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Unit Test (Solutions): Real Numbers | Mathematics (Maths) Class 10 PDF Download

Time: 1 hour

M.M. 30

Attempt all questions.

  • Question numbers 1 to 5 carry 1 mark each.
  • Question numbers 6 to 8 carry 2 marks each.
  • Question numbers  9 to 11 carry 3 marks each.
  • Question number 12 & 13 carry 5 marks each.

Q1: Which of the following numbers is irrational?  (1 Mark)  
(a) √25
(b) 3.14
(c) 0.333...
(d) -7

Ans: (c)
An irrational number is a number that cannot be expressed as a fraction of two integers. Option c) 0.333... is an example of an irrational number as it represents a non-repeating and non-terminating decimal (1/3), making it irrational.

Q2: What is the value of (5² + 12²)?  (1 Mark)  
(a) 169
(b) 144
(c) 25
(d) 169√2

Ans: (a)
The given expression is (5² + 12²) = (25 + 144) = 169.

Q3: Which one is not a prime number?  (1 Mark)  
(a) 1
(b) 2
(c) 3
(d) 5

Ans: (a)
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. However, 1 does not meet this criteria, as it only has one positive divisor.

Q4: State whether "√16" is a rational number or not.  (1 Mark)  
Ans:  
"√16" is a rational number.
√16 = 4, which is a rational number since it can be expressed as the fraction 4/1.

Q5: Without performing the actual division, state whether "1089" is divisible by "9" or not.  (1 Mark)
Ans: 
"1089" is divisible by "9".
A number is divisible by 9 if the sum of its digits is divisible by 9. Here, 1 + 0 + 8 + 9 = 18, and 18 is divisible by 9.

Q6: Write a rational number between √5 and √6.
Ans: √5 = 2.236… and √6 = 2.449…
∴ A rational no between 2.24 and 2.44 (approximately) is 2.3 or 2.31 or 2.32 etc.
Note: Take the lower limit slightly greater than √5 and upper limit slightly lesser than √6 .
⇒ One number between √5 and √6 = 2.3

Q7: Express 0.37 as a fraction in its simplest form.  (2 Marks)
Ans: 
0.37 can be expressed as the fraction 37/100.
To convert a decimal to a fraction, we remove the decimal point and place the digits after the decimal over the appropriate place value (in this case, 37/100). We simplify the fraction, but in this case, it is already in its simplest form.

Q8: Find the LCM (Least Common Multiple) of 15 and 20.  (2 Marks)
Ans: 
The LCM of 15 and 20 is 60.
To find the LCM, we can use the prime factorization method. The prime factorization of 15 is 3 x 5, and the prime factorization of 20 is 2 x 2 x 5. The LCM is the product of the highest powers of all the prime factors involved, which is 2 x 2 x 3 x 5 = 60.

Q9: Prove that the square of any positive integer of the form (5k + 1) is one more than a multiple of 8, where "k" is an integer.  (3 Marks) 
Ans: Let's assume the positive integer be "n" in the form of (5k + 1).
Step 1: Square of "n"
n² = (5k + 1)² = 25k² + 10k + 1 = 5(5k² + 2k) + 1
Step 2: Express (5k² + 2k) as an integer "m"

Let (5k² + 2k) = m (where m is an integer)
Step 3: Express n² in terms of "m"
n² = 5m + 1
Step 4: Prove that n² is one more than a multiple of 8

n² = 5m + 1 = 8k + (5m - 8k + 1)
Since (5m - 8k + 1) is an integer, let's say it equals "p"
n² = 8k + p
Thus, n² is one more than a multiple of 8.

Q10: Find the HCF (Highest Common Factor) of 72 and 96 using the prime factorization method.  (3 Marks) 
Ans: The HCF of 72 and 96 is 24.
To find the HCF, we can use the prime factorization method.
Prime factorization of 72: 72 = 2³ x 3²
Prime factorization of 96: 96 = 2⁵ x 3¹
HCF = Product of the common prime factors with their lowest powers
HCF = 2³ x 3¹ = 8 x 3 = 24

Q11: Given that p is a rational number and q is an irrational number, prove that their sum (p + q) is an irrational number.  (3 Marks) 
Ans: Let's assume p + q = r, where r is a rational number (to reach a contradiction).
Since p is rational, it can be represented as p = a/b, where "a" and "b" are integers and b ≠ 0.
Then, q = r - p
q = r - (a/b)
Now, as q is irrational and r is rational, let's assume r = c/d, where "c" and "d" are integers and d ≠ 0.
So, q = (c/d) - (a/b)
q = (bc - ad)/(bd)
Since both bc and ad are integers (the product of two integers is an integer), let's assume (bc - ad) = x, where "x" is an integer.
q = x/(bd)
Now, q can be expressed as a fraction of two integers "x" and "bd," making it rational. However, this contradicts our assumption that q is irrational.
Therefore, our assumption that r is rational is incorrect. Hence, the sum (p + q) must be irrational.

Q12: A circular garden has a circumference of 44 meters. Find the area of the garden (in square meters) using the formula for the area of a circle. (Take π = 3.14).  (5 Marks) 

Ans: Given the circumference of the circular garden, C = 44 meters.
We know the formula for the circumference of a circle is C = 2πr, where "r" is the radius.
Substitute the given value of the circumference:
44 = 2πr
Now, we need to find the radius "r" first:
r = 44/(2π)
r ≈ 7 meters
Now, the area of the circle can be calculated using the formula A = πr²:
A ≈ 3.14 x (7)²
A ≈ 3.14 x 49
A ≈ 153.86 square meters

Q13: Prove that 5√3 - 3√75 is an irrational number.  (5 Marks)  
Ans:
To prove that 5√3 - 3√75 is an irrational number, we assume the contrary, i.e., let's assume 5√3 - 3√75 is a rational number. So, it can be expressed as 5√3 - 3√75 = p/q, where p and q are co-prime integers (i.e., they have no common factors other than 1) and q ≠ 0.
Now, let's work on simplifying the expression:
5√3 - 3√75
Step 1: Factorize the numbers inside the radicals.
√3 cannot be simplified further as it is a prime number.
√75 = √(5 * 5 * 3) = 5√3
Step 2: Substitute the factorized value back into the expression.
5√3 - 3√75 = 5√3 - 3 * 5√3
Step 3: Combine like terms.
5√3 - 3√75 = (5 - 3)√3 = 2√3
Now, let's express 2√3 as a rational number:
2√3 = p/q
Squaring both sides, we get:
4 * 3 = (p/q)2
12 = p2 / q2
From the above equation, we can see that p2 is a multiple of 12, which means p must also be a multiple of 12 (since 12 is not a prime number). Let's write p as p = 12k, where k is an integer.
Substituting the value of p back into our equation, we get:
12 = (12k)2 / q2
12 = 144k2  / q2
q^2 = 144k2 / 12
q^2 = 12k2
Now, we see that qis also a multiple of 12, which implies that q must also be a multiple of 12.
But this contradicts our initial assumption that p and q are co-prime (i.e., they have no common factors other than 1) because both p and q are divisible by 12. Hence, our initial assumption that 5√3 - 3√75 is rational is incorrect. Therefore, 5√3 - 3√75 must be an irrational number.

The document Unit Test (Solutions): Real Numbers | Mathematics (Maths) Class 10 is a part of the Class 10 Course Mathematics (Maths) Class 10.
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FAQs on Unit Test (Solutions): Real Numbers - Mathematics (Maths) Class 10

1. What are real numbers and how are they classified?
Ans. Real numbers are the set of numbers that include all the rational and irrational numbers. They can be classified into various categories: natural numbers (1, 2, 3, ...), whole numbers (0, 1, 2, ...), integers (..., -2, -1, 0, 1, 2, ...), rational numbers (fractions like 1/2, 3/4), and irrational numbers (numbers that cannot be expressed as a fraction, like √2 and π).
2. How do you perform operations with real numbers?
Ans. Operations with real numbers include addition, subtraction, multiplication, and division. To perform these operations, you follow the standard arithmetic rules. For example, when adding or subtracting, you combine like terms; for multiplication, you use the distributive property if necessary; and for division, you ensure the divisor is not zero.
3. What is the difference between rational and irrational numbers?
Ans. Rational numbers are those that can be expressed as the quotient of two integers (where the denominator is not zero), such as 1/2 or -3. Irrational numbers cannot be expressed as a simple fraction; they have non-repeating, non-terminating decimal expansions, such as √3 or π.
4. Can real numbers be negative?
Ans. Yes, real numbers can be negative. The set of real numbers includes negative integers, negative fractions, and negative irrational numbers. For example, -1, -1/2, and -√2 are all examples of negative real numbers.
5. How do you represent real numbers on a number line?
Ans. Real numbers are represented on a number line as points corresponding to their value. The number line is a straight line where each point represents a real number. Positive numbers are placed to the right of zero, while negative numbers are placed to the left. The distance between points indicates the value of the numbers.
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