CBSE Class 10 Real Numbers Sure Shot Questions Set 06

Question. If n is a natural number, then \( 9^{2n} - 4^{2n} \) is always divisible by
(a) 5
(b) 13
(c) Both (a) and (b)
(d) Neither (a) nor (b)
Answer: (c)

Question. N is a natural number such that when \( N^3 \) is divided by 9, it leaves remainder a. It can be concluded that
(a) a is a perfect square
(b) a is a perfect cube
(c) Both (a) and (b)
(d) Neither (a) nor (b)
Answer: (b)

Question. The remainder of any perfect square divided by 3 is
(a) 0
(b) 1
(c) Either (a) or (b)
(d) Neither (a) nor (b)
Answer: (c)

Question. Find the HCF of 432 and 504 using prime factorization method.
(a) 36
(b) 72
(c) 96
(d) 108
Answer: (b)

Question. If n is any natural number, then \( 6^n - 5^n \) always ends with
(a) 1
(b) 3
(c) 5
(d) 7
Answer: (a)

Question. The LCM of two numbers is 1200. Which of the following cannot be their HCF ?
(a) 600
(b) 500
(c) 200
(d) 400
Answer: (b)

Question. Which of the following is always true ?
(a) The rationalising factor of a number is unique
(b) The sum of two distinct irrational numbers is rational
(c) The product of two distinct irrational numbers is irrational
(d) None of these
Answer: (d)

Question. Find the remainder when the square of any number is divided by 4.
(a) 0
(b) 1
(c) Either (a) or (b)
(d) Neither (a) nor (b)
Answer: (c)

Question. Ashok has two vessels which contain 720 ml and 405 ml of milk respectively. Milk in each vessel is poured into glasses of equal capacity of their brim. find the minimum number of glasses which can be filled with milk.
(a) 45
(b) 35
(c) 25
(d) 30
Answer: (c)

Question. If n is an odd natural number, \( 3^{2n} + 2^{2n} \) is always divisible by
(a) 13
(b) 5
(c) 17
(d) 19
Answer: (a)

Question. If the product of two irrational numbers is rational, then which of the following can be concluded ?
(a) The ratio of the greater and the smaller numbers is an integer
(b) The sum of the numbers must be rational
(c) The excess of the greater irrational number over the smaller irrational number must be rational
(d) None of these
Answer: (d)

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

Question. The sum of LCM and HCF of two numbers is 1260. If their LCM is 900 more than their HCF, find the product of two numbers.
(a) 203400
(b) 194400
(c) 198400
(d) 205400
Answer: (b)

Question. Find the remainder when the square of any prime number greater than 3 is divided by 6.
(a) 1
(b) 3
(c) 2
(d) 4
Answer: (a)

Question. If HCF (72, q) = 12 then how many values can q take ? (Assume q be a product of a power of 2 and a power of 3 only)
(a) 1
(b) 2
(c) 3
(d) 4
Answer: (b)

Question. Find the HCF of 120 and 156 using Euclid’s division algorithm.
(a) 18
(b) 12
(c) 6
(d) 24
Answer: (b)

Question. What is the digit in the tens place in the product of the first 35 even natural numbers ?
(a) 6
(b) 2
(c) 0
(d) 5
Answer: (c)

Question. The LCM of \( \frac{1}{4} \) and \( \frac{2}{5} \) is
(a) 1
(b) \( \frac{1}{10} \)
(c) 2
(d) \( \frac{1}{20} \)
Answer: (c)

Question. The multiplicative inverse of \( (x + 1) + \frac{1}{(x - 1)} \) is
(a) \( \frac{1}{(x+1)} + (x - 1) \)
(b) \( (x - 1) - \frac{1}{(x+1)} \)
(c) \( \frac{x-1}{x^2} \)
(d) \( \frac{x+1}{x^2} \)
Answer: (c)

Question. Find the unit’s digit in the product of the first 50 odd natural numbers.
(a) 0
(b) 5
(c) 7
(d) None
Answer: (b)

Question. There are 20 balls. The balls are numbered consecutively starting from anyone of the numbers from 1 to 20. For any case, the sum of the numbers on all the balls will be a/an
(a) odd number
(b) even number
(c) prime number
(d) Cannot say
Answer: (b)

Question. Which pair of numbers below are twin primes ?
(a) 8 and 9
(b) 2 and 3
(c) 3 and 7
(d) 41 and 43
Answer: (d)

Question. Which of the following values are even ?
(a) 21 + 18 + 9 + 2 + 19
(b) 34 × 28 × 37 × 94 × 12712
(c) 33 × 35 × 37 × 39 × 41 × 43
(d) \( 11 \times 11 \times 11 \times 11 \times 11 \times \dots \)
(e) \( 1^{10} \)
(f) 39 – 24
(1) a,b,c
(2) d,e,f
(3) b
(4) a,b,d,e
Answer: (3)

Question. What is the number in the units place of \( (763)^{84} \) ?
(a) 1
(b) 3
(c) 7
(d) 9
Answer: (a)

Question. If the numbers \( a - b \) and \( a + b \) are twin primes, then a and b are necessarily
(a) Twin primes
(b) Co-primes
(c) Cannot say
(d) None
Answer: (b)

Question. The HCF of all the natural numbers from 200 to 478 is
(a) 2
(b) 1
(c) 478
(d) 3
Answer: (b)

Question. Find the greatest number that divides 59 and 54 leaving remainders 3 and 5 respectively.
(a) 3
(b) 7
(c) 8
(d) 5
Answer: (b)

Question. Find the unit digit in the expansion of \( (44)^{44} + (55)^{55} + (88)^{88} \).
(a) 7
(b) 5
(c) 3
(d) 3
Answer: (a)

Question. Find the digit in the units place of \( (676)^{99} \).
(a) 9
(b) 2
(c) 4
(d) 6
Answer: (d)

Question. The LCM of \( \frac{5}{12}, \frac{6}{5}, \frac{3}{2} \) and \( \frac{4}{17} \) is
(a) 60
(b) \( \frac{1}{60} \)
(c) 180
(d) None
Answer: (a)

Question. Find the number of factors of 1080.
(a) 32
(b) 28
(c) 24
(d) 36
Answer: (a)

Question. If p,q and r are prime numbers such that \( r = q + 2 \) and \( q = p + 2 \), then the number of triplets of the form (p,q,r) is
(a) 0
(b) 1
(c) 2
(d) 3
Answer: (b)

Question. The absolute value of \( 25 - (25 + 10) + 25 \div 125 \times 25 \) is
(a) -5
(b) 3
(c) 15
(d) 5
Answer: (d)

Question. The greatest five digit number exactly divisible by 9 and 13 is
(a) 99945
(b) 99918
(c) 99964
(d) 99972
Answer: (a)

Question. If the number \( 2345p60q \) is exactly divisible by 3 and 5, then the maximum value of \( p + q \) is
(a) 12
(b) 13
(c) 14
(d) 15
Answer: (b)

Question. If \( 1 \le k \le 25 \), how many prime numbers are there which are of the form \( 6k + 1 \) ?
(a) 15
(b) 16
(c) 17
(d) 18
Answer: (b)

Question. If a,b,c and d are four positive real numbers such that sum of a,b, and c is even and the sum of b,c and d is odd, then \( a^2 - d^2 \) is necessarily
(a) odd
(b) even
(c) prime
(d) Either (a) or (b)
Answer: (a)

Question. Mukesh bought 3 apples, 5 bananas and 7 custard apples for certain amount (which is even). The cost of apples, bananas and custard apples could be (in Rs.)
(a) 5,7,9
(b) 9,8,6
(c) 2,4,5
(d) 9,10,11
Answer: (c)

Question. In a class there are 72 boys and 64 girls. If the class is to be divided into least number of groups such that each group contains either only boys or only girls, then how many groups will be formed ?
(a) 17
(b) 34
(c) 24
(d) None
Answer: (a)

Question. The HCF of two numbers, obtained in three steps of division, is 7 and the first 3 quotients are 2,4 and 6 respectively. Find the numbers.
(a) 175, 392
(b) 189, 392
(c) 168, 385
(d) None
Answer: (b)

Question. Find the greatest four digit number which when divided by 18 and 12 leaves a remainder of 4 in each case
(a) 9976
(b) 9940
(c) 9904
(d) 9868
Answer: (a)

Question. Rahul wanted to type of first 180 natural numbers. Find the number of times he had to press the numbered keys.
(a) 384
(b) 432
(c) 416
(d) 448
Answer: (b)

Question. If the seven digit number \( 4567 X 75 \) is divisible by 15 then find the least possible value of X.
(a) 2
(b) 1
(c) 0
(d) 3
Answer: (b)

Question. A rational number can be expressed as a terminating decimal if the denominator has factors
(a) 2 or 5
(b) 2, 3 or 5
(c) 3 or 5
(d) None of these
Answer: (a)

Question. The only prime number which is even is
(a) 2
(b) 4
(c) 6
(d) none of these
Answer: (a)

Question. The value of \( 23.4\bar{3} + 5.\bar{2} \) is
(a) \( \frac{2395}{990} \)
(b) \( \frac{2527}{99} \)
(c) \( \frac{5169}{990} \)
(d) \( \frac{2837}{99} \)
Answer: (d)

Question. If \( 2 = x + \frac{1}{1 + \frac{1}{1 + \frac{1}{3 + \frac{1}{4}}}} \), then value of x is
(a) \( \frac{12}{17} \)
(b) \( \frac{13}{17} \)
(c) \( \frac{18}{17} \)
(d) \( \frac{21}{17} \)
Answer: (d)

Question. If R “Every fraction is a rational number” and T “Every rational number is a fraction”, then which of the following is correct?
(a) R is True and T is False.
(b) R is False and T is True.
(c) Both R and T are True.
(d) Both R and T are False.
Answer: (a)

Question. \( 5.\bar{2} \) is equal to
(a) \( \frac{45}{9} \)
(b) \( \frac{46}{9} \)
(c) \( \frac{47}{9} \)
(d) None of these
Answer: (c)

Question. For any two rational numbers A and B, which of the following properties are correct?
(i) A < B (ii) A = B (iii) A > B
(a) Only (i) and (ii) are correct.
(b) Only (ii) and (iii) are correct.
(c) Only (ii) is correct.
(d) All (i), (ii), (iii) are correct.
Answer: (d)

Question. If \( x = \frac{3 + \sqrt{2}}{3 - \sqrt{2}} \) and \( y = \frac{3 - \sqrt{2}}{3 + \sqrt{2}} \), the value of \( (x + y) \) is
(a) \( \frac{306}{49} \)
(b) \( \frac{484}{49} \)
(c) \( \frac{22}{7} \)
(d) \( \frac{73}{7} \)
Answer: (c)

Question. If x: Every whole number is a natural number and y: 0 is not a natural number, Then which of the following statement is true?
(a) x is false and y is the correct explanation of x.
(b) x is true and y is the correct explanation of x.
(c) x is true and y is false.
(d) Both x and y are true.
Answer: (a)

Question. If \( a = \frac{1}{3 - 2\sqrt{2}}, b = \frac{1}{3 + 2\sqrt{2}} \) then the value of \( a^2 + b^2 \) is
(a) 34
(b) 35
(c) 36
(d) 37
Answer: (a)

Question. If \( a = \frac{1}{3 - 2\sqrt{2}}, b = \frac{1}{3 + 2\sqrt{2}} \) then the value of \( a^3 + b^3 \) is
(a) 194
(b) 196
(c) 198
(d) 200
Answer: (c)

Question. If \( x = (7 + 4\sqrt{3}) \), then the value of \( x^2 + \frac{1}{x^2} \) is
(a) 193
(b) 194
(c) 195
(d) 196
Answer: (b)