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Trigonometry and standard angles.
Calculus basics for physics.
Vector addition and resolution.
Dot product and cross product.
A comprehensive mock covering all subtopics — trigonometry, calculus basics, vector addition, resolution, dot and cross products. Ideal after reading the notes and working through solved examples.
1. A vector quantity is one which has:
Explanation: A vector quantity has both magnitude and direction. Examples: velocity, force, displacement. Scalars have only magnitude.
2. Which of the following is a scalar quantity?
Explanation: Work = F·d·cosθ is a scalar (dot product of two vectors). Displacement, velocity, and electric field are all vectors.
3. The value of sin 30° is:
Explanation: sin 30° = 1/2. Standard values: sin 0° = 0, sin 30° = 1/2, sin 45° = 1/√2, sin 60° = √3/2, sin 90° = 1.
4. The value of cos 60° is:
Explanation: cos 60° = 1/2. Standard values: cos 0° = 1, cos 30° = √3/2, cos 45° = 1/√2, cos 60° = 1/2, cos 90° = 0.
5. Two forces of 3 N and 4 N act at right angles to each other. The resultant force is:
Explanation: For perpendicular forces: R = √(3² + 4²) = √25 = 5 N. This is the classic 3-4-5 right triangle.
6. If two equal vectors of magnitude A are inclined at angle θ to each other, the magnitude of their resultant is:
Explanation: For two equal vectors of magnitude A at angle θ: R = A√(2+2cosθ) = A√(4cos²(θ/2)) = 2A cos(θ/2).
7. The derivative of xⁿ with respect to x is:
Explanation: By the power rule: d(xⁿ)/dx = nxⁿ⁻¹. For example, d(x³)/dx = 3x².
8. The integration of a constant k with respect to x is:
Explanation: ∫k dx = kx + C, where C is the constant of integration.
9. A vector A of magnitude 10 N makes an angle of 30° with the x-axis. Its x-component is:
Explanation: x-component = A cos30° = 10 × (√3/2) = 5√3 N. y-component = A sin30° = 5 N.
10. A unit vector has magnitude:
Explanation: A unit vector has magnitude exactly 1. It is used to indicate direction only. â = A/|A|.
11. The derivative of sin x with respect to x is:
Explanation: d(sin x)/dx = cos x. This is a fundamental derivative formula in calculus.
12. The resultant of two vectors is minimum when the angle between them is:
Explanation: R = √(A² + B² + 2AB cosθ). When θ = 180°, cosθ = −1, R = |A − B| which is minimum.
13. sin²θ + cos²θ equals:
Explanation: sin²θ + cos²θ = 1 is the fundamental Pythagorean trigonometric identity, valid for all values of θ.
14. Two vectors are equal if they have:
Explanation: Two vectors are equal if and only if they have the same magnitude AND the same direction.
15. ∫x dx equals:
Explanation: ∫x dx = x²/2 + C by the power rule for integration: ∫xⁿ dx = xⁿ⁺¹/(n+1) + C.
16. The dot product A·B of two vectors A and B inclined at angle θ is:
Explanation: A·B = AB cosθ. The dot product gives a scalar. It is maximum when θ = 0° and zero when θ = 90°.
17. The cross product A × B of two vectors A and B inclined at angle θ has magnitude:
Explanation: |A × B| = AB sinθ. The cross product gives a vector perpendicular to both A and B.
18. A force of 100 N acts at 60° to the horizontal. Its vertical component is:
Explanation: Vertical component = F sin60° = 100 × (√3/2) = 50√3 N.
19. If displacement s = 5t² + 3t + 2, the velocity at t = 2 s is:
Explanation: v = ds/dt = 10t + 3. At t = 2: v = 10(2) + 3 = 23 m/s.
20. If v = 4t³ − 3t² + 2, the acceleration at t = 1 s is:
Explanation: a = dv/dt = 12t² − 6t. At t = 1: a = 12 − 6 = 6 m/s².
21. Vectors A = 3î + 4ĵ and B = 1î + 2ĵ. The magnitude of A + B is:
Explanation: A + B = 4î + 6ĵ. Magnitude = √(16+36) = √52 = 2√13.
22. The work done by force F = 3î + 4ĵ N through displacement s = 2î + 5ĵ m is:
Explanation: W = F·s = (3)(2) + (4)(5) = 6 + 20 = 26 J.
23. The area under a velocity-time graph represents:
Explanation: s = ∫v dt, which is the area under the v-t graph. This gives the displacement.
24. A vector has components 3 units along x-axis and 4 units along y-axis. Its magnitude and angle with x-axis are:
Explanation: Magnitude = √(9+16) = 5 units. Angle = tan⁻¹(4/3) ≈ 53° with x-axis.
25. If A and B are parallel vectors, then A × B equals:
Explanation: When θ = 0° (parallel), |A × B| = AB sin0° = 0. The cross product of parallel vectors is the zero vector.
26. The slope of a position-time graph at any instant gives:
Explanation: v = ds/dt. The derivative of position with respect to time is velocity, which equals the slope of the position-time graph.
27. ∫sin x dx equals:
Explanation: ∫sin x dx = −cos x + C. Verified: d(−cos x)/dx = sin x.
28. The unit vector along the direction of A = 3î + 4ĵ is:
Explanation: |A| = √(9+16) = 5. Unit vector â = A/|A| = (3î + 4ĵ)/5.
29. A projectile is launched at 45°. Its horizontal and vertical components of initial velocity u are:
Explanation: uₓ = u cos45° = u/√2, uᵧ = u sin45° = u/√2. At 45°, both components are equal: u/√2 each.
30. If acceleration a = 6t, and v = 0 at t = 0, then v at t = 3 s is:
Explanation: v = ∫a dt = ∫6t dt = 3t² + C. At t=0, v=0 → C=0. At t=3: v = 3(9) = 27 m/s.
31. If |A + B| = |A − B|, then the angle between A and B is:
Explanation: |A+B|² = A²+B²+2AB cosθ and |A−B|² = A²+B²−2AB cosθ. For equality: 4AB cosθ = 0 → cosθ = 0 → θ = 90°.
32. The resultant of two forces P and Q is R. If Q is doubled, the new resultant is perpendicular to P. Then:
Explanation: If new resultant (with 2Q) is perpendicular to P, using the triangle law and the condition of perpendicularity leads to Q = R.
33. The displacement of a particle is given by x = A sin(ωt). The maximum velocity is:
Explanation: v = dx/dt = Aω cos(ωt). Maximum value of cos is 1, so v_max = Aω.
34. A particle's velocity is v = 3t² − 12t + 9. When is the acceleration zero?
Explanation: a = dv/dt = 6t − 12. Setting a = 0: 6t = 12 → t = 2 s.
35. The scalar triple product A·(B × C) represents:
Explanation: The scalar triple product A·(B × C) gives the volume of the parallelepiped formed by vectors A, B, and C as edges.
36. î × ĵ equals:
Explanation: î × ĵ = k̂ (right-hand rule). The cyclic order is î → ĵ → k̂ → î.
37. The distance covered by a particle with v = 2t from t = 0 to t = 4 s is:
Explanation: s = ∫₀⁴ 2t dt = [t²]₀⁴ = 16 − 0 = 16 m.
38. For what angle θ is the range of a projectile (R = u²sin2θ/g) maximum?
Explanation: R = (u²/g)sin2θ is maximum when sin2θ = 1, i.e., 2θ = 90°, θ = 45°.
39. If A = 2î + 3ĵ − k̂ and B = î − 2ĵ + 3k̂, then A·B is:
Explanation: A·B = (2)(1) + (3)(−2) + (−1)(3) = 2 − 6 − 3 = −7.
40. If v = u + at (where u, a are constants), then the displacement from t = 0 to t = T is:
Explanation: s = ∫₀ᵀ (u + at) dt = [ut + at²/2]₀ᵀ = uT + (1/2)aT². This is the standard kinematic equation.
41. A × B = −(B × A) illustrates which property of cross product?
Explanation: The cross product is anti-commutative: A × B = −(B × A). Reversing the order of vectors reverses the direction of the resultant.
42. The velocity of a particle is v = A cos(ωt). The acceleration is:
Explanation: a = dv/dt = d[A cos(ωt)]/dt = −Aω sin(ωt).
43. The function f(x) = x² − 4x + 5 has a minimum value of:
Explanation: f'(x) = 2x − 4 = 0 → x = 2. f(2) = 4 − 8 + 5 = 1. Since f''(x) = 2 > 0, this is a minimum.
44. The angle between vectors A = î + ĵ and B = î − ĵ is:
Explanation: A·B = (1)(1) + (1)(−1) = 0. Since dot product is zero, the vectors are perpendicular: θ = 90°.
45. The slope of a v-t graph gives:
Explanation: a = dv/dt. The slope of the velocity-time graph = dv/dt = acceleration.
46. The scalar product of a vector with itself, A·A, equals:
Explanation: A·A = |A||A|cos0° = A². The dot product of a vector with itself equals the square of its magnitude.
47. The process of splitting a vector into its rectangular components is called:
Explanation: Resolution of vectors is the process of finding components of a vector along two perpendicular directions.
48. If position vector r = 3t²î + 2tĵ, then velocity at t = 1 s is:
Explanation: v = dr/dt = 6tî + 2ĵ. At t = 1: v = 6î + 2ĵ m/s.
49. Three forces in equilibrium must be:
Explanation: For three forces to be in equilibrium, they must be coplanar (in the same plane) and concurrent (meeting at a point).
50. The derivative of a constant is:
Explanation: d(c)/dx = 0 for any constant c. The graph of a constant is a horizontal line with zero slope.
51. Two forces of equal magnitude F make an angle of 120° with each other. The magnitude of their resultant is:
Explanation: R = √(F²+F²+2F²cos120°) = √(2F²−F²) = F.
52. ∫₀^π sin x dx equals:
Explanation: ∫₀^π sin x dx = [−cos x]₀^π = −cosπ − (−cos0) = 1 + 1 = 2.
53. The magnitude of cross product A × B = AB sinθ is maximum when θ is:
Explanation: |A × B| = AB sinθ is maximum when sinθ = 1, i.e., θ = 90°.
54. A negative vector of vector A has:
Explanation: The negative vector (−A) has the same magnitude as A but points in the exactly opposite direction.
55. If a ball is thrown horizontally with speed u from height h, the time to reach ground is:
Explanation: Vertical motion: h = (1/2)gt². So t = √(2h/g).
56. The product rule of differentiation states d(uv)/dx =:
Explanation: The product rule: d(uv)/dx = u·(dv/dx) + v·(du/dx).
57. The displacement vector from point A(1,2) to point B(4,6) is:
Explanation: Displacement = B − A = (4−1)î + (6−2)ĵ = 3î + 4ĵ. Its magnitude = 5 units.
58. For the equation of motion v² = u² + 2as, the differential form is:
Explanation: a = v(dv/ds), so v dv = a ds. Integrating both sides: v²/2 = as + C, giving v² = u² + 2as.
59. The angle between the unit vectors î and ĵ is:
Explanation: î and ĵ are unit vectors along x and y axes respectively, which are perpendicular. The angle between them is 90°.
60. If A × B = 0 and A·B = 0, then:
Explanation: A×B = 0 means parallel or zero; A·B = 0 means perpendicular or zero. Both simultaneously require at least one to be a zero vector.