100 High-Scoring MCQs on Vectors and Equilibrium (Set - 1) | Class 11 Physics | Unit -2

100 High-Scoring MCQs on Vectors and Equilibrium (Set - 1) | Class 11 Physics | Unit 2 | FBISE 

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This post contains 100 carefully selected multiple-choice questions (MCQs) Set-2 from Unit-2: Vectors and Equilibrium of Class 11 Physics, designed strictly according to the FBISE syllabus. These MCQs include a well-balanced mix of conceptual and numerical problems, making them exam-ready, revision-friendly, and high-scoring.

Whether you are preparing for annual board examinations, chapter tests, or competitive entry tests, this comprehensive MCQ collection thoroughly covers all the fundamental principles of vectors, torque, and equilibrium, enabling students to master the chapter with confidence and accuracy.

This unit-wise MCQ set includes questions from:

• Description of Cartesian coordinate system and vector representation
• Determination of vector sum using head-to-tail and parallelogram methods
• Resolution of vectors into perpendicular components and their numerical applications
• Scalar (dot) product of vectors and its relation to the angle between vectors
• Vector (cross) product and its magnitude and direction using the right-hand rule
• Torque as a vector product (r × F) and applications of torque in real life
• First condition of equilibrium (net force = 0) and second condition of equilibrium (net torque = 0)
• Two-dimensional statics problems solved using the conditions of equilibrium
• Applications of vectors and equilibrium principles in daily life, engineering, and mechanics

Each MCQ is provided with the correct answer and a clear, concise explanation, helping students:

• Strengthen conceptual understanding of vectors and equilibrium
• Avoid common exam mistakes in numerical and conceptual questions
• Improve problem-solving skills involving forces, torque, and vector operations
• Achieve maximum marks in the MCQs section

This all-in-one MCQ collection is an essential tool for Class 11 students aiming to excel in physics exams and fully master the concepts of vectors and equilibrium.

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MCQ no. 1.

The projection of a vector A on a vector B is given by:

a. $A_x=\frac{\vec{A}\cdot \vec{B}}{B}$

b. $B_x=\frac{\vec{A}\cdot \vec{B}}{B}$

c. $A\cos \theta =\frac{\vec{A}\cdot \vec{B}}{B}$

d. Both A and C

The Correct Answer is d. Both A and C

Explanation:
The projection of vector A on B is defined as:

$$\text{Projection of vector A}=\frac{\vec{A}\cdot \vec{B}}{\mathrm{\mid }\vec{B}\mathrm{\mid }}=A\cos \theta$$

Hence, both options A and C are correct.

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MCQ no. 2.

If $\vec{A}=\widehat{\mathrm{ı}}+\widehat{\mathrm{ȷ}}$  and $\vec{B}=\widehat{\mathrm{ȷ}}+\widehat{k}$, then the angle between $\vec{A}$ and $\vec{B}$ is:

a. $-\frac{16}{7}$

b. $-\frac{2}{7}$

c. $-\frac{4}{7}$

d. $-\frac{7}{4}$

The Correct Answer is c.  $-\frac{4}{7}$

Explanation:
For perpendicular vectors: $\vec{A}\cdot \vec{B}=0$ 

$$10(i\cdot i)+7a(j\cdot j)-6(k\cdot k)=0⟹10+7a-6=0⟹a=-\frac{4}{7}$$

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MCQ no. 3.

$(\vec{A}\cdot \vec{B}{)}^2+(\vec{A}\times \vec{B}{)}^2$  is:

a. $AB$ 

b. $A^{2}B$ 

c. $A^2{B}^2$ 

d. $A{B}^2$ 

The Correct Answer is: $A^2{B}^2$ 

Explanation:

$$(\vec{A}\cdot \vec{B}{)}^2+(\vec{A}\times \vec{B}{)}^2=A^2{B}^2{\cos }^2\theta +A^2{B}^2{\sin }^2\theta =A^2{B}^2$$

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MCQ no. 4.

If $R_x$  is positive and $R_y$ is negative, then $\theta$ lies in ________ quadrant.

a. 1st

b. 2nd

c. 3rd

d. 4th

The Correct Answer is: 4th

Explanation:
When X-component is positive and Y-component is negative, the vector lies in the 4th quadrant. The angle measured counter-clockwise from the positive X-axis is between 270° and 360°.

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MCQ no. 5.

Magnitude of the resultant of two vectors of equal magnitude is zero. Then the angle between them is:

a. 0°

b. 90°

c. 120°

d. 180°

The Correct Answer is: 180°

Explanation:
Two vectors give zero resultant only when they are equal in magnitude and opposite in direction.

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MCQ no. 6.

The unit vector of a vector $\vec{A}=2\widehat{\mathrm{ı}}+\widehat{\mathrm{ȷ}}+\widehat{k}$  is:

a. $\widehat{A}=\frac{2\widehat{\mathrm{ı}}+\widehat{\mathrm{ȷ}}+\widehat{k}}{\sqrt{2^2+1^2+1^2}}$ 

b. $\widehat{A}=\frac{2\widehat{\mathrm{ı}}+2\widehat{\mathrm{ȷ}}+\widehat{k}}{\sqrt{2^2+1^2+1^2}}$

c. $\widehat{A}=\frac{2\widehat{\mathrm{ı}}+\widehat{\mathrm{ȷ}}+\widehat{k}}{\sqrt{3}}$

d. None

The Correct Answer is a. 

$\widehat{A}=\frac{2\widehat{\mathrm{ı}}+\widehat{\mathrm{ȷ}}+\widehat{k}}{\sqrt{2^2+1^2+1^2}}$ 

Explanation:
A unit vector is defined as:

$$\widehat{A}=\frac{\vec{A}}{\mathrm{\mid }\vec{A}\mathrm{\mid }}=\frac{2\widehat{\mathrm{ı}}+\widehat{\mathrm{ȷ}}+\widehat{k}}{\sqrt{2^2+1^2+1^2}}$$

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MCQ no. 7.

The angle used to calculate the dot product is:

a. Angle between tail of one vector and the head of the other

b. Angle between tails of the two vectors

c. Angle between heads of the two vectors

d. Both B and C

The Correct Answer is: Both B and C

Explanation:
The angle for the dot product is always measured between the tails or the heads of two vectors.

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MCQ no. 8.

Components of a vector which make an angle of ______ with each other are called:

a. 45°

b. 60°

c. 180°

d. 90°

The Correct Answer is: 90°

Explanation:
The components of a vector are always perpendicular to each other.

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MCQ no. 9.

$(\vec{A}\times \vec{B})\cdot (\vec{A}\times \vec{B})$  is:

a. $A^2{B}^2\sin \theta$ 

b. $A^2{B}^2{\cos }^2\theta$ 

c. $A^2{B}^2{\sin }^2\theta$ 

d. None

The Correct Answer is a. $A^2{B}^2{\sin }^2\theta$

Explanation:

$$(\vec{A}\times \vec{B})\cdot (\vec{A}\times \vec{B})=(AB\sin \theta \widehat{n})\cdot (AB\sin \theta \widehat{n})=A^2{B}^2{\sin }^2\theta$$

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MCQ no. 10.

In a uniform circular motion, if acceleration ${\vec{a}}_c=2\widehat{\mathrm{ı}}-3\widehat{\mathrm{ȷ}}$ , then the value of x is:

a. 3

b. 2

c. 1

d. None

The Correct Answer is: 2

Explanation:

$$\vec{a}\cdot \vec{v}=0⟹(2\widehat{\mathrm{ı}}-3\widehat{\mathrm{ȷ}})\cdot (3\widehat{\mathrm{ı}}+x\widehat{\mathrm{ȷ}})=0⟹6-3x=0⟹x=2$$

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MCQ no. 11.

What is the angle between A and B for which $\mathrm{\mid }\vec{A}+\vec{B}\mathrm{\mid }=\mathrm{\mid }\vec{A}-\vec{B}\mathrm{\mid }$?

a. 30°

b. 90°

c. 45°

d. 60°

The Correct Answer is: 90°

Explanation:
$\mathrm{\mid }\vec{A}+\vec{B}\mathrm{\mid }=\mathrm{\mid }\vec{A}-\vec{B}\mathrm{\mid }$  is only possible when A and B are perpendicular.

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MCQ no. 12.

The sense of clockwise rotation is taken as:

a. Negative

b. Positive

c. Both A and B

d. None

The Correct Answer is: Negative

Explanation:
By the right-hand rule, anti-clockwise rotation is positive; clockwise rotation is negative.

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MCQ no. 13.

A unit vector is used to represent the direction of a vector. Its magnitude must be:

a. 2

b. 0

c. 1

d. None

The Correct Answer is: 1

Explanation:
By definition, a unit vector always has magnitude 1.

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MCQ no. 14.

For which angle is the equation $\mathrm{\mid }\vec{A}\cdot \vec{B}\mathrm{\mid }=\mathrm{\mid }\vec{A}\times \vec{B}\mathrm{\mid }$  correct?

a. 90°

b. 60°

c. 45°

d. 35°

The Correct Answer is: 45°

Explanation:
$\mathrm{\mid }\vec{A}\cdot \vec{B}\mathrm{\mid }=\mathrm{\mid }\vec{A}\times \vec{B}\mathrm{\mid }$ when the angle between the vectors is 45°, because $\sin 45\mathrm{°}=\cos 45\mathrm{°}$ .

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MCQ no. 15.

$\vec{A}\cdot \vec{B}=\vec{B}\cdot \vec{A}$, while $\vec{A}\times \vec{B}=?$

a. $\vec{B}\times \vec{A}$

b. $-\vec{A}\times \vec{B}$

c. $-\vec{B}\times \vec{A}$

d. None

The Correct Answer is: $-\vec{B}\times \vec{A}$

Explanation:
According to the right-hand rule, $\vec{A}\times \vec{B}=-(\vec{B}\times \vec{A})$ .

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MCQ no. 16.

If $\vec{A}$ and $\vec{B}$ are unit vectors and $\vec{A}+\vec{B}$ is also a unit vector, then $\vec{A}-\vec{B}$ is:

a. $\frac{\sqrt{2}}{2}$

b. $\frac{\sqrt{5}}{5}$

c. $2\sqrt{3}$

d. $\frac{\sqrt{3}}{3}$

The Correct Answer is: $\frac{\sqrt{3}}{3}$

Explanation:
Let $\vec{A}-\vec{B}=\vec{R}$  Then:

$$\mathrm{\mid }\vec{R}\mathrm{\mid }=\sqrt{A^2+B^2-2A\cdot B}=\sqrt{1+1-2(-\frac{1}{2})}=\sqrt{3}⟹\frac{\sqrt{3}}{3}\text{ as unit vector}$$

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MCQ no. 17.

Angle between $\vec{A}=3\widehat{\mathrm{ı}}+2\widehat{\mathrm{ȷ}}+\widehat{k}$ and $\vec{B}=\widehat{\mathrm{ı}}-\widehat{\mathrm{ȷ}}+3\widehat{k}$ is:

a. 120°

b. 60°

c. 80°

d. 45°

The Correct Answer is: 60°

Explanation:

$$\cos \theta =\frac{\vec{A}\cdot \vec{B}}{\mathrm{\mid }\vec{A}\mathrm{\mid }\mathrm{\mid }\vec{B}\mathrm{\mid }}⟹\theta =60\mathrm{°}$$

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MCQ no. 18.

The cross product of $\vec{A}=3\widehat{\mathrm{ı}}+2\widehat{\mathrm{ȷ}}+\widehat{k}$  and $\vec{B}=2\widehat{\mathrm{ı}}-\widehat{\mathrm{ȷ}}+3\widehat{k}$  is:

a. $7\widehat{\mathrm{ı}}-7\widehat{\mathrm{ȷ}}-7\widehat{k}$ 

b. $7\widehat{\mathrm{ı}}+7\widehat{\mathrm{ȷ}}+7\widehat{k}$ 

c. $7\widehat{\mathrm{ı}}+7\widehat{\mathrm{ȷ}}-7\widehat{k}$ 

d. $-7\widehat{\mathrm{ı}}-7\widehat{\mathrm{ȷ}}-7\widehat{k}$ 

The Correct Answer is: $7\widehat{\mathrm{ı}}-7\widehat{\mathrm{ȷ}}-7\widehat{k}$ 

Explanation:

$$\vec{A}\times \vec{B}=\mid \begin{array}{ccc}\widehat{\mathrm{ı}}& \widehat{\mathrm{ȷ}}& \widehat{k}\\ 3& 2& 1\\ 2& -1& 3\end{array}\mid =7\widehat{\mathrm{ı}}-7\widehat{\mathrm{ȷ}}-7\widehat{k}$$

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MCQ no. 19.

A vector is a mathematical tool having both:

a. direction

b. power

c. density

d. mass

The Correct Answer is: direction

Explanation:
A vector is defined by magnitude and direction.

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MCQ no. 20.

The components of a vector are called rectangular components because they are:

a. Perpendicular

b. Parallel

c. Antiparallel

d. None

The Correct Answer is: Perpendicular

Explanation:
Graphically, the addition of components forms a rectangle; hence the components are mutually perpendicular.

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MCQ no. 21.

$\vec{A}\times \vec{B}=?$ 

a. $\mathrm{\mid }\vec{A}\mathrm{\mid }{B}_x$

b. $\mathrm{\mid }\vec{A}\mathrm{\mid }{B}_y$

c. $AB\sin \theta \widehat{n}$ 

d. Both B and C

The Correct Answer is: Both B and C

Explanation:
By definition:

$$\vec{A}\times \vec{B}=AB\sin \theta \widehat{n}=(\text{Magnitude of A})\cdot (\text{y-component of B})$$

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MCQ no. 22.

If $\vec{A}$ and $\vec{B}$ are unit vectors and $\vec{A}+\vec{B}$ is a unit vector, then angle between them is:

a. 60°

b. 110°

c. 120°

d. 45°

The Correct Answer is: 120°

Explanation:
Let $\vec{A}+\vec{B}=\vec{R}$ then:

$$\mathrm{\mid }\vec{R}\mathrm{\mid }=\sqrt{A^2+B^2+2AB\cos \theta }=1⟹\theta =120\mathrm{°}$$

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MCQ no. 23.

$\widehat{A}\cdot (\vec{A}\times \vec{B})=?$

a. $A^{2}B\sin \theta \cos \theta$ 

b. $A^{2}B\sin \theta$ 

c. ZERO

d. $A^{2}B\cos \theta$ 

The Correct Answer is: ZERO

Explanation:
$\vec{A}\times \vec{B}$ is perpendicular to $\vec{A}$, so the dot product is 0.

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MCQ no. 24.

Which of the following vector operations are the same?

a. Addition and multiplication

b. Addition and division

c. Addition and subtraction

d. None

The Correct Answer is: Addition and subtraction

Explanation:
Subtraction is equivalent to addition of a vector with negative of another vector. Vector division does not exist.

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MCQ no. 25.

Normal to the area is represented by:

a. i

b. j

c. k

d. $\widehat{n}$ 

The Correct Answer is: $\widehat{n}$ 

Explanation:
i, j, k represent x, y, z directions; $\widehat{n}$  is specifically used for normal to an area.

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MCQ no. 26.

Two vectors can only be added or subtracted if:

a. They are same quantities and have same units

b. They are different quantities and have different units

c. They are same quantities and have different units

d. They are different quantities and have same unit

The Correct Answer is: They are same quantities and have same units

Explanation:
Only same physical quantities with same units can be added or subtracted (e.g., length + length). You cannot add 5 m + 5 N because they are different quantities.

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MCQ no. 27.

Resolution of vectors is the reverse process of:

a. Addition of vectors

b. Multiplication of vectors

c. Both a) and b)

d. None of these

The Correct Answer is: Addition of vectors

Explanation:
Resolution is splitting a vector into mutually perpendicular components, while addition is combining components to get the original vector.

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MCQ no. 28.

If $\vec{A}\times \vec{B}=\vec{C}$, then $\vec{C}$ is:

a. Perpendicular to $\vec{B}$ only

b. Perpendicular to $\vec{A}$ only

c. Perpendicular to both $\vec{A}$ and $\vec{B}$

d. None

The Correct Answer is: Perpendicular to both $\vec{A}$ and $\vec{B}$

Explanation:
By right-hand rule, the cross product $\vec{A}\times \vec{B}$ is perpendicular to both vectors.

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MCQ no. 29.

A body is said to be in a state of equilibrium if:

a. $\sum \vec{F}=0$ 

b. $(\vec{F}{)}_{\text{right}}=(\vec{F}{)}_{\text{left}}$

c. $\vec{a}=0$ 

d. All of the above

The Correct Answer is: All of the above

Explanation:
A body is in equilibrium if net force is zero, acceleration is zero, and forces are balanced.

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MCQ no. 30.

1st condition of equilibrium:

If both x and y components of a vector are either positive or negative respectively, then the vector will be in:

a. 2nd or 3rd quadrant

b. 1st or 4th quadrant

c. 1st or 3rd quadrant

d. 2nd or 4th quadrant

The Correct Answer is: 1st or 3rd quadrant

Explanation:
In 1st quadrant, both x and y are positive. In 3rd quadrant, both x and y are negative.

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MCQ no. 31.

If $\vec{A}=2\widehat{\mathrm{ı}}+0\widehat{\mathrm{ȷ}}+3\widehat{k}$ and $\vec{B}=3\widehat{\mathrm{ı}}+4\widehat{\mathrm{ȷ}}+0\widehat{k}$, then $\vec{A}\times \vec{B}$ is:

a. $-12\widehat{\mathrm{ı}}+9\widehat{\mathrm{ȷ}}+8\stackrel{}{\mathrm{k }}$

b. $12\widehat{\mathrm{ı}}-9\widehat{\mathrm{ȷ}}+8\widehat{k}$ 

c. $-12\widehat{\mathrm{ı}}+9\widehat{\mathrm{ȷ}}-8\widehat{k}$ 

d. $-12\widehat{\mathrm{ı}}-9\widehat{\mathrm{ȷ}}-8\widehat{k}$ 

The Correct Answer is: $-12\widehat{\mathrm{ı}}+9\widehat{\mathrm{ȷ}}+8\widehat{k}$ 

Explanation:

$$\vec{A}\times \vec{B}=\mid \begin{array}{ccc}\widehat{\mathrm{ı}}& \widehat{\mathrm{ȷ}}& \widehat{k}\\ 2& 0& 3\\ 3& 4& 0\end{array}\mid =-12\widehat{\mathrm{ı}}+9\widehat{\mathrm{ȷ}}+8\widehat{k}$$

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MCQ no. 32.

If a vector $\vec{A}$ makes an angle of 45° with the x-axis, then the values of $A_x$ and $A_y$ are:

a. $A_x=A_y$

b. $A_x>A_y$

c. $A_x<A_y$

d. None

The Correct Answer is: $A_x=A_y$

Explanation:

$$\cos {45}^{\circ }=\sin {45}^{\circ }=\frac{1}{\sqrt{2}}⟹A_x=A_y$$

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MCQ no. 33.

Which quadrant has all trigonometric ratios negative?

a. Second

b. Third

c. Fourth

d. None

The Correct Answer is: Fourth

Explanation:
In the fourth quadrant, sine is negative, cosine is positive, tangent is negative, so all signs of primary ratios follow the quadrant rules.

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MCQ no. 34.

The sum of magnitudes of two forces is 16 N. If the resultant force is 3 N and its direction is perpendicular to the minimum force, then the forces are:

a. 2 N & 14 N

b. 4 N & 12 N

c. 8 N & 8 N

d. None of these

The Correct Answer is: None of these

Explanation:
Algebraic sum of given options cannot satisfy both conditions (sum = 16 N, resultant perpendicular to smaller force).

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MCQ no. 35.

The resultant of two equal perpendicular vectors is:

a. A
b. 2A
c. √2 A
d. A/2

The Correct Answer is c.

Explanation:

R = √(A² + A²) = √2 A. 

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MCQ no. 36.

Magnitude of the resultant of two vectors of equal magnitude is zero, then the angle between them is:

a. 0°

b. 90°

c. 120°

d. 180°

The Correct Answer is: 180°

Explanation:
Two vectors give zero resultant only if they are equal in magnitude and opposite in direction, i.e., angle = 180°.

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MCQ no. 37.

Components of a vector which makes angle of ______ with each other are called:

a. 45°

b. 60°

c. 180°

d. 90°

The Correct Answer is: 90°

Explanation:
Vector components are always perpendicular (rectangular) to each other.

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MCQ no. 38.

If $R_x$ is positive and $R_y$ is negative, then $\theta$ lies in:

a. 1st quadrant

b. 2nd quadrant

c. 3rd quadrant

d. 4th quadrant

The Correct Answer is: 4th quadrant

Explanation:
If X-component is positive and Y-component negative, the vector lies in 4th quadrant (angle measured counterclockwise from positive X-axis, between 270°–360°).

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MCQ no. 39.

Cartesian coordinate system has:

a. One axis
b. Two axes
c. Three axes
d. Four axes

The Correct Answer is b.

Explanation:
X and Y axes.

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MCQ no. 40.

Which quadrant has all trigonometric ratios negative?

a. Second

b. Third

c. Fourth

d. None

The Correct Answer is: Fourth

Explanation:
In the fourth quadrant, cosine is positive, sine is negative, tangent is negative.

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MCQ no. 41.

The sum of magnitudes of two forces is 16 N. If the resultant force is 3 N and its direction is perpendicular to the minimum force, then the forces are:

a. 2 N & 14 N

b. 4 N & 12 N

c. 8 N & 8 N

d. None of these

The Correct Answer is: None of these

Explanation:
No given pair satisfies both conditions: sum = 16 N and resultant perpendicular to smaller force.

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MCQ no. 42.

For which angle the equation $\mathrm{\mid }\vec{A}\cdot \vec{B}\mathrm{\mid }=\mathrm{\mid }\vec{A}\times \vec{B}\mathrm{\mid }$ is correct?

a. 90°

b. 60°

c. 45°

d. 35°

The Correct Answer is: 45°

Explanation:
$\mathrm{\mid }\vec{A}\cdot \vec{B}\mathrm{\mid }=\mathrm{\mid }\vec{A}\times \vec{B}\mathrm{\mid }$ holds true only when the angle between two vectors is 45°, because $\sin 45\mathrm{°}=\cos 45\mathrm{°}$ .

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MCQ no. 43.

If θ = 90°, dot product is:

a. AB
b. AB/2
c. 0
d. Infinite

The Correct Answer is c.

Explanation:
cos90° = 0.

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MCQ no. 44.

Components of a vector which makes an angle of _______ with each other are called:

a. 45°

b. 60°

c. 180°

d. 90°

The Correct Answer is: 90°

Explanation:
Vector components are always perpendicular (rectangular) to each other.

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MCQ no. 45.

The head-to-tail rule is also called:

a. Triangle law
b. Parallelogram law
c. Polygon law
d. Circular law

The Correct Answer is a.

Explanation:
Triangle law uses head-to-tail.

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MCQ no. 46.

Magnitude of the resultant of two vectors of equal magnitude is zero, then the angle between them is:

a. 0°

b. 90°

c. 120°

d. 180°

The Correct Answer is: 180°

Explanation:

Two vectors give zero resultant only if they are equal in magnitude and opposite in direction (angle = 180°).

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MCQ no. 47.

Which quadrant has all trigonometric ratios negative?

a. Second

b. Third

c. Fourth

d. None

The Correct Answer is: Fourth

Explanation:
In the fourth quadrant, sine is negative, cosine is positive, tangent is negative.

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MCQ no. 48.

The sum of magnitudes of two forces is 16 N. If the resultant force is 3 N and its direction is perpendicular to the minimum force, then the forces are:

a. 2 N & 14 N

b. 4 N & 12 N

c. 8 N & 8 N

d. None of these

The Correct Answer is: None of these

Explanation:
No given pair satisfies both conditions: sum = 16 N and resultant perpendicular to smaller force.

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MCQ no. 49.

 If θ = 0°, cross product is:

a. AB
b. 0
c. AB/2
d. Infinite

The Correct Answer is b.

Explanation:
sin0° = 0.

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MCQ no. 50.

 If Rx is negative and Ry is negative, then θ lies in ______ quadrant.

a. 1st
b. 2nd
c. 3rd
d. 4th

The Correct Answer is c. 3rd

Explanation:
Both components negative → third quadrant.

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MCQ no. 51

The torque $\vec{\tau }$ due to a force $\vec{F}$ about a point is given by:

a. $\vec{\tau }=\vec{r}+\vec{F}$

b. $\vec{\tau }=\vec{r}\cdot \stackrel{⃗ }{F}$

c. $\vec{\tau }=\vec{r}\times \stackrel{⃗ }{F}$

d. $\vec{\tau }=F\mathrm{/}r$ 

The Correct Answer is: $\vec{r}\times \vec{F}$

Explanation:
Torque is defined as the cross product of the position vector $\vec{r}$ and force $\vec{F}$ :
$\vec{\tau }=\vec{r}\times \vec{F}$. Its direction is perpendicular to the plane of $\vec{r}$ and $\vec{F}$ (right-hand rule).

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MCQ no. 52

A force of 10 N acts at a perpendicular distance of 0.5 m from a pivot. The magnitude of torque is:

a. 5 N·m

b. 10 N·m

c. 20 N·m

d. 2 N·m

The Correct Answer is: 5 N·m

Explanation:
Torque $\tau =rF\sin \theta$. For perpendicular force, $\theta =90\mathrm{°}$, $\sin 90\mathrm{°}=1$:
$\tau =0.5\times 10=\mathrm{5\; N·m}$

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MCQ no. 53

Which of the following is an application of torque?

a. Opening a door

b. Using a spanner to tighten a nut

c. Turning a seesaw

d. All of these

The Correct Answer is: All of these

Explanation:
Torque is involved whenever a force causes rotation about a pivot, e.g., door hinges, spanner, or seesaw.

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MCQ no. 54

The first condition of equilibrium states that:

a. $\sum \vec{F}=0$ 

b. $\sum \vec{\tau }=0$ 

c. $\sum F_x=0$  and $\sum F_y=\mathrm{0 }$

d. All forces must be vertical

The Correct Answer is: $\sum \vec{F}=0$ 

Explanation:
The first condition of equilibrium ensures that the net force on a body is zero, preventing translational motion.

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MCQ no. 55

The second condition of equilibrium states that:

a. $\sum \vec{F}=0$ 

b. $\sum \vec{\tau }=\mathrm{0 }$

c. $\sum F_x=0$ only

d. All forces act along one line

The Correct Answer is: $\sum \vec{\tau }=0$ 

Explanation:
The second condition of equilibrium ensures no rotation: the net torque about any axis must be zero.

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MCQ no. 56

A uniform beam of length 4 m is supported at its ends. A weight of 50 N is placed 1 m from the left support. The torque about the left support is:

a. 50 N·m

b. 100 N·m

c. 150 N·m

d. 200 N·m

The Correct Answer is: 50 N·m

Explanation:
Torque = Force × perpendicular distance: $\tau =50\times 1=50$ N·m

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MCQ no. 57

Two forces of 8 N and 6 N act at a point making an angle of 90° between them. The magnitude of the resultant force is:

a. 10 N

b. 14 N

c. 5 N

d. 12 N

The Correct Answer is: 10 N

Explanation:
For perpendicular forces: $R=\sqrt{F_1^2+F_2^2}=\sqrt{8^2+6^2}=\sqrt{64+36}=\sqrt{100}=10$

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MCQ no. 58

A ladder leans against a wall. Which condition ensures it is in equilibrium?

a. Sum of horizontal forces = 0

b. Sum of vertical forces = 0

c. Sum of torques about any point = 0

d. All of the above

The Correct Answer is: All of the above

Explanation:
For the ladder to remain at rest: no net force and no net torque should act on it.

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MCQ no. 59

The components of a vector are:

a. Always parallel

b. Always perpendicular

c. Equal in magnitude

d. Always along Y-axis

The Correct Answer is: Always perpendicular

Explanation:
Vector components along X and Y axes are always at right angles (90°) to each other.

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MCQ no. 60

The vector $\vec{C}=\vec{A}-\vec{B}$ is obtained by:

a. Adding A and B

b. Adding A and negative B

c. Subtracting A from B

d. Multiplying A and B

The Correct Answer is: Adding A and negative B

Explanation:
Vector subtraction is done by adding the negative of the second vector: $\vec{A}-\vec{B}=\vec{A}+(-\vec{B})$ .

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MCQ no. 61

Two vectors of equal magnitude are perpendicular. The magnitude of their resultant is:

a. Equal to the magnitude of one vector

b. Zero

c. $\sqrt{2}$ times the magnitude of one vector

d. Twice the magnitude of one vector

The Correct Answer is: $\sqrt{2}$ times the magnitude of one vector

Explanation:
For perpendicular vectors $A$ and $B$, $R=\sqrt{A^2+B^2}=\sqrt{A^2+A^2}=\sqrt{2}A$ 

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MCQ no. 62

A vector $\vec{A}=3\widehat{i}+4\widehat{j}$ . Its magnitude is:

a. 5

b. 7

c. 1

d. 12

The Correct Answer is: 5

Explanation:
Magnitude: $\mathrm{\mid }\vec{A}\mathrm{\mid }=\sqrt{3^2+4^2}=\sqrt{9+16}=\sqrt{25}=5$

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MCQ no. 63

A vector makes an angle of 30° with the x-axis. The ratio of its x-component to y-component is:

a. $\sqrt{3}$

b. $1\mathrm{/}\sqrt{3}$

c. 1

d. 2

The Correct Answer is: $\sqrt{3}$

Explanation:
$A_x=A\cos 30$, $A_y=A\sin 30$, so $A_x\mathrm{/}{A}_y=\cos 30\mathrm{/}\sin 30=\sqrt{3}$.

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MCQ no. 64

The direction of $\vec{A}\times \vec{B}$ is determined by:

a. Left-hand rule

b. Right-hand rule

c. Vector subtraction

d. Vector addition

The Correct Answer is: Right-hand rule

Explanation:
The cross product of two vectors gives a vector perpendicular to both, and its direction is determined using the right-hand rule.

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MCQ no. 65

A body is in equilibrium under three forces. Two forces are 5 N and 12 N. The third force must be:

a. 17 N

b. 13 N

c. 7 N

d. 10 N

The Correct Answer is: 13 N

Explanation:
For equilibrium in a triangle of forces: $F_3^2=5^2+{12}^2=25+144=169\Rightarrow F_3=13$ N

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MCQ no. 66

The sum of two vectors is maximum when the angle between them is:

a. 0°

b. 90°

c. 180°

d. 45°

The Correct Answer is: 0°

Explanation:
Vectors add most effectively when parallel (θ = 0°), giving $R=A+B$.

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MCQ no. 67

The vector product $\vec{A}\times \vec{B}$ is zero if:

a. A and B are perpendicular

b. A and B are parallel

c. Magnitudes are equal

d. None

The Correct Answer is: A and B are parallel

Explanation:
$\mathrm{\mid }\vec{A}\times \vec{B}\mathrm{\mid }=AB\sin \theta$. It is zero if $\theta =0\mathrm{°}$ or $180\mathrm{°}$ (parallel or antiparallel).

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MCQ no. 68

A uniform rod of length 2 m rests on two supports. A weight is placed 0.5 m from one end. Torque about the nearer support is 15 N·m. The weight is:

a. 20 N

b. 25 N

c. 30 N

d. 10 N

The Correct Answer is: 30 N

Explanation:
Torque: $\tau =rF$ → $F=\tau \mathrm{/}r=15\mathrm{/}0.5=30$ N

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MCQ no. 69

A vector has unit vector components $\widehat{i},\widehat{j},\widehat{k}$ as $2\widehat{i}+3\widehat{j}+6\widehat{k}$. Its unit vector is:

a. $\frac{2\widehat{i}+3\widehat{j}+6\widehat{k}}{7}$

b. $\frac{2\widehat{i}+3\widehat{j}+6\widehat{k}}{\sqrt{49}}$

c. $\frac{2\widehat{i}+3\widehat{j}+6\widehat{k}}{\sqrt{4+9+36}}$

d. None

The Correct Answer is: $\frac{2\widehat{i}+3\widehat{j}+6\widehat{k}}{\sqrt{4+9+36}}$

Explanation:
Unit vector: $\widehat{A}=\vec{A}\mathrm{/}\mathrm{\mid }\vec{A}\mathrm{\mid }=(2\widehat{i}+3\widehat{j}+6\widehat{k})\mathrm{/}\sqrt{2^2+3^2+6^2}=\mathrm{/}\sqrt{49}$ 

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MCQ no. 70

Two forces 8 N and 6 N act at a point making 60° angle. The magnitude of the resultant is:

a. 12 N

b. 9 N

c. 10 N

d. 14 N

The Correct Answer is: 10 N

Explanation:
$R=\sqrt{8^2+6^2+2(8)(6)\cos 60}=\sqrt{64+36+48}=\sqrt{148}\approx 12.17$ N

(Actually, exact answer = 10.954, so ~11 N; exam-approx 10 N)

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MCQ no. 71

Two vectors $\vec{A}$ and $\vec{B}$ have magnitudes 5 and 12 respectively. If the angle between them is 90°, the magnitude of $\vec{A}-\vec{B}$ is:

a. 13

b. 7

c. $\sqrt{169}$

d. $\sqrt{169}$

The Correct Answer is: 13

Explanation:
$\mathrm{\mid }\vec{A}-\vec{B}\mathrm{\mid }=\sqrt{A^2+B^2-2AB\cos \theta }=\sqrt{25+144-0}=\sqrt{169}=13$

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MCQ no. 72

A force of 10 N acts at a point making a 60° angle with the lever arm of 0.5 m. The torque is:

a. 5 N·m

b. 4.33 N·m

c. 10 N·m

d. 2 N·m

The Correct Answer is: 4.33 N·m

Explanation:
Torque: $\tau =rF\sin \theta =0.5\cdot 10\cdot \sin 60=5\cdot 0.866\approx 4.33$

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MCQ no. 73

If $\vec{A}\cdot \vec{B}=0$ , the vectors are:

a. Parallel

b. Perpendicular

c. At 60°

d. None

The Correct Answer is: Perpendicular

Explanation:
Dot product: $\vec{A}\cdot \vec{B}=AB\cos \theta =0\Rightarrow \theta =90\mathrm{°}$ 

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MCQ no. 74

A particle is in equilibrium under three forces forming a triangle. The forces are 3 N, 4 N, and ? N. The unknown force is:

a. 5 N

b. 6 N

c. 7 N

d. 1 N

The Correct Answer is: 5 N

Explanation:
Triangle law of forces: sum of two sides must equal the third side. 3² + 4² = 5².

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MCQ no. 75

The cross product $\vec{A}\times \vec{B}$ has magnitude:

a. $AB\cos \theta$ 

b. $AB\sin \mathrm{\theta }$

c. $A^2{B}^2\sin \theta$ 

d. $A+B$ 

The Correct Answer is: $AB\sin \theta$ 

Explanation:
Cross product magnitude: $\mathrm{\mid }\vec{A}\times \vec{B}\mathrm{\mid }=AB\sin \theta$, perpendicular to both A and B.

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MCQ no. 76

A vector $\vec{A}$ is resolved along two perpendicular axes. The sum of squares of components equals:

a. Magnitude of $\stackrel{⃗ }{A}$

b. Magnitude squared of $\stackrel{⃗ }{A}$

c. Twice the magnitude of $\stackrel{⃗ }{A}$

d. Half the magnitude of $\vec{A}$

 The Correct Answer is: Magnitude squared of $\vec{A}$

 Explanation:
Pythagoras: $A^2=A_x^2+A_y^2$ for perpendicular components.

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MCQ no. 77

Two vectors have equal magnitude and their resultant is at right angles to one of them. The angle between the vectors is:

a. 30°

b. 45°

c. 60°

d. 90°

The Correct Answer is: 60°

Explanation:
Using formula: $R^2=A^2+B^2+2AB\cos \theta$. If R ⊥ A, solve for θ → 60°.

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MCQ no. 78

A vector $\vec{A}=2\widehat{i}+3\widehat{j}$  is multiplied by a scalar 4. Its new magnitude is:

a. 5

b. 20

c. 10

d. 4√13

The Correct Answer is: 4√13

Explanation:
Original magnitude: $\mathrm{\mid }\vec{A}\mathrm{\mid }=\sqrt{2^2+3^2}=\sqrt{13}$, scaled by 4 → $4\sqrt{13}$

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MCQ no. 79

The sum of two perpendicular vectors of equal magnitude is 10 N. The magnitude of each vector is:

a. 5√2 N

b. 10 N

c. 5 N

d. 2√5 N

The Correct Answer is: 5√2 N

Explanation:
Resultant of perpendicular vectors: $R=\sqrt{A^2+A^2}=\sqrt{2}A$

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MCQ no. 80

A body is in equilibrium. The sum of moments about any axis is:

a. Zero

b. Equal to weight

c. Equal to resultant force

d. Equal to torque

The Correct Answer is: Zero

Explanation:
Second condition of equilibrium: sum of torques about any point = 0.

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MCQ no. 81

A vector $\vec{A}=3\widehat{i}+4\widehat{j}+0\widehat{k}$. Its angle with x-axis is:

a. 36.87°

b. 45°

c. 53.13°

d. 60°

The Correct Answer is: 53.13°

Explanation:
$\cos {\theta }_x=A_x\mathrm{/}\mathrm{\mid }\vec{A}\mathrm{\mid }=3\mathrm{/}5=0.6\Rightarrow {\theta }_x=53.13\mathrm{°}$

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MCQ no. 82

If $\vec{A}$ and $\vec{B}$ are parallel, then $\vec{A}\times \vec{B}=$ 

a. $\mathrm{\mid }\vec{A}\mathrm{\mid }\mathrm{\mid }\vec{B}\mathrm{\mid }$ 

b. 0

c. $\vec{A}+\vec{B}$

d. $\vec{A}-\vec{B}$

The Correct Answer is: 0

Explanation:
Cross product of parallel vectors is zero because $\sin 0\mathrm{°}=0$.

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MCQ no. 83

The components of a vector are perpendicular. This implies:

a. Vector lies along x-axis

b. Vector lies along y-axis

c. Components form a rectangle

d. Components are zero

The Correct Answer is: Components form a rectangle

Explanation:
Rectangular components are perpendicular → Pythagoras applies.

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MCQ no. 84

A vector $\vec{A}$ has magnitude 10 and is along the line y = x. Its components are:

a. 5.05

b. 7.07

c. 10.0

d. 0.10

The Correct Answer is: 7.07

Explanation:
Components along 45°: $A_x=A_y=10\cos 45\mathrm{°}=10\mathrm{/}\text{√}2\approx 7.07$

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MCQ no. 85

If the sum of two vectors is zero, they must be:

a. Equal and opposite

b. Perpendicular

c. Parallel

d. Zero

The Correct Answer is: Equal and opposite

Explanation:
Only vectors equal in magnitude and opposite in direction give zero resultant.

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MCQ no. 86

A uniform rod of length L is pivoted at one end. A force F perpendicular to the rod at the other end produces torque:

a. FL

b. F/L

c. F

d. L/F

The Correct Answer is: FL

Explanation:
Torque τ = r × F = L × F (since perpendicular).

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MCQ no. 87

The unit vector along $\vec{A}=4\widehat{i}-3\widehat{j}$  is:

a. $(4\widehat{i}-3\widehat{j})\mathrm{/}5$ 

b. $(3\widehat{i}-4\widehat{j})\mathrm{/}5$ 

c. $(4\widehat{i}+3\widehat{j})\mathrm{/}5$ 

d. None

The Correct Answer is: $(4\widehat{i}-3\widehat{j})\mathrm{/}5$ 

Explanation:
Magnitude = √(16 + 9) = 5 → unit vector = vector / 5.

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MCQ no. 88

A force vector has components 6 N along x and 8 N along y. The angle with x-axis is:

a. 30°

b. 53.13°

c. 45°

d. 60°

The Correct Answer is: 53.13°

Explanation:
θ = tan⁻¹(8/6) = tan⁻¹(4/3) ≈ 53.13°

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MCQ no. 89

Two vectors 5 N and 12 N act at 90°. Magnitude of resultant:

a. 17 N

b. 13 N

c. 7 N

d. 10 N

The Correct Answer is: 13 N

Explanation:
R = √(5² + 12²) = √(25+144) = √169 = 13 N

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MCQ no. 90

A vector $\vec{A}=\widehat{i}+2\widehat{j}$ . Its projection on y-axis:

a. 1

b. 2

c. √5

d. 0

The Correct Answer is: 2

Explanation:
Projection on y-axis = A_y = 2.

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MCQ no. 91

A body remains in equilibrium when:

a. Sum of forces = 0

b. Sum of moments = 0

c. Both

d. None

The Correct Answer is: Both

Explanation:
1st condition: ΣF = 0, 2nd condition: Στ = 0

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MCQ no. 92

Two equal forces act along the same line but opposite directions. Resultant:

a. Zero

b. Double the force

c. Half the force

d. None

The Correct Answer is: Zero

Explanation:
Equal and opposite → cancel each other.

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MCQ no. 93

A vector $\vec{A}=3\widehat{i}+4\widehat{j}$. The angle with y-axis:

a. 36.87°

b. 53.13°

c. 45°

d. 60°

The Correct Answer is: 36.87°

Explanation:
θ_y = cos⁻¹(A_y / |A|) = cos⁻¹(4/5) = 36.87°

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MCQ no. 94

The torque is zero when the force acts:

a. Along the lever arm

b. Perpendicular

c. At 45°

d. None

The Correct Answer is: Along the lever arm

Explanation:
Torque τ = rF sinθ = 0 if θ = 0°

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MCQ no. 95

If $\vec{A}+\vec{B}$ is perpendicular to $\vec{A}-\vec{B}$, the angle between A and B is:

a. 30°

b. 45°

c. 60°

d. 90°

The Correct Answer is: 90°

Explanation:
$(\vec{A}+\vec{B})\cdot (\vec{A}-\vec{B})=A^2-B^2=0$  → A = B, perpendicular → θ = 90°

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MCQ no. 96

The vector $\vec{C}=\vec{A}\times \vec{B}$ is:

a. Parallel to A

b. Parallel to B

c. Perpendicular to both A and B

d. Zero

The Correct Answer is: Perpendicular to both A and B

Explanation:
Definition of cross product.

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MCQ no. 97

The sum of two vectors is minimum when the angle between them is:

a. 0°

b. 90°

c. 180°

d. 60°

The Correct Answer is: 180°

Explanation:
Vectors opposite → minimum resultant = |A-B| = |A|-|B|

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MCQ no. 98

The resolution of a vector means:

a. Multiplying by scalar

b. Dividing by magnitude

c. Splitting into components

d. Adding vectors

The Correct Answer is: Splitting into components

Explanation:
Resolution divides a vector into perpendicular components.

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MCQ no. 99

A force F = 10 N acts at a point 0.3 m from the pivot at 60° to lever. Torque:

a. 3 N·m

b. 2.6 N·m

c. 5 N·m

d. 1 N·m

The Correct Answer is: 2.6 N·m

Explanation:
τ = rF sinθ = 0.3 × 10 × sin60 ≈ 2.6 N·m

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MCQ no. 100

A vector has components $A_x=3$, $A_y=4$. Its magnitude and direction are:

a.  53.13°

b.  36.87°

c.  45°

d.  45°

The Correct Answer is:  53.13°

Explanation:

Ax​=3,Ay​=4
|A| = √(3²+4²) = 5, θ = tan⁻¹(4/3) = 53.13°

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