Yearly Archives: 2021

In This Post we are  providing Chapter-14 Oscillations NCERT MCQ for Class 11 Physics which will be beneficial for students. These solutions are updated according to 2021-22 syllabus. These MCQS  can be really helpful in the preparation of Board exams and will provide you with a brief knowledge of the chapter.

NCERT MCQ ON OSCILLATIONS

Question 1: Select the incorrect statement(s) from the following.

I. A simple harmonic motion is necessarily periodic.
II. A simple harmonic motion may be oscillatory
III. An oscillatory motion is necessarily periodic

• a) II and III
• b) I and II
• c) I only
• d) I and III

Answer: II and III

Question 2: Suppose a tunnel is dug along a diameter of the earth. A particle is dropped from a point, a distance h directly above the tunnel, the motion of the particle is

• a) oscillatory
• b) simple harmonic
• c) parabolic
• d) non-periodic

Answer: oscillatory

Question 3: The graph plotted between the velocity and displacement from mean position of a particle executing SHM is

• a) ellipse
• b) straight line
• c) circle
• d) parabola

Answer: ellipse

Question 4: A body executing linear simple harmonic motion has a velocity of 3 m/s when its displacement is 4 cm and a velocity of 4 m/s when its displacement is 3 cm. What is the amplitude of oscillation ?

• a) 5 cm
• b) 7.5 cm
• c) 10 cm
• d) 12.5 cm

Answer: 5 cm

Question 5: 

Assertion : A particle executing simple harmonic motion comes to rest at the extreme positions .

Reason : The resultant force on the particle is zero at these positions

• a) Assertion is correct, reason is incorrect
• b) Assertion is incorrect, reason is correct
• c) Assertion is correct, reason is correct; reason is not a correct explanation for assertion
• d) Assertion is correct, reason is correct; reason is a correct explanation for assertion

Answer: Assertion is correct, reason is incorrect

Question 6: The total energy of a particle executing S.H.M. is proportional to

• a) square of amplitude of motion
• b) velocity in equilibrium position
• c) frequency of oscillation
• d) displacement from equilibrium position

Answer: square of amplitude of motion

Question 7: Which of the following is true about total mechanical energy of SHM ?

• a) It is never zero
• b) It is always zero
• c) It is zero at extreme position
• d) It is zero at mean position.

Answer: It is never zero

Question 8: 

Assertion : The graph of total energy of a particle in SHM w.r.t. position is a straight line with zero slope.

Reason : Total energy of particle in SHM remains constant throughout its motion.

• a) Assertion is correct, reason is correct; reason is a correct explanation for assertion
• b) Assertion is correct, reason is correct; reason is not a correct explanation for assertion
• c) Assertion is correct, reason is incorrect
• d) Assertion is incorrect, reason is correct.

Answer: Assertion is correct, reason is correct; reason is a correct explanation for assertion

Question 9: A body executes simple harmonic motion. The potential energy (P.E.), the kinetic energy (K.E.) and total energy (T.E.) are measured as a function of displacement x. Which of the following statement is true?

• a) K.E. is maximum when x = 0.
• b) T.E. is zero when x = 0.
• c) K.E. is maximum when x is maximum
• d) P.E. is maximum when x = 0.

Answer: K.E. is maximum when x = 0.

Question 10: The total energy of the particle executing simple harmonic motion of amplitude A is 100 J. At a distance of 0.707 A from the mean position, its kinetic energy is

• a) 50 J
• b) 100 J
• c) 12.5 J
• d) 25 J

Answer: 50 J

Question 11: When the displacement of a particle executing simple harmonic motion is half of its amplitude, the ratio of its kinetic energy to potential energy is

• a) 3 : 1
• b) 1 : 2
• c) 2 : 1
• d) 1 : 3

Answer: 3 : 1

Question 12: If the maximum velocity and acceleration of a particle executing SHM are equal in magnitude, the time period will be

• a) 6.28 sec
• b) 12.56 sec
• c) 3.14 sec
• d) 1.57 sec

Answer: 6.28 sec

Question 13: A simple pendulum oscillates in air with time period T and amplitude A. As the time passes

• a) T remains same and A decreases
• b) T decreases and A is constant
• c) T increases and A is constant
• d) T and A both decrease

Answer: T remains same and A decreases

Question 14: Which of the following will change the time period as they are taken to moon?

• a) A simple pendulum
• b) A torsional pendulum
• c) A physical pendulum
• d) A spring-mass system

Answer: A simple pendulum

Question 15: For an oscillating simple pendulum, the tension in the string is

• a) maximum at mean position
• b) constant throughout the motion
• c) cannot be predicted
• d) maximum at extreme position

Answer: maximum at mean position

In This Post we are  providing Chapter-13 Kinetic Energy NCERT MCQ for Class 11 Physics which will be beneficial for students. These solutions are updated according to 2021-22 syllabus. These MCQS  can be really helpful in the preparation of Board exams and will provide you with a brief knowledge of the chapter.

NCERT MCQ ON KINETIC ENERGY

Question 1: In kinetic theory of gases, it is assumed that molecules

• a) have same mass but negligible volume
• b) have different mass as well as volume
• c) have same volume but mass can be different
• d) have same mass but can have different volume

Answer: have same mass but negligible volume

Question 2: The internal energy of a gram-molecule of an ideal gas depends on

• a) pressure alone
• b) volume alone
• c) temperature alone
• d) both on pressure as well as temperature

Answer: pressure alone

Question 3: The phenomenon of Browninan movement may be taken as evidence of

• a) kinetic theory of matter
• b) electromagnetic theory of radiation
• c) corpuscular theory of light
• d) photoelectric phenomenon

Answer: kinetic theory of matter

Question 4: According to kinetic theory of gases, at absolute zero temperature

• a) molecular motion stops
• b) liquid hydrogen freezes
• c) liquid helium freezes
• d) water freezes

Answer: molecular motion stops

Question 5: At a given temperature the force between molecules of a gas as a function of intermolecular distance is

• a) first decreases and then increases
• b) always increases
• c) always decreases
• d) always constant

Answer: first decreases and then increases 

Question 6: For Boyle’s law to hold, the gas should be

• a) perfect and of constant mass and temperature
• b) real and of constant mass and temperature
• c) perfect and constant temperature but variable mass
• d) real and constant temperature but variable mass

Answer: perfect and of constant mass and temperature

Question 7: Boyle’ law is applicable for an

• a) isothermal process
• b) isochoric process
• c) adiabatic process
• d) isobaric process

Answer: isothermal process

Question 8: The deviation of gases from the behavior of ideal gas is due to

• a) attraction of molecules
• b) absolute scale of temp
• c) covalent bonding of molecules
• d) colourless molecules

Answer: attraction of molecules

Question 9: In a mixture of gases at a fixed temperature

• a) heavier molecule has lower average speed
• b) lighter molecule has lower average speed
• c) heavier molecule has higher average speed
• d) None of these

Answer: heavier molecule has lower average speed

Question 10: The average kinetic energy of gas molecules depends upon which of the following factor?

• a) Temperature of the gas
• b) Nature of the gas
• c) Volume of the gas
• d) None of these

Answer: Temperature of the gas

Question 11: The temperature of a gas is a measure of

• a) the average kinetic energy of the gaseous molecules
• b) the average potential energy of the gaseous molecules
• c) the average distance between the molecules of the gas
• d) the size of the molecules of the gas

Answer: the average kinetic energy of the gaseous molecules

Question 12: In the isothermal expansion of 10g of gas from volume V to 2V the work done by the gas is 575J. What is the root mean square speed of the molecules of the gas at that temperature?

• a) 499m/s
• b) 532m/s
• c) 520m/s
• d) 398m/s

Answer: 499m/s

Question 13: In a diatomic molecules, the rotational energy at a given temperature

• a) obeys Maxwell’s distribution
• b) have the same volue for all molecules
• c) equals the translational kinetic energy for each molecule
• d) None of these

Answer: obeys Maxwell’s distribution

Question 14: Cooking gas containers are kept in a lorry moving with uniform speed. The temperature of the gas molecules inside will.

• a) remains the same
• b) decrease for some and increase for others
• c) decrease
• d) increase

Answer: remains the same

Question 15: Pressure exerted by a gas is

• a) directly proportional to the density of the gas
• b) directly proportional to the square of the density of the gas
• c) inversely proportional to the density of the gas
• d) independent of density of the gas

Answer: directly proportional to the density of the gas

Complex Number

Complex number is of the form a +ib where a is real part and b is imaginary part. Here i = √ -1

 E.g.: 2+ i3 ;  7+ i9 etc

Complex Numbers are used in many scientific fields.

 Two complex numbers are equal if:

• Real parts are equal
• Imaginary parts are equal

E.g. Two complex numbers z1 = a + ib and z2 = c + id are equal if a = c and b = d.

Algebra of a Complex number

Addition of two complex numbers

Let z1 = a + ib and z2 = c + id be any two complex numbers.   Then, z1 + z2 = (a + c) + i (b + d)

 For example, (2 + i3) + (4 +i5) = 6 + i8

The addition of complex numbers satisfies the following properties:

• Closure law : z1 + z2  = complex Number
• Commutative law: z1 + z2 = z2 + z1
• Associative law: (z1 + z2) + z3 = z1 + (z2 + z3).
• Additive identity : z + 0 = z.
• Additive inverse : z + (–z) = 0.

Difference of two complex numbers

Let z1 = a + ib and z2 = c + id be any two complex numbers.   Then, z1 – z2 = (a – c) + i (b – d)

 For example, (6 + i3) – (2 + i) = 4 + i2

Multiplication of two complex numbers

Let z1 = a + ib and z2 = c + id be any two complex numbers.   Then, z1 * z2 = (ac – bd) + i(ad + bc)

 For example, (3 + i5) (2 + i6)  = (3*2 – 5*6 ) + i(3*6 +5*2)  = -24 + i28

The multiplication of complex numbers satisfies the following properties:

• Closure law : z1 * z2  = complex Number
• Commutative law: z1 * z2 = z2 * z1
• Associative law: (z1 * z2) *z3 = z1 * (z2 * z3).
• Multiplicative identity : z * 1 = z.
• Multiplicative inverse : z * (1/z) = 1.    (where z ≠ 0)
• Distributive law  :  z1 (z2 + z3) = z1 z2 + z1 z3

Division of two complex numbers

Given any two complex numbers z1 and z2, where z2 ≠ 0 ,   z1/z2  = z1 * (1/z2)

 For example, let z1 = 2+ 3i and z2 = 2 +2i,

z1* z2 =  (2+ 3i)/ (2+ 2i)

To solve this, we will rationalize the denominator

z1* z2 =  (2+ 3i)/ (2+ 2i)   *  (2- 2i)/ (2- 2i)     =  (-2 + i10) / 8   = -1/4 + i5/4

Power of I

• i² = -1
• i³ = -i
• i⁴ = 1
• i⁵ = i
• i⁶ = -1   etc
• i^-1 = -i
• i^-2 = -1
• i^-3 = i
• i^-4 = 1

Identities

• (z₁ + z₂)² = z₁² + z₂² + 2z₁z₂
• (z₁ – z₂)² = z₁² + z₂² – 2z₁z₂
• (z₁ + z₂)³ = z₁³ + z₂³ + 3z₁z₂² + 3z₁²z₂
• (z₁ – z₂)³ = z₁³ – z₂³ + 3z₁z₂² _– 3z₁²z₂
• z₁² – z₂²  = (z₁ + z₂) (z₁ – z₂)

Refer ExamFear video lessons for Proofs for these identities.

Example: Express (5 – 3i)³ in the form a + ib.

Solution:  (5 – 3i)³ = 5³ – 3 × 52 × (3i) + 3 × 5 (3i)² – (3i)³ = 125 – 225i – 135 + 27i = – 10 – 198i.

Modulus & Conjugate  of a complex Number

Let z = a + ib be a complex number.   Modulus of z, denoted by | z |, is defined to be real number (a² + b² )^1/2 ,  | z | = (a² + b²)^1/2

Numerical: Find the Modulus of (3 – 4i )

Solution:  | z | = (a² + b² )^1/2 = (3² + 4²)^1/2  = 5

Let z = a + ib be a complex number.  The conjugate of z, denoted as �, is the complex number a – ib, i.e., �  = a – ib.

Also Z* � = | Z |²

Or   Z^–1 =   � / | Z |²    ( Useful to find inverse of a complex number)

Numerical: Find the conjugate  of  (3 + 4i )

Solution:  Conjugate � = 3-4i

Numerical: Find inverse of  (3 + 4i )

  Z^–1 =   � / | Z |²    = (3 – 4i)/5     = 3/5 – 4/5i

Argand Plane & Polar representation

Complex numbers can represented in 2 forms

• Argand Plane
• Polar Representation

Argand Plane

The complex number x + iy  can be represented  geometrically as the unique point  P(x, y) in the XY-plane and vice-versa. Plane with complex number assigned to each of its point is called complex or Argand plane.

Let’s plot some points on the graph.

Note: Modulus of the complex number is distance between point P(x, y) to the origin O (0, 0)

Polar representation

Let point P represent z = x + iy.  

Let   x = r cos θ , y = r sin θ and therefore, z = r (cos θ + i sin θ).

 Here – π < θ ≤ π

Point P is uniquely determined by the ordered pair of real numbers (r, θ), called the polar coordinates of the point P.

Numerical: Represent the complex number z =1+ i √3 in the polar form.

Solution:  let z =1+ i √3  = r(cos θ + i sin θ)

 r=| z | = (a² + b² )^1/2      = ((1)² + (√3)²)^1/2     = 2

Comparing real parts of  z =1+ i √3  = r(cos θ + i sin θ)   = 2(cos θ + i sin θ) 

1 = 2 cos θ  

or  cos θ   = ½  

or cos θ    = π/3

Therefore,  polar representation will be  z = r(cos θ + i sin θ) = 2(cos π/3 + i sin π/3)

Quadratic Equation

We have seen of real numbers in the cases where discriminant is non-negative, i.e., ≥ 0,

Let us consider the following quadratic equation: ax² + bx + c = 0 with real coefficients a, b, c and a ≠ 0.

Also, let us assume that the b² – 4ac < 0.

Numerical: Solve x² + x + 1= 0

Solution:  Determinant,  b² – 4ac = 12 – 4 × 1 × 1 = 1 – 4 = – 3

X = (-1 ± I √3)/2

• One key basis for mathematical thinking is deductive reasoning. In contrast to deduction, inductive reasoning depends on working with different cases and developing a conjecture by observing incidences till we have observed each and every case. Thus, in simple language we can say the word ‘induction’ means the generalisation from particular cases or facts.
• Statement: A sentence is called a statement, if it is either true ot false.
• Motivation: Motivation is tending to initiate an action. Here Basis step motivate us for mathematical induciton.
• Principle of Mathematical Induction: The principle of mathematical induction is one such tool which can be used to prove a wide variety of mathematical statements. Each such statement is assumed as P(n) associated with positive integer n, for which the correctness for the case n = 1 is examined. Then assuming the truth of P(k) for some positive integer k, the truth of P (k+1) is established.
• Working Rule:
  Step 1: Show that the given statement is true for n = 1.
  Step 2: Assume that the statement  is true for n = k.
  Step 3: Using the assumption made in step 2, show that the statement is true for n = k  + 1. We have proved the statement is true for n = k. According to step 3, it is also true for k + 1 (i.e., 1 + 1 = 2). By repeating the above logic, it is true for every natural number

Principle of Mathematical Induction
Mathematical induction is one of the techniques, which can be used to prove a variety of mathematical statements which are formulated in terms of n, where n is a positive integer.

Let P(n) be given statement involving the natural number n such that
(i) The statement is true for n = 1, i.e. P(1) is true.
(ii) If the statement is true for n = k (where k is a particular but arbitrary natural number), then the statement is also true for n = k + 1 i.e. truth of P(k) implies that the truth of P(k + 1). Then, P(n) is true for all natural numbers n.

Angle

Angle is a measure of rotation of a given ray about its initial point. The original ray is called the initial side and the final position of ray after rotation is called terminal side of the angle. The point of rotation is called vertex. If the direction of rotation is anti-clockwise, the angle is said to be positive and if the direction of rotation is clockwise, then the angle is negative.

When a ray OA starting from its initial position OA rotates about its end point 0 and takes the final position OB, we say that angle

AOB (written as ∠ AOB) has been formed. The amount of rotation from the initial side to the terminal side is Called the measure of the angle.

Positive and Negative Angles

An angle formed by a rotating ray is said to be positive or negative depending on whether it moves in an anti-clockwise or a clockwise direction, respectively.

Measurement of Angles

There are three system for measuring angles,

1. Sexagesimal System/Degree Measure (English System)

In this system, a right angle is divided into 90 equal parts, called degrees. The symbol 1° is used to denote one degree. Each degree is divided into 60 equal parts, called minutes and one minute is divided into 60 equal parts, called seconds. Symbols 1′ and 1″ are used to denote one minute and one second, respectively.

i.e., 1 right angle = 90°
1° = 60′
1′ = 60″

2. Centesimal System (French System)

In this system, a right angle is divided into 100 equal parts, called ‘grades’. Each grade is subdivided into 100 min and each minute is divided into 100 s.

i.e., 1 right angle = 100 grades = 100g
1g = 100′
1′ = 100″

3. Circular System (Radian System) In this system, angle is measured in radian.

A radian is the angle subtended at the centre of a circle by an arc, whose length is equal to the radius of the circle.

The number of radians in an angle subtended by an arc of circle at the centre is equal to arc/radius.

Relationships

(i) π radian = 180° or 1 radian (180°/π)= 57°16’22” where, π = 22/7 = 3.14159
(ii) 1° = (π/180) rad = 0.01746 rad
(iii) If D is the number of degrees, R is the number of radians and G is the number of grades in an angle θ, then

(iv) θ = l/r where θ = angle subtended by arc of length / at the centre of the circle, r = radius of the circle.

Trigonometric Ratios

Relation between different sides and angles of a right angled triangle are called trigonometric ratios or T-ratios

Trigonometric (or Circular) Functions

Let X’OX and YOY’ be the coordinate axes. Taking 0 as the centre and a unit radius, draw a circle, cutting the coordinate axes at A,B, A’ and B’, as shown in the figure.

Now, the six circular functions may be defined as under
(i) cos θ = x
(ii) sin θ = y
(iii) sec θ = 1/x, x ≠ 0
(iv) cosec θ = 1/y, y ≠ 0
(v) tan θ = y/x, x ≠ 0
(vi) cot θ = x/y, y ≠ 0

Domain and Range

Range of Modulus Functions

sin θ|≤ 1, |cos θ| ≤ 1, |sec θ| ≥ 1, |Cosec θ| ≥ 1 for all values of 0, for which the functions are defined.

Trigonometric Identities

An equation involving trigonometric functions which is true for all those angles for which the functions are defined is called trigonometrical identity. Some identities are

Sign of Trigonometric Ratios

Trigonometric Ratios of Some Standard Angles

Trigonometric Ratios of Some Special Angles

Trigonometric Ratios of Allied Angles

Two angles are said to be allied when their sum or difference is either zero or a multiple of 90°.

The angles — θ, 90° ± θ, 180° ± θ, 270° + θ, 360° —θ etc., are angles allied to the angle θ, if θ is measured in degrees.

Trigonometric Periodic Functions

A function f(x) is said to be periodic, if there exists a real number T> 0 such that f(x + T)= f(x) for all x. T is called the period of the function, all trigonometric functions are periodic.

Maximum and Minimum Values of Trigonometric Expressions

Trigonometric Ratios of Compound Angles

The algebraic sum of two or more angles are generally called compound angles and the angles are known as the constituent angle. Some standard formulas of compound angles have been given below.

Transformation Formulae

Trigonometric Ratios of Multiple Angles

Trigonometric Ratios of Some Useful Angles

 CARTESIAN PRODUCT OF SETS:
 Given two non-empty sets A and B, the set of all ordered pairs (x, y), where x ∈
A and y ∈ B is called Cartesian product of A and B; symbolically, we write A ×
B = {(x, y) | x ∈ A and y ∈ B}.
 Example- A = {1, 2, 3} and B = {4, 5}, then A × B = {(1, 4), (2, 4), (3, 4), (1,
5), (2, 5), (3, 5)} and B × A = {(4, 1), (4, 2), (4, 3), (5, 1), (5, 2), (5, 3)}.
 Two ordered pairs are equal, if and only if the corresponding first elements are
equal and the second elements are also equal.
 If there are p elements in A and q elements in B, then there will be pq elements
in A × B, i.e., if n(A) = p and n(B) = q, then n(A × B) = pq.
 If either P or Q is the null set, then P × Q will also be empty set, i.e., P × Q = φ
 If A and B are non-empty sets and either A or B is an infinite set, then so is A ×
B.
 A × A × A = {(a, b, c): a, b, c ∈ A}. Here (a, b, c) is called an ordered triplet.

Ordered Pair
An ordered pair consists of two objects or elements in a given fixed order.

Equality of Two Ordered Pairs
Two ordered pairs (a, b) and (c, d) are equal if a = c and b = d.

Cartesian Product of Two Sets
For any two non-empty sets A and B, the set of all ordered pairs (a, b) where a ∈ A and b ∈ B is called the cartesian product of sets A and B and is denoted by A × B.
Thus, A × B = {(a, b) : a ∈ A and b ∈ B}
If A = Φ or B = Φ, then we define A × B = Φ

Note:

• A × B ≠ B × A
• If n(A) = m and n(B) = n, then n(A × B) = mn and n(B × A) = mn
• If atieast one of A and B is infinite, then (A × B) is infinite and (B × A) is infinite.

Relations
A relation R from a non-empty set A to a non-empty set B is a subset of the cartesian product set A × B. The subset is derived by describing a relationship between the first element and the second element of the ordered pairs in A × B.
The set of all first elements in a relation R is called the domain of the relation B, and the set of all second elements called images is called the range of R.

Note:

• A relation may be represented either by the Roster form or by the set of builder form, or by an arrow diagram which is a visual representation of relation.
• If n(A) = m, n(B) = n, then n(A × B) = mn and the total number of possible relations from set A to set B = 2^mn

Inverse of Relation
For any two non-empty sets A and B. Let R be a relation from a set A to a set B. Then, the inverse of relation R, denoted by R^-1 is a relation from B to A and it is defined by
R^-1 ={(b, a) : (a, b) ∈ R}
Domain of R = Range of R^-1 and
Range of R = Domain of R^-1.

Functions
A relation f from a set A to set B is said to be function, if every element of set A has one and only image in set B.
In other words, a function f is a relation such that no two pairs in the relation have the first element.

Real-Valued Function
A function f : A → B is called a real-valued function if B is a subset of R (set of all real numbers). If A and B both are subsets of R, then f is called a real function.

Some Specific Types of Functions
Identity function: The function f : R → R defined by f(x) = x for each x ∈ R is called identity function.
Domain of f = R; Range of f = R

Constant function: The function f : R → R defined by f(x) = C, x ∈ R, where C is a constant ∈ R, is called a constant function.
Domain of f = R; Range of f = C

Polynomial function: A real valued function f : R → R defined by f(x) = a₀ + a₁x + a₂x²+…+ a_nxⁿ, where n ∈ N and a₀, a₁, a₂,…….. a_n ∈ R for each x ∈ R, is called polynomial function.

Rational function: These are the real function of type f(x)g(x), where f(x)and g(x)are polynomial functions of x defined in a domain, where g(x) ≠ 0.

The modulus function: The real function f : R → R defined by f(x) = |x|
or

for all values of x ∈ R is called the modulus function.
Domaim of f = R
Range of f = R⁺ U {0} i.e. [0, ∞)

Signum function: The real function f : R → R defined
by f(x) = |x|x, x ≠ 0 and 0, if x = 0
or

is called the signum function.
Domain of f = R; Range of f = {-1, 0, 1}

Greatest integer function: The real function f : R → R defined by f (x) = {x}, x ∈ R assumes that the values of the greatest integer less than or equal to x, is called the greatest integer function.
Domain of f = R; Range of f = Integer

Fractional part function: The real function f : R → R defined by f(x) = {x}, x ∈ R is called the fractional part function.
f(x) = {x} = x – [x] for all x ∈R
Domain of f = R; Range of f = [0, 1)

Algebra of Real Functions
Addition of two real functions: Let f : X → R and g : X → R be any two real functions, where X ∈ R. Then, we define (f + g) : X → R by
{f + g) (x) = f(x) + g(x), for all x ∈ X.

Subtraction of a real function from another: Let f : X → R and g : X → R be any two real functions, where X ⊆ R. Then, we define (f – g) : X → R by (f – g) (x) = f (x) – g(x), for all x ∈ X.

Multiplication by a scalar: Let f : X → R be a real function and K be any scalar belonging to R. Then, the product of Kf is function from X to R defined by (Kf)(x) = Kf(x) for all x ∈ X.

Multiplication of two real functions: Let f : X → R and g : X → R be any two real functions, where X ⊆ R. Then, product of these two functions i.e. f.g : X → R is defined by (fg) x = f(x) . g(x) ∀ x ∈ X.

Quotient of two real functions: Let f and g be two real functions defined from X → R. The quotient of f by g denoted by fg is a function defined from X → R as

Set
A set is a well-defined collection of objects.

Representation of Sets
There are two methods of representing a set

• Roster or Tabular form In the roster form, we list all the members of the set within braces { } and separate by commas.
• Set-builder form In the set-builder form, we list the property or properties satisfied by all the elements of the sets

Set is a collection of well defined objects which are distinct from each other. Sets are usually denoted by capital letters A, B,C,… and elements are usually denoted by small letters a, b,c,…

If ‘a’ is an element of a set A, then we write a ∈ A and say ‘a’ belongs to A or ‘a’ is in A or ‘a’ is a member of A. If ‘a’ does not belongs to A, we write a ∉ A.

Standard Notations

• N : A set of natural numbers.
• W : A set of whole numbers.
• Z : A set of integers.
• Z⁺/Z^– : A set of all positive/negative integers.
• Q : A set of all rational numbers.
• Q⁺/Q^– : A set of all positive/ negative rational numbers.
• R : A set of real numbers.
• R⁺/R^–: A set of all positive/negative real numbers.
• C : A set of all complex numbers.

Methods for Describing a Set

(i) Roster/Listing Method/Tabular Form In this method, a set is described by listing element, separated by commas, within braces.

e.g., A = {a, e, i, o, u}

(ii) Set Builder/Rule Method In this method, we write down a property or rule which gives us all the elements of the set by that rule.

e.g.,A = {x : x is a vowel of English alphabets}

Types of Sets

1. Finite Set A set containing finite number of elements or no element.

2. Cardinal Number of a Finite Set The number of elements in a given finite set is called cardinal number of finite set, denoted by n (A).

3. Infinite Set A set containing infinite number of elements.

4. Empty/Null/Void Set A set containing no element, it is denoted by (φ) or { }.

5. Singleton Set A set containing a single element.

6. Equal Sets Two sets A and B are said to be equal, if every element of A is a member of B and every element of B is a member of A and we write A = B.

7. Equivalent Sets Two sets are said to be equivalent, if they have same number of elements.

If n(A) = ⁿ (B), then A and B are equivalent sets. But converse is not true.

8. Subset and Superset Let A and B be two sets. If every element of A is an element of B, then A is called subset of B and B is called superset of A. Written as A ⊆ B or B ⊇ A

9. Proper Subset If A is a subset of B and A ≠ B, then A is called proper subset of B and we write A ⊂ B.

10. Universal Set (U) A set consisting of all possible elements which occurs under consideration is called a universal set.

11. Comparable Sets Two sets A and Bare comparable, if A ⊆ B or B ⊆ A.

12. Non-Comparable Sets For two sets A and B, if neither A ⊆ B nor B ⊆ A, then A and Bare called non-comparable sets.

13. Power Set (P) The set formed by all the subsets of a given set A, is called power set of A, denoted by P(A).

14. Disjoint Sets Two sets A and B are called disjoint, if, A ∩ B = (φ).

Venn Diagram

In a Venn diagram, the universal set is represented by a rectangular region and a set is represented by circle or a closed geometrical figure inside the universal set.

Operations on Sets

Union of Sets

The union of two sets A and B, denoted by A ∪ B is the set of all those elements, each one of which is either in A or in B or both in A and B.

Intersection of Sets

The intersection of two sets A and B, denoted by A ∩ B, is the set of all those elements which are common to both A and B.

If A₁, A₂,… , An is a finite family of sets, then their intersection is denoted by

Complement of a Set

If A is a set with U as universal set, then complement of a set, denoted by A’ or Ac is the set U – A .

Difference of Sets

For two sets A and B, the difference A – B is the set of all those elements of A which do not belong to B.

Symmetric Difference

For two sets A and B, symmetric difference is the set (A – B) ∪ (B – A) denoted by A Δ B.

Laws of Algebra of Sets

For three sets A, B and C

(i) Commutative Laws
A ∩ B = B ∩ A
A ∪ B = B ∪ A

(ii) Associative Laws
(A ∩ B) ∩ C = A ∩ (B ∩ C)
(A ∪ B) ∪ C = A ∪ (B ∪ C)

(iii) Distributive Laws
A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)

(iv) Idempotent Laws
A ∩ A = A
A ∪ A = A

(v) Identity Laws
A ∪ Φ = A
A ∩ U = A

(vi) De Morgan’s Laws

(a) (A ∩ B) ′ = A ′ ∪ B ′
(b) (A ∪ B) ′ = A ′ ∩ B ′
(c) A – (B ∩ C) = (A – B) ∩ (A- C)
(d) A – (B ∪ C) = (A – B) ∪ ( A – C)

(vii) (a) A – B = A ∩ B’
(b) B – A = B ∩ A’
(c) A – B = A ⇔A ∩ B= (Φ)
(d) (A – B) ∪ B= A ∪ B
(e) (A – B) ∩ B = (Φ)
(f) A ∩ B ⊆ A and A ∩ B ⊆ B
(g) A ∪ (A ∩ B)= A
(h) A ∩ (A ∪ B)= A

(viii) (a) (A – B) ∪ (B – A) = (A ∪ B) – (A ∩ B)
(b) A ∩ (B – C) = (A ∩ B) – (A ∩ C)
(c) A ∩ (B Δ C) = (A ∩ B) A (A ∩ C)
(d) (A ∩ B) ∪ (A – B) = A
(e) A ∪ (B – A) = (A ∪ B)

(ix) (a) U’ = (Φ)
(b) Φ’ = U
(c) (A’ )’ = A
(d) A ∩ A’ = (Φ)
(e) A ∪ A’ = U
(f) A ⊆ B ⇔ B’ ⊆ A’

Important Points to be Remembered

• Every set is a subset of itself i.e., A ⊆ A, for any set A.
• Empty set Φ is a subset of every set i.e., Φ ⊂ A, for any set A.
• For any set A and its universal set U, A ⊆ U
• If A = Φ, then power set has only one element i.e., ⁿ (P(A)) = 1
• Power set of any set is always a non-empty set. Suppose A = {1, 2}, thenP(A) = {{1}, {2},{1,2}, Φ}.(a) A ∉ P(A) (b) {A} ∈ P(A)
• (vii) If a set A has n elements, then P(A) or subset of A has 2_n elements.
• (viii) Equal sets are always equivalent but equivalent sets may not be equal. The set {Φ} is not a null set. It is a set containing one element Φ.

Results on Number of Elements in Sets

• n (A ∪ B) = n(A) + (B)- n(A ∩ B)
• n(A ∪ B) = n(A)+ n(B), if A and B are disjoint.
• n(A – B) = n(A) – n(A ∩ B)
• n(A Δ B) = n(A) + n(B)- 2n(A ∩ B)
• n(A ∪ B ∪ C)= n(A)+ n(B)+ n(C)- n(A ∩ B) – n(B ∩ C)- n(A ∩ C)+ n(A ∩ B ∩ C)
• n (number of elements in exactly two of the sets A, B, C) = n(A ∩ B) + n(B ∩ C) + n (C ∩ A)- 3n(A ∩ B ∩ C)
• n (number of elements in exactly one of the sets A, B, C) = n(A) + n(B) + n(C) – 2n(A ∩ B) – 2n(B ∩ C) – 2n(A ∩ C) + 3n(A ∩ B ∩ C)
• n(A’ ∪ B’)= n(A ∩ B)’ = n(U) – n(A ∩ B)
• n(A’ ∩ B’ ) = n(A ∪ B)’ = n(U) – n(A ∪ B)
• n(B – A) = n(B)- n(A ∩ B)

Ordered Pair

An ordered pair consists of two objects or elements in a given fixed order.

Equality of Ordered Pairs Two ordered pairs (a1, b1) and (a2, b2) are equal iff a1 = a2 and b1 = b2.

Cartesian Product of Sets

For two sets A and B (non-empty sets), the set of all ordered pairs (a, b) such that a ∈ A and b ∈ B is called Cartesian product of the sets A and’ B, denoted by A x B. A x B={(a,b):a ∈ A and b ∈ B}

If there are three sets A, B, C and a ∈ A, be B and c ∈ C, then we form, an ordered triplet (a, b, c). The set of all ordered triplets (a, b, c) is called the cartesian product of these sets A, B and C.

i.e., A x B x C = {(a,b,c):a ∈ A,b ∈ B,c ∈ C}

Properties of Cartesian Product

For three sets A, B and C

• n (A x B)= n(A) n(B)
• A x B = Φ, if either A or B is an empty set.
• A x (B ∪ C)= (A x B) ∪ (A x C)
• A x (B ∩ C) = (A x B) ∩ (A x C)
• A x (B — C)= (A x B) — (A x C)
• (A x B) ∩ (C x D)= (A ∩ C) x (B ∩ D)
• If A ⊆ B and C ⊆ D, then (A x C) ⊂ (B x D)
• If A ⊆ B, then A x A ⊆ (A x B) ∩ (B x A)
• A x B = B x A ⇔ A = B
• If either A or B is an infinite set, then A x B is an infinite set.
• A x (B’ ∪ C’ )’ = (A x B) ∩ (A x C)
• A x (B’ ∩ C’ )’ = (A x B) ∪ (A x C)
²elements in common.
• If ≠ B, then A x B ≠ B x A
• If A = B, then A x B= B x A
• If A ⊆ B, then A x C = B x C for any set C.

In This Post we are  providing Chapter-12 Thermodynamics NCERT MCQ for Class 11 Physics which will be beneficial for students. These solutions are updated according to 2021-22 syllabus. These MCQS  can be really helpful in the preparation of Board exams and will provide you with a brief knowledge of the chapter.

NCERT MCQ ON THERMODYNAMICS

Question 1.
One gram of sample of NH₄NO₃ is decomposed in a bomb calorimeter. The temperature of the calorimeter increases by 6.12 K. The heat capacity of the system is 1.23 kj/deg. What is the molar heat of decomposition of NH₄NO₃?
(a) -7.53 kj/mol
(b) -398.1 kj mol^-1
(c) -16.1 kj/mol
(d) -602 kj/mol.

Answer: (d) -602 kj/mol.

Question 2.
AHr of graphite is 0.23 kj/mol and ∆H_f for diamond is 1.896 kj mol^-1, ∆H_transition from graphite to diamond is
(a) 1.66 kj/mol
(b) 2.1 kj/mol
(c) 2.33 kj/mol
(d) 1.5 kj/mol

Answer: (a) 1.66 kj/mol

Question 3.
The bond energies of C-C, C=C; H-H and C-H linkages are 350, 600, 400 and 410 kj per mole respectively. The heat of hydrogenation of ethylene is
(a) -170 kj mol^-1
(b) -260 kj mol^-1
(c) 400 kj mol^-1
(d) -450 kj mol^-1

Answer: (a) -170 kj mol^-1

Question 4.
Which of the following reaction defines ∆H0f?
(a) C(Diamond) + O₂(g) → CO₂(g)
(b) 12 H₂(g) + 12 F₂(g) → HF(g)
(c) N₂(g) + 3H₂(g) → 2NH₃(g)
(d) CO(g) + 12 O₂(g) → CO₂(g)

Answer: (b) 12 H₂(g) + 12 F₂(g) → HF(g)

Question 5.
One mole of a hon-ideal gas undergoes a change of state (2.0 atm, 3.0 L, 95 K) → (4.0 atm, 5.0 L, 245 K) with a change in internal energy, ∆U = 30.0 L atm. The change in enthalpy (∆H) of the process in L atm is
(a) 44.0
(b) 42.3
(c) 44.0
(d) not defined because pressure is not constant.

Answer: (c) 44.0

Question 6.
Which one of the following statement is false?
(a) Work is a state function
(b) Temperature is a state function
(c) Change in the state is completely defined when the initial final states are specified
(d) Work appears at the boundary of the system

Answer: (a) Work is a state function

Question 7.
Molar heat capacity of water in equilibrium with ice at constant pressure is
(a) zero
(b) infinity
(c) 40.45 kj K^-1 mol^-1
(d) 75.48 JK^-1 mol^-1

Answer: (b) infinity

Question 8.
For the reaction C₃H₈(g) + 5O₂(g) → 3CO₂(g) + 4H₂O(I) at constant temperature ∆H – ∆E is
(a) + RT
(b) – 3RT
(c) + 3RT
(d) -RT

Answer: (b) – 3RT

Question 9.
For which one of the following equations AH_reaction equals ∆H0f for the product?
(a) N₂(g) + O₃(g) → N₂O₃(g)
(b) CH₄(g) + 2Cl₂(g) → CH₂Cl₂(l) + 2HCl(g)
(c) Xe(g) + 2F₂(g) → XeF₄(g)
(d) 2C(g) + O₂(g) → 2CO₂(g)

Answer: (c) Xe(g) + 2F₂(g) → XeF₄(g)

Question 10.
Enthalpy of CH₄ + 12 O₂ → CH₃OH is negative.
If enthalpy of combustion of CH₄ and CH₃OH are x and y respectively then which reaction is correct?
(a) x > y
(b) x < y
(c) x = y
(d) x ≥ y

Answer: (b) x < y

Question 11.
The heat required to raise the temperature of a body by 1 K is called
(a) Specific heat
(b) Thermal capacity
(c) Water equivalent
(d) Molar heat capacity

Answer: (b) Thermal capacity

Question 12.
In a reaction involving only solids and liquids, which of the following is true?
(a) ∆H < ∆E (b) ∆H = ∆E (c) ∆H > ∆E
(d) ∆H = ∆E + RT∆₁₁

Answer: (b) ∆H = ∆E

Question 13.
In which of the following process, the process is always non-feasible?
(a) ∆H > 0, ∆S > 0
(b) ∆H < 0, ∆S > 0
(c) ∆H > 0, ∆S < 0
(d) ∆H < 0, ∆S < 0

Answer: (c) ∆H > 0, ∆S < 0

Question 14.
Internal energy does not include
(a) Nuclear energy
(b) Vibrational energy
(c) Rotational energy
(d) Energy of gravitational pull

Answer: (d) Energy of gravitational pull

Question 15.
Which of the following reactions is endothermic?
(a) N₂ + O₂ → 2NO
(b) H₂ + Cl₂ → 2HCl
(c) H₂SO₄ + 2NaOH → Na₂SO₄ + 2H₁₁O
(d) None of these

Answer: (a) N₂ + O₂ → 2NO

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In This Post we are  providing Chapter-11 Thermal Properties of Matter NCERT MCQ for Class 11 Physics which will be beneficial for students. These solutions are updated according to 2021-22 syllabus. These MCQS  can be really helpful in the preparation of Board exams and will provide you with a brief knowledge of the chapter.

NCERT MCQ ON THERMAL PROPERTIES OF MATTER

Question 1: 

Assertion: The triple point of water is a standard fixed point in modern thermometry.
Reason: The triple point of a substance is unique i.e. it occurs at one particular set of values of pressure and temperature

• a) Assertion is correct, reason is correct; reason is a correct explanation for assertion
• b) Assertion is incorrect, reason is correct
• c) Assertion is correct, reason is correct; reason is not a correct explanation for assertion
• d) None of these

Answer: Assertion is correct, reason is correct; reason is a correct explanation for assertion

Question 2: A metal sheet with a circular hole is heated. The hole

• a) gets larger
• b) gets deformed
• c) gets smaller
• d) None of these

Answer: gets larger

Question 3: A solid ball of metal has a spherical cavity inside it. The balln is heated. The volume of cavity will

• a) increase
• b) decrease
• c) remain unchanged
• d) have its shape changed

Answer: increase

Question 4:

Assertion : Water kept in an open vessel will quickly evaporate on the surface of the moon.
Reason : The temperature at the surface of the moon is much higher than boiling point of the water.

• a) Assertion is correct, reason is incorrect
• b) None of these
• c) Assertion is correct, reason is correct; reason is not a correct explanation for assertion
• d) Assertion is correct, reason is correct; reason is a correct explanation for assertion

Answer: Assertion is correct, reason is incorrect

Question 5: Consider the following statements and select the correct statement(s).

I. Water can never be boiled without heating.
II. Water can be boiled below room temperature by lowering the pressure.
III. On releasing the excess pressure water refreezes into ice.

• a) II and III
• b) I only
• c) II only
• d) I and II

Answer: II and III

Question 6: Two marks on a glass rod 10 cm apart are found to increase their distance by 0.08 mm when the rod is heated from 0°C to 100°C. A flask made of the same glass as that of rod measures a volume of 1000 cc at 0°C. The volume it measures at 100°C in cc is

• a) 1002.4
• b) 1008.2
• c) 1004.2
• d) 1006.4

Answer: 1002.4

Question 7: Mass of water which absorbs or emits the same amount of heat as is done by the body for the same rise or fall in temperature is known as

• a) water equivalent of the body
• b) latent heat capacity of the body
• c) specific heat capacity of the body
• d) thermal capacity of the body

Answer: water equivalent of the body

Question 8: Heat is transmitted from higher to lower temperature through actual mass motion of the molecules in

• a) convection
• b) radiation
• c) conduction
• d) None of these

Answer: convection

Question 9: Good absorbers of heat are

• a) good emitters
• b) non-emitters
• c) poor emitters
• d) highly polished

Answer: good emitters

Question 10: Three bodies A, B and C have equal area which are painted red, yellow and black respectively. If they are at same temperature, then

• a) emissive power of C is maximum
• b) emissive power of A is maximum
• c) emissive power of B is maximum
• d) emissive power of A, B and C are equal.

Answer: emissive power of C is maximum

Question 11: Sweet makers do not clean the bottom of cauldron because

• a) black and rough surface absorbs more heat
• b) absorption power of black and bright surface is more.
• c) transmission power of black and rough surface is more.
• d) emission power of black and bright surface is more

Answer: black and rough surface absorbs more heat

Question 12: 4200 J of work is required for

• a) increasing the temperature of 100 g of water through 10°C
• b) increasing the temperature of 1 kg of water through 10°C
• c) increasing the temperature of 500 g of water through 10°C
• d) increasing the temperature of 10 g of water through 10°C

Answer: increasing the temperature of 100 g of water through 10°C

Question 13: The latent heat of vaporization of a substance is always

• a) greater than its latent heat of fusion
• b) less than its latent heat of fusion
• c) greater than its latent heat of sublimation
• d) equal to its latent heat of sublimation

Answer: greater than its latent heat of fusion

Question 14: Which of the following statements regarding specific heat capacity of a substance are correct ? It depends on

I. mass of substance.
II. nature of substance.
III. temperature of substance.
IV. volume of substance.

• a) II and III
• b) None of these
• c) I and II
• d) III and IV

Answer: II and III

Question 15: A quantity of heat required to change the unit mass of a solid substance, from solid state to liquid state, while the temperature remains constant, is known as

• a) latent heat of fusion
• b) sublimation
• c) latent heat
• d) hoar frost

Answer: latent heat of fusion

In This Post we are  providing Chapter-10 Mechanical Properties of Fluids NCERT MCQ for Class 11 Physics which will be beneficial for students. These solutions are updated according to 2021-22 syllabus. These MCQS  can be really helpful in the preparation of Board exams and will provide you with a brief knowledge of the chapter.

NCERT MCQ ON MECHANICAL PROPERTIES OF FLUIDS

Question 1.
A number of small drops of mercury coalesce adiabatically to form a single drop. The temperature of drop
(a) Increases
(b) Is infinite
(c) Remains unchanged
(d) May decrease or increase depending upon size

Answer: (d) May decrease or increase depending upon size

Question 2.
Surface tension of a soap solution is 1.9 × 10^-2N/m. work done in blowing a bubble of 2.0 cm diameter will be
(a) 7.6 × 10^-6 p J
(b) 15.2 × 10^-6 p J
(c) 1.9 × 10^-6 p J
(d) 1 × 10^-4 p J

Answer: (b) 15.2 × 10^-6 p J

Question 3.
Plants get water through the roots because of
(a) Capillarity
(b) Viscosity
(c) Gravity
(d) Elasticity

Answer: (a) Capillarity

Question 4.
Choose the wrong statement from the following.
(a) Small droplets of a liquid are spherical due to surface tension
(b) Oil rises through the wick due to capillarity
(c) In drinking the cold drinks through a straw, we use the phenomenon of capillarity
(d) Gum is used to stick two surfaces. In this process we use the property of Adhesion

Answer: (c) In drinking the cold drinks through a straw, we use the phenomenon of capillarity

Question 5.
The height of a liquid in a fine capillary tube
(a) Increases with an increase in the density of a liquid
(b) Decreases with a decrease in the diameter of the tube
(c) Decreases with an increase in the surface tension
(d) Increases as the effective value of acceleration due to gravity is decreased

Answer: (d) Increases as the effective value of acceleration due to gravity is decreased

Question 6.
At critical temperature, the surface tension of a liquid
(a) Is zero
(b) Is infinity
(c) Is the same as that at any other temperature
(d) Can not be determined

Answer: (a) Is zero

Question 7.
A capillary tube when immersed vertically in a liquid records a rise of 3 cm. if the tube is immersed in the liquid at an angle of 60° with the vertical, then length of the liquid column along the tube will be
(a) 2 cm
(b) 3 cm
(c) 6 cm
(d) 9 cm

Answer: (c) 6 cm

Question 8.
When the angle of contact between a solid and a liquid is 90°, then
(a) Cohesive force > Adhesive force
(b) Cohesive force < Adhesive force
(c) Cohesive force = Adhesive force
(d) Cohesive force >> Adhesive force

Answer: (c) Cohesive force = Adhesive force

Question 9.
Water rises up to a height of 5 cm in a capillary tube of radius 2 mm. what is the radius of the radius of the capillary tube if the water rises up to a height of 10 cm in another capillary?
(a) 4 mm
(b) 1 mm
(c) 3 mm
(d) 1 cm

Answer: (b) 1 mm

Question 10.
If the surface of a liquid is plane, then the angle of contact of the liquid with the walls of container is
(a) Acute angle
(b) Obtuse angle
(c) 90°
(d) 0°

Answer: (d) 0°

Question 11.
Two soap bubbles have radii in the ratio of 4 : 3. What is the ratio of work done to below these bubbles?
(a) 4 : 3
(b) 16 : 9
(c) 09 : 16
(d) 3 : 4

Answer: (b) 16 : 9

Question 12.
The height of water in a capillary tube of radius 2 cm is 4 cm. what should be the radius of capillary, if the water rises to 8 cm in tube?
(a) 1 cm
(b) 0.1 cm
(c) 2 cm
(d) 4 cm

Answer: (a) 1 cm

Question 13.
Water rises up to a height of 4 cm, in a capillary tube immersed vertically in water. What will be the length of water column in the capillary tube, if the tube is immersed in water, at an angle of 60° with the vertical?
(a) 4 cm
(b) 6 cm
(c) 8 cm
(d) 2 cm

Answer: (c) 8 cm

Question 14.
The surface of water in contact with glass wall is
(a) Plane
(b) concave
(c) convex
(d) Both b and c

Answer: (b) concave

Question 15.
Pressure inside two soap bubbles is 1.01 and 1.02 atmospheres. ratio between their volume is
(a) 102 : 101
(b) (102)3 : (101)3
(c) 8 : 1
(d) 2 : 1

Answer: (c) 8 : 1

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