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Basic Mathematics

127 Questions Start Allen Test
All Exams Allen Objective Errorless
Q26 Allen 5. 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 5. 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 5. 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 5. 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 5. 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 6. 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 6. 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 6. 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 6. 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 6. 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 6. 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 6. 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 6. 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 6. 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 6. 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 6. 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 6. 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 6. 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 6. 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 6. 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 6. 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 6. 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 6. 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 6. 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 6. 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