Free Number Systems 03 Practice Test - 9th Grade
Question 1
34 lies between 12 and 1.
True
False
SOLUTION
Solution : A
To check if a number lies between any two numbers, we first convert the numbers into the decimal form. 34=0.75 ,12=0.5 and we have to check if 0.75 lies between 0.5 and 1 and plot the numbers on the number line as follows :
Therefore, we can say that 34 lies between 12 and 1.
Question 2
A number ′x′was found to be represented in the simplest pq form. The decimal expansion of this number was found to be 5.3333333333..... Find the value of p - q.
11
12
13
14
SOLUTION
Solution : C
Given decimal is 5.3333333......
Let, x=5.3333333......
Therefore, 10x=53.33333333....
Subtracting x from 10x, we get
10x−x=53.33333..−5.33333..=48
⇒9x=48
⇒x=489 =163=pq
So, p−q=16−3
∴ p − q = 13
Question 3
Which of the following represents whole numbers?
Positive integers
Negative integers
Integers
SOLUTION
Solution : A
Integers are constituted by natural numbers, their negatives and 0. Removing the negative numbers from integers would leave us with the whole numbers. Therefore, whole numbers include 0 as well as all the positive integers.
Question 4
The product of two irrational numbers is a/an
Irrational number
Rational number
Real number
Integer
SOLUTION
Solution : A, B, C, and D
The product of two irrational numbers will be either a rational or an irrational number.
Consider the following example: (√3+√2)×(√3−√2)=1.
Both (√3+√2) and (√3−√2) are irrational numbers and their product is 1.
1 is an integer and since, all integers are rational numbers as well, we can infer that product of two irrational numbers may be an integer, a rational number or an irrational number and they are all real numbers.
When we multiply the irrational numbers √2 and √3, we get √6, which is also an irrational number.
Question 5
Which of the following is an irrational number?
14.287628762876.....
15.2323232323......
5.2731687143725186.....
1.33333333...
SOLUTION
Solution : C
Irrational numbers are defined as those numbers which have a non-terminating and non-repeating decimal expansion and hence cannot be represented in the pq form. So, here we see that out of the given options the number 5.2731687143725186..... has a non-terminating and non-repeating decimal expansion as clearly one can interpret that there are no patterns formed in the decimal expansion and the number goes on expanding arbitrarily.
Question 6
The expression √5−√3√5+√3 when simplified reduces to a-b√15.
Then a + b is __________.
15
10
20
5
SOLUTION
Solution : D
In √5−√3√5+√3, the denominator has to be rationalised for simplification. So ,
√5−√3√5+√3×√5−√3√5−√3
=(√5−√3)2√52−√32
=12[(√5)2+(√3)2−2√15]
=4−√15
So, a=4 and b=1.
∴a + b = 4 + 1 = 5
Question 7
SOLUTION
Solution :The integers include positive integers i.e. 1, 2, 3, ... , negative integers i.e. -1, -2, -3, ... and zero. Hence, zero is neither a positive nor a negative integer.
Question 8
Find the value of (√2+√2)(√2−√2).
1
SOLUTION
Solution : A
A common tendency to solve this question is to apply the algebraic identity (a2−b2)=(a+b)(a−b). But by observation, we can see that (√2−√2), which is one of the factors, will result in 0, thus making the final value of the whole expression as 0 i.e.
(√2+√2)(√2−√2)
=(√2+√2)×(0)
=0
Question 9
If 14.287628762876........ can be represented in pq form, then 'p - q ' is ________.
132863
142862
142876
142863
SOLUTION
Solution : A
Let 'x' = 14.287628762876 ------- (i)
then 10000x = 142876.287628762876 --------(ii)
Subtracting (i) from (ii) we get
10000x - x = 142876.287628762876 - 14.287628762876
9999x = 142862
then 'x' = 1428629999
p = 142862, q = 9999
So, p-q = 132863
Question 10
1417×1417 can be represented as ___________.
1417+17
(142)17
(14×14)17
All of the above
SOLUTION
Solution : D
For any positive real number 'a' and integers 'm' and 'n', we define
1. am×an=am+n
2. (am)n=amn
3. (ab)m=ambm and if a=b, the expression becomes (a×a)m=(a2)m=a2m.
So, using first rule, we have
1417+17=1427
Now, using second rule,
(142)17= 1427
Finally, using third rule,
(14×14)17 =1427
Hence, we see that all the three results are same.