#### Chapters

Chapter 2: Physics and Mathematics

Chapter 3: Rest and Motion: Kinematics

Chapter 4: The Forces

Chapter 5: Newton's Laws of Motion

Chapter 6: Friction

Chapter 7: Circular Motion

Chapter 8: Work and Energy

Chapter 9: Centre of Mass, Linear Momentum, Collision

Chapter 10: Rotational Mechanics

Chapter 11: Gravitation

Chapter 12: Simple Harmonics Motion

Chapter 13: Fluid Mechanics

Chapter 14: Some Mechanical Properties of Matter

Chapter 15: Wave Motion and Waves on a String

Chapter 16: Sound Waves

Chapter 17: Light Waves

Chapter 18: Geometrical Optics

Chapter 19: Optical Instruments

Chapter 20: Dispersion and Spectra

Chapter 21: Speed of Light

Chapter 22: Photometry

#### H.C. Verma Concept of Physics Part-1 (2018-2019 Session) by H.C Verma

## Chapter 2: Physics and Mathematics

#### Chapter 2: Physics and Mathematics Exercise Short Answers solutions [Pages 27 - 28]

Is a vector necessarily changed if it is rotated through an angle ?

Is it possible to add two vectors of unequal magnitudes and get zero? Is it possible to add three vectors of equal magnitudes and get zero?

Does the phrase "direction of zero vector" have physical significance? Discuss it terms of velocity, force etc.

Can you add three unit vectors to get a unit vector? Does your answer change if two unit vectors are along the coordinate axes?

Can we have physical quantities having magnitude and direction which are not vectors?

Which of the following two statements is more appropriate?

(a) Two forces are added using triangle rule because force is a vector quantity.

(b) Force is a vector quantity because two forces are added using triangle rule.

Can you add two vectors representing physical quantities having different dimensions? Can you multiply two vectors representing physical quantities having different dimensions?

Can a vector have zero component along a line and still have nonzero magnitude?

Let ε_{1} and ε_{2} be the angles made by \[\vec{A}\] and -\[\vec{A}\] with the positive *X*-axis. Show that tan ε_{1} = tan ε_{2}_{.} Thus, giving tan ε does not uniquely determine the direction of \[\vec{A}\].

Is the vector sum of the unit vectors \[\vec{i}\] and \[\vec{i}\] a unit vector? If no, can you multiply this sum by a scalar number to get a unit vector?

Let \[\vec{A} = 3 \vec{i} + 4 \vec{j}\]. Write a vector \[\vec{B}\] such that \[\vec{A} \neq \vec{B}\], but A = B.

Can you have \[\vec{A} \times \vec{B} = \vec{A} \cdot \vec{B}\] with A ≠ 0 and B ≠ 0 ? What if one of the two vectors is zero?

If \[\vec{A} \times \vec{B} = 0\] can you say that

(a) \[\vec{A} = \vec{B} ,\]

(b) \[\vec{A} \neq \vec{B}\] ?

Let \[\vec{A} = 5 \vec{i} - 4 \vec{j} \text { and } \vec{B} = - 7 \cdot 5 \vec{i} + 6 \vec{j}\]. Do we have \[\vec{B} = k \vec{A}\] ? Can we say \[\frac{\vec{B}}{\vec{A}}\] = k ?

#### Chapter 2: Physics and Mathematics Exercise MCQ solutions [Page 28]

A vector is not changed if

it is rotated through an arbitrary angle

it is multiplied by an arbitrary scalar

it is cross multiplied by a unit vector

it is slid parallel to itself.

Which of the sets given below may represent the magnitudes of three vectors adding to zero?

2, 4, 8

4, 8, 16

1, 2, 1

0.5, 1, 2

The resultant of \[\vec{A} \text { and } \vec{B}\] makes an angle α with \[\vec{A}\] and β with \[\vec{B}\],

α < β

α < β if A < B

α < β if A > B

α < β if A = B

The component of a vector is

always less than its magnitude

always greater than its magnitude

always equal to its magnitude

None of these.

A vector \[\vec{A}\] points vertically upward and \[\vec{B}\] points towards the north. The vector product \[\vec{A} \times \vec{B}\] is

along the west

along the east

zero

vertically downward.

The radius of a circle is stated as 2.12 cm. Its area should be written as

14 cm

^{2}14.1 cm

^{2}14.11 cm

^{2}14.1124 cm

^{2}

#### Chapter 2: Physics and Mathematics Exercise MCQ solutions [Page 28]

A situation may be described by using different sets coordinate axes having different orientation. Which the following do not depended on the orientation of the axis?

(a) the value of a scalar

(b) component of a vector

(c) a vector

(d) the magnitude of a vector.

Let \[\vec{C} = \vec{A} + \vec{B}\]

\[\left| \vec{C} \right|\] is always greater than \[\left| \vec{A} \right|\]

It is possible to have \[\left| \vec{C} \right|\] < \[\left| \vec{A} \right|\] and \[\left| \vec{C} \right|\] < \[\left| \vec{B} \right|\]

C is always equal to A + B

C is never equal to A + B.

Let the angle between two nonzero vectors \[\vec{A}\] and \[\vec{B}\] be 120° and its resultant be \[\vec{C}\].

C must be equal to \[\left| A - B \right|\]

C must be less than \[\left| A - B \right|\]

C must be greater than \[\left| A - B \right|\]

C may be equal to \[\left| A - B \right|\]

The x-component of the resultant of several vectors

(a) is equal to the sum of the *x*-components of the vectors of the vectors

(b) may be smaller than the sum of the magnitudes of the vectors

(c) may be greater than the sum of the magnitudes of the vectors

(d) may be equal to the sum of the magnitudes of the vectors.

The magnitude of the vector product of two vectors \[\left| \vec{A} \right|\] and \[\left| \vec{B} \right|\] may be

(a) greater than AB

(b) equal to AB

(c) less than AB

(d) equal to zero.

#### Chapter 2: Physics and Mathematics Exercise Exercise solutions [Pages 29 - 30]

A vector \[\vec{A}\] makes an angle of 20° and \[\vec{B}\] makes an angle of 110° with the X-axis. The magnitudes of these vectors are 3 m and 4 m respectively. Find the resultant.

Let \[\vec{A} \text { and } \vec{B}\] be the two vectors of magnitude 10 unit each. If they are inclined to the X-axis at angle 30° and 60° respectively, find the resultant.

Add vectors \[\vec{A} , \vec{B} \text { and } \vec{C}\] each having magnitude of 100 unit and inclined to the X-axis at angles 45°, 135° and 315° respectively.

Let \[\vec{a} = 4 \vec{i} + 3 \vec{j} \text { and } \vec{b} = 3 \vec{i} + 4 \vec{j}\]. Find the magnitudes of (a) \[\vec{a}\] , (b) \[\vec{b}\] ,(c) \[\vec{a} + \vec{b} \text { and }\] (d) \[\vec{a} - \vec{b}\].

Refer to figure (2-E1). Find (a) the magnitude, (b) x and y component and (c) the angle with the X-axis of the resultant of \[\overrightarrow{OA} , \overrightarrow{BC} \text { and } \overrightarrow{DE}\].

Two vectors have magnitudes 2 unit and 4 unit respectively. What should be the angle between them if the magnitude of the resultant is (a) 1 unit, (b) 5 unit and (c) 7 unit.

A spy report about a suspected car reads as follows. "The car moved 2.00 km towards east, made a perpendicular left turn, ran for 500 m, made a perpendicular right turn, ran for 4.00 km and stopped". Find the displacement of the car.

A carrom board (4 ft × 4 ft square) has the queen at the centre. The queen, hit by the striker moves to the from edge, rebounds and goes in the hole behind the striking line. Find the magnitude of displacement of the queen (a) from the centre to the front edge, (b) from the front edge to the hole and (c) from the centre to the hole.

A mosquito net over a 7 ft × 4 ft bed is 3 ft high. The net has a hole at one corner of the bed through which a mosquito enters the net. It flies and sits at the diagonally opposite upper corner of the net. (a) Find the magnitude of the displacement of the mosquito. (b) Taking the hole as the origin, the length of the bed as the X-axis, it width as the Y axis, and vertically up as the Z-axis, write the components of the displacement vector.

Suppose \[\vec{a}\] is a vector of magnitude 4.5 units due north. What is the vector (a) \[3 \vec{a}\], (b) \[- 4 \vec{a}\] ?

Two vectors have magnitudes 2 m and 3m. The angle between them is 60°. Find (a) the scalar product of the two vectors, (b) the magnitude of their vector product.

Let A_{1} A_{2} A_{3} A_{4} A_{5} A_{6} A_{1} be a regular hexagon. Write the *x*-components of the vectors represented by the six sides taken in order. Use the fact the resultant of these six vectors is zero, to prove that

cos 0 + cos π/3 + cos 2π/3 + cos 3π/3 + cos 4π/3 + cos 5π/3 = 0.

Use the known cosine values to verify the result.

Let \[\vec{a} = 2 \vec{i} + 3 \vec{j} + 4 \vec{k} \text { and } \vec{b} = 3 \vec{i} + 4 \vec{j} + 5 \vec{k}\] Find the angle between them.

Prove that \[\vec{A} . \left( \vec{A} \times \vec{B} \right) = 0\].

If \[\vec{A} = 2 \vec{i} + 3 \vec{j} + 4 \vec{k} \text { and } \vec{B} = 4 \vec{i} + 3 \vec{j} + 2 \vec{k}\] find \[\vec{A} \times \vec{B}\].

If \[\vec{A} , \vec{B} , \vec{C}\] are mutually perpendicular, show that \[\vec{C} \times \left( \vec{A} \times \vec{B} \right) = 0\] Is the converse true?

A particle moves on a given straight line with a constant speed ν. At a certain time it is at a point P on its straight line path. O is a fixed point. Show that \[\vec{OP} \times \vec{\nu}\] is independent of the position P.

The force on a charged particle due to electric and magnetic fields is given by \[\vec{F} = q \vec{E} + q \vec{\nu} \times \vec{B}\].

Suppose \[\vec{E}\] is along the X-axis and \[\vec{B}\] along the Y-axis. In what direction and with what minimum speed ν should a positively charged particle be sent so that the net force on it is zero?

Give an example for which \[\vec{A} \cdot \vec{B} = \vec{C} \cdot \vec{B} \text{ but } \vec{A} \neq \vec{C}\].

Draw a graph from the following data. Draw tangents at *x* = 2, 4, 6 and 8. Find the slopes of these tangents. Verify that the curve draw is *y* = 2*x*^{2} and the slope of tangent is \[\tan \theta = \frac{dy}{dx} = 4x\]

\[\begin{array}x & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ y & 2 & 8 & 18 & 32 & 50 & 72 & 98 & 128 & 162 & 200\end{array}\]

A curve is represented by y = sin x. If x is changed from \[\frac{\pi}{3}\text{ to }\frac{\pi}{3} + \frac{\pi}{100}\] , find approximately the change in y.

The electric current in a charging R−C circuit is given by i = i_{0}_{ }e^{−t}^{/RC} where i_{0}, R and C are constant parameters of the circuit and t is time. Find the rate of change of current at (a) t = 0, (b) t = RC, (c) t = 10 RC.

The electric current in a discharging R−C circuit is given by i = i_{0} e^{−t}^{/RC} where i_{0}, R and C are constant parameters and t is time. Let i_{0} = 2⋅00 A, R = 6⋅00 × 10^{5} Ω and C = 0⋅500 μF. (a) Find the current at t = 0⋅3 s. (b) Find the rate of change of current at at 0⋅3 s. (c) Find approximately the current at t = 0⋅31 s.

Find the area bounded under the curve y = 3x^{2} + 6x + 7 and the X-axis with the ordinates at x = 5 and x = 10.

Find the area enclosed by the curve *y* = sin *x* and the X-axis between *x* = 0 and *x* = π.

Find the area bounded by the curve *y* = *e*^{−x}, the X-axis and the Y-axis.

A rod of length L is placed along the X-axis between x = 0 and x = L. The linear density (mass/length) ρ of the rod varies with the distance x from the origin as ρ = a + bx. (a) Find the SI units of a and b. (b) Find the mass of the rod in terms of a, b and L.

The momentum p of a particle changes with time t according to the relation

\[\frac{dp}{dt} = \left( 10 N \right) + \left( 2 N/s \right)t\] If the momentum is zero at t = 0, what will the momentum be at t = 10 s?

The changes in a function* y* and the independent variable *x* are related as

\[\frac{dy}{dx} = x^2\] . Find *y* as a function of* x*.

Write the number of significant digits in (a) 1001, (b) 100.1, (c) 100.10, (d) 0.001001.

A metre scale is graduated at every millimetre. How many significant digits will be there in a length measurement with this scale?

Round the following numbers to 2 significant digits.

(a) 3472, (b) 84.16. (c)2.55 and (d) 28.5

The length and the radius of a cylinder measured with a slide callipers are found to be 4.54 cm and 1.75 cm respectively. Calculate the volume of the cylinder.

The thickness of a glass plate is measured to be 2.17 mm, 2.17 mm and 2.18 mm at three different places. Find the average thickness of the plate from this data.

The length of the string of a simple pendulum is measured with a metre scale to be 90.0 cm. The radius of the bod plus the length of the hook is calculated to be 2.13 cm using measurements with a slide callipers. What is the effective length of the pendulum? (The effective length is defined as the distance between the point of suspension and the centre of the bob.)

## Chapter 2: Physics and Mathematics

#### H.C. Verma Concept of Physics Part-1 (2018-2019 Session) by H.C Verma

#### Textbook solutions for Class 12

## H.C. Verma solutions for Class 12 Physics chapter 2 - Physics and Mathematics

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Concepts covered in Class 12 Physics chapter 2 Physics and Mathematics are Fundamental Forces in Nature, Physics, Technology and Society, Concept of Physics, Scope and Excitement of Physics, Nature of Physical Laws.

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