Number Systems - 2 - Free MCQ Practice Test with solutions, Class 9 Maths

There are 10 prime numbers between 0 and 30 which are: 2,3,5,7,11,13,17,19,23,29. Number one is not a prime number.
Now, for 1, the number of positive divisors or factors is only one i.e. 1 itself.  

Correct Answer:
a) √5 and √6
We are asked to find two irrational numbers between 2 and 2.5.

Let’s check the values:

• √2 = 1.414 (irrational but not between 2 and 2.5)

• √4 = 2 (not between 2 and 2.5, and also rational)

• √5 ≈ 2.236 (irrational and between 2 and 2.5 )

• √6 ≈ 2.449 (irrational and between 2 and 2.5 )

• √9 = 3 (greater than 2.5 )

So, √5 and √6 are both irrational and lie between 2 and 2.5, hence option (a) is correct.

As the bar is under two digits multiply with 100

100x = 2.7435*100

100x = 274.35

now subtract with x terms

that gives

99x = 274.35 - 2.7435

99 x = 271.61

x = 271.61/99

x = 27161/9900

To solve it, we can cancel the recurring part after the decimal by multiplying x by 10 and 1000 and then subtracting the 10x from 1000x.

√4 = 2, √9 = 3, √16 = 4 but √2 = 1.414 it is a non-terminating and non-repeating number and such numbers are called irrational numbers. So, √2 is not a rational number.

(2√3 +√3) = 3√3       ( By addition of like terms )

So option C is correct answer. 

63 is divisible by 7 for any power, so required remainder will depend on the power of 4
Required remainder 

find remainder when 99/4 ( by 4 because to check periodicity)

Step 1: Let x Represent the Repeating Decimal

Let:
x = 5.2222... (where 2 repeats forever)

Step 2: Multiply by 10 to Shift the Decimal Point

Multiply both sides by 10 to move the decimal point one place to the right:
10x = 52.2222...

Step 3: Subtract the Original Equation from the New One

Now subtract the original equation (x = 5.2222...) from this new equation:
10x - x = 52.2222... - 5.2222...
9x = 47

Step 4: Solve for x

x = 47 / 9

Final Answer

The exact value of 5.2222... (or ) is 47/9

The resulting number is always divisible by 9. e.g.,

Consider 47.

Sum of its digits is 4+7=11

So,47−11=36 which is exactly divisible by 9

a = 1/ (3-2√2) 

b =  1/ ( 3+2√2) 

∴ ab =  1/(3-2√2) * 1/ (3+2√2) 

=>  1/ (9-8) 

=> 1 

We know; 

(a+b)² = a² + b² + 2ab

And, a²+ b² = (a+b)² - 2ab

=> [ 1/(3-2√2) + 1/(3+2√2) ]² -2*1

=> [3+2√2+3-2√2 / 1]^2 -2 

=> (6)² - 2 

=>  36 - 2
=>  34

2ⁿ(2⁴-2) + 2^-3
2ⁿx2x2³
14 + 2^-3
2⁴
7/8+1/8
1

Given expression

Step 1: Apply exponent rule

Step 2: Simplify the numerator

So exponent becomes:

Step 3: Factor and simplify

Final Answer:
  
                         

[(x)²]^1/3]1/4

= [(x²)^1/12] (as 1/3 * 1/4 = 1/12)

= x^1/6 (as 2/12 = 1/6)

4√6*3√24=4√6*3√2*√2*√*2*√3

=4√6*6√6

4*6*√6*√6=4*6*6=24*6=144