CBSE Class 10 Mathematics Real Numbers MCQs

More Questions:

More MCQs for Class 10 Mathematics Real Numbers............

Question : 7×11×13× 6 is:
a) a prime number
b) a composite number
c) an even number
d) None
Answer : B

Question : If a < b < c < d < e are consecutive positive integers, such that b + c+ d is a perfect square and a +b + c + d + e is a perfect cube. What is the smallest possible value of c?
a) 675
b) 576
c) 475
d) 384
Answer : A

Question : If we divide a two-digit number by the sum of its digits we get 4 as quotient and 3 as remainder.
Now if we divide that two digit number by the product of its digits, we get 3 as quotient and 5 as remainder. The two digit number is:
a) Even
b) Odd prime
c) Odd composite
d) Odd
Answer : B

Question : Four positive integers sum to 125. If the first of these numbers is increased by 4, the second is decreased by 4, the third is multiplied by 4 and the fourth is divided by 4, we find four equal numbers, then four original integers are:
a) 16, 24, 5, 80
b) 8, 22, 38, 57
c) 7, 19, 46, 53
d) 12, 28, 40, 45
Answer : A

Question : What is the least number which when divided by 42, 72 and 84 leaves the remainder 25, 55 and 67 respectively?
a) 521
b) 512
c) 504
d) 487
Answer : D

Question : y, z are integers, then the value of x + y - z is:
a) 6
b) 8
c) 7
d) -2
Answer : C

Question : A positive integer n has 60 divisors and 7 n has 80 divisors. What is the greatest integer k such that 7k divides n?
a) 0
b) 1
c) 2
d) 3
Answer : D

Question : What is the largest integer that is a divisor of (n +1) (n+3) (n+5) (n+7) (n+9) for all positive even integers n?
a) 3
b) 5
c) 11
d) 15
Answer : B

Question : The sum of 18 consecutive positive integers is a perfect square. The smallest possible value of this sum is…
a) 169
b) 225
c) 289
d) 361
Answer : B

Question : When the decimal point of a certain positive decimal number is moved two places to the right, the new number is two times the sum of the original number and the reciprocal of the original number. The product of 42 and the original number is:
a) 3
b) 6
c) 7
d) 14
Answer : C

Question : The sum of seven consecutive natural numbers is 1617. How many of these are prime?
a) 4
b) 5
c) 2
d) 7
Answer : A

Question : A number when divided by 342 gives a remainder 47. When the same number is divided by 19, what would be the remainder?
(A 9
b) 8
c) 18
d) 11
Answer : C

Question : If xyz + xy + xz + yz + x + y + z = 384where x, y, z are positive integers, then the value of x + y + z is:
a) 23
b) 18
c) 20
d) 35
Answer : A

Question : Let x be the greatest number by which if we divide 366, 513 and 324, then in each case the remainder is the same. The sum of digit of x is:
a) 3
b) 4
c) 5
d) 7
Answer : B

Question : If a number m is divided by 5 leaves a remainder 2, while another number n is divided by 5 leaves a remainder 4, then the remainder when (m + n) is divided by 5 is:
a) 1
b) 2
c) 3
d) 4
Answer : B

Question : Among the numbers 2²⁵⁰ 3²⁰⁰ 4¹⁵⁰ 5¹⁰⁰ 2 ,3 ,4 and 5 the greatest is
a) 2 250
b) 3 200
c) 4 150
d) 5 100
Answer : A

Question : The sum of the digit in (10⁴ⁿ²+8 +1)² , , where ‘n’ is a positive integer, is……….
a) 4
b) 4n
c) 2+2n
d) 4n²
Answer : C

Question : The ascending order of √2 ³√4, ⁴√6 is:
a)  √2, ³√4, ⁴√6
b)  √2, ⁴√6, ³√4
c) 3√4 √2, 4√6
d) ⁴√6, ³√4, √2
Answer : B

Question : The smallest number which when increased by 17 is exactly divisible by both 520 and 468 is:
a) 4697
b) 4663
c) 4656
d) 4680
Answer : B

Question : What is the smallest number which when increased by 5 is completely divisible by 8, 11 and 24?
a) 259
b) 355
c) 255
d) None of these
Answer : A

Question : Find the least number that when divided by 16, 18 and 20 leaves a remainder of 4 in each case, but is completely divisible by 7.
a) 2800
b) 2882
c) 2884
d) None of these
Answer : C

Question : What least number should be subtracted from 1856 so that remainder when divided by 7, 12 and 16 is 4?
a) 170
b) 172
c) 174
d) None of these
Answer : B

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) 40
b) 42
c) 44
d) None of these
Answer : A

Question : L. C. M. of two prime numbers x & y (x > y) is 161. The value of 3y – x is.
a) -1
b) -2
c) -3
d) None of these
Answer : B

Question : Find the greatest number that will divide 43, 91 & 183 so as to leave the same remainder in each case.
a) 4
b) 3
c) 2
d) None of these
Answer : A

Question : The least multiple of 7 which leaves a remainder of 4 when divided by 6, 9, 12 and 18 is.
a) 362
b) 365
c) 364
d) None of these
Answer : C

Question : The greatest number which can divide 1356, 1868, 2764 leaving same remainder 12 in each case is?
a) 63
b) 64
c) 65
d) None of these
Answer : B

Question : The H.C.F & L.C.M of two numbers are 13 and 455 respectively. If one of the numbers lies b/w 75 and 125, find the numbers.
a) 91
b) 95
c) 97
d) None of these
Answer : A

Question : If the L.C.M of the polynomials (y – 3)^a (2y + 1)^b (y + 13)⁷ and (y – 3)⁴(2y + 1)⁹ (y + 13)^c is (y –3)⁶ (2y + 1)¹⁰ (y+ 13)⁷, then the least value of a + b + c is.
a) 16
b) 18
c) 20
d) None of these
Answer : A

Question : The H.C.F of the polynomials 9(x + a)p (x – b)q (x + c)r and 12(x + a) p+3 (x- b)q – 3 (x + c)r + 2 is 3(x+ a)6 (x –b)6 (x + c)6, then the value of p + q – r is.
a) 10
b) 11
c) 9
d) None of these
Answer : C

Question : If the H.C.F of 8x³ya and 12xby² in 4xayb, then find the maximum value of a + b.
a) 6
b) 4
c) 8
d) None of these
Answer : B

Question : How many integers (a, b) exist such that the product of a, b and H.C.F (a, b) = 1080.
a) 12
b) 9
c) 10
d) None of these
Answer : B

Question : Find the smallest number that leaves a remainder of 4 on division by 5, 5 on division by 6, 6 on division by 7, 7 on division by 8 and 8 on division by 9?
a) 9125
b) 20779
c) 9711
d) 2519
Answer : D

Question : There are two numbers such that a>b, H.C.f (a, b) = h and Lcm (a, b) = l, what is the L.C.M of a – b and b.
a) (a b)b /h
b) a – b
c) b/h
d) None of these
Answer : A

Question : How many pairs of positive integers x, y exist such that H.C.F of x, y = 35 and the sum of x & y = 1085?
a) 10
b) 2
c) 3
d) 15
Answer : D

Question : How many pairs of positive integers x, y exist such that H.C.F (x, y) + L.C.M (x, y) = 91?
a) 8
b) 2
c) 6
d) None of these
Answer : A

Question : The number .211 2111 21111 211111… is a:

a) terminating decimal

b) non-terminating repeating decimal

c) non-terminating decimal which is non-repeating

d) None of these 

Answer : C

 

Question : If mⁿ = 32 , where m and n are positive integers, then the value of n^mn is:

a) 32

b) 25

c) 5¹⁰

d) 5²⁵ 

Answer : C

 

Question : Any one of the numbers a, a +2 and a + 4 is a multiple of:

a) 2

b) 3

c) 5

d) 7 

Answer : B

 

Question : If p is a prime number and p divides k² , then p divides:

a) 2k²

b) k

c) 3k

d) None of these 

Answer : B

 

Question : If the HCF of 85 and 153 is expressible in the form of 85n -153 then, the value of n is:

a) 3

b) 2

c) 4

d) 1  

Answer : B

 

Question : Given that LCM (91, 26) = 182 then, HCF (91, 26) is:

a) 13

b) 26

c) 7

d) 9

Answer : A

 

Question : Out of the four numbers

 the rational number is :

a) i

b) ii

c) iii

d) iv

Answer : B

1. Natural numbers

1, 2, 3, 4,... etc, are called natural numbers denoted by N.

2. Whole numbers
All natural numbers together with zero are called whole numbers denoted by W.
W = 0, 1, 2, 3, 4,....

3. Integers (Z)
... – 4, – 3, – 2, –1, 0, 1, 2, 3, 4, ....
(i) – 1, –2, – 3, – 4,..... are called negative integers.
(ii) 1, 2, 3, 4, ... are called positive integers.
Note: Zero is neither positive not negative.

4. The numbers of the form a/b, where a and b are natural numbers are called fractions.
e.g., 3/5 , 7/11 , 13/213 , .... etc.

5. The numbers of the form p/q, where p and q are integers and q ¹ 0 are called rational numbers.
e.g., -3/5 , 7/-11 , -13/-213 , .... etc.

6. If the denominator of a rational number has no prime factors other than 2 (or) 5 then and only then it is expressible as a terminating decimal.

7. Every rational number is always expressible in the form of terminating (or) a repeating decimal.

8. Numbers which when expressed in decimal form are expressible neither in terminating nor in repeating decimals are known as irrational numbers.
e.g., π, √2, 0.232332333... etc.

9. Some results on irrational numbers
(a) The – ve of an irrational number is an irrational number.
(B) The sum of a rational and an irrational number is an irrational number
(C) The product of a non-zero rational number with an irrational number is always an irrational number.

10. Real numbers
The totality of all rational and all irrational numbers forms the set R of all real numbers.

11. Properties of all real numbers
(a) Closure property of addition
The sum of two real numbers is always a real number.
(b) Commutative law of addition a + b = b + a, ∀ real numbers ‘a’ and ‘b’.
(c) Associative law for addition (a + b) + c = a + (b + c), ∀ real numbers a, b and c.
(d) Existence of additive identity
zero is the additive identity a + 0 = 0 + a = a, ∀ real numbers a.
(e) Existence of additive inverse
for each real number ‘a’ there exists a real number ‘–a’ such that
a + (–a) = (–a) + a = 0
(f) Closure property for multiplication
The product of two real numbers is a real numbers.
(g) Commutative law of multiplication ab = ba, ∀ real numbers a and b.
(h) Associative law of multiplication (ab)c = a(bc), ∀ real numbers a, b and c.
(i) Existence of multiplicative identity
1 is called the multiplicative identity.
1. a = a. 1 = a, ∀ real numbers a.
(j) Existence of multiplicative inverse
Every non-zero real number ‘a’ has its multiplicative inverse 1/a.
(k) Distributive law of multiplication over addition a(b+c) = ab + ac, ∀ real number a, b and c.

12. Zero is a real number which has no multiplicative inverse.

Question. A rational number between 1/4 and 1/3 is
(A) 7/24
(B) 0.29
(C) 13/48
(D) all the above

Answer: D

Question. An irrational number is
(A) a terminating and non-repeating decimal
(B) a nonterminating and non-repeating decimal
(C) a terminating and repeating decimal
(D) a nonterminating and repeating decimal

Answer: B

Question. Which of the following statements 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) 22/7 is a rational number
(D) None of these

Answer: D

Question. The set of real numbers does not satisfy the property of
(A) multiplicative inverse
(B) additive inverse
(C) multiplicative identity
(D) none of these

Answer: D

Question. Which step in the following problem is wrong?
a = b = 1; a = b
step-1 = a² = ab
step-2 = a² – b² = ab – b²
step-3 = (a + b) (a – b) = b(a – b)
step-4 = a + b = b(a-b)/a-b
a + b = b
1 +1 = 1
2 = 1
(A) step-4
(B) step- 3
(C) step-32
(D) step-1

Answer: A

Question. If ‘m’ is an irrational number then ‘2m’ is
(A) a rational number
(B) an irrational number
(C) a whole number
(D) a natural number

Answer: B

Question. The value of √3 is
(A) 1.414
(B) 2.256
(C) 1.732
(D) none

Answer: C

Question. The sum of a rational and an irrational number is
(A) an irrational number
(B) a rational number
(C) an integer
(D) a whole number

Answer: A

Question. The product of two irrationals is
(A) a rational number
(B) an irrational number
(C) either A or B
(D) neither A nor B

Answer: C

Question. 

(A) 133/99
(B) 371/90
(C) 5169/990
(D) 5411/990

Answer: D

Question. The value of 

(A) 40/31
(B) 4/9
(C) 1/8
(D) 31/40

Answer: C

Question. The sum of the additive inverse and multiplicative inverse of 2 is
(A) 3/2
(B) -3/2
(C) 1/2
(D) -1/2

Answer: B

Question. If √6 = 2.449 then the value of 3√2/2√3 is close to
(A) 1.225
(B) 0.816
(C) 0.613
(D) 2.449

Answer: A

Question. The value of √5√5√5√5...... is
(A) 0
(B) 5
(C) can’t be determined
(D) none

Answer: B

Question. The greatest among the following is 

(A) I
(B) IV
(C) II
(D) III

Answer: D

Question. π is
(A) rational
(B) irrational
(C) imaginary
(D) an integer

Answer: B

Question. The set of all irrational numbers is closed for
(A) addition
(B) multiplication
(C) division
(D) none of these

Answer: D

Question. The additive inverse of -a/b is
(A) a/b
(B) b/a
(C) -b/a
(D) -a/b

Answer: A

Question. Multiplicative inverse of ‘O’ is
(A) 1/0
(B) 0
(C) does not exist
(D) none of these

Answer: C

Question. Which of the following statement is false?
(A) Every fraction is a rational number
(B) Every rational number is a fraction
(C) Every integer is a rational number
(D) All the above

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. Express 0.75 as rational number
(A) 75/99
(B) 75/90
(C) 3/4
(D) none

Answer: C

Question. Express 

(A) 75/90
(B) 23/33
(C) 3/4
(D) none

Answer: B

Question. Express 

(A) 358/1000
(B) 358/999
(C) 355/990
(D) All

Answer: C

Question. Which of the following statements is true? 

Answer: A

Question. If 3/7 + x+(-8/21) + 5/22 = -125/462 , then x is
(A) 125/462
(B) -124/462
(C) -125/462
(D) 124/462

Answer: C

Question. Consider the following statements 

(A) 1 alone is correct
(B) 2 alone is correct
(C) both 1 and 2 are false
(D) both 1 and 2 are correct

Answer: D

Question. The value of x for which the two rational numbers 3/7,x/42 are equivalent is
(A) 18
(B) 15
(C) 12
(D) 10

Answer: A

Question.

(A) 133/99
(B) 371/90
(C) 5169/990
(D) 5411/990

Answer: D

Question. To satisfy the equation a+b/a = b/a+b , a and b must be
(A) both rational
(B) both real, but not rational
(C) one real, other non real
(D) either one ral or both non-real

Answer: D

Question. Match the following:
1. Natural numbers (p) 0, 1, 2, 3, 4, 5 ...etc
2. Whole numbers (q) 1, 2, 3, 4, 5,....etc.
3. Integers (r) 3/-5 , -7/9 , 13/9 , -11/36 etc.
4. Rational numbers (s) ....., – 4, –3, –2, –1, 0, 1, 2
(A) 1q, 2p, 3s, 4r
(B) 1p, 2q, 3r, 4s
(C) 1p, 2s, 3q, 4r
(D) 1q, 2p, 3r, 4s

Answer: A

Question. What is the value of 1+√2/1-√2 + 1-√2/1+√2 ?
(A) – 6
(B) √2
(C) √2/2
(D) 4

Answer: A

Question. 

Answer: D

Question. If x = -1/3 and y = 2/7 , then | x + y | is
(A) ≥ | x | + | y |
(B) = | x | + | y |
(C) ≤ | x | + | y |
(D) = x + y

Answer: C

Question. The set of positive real numbers does not satisfy the property of
(A) Multiplicative inverse
(B) Additive inverse
(C) Multiplicative identity
(D) None of these

Answer: D

Question. What is the value of the fraction, 

(A) 1.5
(B) 2.25
(C) 2.5
(D) 2.6

Answer: B

Question. 

Answer: D

Question. The expresion 1= 1/1+√3 + 1/1-√3 is equal to
(A) 1 – √3
(B) 1
(C) – √3
(D) √3

Answer: A

Question. The value of expression 2/3 + 5/11 + -1/3 + -3/11 is equal to
(A) 1/33
(B) 2/33
(C) 10/33
(D) 17/33

Answer: D

Question. What is the value of 

(A) 1/4
(B) 11/31
(C) 22/31
(D) 1/2

Answer: C

Question. Which of the following expressions is not equal to the others? 

Answer: C

Question. If x = -2/7 , then | – x | is equal to
(A) | x |
(B) x
(C) 0
(D) 1

Answer: A 

Question. The rational number which is not lying between 5/16 and 1/2 is
(A) 3/8
(B) 7/16
(C) 1/4
(D) 13/32

Answer: C

Question. Which property is used in the identity given below
x + y = y + x
(A) Associative property of addition
(B) Commutative property of addition
(C) Distributive property
(D) Reflexive property

Answer: B

Question. Which of the following illustrates the inverse property of addition?
(A) 3 – 3 = 0
(B) 3 + 3 = 6
(C) 3 + 0 = 0
(D) 3 – 0 = 3

Answer: A

Question. 

(A) 28.5
(B) 27.5
(C) 30.5
(D) 29.5

Answer: D

Question. The difference between the largest and the smallest of the fractions 5/8 , 7/12 , 1/3 , 2/5 is
(A) 1/4
(B) 5/24
(C) 7/24
(D) 13/21

Answer: C

Question. Which of the following statements is false?
(A) π is an irrational number
(B) The set of real numbers is closed for addition
(C) The set of irrational numbers is closed for multiplication
(D) The set of a real numbers is closed for multiplication

Answer: C

Question. Which of the following is an irrational number?
(A) 4/3
(B) √24 
(C) √81 
(D) – 4.07

Answer: B

Question. Additive inverse of 3/-4 is
(A) 3/4
(B) 1/4
(C) 3
(D) 0

Answer: A

Question. 

Answer: D

Question. The standard form of 192/-168 is
(A) -2/3
(B) -8/7
(C) -1/7
(D) -6/7

Answer: B

Question. The number which is subtracted from 27/13 to get -3/7 is
(A) 228/91
(B) 1/91
(C) 200/91
(D) 198/91

Answer: A

Question. √2-1/√2+1 = ......
(A) 3 – 2√2
(B) 3 + 3√2
(C) 1/√2
(D) 1.632

Answer: A

Question. -24/36 is a rational number with denominator is
(A) 3
(B) 4
(C) 1
(D) 2

Answer: A

Question. √5-√3/√5+√3 is equal to
(A) 4 + √15
(B) 4 – √15
(C) 1/2
(D) 1

Answer: B

Question. The unit digit in the product (327)¹⁷³ is
(A) 9
(B) 7
(C) 1
(D) 3

Answer: B

Question. The unit digit in the product (7⁷¹ × 6⁵⁹ × 3⁶⁵) as:
(A) 6
(B) 2
(C) 4
(D) 1

Answer: C

Question. The unit digit in (7²⁷ – 3¹⁴) is
(A) 0
(B) 7
(C) 4
(D) 6

Answer: C

Question. If -1≤ x ≤ 3 and 1≤ y ≤ 3, then the maximum value of (3y – 4x) is
(A) 18
(B) 13
(C) 5
(D) –6

Answer: B

Question. If x is negative
(A) | x | = x
(B) | x | = x
(C) | x |= 1/x
(D) | x|= -1/x

Answer: B

Question. 235.235235235... is a/an:
(A) integer
(B) whole number
(C) rational number
(D) irrational number

Answer: C

Question. Find the value of ‘a’ and ‘b’ if 

(A) 2, 11
(B) 11, 2
(C) 1, 1
(D) 2, 1

Answer: B

Question. If √[0.04x 0.4x x] = 0.4x 0.04x√y then the value of x/y is:
(A) 0.0016
(B) 0.16
(C) 0.016
(D) None of these

Answer: C

Question. The unit in the product (281 × 15 × 16 × 18) is:
(A) 6
(B) 0
(C) 5
(D) 8

Answer: B

Question. The unit digit in the product (127)¹⁷⁰ is
(A) 3
(B) 9
(C) 7
(D) 3

Answer: B

Question. If x/y = 3/5 , then the value of x-y/x+y is
(A) -1/4
(B) 1/4
(C) 1/6
(D) – 6

Answer: A

Question. The LCM of two numbers is 198 and their product is 1188. Find their HCF
a. 18
b. 6
c. 66
d. None of above

Answer : B

Question. Two natural numbers whose sum is 85 and the least common multiple is 102 are:
(a) 30 and 55
(b) 17 and 68
(c) 35 and 55
(d) 51 and 34

Answer : D

Question. The values of x and y in the given figure are:
(a) x = 10; y = 14
(b) x = 21; y = 84
(c) x = 21; y = 25
(d) x = 10; y = 40

Answer : B

Question. What is the greatest possible speed at which a man can walk 52 km and 91 km in an exact number of minutes?
(a) 17 m/min
(b) 7 m/min
(c) 13 m/min
(d) 26 m/min

Answer : C

Question. The HCF and LCM of two numbers are 33 and 264 respectively. When the first number is completely divided by 2 the quotient is 33. The other number is:
(a) 66
(b) 130
(c) 132
(d) 196

Answer : C

Question. Pairs of natural numbers whose least common multiple is 78 and the greatest common divisor is 13 are:
(a) 58 and 13 or 16 and 29
(b) 68 and 23 or 36 and 49
(c) 18 and 73 or 56 and 93
(d) 78 and 13 or 26 and 39

Answer : D

Question. What will be the least possible number of the planks, if three pieces of timber 42 m, 49 m and 63 m long have to be divided into planks of the same length?
(a) 5
(b) 6
(c) 7
(d) none of these

Answer : C

Question. If two positive integers and are expressible in terms of primes as a = p⁶ q³ and b = p⁴ q⁸, then which of the following is true?
a. HCF = p⁵q² × LCM
b. LCM = p²q⁵ × HCF
c. LCM = p⁵q² × HCF
d. HCF = p²q⁵ × LCM

Answer : B

Question. If A = 2n + 13, B = n + 7, where n is a natural number then HCF of A and B is:
(a) 2
(b) 1
(c) 3
(d) 4

Answer : B

Question. 4 Bells toll together at 9.00 am. They toll after 7, 8, 11 and 12 seconds respectively. How many times will they toll together again in the next 3 hours?
(a) 3
(b) 4
(c) 5
(d) 6

Answer : C

Direction: In the following questions, a statement of Assertion (A) is followed by a statement of Reason (R). Mark the correct choice as:
(a) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
(b) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).
(c) Assertion (A) is true but Reason (R) is false.
(d) Assertion (A) is false but Reason (R) is true.

Question. Assertion: The H.C.F. of two numbers is 16 and their product is 3072. Then their L.C.M. = 162.
Reason: If a and b are two positive integers, then H.C.F. × L.C.M. = a × b.
Answer : D

Question. Assertion: For any two positive integers p and q, HCF (p, q) × LCM (p, q) = p × q
Reason: If the HCF of two numbers is 5 and their product is 150, then their LCM is 4
Answer : C

Question. Assertion: A number N when divided by 15 gives the remainder 2. Then the remainder is same when N is divided by 5.
Reason: √3 is an irrational number.
Answer : B

Question. Assertion: 2 is an example of a rational number.
Reason: The square roots of all positive integers are irrational numbers.
Answer : C

Question. Assertion: Denominator of 34.12345. When expressed in the form p/q, q ≠ 0, is of the form 2m × 5n, where m and n are non-negative integers.
Reason: 34.12345 is a terminating decimal fraction.
Answer : A