Teams A,B,C went for playing a tug of war game. Teams A,B,C have attached a rope to a metal ring and is trying to pull the ring into their own area. Team A pulls with force F_1=6 i+0 j kN, Team B pulls with force F_2=-4 i+4 j kN, Team C pulls with force F_3=-3 i-3 j kN, (i) What is the magnitude of the force of Team A? (ii) Which team will win the game? (iii) Find the magnitude of the resultant force exerted by the teams. (iii) In what direction is the ring getting pulled?

(i) 6 kN; (ii) Team B; (iii)(a) 2 kN; (iii)(b) along - i+ j, at 3π/4 with positive x-axis

MathsCBSE Class 12 · Sample paper, session 2023-24 · 4 marks · Case-based

CBSE Class 12 Mathematics sample paper 2023-24: question 37

From the official CBSE sample question paper for 2023-24, section E, question 37, for 4 marks. Try it first; the hints and our step-by-step solution are below.

Q37 · 4 marks · Case-basedCompetency-basedVector Algebra

Teams $A, B, C$ went for playing a tug of war game. Teams $A, B, C$ have attached a rope to a metal ring and is trying to pull the ring into their own area.
Team $A$ pulls with force $F_{1} = 6 \hat{i} + 0 \hat{j}$ kN,
Team $B$ pulls with force $F_{2} = - 4 \hat{i} + 4 \hat{j}$ kN,
Team $C$ pulls with force $F_{3} = - 3 \hat{i} - 3 \hat{j}$ kN,

(i)
What is the magnitude of the force of Team $A$ ?
1 mark
(ii)
Which team will win the game?
1 mark
(iii)
Find the magnitude of the resultant force exerted by the teams.
2 marks
OR
(iii)
In what direction is the ring getting pulled?
2 marks

Figure in the paper: Ring at centre with three force vectors: a=6i+0j pointing right towards A; b=-4i+4j pointing up-left towards C-side direction labelled B (lower left, arrow towards B); c=-3i-3j pointing towards C (upper left); dashed reference lines through the ring. See it in the official paper.

Source: official sample paper from cbseacademic.nic.in (PDF). See it in the full Sample paper 2023-24.

Hints

Open one hint at a time and try again before the next.

Hint 1

Magnitude of $a \hat{i} + b \hat{j}$ is $a^{2} + b^{2}$ .

Hint 2

Add all three forces; the team in whose direction the resultant points wins.

Hint 3

For direction, find the angle the resultant makes with the positive x-axis, noting which quadrant it lies in.

Step-by-step solution

Each step shows the marks it earns, as an examiner would award them. Checked by an independent re-solve from the question paper.

(i) $∣ F_{1} ∣ = 6^{2} + 0^{2} = 6$ kN.
1 mark
(ii) $F_{1} + F_{3} = 3 \hat{i} - 3 \hat{j} = 3 (\hat{i} - \hat{j})$ and $F_{2} = - 4 (\hat{i} - \hat{j})$ , so they are opposite (unlike) vectors. $∣ F_{2} ∣ = 42 > ∣ F_{1} + F_{3} ∣ = 32$ , so the net pull is towards B. Team B will win.
1 mark
(iii)(a) $F = F_{1} + F_{2} + F_{3} = (6 - 4 - 3) \hat{i} + (0 + 4 - 3) \hat{j} = - \hat{i} + \hat{j}$ .
1 mark
(iii)(a) $∣ F ∣ = (- 1)^{2} + 1^{2} = 2$ kN.
1 mark
(iii)(b) Resultant $F = - \hat{i} + \hat{j}$ , which lies in the second quadrant.
1 mark
(iii)(b) Angle with the positive $x$ -axis: $θ = π - tan^{- 1} (\frac{1}{1}) = π - \frac{π}{4} = \frac{3 π}{4}$ . The ring is pulled along $- \hat{i} + \hat{j}$ , i.e. at $13 5^{\circ}$ to the positive $x$ -axis (towards B).
1 mark

Answer: (i) 6 kN; (ii) Team B; (iii)(a) $2$ kN; (iii)(b) along $- \hat{i} + \hat{j}$ , at $3 π /4$ with positive x-axis

Where the marks are

(i) Magnitude of $F_{1}$ is 6 kN (1 mark)
(ii) Compares $F_{2}$ with $F_{1} + F_{3}$ and concludes B wins (1 mark)
(iii)(a) Resultant $- \hat{i} + \hat{j}$ (1 mark)
(iii)(a) Magnitude $2$ kN (1 mark)
(iii)(b) Resultant $- \hat{i} + \hat{j}$ (1 mark)
(iii)(b) Direction $3 π /4$ with positive x-axis (1 mark)

Revise the chapter

Chapter: Vector Algebra (notes, formulas, where marks are lost and every past question)

Practise this chapterBoard-style questions from Vector Algebra, one at a time, marked against the answer key and marking points.Back to the full paperSample paper 2023-24: every question, by section.

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CBSE Class 12 Mathematics sample paper 2023-24: question 37

Hints

Step-by-step solution

Revise the chapter

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