Chapter 2: Mathematical Methods - Solutions

1. Choose the correct option.

i) The resultant of two forces 10 N and 15 N acting along + x and - x-axes respectively, is

Correct Option: (C) 5 N along + x-axis

ii) For two vectors to be equal, they should have the

Correct Option: (C) same magnitude and direction

iii) The magnitude of scalar product of two unit vectors perpendicular to each other is

Correct Option: (A) zero

iv) The magnitude of vector product of two unit vectors making an angle of 60° with each other is

Correct Option: (A) 1

v) If AB, C are three vectors, then which of the following is not correct?

Correct Option: (C) A B B A

2. Answer the following questions.

i) Show that a = i - j is a unit vector.

Magnitude: |a| = √(1² + (-1)²) = √2. Since |a| = 1, it is a unit vector.

ii) If v₁ = 3i + 4j + k and v₂ = i - j - k, determine the magnitude of v₁ + v₂.

v₁ + v₂ = (3+1)i + (4-1)j + (1-1)k = 4i + 3j. Magnitude = √(4² + 3²) = 5.

iii) For v₁ = 2i - 3j and v₂ = 6i + 5j, determine the magnitude and direction of v₁ + v₂.

v₁ + v₂ = (2+6)i + (-3+5)j = 8i + 2j. Magnitude = √(8² + 2²) = √68 = 2√17. Direction: θ = tan^-1(2/8) = tan^-1(1/4).

iv) Find a vector which is parallel to v = i - 2j and has a magnitude 10.

Unit vector = (1/√5)(i - 2j). Vector with magnitude 10: 10(1/√5)(i - 2j) = (10/√5)i - (20/√5)j.

v) Show that vectors a = 2i + 5j - 6k and b = i + 5j - 3k are parallel.

Cross product a × b = |i j k|
|2 5 -6|
|1 5 -3| = 0. Since a × b = 0, they are parallel.

3. Solve the following problems.

i) Determine a × b, given a = 2i + 3j and b = 3i + 5j.

a × b = |i j k|
|2 3 0|
|3 5 0| = (0 - 0)i - (0 - 0)j + (10 - 9)k = k.

ii) Show that vectors a = 2i + 3j + 6k, b = 3i - 6j + 2k and c = 6i + 2j - 3k are mutually perpendicular.

a · b = 0, a · c = 0, b · c = 0. Hence, they are mutually perpendicular.

iii) Determine the vector product of v₁ and v₂, where v₁ = 2i + 3j - k and v₂ = i + 2j - 3k.

v₁ × v₂ = |i j k|
|2 3 -1|
|1 2 -3| = (3 - 2)i - (-6 - 1)j + (4 - 3)k = 7i + 5j + k.

iv) Given v₁ = 5i + 2j and v₂ = ai - 6j are perpendicular to each other, determine the value of a.

For perpendicular vectors, v₁ · v₂ = 0. Hence, 5a - 12 = 0, a = 12/5.

v) Obtain derivatives of the following functions:

(i) d/dx [x sin x] = sin x + x cos x,
(ii) d/dx [x⁴ + cos x] = 4x³ - sin x,
(iii) d/dx [x/sin x] = (sin x - x cos x)/sin² x.

vi) Using the rule for differentiation for the quotient of two functions, prove that d/dx [x/sin x] = (sin x - x cos x)/sin² x.

Using quotient rule: d/dx [u/v] = (v du - u dv)/v². Here, u = x, v = sin x.
du = 1, dv = cos x. Thus, d/dx [x/sin x] = (sin x * 1 - x * cos x)/sin² x.

vii) Evaluate the following integral:

(i) ∫₀² sin x dx = -cos x |₀² = -cos(2) + cos(0) = 1.
(ii) ∫₁⁵ x dx = [x²/2]|₁⁵ = (25/2) - (1/2) = 12.