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Page No. 27

**Ques 1:**** ****Is a vector necessarily changed if it is rotated through an angle? ****Ans:** Yes. A vector is defined by its magnitude and direction, so a vector can be changed by changing its magnitude and direction. If we rotate it through an angle, its direction changes and we can say that the vector has changed.

Example: Let us add two vectors of unequal magnitudes acting in opposite directions. The resultant vector is given by

If two vectors are exactly opposite to each other, then

From the above equation, we can say that the resultant vector is zero (R = 0) when the magnitudes of the vectors are equal (A = B) and both are acting in the opposite directions.

Yes, it is possible to add three vectors of equal magnitudes and get zero.

Lets take three vectors of equal magnitudes given these three vectors make an angle of 120° with each other. Consider the figure below:

Lets examine the components of the three vectors.

Hence, proved.

Ans:

For any vector , assume that

Again, for any real number λ we have:

The significance of a zero vector can be better understood through the following examples:

The displacement vector of a stationary body for a time interval is a zero vector.

Similarly, the velocity vector of the stationary body is a zero vector.

When a ball, thrown upward from the ground, falls to the ground, the displacement vector is a zero vector, which defines the displacement of the ball.

No, the answer does not change if two unit vectors are along the coordinate axes. Assume three unit vectors along the positive

The magnitudes of the three unit vectors are the same, but their directions are different.

So, the resultant of is a zero vector.

Now, (Using the property of zero vector)

∴ The resultant of three unit vectors is a unit vector

Page No. 28

(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.

Example: Consider a two dimensional vectorThis vector has zero components along a line lying along the Y-axis and a nonzero component along the X-axis. The magnitude of the vector is also nonzero.

Now, magnitude of

Ans: The direction ofis opposite to So, if vector make the angles ε1 and ε2 with the X-axis, respectively, then ε1 is equal to ε2 as shown in the figure:

Here, tan ε

Because these are alternate angles.

Thus, giving tan ε does not uniquely determine the direction of

Ans:

Magnitude of the resultant vector is given by

Yes, we can multiply this resultant vector by a scalar number to get a unit vector.

Ans:

Ans:

Both the conditions can be satisfied:

(a) i.e., the two vectors are equal in magnitude and parallel to each other

(b) i.e., the two vectors are unequal in magnitude and parallel or anti parallel to each other

Ans:

However, we cannot say that because a vector cannot be divided by other vectors, as vector division is not possible.

(a) it is rotated through an arbitrary angle

(b) it is multiplied by an arbitrary scalar

(c) it is cross multiplied by a unit vector

(d) it is slid parallel to itself.

A vector is defined by its magnitude and direction. If we slide it to a parallel position to itself, then none of the given parameters, which define the vector, will change.

Let the magnitude of a displacement vector directed towards the north be 5 metres. If we slide it parallel to itself, then the direction and magnitude will not change.

(a) 2, 4, 8

(b) 4, 8, 16

(c) 1, 2, 1

(d) 0.5, 1, 2

(a) α < β

(b) α < β if A < B

(c) α < β if A > B

(d) α < β if A = B

Ans:

(c) α < β if

The resultant of

Thus, α < β if

(a) always less than its magnitude

(b) always greater than its magnitude

(c) always equal to its magnitude

(d) None of these.

All the given options are incorrect. The component of a vector may be less than, greater than or equal to its magnitude, depending upon the vector and its components.

(a) along the west

(b) along the east

(c) zero

(d) vertically downward.

The vector product will point towards the west. We can determine this direction using the right hand thumb rule.

(b) 14.1 cm

(c) 14.11 cm

(d) 14.1124 cm

Area of a circle,

On putting the values, we get:

The rules to determine the number of significant digits says that in the multiplication of two or more numbers, the number of significant digits in the answer should be equal to that of the number with the minimum number of significant digits. Here, 2.12 cm has a minimum of three significant digits. So, the answer must be written in three significant digits.

(a) the value of a scalar

(b) component of a vector

(c) a vector

(d) the magnitude of a vector.

(c) a vector

(d) the magnitude of a vector

The value of a scalar, a vector and the magnitude of a vector do not depend on a given set of coordinate axes with different orientation. However, components of a vector depend on the orientation of the axes.

(a) is always greater than

(b) It is possible to have

(c)

(d)

Ans:

Statements (a), (c) and (d) are incorrect.

Given:

Here, the magnitude of the resultant vector may or may not be equal to or less than the magnitudes of or the sum of the magnitudes of both the vectors if the two vectors are in opposite directions.

(a) C must be equal to

Here, we have three vector A, B and C.

Using the resultant property we get:

Since cosine is negative in the second quadrant, C must be less than

(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.

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

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

The x-component of the resultant of several vectors cannot be greater than the sum of the magnitudes of the vectors.

(b) equal to AB

(c) less than AB

(d) equal to zero.

Ans:

(c) less than AB

(d) equal to zero.

The magnitude of the vector product of two vectors may be less than or equal to AB, or equal to zero, but cannot be greater than AB.

Ques 26: A vector makes an angle of 20° and makes an angle of 110° with the X-axis. The magnitudes of these vectors are 3 m and 4 m respectively. Find th

Ans:

From the above figure, we have:

Angle between = 110° − 20° = 90°

Magnitude of the resultant vector is given by

Let β be the angle between

=10 units

The magnitude of the resultant vector is given by

Let β be the angle between

Angle made by the resultant vector with the X-axis = 15° + 30° = 45°

∴ The magnitude of the resultant vector is 17.3 and it makes angle of 45° with the X-axis.

Now,

Resultant y-component

Magnitude of the resultant

Angle made by the resultant vector with the

∴ The magnitude of the resultant vector is 100 units and it makes an angle of 45° with the

Ans:

Given:

(a) Magnitude of is given by

(b) Magnitude of is given by

∴ Magnitude of vector is given by

∴ Magnitude of vector is given by

Ans:

x-component of

x-component of

x-component of

y-component of

y-component of

y-component of

x-component of resultant R

y-component of resultant R

If it makes an angle α with the positive x-axis, then

Ans:

(a) If the resultant vector is 1 unit, then

Squaring both sides, we get:

Hence, the angle between them is 180°.

(b) If the resultant vector is 5 units, then

Squaring both sides, we get:

Hence, the angle between them is 90°.

(c) If the resultant vector is 7 units, then

Squaring both sides, we get:

Hence, the angle between them is 0°.

Magnitude of is given by

Hence, the displacement of the car is 6.02 km along the direction with positive the

Ans:

Consider that the queen is initially at point A as shown in the figure.

Let AB be x ft.

So, DE = (2 −- x) ft

In ∆ABC, we have:

Also, in ∆DCE, we have:

From (i) and (ii), we get:

(a) In ∆ABC, we have:

(b) In ∆CDE, we have:

CD = 4 ft

(c) In ∆AGE, we have:

Ans:

(a) Magnitude of displacement=

(b) The components of the displacement vector are 7 ft, 4 ft and 3 ft along the X, Y and Z-axes, respectively.

Case (a):

is a vector of magnitude 13.5 units due north.

Case (b):

is a vector of magnitude 6 units due south.

Angle between the vectors, θ = 60°

(a) The scalar product of two vectors is given by

(b) The vector product of two vectors is given by

So,

[As the resultant is zero, the x-component of resultant Rx is zero]

Ans:

Using scalar product, we can find the angle between vectors

i.e.,

Ans:

is a vector which is perpendicular to the plane containing

This implies that it is also perpendicular to We know that the dot product of two perpendicular vectors is zero.

Hence, proved.

Ans:

The vector product of can be obtained as follows:

Ans:

∴ The angle between is either 0° or 180°.

i.e.,

However, the converse is not true. For example, if two of the vectors are parallel, then also,

So, they need not be mutually perpendicular.

Ans:

This product is always equal to the perpendicular distance from point O. Also, the direction of this product remains constant.

So, irrespective of the the position of the particle, the magnitude and direction of remain constant.

is independent of the position P.

So, the direction of should be opposite to the direction of Hence, should be along the positive z-direction.

Again,

For ν to be minimum, θ=90° and, thus

So, the particle must be projected at a minimum speed of along the

Suppose that is perpendicular to is along the west direction.

Also, is perpendicular to are along the south and north directions, respectively.

is perpendicular to so there dot or scalar product is zero.

i.e.,

is perpendicular to so there dot or scalar product is zero.

i.e.,

Hence, proved. **Ques 44: Draw a graph from the following data. Draw tangents at x = 2, 6 and 8. Find the slopes of these tangents. Verify that the curve draw is y = 2x2 and the slope of tangent is ****Ans: ****Note: **Students should draw the graph *y* = 2*x*^{2} on a graph paper for results.

To find a slope at any point, draw a tangent at the point and extend the line to meet the *x*-axis. Then find tan θ as shown in the figure.

The above can be checked as follows:

Slope

Here,* x* = *x*-coordinate of the point where the slope is to be measured **Ques 45: A curve is represented by y = sin x. If x is changed from find approximately the change in y. **

Now, consider a small increment ∆

Then

Here, ∆

Subtracting (ii) from (i), we get:

∆

= 0.0157

∴ Rate of change of current

On applying the conditions given in the questions, we get:

**(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. **

i

Here,

= 5 × 10

On substituting the values of

According to the question, we have:

(a) current at

(b) rate of change of current at

When

(c) approximate current at

Page No.30

The area bounded by the curve and the X-axis with coordinates

The required area can found by integrating

=1+1=2 sq. unit

Ans:

When

When

So, the required area can be determined by integrating the function from 0 to ∞.

=−[0−1]=1 sq. unit

So, the SI unit of ρ is kg/m.

(a)

SI unit of

SI unit of

(From the principle of homogeneity of dimensions)

(b) Let us consider a small element of length

= ρ

= (

∴ Mass of the rod = ∫ d

Ans:

Momentum is zero at time,

Now,

On integrating the above equation, we get:

Integrating of both sides, we get:

∫

where

∴ y as a function of x is represented by

Number of significant digits = 4

(b) 100.1

Number of significant digits = 4

(c) 100.10

Number of significant digits = 5

(d) 0.001001

Number of significant digits = 4

i.e., 1 m = 1000 mm

The minimum number of significant digits may be one (e.g., for measurements like 4 mm and 6 mm) and the maximum number of significant digits may be 4 (e.g., for measurements like 1000 mm). Hence, the number of significant digits may be 1, 2, 3 or 4.

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

Ans:

∴ The value becomes 3500.

(b) 84

(c) 2.6

(d) 29

Radius of the cylinder,

Volume of the cylinder, V = π

= (π) × (4.54) × (1.75)^{2}

The minimum number of significant digits in a particular term is three. Therefore, the result should have three significant digits, while the other digits should be rounded off.

∴ Volume, V = πr^{2}l

= (3.14) × (1.75) × (1.75) × (4.54)

= 43.6577 cm^{3}

Since the volume is to be rounded off to 3 significant digits, we have:*V* = 43.7 cm^{3} **Ques 58: 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.****Ans: **Average thickness

=2.1733 mm

∴ Rounding off to three significant digits, the average thickness becomes 2.17 mm. **Ques 59: 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.) ****Ans: **Consider the figure shown below:

Actual effective length = (90.0 + 2.13) cm

However, in the measurement 90.0 cm, the number of significant digits is only two.

So, the effective length should contain only two significant digits.

i.e., effective length = 90.0 + 2.13 = 92.1 cm.** **

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