Prime Factorisation

Q. Which among the following is/are the prime factors of 125?

1. 1
2. 25
3. 125
4. 5

Q. Q- proove that root3-root7 is an irrational number. Q-show that there is no positive integer n for which root n-1+root n+1is rational

Q.

The product of two coprime numbers is $117,$ then their $L.C.M.$ is

1. $9$

2. $13$

3. $39$

4. $117$

Q. Question 9
Simplify the following
$\left(i\right)\sqrt{45}-3\sqrt{20}+4\sqrt{5}\left(ii\right)\frac{\sqrt{24}}{8}+\frac{\sqrt{54}}{9}\left(iii\right)4\sqrt{12}\times 7\sqrt{6}\left(iv\right)4\sqrt{28}+3\sqrt{7}+\sqrt[3]{37}\left(v\right)3\sqrt{3}+2\sqrt{27}+\frac{7}{\sqrt{3}}\left(vi\right)(\sqrt{3}-\sqrt{2}{)}^2(vii)\sqrt[4]{481}-8\sqrt[3]{3216}+15\sqrt[5]{532}+\sqrt{225}(viii)\frac{3}{\sqrt{8}}+\frac{1}{\sqrt{2}}(ix)\frac{2\sqrt{3}}{3}-\frac{\sqrt{3}}{6}$

Q. numbers are the numbers that have only two factors, one and the number itself.

1. Composite
2. Prime
3. Whole

Q. Simplify the following:
$\left(i\right)\sqrt{45}-3\sqrt{20}+4\sqrt{5}\left(ii\right)\frac{\sqrt{24}}{8}+\frac{\sqrt{54}}{9}\left(iii\right)4\sqrt{12}\times 7\sqrt{6}$

Q.

Factorize the following expression: $10a^2-15b^2+20c^2$

Q.

The last two digits in $X=\sum _{k=1}^{100}k!$are?

1. $10$

2. $11$

3. $12$

4. $13$

Q.

If $132300=2^2\times 3^3\times 5^2\times a^b,$ then which of the following is true ?

1. $a+b=8$

2. $a=b$

3. $a=2b$

4. LCM of $a$ and $b$ is 14.

Q. An hexagonal pyramid is 20m high. Side of the base is 5m. The volume of the pyramid is

Q. Use Euclid's Division Lemma, prove that for any positive integer $n,n^3-n$ is divisible by $6$.

Q. In the following equations find which variables $x,y,z,$ etc., represent rational or irrational
numbers,
(i) $x^2=5$

Q.

${105}^2$ is equivalent to ___

Q.

While arranging oranges in heaps of$24,32and48$, a fruit vendor notices that $8$ oranges are always left. What is the smallest number of oranges he has?

1. $96$

2. $104$

3. $88$

4. $124$

Q.

Find the greatest number that will divide $95,114$ and $129$ leaving a remainder $5,4$ and $3$ respectively.

Q.

Give three pairs of prime numbers whose difference is $2$.

Q. If LCM of two numbers is the greater of the two numbers, then its HCF

1. Will always be 1
2. Will be$\frac{LCM}{Smallernumber}$
3. Will be equal to LCM
4. Will be the smaller number

Q. how a irrational number can be cons†an t in a polynomial as it is non terminating and non repeating

Q.

Prove that the product of three consecutive positive integer is divisible by $6$.

Q. Simplify the following:
$\left(i\right)\sqrt{45}-3\sqrt{20}+4\sqrt{5}\left(ii\right)\frac{\sqrt{24}}{8}+\frac{\sqrt{54}}{9}\left(iii\right)4\sqrt{12}\times 7\sqrt{6}$

Q.

If LCM of two numbers is equal to the greater number, then its HCF ____.

1. will be the smaller number

2. will be equal to LCM

3. is 1

4. is 0

Q. Express in the simplest form

1. $-3ab.\sqrt[3]{34a}$
2. $3ab.\sqrt[3]{34a}$
3. $3ab.\sqrt[3]{3a}$
4. Not possible

Q.

Question 5
The largest number which divides 70 and 125, leaving remainders 5 and 8 respectively, is

A) 13
B) 65
C) 875
D) 1750

Q. Q- proove that root3-root7 is an irrational number. Q-show that there is no positive integer n for which root n-1+root n+1is rational

Q.

If $132300=2^2\times 3^3\times 5^2\times a^b,$ then which of the following is true ?

1. $a=b$

2. $a=2b$

3. LCM of $a$ and $b$ is 14.

4. $a+b=8$

Q. $\sqrt{x^{-1}y}.\sqrt{y^{-1}z}.\sqrt{z^{-1}x}=$

1. 2
2. 4
3. 1
4. 3

Q. For any natural numbers ( m, n ) , prove that ( m³ + n³ + 4 ) cannot be a perfect cube .

Q.

If LCM of two numbers is equal to the greater number, then its HCF ____.

1. will be the smaller number

2. will be equal to LCM

3. is 0

4. is 1

Q.

If $132300=2^2\times 3^3\times 5^2\times a^b,$ then which of the following is true ?

1. $a+b=8$

2. LCM of $a$ and $b$ is 14.

3. $a=2b$

4. $a=b$

Q. If $1176=2^a{3}^b{7}^c$, find a, b and c.