HC Verma solutions for Concepts of Physics Vol. 1 [English] Class 11 and 12 chapter 2 - Physics and Mathematics [Latest edition]

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?

Which of the following two statements is more appropriate?
(a) Two forces are added using the triangle rule because force is a vector quantity.
(b) Force is a vector quantity because two forces are added using the triangle rule.

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

Let ε₁ and ε₂ be the angles made by  \[\vec{A}\] and -\[\vec{A}\] with the positive X-axis. Show that tan ε₁ = 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 ?

A vector is not changed if

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

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

The component of a vector is 

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

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

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

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

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.

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 \[\overrightarrow{OA}, \overrightarrow{BC} \text { and } \overrightarrow{DE}\].

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

Let A₁ A₂ A₃ A₄ A₅ A₆ A₁ 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 = 2x² 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 = i₀ e^−t/RC where i₀, 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 = i₀ e^−t/RC where i₀, R and C are constant parameters and t is time. Let i₀ = 2⋅00 A, R = 6⋅00 × 10⁵ 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 = 3x² + 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.)