Class 10 Practice – Level 3 – Set 1 – Real Numbers

Total questions: 15

If $ \displaystyle {{a}^{2}}+{{b}^{2}}$ is divisible by 5, then both $ \displaystyle a$ and $ \displaystyle b$ are divisible by 5. What can be said about this statement?

Topic: Euclid's Division Lemma

True

False

Cannot be determined

None of the above

None

$ \displaystyle \sqrt{2}+\sqrt{3}$ is irrational. What can be said about this statement?

Topic: Fundamental Theorem of Arithmetic

Always true

Sometimes true

False

None of the above

None

If $ \displaystyle \sqrt{a}\text{ and }\sqrt{b}$ are irrational numbers, under what condition is $ \displaystyle \sqrt{a}+\sqrt{b}$ a rational number?

Topic: Properties of Irrational Numbers

$ \displaystyle a=b$

$ \displaystyle a={{b}^{2}}$

$ \displaystyle a={{b}^{3}}$

$ \displaystyle a=2b$

None

Find the general form of all numbers that gives a remainder of 1 when divided by 3.

Topic: Euclid's Division Lemma

$ \displaystyle 3k+1$

$ \displaystyle 3k+2$

$ \displaystyle 3k$

$ \displaystyle 3k+4$

None

Prove that if a number is in the form of $ \displaystyle 4n+1$, it can never be a perfect square.

Topic: Fundamental Theorem of Arithmetic

True

False

Sometimes true

None of the above

None

If $ \displaystyle \sqrt{3}+2\text{ and }\sqrt{3}-2$ are added, what type of number is the result?

Topic: Properties of Irrational Numbers

Rational

Irrational

Prime

Composite

None

The square of any odd integer can be represented in the form of $ \displaystyle 8k+1.$ Is it true or false?

Topic: Euclid's Division Lemma

True

False

Sometimes true

None of the above

None

If the product of three consecutive integers is always divisible by 6, then prove that their sum is divisible by 3.

Topic: Fundamental Theorem of Arithmetic

True

False

Sometimes true

None of the above

None

Prove that $ \displaystyle \sqrt{2}\times \sqrt{3}$ is an irrational number.

Topic: Properties of Irrational Numbers

True

False

Sometimes true

None of the above

None

10.

The sum of the square of two consecutive odd numbers is divisible by 4. What can be said about this statement?

Topic: Euclid's Division Lemma

True

False

Sometimes true

None of the above

None

11.

If $ \displaystyle a$ and $ \displaystyle b$ are positive integers such that $ \displaystyle {{a}^{2}}+{{b}^{2}}={{c}^{2}},$ prove that at least one of $ \displaystyle a$ or $ \displaystyle b$ is even.

Topic: Fundamental Theorem of Arithmetic

True

False

Sometimes true

None of the above

None

12.

If $ \displaystyle x=\sqrt{7}+\sqrt{{5,}}$ prove that $ \displaystyle {{x}^{2}}$ is irrational.

Topic: Properties of Irrational Numbers

True

False

Sometimes true

None of the above

None

13.

Every integer of the form $ \displaystyle \text{6}n+5$ is either a prime number or a multiple of 5. What can be said about this statement?

Topic: Euclid's Division Lemma

True

False

Sometimes true

None of the above

None

14.

The difference between the squares of two odd numbers is always divisible by 8. Is it true or not?

Topic: Fundamental Theorem of Arithmetic

True

False

Sometimes true

None of the above

None

15.

The cube root of 2 is irrational. Is it true or not?

Topic: Properties of Irrational Numbers

True

False

Sometimes true

None of the above

None

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Class 10 Practice – Level 3 – Set 1 – Real Numbers

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