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Practice each question, then open the answer or worked solution.
Every rational number is
(A) a natural number
(B) an integer
(C) a real number
(D) a whole number
(C) a real number
Explanation:
We know that rational and irrational numbers taken together are known as real numbers. Therefore, every real number is either a rational number or an irrational number. Hence, every rational number is a real number.
Hence, (C) is the correct option.
Between two rational numbers
(A) there is no rational number
(B) there is exactly one rational number
(C) there are infinitely many rational numbers
(D) there are only rational numbers and no irrational numbers
(C) there are infinitely many rational numbers
Explanation:
Between two rational numbers, there is infinitely many rational numbers.
Hence, (C) is the correct option.
Decimal representation of a rational number cannot be
(A) terminating
(B) non-terminating
(C) non-terminating repeating
(D) non-terminating non-repeating
(D) non-terminating non-repeating
Explanation:
The decimal representation of a rational number cannot be non-terminating and non-repeating.
Hence, (D) is the correct option.
The product of any two irrational numbers is
(A) always an irrational number
(B) always a rational number
(C) always an integer
(D) sometimes rational, sometimes irrational
(D) sometimes rational, sometimes irrational
Explanation:
The product of any two irrational numbers is sometimes rational and sometimes irrational.
Hence, (D) is the correct option.
The decimal expansion of the number √2 is
(A) a finite decimal
(B) 1.41421
(C) non-terminating recurring
(D) non-terminating non-recurring
(D) non-terminating non-recurring
Explanation:
The decimal expansion of the number √2 = 1.41421356237…
Hence, (D) is the correct option.
Which of the following is irrational?
(C) √7
Explanation:
(A) √4/√9 = 2/3
(B) √12/√3 = 2√3/√3 = 2
(C) √7 = 2.64575131106
(D) √81 = 9
Here, (C) √7 = 2.64575131106 is a non-terminating decimal expansion.
Hence, (C) is the correct option.
Which of the following is irrational?
(D) 0.4014001400014…
Explanation:
A number is irrational if and only if its decimal representation is non-terminating and non-recurring.
(A) is a terminating decimal and, therefore, cannot be an irrational number.
(B) is a non-terminating and recurring decimal and, therefore, cannot be irrational.
(C) is a non-terminating and recurring decimal and, therefore, cannot be irrational.
(D) is a non-terminating and non-recurring decimal and therefore is an irrational number.
Hence, (D) is the correct option.
A rational number between √2 and √3 is
(A) (√2+√3)/2
(B) (√2. √3)/2
(C) 1.5
(D) 1.8
(C) 1.5
Explanation:
√2 =1.4142135…. and √3 =1.732050807….
(A) (√2+√3)/2 = 1.57313218497… is a non-terminating and non-recurring decimal and, therefore, is an irrational number.
(B) (√2. √3)/2 = 1.22474487139… is a non-terminating and non-recurring decimal and, therefore, is an irrational number.
(C) 1.5 is a terminating decimal and, therefore, is a rational number.
(D) 1.8 is a terminating decimal and, therefore, is a rational number.
Here both 1.5 and 1.8 are rational numbers. But, 1.8 does not lie in between √2 =1.4142135…. and √3 =1.732050807…. Whereas 1.5 lies in between √2 =1.4142135…. and √3 =1.732050807….
Hence, (C) is the correct option.
The value of 1.999… in the form p/q, where p and q are integers and q ≠ 0, is
(A) 19/10
(B) 1999/1000
(C) 2
(D) 1/9
(C) 2
Explanation:
(A) 19/10 = 1.9
(B) 1999/1000= 1.999
(C) 2
(D) 1/9 = 0.111….
Let x = 1.9999….. — (1)
Multiply equation (1) with 10
10x = 19.9999….. — (2)
Subtract equation (1) from equation(2),
We get,
9x = 18
x = 18 / 9
x = 2
Therefore,
x = 1.9999… = 2
Hence, (C) is the correct option.
2√3 + √3 is equal to
(A) 2√6
(B) 6
(C) 3√3
(D) 4√6
(C) 3√3
Explanation:
2√3 + √3
Taking √3 common,
We get,
√3(2+1) = √3(3) = 3√3
Hence, (C) is the correct option.