Number Systems Test

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Number Systems Test - 26

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Weekly Quiz Competition

Question 1
1 / -0

State which of the following statements is/are true?
I. Numerator and denominator of a positive rational number need not to have like signs.
II. Numerator and denominator of a negative rational number should have like signs.

A
Only I

B
Only II

C
Both I and II

D
Neither I nor II

Solution

If both the numerator and denominator has same sign, then the fraction is a positive rational number.
If the numerator and denominator have different signs, then the fraction is a negative rational number.
Question 2
1 / -0

The fact that $$7\sqrt{5}$$ is irrational is because

A
Product of two irrational numbers in rational

B
Product of a rational and an irrational number is rational

C
Product of two rational numbers is rational

D
Product of a rational and an irrational number is irrational

Solution

$$7$$ is a rational number and $$\sqrt5$$ is an irrational number

Product of a rational and irrational number is irrational, so $$7\sqrt5$$ is irrational.
Question 3
1 / -0

Which of the following rational numbers is positive?

A
$$\dfrac{-7}{9}$$

B
$$\dfrac{-8}{2}$$

C
$$\dfrac{6}{5}$$

D
$$\dfrac{3}{-2}$$

Solution

A rational number is positive only when both its numerator and denominator are positive.
We can see that only $$\dfrac{6}{5}$$ has positive numerator and denominator. Rest all the given options have either negative numerator or negative denominator.
Therefore, only $$\dfrac{6}{5}$$ is a positive rational number.
Hence, option C is correct.
Question 4
1 / -0

Subtraction of two irrational numbers is:

A
a rational

B
an irrational an integer

C
either rational or irrational

D
Can't determine

Solution

The subtraction of two irrational number can be rational or irrational depending on if the irrational part is same or different in both the irrational numbers respectively.
For instance, $$(2+\sqrt3) - (1+\sqrt3) = 1 $$, which is rational.
Whereas, $$\sqrt{3} - \sqrt{2}$$ is irrational.
Question 5
1 / -0

Any operation between one non-zero rational and one irrational number always gives:

A
a rational number

B
an irrational number

C
an integer

D
a fraction
Solution
Any operation between one non-zero rational and one irrational number always gives an irrational number
The sum of a rational number and an irrational number is irrational.
The difference between a rational number and an irrational number is irrational.
The product of a non-zero rational number and an irrational number is irrational.
The division of a non-zero rational number and an irrational number is irrational.
Question 6
1 / -0

The conjugate of the surd $$\sqrt{5}-\sqrt{2}$$ ?

A
$$\sqrt{5}+\sqrt2$$

B
$$-\sqrt{5}+\sqrt2$$

C
Either $$(A)$$ or $$(B)$$

D
None of these

Solution

The sum and difference of two quadratic surds is called as conjugate to each other.

In the given case, difference of two surds $$\sqrt{5}$$ and $$\sqrt2$$ is given. Hence, the conjugate will be sum of these two surds, i.e.
$$\sqrt5+\sqrt2$$.
Option A is the correct answer.
Question 7
1 / -0

$$\dfrac {p}{q}$$ is a rational number when

A
$$p=0,q\neq0$$

B
$$p=0,q=0$$

C
$$p\ne 0,q=0$$

D
$$p=1,q=0$$

Solution

By the definition of rational number, $$\dfrac{p}{q}$$ is rational where $$p$$ and $$q$$ are integers and $$q$$ is not equal to zero.
In all three options except A, $$q=0$$.
Hence, $$p=0, q\ne 0$$ is true.
Question 8
1 / -0

Add : $$6 + 3\sqrt 3 $$ and $$8 + 4\sqrt 3 $$

A
$$1+7\sqrt{3}$$

B
$$8+7\sqrt{3}$$

C
$$14+7\sqrt{3}$$

D
$$4+7\sqrt{3}$$

Solution

$$6+3\sqrt{3}+8+4\sqrt{3}$$

$$=6+8+3\sqrt{3}+4\sqrt{3}$$

$$=14+3\sqrt{3}+4\sqrt{3}$$

$$=14+\sqrt{3}(3+4)$$

$$=14+\sqrt{3}\times 7$$

$$=14+7\sqrt{3}$$
Question 9
1 / -0

If $$\sqrt{a}$$ is an irrational number, what is a?

A
Rational

B
Irrational

C
$$0$$

D
Real

Solution

Consider the given irrational number$$\sqrt{a}$$ ,

Definition of rational number- which number can be write in the form of $$\dfrac{p}{q}$$ but $$q\ne 0$$ is called rational number.

Hence, $$a=\dfrac{a}{1}$$

That why $$a$$ is rational number

Hence, this is the answer.
Question 10
1 / -0

The number $$23+\sqrt{7}$$ is

A
Natural number

B
Irrational number

C
Rational number

D
None of these

Solution

As we've $$\sqrt{7}$$ is an irrational number and $$23$$ is a rational number then the sum of an irrational number and a rational number is again an irrational number.

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Number Systems Test - 26

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Number Systems Test - 26

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Question 1

Only I

Only II

Both I and II

Neither I nor II

Solution

Question 2

Product of two irrational numbers in rational

Product of a rational and an irrational number is rational

Product of two rational numbers is rational

Product of a rational and an irrational number is irrational

Solution

Question 3

$$\dfrac{-7}{9}$$

$$\dfrac{-8}{2}$$

$$\dfrac{6}{5}$$

$$\dfrac{3}{-2}$$

Solution

Question 4

a rational

an irrational an integer

either rational or irrational

Can't determine

Solution

Question 5

a rational number

an irrational number

an integer

a fraction

Solution

Question 6

$$\sqrt{5}+\sqrt2$$

$$-\sqrt{5}+\sqrt2$$

Either $$(A)$$ or $$(B)$$

None of these

Solution

Question 7

$$p=0,q\neq0$$

$$p=0,q=0$$

$$p\ne 0,q=0$$

$$p=1,q=0$$

Solution

Question 8

$$1+7\sqrt{3}$$

$$8+7\sqrt{3}$$

$$14+7\sqrt{3}$$

$$4+7\sqrt{3}$$

Solution

Question 9

Rational

Irrational

$$0$$

Real

Solution

Question 10

Natural number

Irrational number

Rational number

None of these

Solution

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