Number System Test

Result

Number System Test - 6

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Weekly Quiz Competition

Question 1
1 / -0

If , then what is the value of x + y?

A
5

B
6

C
7

D
8

Solution

Simplifying the equation:

Equating the exponents:
4x = 8^{x = 2
And
2y = 6
y = 3
Thus, x + y = 2 + 3 = 5}
Question 2
1 / -0

If A = , then A also equals

A
1

B

C

D

Solution

Simplifying:

A =

A =

A =

A =

A =

A =
Question 3
1 / -0

Write in the form of .

A

B

C

D

Solution

Let = x

Multiplying both sides by 10,

= 10x ... (i)

Again multiplying both sides by 10,

= 100x ... (ii)

Subtracting equation (i) from equation (ii),

514 = 90x

x = =
Question 4
1 / -0

If x = , then what is the value of ?

A

B

C

D

Solution

x =

Rationalising the denominator:
Question 5
1 / -0

Simplify:

A

B

C

D

Solution

Rationalising the denominator:

=

=

=

=

=
Question 6
1 / -0

The rationalising factor of is

A

B

C

D

Solution

The rationalising factor of is as described below:

=

=

= xyz
Question 7
1 / -0

If -5 < x < -4, then which of the following is true?

A
x =

B
x =

C
x =

D
x =

Solution

Decimal expansion of the rational numbers is as follows:
= -4.25
= -3.34
= -3.6
= -5.34
= -1.34

Thus, x = as
Question 8
1 / -0

Which of the following is an irrational number that is greater than but less than?

A

B

C

D

Solution

=

Thus, this is the correct option.
Question 9
1 / -0

On simplifying, what will be the denominator of ?

A
2x² + 2y²

B
2x²

C
y²

D
2y²

Solution

The denominator will be the product of and .
Using identity (a + b)(a – b) = a² – b²;
Question 10
1 / -0

Which of the following options shows the correct ascending order of the numbers?

A

B

C

D

Solution

LCM of 4, 3 and 6 = 12

Thus, is the correct ascending order.
Question 11
1 / -0

Which is the smallest number among these?

A

B

C

D
All are same.

Solution
Question 12
1 / -0

What are the values of x and y if ?

A
x = 2, y = 2

B
x = 3, y = 2

C
x = 2, y = 3

D
x = 1, y = 4

Solution

Equating the exponents:
x + 2 = 5
x = 3
y + 1 = 3
y = 2
Question 13
1 / -0

What is the value of ?

A
3¹³

B
36¹²

C
3 × 36¹²

D
4 × 36¹²

Solution
Question 14
1 / -0

If , then what are the values of , respectively?

A
0, 38

B
, 36

C
, 38

D
, 19

Solution
Question 15
1 / -0

Which of the following statements is false?

A
Every integer is a real number.

B
Every natural number is a whole number.

C
Every rational number is an integer.

D
Some fractions can be non-terminating but repeating decimals.

Solution

(i) A real number is any positive or negative number. This includes all integers and all rational and irrational numbers. Thus, the statement in option 1 is true.

(ii) Whole numbers are collection of all natural numbers and 0. Thus, the statement in option 2 is true.
(iii) The statement in option 3 is false as some rational numbers like are not integers.
= 3.3
(iv) Some fractions like can be non-terminating but repeating decimals. Thus, the statement in option 4 is also true.
Question 16
1 / -0

If , find the values of 'a' and 'b'.

A
a = 2, b = 2

B
a = 2, b = -2

C
a = -2, b = 2

D
a = 3, b = 2

Solution

Simplifying LHS:
After rationalising the given expression, we get
Question 17
1 / -0

Which of the following fractions is a terminating decimal?

A

B

C

D

Solution

(i) Decimal expansion of : 0.10714... (non-terminating, non-repeating)

(ii) Decimal expansion of : 0.555... (non-terminating, repeating)

(iii) Decimal expansion of : 0.189189... (non-terminating, repeating)

(iv) The prime factorisation of 80 (denominator) is 2⁴ × 5 (in the form of 2^a × 5^b). Thus, it is a terminating decimal.

= 0.1125
Question 18
1 / -0

The number 12.43434343... is written in the form of. What is the value of p – 7q?

A
1132

B
938

C
538

D
278

Solution

Let x = 12.43 ... (i)
Multiplying both sides by 100,
1243.43 = 100x ... (ii)

Subtracting equation (i) from equation (ii),
1231 = 99x
x =
p = 1231
q = 99
p – 7q = 1231 – 693 = 538
Question 19
1 / -0

For which of the following is the relation -x > y always true?

A
x = positive integer and y = negative integer

B
x = negative integer and y = negative integer

C
x = positive integer and y = positive integer

D
x = negative integer and y = positive integer

Solution

For the given relation to be always true, final value of the left-hand side should be positive and final value of the right-hand side should be negative.
If x and y are both negative integers, then
-(-x) > -y
x > -y, which is always true.
Question 20
1 / -0

_If_{, then what is the value of}

A
14

B
16

C
18

D
20

Solution
Question 21
1 / -0

Find the value of .

A
0

B
-4

C
-2

D
2

Solution
Question 22
1 / -0

Which of the following statements is false?

A
The sum of two rational numbers is always a rational number.

B
The product of two irrational numbers is always an irrational number.

C
The division of an integer by an integer may not yield another integer.

D
The sum of a rational number and an irrational number is always an irrational number.

Solution

(i) Adding two rational numbers produces another rational number like (1/2) + (1/2) = 1, where 1/2 is a rational number and 1 is also a rational number.
(ii) The product of two irrational numbers like is 2, which is a rational number. Thus, the statement given in this option is false.
(iii) The division of an integer by an integer may not yield another integer. This statement is true as 3 ÷ 2 = 1.5, which is not an integer.
(iv) The sum of a rational number and an irrational number is always an irrational number. For example, the sum of 1 and is 2.41421..., which is an irrational number.

Question 23

1 / -0

Fill in the blanks.

(i) All P numbers are Q numbers.
(ii) The denominator of a terminating decimal number can be expressed in the form of R , where m and n are positive integers.
(iii) The decimal expansion of is S in nature.

	P	Q	R	S
A	whole	natural	5^m2ⁿ	terminating
B	natural	whole	3^m2ⁿ	terminating
C	whole	natural	3^m2ⁿ	non-terminating
D	natural	whole	5^m2ⁿ	non-terminating

A

B

C

D

Solution

(i) Whole numbers are collection of all natural numbers and zero. Thus, all natural numbers are also whole numbers.
(ii) Fractions with terminating decimal expansions are those whose denominator in its lowest form can be expressed as a product of powers of 2 and 5.
(iii) = 1.501501501 (non-terminating but repeating)

Question 24
1 / -0

Read the statements carefully and mark the correct option accordingly:

Statement 1: cannot be represented on a number line.
Statement 2: 0.3 can be represented on a number line.

A
Both statements 1 and 2 are true.

B
Statement 1 is true and statement 2 is false.

C
Statement 1 is false and statement 2 is true.

D
Both statements 1 and 2 are false.

Solution

Since both and 0.3 are real numbers, both can be represented on a number line.
Therefore, statement 1 is false and statement 2 is true.
Question 25
1 / -0

Match the columns:

A
(a) - (i), (b) - (iii), (c) - (ii), (d) - (iv)

B
(a) - (iv), (b) - (iii), (c) - (ii), (d) - (i)

C
(a) - (iv), (b) - (ii), (c) - (iii), (d) - (i)

D
(a) - (i), (b) - (ii), (c) - (iii), (d) - (iv)

Solution

(a)
2x - 6 = 7x + 14
-5x = 20
x = -4

Equating exponents:
x - 3 = 5
or x = 8

(c) 3^2x - 3^{(2x - 2)} = 8
3^2x(1 - 3^-2) = 8

3^2x = 9 = 3²
2x = 2
x = 1

(d) 6^{(x + 2)} = 1296
6^{(x + 2)} = 6⁴
x + 2 = 4
x = 2