CBSE Class 10 Mathematics Real Numbers MCQs Set J

Question. For \( P \in \mathbb{N} \), \( 3^{4P} - 2^{4P} \) is always divisible by ____
(a) 15
(b) 5
(c) 13
(d) Both (b) and (c)
Answer: D

Question. The greatest number of 5 digits exactly divisible by 15, 24 and 36 is ____
(a) 99620
(b) 99720
(c) 99968
(d) 99960
Answer: B

Question. The greatest number of 6 digits exactly divisible by all the numbers between 1 and 10 (both inclusive) is _________
(a) 997920
(b) 999768
(c) 999660
(d) None of the options
Answer: A

Question. The smallest three digit number which leaves remainders 8 and 12 when divided by 28 and 32 respectively, is _________
(a) 102
(b) 222
(c) 202
(d) 204
Answer: D

Question. If \( P = \sqrt{n-1} + \sqrt{n+1} \) where \( n \) is a positive integer then the value of P is
(a) a rational number
(b) not a rational number
(c) an integer
(d) a natural number
Answer: B

Question. The largest number that will divide 398, 606 and 474 leaving remainders 7, 11 and 15 respectively is ________
(a) 52
(b) 26
(c) 17
(d) 18
Answer: C

Question. Which one among the following statements is true?
(a) The remainder when the square of any number is divided by 4 is 1 or 0.
(b) There is no natural number for which \( 4^n \) ends with digit zero.
(c) A positive integer n is prime, if no prime p less than or equal to \( \sqrt{n} \) divides n.
(d) All the above
Answer: D

Question. The unit value of \( 6^{100} - 5^{100} \) is ______
(a) 0
(b) 1
(c) 2
(d) 3
Answer: B

Question. For any odd natural number n, \( (\sqrt{3})^{4n} + (\sqrt{2})^{4n} \) is always divisible by ______
(a) 5
(b) 7
(c) 17
(d) 13
Answer: D

Question. If I is a positive integer then \( (I)^2 \) will be in the form of ————
(a) 4m for some integer m
(b) 8m for some integer m
(c) 4m + 1 for some integer m
(d) Both (a) and (c)
Answer: D

Question. Which among the following statements is not true?
(a) The square of any odd integer is of the form 4q + 1, for some integer q.
(b) For any odd integer p, \( p^2 - 1 \) is divisible by 8.
(c) If p and q are both odd positive integers, then \( p^2 + q^2 \) is even and divisible by 4.
(d) For any natural number n, \( 12^n \) cannot end with the digit 0 or 5.
Answer: C

Question. For any natural number n, \( (2n + 1)^2 - 1 \) is always divisible by ______
(a) 2
(b) 4
(c) 8
(d) All the options
Answer: D

Question. Which of the following statements is always true?
(a) The sum or difference of a rational and an irrational number is rational.
(b) Every irrational number is a surd.
(c) The product or quotient of a non-zero rational number and an irrational number is irrational.
(d) None of the options
Answer: C

Question. The value of \( (27)^{3p} - (13)^{3p} \) ends in ______ (where p is a natural number)
(a) 0
(b) 4
(c) 6
(d) Either (b) or (c)
Answer: D

Question. If \( P = 1 \cdot 3 \cdot 5 \cdot 7 \dots 21 \) and \( Q = 2 \cdot 4 \cdot 6 \cdot 8 \cdot 10 \dots 22 \) then HCF of P and Q is ________
(a) 12375
(b) 14175
(c) 825
(d) 925
Answer: B

Question. In a seminar, the numbers of participants in science, English and Mathematics are 144, 180 and 192 respectively. Find the minimum number of rooms required if in each room the same number of participants are to be seated and all of them being in the same subject.
(a) 38
(b) 40
(c) 43
(d) 45
Answer: C

Question. Without actually performing the Song division, choose which among the following rational numbers will not have a terminating decimal expansion.
(a) \( \frac{123}{16} \)
(b) \( \frac{351}{2^7 \times 5^8 \times 7^{18}} \)
(c) \( \frac{32}{2^8 \times 5^9} \)
(d) \( \frac{833}{49 \times 2^7} \)
Answer: B

Question. The largest number that divides 588, 1999 and 1650 leaving 3, 10 and 12 respectively is ______
(a) 117
(b) 109
(c) 27
(d) 43
Answer: A

Question. The decimal expansion of the rational number \( \frac{12879}{1250} \) will terminate after:
(a) One decimal places
(b) Two decimal places
(c) Three decimal places
(d) Four decimal places
Answer: D

Question. Find the greatest prime factor in 527527.
(a) 17
(b) 11
(c) 13
(d) 31
Answer: D

Question. If p is a single digit natural number and the unit digits of \( p^4 \) and p are same, then how many possibilities p can assume?
(a) 2
(b) 3
(c) 4
(d) 5
Answer: B

Question. The sum of LCM and HCF of two numbers is 29610. If their LCM is 140 times the HCF of the numbers then which among the following can be one of the numbers?
(a) 330
(b) 1470
(c) 525
(d) 462
Answer: B

Question. The value of \( (22)^{3^m} + (28)^{3^m} \) ends in ______ \( \{ M \in N \} \).
(a) 8
(b) 2
(c) 6
(d) 0
Answer: D

Question. If LCM and HCF of two numbers are 324 and 18 respectively, then how many such pairs of numbers are possible?
(a) 0
(b) 1
(c) 2
(d) 3
Answer: C

Question. If \( p = \sqrt{11} + \sqrt{5} \), \( q = \sqrt{14} + \sqrt{2} \) and \( r = \sqrt{13} + \sqrt{3} \) then which one of the following holds true?
(a) p > q > r
(b) p < q < r
(c) p > r > q
(d) p < r < q
Answer: C

Question. The number of ways, in which 360 can be resolved in two factors, is ______
(a) 24
(b) 18
(c) 12
(d) 15
Answer: C

Question. If \( u = \sqrt[16]{7} + \sqrt[16]{5} \), \( v = \sqrt{7} + \sqrt{5} \), \( w = \sqrt[8]{7} + \sqrt[8]{5} \), \( x = \sqrt[16]{7} - \sqrt[16]{5} \), and \( y = \sqrt[4]{7} + \sqrt[4]{5} \), then which one of the following is a rational number?
(a) uvxy
(b) uvwxy
(c) uxwy
(d) vwxy
Answer: B

Question. A mason has to fit two bathrooms with square marble tiles of the largest possible size. The dimensions of each such bathroom are 12 fts and 10 fts. If the size of the tiles in inches has to be taken then number of such tiles required is ____
(a) 15
(b) 30
(c) 60
(d) 80
Answer: C

Question. If HCF of 374 and 255 is H and H = 255m + 374n then the value of m - n is equal to _____
(a) 3
(b) 4
(c) 5
(d) 1
Answer: C

Question. Choose which one among the following statement is incorrect?
(a) HCF of two co-primes a and b is 1.
(b) LCM of two co-primes m and n is mn.
(c) By using Euclid's division lemma for two numbers 155 and 345, we get 345 = 155 × 2 + 35.
(d) None of the options
Answer: D

Question. If LCM and HCF of two numbers are 3003 and 21 respectively, then how many such numbers are possible?
(a) 0
(b) 1
(c) 2
(d) 3
Answer: C

Question. The largest number which divides 1288 and 2915 and leaves the remainders 1 and 8 respectively, is H and it satisfies the expression, H = 45m + 288n. Find the value of m + n.
(a) 11
(b) 15
(c) 13
(d) 10
Answer: A

Question. The smallest number, which when increased by 19 is exactly divisible by both 2079 and 1404, is _______
(a) 6200
(b) 625
(c) 6218
(d) 3208
Answer: C

Question. \( \sqrt{\frac{7+4\sqrt{3}}{2}} \) equals to ________
(a) \( \sqrt{2} + \sqrt{6} \)
(b) \( \frac{2\sqrt{2} + \sqrt{6}}{2} \)
(c) \( \frac{\sqrt{2} + \sqrt{6}}{2} \)
(d) \( \frac{\sqrt{3} + 2}{4} \)
Answer: B

Question. If HCF of the numbers (3600, x) = 20 then how many values are possible for x? (where it is assumed that x is a product of a power of 2 and a power of 5 only)
(a) One
(b) Two
(c) Three
(d) Four
Answer: B

Question. The number of ways, in which 576 can be resolved into two factors, is ________
(a) 8
(b) 9
(c) 10
(d) 11
Answer: D

Question. Four runners P, Q, R and S start running around a circular track simultaneously. If they complete one round in 16, 12, 24, 18 minutes respectively, after how much time they will meet next?
(a) 2 hours 20 minutes
(b) 2 hours
(c) 3 hours 18 minutes
(d) 2 hours 24 minutes
Answer: D

Question. If LCM and HCF of two numbers are equal, then the numbers will be _________
(a) Composite
(b) Prime
(c) Equal
(d) Co-prime
Answer: C

Question. If the product of two numbers is 149058 and HCF of these numbers is 21 then how many pairs of these numbers are possible?
(a) 1
(b) 2
(c) 3
(d) 4
Answer: B

Question. If \( A = 14 + (1 \times 2 \times 3 \times 4 \times 5 \dots 10 \times 14) \) and \( B = 19 + (1 \times 2 \times 3 \times 4 \times 5 \dots 10 \times 19) \) then which one of the following is/are correct?
(i) B - A is a prime number.
(ii) B + A is a composite number.
(iii) A is a composite number.
(iv) B is a prime number.
(a) Both (i) and (ii)
(b) Both (ii) and (iii)
(c) Both (iii) and (iv)
(d) All (i), (ii), (iii) and (iv)
Answer: B