CBSE Class 10 Real Numbers Sure Shot Questions Set 07

Question. If \( x = 7 + 4\sqrt{3} \), then the value of \( \sqrt{x} + \frac{1}{\sqrt{x}} \) is
(a) 8
(b) 6
(c) 5
(d) 4
Answer: (d)

Question. \( 8 - 8 \times \frac{2\frac{1}{5} - 1\frac{2}{7}}{2 - \frac{1}{6 - \frac{1}{6}}} \) is equal to
(a) 6
(b) 2
(c) 4
(d) 8
Answer: (c)

Question. The rational form of \( 2.74\bar{35} \) is
(a) \( \frac{27161}{9999} \)
(b) \( \frac{27}{99} \)
(c) \( \frac{27161}{9900} \)
(d) \( \frac{27161}{9000} \)
Answer: (c)

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

Question. The value of \( \frac{1}{\sqrt{3} + \sqrt{2} - 1} \) on simplifying upto 3 decimal places, given that \( \sqrt{2} = 1.4142 \) and \( \sqrt{6} = 2.4495 \) is
(a) 0.166
(b) 0.366
(c) 0.466
(d) 0.566
Answer: (b)

Question. \( \pi \) is
(a) Rational
(b) Irrational
(c) Imaginary
(d) An integer
Answer: (b)

Question. An irrational number is
(a) A terminating and nonrepeating decimal
(b) A nonterminating and nonrepeating decimal
(c) A terminating and repeating decimal
(d) A nonterminating and repeating decimal
Answer: (b)

Question. Which of the following statement is true ?
(a) Every point on the number line represents a rational number
(b) Irrational numbers cannot be represented by points on the number line
(c) \( \frac{22}{7} \) is a rational number
(d) None of these
Answer: (c)

Question. The sum of rational and irrational number is always
(a) Rational
(b) Irrational
(c) Both
(d) Can’t say
Answer: (b)

Question. The product of rational and irrational number is always
(a) Rational
(b) Irrational
(c) Both
(d) Can’t say
Answer: (b)

Question. Rational number between \( \sqrt{2} \) and \( \sqrt{3} \) is
(a) \( \frac{\sqrt{2} + \sqrt{3}}{2} \)
(b) \( \frac{\sqrt{2} \times \sqrt{3}}{2} \)
(c) 1.5
(d) 1.8
Answer: (c)

Question. The number \( \frac{5 - \sqrt{5}}{5 + \sqrt{5}} \) is
(a) Rational
(b) Irrational
(c) Both
(d) Can’t say
Answer: (b)

Question. \( 0.23 + 0.22 = ? \)
(Note: OCR shows repeating bar \( 0.\overline{23} + 0.\overline{22} \))
(a) 0.45
(b) \( 0.\overline{43} \)
(c) 0.45
(d) 0.45
Answer: (b)

Question. \( 0.01\bar{8} \) can be expressed in the rational form as
(a) \( \frac{18}{1000} \)
(b) \( \frac{18}{990} \)
(c) \( \frac{18}{9900} \)
(d) \( \frac{18}{999} \)
Answer: (b)

Question. The equivalent rational form of \( 17.\bar{6} \) is
(a) \( \frac{53}{3} \)
(b) \( \frac{88}{5} \)
(c) \( \frac{44}{25} \)
(d) None of these
Answer: (a)

Question. \( \frac{961}{625} \) is a
(a) Terminating decimal
(b) Nonterminating decimal
(c) Cannot be determined
(d) None of these
Answer: (a)

Question. When simplified, the product \( (1 - \frac{1}{3})(1 - \frac{1}{4})(1 - \frac{1}{5}) \dots (1 - \frac{1}{n}) \) equals ;
(a) \( \frac{1}{n} \)
(b) \( \frac{2}{n} \)
(c) \( \frac{2(n-1)}{n} \)
(d) \( \frac{2}{n(n+1)} \)
Answer: (b)

Question. Which of the following has most number of divisors ?
(a) 99
(b) 101
(c) 176
(d) 182
Answer: (c)

Question. A number n is said to be perfect if the sum of all its divisors (excluding n itself) is equal to n. An example of perfect number is
(a) 6
(b) 9
(c) 15
(d) 21
Answer: (a)

Question. The H.C.F. of \( 2^2 \times 3^3 \times 5^5, 2^3 \times 3^2 \times 5^2 \times 7 \) and \( 2^4 \times 3^4 \times 5 \times 7^2 \times 11 \) is
(a) \( 2^2 \times 3^2 \times 5 \)
(b) \( 2^2 \times 3^2 \times 5 \times 7 \times 11 \)
(c) \( 2^4 \times 3^4 \times 5^5 \)
(d) \( 2^4 \times 3^4 \times 5^5 \times 7 \times 11 \)
Answer: (a)

Question. Which of the following is a pair of co-primes ?
(a) (16, 62)
(b) (18, 25)
(c) (21, 35)
(d) (23, 92)
Answer: (b)

Question. The L.C.M. of \( 2^3 \times 3^2 \times 5 \times 11, 2^4 \times 3^4 \times 5^2 \times 7 \) and \( 2^5 \times 3^3 \times 5^3 \times 7^2 \times 11 \) is
(a) \( 2^3 \times 3^2 \times 5 \)
(b) \( 2^5 \times 3^4 \times 5^3 \)
(c) \( 2^3 \times 3^2 \times 5 \times 7 \times 11 \)
(d) \( 2^5 \times 3^4 \times 5^3 \times 7^2 \times 11 \)
Answer: (d)

Question. The least number which should be added to 2497 so that the sum is exactly divisible by 5, 6, 4 and 3 is
(a) 3
(b) 13
(c) 23
(d) 33
Answer: (c)

Question. The least number which is a perfect square and is divisible by each of the numbers 16, 20 and 24, is
(a) 1600
(b) 3600
(c) 6400
(d) 14400
Answer: (b)

Question. The smallest number which when diminished by 7, is divisible by 12, 16, 18, 21 and 28 is
(a) 1008
(b) 1015
(c) 1022
(d) 1032
Answer: (b)

Question. The least number which when increased by 5 is divisible by each one of 24, 32, 36 and 54, is
(a) 427
(b) 859
(c) 869
(d) 4320
Answer: (b)

Question. The least number, which when divided by 12, 15, 20 and 54 leaves in each case a remainder of 8, is
(a) 504
(b) 536
(c) 544
(d) 548
Answer: (d)

Question. The largest four-digit number which when divided by 4, 7 or 13 leaves a remainder of 3 in each case, is
(a) 8739
(b) 9831
(c) 9834
(d) 9893
Answer: (c)

Question. The least multiple of 7, which leaves a remainder of 4, when divided by 6, 9, 15 and 18 is
(a) 74
(b) 94
(c) 184
(d) 364
Answer: (d)

Question. The least number, which when divided by 48, 60, 72, 108 and 140 leaves 38, 50, 62, 98 and 130 as remainders respectively, is
(a) 11115
(b) 15110
(c) 15120
(d) 15210
Answer: (b)

Question. Find the least multiple of 23, which when divided by 18, 21 and 24 leaves remainders 7, 10 and 13 respectively.
(a) 3002
(b) 3013
(c) 3024
(d) 3036
Answer: (b)

Question. The least number which when divided by 5, 6, 7 and 8 leaves a remainder 3, but when divided by 9 leaves no remainder, is
(a) 1677
(b) 1683
(c) 2523
(d) 3363
Answer: (b)

Question. The product of two numbers is 960. If H.C.F. is 8, then the numbers are
(a) 24,40
(b) 8, 120 or 24, 40
(c) 8, 140
(d) none of these
Answer: (b)

Question. The least number divisible by 12, 15, 20, and is perfect square is
(a) 900
(b) 400
(c) 36
(d) 256
Answer: (a)

Question. The largest number which divides 133 and 245 leaving a remainder 5 is
(a) 17
(b) 15
(c) 8
(d) 16
Answer: (d)

Question. The H.C.F,. of two numbers is \( \frac{1}{5} \) of their L.C.M. If the product of two number is 720, then the H.C.F. of the numbers is
(a) 13
(b) 12
(c) 14
(d) 18
Answer: (b)

Question. The L.C.M. of two numbers is 39780 and their ratio is 13 : 15 then the numbers are
(a) 273,315
(b) 2652,3060
(c) 516, 685
(d) none
Answer: (b)

Question. The L.C.M. of two numbers is 14 times of their H.C.F. The sum of L.C.M. and H.C.F. is 600. If one of the number is 80, then other is
(a) 280
(b) 218
(c) 25
(d) 45
Answer: (a)

Question. Four bells begin to toll together and toll respectively at intervals of 5, 6, 8 and 12 seconds. How many times will they toll together in an hour excluding the one at the start
(a) 30
(b) 19
(c) 13
(d) 5
Answer: (b)

Question. Two ropes of length 28 m and 36 m are to be cut into bits of same length. The greatest possible length of each is
(a) 7
(b) 3
(c) 4
(d) 5
Answer: (c)

Question. 28 mango trees, 42 apple trees and 21 orange trees have to be planted in rows such that each row contains the same number of trees of one variety only. The minimum number of rows in which the above trees may be planted
(a) 13
(b) 12
(c) 11
(d) 10
Answer: (a)

Question. A heap is to be formed with lots of 8, 10 and 15 pebbles of different colours. The smallest number of pebbles in the heap is
(a) 121
(b) 120
(c) 110
(d) 8
Answer: (b)

Question. Three measuring tapes are 64 cm, 72 cm, 96 cm, the shortest length that can measured by any one of the tapes exactly is
(a) 576
(b) 120
(c) 8
(d) 10.
Answer: (a)

Question. The L.C.M. of two numbers is 48. The numbers are in the ratio 2 : 3. The sum of the numbers is
(a) 28
(b) 32
(c) 40
(d) 64
Answer: (c)

Question. \( \frac{(x^{a+b})^2 (x^{b+c})^2 (x^{c+a})^2}{(x^a \cdot x^b \cdot x^c)^4} = \)
(a) -1
(b) 0
(c) 1
(d) None
Answer: (c)

Question. \( \frac{2^{n+4} - 2(2^n)}{2(2^{n+3})} + 2^{-3} \) is equal to
(a) \( 2^{n+1} \)
(b) \( -2^{n+1} + \frac{1}{8} \)
(c) \( \frac{9}{8} - 2^n \)
(d) 1
Answer: (d)

Question. The value of \( \left[ (x^{a-a^{-1}})^{\frac{1}{a-1}} \right]^{\frac{a}{a+1}} = \)
(a) x
(b) 1/x
(c) \( x^a \)
(d) \( 1/x^a \)
Answer: (a)

Question. If \( a = \frac{2 + \sqrt{3}}{2 - \sqrt{3}}, b = \frac{2 - \sqrt{3}}{2 + \sqrt{3}} \), then the value of \( a + b \) is
(a) 14
(b) -14
(c) \( 8\sqrt{3} \)
(d) \( -\sqrt{3} \)
Answer: (a)

Question. The smallest number by which 3600 can be divided to make it a perfect cube is
(a) 9
(b) 50
(c) 300
(d) 450
Answer: (d)

Question. The remainder when \( 7^{84} \) is divided by 342 is
(a) 0
(b) 1
(c) 49
(d) 341
Answer: (b)