Monthly Archives: December 2021

In This Post we are  providing Chapter-5 COMPLEX NUMBERS AND QUADRATIC EQUATIONS NCERT MOST IMPORTANT QUESTIONS for Class 11 MATHS 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 MOST IMPORTANT QUESTIONS ON COMPLEX NUMBERS AND QUADRATIC EQUATIONS

1. Evaluate i^-39

Ans. 

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2. Solved the quadratic equation 

Ans. 

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3. If = 1, then find the least positive integral value of m.

Ans. 

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4. Evaluate (1+ i)⁴

Ans. 

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5. Find the modulus of 

Ans. Let z = 

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6. Express in the form of a + ib. (1+3i)^-1

Ans. 

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7. Explain the fallacy in -1 = i. i. = 

Ans.  is okay but

 is wrong.

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8. Find the conjugate of 

Ans. Let z = 

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9. Find the conjugate of – 3i – 5.

Ans. Let z = 3i – 5

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10. Let z₁ = 2 – i, z₂ = -2+i Find Re 

Ans. z₁ z₂ = (2 – i)(-2 + i)

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11. Express in the form of a + ib (3i-7) + (7-4i) – (6+3i) + i²³

Ans. Let

Z = 

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12. Find the conjugate of 

Ans. 

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13. Solve for x and y, 3x + (2x-y) i= 6 – 3i

Ans. 3x = 6

x = 2

2x – y = – 3

2 × 2 – y = – 3

– y = – 3 – 4

y = 7

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14. Find the value of 1+i² + i⁴ + i⁶ + i⁸ + —- + i²⁰

Ans.

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15. Multiply 3-2i by its conjugate.

Ans.Let z = 3 – 2i

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In This Post we are  providing Chapter-4 PRINCIPLE ON MATHEMATICAL INDUCTION NCERT MOST IMPORTANT QUESTIONS for Class 11 MATHS 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 MOST IMPORTANT QUESTIONS ON PRINCIPLE ON MATHEMATICAL INDUCTION

Q1. Give an example of a statement P(n) which is true for all n≥ 4 but P(l), P(2) and P(3) are not true. Justify your answer

Sol. Consider the statement P(n): 3n < n!

For n = 1, 3 x 1 < 1!, which is not true
For n = 2, 3 x 2 < 2!, which is not true
For n = 3, 3 x 3 < 3!, which is not true
For n = 4, 3 x 4 < 4!, which is true
For n = 5, 3 x 5 < 5!, which is true

Q2. Prove that number of subsets of a set containing n distinct elements is 2″, for all n ∈
Sol: Let P(n): Number of subset of a set containing n distinct elements is 2″, for all ne N.
For n = 1, consider set A = {1}. So, set of subsets is {{1}, ∅}, which contains 2¹ elements.
So, P(1) is true.
Let us assume that P(n) is true, for some natural number n = k.
P(k): Number of subsets of a set containing k distinct elements is 2^k To prove that P(k + 1) is true,
we have to show that P(k + 1): Number of subsets of a set containing (k + 1) distinct elements is 2^k+1
We know that, with the addition of one element in the set, the number of subsets become double.
Number of subsets of a set containing (k+ 1) distinct elements = 2×2^k = 2^k+1
So, P(k + 1) is true. Hence, P(n) is true.

Q3. 4ⁿ – 1 is divisible by 3, for each natural number
Sol: Let P(n): 4ⁿ – 1 is divisible by 3 for each natural number n.
Now, P(l): 4¹ – 1 = 3, which is divisible by 3 Hence, P(l) is true.
Let us assume that P(n) is true for some natural number n = k.
P(k): 4^k – 1 is divisible by 3
or               4^k – 1 = 3m, m∈ N  (i)
Now, we have to prove that P(k + 1) is true.
P(k+ 1): 4^k+1 – 1
= 4^k-4-l
= 4(3m + 1) – 1  [Using (i)]
= 12 m + 3
= 3(4m + 1), which is divisible by 3 Thus, P(k + 1) is true whenever P(k) is true.
Hence, by the principle of mathematical induction P(n) is true for all natural numbers n.

Q4. 2³ⁿ – 1 is divisible by 7, for all natural numbers
Sol: Let P(n): 2³ⁿ – 1 is divisible by 7
Now, P( 1): 2³ — 1 = 7, which is divisible by 7.
Hence, P(l) is true.
Let us assume that P(n) is true for some natural number n = k.
P(k): 2^3k – 1 is divisible by 7.
or               2^3k -1 = 7m, m∈ N  (i)
Now, we have to prove that P(k + 1) is true.
P(k+ 1): 2^3(k+1)– 1
= 2^3k.2³– 1
= 8(7 m + 1) – 1
= 56 m + 7
= 7(8m + 1), which is divisible by 7.
Thus, P(k + 1) is true whenever P(k) is true.
So, by the principle of mathematical induction P(n) is true for all natural numbers n.

Q5. n³ – 7n + 3 is divisible by 3, for all natural numbers
Sol: Let P(n): n³ – 7n + 3 is divisible by 3, for all natural numbers n.
Now P(l): (l)³ – 7(1) + 3 = -3, which is divisible by 3.
Hence, P(l) is true.
Let us assume that P(n) is true for some natural number n = k.
P(k) = K³ – 7k + 3 is divisible by 3
or K³ – 7k + 3 = 3m, m∈ N         (i)
Now, we have to prove that P(k + 1) is true.
P(k+ 1 ):(k + l)³ – 7(k + 1) + 3
= k³ + 1 + 3k(k + 1) – 7k— 7 + 3 = k³ -7k + 3 + 3k(k + l)-6
= 3m + 3[k(k+l)-2]  [Using (i)]
= 3[m + (k(k + 1) – 2)], which is divisible by 3 Thus, P(k + 1) is true whenever P(k) is true.
So, by the principle of mathematical induction P(n) is true for all natural numbers n

Q6. 3²ⁿ – 1 is divisible by 8, for all natural numbers
Sol: Let P(n): 3²ⁿ – 1 is divisible by 8, for all natural numbers n.
Now, P(l): 3² – 1 = 8, which is divisible by 8.
Hence, P(l) is true.
Let us assume that, P(n) is true for some natural number n = k.
P(k): 3^2k – 1 is divisible by 8
or               3^2k -1 = 8m, m ∈ N  (i)
Now, we have to prove that P(k + 1) is true.
P(k+ 1): 3^2(k+1)– l
= 3^2k • 3² — 1
= 9(8m + 1) – 1     (using (i))
= 72m + 9 – 1
= 72m + 8
= 8(9m +1), which is divisible by 8 Thus P(k + 1) is true whenever P(k) is true.
So, by the principle of mathematical induction P(n) is true for all natural numbers n.

Q7. For any natural number n, 7ⁿ – 2ⁿ is divisible by 5.
Sol: Let P(n): 7ⁿ – 2ⁿ is divisible by 5, for any natural number n.
Now, P(l) = 7¹-2¹ = 5, which is divisible by 5.
Hence, P(l) is true.
Let us assume that, P(n) is true for some natural number n = k.

.’.  P(k) = 7^k -2^k is divisible by 5
or  7^k – 2^k = 5m, m∈ N                                                                           (i)
Now, we have to prove that P(k + 1) is true.
P(k+ 1): 7^k+1 -2^k+1
= 7^k-7-2^k-2
= (5 + 2)7^k -2^k-2
= 5.7^k + 2.7^k-2-2^k
= 5.7^k + 2(7^k – 2^k)
= 5 • 7^k + 2(5 m)     (using (i))
= 5(7^k + 2m), which divisible by 5.
Thus, P(k + 1) is true whenever P(k) is true.
So, by the principle of mathematical induction P(n) is true for all natural numbers n.

Q8. For any natural number n, xⁿ -yⁿ is divisible by x -y, where x and y are any integers with x ≠y
Sol: Let P(n) : xⁿ – yⁿ is divisible by x – y, where x and y are any integers with x≠y.
Now, P(l): x¹ -y¹ = x-y, which is divisible by (x-y)
Hence, P(l) is true.
Let us assume that, P(n) is true for some natural number n = k.
P(k): x^k -y^k is divisible by (x – y)
or   x^k-y^k = m(x-y),m ∈ N …(i)
Now, we have to prove that P(k + 1) is true.
P(k+l):x^k+l-y^k+l
= x^k-x-x^k-y + x^k-y-y^ky
= x^k(x-y) +y(x^k-y^k)
= x^k(x – y) + ym(x – y)  (using (i))
= (x -y) [x^k+ym], which is divisible by (x-y)
Hence, P(k + 1) is true whenever P(k) is true.
So, by the principle of mathematical induction P(n) is true for any natural number n.

Q9. n³ -n is divisible by 6, for each natural number n≥
Sol: Let P(n): n³ – n is divisible by 6, for each natural number n> 2.
Now, P(2): (2)³ -2 = 6, which is divisible by 6.
Hence, P(2) is true.
Let us assume that, P(n) is true for some natural number n = k.
P(k): k³ – k is divisible by 6
or    k³ -k= 6m, m∈ N       (i)
Now, we have to prove that P(k + 1) is true.
P(k+ 1): (k+ l)³-(k+ 1)
= k³+ 1 +3k(k+ l)-(k+ 1)
= k³+ 1 +3k² + 3k-k- 1 = (k³-k) + 3k(k+ 1)
= 6m + 3 k(k +1)  (using (i))
Above is divisible by 6.   (∴ k(k + 1) is even)
Hence, P(k + 1) is true whenever P(k) is true.
So, by the principle of mathematical induction P(n) is true for any natural number n,n≥ 2.

Q10. n(n² + 5) is divisible by 6, for each natural number
Sol: Let P(n): n(n² + 5) is divisible by 6, for each natural number.
Now P(l): 1 (l² + 5) = 6, which is divisible by 6.
Hence, P(l) is true.
Let us assume that P(n) is true for some natural number n = k.
P(k): k( k² + 5) is divisible by 6.
or K (k²+ 5) = 6m, m∈ N         (i)
Now, we have to prove that P(k + 1) is true.
P(K+l):(K+l)[(K+l)² + 5]
= (K + l)[K² + 2K+6]
= K³ + 3 K² + 8K + 6
= (K² + 5K) + 3 K² + 3K + 6 =K(K² + 5) + 3(K² + K + 2)
= (6m) + 3(K² + K + 2)        (using (i))
Now, K² + K + 2 is always even if A is odd or even.
So, 3(K² + K + 2) is divisible by 6 and hence, (6m) + 3(K² + K + 2) is divisible by 6.
Hence, P(k + 1) is true whenever P(k) is true.
So, by the principle of mathematical induction P(n) is true for any natural number n.

Q11. n² < 2ⁿ, for all natural numbers n ≥
Sol: Let P(n): n² < 2ⁿ for all natural numbers n≥ 5.
Now P(5): 5² < 2⁵ or 25 < 32, which is true.
Hence, P(5) is true.
Let us assume that P(n) is true for some natural number n = k.
∴ P(k): k² < 2^k  (i)
Now, to prove that P(k + 1) is true, we have to show that P(k+ 1): (k+ l)² <2^k+1
Using (i), we get
(k + l)² = k² + 2k + 1 < 2^k + 2k + 1         (ii)
Now let, 2^k + 2k + 1 < 2^k+1     (iii)
∴ 2^k + 2k + 1 < 2 • 2^k
2k + 1 < 2^k, which is true for all k > 5 Using (ii) and (iii), we get (k + l)² < 2^k+1 Hence, P(k + 1) is true whenever P(k) is true.
So, by the principle of mathematical induction P(n) is true for any natural number n,n≥ 5.

Q12. 2n<(n + 2)! for all natural numbers
Sol: Let P(n): 2n < (n + 2)! for all natural numbers n.
P( 1): 2 < (1 + 2)! or 2 < 3! or 2 < 6, which is true.
Hence,P(l) is true.
Let us assume that P(n) is true for some natural number n = k.
P(k) :2k<(k + 2)!  (i)
To prove that P(k + 1) is true, we have to show that
P(k + 1): 2(k+ 1) < (k + 1 + 2)!
or 2(k+ 1) < (k + 3)!
Using (i), we get
2(k + 1) = 2k + 2<(k+2) !  +2  (ii)
Now let, (k + 2)! + 2 < (k + 3)!  (iii)
=>  2 < (k+ 3)! – (k+2) !
=> 2 < (k + 2) ! [k+ 3-1]
=>2<(k+ 2) ! (k + 2), which is true for any natural number.
Using (ii) and (iii), we get 2(k + 1) < (k + 3)!
Hence, P(k + 1) is true whenever P(k) is true.
So, by the principle of mathematical induction P(n) is true for any natural number n.

Q13. 2 + 4 + 6+… + 2n = n² + n, for all natural numbers
Sol: Let P(n) :2 + 4 + 6+ …+2 n = n² + n
P(l): 2 = l² + 1 = 2, which is true
Hence, P(l) is true.
Let us assume that P(n) is true for some natural number n = k.
∴ P(k): 2 + 4 + 6 + .,.+2k = k² + k  (i)
Now, we have to prove that P(k + 1) is true.
P(k + l):2 + 4 + 6 + 8+ …+2k+ 2 (k +1)
= k² + k + 2(k+ 1)  [Using (i)]
= k² + k + 2k + 2
= k² + 2k+1+k+1
= (k + 1)² + k+ 1
Hence, P(k + 1) is true whenever P(k) is true.
So, by the principle of mathematical induction P(n) is true for any natural number n.

Q14. 1 + 2 + 2² + … + 2ⁿ = 2^n +1 – 1 for all natural numbers
Sol: Let P(n): 1 + 2 + 2² + … + 2ⁿ = 2^n +1 – 1, for all natural numbers n
P(1): 1 =2^{0 + 1} — 1 = 2 — 1 = 1, which is true.
Hence, ,P(1) is true.
Let us assume that P(n) is true for some natural number n = k.

P(k): l+2 + 2²+…+2^k = 2^k+1-l              (i)

Now, we have to prove that P(k + 1) is true.

P(k+1): 1+2 + 2²+ …+2^k + 2^k+1
= 2^{k +1} – 1 + 2^k+1  [Using (i)]
= 2.2^k+l– 1 = 1
₌ 2^(k+1)+1-1
Hence, P(k + 1) is true whenever P(k) is true.
So, by the principle of mathematical induction P(n) is true for any natural number n.

Q15. 1 + 5 + 9 + … + (4n – 3) = n(2n – 1), for all natural numbers
Sol: Let P(n): 1 + 5 + 9 + … + (4n – 3) = n(2n – 1), for all natural numbers n.
P(1): 1 = 1(2 x 1 – 1) = 1, which is true.
Hence, P(l) is true.
Let us assume that P(n) is true for some natural number n = k.
∴ P(k):l+5 + 9 +…+(4k-3) = k(2k-1)  (i)
Now, we have to prove that P(k + 1) is true.
P(k+ 1): 1 + 5 + 9 + … +  (4k- 3) + [4(k+ 1) – 3]
= 2k² -k+4k+ 4-3
= 2k² + 3k + 1
= (k+ 1)( 2k + 1)

= (k+l)[2(k+l)-l]

Hence, P(k + 1) is true whenever P(k) is true.

So, by the principle of mathematical induction P(n) is true for any natural number n.

In This Post we are  providing Chapter- 2 TRIGNOMETRIC FUNCTIONS NCERT MOST IMPORTANT QUESTIONS for Class 11 MATHS 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 MOST IMPORTANT QUESTIONS ON TRIGNOMETRIC FUNCTIONS

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1.Convert into radian measures. -37⁰ 30’

Ans. – 37⁰ 30’ = –

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2.Prove Sin (n+1) x Sin (n+2) x + Cos (n+1) x. Cos (n+2) x = Cos x 

Ans.L.H.S = Cos (n+1) x Cos (n+2) x + Sin (n+1) x Sin (n+2) x

=Cos 

=Cos x

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3.Find the value of Sin 

Ans. Sin = Sin 

= sin 

= Sin 

=

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4.Find the principal solution of the eq. tan x = 

Ans. tan x = 

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5.Convert into radian measures. 

Ans. 5⁰ 37¹ 30¹¹ = 5⁰ + 

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6.Evaluate 2 Sin 

Ans.2 Sin 

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6.Find the solution of Sin x = 

Ans.Sin x = 

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7.Prove that 

Ans.L. H. S = tan 36⁰

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8.Find the value of tan 

Ans.

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9.Prove Cos 4x = 1 – 8 Sin² x. Cos²x

Ans. L. H. S = Cos 4x

10.Find the value of Cos (- 1710⁰).

Ans. Cos (-1710⁰) = Cos (1800-90)[Cos (-θ) = Cos θ

= Cos [5 360 +90]

= Cos  = 0

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11.A wheel makes 360 revolutions in 1 minute. Through how many radians does it turn in 1 second.

Ans.N. of revolutions made in 60 sec. = 360

N. of revolutions made in 1 sec = 

Angle moved in 6 revolutions = 2 π 6 = 12 π

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12.If in two circles, arcs of the same length subtend angles 60⁰ and 75⁰ at the centre find the ratio of their radii.

Ans.

(1)÷ ( 2)

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13.Prove that Cos 6x= 32 Cos⁶x – 48 Cos⁴ x + 18 Cos² x-1

Ans.L.H.S. = Cos 6x

= 

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14.Solve Sin2x-Sin4x+Sin6x=o

Ans.

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15.Prove that (Cos x + Cos y)² + (Sin x – Sin y)² = 4 Cos² 

Ans. L. H. S = (Cos x + Cos y)² + (Sin x – Sin y)²

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In This Post we are  providing Chapter-2 RELATIONS AND FUNCTIONS NCERT MOST IMPORTANT QUESTIONS for Class 11 MATHS 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 MOST IMPORTANT QUESTIONS ON RELATIONS AND FUNCTIONS

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1. If P = {1,3}, Q = {2,3,5}, find the number of relations from A to B

Ans. = 64

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2. If A = {1,2,3,5} and B = {4,6,9}, R = {(x, y) : |x – y| is odd, x ∈ A, y ∈ B} Write R in roster form

Ans. Function

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3.

Let f and g be two real valued functions, defined by, f(x) = x2, g(x) = 3x +2.

Ans. Not a function

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4. Find the domain of the relation, R = { (x, y) : x, y ϵ Z, xy = 4} Find the range of the following relations : (Question-23, 24)

Ans. {–4, –2, –1,1,2,4}

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5.If and form the sets and are these two Cartesian products equal?

Ans. Given and by definition of cartesion product, we set

and 

By definition of equality of ordered pains the pair is not equal to the pair therefore 

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6.If and are finite sets such that and find the number of relations from to 

Ans. Linen 

the number of subsets of 

then the number of subsets of 

Since every subset of is a relation from A to B therefore the number of relations from A to B = 2^mk

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7.Let be a function from z to z defined by for same integers a and b determine a and b.

Ans. Given 

Since 

Subtracting (i) from(ii) we set a=2

Substituting a=2 is (ii) we get 2+b=1

b = -1

Hence a = 2, b = -1

8.Express as the set of ordered pairs

Ans. Since and 

Put 

For anther values of we do not get 

Hence the required set of ordered peutes is 

9.Function is defined by find 

Ans. 

10.Let and be the relation, is one less than from to then find domain and Range of 

Ans. Given and is the relation ‘is one less than’ from to therefore 

Domain of and range of 

11.Let be a relation from to define by.

Is the following true implies 

Ans. No; let As so but so 

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12.Let be the set of natural numbers and the relation be define in by =what is the domain, co domain and range of? Is this relation a function?

Ans. Given 

Domain of co domain of and Range of is the set of even natural numbers.

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13. Let A = {1,2,3,4}, B = {1,4,9,16,25} and R be a relation defined from A to B as, R = {(x, y) : x ϵ A, y ϵ B and y = x²}

(a) Depict this relation using arrow diagram.

(b) Find domain of R.

(c) Find range of R.

(d) Write co-domain of R.

Ans.

(b) {1,2,3,4}

(c) {1,4,9,16}

(d) {1,4,9,16,25}

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14. Let R = { (x, y) : x, y ϵ N and y = 2x} be a relation on N. Find :

(i) Domain

(ii) Codomain

(iii) Range

Is this relation a function from N to N

Ans. (i) N

(ii) N

(iii) Set of even natural numbers

yes, R is a function from N to N.

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15. Find the domain and range of, f(x) = |2x – 3| – 3

Ans. Domain is R

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In This Post we are  providing Chapter-1 SETS NCERT MOST IMPORTANT QUESTIONS for Class 11 MATHS 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 MOST IMPORTANT QUESTIONS ON SETS

Q1.If A = { 1,2,3,4,5,6}, B = {2,4,6, 8} then find A – B

Ans. We are given the sets A ={ 1,2,3,4,5,6}, B = {2,4,6, 8}

A – B = { 1,2,3,4,5,6}- {2,4,6, 8} = {1, 3, 5}

Q2. Let A and B be two sets containing 3 and 6 elements respectively. Find the maximum and the minimum number of elements in A ∪ B.

Ans.There may be the case when atleast 3 elements are common between both sets

Let a set A = {a, b, c} and B = {a, b, c, d, e, f}

∴ A ∪ B = {a, b, c, d, e, f} implies that the minimum number of elements in A ∪ B are = 6

There may be the case when there are no any elements are common between both sets

lf A = {a, b, c}, B = { d, e, f, g, h, i}

A ∪ B = {a, b, c, d, e, f,g,h,i} implies that the maximum number of elements in A ∪ B are = 9

Q3.If A = {(x,y) : x² + y²= 25  where x, y ∈ W } write a set of all possible ordered pair .

Ans.  We are given the set A = {(x,y) : x² + y²= 25  where x, y ∈ W }

All possible ordered pair of set A are following

For x = 0,y =5, x=3,y=4,for x =4, y =3,for x=5,y =0

A = {(0,5),(3,4),(4,3),(5,0)}

Q4.If A = {1,2,3}, B = {4, 5, 6} and C ={5} verify that A ∪ ( B ∩ C) = (A ∪ B) ∩ A ∪ C.

We are given the sets A = {1,2,3}, B = {4, 5, 6} ,C = {5}

B ∩ C = {5}

LHS

A ∪ ( B ∩ C) = {1,2,3,5}

(A ∪ B) and A ∪ C

A ∪ B = {1,2,3,4,5,6} and A ∪ C = {1,2,3,5}

RHS

(A ∪ B) ∩ A ∪ C = {1,2,3,5}

Therefore

A ∪ ( B ∩ C) = (A ∪ B) ∩ A ∪ C, Hence proved

Q5.From the adjoining Venn diagram, write the value of the following.

(a) A ‘

(b) B’

(c) (A ∩ B)’

Ans. From the venn diagram ,we have

U = {1,2,3,4………15}

A = {7,9.11}. B ={11,12,13,14}

A’ = U – A = {1,2,3,4………15} – {7,9.11} = {1,2,3,4,5,6,8,10,12,13,14,15}

B’ = U – B = {1,2,3,4………15} – {11,12,13,14}= {1,2,3,4,5,6,7,8,9,10,,15}

We have,(A ∩ B) = 11

(A ∩ B)’ = U – (A ∩ B) = {1,2,3,4………15} – {11} = (1,2,3,4,5,6,7,8,9,10,12,13,14,15}

Q6. If P(A) = P(B) show that A = B.

Ans. P(A} and P(B) implies that both are power sets of A and B respectively

Every set is an element of its power set , so A ∈ P(A)

Since, we are given that

P(A) = P(B)

Therefore, A ∈ P(B)

Indicates that every element of A belongs to the set B

So, A ⊂ B……(i)

Similarly B ∈ P(B)

Since, we are given that

P(A) = P(B)

Indicates that every element of B belongs to the set A

So, B ⊂ A……(ii)

From (i) and (ii), we get

A = B, Hence proved

Q7. Let A  and B be sets ; if A∩X = B∩X = ∅ and A∪X = B∪X for some set X. Show that A=B.

Ans. We are given that A∩X = B∩X = ∅ and A∪X = B∪X

To prove   A=B

Proof.  A∪X = B∪X (given)

Multiplying both sides by A∩

A∩ (A∪X) =A∩ (B∪X)

Using distributive property

(A∩ A) ∪ (A ∩ X) = (A∩ B )∪ (A∩ X)

A∩Φ = (A∩ B) ∪ Φ

A = (A∩ B) …………(i)

A∪X = B∪X

Multiplying both sides by B∩

B∩(A∪X) = B∩(B∪X)

(B∩ A) ∪ (B ∩ X) = (B∩ B )∪ (B∩ X)

(B∩ A) ∪φ = B ∪ φ

B = (B∩ A)

B = (A∩ B) …………(ii)

From (i) and (ii)

A = B, Hence proved

Q8.If A ={1,2,3,4,5},then write the proper subsets of A.

Ans. The number of elements in the given sets A ={1,2,3,4,5} are =5

The number of proper subsets of any set are = 2ⁿ – 1

   ^{Where n = number of elements = 5}

  ^{The number of proper subsets of any set are =}  2⁵ – 1 =32 – 1 = 31

Q9.Write the following sets in the Roster form

(i) A={x : x ∈ R, 2x+11 =15}

(ii)B={x |x² =x, x ∈ R}

(iii)C={x = x is a positive factor of the prime number p}

Ans.(i)We have, A={x : x ∈ R, 2x+11 =15}

2x+1= 15 ⇒x= 2

∴ A = {2}

(ii)We have,B={x |x² =x, x ∈ R}

∴ x² = x ⇒ x²-x= 0 ⇒x(x-1)= 0 ⇒x=0,1

∴ B ={0,1}

(iii)We have, C={x = x is a positive factor of the prime number p}

Sice positive factors of a prime u∪mer are 1 ad the number itself,we have

C={1,p}

Q10.For all sets A,B and C show that (A – B) ∩(A – C) = A – (B ∪ C).

Ans. Considering that

x ∈ (A – B)∩(A – C)

⇒ x ∈ (A – B) and x ∈ (A – C)

⇒ (x ∈ A and x∉ B) and (x ∈ A and x∉ C)

⇒ (x ∈ A ) and (x ∉B and  x∉ C )

⇒(x ∈ A ) x∉ (B ∪C)∈

⇒x ∈ A – (B ∪C)

⇒(A – B) ∩(A – C)⊂A – (B ∪ C)…….(i)

Now,Considering that

y ∈A – (B ∪ C)

⇒ y ∈A and y ∉  (B ∪ C)

⇒y ∈A and (y ∉B and y ∉ C)

⇒(y ∈A and y ∉B) and (y ∈A and y ∉ C)

⇒y ∈ (A – B) and y ∈ (A – C)

⇒y ∈ (A – B) ∩ y ∈ (A – C)

⇒A – (B ∪ C) ⊂(A – B) ∩(A – C)………(ii)

From (i) and (ii)

(A – B) ∩(A – C)= A – (B ∪ C), Hence proved

Q11.Let A,B and C be the sets such that A∪B = A∪C and A ∩B = A ∩C,show that B = C.

Ans. According to question, A ∪ B = A ∪ C and A ∩ B = A ∩ C

To show, B = C

Let us assume, x ∈ B So, x ∈ A ∪ B

x ∈ A ∪ C

Hence, x ∈ A or x ∈ C

when x ∈ A, then x ∈ B

∴ x ∈ A ∩ B

As, A ∩ B = A ∩ C

So, x ∈ A ∩ C

∴ x ∈ A or x ∈ C

x ∈ C

∴ B ⊂ C

similarly, it can be shown that C ⊂ B

Hence, B = C

Q12.Show that for any sets A and B

A = (A∩B)∪(A-B) 

Ans.We have to prove

A = (A∩B)∪(A-B)

Taking RHS and solving it

(A∩B)∪(A-B)

Using the property

A -B = A -(A∩B)

A-B = A∩B’

=  (A∩B) ∪ (A∩B’)

Applying distributive property

A∩(B∪C) = (A∩B) ∪(A∩C)

Replacing C = B’ in LHS

A-B = A∩(B∪B’)

= A∩ (U) [since B∪B’ = U]

= A [since A∩ U = A]

= RHS, Hence proved

Q13. Write the following sets in the roster form:

(i) A = {x : x ∈ R, 2x + 15 = 15}

(ii) B = {x : x² = x, x ∈ R}

Ans(i) It is given to us that set

A = {x : x ∈ R, 2x + 11 = 15}

2x + 11 = 15

2x = 15 – 11

2x = 4

x = 2

Therefore in roster form it is written as  A = {2}

(ii) It is given to us that

B = {x : x² =x, x ∈ R}

x² = x

x² – x = 0

x(x – 1) = 0

x = 0, x = 1

Therefore in roster form it is written as

B = {0, 1}

Q14. If  A and B are subsets of the universal set U, then show that 

(i) A ⊂ A ∪ B

Ans. Let’s prove that

A ⊂ A ∪ B

Let  x ∈ A or x ∈ B

If x ∈ A then x ∈ A ∪ B

Hence A ⊂ A ∪ B

Q15. A,B and C are subsets of universal set if A = {2,4,6,8,12,20}, B ={3,6.9.12.15},C ={5,10,15,20} and U is the set of all whole numbers, draw a venn diagram showing the relation of  U,A,B and C.

Ans.

Question And Answer:

Q.1Bar diagram is a
(a) one-dimensional diagram
(b) two-dimensional diagram
(c) diagram with no dimension
(d) None of these
ANSWER:
(a) Bar diagrams are one-dimensional diagrams. Though these are represented on a plane of two axis in form of rectangular bars, the width is of no consequence and only the length depicts the frequency.

Q.2 Data represented through a histogram can help in finding graphically the
(a) mean
(b) mode
(c) median
(d) All of these
ANSWER:
(b) Histogram gives value of mode of the frequency distribution graphically through the highest rectangle.

Q.3 Ogives can be helpful in locating graphically the
(a) mode
(b) mean
(c) median
(d) None of these
ANSWER:
(c) Intersection point of the less than and more than ogives gives the median.

Q.4 Data represented through arithmetic line graph help in understanding
(a) long term trend
(b) cyclicity in data
(c) seasonality in data
(d) All of the above
ANSWER:
(a) Arithmetic line graph helps in understanding the trend, periodicity, etc in a long term time series data.

Q.5 Width of bars in a bar diagram need not be equal. (True/False)
ANSWER:
False
Bar diagram comprises a group of equispaced and equiwidth rectangular bars for each class or category of data.

Q.6 Width of rectangles in a histogram should essentially be equal. (True/False)
ANSWER:
False
If the class intervals are of equal width, the area of the rectangles are proportional to their respective frequencies and width of rectangles will be equal. However, sometimes it is convenient or necessary to use varying width of class intervals and hence unequal width of rectangles.

Q.7 Histogram can only be formed with continuous classification of data. (True/False)
ANSWER:
True
A histogram is never drawn for a discrete variable/data. If the classes are not continuous they are first converted into continuous classes.

Q.8 Histogram and column diagram are the same method of presentation of data. (True/False)
ANSWER:
False
Histogram is a two dimensional diagram drawn for continuous data and the rectangles do not have spaces in between while column diagram is one dimensional with space in between every column (bar).

Q.9 Mode of a frequency distribution can be known graphically with the help of histogram. (True/False)
ANSWER:
True
Histogram gives value of mode of the frequency distribution graphically through the highest rectangle.

Q.10 Median of a frequency distribution cannot be known from the ogives. (True/False)
ANSWER:
False
Intersection-point of the less than and more than ogives gives the median.

Q.11What kind of diagrams are more effective in representing the following?
(a) Monthly rainfall in a year
(b) Composition of the population of Delhi by religion
(c) Components of cost in a factory
ANSWER:
(a) The monthly rainfall in a year can be best represented by a bar diagram as only one variable i.e., monthly rainfall is to be presented diagrammatically. The rainfall is plotted on Y-axis in the corresponding month that is plotted on the X-axis.
(b) Composition of the population of Delhi by religion can be represented by a component bar diagram. A component bar diagram shows the bar and its sub-divisions into two or more components. Thus, the total population can be sub divided in terms of religion and presented through a component bar diagram.
(c) Different components of cost in a factory can most effectively be depicted through a pie chart. The circle represents the total cost and various components of costs are shown by different portions of the circle drawn according to percentage of total cost each component covers.

Q.12 Suppose you want to emphasise the increase in the share of urban non-workers and lower level of urbanisation in India as shown in Example 4.2. How would you do it in the tabular form?
ANSWER:
Share of urban workers and non workers in India

	Location
Sex	Worker in urban (in crore)	Non-worker in urban (in crore)	Total
Male	50	70	120
Femal	25	50	75
Total	75	120	195

Q.13 How does the procedure of drawing a histogram differ when class intervals are unequal in comparison to equal class intervals in a frequency table?
ANSWER:
A histogram is a set of rectangles with bases as the intervals between class boundaries (along X-axis) and with areas proportional to the class frequency. If the class intervals are of equal width, the area of the rectangles are proportional to their respective frequencies.

However, sometimes it is convenient or at times necessary, to use varying width of class intervals. For graphical representation of such data, height for area of a rectangle is the quotient of height i.e., frequency and base i.e., width of the class interval. When intervals are equal, all rectangles have the same base and area can conveniently be represented by the frequency of the interval.

But, when bases vary in their width, the heights of rectangles are to be adjusted to yield comparable measurements by dividing class frequency by width of the class interval instead of absolute frequency. This gives us the frequency density for the purpose of comparison.
Thus  Frequency density ( Height of rectangle )= Class Frequency  Width of the class interval 

Q.14 The Indian Sugar Mills Association reported that, ‘sugar production during the first fortnight of December, 2001 was about 3,87,000 tonnes, as against 3,78,000 tonnes during the same fortnight last year (2000). The off-take of sugar from factories during the first fortnight of December, 2001 was 2,83,000 tonnes for internal consumption and 41,000 tonnes for exports as against 1,54,000 tonnes for internal consumption and nil for exports during the same fortnight last season.’
(i) Present the data in tabular form.
(ii) Suppose you were to present these data in diagrammatic form which of the diagrams would you use and why?
(iii) Present these data diagrammatically.
ANSWER:
(i) Data in tabular form.
Sugar Production in India

	Total Production (tonnes)	Off-take for Internal Consumption (tonnes)	Off-take for Exports (tonnes)
December 2000	378000	154000	—
December 2001	387000	283000	41000

(ii) The data can effectively be presented diagrammatically using the multiple bar diagram. This is because multiple bar diagrams are used for comparing two or more sets of data for different years or classes, etc.

Q.15 The following table shows the estimated sectoral real growth rates (percentage change over the previous year) in GDP at factor cost.

Represent the data as multiple time-series graphs.
ANSWER:

Question And Answer:

Q.1Which of the following alternatives is true?
(i) The class mid-point is equal to
(a) the average of the upper class limit and the lower class limit
(b) the product of upper class limit and the lower class limit
(c) the ratio of the upper class limit and the lower class limit
(d) None of the above
ANSWER:
(a) The class mid-point is the middle value of a class. It lies halfway between the lower class limit and the upper class limit of a class and is calculated as
Class Mid-Point or Class Mark =  (Upper Class Limit + Lower Class Limit) 2

(ii) The frequency distribution of two variables is known as
(a) Univariate Distribution
(b) Bivariate Distribution
(c) Multivariate Distribution
(d) None of the above
ANSWER:
(b) Bi means two and hence the frequency distribution of two variables is known as Bivariate Distribution.

(iii) Statistical calculation in classified data are based on
(a) the actual values of observations
(b) the upper class limits
(c) the lower class limits
(d) the class mid-points
ANSWER:
(d) The class mid-points of each class is used to represent the class and therefore, it is used in further calculations after the raw data are grouped into classes

(iv) Under exclusive method,
(a) the upper class limit of a class is excluded in the class interval
(b) the upper class limit of a class is included in the class interval
(c) the lower class limit of a class is excluded in the class interval
(d) the lower class limit of a class is included in the class interval
ANSWER:
(a) Under the exclusive method we form classes in such a way that the lower limit of a class coincides with the upper class limit of the previous class. Under the method, the upper class limit is excluded but the lower class limit of a class is included in the interval.

(v) Range is the
(a) difference between the largest and the smallest observations
(b) difference between the smallest and the largest observations
(c) average of the largest and the smallest observations
(d) ratio of the largest to the smallest observation
ANSWER:
(a) The variation in variable’s value are captured by its range. The range is the difference between the largest and the smallest values of the variable. A large range indicates that the values of the variable are widely spead.

Q.2 Can there be any advantage in classifying things? Explain with an example from your daily life.
ANSWER:
Classification refers to arranging or organising similar things into groups or classes. Classification of objects or things saves our valuable time and effort. Classification is done to group things in such a way that each group consists of similar items, e.g., we classify our wardrobe into different types of clothes or dresses according to the occasions on which they are to be worn. We put party wears, school uniform, casual daily wears and night wears separately. This helps us in an orderly arrangement of clothes and we can easily fetch the clothes we want at a particular time without searching through the whole wardrobe. Thus, it is evident that classification saves time and labour and helps to produce the desired results.

Q.3 What is a variable? Distinguish between a discrete and a continuous variable.
ANSWER:
A measurable characteristic which takes different values at different points of time and in different circumstance is called a variable as it keeps varying. Different varibles vary differently and depending on the way they vary, they are broadly classified into two types

S.N.	Discrete Variable	Continuous Variable
(i)	A discrete variable can take only whole numbers.	A continuous variable can take any numerical value.
(ii)	Discrete varibles increase in finite jumps from one value to another and cannot take any intermediate value between them.	Continuous variables can take any conceivable value and can be broken into infinite gradations.
(iii)	Examples-number of workers in a factory, number of residents in a colony, etc.	Examples-height, weight, distance, etc.

Q.4 Explain the ‘exclusive’ and ‘inclusive’ methods used in classification of data.
ANSWER:
Exclusive Method In this method, the classes are formed in such a way that the upper class limit of one class becomes the lower class limit of the next class. Continuity of the data is maintained in this method. Under this method, the upper class limit is excluded but the lower class limit of a class is included in the interval.

According to this method, an observation that is exactly equal to the upper class limit would not be included in that class but would be included in the next class. On the other hand, if it were equal to the lower class limit then it would be included in that class, e.g., if the class intervals are 0-5, 5-10, 15¬20 and so on, a value of 10 would be included in the 10-15 and not in the interval 5-10.

Inclusive Method The inclusive method does not exclude the upper class limit in a class interval. It inlcludes the upper class in a class. Thus, both class limits are parts of the class interval, e.g., the class intervals of 0-5, 6-10, 11-15, and so on are inclusive.

Q.5 Use the data in Table 3.2 that relate to monthly household expenditure (in ₹) on food of 50 households and
(i) Obtain the range of monthly household expenditure on food.
(ii) Divide the range into appropriate number of class intervals and obtain the frequency distribution of expenditure.
(iii) Find the number of households whose monthly expenditure on food is

• less than ₹ 2,000
• more than ₹ 3,000
• between ₹ 1,500 and ₹ 2,500

ANSWER:
(i) Range = Largest Value – Smallest Value
Highest Value = 5090
Lowest Value = 1007
So, Range = 5090 – 1007 = 4083

(iii) (a) Number of households whose monthly expenditure on food is less than ₹ 2000
= 20 + 13 = 33
(b) Number of hoseholds whose monthly expenditure on food is more than ₹ 3000
= 2 + 1 + 2 + 0 + 1 = 6
(c) Number of households whose expenditure on food is between ₹ 1500 and ₹ 2500
= 13 + 6= 19

Q.6 In a city, 45 families were surveyed for the number of domestic appliances they used. Prepare a frequency array based on their replies as recorded below.
1 3 2 2 2 2 1 2 1 2 2 3 3 3 3
3 3 2 3 2 2 6 1 6 2 1 5 1 5 3
2 4 2 7 4 2 4 3 4 2 0 3 1 4 3
ANSWER:

No. of Domestic Appliances	No. of Households
0	1
1	7
2	15
3	12
4	5
5	2
6	2
7	1
Total	45

Q.7 What is loss of information’ in classified data?
ANSWER:
Classification of data as a frequency distribution summarises the raw data making it concise and comprehensible but it does not show the details that are found in raw data. Once, the data are grouped into classes, an individual observation has no significance in further statistical calculations.

All values in a class interval are assumed to be equal to the middle value of the class interval instead of their actual value which causes considerable loss of information. It not only save our time but also our energy, which would otherwise be utilised in searching from entire things.

Q.8 Do you agree that classified data is better than raw data?
ANSWER:
The raw data is usually large and fragmented and it is very difficult to draw any meaningful conclusion from them. Classification makes the raw data comprehensible by summarising them into groups. When facts of similar characteristics are placed in the same class, it enables one to locate them easily, analyse them, make comparison and draw inferences.

Q.9 Distinguish between univariate and bivariate frequency distribution.
ANSWER:
The term “uni” stands for one and thus the frequency distribution of a single variable is called a Univariate Distribution, e.g., the fequency distribution of age of students in a class is univariate as its gives the distribution of a single variable i.e., age. On the other hand “bi” means two and a Bivariate Frequency Distribution is the frequency distribution of two variables, e.g., the frequency distribution of two varibles, e.g., like price of good and sales of the good is a bivariate distribution.

Q.10 Prepare a frequency distribution by inclusive method taking class interval of 7 from the following data

ANSWER:

Question And Answer:

Q.1 Frame at least four appropriate multiple choice options for following questions
(i) Which of the following is the most important when you buy a new dress?
(ii) How often do you use computers?
(iii) Which of the newspapers do you read regularly?
(iv) Rise in the price of petrol is justified.
(v) What is the monthly income of your family?
ANSWER:
(i) Which of the following is the most important when you buy a new dress?

• Price of the dress
• Fabric of the dress
• Colour of the dress
• Brand of the dress

(ii) How often do you use computers?

• At least once a day
• At least once a week
• At least once in fortnight
• Occasionally

(iii) Which of the newspapers do you read regularly?

• Times of India
• Hindustan Times
• Indian Express
• The Hindu

(iv) Rise in the price of petrol is justified.

• Strongly agree
• Strongly disagree
• Neither agree nor disagree
• Somewhat agree

(v) What is the monthly income of your family?

• Less than ₹ 10,000
• More than ₹ 10,000 but less than ₹ 25,000
• More than ₹ 25,000 but less than ₹ 50,000
• More than ₹ 50,000

Q.2 Frame five two-way questions (with Tes’ or ‘No’).
ANSWER:

• Are you an Indian? (Yes/No)
• Do you live in Delhi? (Yes/No)
• Are you graduate? (Yes/No)
• Do you know swimming? (Yes/No)
• Have you ever been convicted by a court of law? (Yes/No)

Q.3 State whether the following statement are true or false.
(i) There are many sources of data. (True/False)
ANSWER:
False
There are mainly two sources of data : Primary and Secondary.

(ii) Telephone survey is the most suitable method of collecting data, when the population is literate and spread over a large area. (True/False)
Answer:
False
Mailing questionnaires would be more suitable as the population is literate. Telephonic survey is most suitable in case of illiterate population spread over a large area.

(iii) Data collected by investigator is called the secondary data. (True/False)
Answer:
False
Investigator may collect the data by conducting an enquiry or an investigation. Such data are called primary data, as they are based on first hand information.

(iv) There is a certain bias involved in the non-random selection of samples. (True/False)
Answer:
True
In a non-random sampling method all the units of the population do not have an equal chance of being selected and convenience or judgement of the investigator may create a bias

(v) Non-sampling errors can be minimised by taking large samples. (True/False)
Answer:
False
It is difficult to minimise non-sampling error even by taking a large sample as they include Errors in Data Acquisition, Non-Response Errors and Sampling bias.

Q.4 What do you think about the following questions. Do you find any problem with these questions? If yes, how?
(i) How far do you live from the closest market?
ANSWER:
This question is ambiguous people will not be able to answer this question as the different measures of distance like meters, kilometers, yards etc will complicate the analysis. It should be made specific as Flow many kilometers away is your home from the closest market?

• Less than 5 km
• Between 5-10 km
• More than 10 km

(ii) If plastic bags are only 5 per cent of our garbage, should it be banned?
Answer:
This question is a leading question, which gives a clue about how the respondent should answer by trying to point that 5% is a small percentage which can be tolerated. Better question would be Do you think plastic bags should be banned? (Yes)

(iii) Wouldn’t you be opposed to increase in price of petrol?
Answer:
This question comprises of two negatives which creates confusion to the respondents and may lead to biased response. Better question would be Would you opposed the increase in price of petrol?

(iv)
(a) Do you agree with the use of chemical fertilizers?
(b) Do you use fertilizers in your fields?
(c) What is the yield per hectare in your field?
Answer:
The order or sequence of questions is incorrect. The series of questions should move from general to specific. The correct order would be
(a) What is the yield per hectare in your field?
(b) Do you use fertilizers in your fields?
(c) Do you agree with the use of chemical fertilizers?

Q.5 You want to research on the popularity of vegetable atta noodles among children. Design a suitable questionnaire for collecting this information.
ANSWER:
Questionnaire
Name ………………………………….
Age …………………………………………
Address …………………………………
……………………………………………….
………………………………………………
Gender: Male □ Female □
Answer:
Questionnaire
Name ………………………………….
Age …………………………………………
Address …………………………………
……………………………………………….
………………………………………………
Gender: Male □ Female □
1.Do you eat noodles?
(a) Yes □
(b) No □

2.Do you like noodles more than other snacks?
(a) Yes □
(b) No □

3.How many packets do you consume in a month?
(a) Less than 2 packets □
(b) 3-5 packets □
(c) 5-8 packets □
(d) More than 8 packets □

4.Do you prefer atta noodles over maida noodles?
(a) Yes □
(b) No □

5.Do you like vegetables in your noodles?
(a) Yes □
(b) No □

6.Do you think more vegetables should be added in vegetable atta noodles?
(a) Yes □
(b) No □

Q.7 Which vegetables according to you should be added in vegetable atta noodles?
…………………………………………………

8.Do you think it should be spicier?
(a) Yes □
(b) No □

9.When do you prefer to have vegetable atta noodles?
(a) In breakfast □
(b) In lunch □
(c) As evening snacks □
(d) In dinner □

10.Do your parents also like vegetable atta noodles?
(a) Yes □
(b) No □

Q.6 In a village of 200 farms, a study was conducted to find the cropping pattern. Out of the 50 farms surveyed, 50% grew only wheat. Identify the population and the sample here.
ANSWER:
The population or the Universe in statistics means totality of the items under study. It is a group to which the results of the study are intended to apply. In this case, the population is 200 farms in the village.

A sample refers to a group or section of the population from which information is to be obtained. A good sample (representative sample) is generally smaller than the population and is capable of providing reasonably accurate information about the population. In this case, the sample is 50 farms which are surveyed.

Q.7Give two examples each of sample, population and variable.
ANSWER:
Example 1 A study was conducted to know the average weight of students of class seventh in Delhi. The total number of students in class seventh was 2860. Out of these 200 students were randomly selected and their weight was recorded.
In this example

• Population is, the no of students of class seventh in Delhi, the total number of which is equal to 2860.
• Sample is, the 200 students selected whose weight was recorded.
• Variable under study, is the weight of the students.

Example 2 A person suffering from weakness and fatigue was advised by the doctor to have his blood test done for detection of anaemia. The pathologist took 2 ml of his blood for the test and tested the haemoglobin level in the blood.
In this example

• Population is the total amount of blood in the person’s body.
• Sample is, the 2 ml blood tested.
• Variable under study, is the haemoglobin in the blood sample.

Q.8 Which of the following methods give better results and why?
(a) Census
(b) Sample
ANSWER:
(b) In terms of accuracy of results, census is better as it studies all the units of population but this method is very time consuming, expensive and sometimes not feasible to use. Hence, sampling is better due to following reasons

• Economical Sampling involves study, of a fraction of population and hence the cost involved In sampling is relatively low.
• Time Saving Huge amount of time is required to conduct a census survey while sample studies do not take that much time.
• Lesser Effort As only a part of the population is studied, it entails lesser effort on the part of the investigator than that required in census.
• Considerable Accuracy Results from sampling may not be as accurate as in case of sampling but the level of accuracy of these results can be established through statistical tests of significance and hence can be applied in general to the whole population if found significant.

Q.9 Which of the following errors is more serious and why?
(a) Sampling error
(b) Non-sampling error
ANSWER:
(b) Sampling error refers to the difference between the sample estimate and the actual value of a population characteristic. This type of error occurs when one makes an observation from the sample taken from the population. It is possible to reduce the magnitude of sampling error by taking a larger sample.

Non-sampling errors are more serious than sampling errors because a sampling error can be minimised by taking a larger sample but it is difficult to minimise non-sampling error, even by taking a large sample. Even a Census can contain non-sampling errors. These include errors in data acquisition, non-response errors and sampling bias.

Q.10 Suppose there are 10 students in your class. You want to select three out of them. How many samples are possible?
ANSWER:
In general, you use combinations to determine the number of ways you can select a sample of size n from a population of size N. The formula for the number of such combinations is
N! (n!) (N – n)!
where N! (spoken “Nfactorial”) equals N(N — 1)(N – 2)…(3)(2)(1)
(e.g., 5! = (5) (4) (3) (2) (1) = 120
In this problem, our population size is N = 10 students, and our sample size is n = 3 students. Number of samples possible can be calculated as follows
Number of samples = 10! (3!) (10 – 3)!
= 10!(3)!(7)! = 10×9×8×7!3×2×1×7!
= 120 possible random samples

Q.11Discuss how you would use the lottery method to select 3 students out of 10 in your class?
ANSWER:
A representative (random) sample of 3 students can be taken out of 10 through lottery method. The names of all the 10 students of the class are written on 10 separate pieces of paper of equal size and all the slips are folded in a similar manner. These slips are then mixed well and 3 slips with these names are selected one by one so that all the students have equal chance of being selected in the sample.

Q.12 Does the lottery method always give you a random sample? Explain.
ANSWER:
Lottery method always gives a random sample if it is used in the proper manner without any bias. If the slips are prepared properly and drawn out one by one so that all the slips have equal chance of being selected in the sample, it will definitely give a random sample. But, if the slips are not mac . of identical size and identification is possible of the names or numbers on the slips, the selection will become biased.

Similarly, if the same name or number is written on more than one slip and if some name or number is missed then also the chances of selection of different units of population in the sample will not be equal. In such cases even lottery method will not give random sample.

Q.13 Explain the procedure of selecting a random sample of 3 students out of 10 in your class, by using random number tables.
ANSWER:
Random number tables have been devised to guarantee equal probability of selection of every individual unit in the population according to their listed serial number in the sampling frame. They are available either in a published form or can be generated by using appropriate software packages.
The procedure of selecting a random sample of 3 students out of 10 in a class, by using random number tables is as follows

• Assign a specific number between 1 and 10 to all the 10 students.
• Here, the largest serial number is 10 which is a two digit number and therefore we consult two digit random numbers in sequence.
• We can start using the table from anywhere, i.e., from any page, column, row or point and select the first number randomly. We need to select a sample of 3 students out of 10 total students.
• We will select two more numbers from the table according to sequence. We will skip the random numbers greater than 10 since there is no student number greater than 10. Thus, the 3 selected students are with serial numbers.

Q.14 Do samples provide better results than surveys? Give reasons for your answer.
ANSWER:
A survey, which includes every element of the population, is known as Census or the Method of Complete Enumeration. On the other hand, when a part of the population is studied and predictions are made about the population based on this part, it is called sampling.
In terms of accuracy of results, census is better as it studies all the units of population but this method is very time consuming, expensive and sometimes not feasible to use. Hence, sampling is better due to following reasons

• Economical Sampling involves study of a fraction of population and hence the cost involved in sampling is relatively low. Census costs are high especially in case of large population with wide coverage in terms of area.
• Time Saving Huge amount of time is required to conduct a census survey if the population size is large or spread over a wide area while sample studies do not take that much time to be conducted.
• Lesser Effort As only a part of the population is studied, it entails lesser effort on the part of the investigator than that required in census.
• Inappropriateness of Census In certain case, when the population is infinite or exhaustible, census cannot be done and hence sampling is the only choice, e.g., one cannot burn all the units of coal available to know their calorific value; sample is the only means of testing it.
• Considerable Accuracy Results from sampling may not be as accurate as in case of sampling but the level of accuracy of these results can be established through statistical tests of significance and hence can be applied in general to the whole population if found significant.

Question And Answer:

Q.1Mark the following statements as true or false.
(i) Statistics can only deal with quantitative data.
(ii) Statistics solves economic problems.
(iii) Statistics is of no use to Economics without data.
ANSWER:
(i) False Statistics deals with both quantitative data as well as with qualitative data. Qualitative data describes the attributes.
(ii) True Economists use Statistics as a tool to understand and evaluate an economic problem by analysing past data. Statistical tools help economists to identify causes of an economic problem and devise policies accordingly.
(iii) True Data is the raw material for economic analysis. Statistical analysis of economic variables cannot be undertaken without having any data.

Q.2 Make a list of activities that constitute the ordinary business of life. Are these economic activities?
ANSWER:
The following are the activities that constitute the ordinary business of life

• Buying of goods and services.
• Rendering services to a company by employees and workers.
• Selling of goods and services.
• Production process carried out by a firm.

Yes, the above mentioned activities are regarded as economic activities. This is because, these activities are undertaken for monetary gain and are thus economic activities.

Q.3 The government and policy makers use statistical data to formulate suitable policies of economic development’. Illustrate with two examples.
ANSWER:
The statistical data provide the base for the government and the policy makers to formulate policies. The statistical data not only help them to analyse and evaluate the outcomes of the past policies but also assist them to take corrective measures and to formulate new policies. Statistical data also help the government to ascertain the relationship between economic variables and form policies accordingly.

For example, if Indian Government aims at increasing the national output, then it formulates its investment expenditure policy based on the capital output ratio in the past few years. Another example could be the preparation of monetary policy. The previous data of inflation and economic growth are taken into consideration for estimating the money supply required in the next period.

“Q.4 You have unlimited wants and limited resources to satisfy them.” Explain by giving two examples.
ANSWER:
The problem of scarcity is the most basic economic problem. Human wants are unlimited and resources to satisfy these wants are limited and these limited resources have alternative uses. ‘Scarcity of resources’ implies that there are unlimited wants to be fulfilled by limited resources which leads to lesser supply of resources as compared to demand for them.

The basic concern of an economy is to allocate the scarce resources to the best possible use in order to satisfy maximum wants. The limited resources have alternative uses which along with problem of scarcity makes it necessary for an economy to make a choice among various alternatives.

For example, an economy endowed with a given level of resources has to make a choice between the production of capital goods and consumer goods. The choice of the economy (i.e., what to produce and in what quantities) depends on the need of the economy. While the production of consumer goods will hamper the capital formation in the country for future production, the production of capital goods will not provide sufficient goods for consumption to the present population

The same problem of scarcity can be felt at an individual level, e.g., with a given amount of money say, ? 10,000, one cannot buy a refrigerator and a washing machine simultaneously. Thus, the individual needs to make a choice between the alternatives according to his/her priority.

Q.5 How will you choose the wants to be satisfied?
ANSWER:
An individual may have unlimited wants but these wants are in an order of priority according to their intensity. The wants of highest intensity will be fulfilled first as they provide the highest satisfaction or utility to the individual and hence, the individual attaches the top most priority to these wants.

Further, the choice of want also depends on the need or priority in the given situation, availability of the goods and services which can satisfy the wants and the purchasing power to realise a particular want. Thus, depending on all these conditions, we can say that an individual having a limited budget will fulfil a particular need that would provide him/her the highest possible satisfaction in the given income and given prices of the goods and services required to satisfy the wants. satisfaction in the given income and given prices of the goods and services required to satisfy the wants.

Q.6 What are your reasons for studying Economics?
ANSWER:
Human wants are unlimited and resources to satisfy these wants are limited and these limited resources have alternative uses. The basic concern of economics is to allocate the scarce resources to the best possible use in order to derive maximum benefit from the scarce resources. Due to the scarcity of resources having alternative uses, an economy needs to allocate the scarce resources to the areas with maximum possible and optimum returns. The following are the reasons that make the study of economics important
(i) To Study the Consumer Behaviour The theory of consumer behaviour in Economics deals with the study of the behaviour of the consumers in different types of market situations. This theory helps us understand how a rational consumer makes his/her decisions to get the maximum possible satisfaction in the given income and given prices of the goods and services.

(ii) To Study the Production Theory The theory of production studies the production decisions of the producers in different types of market. The theory explains how a producer takes production decisions related to maximisation of output in given cost or the minimisation of cost for a given level of output. The theory highlights how a producer combines different inputs (given their prices) in order to minimise the cost of production and to maximise the profits.

(iii) To Study the Distribution of Income The study of Economics makes us aware about the distribution of national income. In other words, it tells us how the income arising from the total production in an economy is distributed in the form of wage, rent, interest and profit to different factor owners (like labour, land, capital and entrepreneur).

(iv) To Study the Macroeconomic Problems Faced by an Economy Economics proves to be the most powerful tool to understand and analyse the root cause of basic macroeconomic problems faced by an economy like poverty, unemployment, inflation, recession etc. Economics helps us not only in understanding the interrelationship among these problems but also to take various corrective measures.

Q.7 Statistical methods are no substitute for common sense. Comment.
ANSWER:
It is absolutely true that statistical methods are no substitute for common sense. Statistical data should not be believed blindly as they can be misinterpreted or misused. The statistical data may involve personal bias or may be subject to manipulations for one’s own selfish motive.

Statistical data and methods are subject to the errors committed by an investigator while surveying and collecting data. Thus, one should use his/her common sense while working with the statistical methods.

This point can be understand with the help of an example A person who wanted to cross a river with his family but did not know how to swim. He knew the average depth of the river to be 125 cm. His height was 175 cm, that of his wife was 152 cm and his two children measured 120 cm and 90 cm respectively in height.

He calculated the average height of his family and found it to be around 134 cm. He analysed that the average depth of the river was less than the average height of his family and concluded that they all could cross the river safely on foot. This resulted in drowning of his children. This example proves that common sense must supersede statistical methods.

Question And Answer:

Q.1Mention some examples of regional and economic groupings.
ANSWER: Every country aims to strengthen its own domestic territory. The nations are forming regional and global economic groupings such as:

1. SAARC. It has 8 countries of South Asia.
2.  EU has 25 independent states based on European Communities.
3.  ASEAN. It has 5 countries of South East Asia.
4.  G-8 (Group of Eight). It has 8 countries.
5.  G-20 (Group of Twenty). It consists of 19 world’s largest economies.

Q.2 What are the various means by which countries are trying to strengthen their own domestic economies?
ANSWER:  Countries are trying to strengthen their own domestic economies by:

1.  forming regional apd global economic groupings like SAARC, EU, ASEAN, G-8, G-20, etc.
2.  By having economic reforms.

Q.3 What similar development strategies have India and Pakistan followed for their respective developmental paths?
ANSWER: Similar developmental strategies of India and Pakistan are:

1.  India has the largest democracy of the world. Pakistan has authoritarian militarist political power structure.
2.  Both India and Pakistan followed a mixed economy approach. Both countries created a large public sector and planned to raise public expenditure on social development.

Q.4 Explain the Great Leap Forward campaign of China as initiated in 1958.
ANSWER:  Communist China or the People’s Republic of China, as it is formally known, came into being in 1949. There is only one party, i.e., the Communist Party of China that holds the power there. All the sectors of economy including various enterprises and all land owned by individuals was brought under governmental control. A programme called ‘The Great Leap Forward’ was launched in 1958. Its aim was to industrialise the country on a large scale and in as short a time as possible. For this, people were eyeji encouraged to set up industries in their backyards. In villages, village Communes or cooperatives were set up. Communes means collective cultivation of land. Around 26000 communes covered almost all the farm population in 1958.
The Great Leap Forward programme faced many problems. These were:

1.  In the earlier phase, a severe drought occurred in China and it killed some 3 crore people.
2.  Soviet Russia was a comrade to communist China, but they had border dispute. As a result, Russia withdrew its professionals who had been helping China in its industrialisation bid.

Q.5 China’s rapid industrial growth can be traced back to its reforms in 1978. Do you agree? Elucidate.
ANSWER: Starting 1978, several reforms were introduced in phases in China. First, agriculture, foreign trade and investment sectors were taken up. Commune lands were divided into small plots. These were allotted to individual households for cultivation.
The reforms were expanded to industrial sector. Private firms were allowed to set up manufacturing units. Also, local collectives or cooperatives could produce goods. This meant competition between the newly sanctioned private sector and the old state-owned enterprises.
This kind of reform in China brought in the necessity of dual pricing. This meant the farmers and industrial units were to buy and sell fixed quantities of raw material and products on the basis of prices fixed by the government. As production increased, the material transacted through the open market also rose in quantity. Special Economic Zones (SEZs) were set up in China to attract foreign investors.

Q.6 Describe the path of developmental initiatives taken by Pakistan for its economic development.
ANSWER:  The developmental initiatives taken by Pakistan were:

1.  In the late 1950s and 1960s, Pakistan introduced a variety of regulated policy framework (for import substitution industrialisation). The policy combined tariff protection for manufacturing of consumer goods together with direct import controls on competing imports.
2.  The introduction of Green Revolutioned led to mechanisation of agriculture. It finally led to a rise in the production of foodgrains. This changed the agrarian structure dramatically.
3.  In the 1970s, nationalisation of capital good industries took place.
4. In 1988, structural reforms were introduced. The thrust areas were denationalisation and en¬couragement to private sector.
5.  Pakistan received financial support from western nations and remittances from emigrants to the Middle East. It helped in raising economic growth of the country.

Q.7 What is the important implication of ‘one child norm’ in China?
ANSWER:  One-child norm introduced in China in the late 1970s is the major reason for low population growth. It is stated that this measure led to a decline in the sex ratio, that is, the proportion of females per 1000 males.

Q.8 Mention the salient demographic indicators of China, Pakistan and India.
ANSWER:  We shall compare some demographic indicators of India, China and Pakistan.

1. The population of Pakistan is very small and accounts for roughly about one-tenth of China or India.
2.  Though China is the largest nation geographically among the three, its density is the lowest.
3.  The population growth is highest in Pakistan followed by India and China. One-child norm introduced in China in the late 1970s is the major reason for low population growth. They also state that this measure led to a decline in the sex ratio, that is, the proportion of females per 1000 males.
4.  The sex ratio is low and biased against females in all the three countries. There is strong son- preference prevailing in 11 these countries.
5.  The fertility rate is low in China and very high in Pakistan.
6. Urbanisation is high in both Pakistan and China with India having 28 per cent of its people living in urban areas.

Q.9 Compare and contrast India and China’s sectoral contribution towards GDP. What does it in¬dicate?
ANSWER: Sectoral Distribution of Output and Employment:

1.  Agriculture Sector. China has more proportion of urban people than India. In China in the year 2009, with 54 per cent of its workforce engaged in agriculture, its contribution to GDP is 10 per cent. In India’s contribution of agriculture to GDP is at 17 per cent.
2.  Industry and Service Sectors. In both India and China, the industry and service sectors have less proportion of workforce but contribute more in terms of output. In China, manufacturing contributes the highest to GDP at 46 per cent whereas in India it is the service sector which contributes the highest. Thus, China’s growth is mainly contributed by the manufacturing sector and India’s growth by service sector.

Q.10 Mention the various indicators of human development.
ANSWER: Parameters of human development are:

1.  HDI— (a) Value—higher the better.
   (b) Rank—lower the better.
2.  Life expectancy—higher the better.
3. Adult literacy rate—higher the better.
4.  GDP per capita (PPP US $)—higher the better –
5.  Percentage of population below poverty line (on $1 a day)—lower the better.
6. Infant mortality rate (per 1000 live births)—lower the better.
7.  Maternal mortality rate (per 100,000 live births)—lower the better.
8.  Percentage of population having access to improved sanitation—higher the better.
9.  Percentage of population having access to improved water source—higher the better.
10.  Percentage of population which is undernourished (% of total) – lower the better.

Q.11Define the liberty indicator. Give some examples of liberty indicators.
ANSWER: Liberty indicator has actually been added as a measure of ‘the extent of democratic participation
in social and political decision-making’ but it has not been given any extra weight. Some of the
examples of liberty indicators are : literacy rate, women participation in politics, etc.

Q.12 Evaluate the various factors that led to the rapid growth in economic development in China.
ANSWER: Reforms were initiated jn China in 1978. China did not have any compulsion to introduce reforms.
1. Pre-Reform Period : Failures
(a) There was slow pace of growth and lack of modernisation in the Chinese economy under the Maoist rule.
(b) It was felt that Maoist vision of economic development which was based on decentralisation, self-sufficiency and shunning of foreign technology, goods and capital, had failed.
(c) Despite extensive land reforms, collectivisation, the Great Leap Forward and other initiatives, the per capita grain output in 1978 was the same as it was in the mid-1950s.
Pre-Reform Period: Success
(a) There was existence of infrastructure in the areas of education and health.
(b) There were land reform.
(c) There was decentralised planning and existence of small enterprises.
(d) There was extension of basic health services in rural areas.
(e) Through the commune system, there was more equitable distribution of foodgrains.
2. Post-Reform Period (after 1978): Success
(a) In agriculture, by handing over plots of land to individuals for cultivation, it brought prosperity to a vast number of poor people.
(b) It created conditions for the subsequent phenomenal growth in rural industries and built up a strong support base for more reforms.
(c) More reforms included the gradual liberalisation of prices, fiscal decentralisation, increased autonomy for state owned enterprises (SOEs), the introduction of a diversified banking system, the development of stock markets, the rapid growth of the non-state sector, and the opening to foreign trade and investment.
(d) The restructuring of the economy and resulting efficiency gains have contributed to a more than ten-fold increase in GDP since 1978. Measured on a Purchasing Power Parity (PPP) basis, China in 2005 stood as the second largest economy in the world after the US.
(e) China’s economic growth as measured in terms of GDP on an average is 10.9% per year. In economic size, China is surpassed today only by the US, Japan, Germany and France.
(f) If its present growth trend continues, China is likely to be the world’s largest economic power by any measure by the year 2025.
Comparative Development Experience of India with its Neighbours 11 .IS
(g) China had success when it enforced one-child norm in 1979. The low population growth of China can be attributed to this one factor.
Thus, China’s structural reforms introduced in 1978 in a phased manner offer various lessons from its success story.

Q.13 Group the following features pertaining to the economies of India, China and Pakistan under three heads.

1. One-child norm
2. Low fertility rate
3. High degree of urbanisation
4. Mixed economy
5. Very high fertility rate
6. Large population
7. High density of population
8. Growth due to inanufacturing sector
9. Growth due to service sector

ANSWER:

1. China
2. China
3.  Pakistan and China
4.  India and Pakistan
5. Pakistan
6.  India and China
7.  India
8.  China
9.  India.

Q.14 Give reasons for the slow growth and re-emergence of poverty in Pakistan.
ANSWER:  Reforms were initiated in Pakistan in 1988.
1. Pre-Reform Period : Failure
(a) The proportion of poor in 1960s was more than 40 per cent.
(b) The economy started to stagnate, suffering from the drop in remittances from the Middle East.
(c) A growth rate of over 5% in the 1980s could not be sustained and the budget deficit increased steadily.
(d) At times foreign exchange reserves were as low as 2 weeks of imports.
2. Post-Reform Period (after 1988): Failure
The reform process led to worsening of all the economic indicators.
(a) The growth rate of GDP and its sectoral constituents have fallen in the 1990s.
(b) The proportion of poor declined to 25 per cent in 1980s and started rising again in 1990s. The reasons for the slow-down of growth and re-emergence of poverty in Pakistan’s economy are:
(i) Agricultural growth and food supply situation were based not on an institutionalised process of technical change but on good harvest. When there was a good harvest, the economy was in a good condition; when it was not, the economic indicators showed stagnation or negative trends.
(ii) Fall in foreign exchange earnings coming from remittances from Pakistani workers in the Middle East and the exports of highly volatile agricultural products.
(iii) There was also growing dependence on foreign loans on the one hand and increasing difficulty in paying back the loans on the other.

Q.15 Compare and contrast the development of India, China and Pakistan with respect to some
salient human development indicators.
ANSWER:  It is clear that:

1. China is moving ahead of India .and Pakistan. This is true for many indicators—income indicator such as GDP per capita, or proportion of population below poverty line or health indicators such as mortality rates, access to sanitation, literacy, life expectancy or malnourishment.
2. Pakistan is ahead of India in reducing proportion of people below the poverty line and also its performance in education, sanitation and access to water is better than that of India. Both China and Pakistan are in similar position with respect to the proportion of people below the international poverty rate of $1 a day, whereas the proportion is almost two times higher for India.
3. In China, for one lakh births, only 38 women die whereas in India it is 230 and in Pakistan it is 260.
4. India and Pakistan are ahead of China in providing improved water sources.

Q.16 Comment on the growth rate trends witnessed in China and India in the last two decades.
ANSWER: Growth of Gross Domestic Product (%), 1980-2009 In 1980s, China had remarkable growth rate of 10.3% when India was finding it difficult to maintain a growth rate of even 5%. After two decades, there was a marginal improvement in India’s and China’s growth rate.

Q.17 Fill in the blanks:

1.  First Five Year Plan of commenced in the year 1956. (Pakistan/China)
2.  Maternal mortality rate is high iri (China/Pakistan)
3. Proportion of people below poverty line is more in (India/Pakistan)
4.  Reforms in were introduced in 1978. (China/Pakistan).

ANSWER: (1) Pakistan, (2) Pakistan, (3) India, (4) China.