Physics and Mathematics NEET Notes | EduRev

Physics Class 11

JEE : Physics and Mathematics NEET Notes | EduRev

The document Physics and Mathematics NEET Notes | EduRev is a part of the JEE Course Physics Class 11.
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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.

Ques 2: 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? 
Ans: No, it is not possible to obtain zero by adding two vectors of unequal magnitudes.
Example: Let us add two vectors Physics and Mathematics NEET Notes | EduRevof unequal magnitudes acting in opposite directions. The resultant vector is given by
Physics and Mathematics NEET Notes | EduRev
If two vectors are exactly opposite to each other, then 
Physics and Mathematics NEET Notes | EduRev
From the above equation, we can say that the resultant vector is zero (R = 0) when the magnitudes of the vectors  Physics and Mathematics NEET Notes | EduRevare 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 Physics and Mathematics NEET Notes | EduRevgiven these three vectors make an angle of 120° with each other. Consider the figure below: 
Physics and Mathematics NEET Notes | EduRev
Lets examine the components of the three vectors.
Physics and Mathematics NEET Notes | EduRev
Hence, proved.

Ques 3: Does the phrase "direction of zero vector" have physical significance? Discuss it terms of velocity, force etc.
Ans:
A zero vector has physical significance in physics, as the operations on the zero vector gives us a vector.
For any vector Physics and Mathematics NEET Notes | EduRev, assume that
Physics and Mathematics NEET Notes | EduRev
Again, for any real number λ  we have: 
Physics and Mathematics NEET Notes | EduRev
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.


Ques 4: Can you add three unit vectors to get a unit vector? Does your answer change if two unit vectors are along the coordinate axes? 
Ans: Yes we can add three unit vectors to get a unit vector.
No, the answer does not change if two unit vectors are along the coordinate axes. Assume three unit vectors Physics and Mathematics NEET Notes | EduRevalong the positive x-axis, negative x-axis and positive y-axis, respectively. Consider the figure given below:
Physics and Mathematics NEET Notes | EduRev
The magnitudes of the three unit vectors Physics and Mathematics NEET Notes | EduRevare the same, but their directions are different.
So, the resultant of Physics and Mathematics NEET Notes | EduRevis a zero vector.
Now,Physics and Mathematics NEET Notes | EduRev (Using the property of zero vector)
∴ The resultant of three unit vectors Physics and Mathematics NEET Notes | EduRevis a unit vector Physics and Mathematics NEET Notes | EduRev

Page No. 28
Ques 5: Can we have physical quantities having magnitude and direction which are not vectors?
Ans: Yes, there are physical quantities like electric current and pressure which have magnitudes and directions, but are not considered as vectors because they do not follow vector laws of addition.

Ques 6: 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.

Ans: Two forces are added using triangle rule, because force is a vector quantity. This statement is more appropriate, because we know that force is a vector quantity and only vectors are added using triangle rule.

Ques 7: Can you add two vectors representing physical quantities having different dimensions? Can you multiply two vectors representing physical quantities having different dimensions?
Ans: No, we cannot add two vectors representing physical quantities of different dimensions. However, we can multiply two vectors representing physical quantities with different dimensions.
Example: Torque, Physics and Mathematics NEET Notes | EduRev

Ques 8: Can a vector have zero component along a line and still have nonzero magnitude?
Ans: Yes, a vector can have zero components along a line and still have a nonzero magnitude.
Example: Consider a two dimensional vectorPhysics and Mathematics NEET Notes | EduRevThis 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 Physics and Mathematics NEET Notes | EduRev

Ques 9: Let ε1 and ε2 be the angles made byPhysics and Mathematics NEET Notes | EduRev with the positive X-axis. Show that tan ε1 = tan ε2.Thus, giving tan ε does not uniquely determine the direction of Physics and Mathematics NEET Notes | EduRev
Ans: The direction ofPhysics and Mathematics NEET Notes | EduRevis opposite to Physics and Mathematics NEET Notes | EduRev So, if vector Physics and Mathematics NEET Notes | EduRevmake the angles ε1 and ε2 with the X-axis, respectively, then ε1 is equal to ε2 as shown in the figure:
Physics and Mathematics NEET Notes | EduRev
Here, tan ε1 = tan ε2
Because these are alternate angles.
Thus, giving tan ε does not uniquely determine the direction of  Physics and Mathematics NEET Notes | EduRev


Ques 10: Is the vector sum of the unit vectors Physics and Mathematics NEET Notes | EduRev a unit vector? If no, can you multiply this sum by a scalar number to get a unit vector?
Ans: 
No, the vector sum of the unit vectors Physics and Mathematics NEET Notes | EduRevis not a unit vector, because the magnitude of the resultant ofPhysics and Mathematics NEET Notes | EduRevis not one.
Magnitude of the resultant vector is given by
Physics and Mathematics NEET Notes | EduRev 
Yes, we can multiply this resultant vector by a scalar number Physics and Mathematics NEET Notes | EduRev to get a unit vector.

Ques 11: LetPhysics and Mathematics NEET Notes | EduRev Write a vector Physics and Mathematics NEET Notes | EduRev such that Physics and Mathematics NEET Notes | EduRevbut A = B.
Ans: 
A vector Physics and Mathematics NEET Notes | EduRevsuch that Physics and Mathematics NEET Notes | EduRevbut A = B are as follows:
Physics and Mathematics NEET Notes | EduRev

Ques 12: Can you have Physics and Mathematics NEET Notes | EduRevwith A ≠ 0 and B ≠ 0? What if one of the two vectors is zero?
Ans:
No, we cannot have  Physics and Mathematics NEET Notes | EduRev with A ≠ 0 and B ≠ 0. This is because the left hand side of the given equation gives a vector quantity, while the right hand side gives a scalar quantity. However, if one of the two vectors is zero, then both the sides will be equal to zero and the relation will be valid.

Ques 13: If Physics and Mathematics NEET Notes | EduRev can you say that (a) Physics and Mathematics NEET Notes | EduRev
Ans: If Physics and Mathematics NEET Notes | EduRev  then both the vectors are either parallel or antiparallel, i.e., the angle between the vectors is either Physics and Mathematics NEET Notes | EduRev
Physics and Mathematics NEET Notes | EduRev
Both the conditions can be satisfied:
(a) Physics and Mathematics NEET Notes | EduRevi.e., the two vectors are equal in magnitude and parallel to each other
(b) Physics and Mathematics NEET Notes | EduRevi.e., the two vectors are unequal in magnitude and parallel or anti parallel to each other

Ques 14: Let Physics and Mathematics NEET Notes | EduRevDo we have Physics and Mathematics NEET Notes | EduRev? Can we say Physics and Mathematics NEET Notes | EduRev
Ans:

Physics and Mathematics NEET Notes | EduRev then we have Physics and Mathematics NEET Notes | EduRevby putting the value of scalar k as −1.5-1.5.
However, we cannot say that Physics and Mathematics NEET Notes | EduRevbecause a vector cannot be divided by other vectors, as vector division is not possible. 

Ques 15: A vector is not changed if
(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.

Ans: (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 Physics and Mathematics NEET Notes | EduRevdirected towards the north be 5 metres. If we slide it parallel to itself, then the direction and magnitude will not change.

Ques 16: Which of the sets given below may represent the magnitudes of three vectors adding to zero?
(a) 2, 4, 8
(b) 4, 8, 16
(c) 1, 2, 1
(d) 0.5, 1, 2

Ans: (c) 1, 2, 1
1,2 and 1 may represent the magnitudes of three vectors adding to zero.For example one of the vector of length 1 should make an angle of 135∘ with x axis and the other vector of length 1 makes an angle of 225∘ with x axis. The third vector of length 2 should lie along x axis.

Ques 17: The resultant of Physics and Mathematics NEET Notes | EduRevmakes an angle α with Physics and Mathematics NEET Notes | EduRevand β with Physics and Mathematics NEET Notes | EduRev
(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 A > B
The resultant of two vectors is closer to the vector with the greater magnitude.
Thus, α < β if A > B

Ques 18: The component of a vector is
(a) always less than its magnitude
(b) always greater than its magnitude
(c) always equal to its magnitude
(d) None of these. 

Ans: (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.

Ques 19: A vector Physics and Mathematics NEET Notes | EduRevpoints vertically upward and Physics and Mathematics NEET Notes | EduRevpoints towards the north. The vector product Physics and Mathematics NEET Notes | EduRevis
(a) along the west
(b) along the east
(c) zero
(d) vertically downward.

Ans: (a) along the west
The vector product Physics and Mathematics NEET Notes | EduRev will point towards the west. We can determine this direction using the right hand thumb rule.

Ques 20: The radius of a circle is stated as 2.12 cm. Its area should be written as
(a) 14 cm2
(b) 14.1 cm2
(c) 14.11 cm2
(d) 14.1124 cm

Ans: (b) 14.1 cm2
Area of a circle, Physics and Mathematics NEET Notes | EduRev
On putting the values, we get: 
Physics and Mathematics NEET Notes | EduRev
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.

Ques 21: 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.

Ans: (a) the value of a scalar
(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.

Ques 22: Physics and Mathematics NEET Notes | EduRev
(a) Physics and Mathematics NEET Notes | EduRev is always greater than Physics and Mathematics NEET Notes | EduRev
(b) It is possible to have Physics and Mathematics NEET Notes | EduRev
(c) C is always equal to A + B
(d) C is never equal to A + B. 
Ans: 
(b) It is possible to have Physics and Mathematics NEET Notes | EduRev
Statements (a), (c) and (d) are incorrect.
Given:  Physics and Mathematics NEET Notes | EduRev
Here, the magnitude of the resultant vector may or may not be equal to or less than the magnitudes of Physics and Mathematics NEET Notes | EduRevor the sum of the magnitudes of both the vectors if the two vectors are in opposite directions.

Ques 23: Let the angle between two nonzero vectors Physics and Mathematics NEET Notes | EduRevbe 120° and its resultant be Physics and Mathematics NEET Notes | EduRev
(a) C must be equal to Physics and Mathematics NEET Notes | EduRev
(b) C must be less than Physics and Mathematics NEET Notes | EduRev
(c) C must be greater than  Physics and Mathematics NEET Notes | EduRev 
(d) C may be equal to Physics and Mathematics NEET Notes | EduRev 
Ans: (b) C must be less than Physics and Mathematics NEET Notes | EduRev 
Here, we have three vector A, B and C.
Physics and Mathematics NEET Notes | EduRev
Using the resultant property Physics and Mathematics NEET Notes | EduRevwe get:
Physics and Mathematics NEET Notes | EduRev
Since cosine is negative in the second quadrant, C must be less than Physics and Mathematics NEET Notes | EduRev 

Ques 24:  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.

Ans: (a) is equal to the sum of the x-components 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.

Ques 25: The magnitude of the vector product of two vectors Physics and Mathematics NEET Notes | EduRevmay be
(a) greater than AB
(b) equal to AB
(c) less than AB
(d) equal to zero.
Ans: 
(b) equal to AB
(c) less than AB
(d) equal to zero.
The magnitude of the vector product of two vectors Physics and Mathematics NEET Notes | EduRev  may be less than or equal to AB, or equal to zero, but cannot be greater than AB.

Page No. 29
Ques 26: A vector Physics and Mathematics NEET Notes | EduRev makes an angle of 20° and Physics and Mathematics NEET Notes | EduRevmakes an angle of 110° with the X-axis. The magnitudes of these vectors are 3 m and 4 m respectively. Find th
e resultant.
Ans:

Physics and Mathematics NEET Notes | EduRev 
From the above figure, we have:
Angle between Physics and Mathematics NEET Notes | EduRev = 110° − 20° = 90°
Physics and Mathematics NEET Notes | EduRev
Magnitude of the resultant vector is given by
Physics and Mathematics NEET Notes | EduRev
Let β be the angle between Physics and Mathematics NEET Notes | EduRev
Physics and Mathematics NEET Notes | EduRev

Ques 27: Let Physics and Mathematics NEET Notes | EduRev 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.
Ans: Angle between Physics and Mathematics NEET Notes | EduRev θ = 60° − 30° = 30°
Physics and Mathematics NEET Notes | EduRev=10 units
The magnitude of the resultant vector is given by
Physics and Mathematics NEET Notes | EduRev
Physics and Mathematics NEET Notes | EduRev
Let β be the angle between Physics and Mathematics NEET Notes | EduRev
Physics and Mathematics NEET Notes | EduRev
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. 

Ques 28: Add vectors Physics and Mathematics NEET Notes | EduReveach having magnitude of 100 unit and inclined to the X-axis at angles 45°, 135° and 315° respectively.
Ans: First, we will find the components of the vector along the x-axis and y-axis. Then we will find the resultant x and y-components. 
Physics and Mathematics NEET Notes | EduRev
Physics and Mathematics NEET Notes | EduRev
Now, y-component of Physics and Mathematics NEET Notes | EduRev
y-component of  Physics and Mathematics NEET Notes | EduRev
y-component of  Physics and Mathematics NEET Notes | EduRev
Resultant y-componentPhysics and Mathematics NEET Notes | EduRev
Magnitude of the resultant Physics and Mathematics NEET Notes | EduRev
Physics and Mathematics NEET Notes | EduRev
Angle made by the resultant vector with the x-axis is given by 
Physics and Mathematics NEET Notes | EduRev
∴ The magnitude of the resultant vector is 100 units and it makes an angle of 45° with the x-axis. 

Ques 28: Physics and Mathematics NEET Notes | EduRev (a) Find the magnitudes of Physics and Mathematics NEET Notes | EduRevand (d) Physics and Mathematics NEET Notes | EduRev
Ans:

Given: Physics and Mathematics NEET Notes | EduRev
(a) Magnitude of Physics and Mathematics NEET Notes | EduRevis given by Physics and Mathematics NEET Notes | EduRev
(b)  Magnitude of Physics and Mathematics NEET Notes | EduRevis  given by Physics and Mathematics NEET Notes | EduRev
Physics and Mathematics NEET Notes | EduRev
∴ Magnitude of vector Physics and Mathematics NEET Notes | EduRevis given by Physics and Mathematics NEET Notes | EduRev
Physics and Mathematics NEET Notes | EduRev
∴ Magnitude of vector Physics and Mathematics NEET Notes | EduRevis given by Physics and Mathematics NEET Notes | EduRev

Ques 29: 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 Physics and Mathematics NEET Notes | EduRev
Physics and Mathematics NEET Notes | EduRev
Ans: 
First, let us find the components of the vectors along the x and y-axes. Then we will find the resultant x and y-components.  
x-component of Physics and Mathematics NEET Notes | EduRev
x-component of Physics and Mathematics NEET Notes | EduRev
x-component of Physics and Mathematics NEET Notes | EduRev
y-component of Physics and Mathematics NEET Notes | EduRev
y-component of Physics and Mathematics NEET Notes | EduRev
y-component of Physics and Mathematics NEET Notes | EduRev
x-component of resultant Rx=Physics and Mathematics NEET Notes | EduRev
y-component of resultant Ry = Physics and Mathematics NEET Notes | EduRev
Physics and Mathematics NEET Notes | EduRev
If it makes an angle α with the positive x-axis, then
Physics and Mathematics NEET Notes | EduRev

Ques 30: 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.
Ans:
Let the two vectors be Physics and Mathematics NEET Notes | EduRev
Physics and Mathematics NEET Notes | EduRev
(a) If the resultant vector is 1 unit, then
Physics and Mathematics NEET Notes | EduRev
Squaring both sides, we get:
Physics and Mathematics NEET Notes | EduRev
Hence, the angle between them is 180°.
(b) If the resultant vector is 5 units, then
Physics and Mathematics NEET Notes | EduRev
Squaring both sides, we get:
Physics and Mathematics NEET Notes | EduRev
Hence, the angle between them is 90°.
(c) If the resultant vector is 7 units, then
Physics and Mathematics NEET Notes | EduRev
Squaring both sides, we get:
Physics and Mathematics NEET Notes | EduRev
Hence, the angle between them is 0°.

Ques 31: 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.
Ans: The displacement of the car is represented by  Physics and Mathematics NEET Notes | EduRev
Physics and Mathematics NEET Notes | EduRev
Magnitude of Physics and Mathematics NEET Notes | EduRev is given by 
Physics and Mathematics NEET Notes | EduRev
Hence, the displacement of the car is 6.02 km along the direction Physics and Mathematics NEET Notes | EduRevwith positive the x-axis. 

Ques 32: 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.
Ans: 

Physics and Mathematics NEET Notes | EduRev
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: 
Physics and Mathematics NEET Notes | EduRev
Also, in ∆DCE, we have: 
Physics and Mathematics NEET Notes | EduRev
From (i) and (ii), we get: 
Physics and Mathematics NEET Notes | EduRev
(a) In ∆ABC, we have: 
Physics and Mathematics NEET Notes | EduRev
(b) In ∆CDE, we have:
Physics and Mathematics NEET Notes | EduRev
CD = 4 ft 
Physics and Mathematics NEET Notes | EduRev
(c) In ∆AGE, we have: 
Physics and Mathematics NEET Notes | EduRev

Ques 33: 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.
Ans: 
Displacement vector of the mosquito, Physics and Mathematics NEET Notes | EduRev
Physics and Mathematics NEET Notes | EduRev
(a) Magnitude of displacement= Physics and Mathematics NEET Notes | EduRev
Physics and Mathematics NEET Notes | EduRev
(b) The components of the displacement vector are 7 ft, 4 ft and 3 ft along the X, Y and Z-axes, respectively. 

Ques 34: Suppose  Physics and Mathematics NEET Notes | EduRev is a vector of magnitude 4.5 units due north. What is the vector (a) Physics and Mathematics NEET Notes | EduRev
Ans: Given: Physics and Mathematics NEET Notes | EduRev  is a vector of magnitude 4.5 units due north.
Case (a):
Physics and Mathematics NEET Notes | EduRev
Physics and Mathematics NEET Notes | EduRevis a vector of magnitude 13.5 units due north.
Case (b):
Physics and Mathematics NEET Notes | EduRev
Physics and Mathematics NEET Notes | EduRevis a vector of magnitude 6 units due south.

Ques 35: 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.
Ans: Let the two vectors be Physics and Mathematics NEET Notes | EduRev
Angle between the vectors, θ = 60°
(a) The scalar product of two vectors is given by  Physics and Mathematics NEET Notes | EduRev
Physics and Mathematics NEET Notes | EduRev
(b) The vector product of two vectors is given by Physics and Mathematics NEET Notes | EduRev
Physics and Mathematics NEET Notes | EduRev

Ques 36: 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
Physics and Mathematics NEET Notes | EduRev
Ans: According to the polygon law of vector addition, the resultant of these six vectors is zero.Here, a = b = c = d = e = f (magnitudes), as it is a regular hexagon. A regular polygon has all sides equal to each other.
So, Physics and Mathematics NEET Notes | EduRev
[As the resultant is zero, the x-component of resultant Rx is zero]
Physics and Mathematics NEET Notes | EduRev
Physics and Mathematics NEET Notes | EduRev
Note: Similarly, it can be proven that Physics and Mathematics NEET Notes | EduRev

Ques 37: Let  Physics and Mathematics NEET Notes | EduRev Find the angle between them.
Ans: 
We have:
Physics and Mathematics NEET Notes | EduRev 
Using scalar product, we can find the angle between vectors  Physics and Mathematics NEET Notes | EduRev
i.e., 
Physics and Mathematics NEET Notes | EduRev
Physics and Mathematics NEET Notes | EduRev

Ques 38: Prove that  Physics and Mathematics NEET Notes | EduRev
Ans:
To prove:
Physics and Mathematics NEET Notes | EduRev
Proof: Vector product is given by Physics and Mathematics NEET Notes | EduRev
Physics and Mathematics NEET Notes | EduRevis a vector which is perpendicular to the plane containing Physics and Mathematics NEET Notes | EduRev
This implies that it is also perpendicular to Physics and Mathematics NEET Notes | EduRevWe know that the dot product of two perpendicular vectors is zero. 
Physics and Mathematics NEET Notes | EduRev
Hence, proved. 

Ques 39: Physics and Mathematics NEET Notes | EduRev
Ans: 
Given:  Physics and Mathematics NEET Notes | EduRev
Physics and Mathematics NEET Notes | EduRev
The vector product of Physics and Mathematics NEET Notes | EduRevcan be obtained as follows: 
Physics and Mathematics NEET Notes | EduRev

Ques 40: If Physics and Mathematics NEET Notes | EduRevare mutually perpendicular, show that Physics and Mathematics NEET Notes | EduRevIs the converse true?
Ans: 
Given:  Physics and Mathematics NEET Notes | EduRevare mutually perpendicular. Physics and Mathematics NEET Notes | EduRevis a vector with its direction perpendicular to the plane containing Physics and Mathematics NEET Notes | EduRev
Physics and Mathematics NEET Notes | EduRev
∴ The angle between Physics and Mathematics NEET Notes | EduRevis either 0° or 180°.
i.e., Physics and Mathematics NEET Notes | EduRev
However, the converse is not true. For example, if two of the vectors are parallel, then also,  
Physics and Mathematics NEET Notes | EduRev
So, they need not be mutually perpendicular. 

Ques 41: 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 Physics and Mathematics NEET Notes | EduRev is independent of the position P.
Ans: 
The particle moves on the straight line XX' at a uniform speed ν.
Physics and Mathematics NEET Notes | EduRev
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 Physics and Mathematics NEET Notes | EduRev remain constant. 
Physics and Mathematics NEET Notes | EduRevis independent of the position P. 

Ques 42: The force on a charged particle due to electric and magnetic fields is given by  Physics and Mathematics NEET Notes | EduRev Suppose Physics and Mathematics NEET Notes | EduRevis along the X-axis and Physics and Mathematics NEET Notes | EduRevalong 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?
Ans: According to the problem, the net electric and magnetic forces on the particle should be zero. 
Physics and Mathematics NEET Notes | EduRev 
So, the direction of Physics and Mathematics NEET Notes | EduRevshould be opposite to the direction of Physics and Mathematics NEET Notes | EduRevHence, Physics and Mathematics NEET Notes | EduRevshould be along the positive z-direction.
Again, Physics and Mathematics NEET Notes | EduRev
Physics and Mathematics NEET Notes | EduRev
For ν to be minimum, θ=90° and, thus Physics and Mathematics NEET Notes | EduRev
So, the particle must be projected at a minimum speed of Physics and Mathematics NEET Notes | EduRev along the z-axis. 

Ques 43: Give an example for which Physics and Mathematics NEET Notes | EduRev
Ans:
 To prove: Physics and Mathematics NEET Notes | EduRev
Suppose that Physics and Mathematics NEET Notes | EduRevis perpendicular to  Physics and Mathematics NEET Notes | EduRev is along the west direction. 
Also, Physics and Mathematics NEET Notes | EduRevis perpendicular to Physics and Mathematics NEET Notes | EduRevare along the south and north directions, respectively. 
Physics and Mathematics NEET Notes | EduRev
Physics and Mathematics NEET Notes | EduRevis perpendicular to Physics and Mathematics NEET Notes | EduRev so there dot or scalar product is zero.
i.e., 

Physics and Mathematics NEET Notes | EduRev
Physics and Mathematics NEET Notes | EduRevis perpendicular toPhysics and Mathematics NEET Notes | EduRev  so there dot or scalar product is zero. 
i.e., Physics and Mathematics NEET Notes | EduRev
Physics and Mathematics NEET Notes | EduRev
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 Physics and Mathematics NEET Notes | EduRev
Physics and Mathematics NEET Notes | EduRev
Ans: 
Note: Students should draw the graph y = 2x2 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.
Physics and Mathematics NEET Notes | EduRev
The above can be checked as follows:
Slope Physics and Mathematics NEET Notes | EduRev
Physics and Mathematics NEET Notes | EduRev
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 Physics and Mathematics NEET Notes | EduRev find approximately the change in y. 
Ans: y = sin x   ...(i)
Physics and Mathematics NEET Notes | EduRev
Now, consider a small increment ∆x in x. 
Then y + ∆y = sin (x + ∆x)   ...(ii)
Here, ∆y is the small change in y.
Subtracting (ii) from (i), we get:
y = sin (x + ∆x) − sin x 

Physics and Mathematics NEET Notes | EduRev
 = 0.0157 

Ques 46: The electric current in a charging R−C circuit is given by Physics and Mathematics NEET Notes | EduRevR 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.
Ans: Given:Physics and Mathematics NEET Notes | EduRev 
∴ Rate of change of current Physics and Mathematics NEET Notes | EduRev
Physics and Mathematics NEET Notes | EduRev
On applying the conditions given in the questions, we get: 
Physics and Mathematics NEET Notes | EduRev

Ques 47: The electric current in a discharging R−C circuit is given by Physics and Mathematics NEET Notes | EduRevR and C are constant parameters and t is time. Let 

Physics and Mathematics NEET Notes | EduRev(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. 
Ans: Electric current in a discharging R-C circuit is given by the below equation:
i0 ⋅ et/RC   ...(i)
Here, i0 = 2.00 A
R = 6 × 105 Ω
C = 0.0500 × 10−6 F
    = 5 × 10−7 F 

On substituting the values of RC and i0 in equation (i), we get:
i = 2.0 et/0.3   ...(ii)
According to the question, we have:
(a) current at t = 0.3 s 
Physics and Mathematics NEET Notes | EduRev
(b) rate of change of current at t = 0.3 s 
Physics and Mathematics NEET Notes | EduRev
When t = 0.3 s, we have: 
Physics and Mathematics NEET Notes | EduRev
(c) approximate current at t = 0.31 s 
Physics and Mathematics NEET Notes | EduRev

Page No.30
Ques 48: Find the area bounded under the curve y = 3x2 + 6x + 7 and the X-axis with the ordinates at x = 5 and x = 10.
Ans: The given equation of the curve is y = 3x2 + 6x + 7.
Physics and Mathematics NEET Notes | EduRev
The area bounded by the curve and the X-axis with coordinates x1 = 5 and x2 = 10 is given by Physics and Mathematics NEET Notes | EduRev
Physics and Mathematics NEET Notes | EduRev
Physics and Mathematics NEET Notes | EduRev

Ques 49: Find the area enclosed by the curve y = sin x and the X-axis between x = 0 and x = π.
Ans: The given equation of the curve is y = sin x. 
Physics and Mathematics NEET Notes | EduRev 
The required area can found by integrating y w.r.t x within the proper limits. 
Physics and Mathematics NEET Notes | EduRev
 =1+1=2 sq. unit

Ques 50: Find the area bounded by the curve y = ex, the X-axis and the Y-axis. 
Ans: 
The given function is y = ex.
When x = 0, y = e−0 = 1
When x increases, the value of y decrease. Also, only when x = ∞, = 0
So, the required area can be determined by integrating the function from 0 to ∞. 
Physics and Mathematics NEET Notes | EduRev
=[0−1]=1 sq. unit

Ques 51: 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 ab and L. 
Ans: ρ = mass/length = a + bx
So, the SI unit of ρ is kg/m.
(a)
SI unit of a = kg/m
SI unit of b = kg/m2
(From the principle of homogeneity of dimensions)
(b) Let us consider a small element of length dx at a distance x from the origin as shown in the figure given below: 
Physics and Mathematics NEET Notes | EduRev
dm = mass of the element
      = ρdx
      = (a + dxdx
∴ Mass of the rod = ∫ dm 
Physics and Mathematics NEET Notes | EduRev 

Ques 52: The momentum p of a particle changes with time t according to the relation Physics and Mathematics NEET Notes | EduRev
Ans: 
According to the question, we have: 
Physics and Mathematics NEET Notes | EduRev
Momentum is zero at time, t = 0
Now, dp = [(10 N) + (2 Ns−1)t]dt
On integrating the above equation, we get:
Physics and Mathematics NEET Notes | EduRev
Physics and Mathematics NEET Notes | EduRev

Ques 53: The changes in a function y and the independent variable x are related as Physics and Mathematics NEET Notes | EduRev Find y as a function of x. 
Ans: Changes in a function of y and the independent variable x are related as follows: 
Physics and Mathematics NEET Notes | EduRev
Integrating of both sides, we get:
dy = ∫x2 dx 
Physics and Mathematics NEET Notes | EduRev where c is a constant
∴ y as a function of x is represented by Physics and Mathematics NEET Notes | EduRev

Ques 54: Write the number of significant digits in (a) 1001, (b) 100.1, (c) 100.10, (d) 0.001001.
Ans: (a) 1001
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 


Ques 55: A metre scale is graduated at every millimetre. How many significant digits will be there in a length measurement with this scale?
Ans: The metre scale is graduated at every millimetre.
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. 

Ques 56: Round the following numbers to 2 significant digits.
(a) 3472, (b) 84.16. (c)2.55 and (d) 28.5 
Ans:
(a) In 3472, 7 comes after the digit 4. Its value is greater than 5. So, the next two digits are neglected and 4 is increased by one.
∴ The value becomes 3500.
(b) 84
(c) 2.6
(d) 29 

Ques 57: 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.
Ans: Length of the cylinder, l = 4.54 cm
Radius of the cylinder, r = 1.75 cm
Volume of the cylinder, V = π2l

= (π) × (4.54) × (1.75)2 
Physics and Mathematics NEET Notes | EduRev
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 = πr2l
= (3.14) × (1.75) × (1.75) × (4.54)
= 43.6577 cm3
Since the volume is to be rounded off to 3 significant digits, we have:
V = 43.7 cm3 

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
Physics and Mathematics NEET Notes | EduRev
=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: 
Physics and Mathematics NEET Notes | EduRev
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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