electrochemistry formulas

Formula of rate of appearance
Average rate of reaction
Instantaneous rate of reaction  

rate of appearance average rate of reaction instantaneous rate and rate law of reaction

Formula of first order reaction  

formula of first order reaction 

Arrhenius equation all formulas

All formula of Arrhenius equation 

 Table of rate constant 

Table of formula of rate constant 

Formula of zero order reaction 

Formula of zero order reaction 

What is the difference between the manner in which movement takes place in a sensitive plant and the movement in our legs

Q. No. 12: What is the difference between the manner in which movement takes place in a sensitive plant and the movement in our legs?

Ans:

	Movement in sensitive plants		Movement in our legs
1	The movement in a sensitive plant is a response to stimulus(touch) which is a involuntary action.	1	Movement in our legs is a voluntary action.
2	No special tissue is there for the transfer of information	2	A complete system CNS and PNS is there for the information exchange.
3	Plant cells do not have specialised protein for movements.	3	Animal cells have specialised protein which help muscles to contract.

Compare and contrast nervous and hormonal mechanisms for control and coordination in animals

Q. No. 11:Compare and contrast nervous and hormonal mechanisms for control and coordination in animals.

Ans:

	Nervous control		Hormonal Control
1	It is consist of nerve impulses between PNS, CNS and Brain.	1	It consists of endocrine system which secretes hormones directly into blood.
2	Here response time is very short.	2	Here response time is very long.
3	Nerve impulses are not specific in their action.	3	Each hormone has specific actions.
4	The flow of information is rapid.	4	The flow of information is very slow.

How are involuntary actions and reflex actions different from each other

Q. No. 10: How are involuntary actions and reflex actions different from each other?

Ans:

Involuntary action is the set of muscle movement which do not require thinking. But it is controlled by brain for example beating of heart beat.

While on the other hand, the reflex action is rapid and spontaneous action in response to any stimulus. For example closing of eyes immediately when bright light is focused.

What is the need for a system of control and coordination in an organism

Q. No. 9: What is the need for a system of control and coordination in an organism?

Ans:  There are various organs in an organism. These organs must be carefully controlled and coordinated for the survival of an organisms. In the body of an organism various fluids are secreted from the glands of the endocrine system. These hormones are responsible for the overall growth and development of an organism. All others daily decision that includes voluntary and involuntary action are controlled by central nervous system(CNS). 

How does chemical coordination occur in plants

Q. No. 8: How does chemical coordination occur in plants?

 Ans: Chemical coordination in plants are occurred in plants with the help of fluids secreted in plants known as phytohormones or plant hormones. These hormones regulate the growth of the plants. For example auxin responsible for the growth of the plants and the Cytokinin helps in cell division in the fast growing part of the plant such as plant hormones.

Which signals will get disrupted in case of a spinal cord injury

Q. No. 7:  Which signals will get disrupted in case of a spinal cord injury?

Ans: In case of the spinal cord injury, the signals coming from the nerves as well as the signals coming to the receptors will be disrupted. As both these signals meet in a bundle in spinal cord so there is any spinal cord injury then both these signals are disrupted.  

tan-1√ 3 − sec-1 (− 2) is equal to

tan^-1√ 3 − sec^-1 (− 2) is equal to
Answer

tan^-1√ 3 − sec^-1 (− 2) is equal to

if sin-1x= y, then

if sin^-1x= y, then

Answer: Here we can see that if sin^-1x=y then y must be the principle value branch of the sine inverse so now we know that the principal value branch of the sine inverse is [-pi/2 , pi/2],   so the correct answer is option (B).

Find the values of the cos-1(1/2) + 2 sin-1(1/2)

Find the values of the cos^-1(1/2) + 2 sin^-1(1/2)
Answer

Find the values of the cos^-1(1/2) + 2 sin^-1(1/2)

Find the values of the following: tan-1(1) + cos-1(-1/2) +sin-1(-1/2)

Find the values of the following: tan^-1(1) + cos^-1(-1/2) +sin^-1(-1/2)
Answer

Find the values of the following: tan^-1(1) + cos^-1(-1/2) +sin^-1(-1/2)

Find the principal value of cosec -1(-√2)

Find the principal value of cosec ^-1(-√2)
Answer

Find the principal value of cosec ^-1(-√2)

Find the principal value of cos-1(-1/√2)

Find the principal value of cos^-1(-1/√2)
Answer

Find the principal value of cos^-1(-1/√2)

Find the principal value of cot-1(√3)

Find the principal value of cot^-1(√3)
Answer

Find the principal value of cot^-1(√3)

Find the principal value of sec-1(2/√3)

Find the principal value of sec^-1(2/√3)
Answer

Find the principal value of sec^-1(2/√3)

Find the principal value of cos-1(-1/2)

Find the principal value of cos^-1(-1/2)
Answer 

Find the principal value of tan-1 (-√3)

Find the principal value of tan^-1 (-√3)
Answer 

Find the principal value of tan^-1 (-√3)

Find the principal value of cosec -1 (2)

Find the principal value of cosec ^-1 (2)

Find the principal value of cosec^-1 (2)

Find the principal value of cos-1(√3/2)

Q-2 Find the principal value of cos^-1(√3/2)

Find the principal value of cos^-1(√3/2)

Find the principal values of the following: sin-1(-1/2)

Find the principal values of the following: sin^-1(-1/2)
Answer:

principal values of the function sin^-1(-1/2)

Control and Coordination

S.No.	Control and Coordination
	Intext questions page 119
1	What is the difference between a reflex action and walking?
2	What happens at the synapse between two neurons?
3	Which part of the brain maintains posture and equilibrium of the body?
4	How do we detect the smell of an agarbatti (incense stick)?
5	What is the role of the brain in reflex action?
	Intext questions page 122
1	What are plant hormones?
2	How is the movement of leaves of the sensitive plant different from themovement of a shoot towards light?
3	Give an example of a plant hormone that promotes growth.
4	How do auxins promote the growth of a tendril around a support?
5	Design an experiment to demonstrate hydrotropism.
	Intext questions page 125
1	How does chemical coordination take place in animals?
2	Why is the use of iodised salt advisable?
3	How does our body respond when adrenaline is secreted into the blood?
4	Why are some patients of diabetes treated by giving injections of insulin?
	Exercise Questions
1	Which of the following is a plant hormone?(a) Insulin(b) Thyroxin(c) Oestrogen(d) Cytokinin.
2	The gap between two neurons is called a (a) dendrite. (b) synapse. (c) axon. (d) impulse.
3	The brain is responsible for(a) thinking.(b) regulating the heart beat.(c) balancing the body.(d) all of the above.
4	What is the function of receptors in our body? Think of situations where receptorsdo not work properly. What problems are likely to arise?
5	Draw the structure of a neuron and explain its function.
6	How does phototropism occur in plants?
7	Which signals will get disrupted in case of a spinal cord injury?
8	How does chemical coordination occur in plants?
9	What is the need for a system of control and coordination in an organism?
10	How are involuntary actions and reflex actions different from each other?
11	Compare and contrast nervous and hormonal mechanisms for control andcoordination in animals.
12	What is the difference between the manner in which movement takes place in asensitive plant and the movement in our legs?

relations and functions class 11

S.No.	RELATIONS AND FUNCTIONS
	EXERCISE 2.1
1	If (x/3 +1 , y -2/3) = (5/3 , 1/3), find the values of x and y.
2	If the set A has 3 elements and the set B = {3, 4, 5}, then find the number of elements in (A×B).
3	If G = {7, 8} and H = {5, 4, 2}, find G × H and H × G.
4	State whether each of the following statements are true or false. If the statement is false, rewrite the given statement correctly. (i) If P = {m, n} and Q = { n, m}, then P × Q = {(m, n),(n, m)}. (ii) If A and B are non-empty sets, then A × B is a non-empty set of ordered pairs (x, y) such that x ∈ A and y ∈ B. (iii) If A = {1, 2}, B = {3, 4}, then A × (B ∩ φ) = φ.
5	If A = {–1, 1}, find A × A × A.
6	If A × B = {(a, x),(a , y), (b, x), (b, y)}. Find A and B
7	Let A = {1, 2}, B = {1, 2, 3, 4}, C = {5, 6} and D = {5, 6, 7, 8}. Verify that (i) A × (B ∩ C) = (A × B) ∩ (A × C). (ii) A × C is a subset of B × D.
8	Let A = {1, 2} and B = {3, 4}. Write A × B. How many subsets will A × B have? List them.
9	Let A and B be two sets such that n(A) = 3 and n(B) = 2. If (x, 1), (y, 2), (z, 1) are in A × B, find A and B, where x, y and z are distinct elements.
10	The Cartesian product A × A has 9 elements among which are found (–1, 0) and (0,1). Find the set A and the remaining elements of A × A.
..	EXERCISE 2.2
1	Let A = {1, 2, 3,...,14}. Define a relation R from A to A by R = {(x, y) : 3x – y = 0, where x, y ∈ A}. Write down its domain, codomain and range.
2	Define a relation R on the set N of natural numbers by R = {(x, y) : y = x + 5, x is a natural number less than 4; x, y ∈N}. Depict this relationship using roster form. Write down the domain and the range.
3	A = {1, 2, 3, 5} and B = {4, 6, 9}. Define a relation R from A to B by R = {(x, y): the difference between x and y is odd; x ∈ A, y ∈ B}. Write R in roster form.
4	The Fig2.7 shows a relationship between the sets P and Q. Write this relation (i) in set-builder form (ii) roster form. What is its domain and range?
5	Let A = {1, 2, 3, 4, 6}. Let R be the relation on A defined by {(a, b): a , b ∈A, b is exactly divisible by a}. (i) Write R in roster form (ii) Find the domain of R (iii) Find the range of R.
6	Determine the domain and range of the relation R defined by R = {(x, x + 5) : x ∈ {0, 1, 2, 3, 4, 5}}.
7	Write the relation R = {(x, x3) : x is a prime number less than 10} in roster form.
8	Let A = {x, y, z} and B = {1, 2}. Find the number of relations from A to B.
9	Let R be the relation on Z defined by R = {(a,b): a, b ∈ Z, a – b is an integer}. Find the domain and range of R.
..	EXERCISE 2.3
1	Which of the following relations are functions? Give reasons. If it is a function, determine its domain and range. (i) {(2,1), (5,1), (8,1), (11,1), (14,1), (17,1)} (ii) {(2,1), (4,2), (6,3), (8,4), (10,5), (12,6), (14,7)} (iii) {(1,3), (1,5), (2,5)}.
2	Find the domain and range of the following real functions: (i) f(x) = –| x| (ii) f(x) = root (9 − x2).
3	A function f is defined by f(x) = 2x –5. Write down the values of (i) f (0), (ii) f (7), (iii) f (–3).
4	The function ‘t’ which maps temperature in degree Celsius into temperature in degree Fahrenheit is defined by t(C) = 9/5C + 32. Find (i) t(0) (ii) t(28) (iii) t(–10) (iv) The value of C, when t(C) = 212.
5	Find the range of each of the following functions. (i) f (x) = 2 – 3x, x ∈ R, x > 0. (ii) f (x) = x2 + 2, x is a real number. (iii) f (x) = x, x is a real number.
..	Miscellaneous Exercise on Chapter 2
1	The relation f is defined by The relation g is defined by Show that f is a function and g is not a function. (solution of Question 1 )
2	If f (x) = x2, find f ......
3	Find the domain of the function f (x) ....
4	Find the domain and the range of the real function f defined by f (x) = root (x-1)
5	Find the domain and the range of the real function f defined by f (x) = |x –1| .
6	let f .... be a function from R into R. Determine the range of f.
7	Let f, g : R→R be defined, respectively by f(x) = x + 1, g(x) = 2x – 3. Find f + g, f – g and f/g
8	Let f = {(1,1), (2,3), (0,–1), (–1, –3)} be a function from Z to Z defined by f(x) = ax + b, for some integers a, b. Determine a, b.
9	Let R be a relation from N to N defined by R = {(a, b) : a, b ∈N and a = b2}. Are the following true? (i) (a,a) ∈ R, for all a ∈ N (ii) (a,b) ∈ R, implies (b,a) ∈ R (iii) (a,b) ∈ R, (b,c) ∈ R implies (a,c) ∈ R. Justify your answer in each case.
10	Let A ={1,2,3,4}, B = {1,5,9,11,15,16} and f = {(1,5), (2,9), (3,1), (4,5), (2,11)} Are the following true? (i) f is a relation from A to B (ii) f is a function from A to B. Justify your answer in each case.
11	Let f be the subset of Z × Z defined by f = {(ab, a + b) : a, b ∈ Z}. Is f a function from Z to Z? Justify your answer.
12	Let A = {9,10,11,12,13} and let f : A→N be defined by f (n) = the highest prime factor of n. Find the range of f.

ncert solution for class 11 maths chapter 1 sets theory

All definitions and concept of sets theory. 

S.No.	Sets
	Exercise 1.1
1	Which of the following are sets ? Justify your asnwer.(i) The collection of all the months of a year beginning with the letter J.(ii) The collection of ten most talented writers of India.(iii) A team of eleven best-cricket batsmen of the world.(iv) The collection of all boys in your class.(v) The collection of all natural numbers less than 100.(vi) A collection of novels written by the writer Munshi Prem Chand.(vii) The collection of all even integers. (viii) The collection of questions in this Chapter.(ix) A collection of most dangerous animals of the world.
2	(ix) A collection of most dangerous animals of the world.2. Let A = {1, 2, 3, 4, 5, 6}. Insert the appropriate symbol ∈ or ∉ in the blankspaces:(i) 5. . .A (ii) 8 . . . A (iii) 0. . .A(iv) 4. . . A (v) 2. . .A (vi) 10. . .A
3	Write the following sets in roster form:(i) A = {x : x is an integer and –3 < x < 7}(ii) B = {x : x is a natural number less than 6}(iii) C = {x : x is a two-digit natural number such that the sum of its digits is 8}(iv) D = {x : x is a prime number which is divisor of 60}(v) E = The set of all letters in the word TRIGONOMETRY(vi) F = The set of all letters in the word BETTER
4	Write the following sets in the set-builder form :(i) (3, 6, 9, 12} (ii) {2,4,8,16,32} (iii) {5, 25, 125, 625}(iv) {2, 4, 6, . . .} (v) {1,4,9, . . .,100}
5	List all the elements of the following sets :(i) A = {x : x is an odd natural number}(ii) B = {x : x is an integer,12– < x <92 }(iii) C = {x : x is an integer, x2 ≤ 4}(iv) D = {x : x is a letter in the word “LOYAL”}(v) E = {x : x is a month of a year not having 31 days}(vi) F = {x : x is a consonant in the English alphabet which precedes k }.
6	Match each of the set on the left in the roster form with the same set on the rightdescribed in set-builder form:(i) {1, 2, 3, 6} (a) {x : x is a prime number and a divisor of 6}(ii) {2, 3} (b) {x : x is an odd natural number less than 10}(iii) {M,A,T,H,E,I,C,S} (c) {x : x is natural number and divisor of 6}(iv) {1, 3, 5, 7, 9} (d) {x : x is a letter of the word MATHEMATICS}.
	EXERCISE 1.2
1	Which of the following sets are finite or infinite (i) The set of months of a year (ii) {1, 2, 3, . . .} (iii) {1, 2, 3, . . .99, 100} (iv) The set of positive integers greater than 100 (v) The set of prime numbers less than 99
2	State whether each of the following set is finite or infinite: (i) The set of lines which are parallel to the x-axis (ii) The set of letters in the English alphabet (iii) The set of numbers which are multiple of 5 (iv) The set of animals living on the earth (v) The set of circles passing through the origin (0,0)
3	In the following, state whether A = B or not: (i) A = { a, b, c, d } B = { d, c, b, a } (ii) A = { 4, 8, 12, 16 } B = { 8, 4, 16, 18} (iii) A = {2, 4, 6, 8, 10} B = { x : x is positive even integer and x ≤ 10} (iv) A = { x : x is a multiple of 10}, B = { 10, 15, 20, 25, 30, . . . }
4	Are the following pair of sets equal ? Give reasons. (i) A = {2, 3}, B = {x : x is solution of x2 + 5x + 6 = 0} (ii) A = { x : x is a letter in the word FOLLOW} B = { y : y is a letter in the word WOLF}
5	From the sets given below, select equal sets :A = { 2, 4, 8, 12}, B = { 1, 2, 3, 4}, C = { 4, 8, 12, 14}, D = { 3, 1, 4, 2}E = {–1, 1}, F = { 0, a}, G = {1, –1}, H = { 0, 1}
	EXERCISE 1.3
1	Make correct statements by filling in the symbols ⊂ or ⊄ in the blank spaces :(i) { 2, 3, 4 } . . . { 1, 2, 3, 4,5 } (ii) { a, b, c } . . . { b, c, d }(iii) {x : x is a student of Class XI of your school}. . .{x : x student of your school}(iv) {x : x is a circle in the plane} . . .{x : x is a circle in the same plane withradius 1 unit}(v) {x : x is a triangle in a plane} . . . {x : x is a rectangle in the plane}(vi) {x : x is an equilateral triangle in a plane} . . . {x : x is a triangle in the same plane}(vii) {x : x is an even natural number} . . . {x : x is an integer}
2	Examine whether the following statements are true or false:(i) { a, b } ⊄ { b, c, a }(ii) { a, e } ⊂ { x : x is a vowel in the English alphabet}(iii) { 1, 2, 3 } ⊂ { 1, 3, 5 }(iv) { a }⊂ { a, b, c }(v) { a }∈ { a, b, c }(vi) { x : x is an even natural number less than 6} ⊂ { x : x is a natural numberwhich divides 36}
3	Write down all the subsets of the following sets(i) {a} (ii) {a, b} (iii) {1, 2, 3} (iv) φ
4	How many elements has P(A), if A = φ?
5	Write the following as intervals : (i) {x : x ∈ R, – 4 < x ≤ 6} (ii) {x : x ∈ R, – 12 < x < –10} (iii) {x : x ∈ R, 0 ≤ x < 7} (iv) {x : x ∈ R, 3 ≤ x ≤ 4}
6	Write the following intervals in set-builder form : (i) (– 3, 0) (ii) [6 , 12] (iii) (6, 12] (iv) [–23, 5)
7	What universal set(s) would you propose for each of the following : (i) The set of right triangles. (ii) The set of isosceles triangles
8	Given the sets A = {1, 3, 5}, B = {2, 4, 6} and C = {0, 2, 4, 6, 8}, which of the following may be considered as universal set (s) for all the three sets A, B and C(i) {0, 1, 2, 3, 4, 5, 6}(ii) φ(iii) {0,1,2,3,4,5,6,7,8,9,10}(iv) {1,2,3,4,5,6,7,8}
	EXERCISE 1.4
1	Find the union of each of the following pairs of sets :(i) X = {1, 3, 5} Y = {1, 2, 3}(ii) A = [ a, e, i, o, u} B = {a, b, c}(iii) A = {x : x is a natural number and multiple of 3}B = {x : x is a natural number less than 6}(iv) A = {x : x is a natural number and 1 < x ≤ 6 }B = {x : x is a natural number and 6 < x < 10 }(v) A = {1, 2, 3}, B = φ
2	Let A = { a, b }, B = {a, b, c}. Is A ⊂ B ? What is A ∪ B ?
3	If A and B are two sets such that A ⊂ B, then what is A ∪ B ?
4	If A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, C = {5, 6, 7, 8 }and D = { 7, 8, 9, 10 }; find (i) A ∪ B (ii) A ∪ C (iii) B ∪ C (iv) B ∪ D(v) A ∪ B ∪ C (vi) A ∪ B ∪ D (vii) B ∪ C ∪ D
5	Find the intersection of each pair of sets of question 1 above.
6	If A = { 3, 5, 7, 9, 11 }, B = {7, 9, 11, 13}, C = {11, 13, 15}and D = {15, 17}; find(i) A ∩ B (ii) B ∩ C (iii) A ∩ C ∩ D(iv) A ∩ C (v) B ∩ D (vi) A ∩ (B ∪ C)(vii) A ∩ D (viii) A ∩ (B ∪ D) (ix) ( A ∩ B ) ∩ ( B ∪ C )(x) ( A ∪ D) ∩ ( B ∪ C)
7	If A = {x : x is a natural number }, B = {x : x is an even natural number} C = {x : x is an odd natural number}andD = {x : x is a prime number }, find (i) A ∩ B (ii) A ∩ C (iii) A ∩ D (iv) B ∩ C (v) B ∩ D (vi) C ∩ D
8	Which of the following pairs of sets are disjoint (i) {1, 2, 3, 4} and {x : x is a natural number and 4 ≤ x ≤ 6 } (ii) { a, e, i, o, u } and { c, d, e, f } (iii) {x : x is an even integer } and {x : x is an odd integer}
9	If A = {3, 6, 9, 12, 15, 18, 21}, B = { 4, 8, 12, 16, 20 },C = { 2, 4, 6, 8, 10, 12, 14, 16 }, D = {5, 10, 15, 20 }; find(i) A – B (ii) A – C (iii) A – D (iv) B – A(v) C – A (vi) D – A (vii) B – C (viii) B – D(ix) C – B (x) D – B (xi) C – D (xii) D – C
10	If X= { a, b, c, d } and Y = { f, b, d, g}, find(i) X – Y (ii) Y – X (iii) X ∩ Y
11	If R is the set of real numbers and Q is the set of rational numbers, then what is R – Q?
12	State whether each of the following statement is true or false. Justify your answer. (i) { 2, 3, 4, 5 } and { 3, 6} are disjoint sets. (ii) { a, e, i, o, u } and { a, b, c, d }are disjoint sets. (iii) { 2, 6, 10, 14 } and { 3, 7, 11, 15} are disjoint sets. (iv) { 2, 6, 10 } and { 3, 7, 11} are disjoint sets.
	EXERCISE 1.5
1	Let U = { 1, 2, 3, 4, 5, 6, 7, 8, 9 }, A = { 1, 2, 3, 4}, B = { 2, 4, 6, 8 } andC = { 3, 4, 5, 6 }. Find (i) A′ (ii) B′ (iii) (A ∪ C)′ (iv) (A ∪ B)′ (v) (A′)′(vi) (B – C)′
2	If U = { a, b, c, d, e, f, g, h}, find the complements of the following sets :(i) A = {a, b, c} (ii) B = {d, e, f, g}(iii) C = {a, c, e, g} (iv) D = { f, g, h, a}
3	Taking the set of natural numbers as the universal set, write down the complementsof the following sets:(i) {x : x is an even natural number} (ii) { x : x is an odd natural number }(iii) {x : x is a positive multiple of 3} (iv) { x : x is a prime number }(v) {x : x is a natural number divisible by 3 and 5}(vi) { x : x is a perfect square } (vii) { x : x is a perfect cube}(viii) { x : x + 5 = 8 } (ix) { x : 2x + 5 = 9}(x) { x : x ≥ 7 } (xi) { x : x ∈ N and 2x + 1 > 10 }
4	If U = {1, 2, 3, 4, 5, 6, 7, 8, 9 }, A = {2, 4, 6, 8} and B = { 2, 3, 5, 7}. Verify that (i) (A ∪ B)′ = A′ ∩ B′ (ii) (A ∩ B)′ = A′ ∪ B′
5	Draw appropriate Venn diagram for each of the following : (i) (A ∪ B)′, (ii) A′ ∩ B′, (iii) (A ∩ B)′, (iv) A′ ∪ B′
6	Let U be the set of all triangles in a plane. If A is the set of all triangles with at least one angle different from 60°, what is A′?
7	Fill in the blanks to make each of the following a true statement : (i) A ∪ A′ = . . . (ii) φ′ ∩ A = . . . (iii) A ∩ A′ = . . . (iv) U′ ∩ A = . . .
	EXERCISE 1.6
1	If X and Y are two sets such that n ( X ) = 17, n ( Y ) = 23 and n ( X ∪ Y ) = 38,find n ( X ∩ Y ).
2	If X and Y are two sets such that X ∪ Y has 18 elements, X has 8 elements andY has 15 elements ; how many elements does X ∩ Y have?
3	In a group of 400 people, 250 can speak Hindi and 200 can speak English. Howmany people can speak both Hindi and English?
4	If S and T are two sets such that S has 21 elements, T has 32 elements, and S ∩ T has 11 elements, how many elements does S ∪ T have?
5	If X and Y are two sets such that X has 40 elements, X ∪ Y has 60 elements andX ∩ Y has 10 elements, how many elements does Y have?
6	In a group of 70 people, 37 like coffee, 52 like tea and each person likes at leastone of the two drinks. How many people like both coffee and tea?
7	In a group of 65 people, 40 like cricket, 10 like both cricket and tennis. How many like tennis only and not cricket? How many like tennis?
8	In a committee, 50 people speak French, 20 speak Spanish and 10 speak bothSpanish and French. How many speak at least one of these two languages?
.....	Miscellaneous Exercise on Chapter 1
1	Decide, among the following sets, which sets are subsets of one and another: A = { x : x ∈ R and x satisfy x2 – 8x + 12 = 0 }, B = { 2, 4, 6 }, C = { 2, 4, 6, 8, . . . }, D = { 6 }.
2	In each of the following, determine whether the statement is true or false. If it istrue, prove it. If it is false, give an example.(i) If x ∈ A and A ∈ B , then x ∈ B(ii) If A ⊂ B and B ∈ C , then A ∈ C(iii) If A ⊂ B and B ⊂ C , then A ⊂ C(iv) If A ⊄ B and B ⊄ C , then A ⊄ C(v) If x ∈ A and A ⊄ B , then x ∈ B(vi) If A ⊂ B and x ∉ B , then x ∉ A
3	Let A, B, and C be the sets such that A ∪ B = A ∪ C and A ∩ B = A ∩ C. Showthat B = C.
4	Show that the following four conditions are equivalent :(i) A ⊂ B(ii) A – B = φ (iii) A ∪ B = B (iv) A ∩ B = A
5	Show that if A ⊂ B, then C – B ⊂ C – A.
6	Assume that P ( A ) = P ( B ). Show that A = B
7	Is it true that for any sets A and B, P ( A ) ∪ P ( B ) = P ( A ∪ B )? Justify youranswer.
8	Show that for any sets A and B, A = ( A ∩ B ) ∪ ( A – B ) and A ∪ ( B – A ) = ( A ∪ B )
9	Using properties of sets, show that (i) A ∪ ( A ∩ B ) = A (ii) A ∩ ( A ∪ B ) = A.
10	Show that A ∩ B = A ∩ C need not imply B = C.
11	Let A and B be sets. If A ∩ X = B ∩ X = φ and A ∪ X = B ∪ X for some set X, show that A = B.
12	Find sets A, B and C such that A ∩ B, B ∩ C and A ∩ C are non-emptysets and A ∩ B ∩ C = φ.
13	In a survey of 600 students in a school, 150 students were found to be taking teaand 225 taking coffee, 100 were taking both tea and coffee. Find how manystudents were taking neither tea nor coffee?
14	In a group of students, 100 students know Hindi, 50 know English and 25 knowboth. Each of the students knows either Hindi or English. How many studentsare there in the group?
15	In a survey of 60 people, it was found that 25 people read newspaper H, 26 readnewspaper T, 26 read newspaper I, 9 read both H and I, 11 read both H and T,8 read both T and I, 3 read all three newspapers. Find:(i) the number of people who read at least one of the newspapers.(ii) the number of people who read exactly one newspaper.
16	In a survey it was found that 21 people liked product A, 26 liked product B and29 liked product C. If 14 people liked products A and B, 12 people liked productsC and A, 14 people liked products B and C and 8 liked all the three products.Find how many liked product C only.

Electrostatic Potential and Capacitance

S.No.	Electrostatic Potential and Capacitance
	Exercise Quetions
1	Two charges 5 × 10–8 C and –3 × 10–8 C are located 16 cm apart. At what point(s) on the line joining the two charges is the electric potential zero? Take the potential at infinity to be zero.
2	A regular hexagon of side 10 cm has a charge 5 μC at each of its vertices. Calculate the potential at the centre of the hexagon.
3	Two charges 2 μC and –2 μC are placed at points A and B 6 cm apart. (a) Identify an equipotential surface of the system. (b) What is the direction of the electric field at every point on this surface?
4	A spherical conductor of radius 12 cm has a charge of 1.6 × 10–7C distributed uniformly on its surface. What is the electric field (a) inside the sphere (b) just outside the sphere (c) at a point 18 cm from the centre of the sphere?
5	A parallel plate capacitor with air between the plates has a capacitance of 8 pF (1pF = 10–12 F). What will be the capacitance if the distance between the plates is reduced by half, and the space between them is filled with a substance of dielectric constant 6?
6	Three capacitors each of capacitance 9 pF are connected in series. (a) What is the total capacitance of the combination? (b) What is the potential difference across each capacitor if the combination is connected to a 120 V supply?
7	Three capacitors of capacitances 2 pF, 3 pF and 4 pF are connected in parallel. (a) What is the total capacitance of the combination? (b) Determine the charge on each capacitor if the combination is connected to a 100 V supply.
8	In a parallel plate capacitor with air between the plates, each plate has an area of 6 × 10–3 m2 and the distance between the plates is 3 mm. Calculate the capacitance of the capacitor. If this capacitor is connected to a 100 V supply, what is the charge on each plate of the capacitor?
9	Explain what would happen if in the capacitor given in Exercise 2.8, a 3 mm thick mica sheet (of dielectric constant = 6) were inserted between the plates, (a) while the voltage supply remained connected. (b) after the supply was disconnected.
10	A 12pF capacitor is connected to a 50V battery. How much electrostatic energy is stored in the capacitor?
11	A 600pF capacitor is charged by a 200V supply. It is then disconnected from the supply and is connected to another uncharged 600 pF capacitor. How much electrostatic energy is lost in the process?
	Additional Excercises
12	A charge of 8 mC is located at the origin. Calculate the work done in taking a small charge of –2 × 10–9 C from a point P (0, 0, 3 cm) to a point Q (0, 4 cm, 0), via a point R (0, 6 cm, 9 cm).
13	A cube of side b has a charge q at each of its vertices. Determine the potential and electric field due to this charge array at the centre of the cube.
14	Two tiny spheres carrying charges 1.5 μC and 2.5 μC are located 30 cm apart. Find the potential and electric field: (a) at the mid-point of the line joining the two charges, and (b) at a point 10 cm from this midpoint in a plane normal to the line and passing through the mid-point.
15	A spherical conducting shell of inner radius r1 and outer radius r2 has a charge Q. (a) A charge q is placed at the centre of the shell. What is the surface charge density on the inner and outer surfaces of the shell? (b) Is the electric field inside a cavity (with no charge) zero, even if the shell is not spherical, but has any irregular shape? Explain.
16	(a) Show that the normal component of electrostatic field has a discontinuity from one side of a charged surface to another given by 2 1 0 ( ) ˆ σ ε E − E n = where ˆn is a unit vector normal to the surface at a point and σ is the surface charge density at that point. (The direction of ˆn is from side 1 to side 2.) Hence show that just outside a conductor, the electric field is σ ˆn /ε0. (b) Show that the tangential component of electrostatic field is continuous from one side of a charged surface to another. [Hint: For (a), use Gauss’s law. For, (b) use the fact that work done by electrostatic field on a closed loop is zero.]
17	A long charged cylinder of linear charged density λ is surrounded by a hollow co-axial conducting cylinder. What is the electric field in the space between the two cylinders?
18	In a hydrogen atom, the electron and proton are bound at a distance of about 0.53 Å: (a) Estimate the potential energy of the system in eV, taking the zero of the potential energy at infinite separation of the electron from proton. (b) What is the minimum work required to free the electron, given that its kinetic energy in the orbit is half the magnitude of potential energy obtained in (a)? (c) What are the answers to (a) and (b) above if the zero of potential energy is taken at 1.06 Å separation?
19	If one of the two electrons of a H2 molecule is removed, we get a hydrogen molecular ion H+ 2. In the ground state of an H+ 2, the two protons are separated by roughly 1.5 Å, and the electron is roughly 1 Å from each proton. Determine the potential energy of the system. Specify your choice of the zero of potential energy.
20	Two charged conducting spheres of radii a and b are connected to each other by a wire. What is the ratio of electric fields at the surfaces of the two spheres? Use the result obtained to explain why charge density on the sharp and pointed ends of a conductor is higher than on its flatter portions.
21	Two charges –q and +q are located at points (0, 0, –a) and (0, 0, a), respectively. (a) What is the electrostatic potential at the points (0, 0, z) and (x, y, 0) ? (b) Obtain the dependence of potential on the distance r of a point from the origin when r/a >> 1. (c) How much work is done in moving a small test charge from the point (5,0,0) to (–7,0,0) along the x-axis? Does the answer change if the path of the test charge between the same points is not along the x-axis?
22	Figure 2.34 shows a charge array known as an electric quadrupole. For a point on the axis of the quadrupole, obtain the dependence of potential on r for r/a >> 1, and contrast your results with that due to an electric dipole, and an electric monopole (i.e., a single charge)
23	An electrical technician requires a capacitance of 2 μF in a circuit across a potential difference of 1 kV. A large number of 1 μF capacitors are available to him each of which can withstand a potential difference of not more than 400 V. Suggest a possible arrangement that requires the minimum number of capacitors.
24	What is the area of the plates of a 2 F parallel plate capacitor, given that the separation between the plates is 0.5 cm? [You will realise from your answer why ordinary capacitors are in the range of μF or less. However, electrolytic capacitors do have a much larger capacitance (0.1 F) because of very minute separation between the conductors.]
25	Obtain the equivalent capacitance of the network in Fig. 2.35. For a 300 V supply, determine the charge and voltage across each capacitor.
26	The plates of a parallel plate capacitor have an area of 90 cm2 each and are separated by 2.5 mm. The capacitor is charged by connecting it to a 400 V supply. (a) How much electrostatic energy is stored by the capacitor? (b) View this energy as stored in the electrostatic field between the plates, and obtain the energy per unit volume u. Hence arrive at a relation between u and the magnitude of electric field E between the plates.
27	A 4 μF capacitor is charged by a 200 V supply. It is then disconnected from the supply, and is connected to another uncharged 2 μF capacitor. How much electrostatic energy of the first capacitor is lost in the form of heat and electromagnetic radiation?
28	Show that the force on each plate of a parallel plate capacitor has a magnitude equal to (½) QE, where Q is the charge on the capacitor, and E is the magnitude of electric field between the plates. Explain the origin of the factor ½.
29	A spherical capacitor consists of two concentric spherical conductors, held in position by suitable insulating supports (Fig. 2.36). Show that the capacitance of a spherical capacitor is given by 0 1 2 1 2 4 – r r C r r πε = where r1 and r2 are the radii of outer and inner spheres, respectively.
30	A spherical capacitor has an inner sphere of radius 12 cm and an outer sphere of radius 13 cm. The outer sphere is earthed and the inner sphere is given a charge of 2.5 μC. The space between the concentric spheres is filled with a liquid of dielectric constant 32. (a) Determine the capacitance of the capacitor. (b) What is the potential of the inner sphere? (c) Compare the capacitance of this capacitor with that of an isolated sphere of radius 12 cm. Explain why the latter is much smaller.
31	Answer carefully: (a) Two large conducting spheres carrying charges Q1 and Q2 are brought close to each other. Is the magnitude of electrostatic force between them exactly given by Q1 Q2/4πε0r 2, where r is the distance between their centres? (b) If Coulomb’s law involved 1/r3 dependence (instead of 1/r2), would Gauss’s law be still true ? (c) A small test charge is released at rest at a point in an electrostatic field configuration. Will it travel along the field line passing through that point? (d) What is the work done by the field of a nucleus in a complete circular orbit of the electron? What if the orbit is elliptical? (e) We know that electric field is discontinuous across the surface of a charged conductor. Is electric potential also discontinuous there? (f ) What meaning would you give to the capacitance of a single conductor? (g) Guess a possible reason why water has a much greater dielectric constant (= 80) than say, mica (= 6).
32	A cylindrical capacitor has two co-axial cylinders of length 15 cm and radii 1.5 cm and 1.4 cm. The outer cylinder is earthed and the inner cylinder is given a charge of 3.5 μC. Determine the capacitance of the system and the potential of the inner cylinder. Neglect end effects (i.e., bending of field lines at the ends).
33	A parallel plate capacitor is to be designed with a voltage rating 1 kV, using a material of dielectric constant 3 and dielectric strength about 107 Vm–1. (Dielectric strength is the maximum electric field a material can tolerate without breakdown, i.e., without starting to conduct electricity through partial ionisation.) For safety, we should like the field never to exceed, say 10% of the dielectric strength. What minimum area of the plates is required to have a capacitance of 50 pF?
34	Describe schematically the equipotential surfaces corresponding to (a) a constant electric field in the z-direction, (b) a field that uniformly increases in magnitude but remains in a constant (say, z) direction, (c) a single positive charge at the origin, and (d) a uniform grid consisting of long equally spaced parallel charged wires in a plane.
35	In a Van de Graaff type generator a spherical metal shell is to be a 15 × 106 V electrode. The dielectric strength of the gas surrounding the electrode is 5 × 107 Vm–1. What is the minimum radius of the spherical shell required? (You will learn from this exercise why one cannot build an electrostatic generator using a very small shell which requires a small charge to acquire a high potential.)
36	A small sphere of radius r1 and charge q1 is enclosed by a spherical shell of radius r2 and charge q2. Show that if q1 is positive, charge will necessarily flow from the sphere to the shell (when the two are connected by a wire) no matter what the charge q2 on the shell is.
37	Answer the following: (a) The top of the atmosphere is at about 400 kV with respect to the surface of the earth, corresponding to an electric field that decreases with altitude. Near the surface of the earth, the field is about 100 Vm–1. Why then do we not get an electric shock as we step out of our house into the open? (Assume the house to be a steel cage so there is no field inside!) (b) A man fixes outside his house one evening a two metre high insulating slab carrying on its top a large aluminium sheet of area 1m2. Will he get an electric shock if he touches the metal sheet next morning? (c) The discharging current in the atmosphere due to the small conductivity of air is known to be 1800 A on an average over the globe. Why then does the atmosphere not discharge itself completely in due course and become electrically neutral? In other words, what keeps the atmosphere charged? (d) What are the forms of energy into which the electrical energy of the atmosphere is dissipated during a lightning? (Hint: The earth has an electric field of about 100 Vm–1 at its surface in the downward direction, corresponding to a surface charge density = –10–9 C m–2. Due to the slight conductivity of the atmosphere up to about 50 km (beyond which it is good conductor), about + 1800 C is pumped every second into the earth as a whole. The earth, however, does not get discharged since thunderstorms and lightning occurring continually all over the globe pump an equal amount of negative charge on the earth.)