Units & Vectors PDF Past Paper

Q1. A force F is applied to a surface of area A, and the pressure P exerted is given by F​ the formula P = A. What is the unit of pressure P ? Options: 2 (a) kg m/s (b) kg m−1 s−2 (c) kg m2 s−2...

Q1. A force F is applied to a surface of area A, and the pressure P exerted is given by F​ the formula P = A. What is the unit of pressure P ? Options: 2 (a) kg m/s (b) kg m−1 s−2 (c) kg m2 s−2 (d) kg m−2 s−1 Answer: (b) kg m−1 s−2 Q2. The gravitational potential energy U of an object of mass m at height h in a gravitational field with acceleration g is given by U = mgh. What is the unit of gravitational potential energy U ? Options: (a) kg m/s2 (b) kg m2 s−2 (c) kg m/s (d) kg m−1 s−2 Answer: (b) kg m2 s−2 Q3. The drag force Fd ​ on an object moving through a fluid is given by Fd ​ = 1​ ​ 2 2 Cd ρAv , where Cd ​ is the drag coefficient (dimensionless), ρ is the density of the fluid, A is the cross-sectional area, and v is the velocity. What is the unit of ρ, the density of the fluid? Options: (a) kg m−2 (b) kg m−3 (c) kg m/s (d) kg m/s2 Answer: (b) kg m−3 Q4. The kinetic energy K of an object is given by K = 12 ​mv 2 , where m is mass and v is velocity. What is the unit of kinetic energy K ? Options: (a) kg m2 s−2 (b) kg m/s (c) kg m/s2 (d) kg m2 s−1 Answer: (a) kg m2 s−2 Q5. The frequency f of a wave is related to the wave speed v and wavelength λ by the equation f = vλ ​. What is the unit of frequency f ? Options: (a) s (b) m/s (c) Hz (or s−1 ) (d) m−1 Answer: (c) Hz (or s−1 ) Q6. Given two displacement vectors: A = 5^i + 7^j ,​ B = −3^i + 2^j ​ What is the magnitude of the resultant vector R = A + B? Options: (a) 10 (b) 65​ (c) 6 (d) 85​ Answer: (d) 85​ Q7. The two displacement vectors are: A = 4^i − 3^j ,​ B = 6^i + 1^j ​ What is the angle θ of the resultant vector with respect to the x-axis? Options: (a) 30∘ (b) 11.31∘ (c) 21.80∘ (d) 45∘ Answer: (c) 21.80∘ Q8. Two vectors are given by: A = 2^i + 3^j ,​ B = −3^i + 5^j ​ What is the x-component of the resultant vector R = A + B? Options: (a) −1 (b) 5 (c) 1 (d) −2 Answer: (a) −1 Q9. Two displacement vectors are: A = 6^i + 4^j ,​ B = 2^i − 7^j ​ What is the magnitude of the resultant vector R? Options: (a) 85​ (b) 50​ (c) 58​ (d) 10 Answer: (c) 58​ Q10. If the vectors are: A = 3^i − 6^j ,​ B = 8^i + 2^j ​ What is the y-component of the resultant vector R = A + B? Options: (a) 2 (b) −4 (c) −3 (d) −8 Answer: (b) −4 Q11. A hiker begins a trip by first walking 30 km due east from her car. On the second day, she walks 50 km in a direction 45° north of east. Determine the magnitude of the hiker’s resultant displacement. A. 60.32 km B. 66.17 km C. 70.50 km D. 72.45 km Answer: B. 66.17 km Q12. A hiker first walks 20 km south from her starting point. On the second day, she walks 35 km in a direction 30° west of north. What is the magnitude of her resultant displacement from the starting point? A. 40.12 km B. 38.79 km C. 36.67 km D. 34.89 km Answer: C. 36.67 km Q13. A hiker walks 15 km west on the first day of a trip. On the second day, she walks 25 km at an angle of 60° south of west. What is the magnitude of her resultant displacement from her starting point? A. 30.89 km B. 28.58 km C. 27.04 km D. 26.22 km Answer: C. 27.04 km Q14. A hiker walks 12 km north on the first day, then 40 km in a direction 30° east of north on the second day. What is the magnitude of the hiker's resultant displacement? A. 44.38 km B. 47.13 km C. 45.99 km D. 46.27 km Answer: D. 46.27 km Q15. A hiker walks 22 km northeast (45° north of east) on the first day. On the second day, she walks 18 km at 60° south of east. What is the magnitude of her resultant displacement? A. 30.56 km B. 27.42 km C. 25.89 km D. 24.33 km Answer: B. 27.42 km 16. Find the angle between the vectors A = 4^i + ^j ​ and B = 3^i − 2^j.​ a) 45∘ b) 57∘ c) 67∘ d) 73∘ Answer: c) 67∘ 17. Find the angle between the vectors A = 2^i + 5^j ​ and B = 4^i + 2^j.​ a) 34∘ b) 52∘ c) 61∘ d) 74∘ Answer: b) 52∘ 18. Find the angle between the vectors A = −^i + 3^j ​ and B = 2^i − 2^j.​ a) 104∘ b) 90∘ c) 116∘ d) 127∘ Answer: c) 116∘ 19. Find the angle between the vectors A = 3^i + 4j^ and B = ^i − ^j.​ a) 81∘ b) 92∘ c) 98∘ d) 110∘ Answer: a) 81∘ 20. Find the angle between the vectors A = 5^i − 2^j ​ and B = −3^i + 6^j.​ a) 145∘ b) 136∘ c) 120∘ d) 152∘ Answer: b) 136∘ 21. Given the vectors A = 4^i + 2^j ​ and B = 3^i + ^j ,​ what is the dot product A ⋅ B? a) 10 b) 14 c) 12 d) 16 Answer: b) 14 22. If A = 6^i − 3^j ​ and B = 2^i + 5^j ,​ what is A ⋅ B? a) -3 b) 12 c) -9 d) -27 Answer: a) -3 23. For the vectors A = 5^i + 7^j ​ and B = ^i + 4^j ,​ what is the value of A ⋅ B? a) 38 b) 33 c) 25 d) 29 Answer: b) 33 24. Find the dot product of A = −3^i + 2^j ​ and B = 4^i − 6^j.​ a) -24 b) 12 c) -20 d) -18 Answer: c) -20 25. Given the vectors A = 2.5^i + 4^j ​ and B = −1^i + 3^j ,​ what is the dot product A ⋅ B? a) 9.5 b) 11.5 c) 13 d) 15 Answer: a) 9.5 Q26. Given a vector with initial point (x1 ​, y1 ​) = (3, −5) and terminal point (x2 ​, y2 ​) = (−2, 4), what are the x- and y -components of this vector? A) Ax ​ = −5, Ay ​ = 9 B) Ax ​ = 5, Ay ​ = −9 C) Ax ​ = −5, Ay ​ = −9 D) Ax ​ = 9, Ay ​ = 5 Answer: A) Ax ​ = −5, Ay ​ = 9 Q27. If a vector has coordinates at points (x1 ​, y1 ​) = (−3, 2) and (x2 ​, y2 ​) = (4, −6), calculate the vector's x- component. A) Ax ​ = 7 B) Ax ​ = −7 C) Ax ​ = 1 D) Ax ​ = −1 Answer: A) Ax ​ = 7 Q28. A vector starts at point (x1 ​, y1 ​) = (0, −1) and ends at (x2 ​, y2 ​) = (−3, 5). What is the y -component of this vector? A) Ay ​ = −6 B) Ay ​ = 6 C) Ay ​ = −4 D) Ay ​ = 4 Answer: B) Ay ​ = 6 Q29. For a vector going from (x1 ​, y1 ​) = (−1, −3) to (x2 ​, y2 ​) = (5, 2), calculate the values of both x- and y - components. A) Ax ​ = 6, Ay ​ = −5 B) Ax ​ = −6, Ay ​ = 5 C) Ax ​ = 6, Ay ​ = 5 D) Ax ​ = −6, Ay ​ = −5 Answer: C) Ax ​ = 6, Ay ​ = 5 Q30. What are the x- and y -components for a vector that starts at (x1 ​, y1 ​) = (2, 3) and ends at (x2 ​, y2 ​) = (−3, −1)? A) Ax ​ = −5, Ay ​ = −4 B) Ax ​ = 5, Ay ​ = 4 C) Ax ​ = −1, Ay ​ = −1 D) Ax ​ = −5, Ay ​ = 4 Answer: D) Ax ​ = −5, Ay ​ = −4 Question 31 Given vectors A, B, C, and D with magnitudes A = 80 m, B = 200 m, C = 120 m, and D = 250 m, and the following directions: A is along the positive x-axis. B is along the negative y-axis. C makes an angle of 45° with the positive x-axis. D makes an angle of 135° with the positive x-axis. Find the resultant displacement vector R. Choices: A) R = −250i + 150j B) R = −100i + 200j C) R = −200i − 250j D) R = −280i − 200j Answer: C) R = −200i − 250j Question 32 Given vectors P, Q, R, and S with magnitudes P = 50 m, Q = 150 m, R = 100 m, and S = 180 m, and the following directions: P is along the positive y-axis. Q is along the positive x-axis. R makes an angle of 60° with the positive x-axis. S makes an angle of 150° with the positive x-axis. Find the resultant displacement vector V. Choices: A) V = 0i + 380j B) V = 50i + 230j C) V = 25i + 300j D) V = 150i + 50j Answer: B) V = 50i + 230j Question 33 Given vectors M, N, O, and P with magnitudes M = 100 m, N = 80 m, O = 200 m, and P = 150 m, and the following directions: M is along the negative y-axis. N is along the negative x-axis. O makes an angle of 30° with the positive x-axis. P makes an angle of 120° with the positive x-axis. Find the resultant displacement vector W. Choices: A) W = −200i − 250j B) W = −100i − 180j C) W = −150i − 100j D) W = −180i − 80j Answer: A) W = −200i − 250j Question 34 Given vectors X, Y , Z, and W with magnitudes X = 60 m, Y = 100 m, Z = 140 m, and W = 200 m, and the following directions: X is along the positive y-axis. Y is along the positive x-axis. Z makes an angle of 45° with the positive y-axis. W makes an angle of 225° with the positive x-axis. Find the resultant displacement vector U. Choices: A) U = −170i + 200j B) U = −100i + 180j C) U = −60i + 250j D) U = −80i + 150j Answer: A) U = −170i + 200j Question 35 Given vectors E, F, G, and H with magnitudes E = 90 m, F = 110 m, G = 160 m, and H = 220 m, and the following directions: E is along the positive x-axis. F is along the positive y-axis. G makes an angle of 30° with the positive x-axis. H makes an angle of 210° with the positive x-axis. Find the resultant displacement vector T. Choices: A) T = −140i + 150j B) T = −80i + 190j C) T = −60i + 250j D) T = −120i + 200j Answer: B) T = −80i + 190j