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## Tamilnadu Samacheer Kalvi 10th Maths Solutions Chapter 2 Numbers and Sequences Ex 2.10

Multiple choice questions

Question 1.

Euclid’s division lemma states that for positive integers a and b, there exist unique integers q and r such that a = bq + r, where r must satisfy.

(1) 1 < r < b

(2) 0 < r < b

(3) 0 < r < b

(4) 0 < r < b

Answer:

(3) 0 < r < b

Question 2.

Using Euclid’s division lemma, if the cube of any positive integer is divided by 9 then the possible remainders are

(1) 0, 1, 8

(2) 1, 4, 8

(3) 0, 1, 3

(4) 1, 3, 5

Answer:

(1) 0,1,8

Hint:

Cube of any +ve integers 1^{3}, 2^{3}, 3^{3}, 4^{3},. . .

1, 8, 27, 64, 125, 216 …

Remainders when 27, 64, 125 are divided by 9.

Question 3.

If the H.C.F of 65 and 117 is expressible in the form of 65m -117 , then the value of m is ………………….

(1) 4

(2) 2

(3) 1

(4) 3

Answer:

(2) 2

Hint:

117 = 3 × 3 × 13

65 = 5 × 13

H.C.F = 13

65m – 117 = 13 ⇒ 65m = 117 + 13 = 130

m = \(\frac { 130 }{ 65 } \) = 2

The value of m = 2

Question 4.

The sum of the exponents of the prime factors in the prime factorization of 1729 is

(1) 1

(2) 2

(3) 3

(4) 4

Answer:

(3) 3

Hint:

1729 = 7^{1} × 13^{1} × 19^{1}

Question 5.

The least number that is divisible by all the numbers from 1 to 10 (both inclusive) is

(1) 2025

(2) 5220

(3) 5025

(4) 2520

Answer:

(4) 2520

Hint:

∴ L.C.M. of 1,2,3,4,…,10 is 2 × 2 × 3 × 5 × 7 × 2 × 3 = 2520

Question 6.

7^{4k} ≡ …………………. (mod 100)

(1) 1

(2) 2

(3) 3

(4) 4

Answer:

(1) 1

Hint:

7^{4k} ≡ . . . . . (mod 100)

7^{4k} = (7^{4})^{k} ≡ ……….. (mod 100) (7^{4} – 2401)

The value is 1.

Question 7.

Given F_{1} = 1, F_{2} = 3 and Fn = F_{n-1} + F_{n-2} then

(1) 3

(2) 5

(3) 8

(4) 11

Answer:

(4) 11

Answer:

F_{1} = 1, F_{2} = 3

F_{n} = F_{n-1} + F_{n-2}

F_{5} = F_{5-1} + F_{5-2} = F_{4} + F_{3}

= F_{3} + F_{2} + F_{2} + F_{1}

= F_{2} + F_{1} + F_{2} + F_{2} + F_{1}

= 3 + 1 + 3 + 3 + 1 = 11

Question 8.

The first term of an arithmetic progression is unity and the common difference is 4. Which of the following will be a term of this A.P …………..

(1) 4551

(2) 10091

(3) 7881

(4) 13531

Answer:

(3) 7881

Hint:

Here a = 1, d = 4

t_{n} = a + (n – 1) d = 1 + (n – 1) 4

= 1 + 4n – 4

= 4n – 3

4554

(i) 4n – 3 = 4551 ⇒ 4n = 4551 + 3 ⇒ n = \(\frac { 4554 }{ 4 } \) = 1138.5.

It is not a term of A.P.

(ii) 4n – 3 = 10091 ⇒ 4n = 10091 + 3 = 10094

n = \(\frac { 10094 }{ 4 } \) = 2523.5 it is a term of A.P.

(iii) 4n – 3 = 7881 ⇒ 4n = 7881 + 3

n = \(\frac { 7884 }{ 4 } \) = 1971.

∴ 7881 is a term of the A.P.

Question 9.

If 6 times of 6^{th} term of an A.P is equal to 7 times the 7th term, then the 13th term of the A.P. is

(1) 0

(2) 6

(3) 7

(4) 13

Answer:

(1) 0

Hint:

6t_{6} = 7t_{7}

6(a + 5d) = 7(a + 6d)

6a + 30d = 7a + 42d

7a + 42d – 6a – 30d = 0

a + 12d = 0 = t_{13}

Question 10.

An A.P consists of 31 terms. If its 16^{th} term is m, then the sum of all the terms of this A.P. is …………..

(1) 16m

(2) 62m

(3) 31m

(4) \(\frac { 31 }{ 2 } \)m

Answer:

(3) 31m

Hint:

M = 31

t_{16} = m ⇒ a + 15d = m

S_{n} = \(\frac { n }{ 2 } \)[2a + (n – 1)d]

S_{n} = \(\frac { 31 }{ 2 } \)[2a + 30d]= \(\frac { 31 }{ 2 } \) × 2[a + 15d]

= 31 (m) = 31m

Question 11.

In an A.P., the first term is 1 and the common difference is 4. How many terms of the A.P must be taken for their sum to be equal to 120?

(1) 6

(2) 7

(3) 8

(4) 9

Answer:

(3) 8

Hint:

Question 12.

If A = 2^{65} and B = 2^{64} + 2^{63} + 2^{62} + +2^{0} which of the following is true?

(1) B is 264 more than A

(2) A and B are equal

(3) B is larger than A by 1

(4) A is larger than B by 1

Answer:

(4) A is larger than B by 1

Hint:

A = 2^{65}

B = 2^{64} + 2^{63} + 2^{62} + … + 20

B = 2^{0} + 2^{1} + 2^{2} + … + 264

G.P = 1 + 2^{1} + 2^{2} + … + 2^{64} it is a G.P

Here a = 1, r = 2, n = 65

A = 2^{65}, B = 2^{65} – 1

∴ B is smaller.

A is larger than B by 1.

Question 13.

The next term of the sequence \(\frac { 3 }{ 16 } \),\(\frac { 1 }{ 8 } \),\(\frac { 1 }{ 12 } \),\(\frac { 1 }{ 18 } \), ….

(1) \(\frac { 1 }{ 24 } \)

(2) \(\frac { 1 }{ 27 } \)

(3) \(\frac { 2 }{ 3 } \)

(4) \(\frac { 1 }{ 81 } \)

Answer:

(2) \(\frac { 1 }{ 27 } \)

Hint:

Question 14.

If the sequence t_{1}, t_{2}, t_{3}, …………are in A.P. then the sequence t_{6}, t_{12}, t_{18}, …… is ………….

(1) a Geometric progression

(2) an Arithmetic progression

(3) neither an Arithmetic progression nor a Geometric progression

(4) a constant sequence

Answer:

(2) an Arithmetic progression

Hint: t_{1}, t_{2}, t_{3} …. are in A.P

t_{6}, t_{12}, t_{18} …… is also an A.P. (6, 12, 18 …….. is an A.P.)

Question 15.

The value of (1^{3} + 2^{3} + 3^{3} + … + 15^{3}) – (1 + 2 + 3 + … + 15) is

(1) 14400

(2) 14200

(3) 14280

(4) 14520

Answer:

(3) 14280

Hint:

\(\left(\frac{15 \times 16}{2}\right)^{2}-\frac{15 \times 16}{2}\) = (120)^{2} – 120 = 14280