CBSE Class 10 Real Numbers Sure Shot Questions Set D

Question. The decimal expansion of the rational number \( \frac{33}{2^2 \cdot 5} \) will terminate after
(a) one decimal place
(b) two decimal places
(c) three decimal places
(d) more than 3 decimal places
Answer: B
Explanation: The termination of any rational number depends upon the power of 2 in the prime factorization of denominator.

Question. For some integer m, every odd integer is of the form
(a) m
(b) m + 1
(c) 2m
(d) 2m + 1
Answer: D
Explanation: As the number 2m will always be even, so if we add 1 to it then, the number will always be odd.

Question. If two positive integers a and b are written as a = p³q² and b = pq³; p, q are prime numbers, then HCF (a, b) is:
(a) pq
(b) pq²
(c) p³q³
(d) p²q²
Answer: B
Explanation: Since \( a = p \times p \times p \times q \times q \),
\( b = p \times q \times q \times q \)
Therefore H.C.F of a and b = \( pq^2 \)

Question. The product of a non-zero number and an irrational number is:
(a) always irrational
(b) always rational
(c) rational or irrational
(d) one
Answer: A
Explanation: Product of a non-zero rational and an irrational number is always irrational i.e.,
\( \frac{3}{4} \times \sqrt{2} = (\text{rational}) \times (\text{irrational}) = \text{irrational} \).

Question. If the HCF of 65 and 117 is expressible in the form 65 m – 117, then the value of m is
(a) 4
(b) 2
(c) 1
(d) 3
Answer: B
Explanation: By Euclid’s division algorithm,
\( b = aq + r \)
\( 117 = 65 \times 1 + 52 \)
\( 65 = 52 \times 1 + 13 \)
\( 52 = 13 \times 4 + 0 \)
H.C.F(65,117) = 13
Since, H.C.F = 65m – 117
So \( 65m – 117 = 13 \)
\( \Rightarrow 65m = 130 \)
\( \Rightarrow m = 2 \)

Question. The largest number which divides 70 and 125, leaving remainders 5 and 8, respectively, is
(a) 13
(b) 65
(c) 875
(d) 1750
Answer: A
Explanation: Since 5 and 8 are the remainders of 70 and 125, respectively. Thus after subtracting these remainders from the numbers, we have the numbers
\( 65 = (70 − 5), 117 = (125 − 8) \) which is divisible by the required number.
Now required number = H.C.F of (65,117).
\( 117 = 65 \times 1 + 52 \)
\( 65 = 52 \times 1 + 13 \)
\( 52 = 13 \times 4 + 0 \)
H.C.F(65,117) = 13

Question. If two positive integers p and q can be expressed as p = ab² and q = a³b; a, b being prime numbers, then LCM (p, q) is
(a) ab
(b) a²b²
(c) a³b²
(d) a³b³
Answer: C
Explanation:
\( p = a \times b \times b \)
\( q = a \times a \times a \times b \)
Since L.C.M is the product of the greatest power of each prime factor involved in the numbers
Therefore, L.C.M of p and q = \( a^3b^2 \)

Question. The values of the remainder r, when a positive integer a is divided by 3 are:
(a) 0, 1, 2, 3
(b) 0, 1
(c) 0, 1, 2
(d) 2, 3, 4
Answer: C
Explanation: According to Euclid’s division lemma, \( a = 3q + r \), where \( 0 \le r < 3 \)
As the number is divided by 3. So the remainder cannot be greater than divisor 3 also r is an integer. Therefore, the values of r can be 0, 1 or 2.

Question. \( \frac{987}{10500} \) will have
(a) Terminating decimal expansion
(b) Non-Terminating Non repeating decimal expansion
(c) Non-Terminating repeating decimal expansion
(d) None of these
Answer: A
Explanation: After simplification
\[ \frac{987}{10500} = \frac{47}{500} \]
\[ = \frac{47}{5^3 \times 2^2} \]
As the denominator has factor \( 5^3 \times 2^2 \) and which is of the type \( 5^m \times 2^n \), So this is a terminating decimal expansion.

Question. A rational number in its decimal expansion is 327.7081. What would be the prime factors of q when the number is expressed in the p/q form?
(a) 2 and 3
(b) 3 and 5
(c) 2, 3 and 5
(d) 2 and 5
Answer: D
Explanation: This can be explained as,
\( 327.7081 = \frac{3277081}{10000} = \frac{p}{q} \)
\( \therefore q = 10000 = 10^4 \)
\( = (2 \times 5)^4 \)
\( = 2^4 \times 5^4 \)

Question. The least number that is divisible by all the numbers from 1 to 10 (both inclusive) is
(a) 10
(b) 100
(c) 2060
(d) 2520
Answer: D
Explanation: Factors of 1 to 10 numbers.
1 = 1
2 = \( 1 \times 2 \)
3 = \( 1 \times 3 \)
4 = \( 1 \times 2 \times 2 \)
5 = \( 1 \times 5 \)
6 = \( 1 \times 2 \times 3 \)
7 = \( 1 \times 7 \)
8 = \( 1 \times 2 \times 2 \times 2 \)
9 = \( 1 \times 3 \times 3 \)
10 = \( 1 \times 2 \times 5 \)
L.C.M of numbers from 1 to 10 is = \( 1 \times 2 \times 2 \times 2 \times 3 \times 3 \times 5 \times 7 = 2520 \)

Question. \( n^2 – 1 \) is divisible by 8, if n is
(a) an integer
(b) a natural number
(c) an odd integer greater than 1
(d) an even integer
Answer: C
Explanation: n can be even or odd
Case 1: If n is even
\( n = 2k \)
Then
\( a = (2k)^2 − 1 \)
\( a = 4k^2 − 1 \)
For \( k = 1 \)
\( 4(1)^2 − 1 = 3 \), not divisible by 8
Case 2: If n is odd
\( n = 2k + 1 \)
Then
\( a = (2k + 1)^2 − 1 \)
\( a = 4k^2 + 4k + 1 − 1 \)
\( a = 4k^2 + 4k \)
For \( k = 1 \)
\( a = 4(1)^2 + 4(1) = 8 \)
Which is divisible by 8.
Similarly we can check for any integer.

Question. If n is a rational number, then \( 5^{2n} − 2^{2n} \) is divisible by
(a) 3
(b) 7
(c) Both 3 and 7
(d) None of these
Answer: C
Explanation: \( 5^{2n} − 2^{2n} \) is of the form \( a^{2n} − b^{2n} \) which is divisible by both (a + b) and (a – b).
So, \( 5^{2n} − 2^{2n} \) is divisible by both 7, 3.

Question. The H.C.F of 441, 567 and 693 is
(a) 1
(b) 441
(c) 126
(d) 63
Answer: D
Explanation:
693 = \( 3 \times 3 \times 7 \times 11 \)
567 = \( 3 \times 3 \times 3 \times 3 \times 7 \)
441 = \( 3 \times 3 \times 7 \times 7 \)
Therefore H.C.F of 693, 567 and 441 is 63.

Question. On a morning walk, three persons step off together and their steps measure 40 cm, 42 cm and 45 cm, respectively. What is the minimum distance each should walk so that each can cover the same distance in complete steps?
(a) 2520cm
(b) 2525cm
(c) 2555cm
(d) 2528cm
Answer: A
Explanation: We need to find the L.C.M of 40, 42 and 45 cm to get the required minimum distance.
40 = \( 2 \times 2 \times 2 \times 5 \)
42 = \( 2 \times 3 \times 7 \)
45 = \( 3 \times 3 \times 5 \)
L.C.M. = \( 2 \times 3 \times 5 \times 2 \times 2 \times 3 \times 7 = 2520 \)

Question. \( 1000 = 2^x \cdot 5^y \). What is the value of x and y.
Answer: \( x=3, y=3 \)

Question. Which prime numbers will be repeatedly multiplied in prime factorization of 3200.
Answer: 2 and 5

Question. Find the digit at units place of \( 8^n \) if n is 9.
Answer: 8

Question. The prime factors of denominator of fraction \( \frac{14}{160} \) is \( 2^x \cdot 5 \). What is the value of x?
Answer: \( x=4 \)

Question. If H.C.F. of two number is 68 and 85 is 17. What is the L.C.M. of two numbers.
Answer: 340

Question. What is the H.C.F. of 95 and 152?
Answer: 19

Question. Which number when divided by 18 gives the quotient and remainder as 7 and 4.
Answer: 130

Question. When 176 is divided by a number it gives the remainder 5 and quotient 9. What is the number?
Answer: 19

Question. By which smallest irrational number \( \sqrt{27} \) be multiplied so as to get a rational number?
Answer: \( \sqrt{3} \)

Question. What is the product of \( (\sqrt{7} + \sqrt{5}) \) and \( (\sqrt{7} - \sqrt{5}) \)?
Answer: 2

Question. Which rational number is equivalent to \( 0.\overline{7} \)?
Answer: \( \frac{7}{9} \)

Question. What is the sum of \( 0.\overline{3} \) and \( 0.\overline{4} \).
Answer: \( 0.\overline{7} \)

Question. \( 0.\overline{17} = \frac{p}{q} \), where p and q are integers and \( q \neq 0 \). What is the value of \( \frac{p}{q} \)?
Answer: \( \frac{17}{99} \)

Question. Which smallest irrational number should be added to \( (3 + \sqrt{5}) \) to get a rational number?
Answer: \( (-\sqrt{5}) \)

Question. Give the fractional form of \( 1.2\overline{5} \).
Answer: \( 1\frac{23}{90} \)

Question. Which number should be multiplied to \( (\sqrt{5} - \sqrt{3}) \) to get a rational number?
Answer: \( (\sqrt{5} + \sqrt{3}) \)

Question. How much is \( 7\sqrt{5} + 8\sqrt{5} \)?
Answer: \( 15\sqrt{5} \)

Question. What is the value of :– \( \sqrt{5} \times \sqrt{7} \times \sqrt{15} \times \sqrt{21} \)
Answer: 105

Question. Division of 133 by 19 gives remainders = 0. What is the H.C.F. (133, 19)?
Answer: 19

Question. Give the decimal representation of \( \frac{13}{11} \).
Answer: \( 1.\overline{18} \)

Question. If \( \frac{3}{7} \) when written in decimal representation is \( 0.\overline{428571} \) then what is the decimal representation of \( \frac{5}{7} \).
Answer: \( 0.\overline{714285} \)

Question. If \( \frac{6}{7} = 0.\overline{857142} \) then what is the decimal representation of \( \frac{4}{7} \).
Answer: \( 0.\overline{571428} \)

Question. What will be the units digit of \( 7^5 \)?
Answer: 7

Question. Find the digit at units place of \( 4^5 \).
Answer: 4

Question. If unit’s digit of \( 7^3 \) is 3 then what will be the unit’s digit of \( 7^{11} \)?
Answer: 3

Question. What will be the unit’s digit of \( (7 \times 3)^{21} \)?
Answer: 1

Question. Which digit you will get at units place of \( 6^{18} \)?
Answer: 6

Question. Simplify :– \( (2 + \sqrt{3}) + (5 - \sqrt{3}) + (6 + \sqrt{3}) + (7 - \sqrt{3}) \)
Answer: 20

Question. Simplify :– \( (\sqrt{3} - 5) + (6 - 2\sqrt{3}) + (2 + \sqrt{3}) \)
Answer: 3

Question. What is the multiplicative inverse of \( \sqrt{5} - 2 \).
Answer: \( \sqrt{5} + 2 \)

Question. Simplify :– \( \frac{(\sqrt{7} - \sqrt{3})(\sqrt{7} + \sqrt{3})}{(\sqrt{7} + 3) + (3 - \sqrt{7})} \)
Answer: 2

Question. Express as a rational number :– \( \frac{(4 - \sqrt{3})(4 + \sqrt{3})}{(\sqrt{5} - 2)(\sqrt{5} + 2)} \)
Answer: 13

Question. If L.C.M. of two numbers 16 & 28 is 112, then what is the H.C.F. of these numbers.
Answer: 4

Question. Find the square of \( (2 + \sqrt{3}) \).
Answer: \( 7 + 4\sqrt{3} \)

Question. Simplify :– \( (3 + \sqrt{5})^2(3 - \sqrt{5})^2 \)
Answer: 16

Question. What is the H.C.F. of 152 and 171?
Answer: 19

Question. If H.C.F. of two numbers 420 and 441 is 21 then find the L.C.M. of these two numbers?
Answer: 8820

Question. H.C.F. and L.C.M. of two numbers are 19 and 380 respectively. If one of the numbers is 95, what is the other number?
Answer: 76