1. The decimal expansion of 22/7 is
(a) Terminating
(b) Non-terminating and repeating
(c) Non-terminating and Non-repeating
(d) None of the above
Answer: (b) Non-terminating and repeating
Explanation: 22/7 = 3.14285714286..
2. For some integer n, the odd integer is represented in the form of:
(a) n
(b) n + 1
(c) 2n + 1
(d) 2n
Answer: (c) 2n + 1
Explanation: Since 2n represents the even numbers, hence 2n + 1 will always represent the odd numbers. Suppose if n = 2, then 2n = 4 and 2n + 1 = 5.
3. HCF of 26 and 91 is:
(a) 15
(b) 13
(c) 19
(d) 11
Answer: (b) 13
Explanation: The prime factorisation of 26 and 91 is;
26 = 2 x 13
91 = 7 x 13
Hence, HCF (26, 91) = 13
4. Which of the following is not irrational?
(a) (3 + √7)
(b) (3 – √7)
(c) (3 + √7) (3 – √7)
(d) 3√7
Answer: (c) (3 + √7) (3 – √7)
Explanation: If we solve, (3 + √7) (3 – √7), we get;
(3 + √7) (3 – √7) = 32 – (√7)2 = 9 – 7 = 2 [By a2 – b2 = (a – b) (a + b)]
5. The addition of a rational number and an irrational number is equal to:
(a) rational number
(b) Irrational number
(c) Both
(d) None of the above
Answer: (b) Irrational number
The addition of a rational number and an irrational number is equal to irrational number.
6. The multiplication of two irrational numbers is:
(a) irrational number
(b) rational number
(c) Maybe rational or irrational
(d) None
Answer: (c) Maybe rational or irrational
The multiplication of two irrational numbers is maybe rational or irrational.
7. If set A = {1, 2, 3, 4, 5,…} is given, then it represents:
(a) Whole numbers
(b) Rational Numbers
(c) Natural numbers
(d) Complex numbers
Answer: (c) Natural numbers
If set A = {1, 2, 3, 4, 5,…} is given, then it represents natural numbers.
8. If p and q are integers and is represented in the form of p/q, then it is a:
(a) Whole number
(b) Rational number
(c) Natural number
(d) Even number
Answer: (b) Rational number
If p and q are integers and is represented in the form of p/q, then it is a rational number.
9. The largest number that divides 70 and 125, which leaves the remainders 5 and 8, is:
(a) 65
(b) 15
(c) 13
(d) 25
Answer: (c) 13
Explanation: 70 – 5 = 65 and 125 – 8 = 117
HCF (65, 117) is the largest number that divides 70 and 125 and leaves remainder 5 and 8.
HCF (65, 117) = 13
10. The least number that is divisible by all the numbers from 1 to 5 is:
(a) 70
(b) 60
(c) 80
(d) 90
Answer: (b) 60
Explanation: The least number will be LCM of 1, 2, 3, 4, 5.
Hence, LCM (1, 2, 3, 4, 5) = 2 x 2 x 3 x 5 = 60
11. The sum or difference of two irrational numbers is always
(a) rational
(b) irrational
(c) rational or irrational
(d) not determined
Answer: (b) irrational
12. The decimal expansion of the rational number 23/(22 . 5) will terminate after
(a) one decimal place
(b) two decimal places
(c) three decimal places
(d) more than 3 decimal places
Answer: (b) two decimal places
Explanation:
23/(22 . 5) = (23 × 5)/(22 . 52) = 115/(10)2 = 115/100 = 1.15
Hence, 23/(22 . 5) will terminate after two decimal places.
13. Euclid’s division lemma states that for two positive integers a and b, there exist unique integers q and r such that a = bq + r, where r must satisfy
(a) 1 < r < b
(b) 0 < r ≤ b
(c) 0 ≤ r < b
(d) 0 < r < b
Correct option: (c) 0 ≤ r < b
Euclid’s division lemma states that for two positive integers a and b, there exist unique integers q and r such that a = bq + r, where r must satisfy 0 ≤ r < b.
14. For some integer m, every even integer is of the form
(a) m
(b) m + 1
(c) 2m
(d) 2m + 1
Answer: (c) 2m
For some integer m, every even integer is of the form 2m.
15. Using Euclid’s division algorithm, the HCF of 231 and 396 is
(a) 32
(b) 21
(c) 13
(d) 33
Answer: (d) 33
Explanation:
396 > 231
Using Euclid’s division algorithm,
396 = 231 × 1 + 165
231 = 165 × 1 + 66
165 = 66 × 2 + 33
66 = 33 × 2 + 0
16. If the HCF of 65 and 117 is expressible in the form 65m – 117, then the value of m is
(a) 4
(b) 2
(c) 1
(d) 3
Answer: (b) 2
Explanation:
117 > 65
117 = 65 × 1 + 52
65 = 52 × 1 + 13
52 = 13 × 4 + 0
Therefore, HCF(65, 117) = 13
According to the given,
65m – 117 = 13
65m = 117 + 13
65m = 130
m = 130/65 = 2
17. The prime factorisation of 96 is
(a) 25 × 3
(b) 26
(c) 24 × 3
(d) 24 × 32
Answer: (a) 25 × 3
Explanation:
The prime factorisation of 96 is:
96 = 2 × 2 × 2 × 2 × 2 × 3 = 25 × 3
18. n² – 1 is divisible by 8, if n is
(a) an integer
(b) a natural number
(c) an odd integer
(d) an even integer
Answer: (c) an odd integer
Explanation:
We know that an odd number in the form (2Q + 1) where Q is a natural number ,
so, n² -1 = (2Q + 1)² -1
= 4Q² + 4Q + 1 -1
= 4Q² + 4Q
Substituting Q = 1, 2,…
When Q = 1,
4Q² + 4Q = 4(1)² + 4(1) = 4 + 4 = 8 , it is divisible by 8.
When Q = 2,
4Q² + 4Q = 4(2)² + 4(2) =16 + 8 = 24, it is also divisible by 8 .
When Q = 3,
4Q² + 4Q = 4(3)² + 4(3) = 36 + 12 = 48 , divisible by 8
It is concluded that 4Q² + 4Q is divisible by 8 for all natural numbers.
Hence, n² -1 is divisible by 8 for all odd values of n.
19. For any two positive integers a and b, HCF (a, b) × LCM (a, b) =
(a) 1
(b) (a × b)/2
(c) a/b
(d) a × b
Answer: (d) a × b
For any two positive integers a and b, HCF (a, b) × LCM (a, b) = a × b.
20. The values of the remainder r, when a positive integer a is divided by 3 are
(a) 0, 1, 2
(b) Only 1
(c) Only 0 or 1
(d) 1, 2
Answer: (a) 0, 1, 2
Explanation:
According to Euclid’s division lemma,
a = 3q + r, where 0 ≤ r < 3 and r is an integer.
Therefore, the values of r can be 0, 1 or 2.
Thanks for the comprehensive overview. Very helpful!