Q.1. Which of the following is an irrational number?
(a)
(b) √3
(c) 1/2
(d)
Ans.
An irrational number is a number that cannot be expressed as a fraction of two integers and has an infinite nonrepeating decimal representation. Let's evaluate the options:
(a) √49/64: This is a rational number because both the numerator and denominator are perfect squares, and their square root can be expressed as a fraction of integers. √49/64 = 7/8.
(b) √3: This is an irrational number because the square root of 3 cannot be expressed as a fraction of integers, and its decimal representation goes on infinitely without repeating.
(c) 1/2: This is a rational number because it can be expressed as a fraction of integers.
(d) √1/4: This is a rational number because √1/4 = 1/2, and the negative sign only changes the sign of the rational number.
So, the irrational number among the given options is: (b) √3
Q.2. The numberin p/q form is
(a) 267/1000
(b) 26/10
(c) 241/900
(d) 241/999
Ans. (c)
Solution:
let x be the p/q form, x =
multiply both side by 100,
100 x = ...(i)
multiply both side by 10
1000 x = ....(ii)
Subtract (ii)  (i)
1000 x  100 x =
900 x = 241
⇒ x = 241/900
Hence, option (c) is correct
Q.3. Every point on the number line represents, which of the following numbers?
(a) Natural numbers
(b) Irrational number
(c) Rational number
(d) Real number
Ans.
Every point on the number line represents a: Real number
The number line includes all types of numbers, such as natural numbers, whole numbers, integers, rational numbers (including fractions), irrational numbers (like the square root of 2 or π), and even imaginary numbers. All these numbers together form the set of real numbers, which is represented on the number line. Therefore the correct option is (d).
Q.4. Which of the following is not a surd ?
(a) √6
(b) √7
(c) ∛343
(d) √11
Ans.
A surd is an irrational number that is expressed as the root of a nonperfect square or a nonperfect cube. An irrational number cannot be expressed as a fraction or a repeating decimal.
Let's go through each option again:
(a) √6: This is a surd because 6 is not a perfect square, and √6 is an irrational number.
(b) √7: This is a surd because 7 is not a perfect square, and √7 is an irrational number.
(c) ∛343: This is not a surd. 343 is a perfect cube because it can be expressed as 7^3, and ∛343 = 7, which is a rational number.
(d) √11: This is a surd because 11 is not a perfect square, and √11 is an irrational number.
So, the correct answer is: (c) ∛343
Q.5. Show that 3√5 is an irrational number.
Ans.
Let us assume 3√5 is a rational number,
3√5 = p/q, where p and q are coprimes,
√5 = p/3q
Clearly, √5 is irrational, while number on right q ≠ 0 are rational
∴ Irrational = Rational
But above deduced can't be right. Therefore our supposition is wrong making 3√5 an irrational number.
Q.6. What is the decimal form of the following no's.
(a) 18/11
(b) 3/26
(c) 1/17
(d) 2/13
Ans.
(a) 18/11 = 1.63636363...
(b) 3/26 = 0.11538461538
(c) 1/17 = 0.05882352941
(d) 2/13 = 0.15384615384
Q.7. The conjugate pair of 2 + √3 is
(a) 2  √3
(b) 2 + √3
(c) 2√3
(d) 4  √3
Ans.
To find the conjugate pair of the given number, 2 + √3, we need to change the sign of the square root term (√3). The conjugate pair is obtained by replacing the "+" with "" or vice versa. So, the conjugate pair of 2 + √3 is 2  √3.
Q.8. Simplify:
Ans.
Q.9. Rationalise:
Ans.
Q.10. Find the value of
Ans.
= 5+4  4√5  5  4  4√5 = 8√5
Q.11. If ,
find the value of a & b.
Ans.
Rationalising LHS
∴ a = 11/7 and b = 6/7
Q.12. Evaluate:
Ans.
Q.13. If ,
find the value of x^{3}  2x^{2}  7x+5
Ans.
x = 2 + √3
x^{3}  2x^{2}  7x + 5
= x(x^{2 } 2x  7) + 5
= (2 + √3) [(2 + √3)^{2}  2(2 + √3)  7 ]+5
= (2 + √3) [4 + 3+4√3  4 + 2√3  7]+5
= (2 + √3)(4 + 2√3) + 5
= (2 + √3) x 2(2+√3)+5
= 2[(√3 + 2)(√3  2)] + 5
= 2[(√3)^{2 } 2^{2}^{ }]+5
= 2[34]+5
= 2(1) + 5
= 2+5
= 3
Q.14. Express in p/q form.
Ans.
let x be the p/q form,
so, x =
10x =
1000x =
1000x  10x = 
990x = 15555
x= 15555/990
= 1037/66
Q.15. Insert five rational no's between 3/5 and 4/5.
Ans.
3/5 and 4/5
30/50 and 40/50
∴ pick any five number between 30 and 40
31/50, 32/50, 36/50, 37/50, 39/50
Q.16. Insert 3 irrational number between 2.6 and 3.8
Ans.
2.6 and 3.8
irrational numbers are non repeating non  terminating
2.61010010001.....
2.802002000200002......
3.604004000400004.......
Q.17. If 27^{x} = 9/3^{x }, find x
Ans.
27^{x} = 9/3^{x}
⇒ 3^{3x} = (3)^{2}/(3)^{x}
⇒ 3^{3x} = 3^{2x}
⇒ 3x = 2x
⇒ 4x = 2
⇒ x = 4/2
⇒ x = 0.5
Q.18. Write the value of
Ans.
= 15
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HOTS Questions: Number System Doc  1 pages 
Short Question Answers: Number System Doc  3 pages 
Detailed Explanation: Real Numbers Video  62:04 min 
1. What is the number system? 
2. What are the different types of number systems? 
3. How does the binary number system work? 
4. What is the significance of the decimal number system? 
5. How is the hexadecimal number system used in computing? 
48 videos387 docs65 tests

HOTS Questions: Number System Doc  1 pages 
Short Question Answers: Number System Doc  3 pages 
Detailed Explanation: Real Numbers Video  62:04 min 

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