Real Numbers Test

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Real Numbers Test - 16

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Weekly Quiz Competition

Question 1
1 / -0

If the L.C.M. of two numbers 240 and 360 is 720, find their H.C.F.

A
360

B
240

C
120

D
60

Solution

Factors of $360 = 2 \times 2 \times 2 \times 3 \times 3 \times 5$
Factor of $240 = 2 \times 2 \times 2 \times 2 \times 3 \times 5$
Common Factor $= 2 \times 2 \times 2 \times 3 \times 5$
Thus, HCF of $360$ and $240 = 120$
Hence, the correct option is (C).
Question 2
1 / -0

Which of the following are non-terminating decimal expansions?
(i)$\frac{35}{5^{2} \times 7 \times 2^{3}}$
(ii) $\frac{7}{2^{2} \times 5^{6}}$
(iii) $\frac{2}{5^{2} \times 7}$
(iv)$\frac{5}{2^{2} \times 5 \times 3}$

A
Only (i) and (ii)

B
Only (i), (iii) and (iv)

C
Only (iii) and (iv)

D
Only (i), (ii) and (iii)

Solution

A rational number can be expressed as a terminating decimal if the denominator has the factors of 2 or 5 only.
A rational number can be expressed as a non-terminating repeating decimal if the denominator has the factors other than 2 or 5.
Only (iii) and (iv) are non-terminating decimal expansions.
Hence, the correct option is (C).
Question 3
1 / -0

If `a` is an irrational number, which of the following numbers is definitely a rational number?

A
$\frac{a}{a}$

B
1 + a

C
1 - a

D
a²

Solution

A rational number is a number that can be written in the form $\frac{p}{q},$ where $p$ and $q$ are integers and $q \neq 0$.
Here, $a$ is an irrational number, $\frac{a}{a}$ will form a rational number.
Hence, the correct option is (A).
Question 4
1 / -0

Which of the following is not a composite number?

A
$17 \times 5 \times 3-17 \times 3 \times 2 \times 3$

B
$17 \times 5+3 \times 2+10$

C
$2 \times 3 \times 5+2 \times 7 \times 11$

D
$23 \times 43 \times 7+11 \times 17 \times 19 \times 7$

Solution

A composite number is a positive integer that has at least one positive divisor other than one or the number itself.
(A) $17 \times 5 \times 3-17 \times 3 \times 2 \times 3=255-306=-51$
(B) $17 \times 5+3 \times 2+10=101$
(C) $2 \times 3 \times 5+2 \times 7 \times 11=184$
(D) $23 \times 43 \times 7+11 \times 17 \times 19 \times 7=6923+24,871=31,794$
Hence, the correct option is (B).
Question 5
1 / -0

What is the greatest number that divides the numbers 38, 45 and 52 and leaves remainders 2, 3 and 4, respectively?

A
6

B
8

C
10

D
4

Solution

Number $38,45$ and $52$ leaves $2,3$ and $4$ as remainder respectively.
$38-2=36$
$45-3=42$
$52-4=48$
HCF $(36,42,48)$
Factors of $ 36: 2 \times 2 \times 3 \times 3$
Factors of $ 42: 2 \times 3\times 7$
Factors of $48: 2 \times 2 \times 2 \times 2 \times 3$
HCF of $(36,42,48)=2\times 3=6$
Thus, $6$ is the greatest number that will divide $38,45$ and $48$ leaving $2,3$ and $4$ as remainder respectively.
Hence, the correct option is (A).
Question 6
1 / -0

If $H$. $C$. $F$. of two numbers, $2^{5} \times 3^{2} \times 5$ and $2^{4} \times 3^{3} \times 5$, is of the form $2^{a} \times 3^{b} \times 5^{c} \times 7^{d}$, then what are the values of $a, b, c$ and $d$, respectively?

A
2, 4, 0 and 1

B
2, 4, 1 and 0

C
4, 2, 0 and 1

D
4, 2, 1 and 0

Solution

The lowest power of each number is their respective values.
Lowest power of value is:
$2^a=2^4$
$3^b=3^2$
$5^c=5^1$
$7^d=7^0$
Values of $a,b,c$ and $d=4,2,1$ and $0$.
Hence, the correct option is (D).
Question 7
1 / -0

If $M=77 \times 144 \times 45,$ which of the following groups gives the prime factors of $M $?

A
2, 3, 5, 7, 11, 13

B
2, 3, 5, 19

C
2, 3, 5, 7, 11, 13, 14

D
2, 3, 5, 7, 11

Solution

$M=77\times 144\times 45$
$=7 \times 11 \times 12 \times 12 \times 9 \times 5$
$=7 \times 11 \times 2 \times 2 \times 3 \times 2 \times 2 \times 3 \times 3 \times 3 \times 5$
So, prime factor of $M$ are $2,3,5,7$ and $11$.
Hence, the correct option is (D).
Question 8
1 / -0

When three persons step off together, their steps measure 15 cm, 25 cm and 40 cm. What is the minimum distance that each person should walk in order to cover the same distance in integral number of steps?

A
5 cm

B
250 cm

C
1500 cm

D
600 cm

Solution

The minimum distance covered by each of them in complete steps = LCM of the measures of their steps
$=$LCM$(15,25,40)$
$40=2\times 2\times 2\times 5=2^3 \times 5$
$15=3\times 5$
$25=5\times 5$
$\Rightarrow$ LCM $(15,25,40)=2^3 \times 3\times 5^2=600$
Hence, the correct option is (D).
Question 9
1 / -0

Which of the following fractions will have a terminating decimal expansion?

A
$\frac{25}{3^{2} \times 2^{3}}$

B
$\frac{41}{2^{2} \times 7^{3}}$

C
$\frac{69}{3^{2} \times 2^{2} \times 5^{2}}$

D
$\frac{343}{2^{2} \times 5^{2} \times 7^{3}}$

Solution

A rational number can be expressed as a terminating decimal if the denominator has the factors of $2$ or $5$ only .
A rational number can be expressed as a non-terminating repeating decimal if the denominator has the factors other than $2$ or $5$.
Terminating decimal expansion $=\frac{343}{2^{2} \times 5^{2} \times 7^{3}}=\frac{1}{2^2\times 5^2}$
Only option (D) have only factors of $2$ and $5$ in their denominator.
Hence, the correct option is (D).
Question 10
1 / -0

Three boys A, B and C started running around a circular stadium at the same time and in the same direction. 'A' completes one round of the stadium in 6 minutes, 'B' in 5 minutes and 'C' in 20 minutes (all started running from the same point). After what time will they meet again at the starting point for the first time?

A
1 hour

B
2 hours

C
1 hour and 6 minutes

D
2 hours and 6 minutes

Solution

A complete his round in $6\times 60 = 360$ seconds
B completes his round in $5\times 60 = 300$ seconds
C completes his round in $20\times 60 = 1200$ seconds.
LCM of $360,300$ and $1200$
$360=2\times 2\times 2\times 3\times 3\times 5$
$300=2\times 2\times 3\times 5\times 5$
$1200=2\times 2\times 2\times 2\times 3\times 5\times 5$
Required LCM $= 2\times 2\times 2\times 2\times 3\times 3\times 5\times 5 = 3600$ seconds $= 60$ minutes $= 1$ hour.
Hence, the correct option is (A).
Question 11
1 / -0

Which of the following are irrational numbers?
(i) $\quad(\sqrt{3}+\sqrt{2})^{2}$
(ii) $\sqrt{3+5+2 \sqrt{15}}$
(iii) $\quad \sqrt{(\sqrt{2})^{2}+\sqrt{14}^{2}}$
(iv) $\sqrt{(\sqrt{13}+2 \sqrt{3})(\sqrt{13}-2 \sqrt{3})}$
(v)$ \quad(\sqrt{3}+\sqrt{50})^{2}$
$(\text { vi }) \quad(\sqrt{3}+\sqrt{12}) \div(\sqrt{12}+\sqrt{27})$

A
Only (i), (ii) and (vi)

B
Only (i), (ii) and (v)

C
Only (iii), (iv) and (vi)

D
Only (iii), (v) and (vi)

Solution

An irrational number is a type of real number which cannot be represented as a simple fraction. It cannot be expressed in the form of a ratio.
Only (i), (ii) and (v) are irrational numbers.
Hence, the correct option is (B).
Question 12
1 / -0

Which of the following has a non-terminating non-repeating decimal expansion?

A
$\frac{\sqrt{2} \times \sqrt{3}}{\sqrt{6}}$

B
$\frac{1}{\sqrt{2}}$

C
$\frac{\sqrt{15} \times \sqrt{5}}{\sqrt{3}}$

D
$\frac{17}{5}$

Solution

A non-terminating non-repeating decimal is a decimal number that continues endlessly with no group of digit repeating endlessly.
(A) $\frac{\sqrt{2} \times \sqrt{3}}{\sqrt{6}}$
$=\frac{\sqrt{2} \times \sqrt{3}}{\sqrt{2} \times \sqrt{3}}=1$
Thus, incorrect option.
(B) $\frac{1}{\sqrt2}$
This is non-terminating non-repeating decimal expansion.
Thus, correct option.
(C) $\frac{\sqrt{15} \times \sqrt{5}}{\sqrt{3}}$
$=\frac{\sqrt{5} \times \sqrt{3} \times \sqrt{5}}{\sqrt{3}}=5$
Thus, incorrect option.
(D) $\frac{17}{5}=3.4$
Thus, this is incorrect option.
Non-terminating non-repeating decimal expansion $=\frac{1}{\sqrt{2}}$.
Hence, correct option is (B).
Question 13
1 / -0

Which of the following statements violates the fundamental theorem of factorisation?

A
For some value of $x, 18^{x}$ will end with 5

B
Prime factorisation of a natural number is unique.

C
Every composite number can be expressed as a product of prime numbers.

D
$7 \times 5+3 \times 5+5 \times 3 \times 2$ is a composite number.

Solution

Prime factorization is the decomposition of a composite number into a product of prime numbers. Prime factorization of $18: 2 \times 3^{2}$.
Thus for some value of ${x}, 18^{x}$ will not end with $5 $.
Hence, the correct option is (A).
Question 14
1 / -0

Which of the following statements is true?

A
$\frac{29}{2^{2} \times 5^{2}}$ is a non-terminating decimal

B
$\frac{31}{2^{2} \times 5^{2} \times 3^{3}}$ is a terminating decimal.

C
$\frac{63}{2^{2} \times 5^{3} \times 7}$ is a non-terminating decimal

D
$\frac{93}{3 \times 2^{2} \times 5^{2}}$ is a terminating decimal

Solution

A rational number can be expressed as a terminating decimal if the denominator has the factors of $2$ or $5$ only.
A rational number can be expressed as a non-terminating repeating decimal if the denominator has the factors other than $2$ or $5$.
(A) $\frac{29}{2^{2} \times 5^{2}}$ is a terminating decimal. Thus, it is false.
(B) $\frac{31}{2^{2} \times 5^{2} \times 3^{3}}$ is a non-terminating decimal. Thus, it is false.
(C) $\frac{63}{2^{2} \times 5^{3} \times 7}$ is a terminating decimal. Thus, it is false.
(D) $\frac{93}{3 \times 2^{2} \times 5^{2}}$ is a terminating decimal . Thus, it is true.
$\frac{93}{3 \times 2^{2} \times 5^{2}}=\frac{31}{2^2 \times 5^ 2}$ option (D) have only factors of $2$ and $5$ in their denominator.
Hence, the correct option is (D).
Question 15
1 / -0

Which of the following statements is true?

A
HCF of two prime numbers is an even number.

B
HCF of two prime numbers is 1.

C
HCF of two prime numbers is 0.

D
HCF of two prime numbers is another prime number.

Solution

A prime number is the number which has only two factors,1 and the number itself. Thus 1 is only common factor for two prime numbers making it HCF.
Hence, the correct option is (B).

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Re-Attempt Weekly Quiz Competition

Real Numbers Test - 16

Result

Real Numbers Test - 16

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SHARING IS CARING

Question 1

360

240

120

60

Solution

Question 2

Only (i) and (ii)

Only (i), (iii) and (iv)

Only (iii) and (iv)

Only (i), (ii) and (iii)

Solution

Question 3

\(\frac{a}{a}\)

1 + a

1 - a

a2

Solution

Question 4

\(17 \times 5 \times 3-17 \times 3 \times 2 \times 3\)

\(17 \times 5+3 \times 2+10\)

\(2 \times 3 \times 5+2 \times 7 \times 11\)

\(23 \times 43 \times 7+11 \times 17 \times 19 \times 7\)

Solution

Question 5

6

8

10

4

Solution

Question 6

2, 4, 0 and 1

2, 4, 1 and 0

4, 2, 0 and 1

4, 2, 1 and 0

Solution

Question 7

2, 3, 5, 7, 11, 13

2, 3, 5, 19

2, 3, 5, 7, 11, 13, 14

2, 3, 5, 7, 11

Solution

Question 8

5 cm

250 cm

1500 cm

600 cm

Solution

Question 9

\(\frac{25}{3^{2} \times 2^{3}}\)

\(\frac{41}{2^{2} \times 7^{3}}\)

\(\frac{69}{3^{2} \times 2^{2} \times 5^{2}}\)

\(\frac{343}{2^{2} \times 5^{2} \times 7^{3}}\)

Solution

Question 10

1 hour

2 hours

1 hour and 6 minutes

2 hours and 6 minutes

Solution

Question 11

Only (i), (ii) and (vi)

Only (i), (ii) and (v)

Only (iii), (iv) and (vi)

Only (iii), (v) and (vi)

Solution

Question 12

\(\frac{\sqrt{2} \times \sqrt{3}}{\sqrt{6}}\)

\(\frac{1}{\sqrt{2}}\)

\(\frac{\sqrt{15} \times \sqrt{5}}{\sqrt{3}}\)

\(\frac{17}{5}\)

Solution

a²