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45:00

Unit Test 1: Number System

Class 10 Mathematics (Standard)

Time: 45 Minutes Max. Marks: 25

General Instructions:

  • The question paper consists of 12 questions divided into 5 sections: A, B, C, D, and E.
  • Section A: 5 questions (1-5) of 1 mark each.
  • Section B: 3 questions (6-8) of 2 marks each.
  • Section C: 2 questions (9-10) of 3 marks each.
  • Section D: 1 Long Answer question (11) of 4 marks.
  • Section E: 1 Case Study based question (12) of 4 marks.
  • Use of calculators is not permitted.
SECTION A (1 Mark Each)
[1] 1. If HCF(a, b) = 12 and a × b = 1800, then find LCM(a, b).
[1] 2. The exponent of 2 in the prime factorization of 144 is:
[1] 3. Which of the following numbers is irrational?
[1] 4. The product of a non-zero rational and an irrational number is:
[1] 5. Assertion (A): The number 5 × 7 × 11 + 11 is a composite number.
Reason (R): A composite number has factors other than 1 and itself.
SECTION B (2 Marks Each)
[2] 6. Find the LCM and HCF of 26 and 91 using prime factorization method.
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[2] 7. Check whether 6n can end with the digit 0 for any natural number n.
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[2] 8. Explain why 7 × 11 × 13 + 13 is a composite number.
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SECTION C (3 Marks Each)
[3] 9. Prove that √5 is an irrational number.
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[3] 10. There is a circular path around a sports field. Sonia takes 18 minutes to drive one round of the field, while Ravi takes 12 minutes for the same. Suppose they both start at the same point and at the same time, and go in the same direction. After how many minutes will they meet again at the starting point?
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SECTION D (4 Marks)
[4] 11. Prove that √3 is an irrational number. Hence, show that 5 + 2√3 is an irrational number.
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SECTION E (Case Study - 4 Marks)
[4] 12. Case Study: Seminar Arrangement

A seminar is being conducted by an Educational Organisation, where the participants will be educators of different subjects. The number of participants in Hindi, English, and Mathematics are 60, 84, and 108 respectively.

Based on the above information, answer the following questions:

(i) Find the prime factorization of 60, 84, and 108. (1 Mark)
(ii) In each room, the same number of participants are to be seated and all of them being in the same subject. Find the maximum number of participants that can be accommodated in each room. (2 Marks)
(iii) What is the minimum number of rooms required during the event? (1 Mark)
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TOTAL SCORE

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Solution Key

Compare your answers with the solutions below and check the box if you got it right to update your score.

Q1. Answer: (b) 150

LCM × HCF = Product of numbers. LCM = 1800 / 12 = 150.

Q2. Answer: (a) 4

144 = 12² = (2² × 3)² = 2⁴ × 3². Exponent of 2 is 4.

Q3. Answer: (c) √5

√16=4, 3+√9=6, √2.25=1.5 are all rational. √5 is irrational.

Q4. Answer: (a) Always irrational

Product of non-zero rational and irrational is irrational.

Q5. Answer: (a) Both A and R are true...

5×7×11 + 11 = 11(35+1) = 11×36. It has factors, so it is composite. Reason is correct definition.

Q6. LCM & HCF

26 = 2×13, 91 = 7×13. HCF = 13. LCM = 2×7×13 = 182.

Q7. Check 6n

6n = (2×3)n = 2n × 3n. Since the prime factorization does not contain the pair (2, 5), it cannot end with the digit 0.

Q8. Composite Number

7×11×13 + 13 = 13(7×11 + 1) = 13(78). Since it has factors other than 1 and itself, it is composite.

Q9. Irrationality of √5

Assume √5 = a/b (coprime). a² = 5b² ⇒ 5 divides a. Let a=5c, then 25c²=5b² ⇒ b²=5c² ⇒ 5 divides b. Contradiction. Hence irrational.

Q10. Circular Path

They meet at LCM(18, 12). 18=2×3², 12=2²×3. LCM = 2²×3² = 36 minutes.

Q11. Irrationality of 5 + 2√3

First prove √3 is irrational (standard proof). Then assume 5 + 2√3 = r (rational). √3 = (r-5)/2. LHS is irrational, RHS is rational. Contradiction.

Q12. Case Study

(i) 60=2²×3×5, 84=2²×3×7, 108=2²×3³. (ii) HCF=12 participants/room. (iii) Total rooms = (60+84+108)/12 = 21.