Vectors

01

PYQ 2021

medium

physics ID: jee-main

The angle between vector and is:

1

2

3

4

\textbf{Correct Answer:} (B)

02

PYQ 2021

medium

physics ID: jee-main

Statement I : Two forces and where, when act at an angle to each other, the magnitude of their resultant is, when they act at an angle, the magnitude of their resultant becomes . This is possible only when . Statement II : In the situation given above, and . In the light of the above statements, choose the most appropriate answer from the options given below :

1

Both Statement I and Statement II are true.

2

Both Statement I and Statement II are false.

3

Statement I is true but Statement II is false.

4

Statement I is false but Statement II is true.

03

PYQ 2021

medium

physics ID: jee-main

$The magnitude of vectors, and in the given figure are equal. The direction of with x-axis will be :$

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2

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4

Official Solution

Correct Option: (1)

$Step 1: Understanding the Concept: Vector addition and subtraction are performed by breaking the vectors into their horizontal (x) and vertical (y) components and then finding the angle of the resultant using . Step 2: Key Formula or Approach: 1. 2. Step 3: Detailed Explanation: Let the magnitude of each vector be . From the diagram: - makes with x-axis: . - makes below positive x-axis: . - makes with negative x-axis: . Calculate Resultant : The direction with x-axis is: Step 4: Final Answer: The direction is .$

04

PYQ 2021

easy

physics ID: jee-main

$Assertion A : If A, B, C, D are four points on a semi-circular arc with centre at 'O' such that, then Reason R : Polygon law of vector addition yields$

1

$Both A and R are correct and R is the correct explanation of A.$

2

$Both A and R are correct but R is not the correct explanation of A.$

3

$A is correct but R is not correct.$

4

$A is not correct but R is correct.$

Official Solution

Correct Option: (2)

$Analysis of Reason (R): By the polygon law of vector addition, . This part is correct. The statement implies, which simplifies to . This is true if and only if AD is a diameter of the circle with center O. So, R is a correct statement provided AD is a diameter. Analysis of Assertion (A): The condition means the lengths of the chords AB, BC, and CD are equal. Equal chords in a circle subtend equal angles at the center. Since A, B, C, D are on a semi-circular arc, the total angle . Let . Then, which means . This confirms that the points A and D are at the ends of a diameter, so . Now, let's evaluate the LHS of the assertion's equation, taking O as the origin: LHS = LHS = Substitute : LHS = Now evaluate the RHS: RHS = Since LHS = RHS, the Assertion (A) is correct. Conclusion: Both Assertion (A) and Reason (R) are correct statements. However, Reason (R) does not explain Assertion (A). The reason simply states the polygon law and a geometric fact () without explaining why this fact is true based on the given condition . The assertion's validity depends critically on this condition, which is not used in the reasoning. Therefore, R is not the correct explanation of A.$

05

PYQ 2021

medium

physics ID: jee-main

$The resultant of these forces and is approximately ________ N. [Take . Given and unit vectors along x, y axis]$

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4

Official Solution

Correct Option: (3)

$Step 1: Understanding the Question: We are asked to find the vector sum (resultant) of five forces given in a diagram. Each force is represented by its magnitude and the angle it makes with the coordinate axes. Step 2: Key Formula or Approach: To find the resultant force, we resolve each force vector into its x () and y () components. Then, we sum all the x-components to get the resultant x-component () and sum all the y-components to get the resultant y-component (). The resultant vector is . A vector with magnitude and angle with the positive x-axis has components: and . Step 3: Detailed Explanation: Let's resolve each vector into its components based on the angles given in the diagram: 1. : Magnitude = 20 N, Angle = 30° with +x axis. 2. : Magnitude = 10 N, Angle = 30° with +x axis. 3. : Magnitude = 20 N, Angle = -45° or 315° with +x axis. 4. : Magnitude = 15 N, Angle = 180° + 45° = 225° with +x axis. 5. : Magnitude = 15 N, Angle = 180° - 60° = 120° with +x axis. Now, sum the components: Substitute the given values and : The calculated resultant is . Note on Discrepancy: The calculated result does not match any of the given options. This suggests a potential error in the question data, the diagram, or the options provided in the examination. The official answer key indicates (C) is the correct answer. There is no standard physical interpretation of the provided diagram and values that leads to this answer. Such questions are often marked as bonus in exams. For the purpose of this solution, we acknowledge the discrepancy and select the official answer. Step 4: Final Answer: Based on the provided official answer key, the correct option is (C), despite the direct calculation from the problem statement yielding a different result.$

06

PYQ 2021

medium

physics ID: jee-main

$Match List I with List II. Choose the correct answer from the options given below :$

1

$(a) (iv), (b) (iii), (c) (i), (d) (ii)$

2

$(a) (iii), (b) (ii), (c) (iv), (d) (i)$

3

$(a) (iv), (b) (i), (c) (iii), (d) (ii)$

4

$(a) (i), (b) (ii), (c) (iii), (d) (iv)$

Official Solution

Correct Option: (4)

$Step 1: Understanding the Concept: Vector addition using the Triangle Law: If two vectors are represented by two sides of a triangle in order (head to tail), the third side taken in the opposite order represents the resultant (sum). If all three vectors are in order (forming a cycle), their sum is zero. Step 2: Key Formula or Approach: Rearrange each equation to identify the resultant: 1. 2. 3. 4. Step 3: Detailed Explanation: (a) . In diagram (i), and are in order, and is the closing side in the opposite order. (b) . In diagram (ii), and are in order, and is the resultant. (c) . In diagram (iii), and are in order, and is the resultant. (d) . In diagram (iv), all vectors are in a continuous head-to-tail cycle. Step 4: Final Answer: The matching sequence is (a) (i), (b) (ii), (c) (iii), (d) (iv).$

07

PYQ 2021

medium

physics ID: jee-main

$In an octagon ABCDEFGH of equal side, what is the sum of, if ?$

1

$16i + 24j - 32k$

2

$-16i - 24j - 32k$

3

$16i + 24j + 32k$

4

$16i + 24j - 32k$

Official Solution

Correct Option: (1)

$Step 1: Let be the center of the regular octagon. The sum of vectors from the center to all vertices is zero: . Step 2: Express each vector in terms of the center :,, etc. Step 3: Total Sum . Step 4: Since, the sum of the other 7 vectors is . Step 5: Total Sum . Step 6: .$

08

PYQ 2022

medium

physics ID: jee-main

$If two vectors P = +2m +m & Q = 4 -2 +m are perpendicular to each other, then the value of m will be:$

1

$m = -3$

2

$m = 2$

3

$m = -1$

4

$m = 1$

Official Solution

Correct Option: (2)

$The condition for two vectors to be perpendicular is: Substitute the given vectors: Expanding the dot product: Rearranging: Thus, the value of is 2.$

09

PYQ 2022

medium

physics ID: jee-main

$Two bodies of mass kg and kg have position vectors and, respectively. The magnitude of position vector of centre of mass of this system will be similar to the magnitude of vector.$

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Official Solution

Correct Option: (1)

$So, the correct option is (A):$

10

PYQ 2022

easy

physics ID: jee-main

$If and . The magnitude of component of vector along vector will be ______.$

Official Solution

Correct Option: (1)

$= = =$

11

PYQ 2022

medium

physics ID: jee-main

$Projection of 2 +4 -2 on +2 -α is zero. Then, the value of α will be________.$

Official Solution

Correct Option: (1)

$The projection of one vector will be zero on another vector only when they are perpendicular to each other Let the two vectors be =2 +4 -2 = +2 +α \cdot =0 as A\toshould be perpendicular to \Rightarrow2+8-2α=0 \Rightarrowα=5$

12

PYQ 2023

easy

physics ID: jee-main

$When vector is subtracted from vector, it gives a vector equal to . Then the magnitude of vector will be :$

Official Solution

Correct Option: (1)

13

PYQ 2023

hard

physics ID: jee-main

$If and then, The unit vector in the direction of is The value of is$

Official Solution

Correct Option: (1)

$The correct answer is 4. P \times Q = ∣ ∣ \hat{i} 34 \hat{j} 33 \hat{k} 2 2.5 ∣ ∣ = 3 \frac{i ^}{2} + \frac{j ^}{2} - 3 \hat{k} \Rightarrow \frac{P \times Q}{∣ P \times Q ∣} = \frac{1}{2} (3 \frac{i ^}{2} + \frac{j ^}{2} - 3 \hat{k}) = \frac{1}{4} (3 \hat{i} + \hat{j} - 23 \hat{k}) x = 4$

14

PYQ 2023

easy

physics ID: jee-main

$A vector in x-y plane makes an angle of 30º with y-axis. The magnitude of y-component of vector is . The magnitude of x -component of the vector will be :$

1

$2$

2

3

4

$6$

Official Solution

Correct Option: (1)

$Understanding the Problem A vector makes a 30° angle with the y-axis. The magnitude of the y-component () is . We need to find the magnitude of the x-component (). Solution 1. Relating to the Vector's Magnitude (A): Since the angle is with the y-axis, we use cosine: Given, we have: 2. Solving for the Vector's Magnitude (A): We know . Substituting: Solving for A: 3. Finding the x-Component (): Since the angle is with the y-axis, the x-component uses sine: We found, and . Substituting: Final Answer The magnitude of the x-component of the vector is 2.$

15

PYQ 2023

hard

physics ID: jee-main

$The position vector of a particle related to time t is given by The direction of net force experienced by the particle is :$

1

$Positive x-axis$

2

$Positive y-axis$

3

$Positive z-axis$

4

$In x-y plane$

Official Solution

Correct Option: (2)

$Understanding the Problem We are given the position vector of a particle: We need to find the direction of the net force acting on the particle. Solution 1. Velocity Vector (v): The velocity vector is the first derivative of the position vector with respect to time: 2. Acceleration Vector (a): The acceleration vector is the first derivative of the velocity vector with respect to time (or the second derivative of the position vector): 3. Net Force (F): According to Newton's second law, the net force is given by: Since, we have: 4. Direction of the Force: The acceleration vector indicates that the acceleration is directed along the positive y-axis. Since, the force vector is also directed along the positive y-axis. Final Answer The net force acting on the particle is directed along the positive y-axis.$

16

PYQ 2024

easy

physics ID: jee-main

$If and make an angle with each other, then The integer value of is _____.$

Official Solution

Correct Option: (1)

$To solve for the integer value of, we begin by using the known relation between vectors and their magnitudes: The magnitudes of the vector sums can be expressed as: where . Substituting, these become: Given: Squaring both sides: Substitute the expanded forms: Simplifying: Combine terms: Factor out common terms: Express in terms of . Thus and, substitute in equation: Dividing through by (assuming non-zero): Rearrange: Solve quadratic equation using the formula: Here,,, : Rounding to the nearest integer within the range,, falls as either 2 or 3, since 3 is explicitly within the specified range 3,3, is the answer.$

17

PYQ 2024

easy

physics ID: jee-main

$A vector has magnitude same as that of A = and is parallel to B = . The x and y components of this vector in the first quadrant are x and y respectively where: .$

Official Solution

Correct Option: (1)

$The problem asks to find the x-component of a vector that has the same magnitude as vector and is parallel to vector . This new vector is stated to be in the first quadrant. Concept Used: To solve this problem, we will use the following concepts from vector algebra: 1. Magnitude of a Vector: The magnitude of a vector is given by: 2. Unit Vector: A unit vector in the direction of a non-zero vector is given by: 3. Representation of a Vector: Any vector can be represented as the product of its magnitude and its unit vector (direction): 4. Parallel Vectors: If two vectors are parallel, they have the same direction, which means they share the same unit vector. Step-by-Step Solution: Step 1: Calculate the magnitude of vector . Let the required vector be . The problem states that the magnitude of is the same as the magnitude of . Therefore: Step 2: Find the unit vector in the direction of vector . First, we find the magnitude of : Now, we find the unit vector : Step 3: Construct the required vector . Since is parallel to, it must have the same direction as . So, the unit vector for is . The vector is the product of its magnitude and its unit vector: Substituting the values we found: Step 4: Simplify the expression for to find its components. Step 5: Identify the x and y components of the vector . The vector is . Both components are positive, which confirms the vector is in the first quadrant. The x-component is . The y-component is . The question asks for the value of the x-component. The x-component of this vector is 4 .$

18

PYQ 2024

medium

physics ID: jee-main

$Find the number of significant digits in the value 10.05$

Official Solution

Correct Option: (1)

$The correct answer is : 4$

19

PYQ 2026

medium

physics ID: jee-main

$If position vector is given as and if its signs are reversed, then which of the following physical quantity remains unaffected?$

1

$Velocity$

2

$Displacement$

3

$Acceleration$

4

$Torque$

Official Solution

Correct Option: (4)

$Concept: Reversing the sign of the position vector: represents a change of origin or spatial inversion. The effect on a physical quantity depends on how it is defined in terms of . Step 1: Effect on Different Physical Quantities Velocity: Velocity changes sign, hence it is affected . Displacement: Displacement is directly related to position vector. Thus, displacement is affected . Acceleration: Acceleration also changes sign, hence it is affected . Torque: Torque is defined as: If, then: However, under spatial inversion, force also reverses direction: Thus, So, torque remains unchanged .$

High-Yield Trend

Chapter Questions
19 MCQs

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Concept Used:

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Vectors

High-Yield Trend

Chapter Questions 19 MCQs

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Concept Used:

Step-by-Step Solution:

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Chapter Questions
19 MCQs