9 + Interview Questions in IDENTITIES IN GENERAL AWARENESS Page 1 InterviewSolution

1.	Find the square of 10.01
Answer» 10.01² = (10 + 0.01)² = 10² + 2 × 10 × 0.01 + 0.01² = 100 + 0.2 + 0.0001 = 100.2001

1.

Find the square of 10.01

Answer»

10.01² = (10 + 0.01)²

= 10² + 2 × 10 × 0.01 + 0.01²

= 100 + 0.2 + 0.0001 = 100.2001

Discussion

2.	Find the square of 99. 99
Answer» 99.99² = (100 – 0.01)² = 100² – 2 × 100 × 0.01 + 0.01² 10000 – 2 + 0.0001 = 9998.0001

2.

Find the square of 99. 99

Answer»

99.99² = (100 – 0.01)²

= 100² – 2 × 100 × 0.01 + 0.01²

10000 – 2 + 0.0001 = 9998.0001

Discussion

3.	Find the squares of the given numbers by using simple calculations.(i) 52(ii) 105(iii) \(20\frac{1}{2}\)(iv) 10.2
Answer» (i) 52² = (50 + 2)² = 50² + 2 × 50 × 2 + 2² = 2500 + 200 + 4 = 2704 (ii) 105² = (100 + 5)² = 100² + 2 × 100 × 5 + 5² = 10000 + 1000 + 25 = 11025 (iii) \((20\frac{1}{2})^2\) = \((20+\frac{1}{2})^2\) = 20² + 2 × 20 × \(\frac{1}{2}\) + \((\frac{1}{2})^2\) = 400 + 20 + \(\frac{1}{4}\) = 420\(\frac{1}{4}\) (iv) (10.2)² = (10 + 0.2)² = 10² + 2 × 10 × 0.2 + (0.2)² = 100 + 4 + 0.04 = 104.04

3.

Find the squares of the given numbers by using simple calculations.(i) 52(ii) 105(iii) \(20\frac{1}{2}\)(iv) 10.2

Answer»

(i) 52² = (50 + 2)²

= 50² + 2 × 50 × 2 + 2²

= 2500 + 200 + 4

= 2704

(ii) 105² = (100 + 5)²

= 100² + 2 × 100 × 5 + 5²

= 10000 + 1000 + 25

= 11025

(iii) \((20\frac{1}{2})^2\) = \((20+\frac{1}{2})^2\)

= 20² + 2 × 20 × \(\frac{1}{2}\) + \((\frac{1}{2})^2\)

= 400 + 20 + \(\frac{1}{4}\)

= 420\(\frac{1}{4}\)

(iv) (10.2)² = (10 + 0.2)²

= 10² + 2 × 10 × 0.2 + (0.2)²

= 100 + 4 + 0.04

= 104.04

Discussion

4.	Observe the following operations1 × 4 = (2 × 3) – 2 2 × 5 = (3 × 4) – 2 3 × 6 = (4 × 5) – 2 4 × 7 = (5 × 6) – 2(i) Write next operations of two rows in the same order.(ii) Among the 4 nearest natural numbers. Find the relation- ship between the product of first and last and those in the middle.(iii) Write the principle in algebraic expression and explain the reason
Answer» (i) 5 × 8 = (6 × 7) – 2 6 × 9 = (7 × 8) – 2 (ii) The product of first and last numbers = 2 subtracted from the product of two middle numbers. (iii) The numbers are n, n + 1, n + 2, (n + 3) n (n + 3) = n² + 3n (n + 1) (n + 2) = n² + 3n + 2 difference = 2

4.

Observe the following operations1 × 4 = (2 × 3) – 2 2 × 5 = (3 × 4) – 2 3 × 6 = (4 × 5) – 2 4 × 7 = (5 × 6) – 2(i) Write next operations of two rows in the same order.(ii) Among the 4 nearest natural numbers. Find the relation- ship between the product of first and last and those in the middle.(iii) Write the principle in algebraic expression and explain the reason

Answer»

(i) 5 × 8 = (6 × 7) – 2

6 × 9 = (7 × 8) – 2

(ii) The product of first and last numbers = 2 subtracted from the product of two middle numbers.

(iii) The numbers are n, n + 1, n + 2, (n + 3) n (n + 3) = n² + 3n

(n + 1) (n + 2) = n² + 3n + 2

difference = 2

Discussion

5.	Explain on the basis of algebra that any natural number which is not a multiple of 3 when divided by 3, we get the remainder as 1.
Answer» We can write the natural numbers which are not the multiple of 3 as. 3n + 1, 3n + 2 (3n + 1)² = (3n)² + 6n + 1 = 9n² + 6n + 1 = 3 (3n² + 2n) + 1 When divided by 3, remainder is 1 (3n + 2)² = 9n² + 6n + 4 = 9n² + 6n + 3 + 1 = 3 (3n² + 2n + 1) + 1 When divided by 3, remainder is 1

5.

Explain on the basis of algebra that any natural number which is not a multiple of 3 when divided by 3, we get the remainder as 1.

Answer»

We can write the natural numbers which are not the multiple of 3 as.

3n + 1, 3n + 2

(3n + 1)² = (3n)² + 6n + 1

= 9n² + 6n + 1

= 3 (3n² + 2n) + 1

When divided by 3, remainder is 1

(3n + 2)² = 9n² + 6n + 4

= 9n² + 6n + 3 + 1

= 3 (3n² + 2n + 1) + 1

When divided by 3, remainder is 1

Discussion

6.	Find the squares of the numbers given below.(i) 49(ii) 98(iii) 7\(\frac{3}{4}\)(iv) 9.25
Answer» (i) 49² = (50 – 1)² = 50² – 2 × 50 × 1 + 1 = 2500 – 100 + 1 = 2401 (ii) 98² = (100 – 2)² = 100² – 2 × 100 × 2 + 2² = 10000 – 400 + 4 = 9604 (iii) (7\(\frac{3}{4}\))² = (8 – \(\frac{1}{4}\))² = 8² – 2 × 8 × \(\frac{1}{4}\) + (\(\frac{1}{4}\))² = 64 – 4 + \(\frac{1}{16}\) = 60\(\frac{1}{16}\) (iv) 9.25² = (10 - 75)² = 10² – 2 × 10 × .75 + (.75)² = 100 – 15 + 0.5625 = 85. 5625

6.

Find the squares of the numbers given below.(i) 49(ii) 98(iii) 7\(\frac{3}{4}\)(iv) 9.25

Answer»

(i) 49² = (50 – 1)²

= 50² – 2 × 50 × 1 + 1

= 2500 – 100 + 1

= 2401

(ii) 98² = (100 – 2)²

= 100² – 2 × 100 × 2 + 2²

= 10000 – 400 + 4

= 9604

(iii) (7\(\frac{3}{4}\))² = (8 – \(\frac{1}{4}\))²

= 8² – 2 × 8 × \(\frac{1}{4}\) + (\(\frac{1}{4}\))²

= 64 – 4 + \(\frac{1}{16}\)

= 60\(\frac{1}{16}\)

(iv) 9.25²

= (10 - 75)²

= 10² – 2 × 10 × .75 + (.75)²

= 100 – 15 + 0.5625

= 85. 5625

Discussion

7.	Prove that, the squares of numbers which end in 3 end in 9.
Answer» We can write the numbers end in 3 as 10x + 3 is (10x + 3)² = (10x)² + 60x + 9; So it ends in 9.

7.

Prove that, the squares of numbers which end in 3 end in 9.

Answer»

We can write the numbers end in 3 as 10x + 3 is (10x + 3)²

= (10x)² + 60x + 9; So it ends in 9.

Discussion

8.	Observe the following(1/2)2 + (1 (1/2)2) = 2(1/2), 2 = 2 x 12Explain the common principle in these using algebra?
Answer» (x – \(\frac{1}{2}\))² + (x + \(\frac{1}{2}\))² x² – x + \(\frac{1}{4}\) + x² + x + \(\frac{1}{4}\) 2x² + \(\frac{1}{2}\)

8.

Observe the following(1/2)2 + (1 (1/2)2) = 2(1/2), 2 = 2 x 12Explain the common principle in these using algebra?

Answer»

(x – \(\frac{1}{2}\))² + (x + \(\frac{1}{2}\))²

x² – x + \(\frac{1}{4}\) + x² + x + \(\frac{1}{4}\)

2x² + \(\frac{1}{2}\)

Discussion

9.	Find the difference in the given questions.(i) (125 × 75) – (126 × 74) (ii) (124 × 76) – (126 × 74) (iii) (224 × 176) – (226 × 174) (iv) (10.3 × 9.7) – (10.7 × 9.3) (v) (11.3 × 10.7) – (11.7 × 11.3)
Answer» (i) (125 × 75) – (126 × 74) (100 + 25) (100 – 25) – (100 + 26) (100 – 26) (100² – 25²) – (100² – 26²) = 26² – 25² = (26 + 25) (26 – 25) = 51 (ii) (124 × 76) – (126 × 74) (100 + 24) (100 – 24) – (100 + 26) (100 – 26) (100² – 24²) – (100² – 26)² = 26² – 24² = (26 + 24) (26 – 24) = 50 × 2 = 100 (iii) (224 × 176) – (226 × 174) (200 + 24) (200 – 24) – (200 + 26) (200 – 26) (200² – 24²) – (200² – 26²) = 26² – 24² = (26 + 24) (26 – 24) = 50 × 2 = 100 (iv) (10.3 × 9.7) – (10.7 × 9.3) (10 + 0.3) (10 – 0.3) – (10 + 0.7) (10 – 0.7) (10² – 0.3²) – (10² – 0.7²). = 0.7² – 0.3² = (0.7 + 0.3) (0.7 – 0.3) = 1 × 4 = 4 (v) (11.3 × 10.7) – (11.7 × 11.3) (11 + 0.3) (11 – 0.3) – (11 + 0.7) (11 – 0.7) (11² – 0.3²) – (11² – 0.7²) = 0.7² – 0.3² = (0.7 + 0.3) (0.7 – 0.3) = 1 × 0.4 = 0.4

9.

Find the difference in the given questions.(i) (125 × 75) – (126 × 74) (ii) (124 × 76) – (126 × 74) (iii) (224 × 176) – (226 × 174) (iv) (10.3 × 9.7) – (10.7 × 9.3) (v) (11.3 × 10.7) – (11.7 × 11.3)

Answer»

(i) (125 × 75) – (126 × 74)

(100 + 25) (100 – 25) – (100 + 26) (100 – 26)

(100² – 25²) – (100² – 26²)

= 26² – 25²

= (26 + 25) (26 – 25)

= 51

(ii) (124 × 76) – (126 × 74)

(100 + 24) (100 – 24) – (100 + 26) (100 – 26)

(100² – 24²) – (100² – 26)²

= 26² – 24²

= (26 + 24) (26 – 24)

= 50 × 2

= 100

(iii) (224 × 176) – (226 × 174)

(200 + 24) (200 – 24) – (200 + 26) (200 – 26)

(200² – 24²) – (200² – 26²)

= 26² – 24²

= (26 + 24) (26 – 24)

= 50 × 2

= 100

(iv) (10.3 × 9.7) – (10.7 × 9.3)

(10 + 0.3) (10 – 0.3) – (10 + 0.7) (10 – 0.7)

(10² – 0.3²) – (10² – 0.7²).

= 0.7² – 0.3²

= (0.7 + 0.3) (0.7 – 0.3)

= 1 × 4 = 4

(v) (11.3 × 10.7) – (11.7 × 11.3)

(11 + 0.3) (11 – 0.3) – (11 + 0.7) (11 – 0.7)

(11² – 0.3²) – (11² – 0.7²)

= 0.7² – 0.3²

= (0.7 + 0.3) (0.7 – 0.3)

= 1 × 0.4 = 0.4

Discussion

Explore topic-wise InterviewSolutions in .

Find the square of 10.01

Find the square of 99. 99

Find the squares of the given numbers by using simple calculations.(i) 52(ii) 105(iii) \(20\frac{1}{2}\)(iv) 10.2

Explain on the basis of algebra that any natural number which is not a multiple of 3 when divided by 3, we get the remainder as 1.

Find the squares of the numbers given below.(i) 49(ii) 98(iii) 7\(\frac{3}{4}\)(iv) 9.25

Prove that, the squares of numbers which end in 3 end in 9.

Observe the following(1/2)2 + (1 (1/2)2) = 2(1/2), 2 = 2 x 12Explain the common principle in these using algebra?

Find the difference in the given questions.(i) (125 × 75) – (126 × 74) (ii) (124 × 76) – (126 × 74) (iii) (224 × 176) – (226 × 174) (iv) (10.3 × 9.7) – (10.7 × 9.3) (v) (11.3 × 10.7) – (11.7 × 11.3)