{\vec{\mathbf{A}}}{|\vec{\mathbf{A}}|}$
B.
$\hat{\mathbf{n}} = \vec{\mathbf{A}} |\vec{\mathbf{A}}|$
C.
$\hat{\mathbf{n}} = \frac{|\vec{\mathbf{A}}|}{\vec{\mathbf{A}}}$
D.
None of the above
Q31 Allen ALGEBRA OF VECTORS MCQ
In the given figure, $\vec{a} + \vec{b} + \vec{c}$ is
A.
$2\vec{a}$
B.
$2\vec{b}$
C.
$2\vec{c}$
D.
$\vec{a} + \vec{b}$
Q32 Allen ALGEBRA OF VECTORS MCQ
Vector sum of two forces of 10N and 5N can be:
A.
4N
B.
18N
C.
12N
D.
2N
Q33 Allen ALGEBRA OF VECTORS MCQ
The vector sum of the forces of 10 newton and 8 newton cannot be:
A.
2N
B.
8N
C.
18N
D.
20N
Q34 Allen ALGEBRA OF VECTORS MCQ
Which of the following pair of forces can give a resultant force of 2 N?
A.
1 N and 2 N
B.
1 N and 5 N
C.
2 N and 5 N
D.
1 N and 4 N
Q35 Allen ALGEBRA OF VECTORS MCQ
Two vectors of equal magnitude have a resultant equal to either of them in magnitude. The angle between the vector is:
A.
$60^{\circ}$
B.
$90^{\circ}$
C.
$105^{\circ}$
D.
$120^{\circ}$
Q36 Allen ALGEBRA OF VECTORS MCQ
$\vec{R} = \vec{a} +\vec{b}$ , where $\vec{R},\vec{a}$ and $\vec{b}$ are non-zero vector. If $R = a = b$ , the angle between $\vec{R}$ and $\vec{a}$ is :-
A.
$\frac{\pi}{3}$
B.
$\frac{2\pi}{3}$
C.
$\frac{\pi}{4}$
D.
$\frac{\pi}{6}$
Q37 Allen ALGEBRA OF VECTORS MCQ
If resultant of two vectors $\vec{a}$ and $\vec{b}$ shown in the figure is $\sqrt{7}b$ , then value of $\frac{b}{a}$ is :-
A.
$\frac{1}{2}$
B.
1
C.
2
D.
3
Q38 Allen ALGEBRA OF VECTORS MCQ
If $\vec{A} +\vec{B} = \vec{C}$ and $A + B = C$ , then the angle between $\vec{A}$ and $\vec{C}$ is :
A.
0
B.
$\frac{\pi}{4}$
C.
$\frac{\pi}{2}$
D.
$\pi$
Q39 Allen ALGEBRA OF VECTORS MCQ
Two vectors $\vec{A}$ and $\vec{B}$ are such that $\vec{A} + \vec{B} = \vec{C}$ and A - B = C. Which of the following statements, is correct?
A.
$\vec{A}$ is parallel to $\vec{B}$
B.
$\vec{A}$ is anti-parallel to $\vec{B}$
C.
$\vec{A}$ is perpendicular to $\vec{B}$
D.
$\vec{A}$ and $\vec{B}$ are equal in magnitude
Q40 Allen ALGEBRA OF VECTORS MCQ
In vector diagram shown in figure where $\left(\vec{R}\right)$ is the resultant of vectors $(\vec{A})$ and $(\vec{B})$ . If $R = \frac{B}{2}$ , then value of angle $\theta$ is :
A.
$30^{\circ}$
B.
$45^{\circ}$
C.
$60^{\circ}$
D.
$75^{\circ}$
Q41 Allen ALGEBRA OF VECTORS MCQ
The resultant of $\vec{A}$ and $\vec{B}$ makes an angle $\alpha$ with $\vec{A}$ and $\beta$ with $\vec{B}$ , then:
A.
$\alpha < \beta$
B.
$\alpha < \beta$ if A < B
C.
$\alpha < \beta$ if A > B
D.
$\alpha < \beta$ if A = B
Q42 Allen ALGEBRA OF VECTORS MCQ
If two forces of equal magnitude acting on a particle. The angle between forces is $60^{\circ}$ . The resultant force on the particle is:
A.
at an angle of $30^{\circ}$ with each force.
B.
at an angle of $45^{\circ}$ with one of the force
C.
at an angle of $15^{\circ}$ with one of the force
D.
at an angle lying between $45^{\circ}$ to $60^{\circ}$
Q43 Allen ALGEBRA OF VECTORS MCQ
The resultant of $\vec{A}$ and $\vec{B}$ is perpendicular to $\vec{B}$ . What is the angle between $\vec{A}$ and $\vec{B}$ ?
A.
$\cos^{-1}\left(\frac{A}{B}\right)$
B.
$\cos^{-1}\left(-\frac{B}{A}\right)$
C.
$\sin^{-1}\left(\frac{A}{B}\right)$
D.
$\sin^{-1}\left(-\frac{A}{B}\right)$
Q44 Allen ALGEBRA OF VECTORS MCQ
The resultant of two forces make $30^{\circ}$ and $60^{\circ}$ angles with them and has magnitude 40 N. The magnitudes of two vectors are:
A.
$20\sqrt{3}N, 20N$
B.
$30N, 20\sqrt{3}N$
C.
40N, 30N
D.
$3\sqrt{3}N, 40\sqrt{3}N$
Q45 Allen ALGEBRA OF VECTORS MCQ
Given that $\vec{\mathrm{P}} +\vec{\mathrm{Q}} = \vec{\mathrm{P}} -\vec{\mathrm{Q}}$ . This can be true when:
A.
$\vec{\mathrm{P}} = \vec{\mathrm{Q}}$
B.
$\vec{\mathrm{Q}} = \vec{0}$
C.
Neither $\vec{\mathrm{P}}$ nor $\vec{\mathrm{Q}}$ is a null vector
D.
Data Insufficient
Q46 Allen ALGEBRA OF VECTORS MCQ
The resultant of $\vec{A} \& \vec{B}$ is $\vec{R}_1$ . On reversing the vector $\vec{B}$ , the resultant becomes $\vec{R}_2$ . What is the value of $2(R_1^2 + R_2^2)$ ?
A.
$4(A^2 + B^2)$
B.
$A^2 - B^2$
C.
$2(A^2 + B^2)$
D.
$2(A^2 - B^2)$
Q47 Allen ALGEBRA OF VECTORS MCQ
If vectors $\vec{A}$ and $\vec{B}$ are such that $|\vec{A}+\vec{B}|=\left|\vec{A}\right|=\left|\vec{B}\right|$ , then $|\vec{A}-\vec{B}|$ may be equated to
A.
$\frac{\sqrt{3}}{2}|\vec{A}|$
B.
$|\vec{A}|$
C.
$\sqrt{2}|\vec{A}|$
D.
$\sqrt{3}|\vec{A}|$
Q48 Allen ALGEBRA OF VECTORS MCQ
If the difference of two unit vectors is a unit vector, then the magnitude of their sum is:
A.
$\sqrt{2}$
B.
$\sqrt{3}$
C.
$\frac{1}{\sqrt{2}}$
D.
$\sqrt{5}$
Q49 Allen ALGEBRA OF VECTORS MCQ
In the given figure, O is the centre of the regular pentagon ABCDE. Five forces each of magnitude $F_{0}$ are acted as shown in the figure. The resultant force is :-
A.
$5\mathrm{F}_0$
B.
$5\mathrm{F}_0\cos 72^{\circ}$
C.
$5\mathrm{F}_0\sin 72^{\circ}$
D.
zero
Q50 Allen ALGEBRA OF VECTORS MCQ
Two vectors $\vec{A}$ and $\vec{B}$ lie in a plane, another vector $\vec{C}$ lies in the same plane, then the resultant of these three vectors i.e. $\vec{A} + \vec{B} + \vec{C}$ :
A.
may be zero
B.
cannot be zero
C.
lies in the plane containing $\vec{A}$ & $\vec{B}$
D.
Both 1 & 3