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Physics · Textbook solutions

Mathematical Methods

Every solved example, exercise, and miscellaneous question, in the order the textbook teaches them. · 28 questions

2. Mathematical Methods — worked examples

11 q

Solved Examples

Worked · 11
  1. Solved Ex.2.1
    Express vector AC→\overrightarrow{AC} in terms of vectors AB→\overrightarrow{AB} and CB→\overrightarrow{CB} shown in the following figure.
  2. Solved Ex.2.2
    From the following figure, determine the resultant of four forces A1⃗,A2⃗,A3⃗\vec{A_1}, \vec{A_2}, \vec{A_3} and A4⃗\vec{A_4}.
  3. Solved Ex.2.3
    Water is flowing in a stream with velocity 5 km/hr in an easterly direction relative to the shore. Speed of a boat is relative to still water is 20 km/hr. If the boat enters the stream heading North, with what velocity will the boat actually travel?
  4. Solved Ex.2.4
    Find a unit vector in the direction of the vector 3i^+4j^3\hat{i} + 4\hat{j}
  5. Solved Ex.2.5
    Given a⃗=i^+2j^\vec{a} = \hat{i} + 2\hat{j} and b⃗=2i^+j^\vec{b} = 2\hat{i} + \hat{j}, what are the magnitudes of the two vectors? Are these two vectors equal?
  6. Solved Ex.2.6
    Find the scalar product of the two vectors v1⃗=i^+2j^+3k^\vec{v_1} = \hat{i} + 2\hat{j} + 3\hat{k} and v2⃗=3i^+4j^−5k^\vec{v_2} = 3\hat{i} + 4\hat{j} - 5\hat{k}
  7. Solved Ex.2.7
    The angular momentum L⃗=r⃗×p⃗\vec{L} = \vec{r} \times \vec{p}, where r⃗\vec{r} is a position vector and p⃗\vec{p} is linear momentum of a body. If r⃗=4i^×6j^−3k^\vec{r} = 4\hat{i} \times 6\hat{j} - 3\hat{k} and p⃗=2i^+4j^−5k^\vec{p} = 2\hat{i} + 4\hat{j} - 5\hat{k}, find L⃗\vec{L}
  8. Solved Ex.2.8
    If A⃗=5i^+6j^+4k^\vec{A} = 5\hat{i} + 6\hat{j} + 4\hat{k} and B⃗=2i^−2j^+3k^\vec{B} = 2\hat{i} - 2\hat{j} + 3\hat{k}, determine the angle between A⃗\vec{A} and B⃗\vec{B}.
  9. Solved Ex.2.9
    Given P⃗=4i^−j^+8k^\vec{P} = 4\hat{i} - \hat{j} + 8\hat{k} and Q⃗=2i^−mj^+4k^\vec{Q} = 2\hat{i} - m\hat{j} + 4\hat{k}, find mm if P⃗\vec{P} and Q⃗\vec{Q} have the same direction.
  10. Solved Ex.2.10
    Find the derivatives of the functions. (a) f(x)=x8f(x) = x^{8} (b) f(x)=x3+sin⁡xf(x) = x^{3} + \sin x (c) f(x)=x3sin⁡xf(x) = x^{3}\sin x
  11. Solved Ex.2.11
    Evaluate the following integrals: (a) ∫x8 dx\displaystyle\int x^{8}\,dx (b) ∫25x2 dx\displaystyle\int_{2}^{5} x^{2}\,dx (c) ∫(x+sin⁡x) dx\displaystyle\int (x + \sin x)\,dx

Exercises

17 q

Choose the correct option

Practice · 5
  1. Choose the correct option.
    Ex Q.1 (i)
    The resultant of two forces 10 N and 15 N acting along +x+x and −x-x-axes respectively, is

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  2. Ex Q.1 (ii)
    For two vectors to be equal, they should have the

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  3. Ex Q.1 (iii)
    The magnitude of scalar product of two unit vectors perpendicular to each other is

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  4. Ex Q.1 (iv)
    The magnitude of vector product of two unit vectors making an angle of 60∘60^{\circ} with each other is

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  5. Ex Q.1 (v)
    If A⃗,B⃗\vec{A}, \vec{B} and C⃗\vec{C} are three vectors, then which of the following is not correct?

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Answer the following questions

Practice · 5
  1. Answer the following questions.
    Ex Q.2 (i)
    Show that a⃗=i^−j^2\vec{a} = \dfrac{\hat{i} - \hat{j}}{\sqrt{2}} is a unit vector.
  2. Ex Q.2 (ii)
    If v1⃗=3i^+4j^+k^\vec{v_1} = 3\hat{i} + 4\hat{j} + \hat{k} and v2⃗=i^−j^−k^\vec{v_2} = \hat{i} - \hat{j} - \hat{k}, determine the magnitude of v1⃗+v2⃗\vec{v_1} + \vec{v_2}.
  3. Ex Q.2 (iii)
    For v1⃗=2i^−3j^\vec{v_1} = 2\hat{i} - 3\hat{j} and v2⃗=−6i^+5j^\vec{v_2} = -6\hat{i} + 5\hat{j}, determine the magnitude and direction of v1⃗+v2⃗\vec{v_1} + \vec{v_2}.
  4. Ex Q.2 (iv)
    Find a vector which is parallel to v⃗=i^−2j^\vec{v} = \hat{i} - 2\hat{j} and has a magnitude 10.
  5. Ex Q.2 (v)
    Show that vectors a⃗=2i^+5j^−6k^\vec{a} = 2\hat{i} + 5\hat{j} - 6\hat{k} and b⃗=i^+52j^−3k^\vec{b} = \hat{i} + \dfrac{5}{2}\hat{j} - 3\hat{k} are parallel.

Solve the following problems

Practice · 7
  1. Solve the following problems.
    Ex Q.3 (i)
    Determine a⃗×b⃗\vec{a} \times \vec{b}, given a⃗=2i^+3j^\vec{a} = 2\hat{i} + 3\hat{j} and b⃗=3i^+5j^\vec{b} = 3\hat{i} + 5\hat{j}.
  2. Ex Q.3 (ii)
    Show that vectors a⃗=2i^+3j^+6k^\vec{a} = 2\hat{i} + 3\hat{j} + 6\hat{k}, b⃗=3i^−6j^+2k^\vec{b} = 3\hat{i} - 6\hat{j} + 2\hat{k} and c⃗=6i^+2j^−3k^\vec{c} = 6\hat{i} + 2\hat{j} - 3\hat{k} are mutually perpendicular.
  3. Ex Q.3 (iii)
    Determine the vector product of v1⃗=2i^+3j^−k^\vec{v_1} = 2\hat{i} + 3\hat{j} - \hat{k} and v2⃗=i^+2j^−3k^\vec{v_2} = \hat{i} + 2\hat{j} - 3\hat{k},
  4. Ex Q.3 (iv)
    Given v1⃗=5i^+2j^\vec{v_1} = 5\hat{i} + 2\hat{j} and v2⃗=ai^−6j^\vec{v_2} = a\hat{i} - 6\hat{j} are perpendicular to each other, determine the value of aa.
  5. Ex Q.3 (v)
    Obtain derivatives of the following functions: (i) xsin⁡xx \sin x (ii) x4+cos⁡xx^{4} + \cos x (iii) x/sin⁡xx/\sin x
  6. Ex Q.3 (vi)
    Using the rule for differentiation for quotient of two functions, prove that ddx(sin⁡xcos⁡x)=sec⁡2x\dfrac{d}{dx}\left(\dfrac{\sin x}{\cos x}\right) = \sec^{2} x
  7. Ex Q.3 (vii)
    Evaluate the following integral: (i) ∫0π/2sin⁡x dx\displaystyle\int_{0}^{\pi/2} \sin x\,dx (ii) ∫15x dx\displaystyle\int_{1}^{5} x\,dx