Number System Test

Result

Number System Test - 7

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Weekly Quiz Competition

Question 1
1 / -0

The value of x, if , is:

A
9

B
7

C
6

D
3

Solution

Taking L.H.S,

⇒
⇒
⇒
⇒
⇒
⇒
⇒
Comparing,
L.H.S. = R.H.S.
=
⇒ x – 1 = 8
⇒ x = 8 + 1
⇒ x = 9
Question 2
1 / -0

If a = 3 + , then what is the value of ?

A
104

B
216

C
198

D
298

Solution

a = 3 +
= = =
= 3 + + 3 - = 6

= 216

a³ + + 3 x 1 x (6) = 216
a³ + = 216 - 18 = 198
Question 3
1 / -0

If a = 5 +, then which of following options will give a rational number?

A

B

C
2a

D

Solution

Given: a = 5 + ... (1)

Now,

After rationalising the denominator,

=

=

=

... (2)

Adding (1) and (2),
=

= 5 + 5 = 10,which is a rational number.
Question 4
1 / -0

Which of the following options represents three rational numbers that lie betweenand?

A

B

C

D

Solution

Given:and

As per options, we can make denominator 180.

and

So, all numbers that lie between are rational numbers.

Hence, option 3 is the correct answer.
Question 5
1 / -0

Rationalise:

A

B

C

D

Solution

Given:

=

=

Rationalising the denominator,

= ×

=

=
=

=
Question 6
1 / -0

Which of the following is the second largest surd?

A

B

C

D

Solution

(i)

₌
₌
₌

(ii)

₌
₌
₌

(iii)

₌
₌
=

(iv)

₌
₌
₌

Hence, the second largest surd is:
=
Question 7
1 / -0

The product of which of the following numbers is an irrational number?

A

B

C

D

Solution

(i)

= 3 × 4 - 3 × 2 × 4 ( )

= 12 - 24

= -12 which is not an irrational number

(ii)

2 × -23 × 11 = -506 which is not an irrational number.

(iii)
=

= ; which will give an irrational number because the value of root 3 is an irrational number.
Question 8
1 / -0

What is the value of ?

A
13³¹ × 2²¹

B
2³¹

C
13³¹

D
13¹³

Solution

⇒

⇒

⇒

⇒

⇒

⇒

⇒
Question 9
1 / -0

If x = , then + is equal to

A
6

B
3

C
0

D
1

Solution

x =
Then,
1 + x = 1 +
1 - x = 1 -
Hence, the given expression is:

=
Question 10
1 / -0

Which of the following statements is incorrect?

A
Real numbers can be negative.

B
All whole numbers are natural numbers.

C
All rational numbers are real numbers.

D
Irrational numbers cannot be written as the ratio of two integers.

Solution

Whole numbers are the numbers 0, 1, 2, 3, 4, and so on (the natural numbers and zero). Negative numbers are not considered whole numbers. All natural numbers are whole numbers, but not all whole numbers are natural numbers since zero is a whole number but not a natural number.

Hence, statement 2 is incorrect.
Question 11
1 / -0

What is the rationalising factor of (6 - )²?

A
(6 + )²

B
(6 + )

C
(7 + )

D
(7 + 2)

Solution

Rationalising factor is a term by which a term is multiplied or divided to make the whole term rational.

(6 - )² = 6² + 6 – 12

= 36 + 6 – 12

= 42 – 12

= 6(7 – 2)

So, rationalising factor = 7 + 2

When we multiply 6(7 - 2) by (7 + 2), we get:

6 × (7 - 2) × (7 + 2)

= 6 × (7 - ) × (7 + )

= 6 × (7² - ()²)

= 6 × (49 - 24)

= 6 × (49 - 24) = 6 × 25

= 150; which is a rational number.

So, option 4 is the correct answer.
Question 12
1 / -0

What is the denominator of after rationalising it?

A
191

B
192

C
193

D
194

Solution

We have

=

=

Rationalising,

=

=

=
Question 13
1 / -0

Which of the following numbers does not have a recurring decimal representation?

A

B

C

D

Solution

= 0.22222...
= 0.77777...
= 0.33333...
= 0.6
Therefore,does not have a recurring decimal representation.
Question 14
1 / -0

Find the value of p + 2q from the equation = p – q.

A

B

C

D

Solution

Rationalising the denominator:

= = =

So,

p = and q =

Now, p + 2q = +

= =
Question 15
1 / -0

If , then the value of is

A
29

B
28

C
27

D
26

Solution

So, =

Rationalising,

=

Now, putting the values in the equation:

=

= + 16

= 2 × 2 × 3 + 16

= 12 + 16

= 28
Question 16
1 / -0

If the number y = 1.262626... can be expressed in the form y = , where a and b are positive integers having no common factors, then what is the value of ?

A
13

B
14

C
16

D
17

Solution

We have
y = 1.262626 --- (1)
100y = 126.2626 --- (2)
Subtracting (1) from (2),
100y – y = 125
99y = 125
y =
Therefore, a = 125, b = 99
=
== 13
Question 17
1 / -0

Which of the following represents the sum of three non-terminating, non-recurring decimals?

A
0.0404040... + 0.0707070... + 0.2300230023...

B
0.400400400... + 0.700700700... + 0.230230230...

C
0.000400040004... + 0.0720720720... + 0.230230230...

D
0.040040004000... + 0.0700700070000... + 0.23002300023...

Solution

A non-terminating, non-repeating decimal is a decimal number that continues endlessly, with no group of digits repeating endlessly. Decimals of this type cannot be represented as fractions, and as a result are irrational numbers.
So, in the option 4, the numbers are continuing endlessly and no group is repeating.
However, in all the other options, the groups are repeating.
Question 18
1 / -0

If x = , y = , then the value of x² + y² is equal to

A
12

B
24

C
34

D
36

Solution

If x = = = 3 + 2y = = = 3 - 2 x² + y² = = 9 + 8 + 12 + 9 + 8 - 12 = 34
Question 19
1 / -0

Which of the following irrational numbers is represented by P in the given diagram, if O is the centre and ODS is an isosceles right angled triangle?

A

B

C

D

Solution

In isosceles right angled triangle OSD,

OS² = OD² + DS²

OS² = 3² + 3²

OS² = 9 + 9

OS =

Since OS is the radius of the circle;

OS = OP =
Question 20
1 / -0

Which of the following equations is true?

A

B

C

D

Solution

a (b c) = (a b) c is an associative property, which is true for all natural numbers.
2 (5 3) = (2 5) 3
L.H.S.
2 (5 3) = 2 15 = 30
R.H.S.
(2 5) 3 = 10 3 = 30
L.H.S. = R.H.S.
Hence, option (4) is correct.
Question 21
1 / -0

What would be the answer if is subtracted from the total of the given calculation?

A

B

C

D

Solution
Question 22
1 / -0

Which of the following statements is definitely true regarding rational numbers?

A
Rational numbers are either terminating or non-recurring in nature.

B
A rational number which has either the numerator negative or the denominator negative is called an irrational number.

C
The decimal expansion of a rational number always either terminates after a finite number of digits or begins to reoccur the same finite sequence of digits over and over.

D
Rational numbers are either non-terminating or recurring in nature.

Solution

A rational number is a number that can be written in the form of a numerator upon a denominator. Here, the denominator should not be equal to 0. The numerator and the denominator will be integers. A rational number is of the form, p = numerator, q = denominator; where p and q are integers and q ≠ 0.
Any rational number can be written as either a terminating decimal or a repeating decimal.
Just divide the numerator by the denominator. If the remainders begin to repeat after some point, then we have a repeating decimal.
For example, = 6.333333
Here, p = 19 and q = 3, and the answer is a repeating decimal.
Just divide the numerator by the denominator. If you end up with a remainder of 0, then you have a terminating decimal.
For example, = 6.25
Here, p = 25 and q = 4, and the answer is a terminating decimal.

Question 23

1 / -0

Match the following:

Column I	Column II
(a) If 5(7 – x) = , then the value of x is	(i) -1
(b) If , then the value of x is	(ii)
(c) If , then the value of x is	(iii)
(d) If , then the value of x is	(iv) 0

(a) = (i), (b) = (iii), (c) = (iv), (d) = (ii)

(a) = (ii), (b) = (iii), (c) = (iv), (d) = (i)

(a) = (iii), (b) = (ii), (c) = (iv), (d) = (i)

(a) = (ii), (b) = (iii), (c) = (i), (d) = (iv)

Solution

(a) 5(7 – x) =
5(7 – x) =
5(7 – x) = 9(x + 2)
9x + 5x = 35 – 18
x =

(b)

5x + 16 = 15
5x = 15 – 16
x =
(c)

7^x = 1
7^x = 7⁰
x = 0

(d)

4x + 2 = 3x + 1

x = -1

Question 24
1 / -0

Read the statements carefully and choose the correct option.

Statement 1: In mathematics, a rational number is any number that can be expressed as the quotient or fraction '' of two integers, a numerator 'a' and a non-zero denominator 'b'.
Statement 2: Since 'b' may be less than 1, every integer will be a rational number.

A
Both statements I and II are true.

B
Both statements I and II are false.

C
StatementsI is true and statement II is false.

D
Statements I is false and statement II is true.

Solution

Statement 1: In mathematics, a rational number is any number that can be expressed as the quotient or fraction '' of two integers, a numerator 'a' and a non-zero denominator 'b'.
This is true for the definition of rational numbers.
Statement 2: Since 'b' may be less than 1, every integer will be a rational number.
Since the value of 'b' has to be at least 1, this statement is false.
Question 25
1 / -0

Fill in the blanks.

1. The sum of a non-zero rational number and an irrational number is a/an ____P____ number.
2. If 0.abababab... = , then the value of a × b is ____Q_____.
3. A/An ____R____ cannot be written in the form , where p and q are the integers and q ≠ 0.
4. All whole numbers are ____S____ numbers.

A

P Q R S

irrational 0 irrational number real

B

P Q R S

rational 11 whole number natural

C

P Q R S

irrational 22 rational number irrational

D

P Q R S

irrational 20 integer real

Solution

1. Let a = 2.311311131111… and b = 2.3333.Then, a + b = 4.644644464444…, which is an irrational number.
2. 0.abababab... = --- (1)

Multiply both sides by 100.

Then, ab.ababab = --- (2)
Subtracting equation (1) from equation (2),
ab = = 9

This implies, 10a + b = 9

Since ab is a two-digit number;a = 0 and b = 9

So, a × b = 0
3. A rational number is any number that can be expressed as the quotient or fraction of two integers, a numerator p and a non-zero denominator q, where q ≠ 0.An irrational number is any number that cannot be expressed as the quotient or fraction of two integers, a numerator p and a non-zero denominator q, where q ≠ 0.

4. Since whole numbers are a subset of real numbers, all whole numbers are real numbers.