Class 10 – Mathematics
Chapter 1: Real Numbers
Time: 50 Minutes | Maximum Marks: 35
SECTION A – MCQs (1 × 8 = 8 marks)
1️⃣ If 210 = 21⋅31⋅51⋅712^1 \cdot 3^1 \cdot 5^1 \cdot 7^121⋅31⋅51⋅71, the number of prime factors is:
A) 4 B) 3 C) 5 D) 2
2️⃣ HCF of 154 and 252 using Euclid’s division algorithm:
A) 11 B) 7 C) 21 D) 14
3️⃣ The decimal representation of 1122⋅53\frac{11}{2^2 \cdot 5^3}22⋅5311 is:
A) Terminating B) Non-terminating C) Irrational D) None
4️⃣ Which one is irrational?
A) 0.44444… B) 1321\frac{13}{21}2113 C) 3\sqrt{3}3 D) 5.25
5️⃣ The LCM of 9 and 15 is:
A) 45 B) 5 C) 135 D) 3
6️⃣ If p is prime and p divides a2a^2a2, then p divides:
A) a B) a+1 C) 2a D) a–1
7️⃣ The product 24⋅562^4 \cdot 5^624⋅56 ends with how many zeroes?
A) 4 B) 6 C) 5 D) 8
8️⃣ 75+12=\sqrt{75} + \sqrt{12} =75+12=
A) 11311\sqrt{3}113 B) 737\sqrt{3}73 C) 535\sqrt{3}53 D) 333\sqrt{3}33
SECTION B – Very Short Answer (2 × 4 = 8 marks)
9️⃣ Without actual division, determine whether
1323⋅3⋅5\frac{13}{2^3 \cdot 3 \cdot 5}23⋅3⋅513 is terminating or not.
🔟 Show that 11\sqrt{11}11 is irrational.
1️⃣1️⃣ Use Euclid’s division algorithm to find the HCF of 414 and 252.
1️⃣2️⃣ Express 441004410044100 in prime factorization.
SECTION C – Short Answer (3 × 3 = 9 marks)
1️⃣3️⃣ Find HCF and LCM of 144 and 160 using prime factorization.
1️⃣4️⃣ Find number of trailing zeroes in
75⋅16⋅40⋅12575 \cdot 16 \cdot 40 \cdot 12575⋅16⋅40⋅125
1️⃣5️⃣ Express 0.135‾0.\overline{135}0.135 in the form pq\frac{p}{q}qp.
SECTION D – Case Study (4 marks)
🍕 Pizza Slices Distribution
A restaurant has 420 cheese slices and 378 veggie slices.
They want to make boxes containing only one type of slice
and maximum slices per box.
Answer:
12(a) What will be the maximum slices per box?
12(b) How many boxes for each?
12(c) Which concept used — HCF or LCM?
12(d) Write prime factorization of both.
SECTION E – HOTS (3 marks)
1️⃣7️⃣ Prove that 2+32 + \sqrt{3}2+3 is irrational.
SECTION F – Long Answer (5 marks)
1️⃣8️⃣ A shopkeeper has 960 pens and 1320 pencils.
He wants to make largest identical kits, each having equal number of pens and pencils.
Find:
(a) Maximum items per kit
(b) Number of pens in each kit
(c) Number of pencils in each kit
(d) Total number of kits
(e) Which concept is used?
✨ ANSWER KEY – CHECK & SCORE
SECTION A ✔
1️⃣ A
2️⃣ C
3️⃣ A
4️⃣ C
5️⃣ A
6️⃣ A
7️⃣ C
8️⃣ B
SECTION B ✔
9️⃣ Non-terminating
(Contains prime factor 3 in denominator)
🔟 Proof by contradiction → irrational
1️⃣1️⃣ Euclid: HCF = 18
1️⃣2️⃣44100=22⋅32⋅52⋅7244100 = 2^2 \cdot 3^2 \cdot 5^2 \cdot 7^244100=22⋅32⋅52⋅72
SECTION C ✔
1️⃣3️⃣
HCF = 16, LCM = 1440
1️⃣4️⃣75=3⋅52,16=24,40=23⋅5,125=5375 = 3 \cdot 5^2,\quad 16 = 2^4,\quad 40 = 2^3 \cdot 5,\quad 125 = 5^375=3⋅52,16=24,40=23⋅5,125=53
Product = 27⋅3⋅562^7 \cdot 3 \cdot 5^627⋅3⋅56
Zeroes = min(7,6) = 6
1️⃣5️⃣
Let x=0.135‾x = 0.\overline{135}x=0.1351000x=135+x⇒999x=135⇒x=135999=5371000x = 135 + x \Rightarrow 999x = 135 \Rightarrow x = \frac{135}{999} = \frac{5}{37}1000x=135+x⇒999x=135⇒x=999135=375
SECTION D ✔
Cheese: 420=22⋅3⋅5⋅7420 = 2^2 \cdot 3 \cdot 5 \cdot 7420=22⋅3⋅5⋅7
Veggie: 378=2⋅33⋅7378 = 2 \cdot 3^3 \cdot 7378=2⋅33⋅7
12(a) HCF = 42
12(b) Cheese = 10 boxes, Veggie = 9 boxes
12(c) Concept: HCF
12(d) ✔ Factorization above
SECTION E ✔
1️⃣7️⃣ Assume rational → contradiction
✔ Irrational
SECTION F ✔
960 = 26⋅3⋅52^6 \cdot 3 \cdot 526⋅3⋅5
1320 = 23⋅3⋅5⋅112^3 \cdot 3 \cdot 5 \cdot 1123⋅3⋅5⋅11
HCF = 23⋅3⋅5=1202^3 \cdot 3 \cdot 5 = 12023⋅3⋅5=120
✔ Maximum = 120 items per kit
Pens per kit = 960÷120=8960 ÷ 120 = 8960÷120=8
Pencils per kit = 1320÷120=111320 ÷ 120 = 111320÷120=11
✔ Number of kits = 120
✔ Concept: HCF