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. [Figure: a triangle with vertex A at the lower left, B at the lower right and C at the top. Three directed segments are drawn - AC→\overrightarrow{AC} from A up to C, CB→\overrightarrow{CB} from C down to B, and AB→\overrightarrow{AB} from A across to B.]
  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}. [Figure: an open polygon O-A-B-C-D. A1⃗\vec{A_1} runs from O to A along the base, A2⃗\vec{A_2} from A up to B, A3⃗\vec{A_3} from B to C, and A4⃗\vec{A_4} from C to D. A fifth vector A5⃗\vec{A_5} is drawn from O to D, closing the figure.]
  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
    1. A.
      25 N along +x+x-axis
    2. B.
      25 N along −x-x-axis
    3. C.
      5 N along +x+x-axis
    4. D.
      5 N along −x-x-axis
  2. Ex Q.1 (ii)
    For two vectors to be equal, they should have the
    1. A.
      same magnitude
    2. B.
      same direction
    3. C.
      same magnitude and direction
    4. D.
      same magnitude but opposite direction
  3. Ex Q.1 (iii)
    The magnitude of scalar product of two unit vectors perpendicular to each other is
    1. A.
      zero
    2. B.
      1
    3. C.
      −1-1
    4. D.
      2
  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
    1. A.
      1
    2. B.
      2
    3. C.
      3/23/2
    4. D.
      3/2\sqrt{3}/2
  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?
    1. A.
      A⃗⋅(B⃗+C⃗)=A⃗⋅B⃗+A⃗⋅C⃗\vec{A} \cdot \left(\vec{B} + \vec{C}\right) = \vec{A} \cdot \vec{B} + \vec{A} \cdot \vec{C}
    2. B.
      A⃗⋅B⃗=B⃗⋅A⃗\vec{A} \cdot \vec{B} = \vec{B} \cdot \vec{A}
    3. C.
      A⃗×B⃗=B⃗×A⃗\vec{A} \times \vec{B} = \vec{B} \times \vec{A}
    4. D.
      A⃗×(B⃗+C⃗)=A⃗×B⃗+B⃗×C⃗\vec{A} \times \left(\vec{B} + \vec{C}\right) = \vec{A} \times \vec{B} + \vec{B} \times \vec{C}

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