Class 8 SESSION ENDING EXAMINATION (2026) SAMPLE PAPER-4 WITH ANSWERS

 Class 8 SESSION ENDING EXAMINATION (2026)  SAMPLE PAPER-4 WITH ANSWERS

 SAMPLE Paper 4

SECTION A

1. Digital root of 34761097: a) 37 b) 10 c) 1 d) 91

2. True proportion: a) 4:7::12:21 b) 8:3::24:6 c) 24:8::9:3 d) 7:12::12:71

3. How many m² in 1 km²: a) 100 b) 1,000 c) 10,000 d) 1,000,000

4. Nandini: 25 marbles, 15 white → % white: a) 10% b) 15% c) 60% d) 25%

5. 672 divisible by: a) 3&5 b) 2,3,4 c) 2&3 only d) 2&5

6. Algebraic expression for "sum of cube of two integers": a) 3+x b) 5+x³ c) x³+27 d) (x+1)³

7. Area of rhombus: a) base×height b) ½×base×height c) ½×product of diagonals d) ½×height×sum of parallel sides

8. Primitive Pythagorean triple: a) 2,3,4 b) 3,4,5 c) 9,12,15 d) 18,24,30

9. 100% of m=215 → 10% of m = ? a) 10 b) 2.15 c) 21.5 d) 21

10. Square of odd number always 1 more than multiple of 8: a) always true b) sometimes c) never d) can't say

11. Ratio proportional to 54:81: a) 4:8 b) 12:18 c) 16:27 d) 20:25

12. p invested 4 years at 6% p.a. no compounding → total amount expression: a) p×6×4 b) p×0.6×4 c) p+(p×0.06×4) d) p×1.6×4

13. Diagonal of square side 5 cm: a) 10 cm b) 25 cm² c) 5√2 cm d) 2√5 cm

Assertion-Reason (14-15)
14. A: (a+b)²−(a−b)² always divisible by 4 R: Simplifies to 4ab
15. A: Price increases 20% then decreases 20% → same R: Increase & decrease by same % cancel

SECTION B (2 marks)
16. Middle of 5 consecutive even numbers=5p → express all in terms of p.
17. Find missing side in right triangle (diagram).
18. 104² using identity OR (3a−9b)(3a+9b)
19. Sum of 4 consecutive numbers=34 → find them.
20. Fill: i) 3 is ___% of 300 ii) ___ is 40% of 4 OR Shirt ₹300, 25% discount → Mohan pays?

SECTION C (3 marks)
21. Midday meal: 120 students→15 kg rice, 80 students→? kg OR Car: 48 L→432 km, 20 L→? km.
22. Side of rhombus diagonals 24 units & 70 units.
23. SP of 3 pencils = CP of 5 pencils → profit/loss %?
24. In parallelogram PQRS, find QN (diagram).
25. Smallest number remainder 2 when ÷3, remainder 4 when ÷5.

SECTION D (4 marks)
26. Pattern: i) Draw next figure ii) Basic units in Step 10 iii) Expression for Step y.
27. Mixture 40 kg sand:cement=3:1 → add cement to make ratio 5:2.
28. Lotus stem tip 1 unit above water, bends to touch water 3 units away → depth of lake OR Generate 2 scaled Pythagorean triples from (5,12,13) and 2 more using (n−1)²+(2n−1)=n².

SECTION E (Case-Based)
29. Three shops: A: Buy1 Get1, B: Buy2 Get1, C: Buy3 Get1. Price ₹100/item.
i) Effective price per item in A
ii) % discount each shop
iii) Arrange cheapest to costliest
30. Rectangular path around garden:
a) Name two rectangles used
b) Formula for path area
c) If garden 20m×12m, path width 2m → area of path

d) Does path area change if outer rectangle shifted?

SAMPLE Paper 4 – Complete Solutions with Explanations

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Section A – MCQs with Explanations

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Q1: Digital root of 34761097?
a) 37 b) 10 c) 1 d) 91
✅ Answer: c) 1
Explanation: Sum digits: 3+4+7+6+1+0+9+7=37 → 3+7=10 → 1+0=1.

Q2: Which proportion is true?
a) 4:7 :: 12:21 b) 8:3 :: 24:6 c) 24:8 :: 9:3 d) 7:12 :: 12:71
✅ Answer: a) 4:7 :: 12:21
Explanation: Cross-check: 4×21=84, 7×12=84 ✔.

Q3: How many m² in 1 km²?
a) 100 b) 1,000 c) 10,000 d) 10,00,000
✅ Answer: d) 10,00,000
Explanation: 1 km² = 1,000,000 m².

Q4: Nandini has 25 marbles, 15 white → % white?
a) 10% b) 15% c) 60% d) 25%
✅ Answer: c) 60%
Explanation: 15/25 = 0.6 = 60%.

Q5: 672 divisible by:
a) 3 and 5 b) 2,3, and 4 c) 2 and 3 only d) 2 and 5
✅ Answer: b) 2,3, and 4
Explanation: 672 even → divisible by 2. Sum digits=15 → divisible by 3. Last two digits 72 → divisible by 4.

Q6: Algebraic expression for “sum of cube of two integers”?
a) 3+x b) 5+x³ c) x³+27 d) (x+1)³
✅ Answer: None match exactly — but likely c) x³+27 (if integers are x and 3). But best generic: a³+b³. Not in options.

Q7: Area of rhombus is:
a) Base×height b) ½×base×height c) ½×product of diagonals d) ½×height×sum of parallel sides
✅ Answer: c) ½×product of diagonals
Explanation: Standard formula.

Q8: Which is primitive Baudhayana (Pythagorean) triplet?
a) 2,3,4 b) 3,4,5 c) 9,12,15 d) 18,24,30
✅ Answer: b) 3,4,5
Explanation: Primitive means coprime: 3,4,5 have HCF=1.

Q9: 100% of m = 215 → 10% of m = ?
a) 10 b) 2.15 c) 21.5 d) 21
✅ Answer: c) 21.5
Explanation: m=215, 10% = 21.5.

Q10: Square of any odd number is always 1 more than multiple of 8:
a) Always true b) sometimes true c) never true d) can’t say
✅ Answer: a) Always true
Explanation: Odd number = 2k+1 → (2k+1)² = 4k²+4k+1 = 4k(k+1)+1, k(k+1) even → 8m+1.

Q11: Ratio proportional to 54:81?
a) 4:8 b) 12:18 c) 16:27 d) 20:25
✅ Answer: b) 12:18
Explanation: 54:81 = 2:3, 12:18 = 2:3 ✔.

Q12: p invested for 4 years at 6% p.a., no compounding → total amount?
a) p×6×4 b) p×0.6×4 c) p+(p×0.06×4) d) p×1.6×4
✅ Answer: c) p+(p×0.06×4)
Explanation: Simple interest formula.

Q13: Diagonal of square side 5 cm:
a) 10 cm b) 25 sq cm c) 5√2 cm d) 2√5 cm
✅ Answer: c) 5√2 cm
Explanation: Diagonal = side√2.

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Assertion-Reason

Q14:
A: (a+b)² − (a−b)² always divisible by 4.
R: Expression simplifies to 4ab.
✅ Answer: (a) Both true, R explains A
Explanation: (a+b)²−(a−b)² = 4ab, which is divisible by 4.

Q15:
A: Price increases 20% then decreases 20% → final price same.
R: Increase and decrease by same % cancel each other.
✅ Answer: (d) A false, R false
Explanation: If original=100, after +20% →120, after −20% →96 (not same). They don’t cancel multiplicatively.

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Section B – Short Answers (2 marks)

Q16: Middle of 5 consecutive even numbers = 5p. Express others in terms of p.
✅ Answer: Let numbers: 5p−4, 5p−2, 5p, 5p+2, 5p+4.

Q17: Find missing side in right triangle. Diagram needed.
If sides given, use Pythagoras.

Q18: Compute 104² using identity.
✅ Answer: 104² = (100+4)² = 10000+800+16 = 10,816.
OR (3a−9b)(3a+9b) = 9a²−81b².

Q19: Sum of four consecutive numbers = 34 → numbers?
✅ Answer: Let n, n+1, n+2, n+3 → 4n+6=34 → n=7 → 7,8,9,10.

Q20(i): 3 is ___% of 300.
✅ Answer: (3/300)×100 = 1%.
(ii): ___ is 40% of 4.
✅ Answer: 0.4×4 = 1.6.
OR Shirt ₹300, 25% discount → Mohan pays 75% of 300 = ₹225.

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Section C – Short Answers (3 marks)

Q21: Midday meal: 120 students → 15 kg rice. 80 students → ? kg rice.
✅ Answer: Proportion: 15/120 = x/80 → x = (15×80)/120 = 10 kg.
OR Car: 48 L → 432 km, 20 L → ? km.
Fuel efficiency = 432/48 = 9 km/L → 20 L → 180 km.

Q22: Side of rhombus with diagonals 24 units and 70 units.
✅ Answer: Side = √[(24/2)² + (70/2)²] = √(12²+35²) = √1369 = 37 units.

Q23: SP of 3 pencils = CP of 5 pencils → profit/loss %?
✅ Answer: Let CP per pencil = ₹1, CP of 5=₹5=SP of 3 → SP per pencil=5/3≈1.67.
Profit % = [(5/3 − 1)/1]×100 = (2/3)×100 ≈ 66.67%.

Q24: In parallelogram PQRS, find QN? Diagram needed. If heights given, use area equality.

Q25: Smallest number leaves remainder 2 when ÷3, remainder 4 when ÷5.
✅ Answer: Let N = 3a+2 = 5b+4. Test:
3a+2 = 5b+4 → 3a−5b=2.
Try b=1 → 3a=7 no. b=2 → 3a=12 → a=4 → N=14.
Check: 14÷3=4 rem 2, 14÷5=2 rem 4 ✔. Smallest is 14.

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Section D – Long Answers (4 marks)

Q25 (Pattern):
(i) Draw next figure: likely growing pattern of squares/triangles.
(ii) Basic units in Step 10: if Step n has n²+something, plug n=10.
(iii) Expression for Step y: e.g., y²+2y.

Q26: Mixture 40 kg sand:cement = 3:1 → sand=30 kg, cement=10 kg. Want ratio 5:2 after adding cement.
Let x kg cement added. New cement = 10+x, sand=30.
Ratio 30/(10+x) = 5/2 → 60 = 50+5x → 5x=10 → x=2 kg.

Q27: Lotus stem tip 1 unit above water, bends to touch water 3 units away horizontally → depth?
✅ Answer: Let depth = d, stem length = d+1.
Pythagoras: (d+1)² = d² + 3² → d²+2d+1 = d²+9 → 2d=8 → d=4 units.
OR
(a) Scaled triple from (5,12,13): multiply by k=2 → (10,24,26), k=3 → (15,36,39).
(b) Using (n−1)²+(2n−1)=n²: Choose n=5 → 4²+9=25 → triple (4,3,5)?? Check: (n−1, 2n−1, n)?? Actually 2n−1 odd. For n=5: (4,9,√97) not integer triple. Maybe misprint.

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Section E – Case-Based (4 marks)

Q28: Shop deals:
A: Buy 1 get 1 free → effective price per item = half.
B: Buy 2 get 1 free → effective price per item = (2×price)/3.
C: Buy 3 get 1 free → effective price per item = (3×price)/4.
Price = ₹100.

(i) Shop A effective price = ₹50.
(ii) % discount:
A: discount 50%.
B: pay 200 for 3 → per item 66.67 → discount 33.33%.
C: pay 300 for 4 → per item 75 → discount 25%.
(iii) Cheapest to costliest: A (50), B (~66.67), C (75).

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Q30: Rectangular path around garden:
a) Two rectangles: ABCD (outer), EFGH (inner).
b) Area of path = Area(ABCD) − Area(EFGH).
c) If length=20 m, breadth=12 m, path width=2 m → outer: 24×16 m.
Area path = (24×16)−(20×12) = 384−240 = 144 m².
d) Area of path changes if outer rectangle shifts? Yes, if dimensions change.

Class 8 SESSION ENDING EXAMINATION (2026) SAMPLE PAPER-3 WITH ANSWERS

 Class 8 SESSION ENDING EXAMINATION (2026)  SAMPLE PAPER-3 WITH ANSWERS

 SAMPLE Paper 3

SECTION A

1. How many cm² is 1 m²: a) 100 b) 1000 c) 10000 d) 100000

2. Area of rhombus diagonals 12 cm and 14 cm: a) 64 b) 74 c) 84 d) 94

3. Simplest form of 48/72: a) 6/9 b) 4/6 c) 2/3 d) 12/18

4. Ratio of ₹5 to 50 paise: a) 1:10 b) 5:50 c) 1:1 d) 10:1

5. Which shows proportional relationship: a) Age & height b) Speed & distance (time constant) c) Marks & roll no d) Shoe size & marks

6. Sum of first (n−1) odd numbers: a) 3n b) n² c) n³ d) 4n

7. How many inches in 1 ft: a) 142 b) 145 c) 144 d) 146

8. If 3/5 = y/35, y = ? a) 15 b) 21 c) 25 d) 30

9. Equation aⁿ+bⁿ=cⁿ has no solution in positive integers when: a) n=2 b) n>2 c) n<2 d) n>0

10. Sum of first 10 natural numbers: a) 45 b) 50 c) 55 d) 60

11. Number divisible by 2 and 3: a) 15 b) 16 c) 18 d) 21

12. a(b+c)=ab+ac is: a) associative b) closure c) distributive d) all

13. Cost of 5 pencils=₹30, cost of 7 pencils: a) 6 b) 12 c) 14 d) 21

Assertion-Reason (14-15)
14. A: Product of two naturals always greater than both. R: Multiplication always increases value.
15. A: 5/10=1/2 R: Fraction unchanged when numerator/denominator divided by same nonzero number.

SECTION B (2 marks)
16. Diagonal of square side 5 cm.
17. Digital root of 8-digit number is 5. Digital root of number+10?
18. Compute 96² using identity OR Multiply 3874×101.
19. If 3a7b8 divisible by 44, list possible values of a,b.
20. TV bought at ₹21,000 depreciates 5% yearly → value after 2 years OR Shopkeeper sells pencils: SP of 3 = CP of 5 → profit/loss %?

SECTION C (3 marks)
21. Scooter: 120 km in 3h, train: 120 km in 2h → ratio of speeds OR Parking: 115 cycles, 75 scooters, 45 bikes → ratio cycles:total.
22. Isosceles right triangle hypotenuse=√72 → find other sides.
23. Manju: ₹12,500 at 12% p.a. simple for 3 years; Anju: same at 10% p.a. compounded annually → who pays more interest and by how much?
24. a) Simplify (am+bn)²−(am−bn)² b) If a+b=8, a−b=4 → find a²+b²
25. Find area of ΔXDC if AB=5 cm, AD=4 cm OR Area of trapezium parallel sides 6 in, 14 in, distance 10 in.

SECTION D (4 marks)
26. Side of rhombus diagonals 24 cm & 70 cm OR Area of equilateral triangle side 6 m.
27. Number remainder 3 when ÷7, another remainder 5 when ÷7 → remainder of sum ÷7 OR Newspapers: 312 total, 216 English → ratio English:Hindi & Hindi:Total.
28. Population reduces 10% per decade, current 1250 → after 4 decades.

SECTION E (Case-Based)
29. Hexagonal cake side 4 cm, 10 friends share equally → each gets? Perimeter?
30. Deepa earns ₹150,000/month, spends 1/10 on food, 1/4 on shopping, 1/5 of remaining on kids' education, rest saved.

a) Food expense b) Education expense c) Savings

SAMPLE Paper 3 – Complete Solutions with Explanations

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Section A – MCQs with Explanations

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Q1: How many cm² in 1 m²?
a) 100 b) 1000 c) 10000 d) 100000
✅ Answer: c) 10000
Explanation: 1 m² = (100 cm)² = 10000 cm².

Q2: Area of rhombus with diagonals 12 cm and 14 cm?
a) 64 b) 74 c) 84 d) 94
✅ Answer: c) 84
Explanation: Area = ½ × d₁ × d₂ = ½ × 12 × 14 = 84 cm².

Q3: Simplest form of 48/72?
a) 6/9 b) 4/6 c) 2/3 d) 12/18
✅ Answer: c) 2/3
Explanation: Divide numerator and denominator by 24 → 48/72 = 2/3.

Q4: Ratio of ₹5 to 50 paise?
a) 1:10 b) 5:50 c) 1:1 d) 10:1
✅ Answer: d) 10:1
Explanation: ₹5 = 500 paise. Ratio = 500:50 = 10:1.

Q5: Which shows proportional relationship?
a) Age and height b) Speed and distance (time constant) c) Marks and roll no d) Shoe size and marks
✅ Answer: b) Speed and distance (time constant)
Explanation: If time constant, distance ∝ speed (proportional).

Q6: Sum of first (n−1) odd numbers?
a) 3n b) n² c) n³ d) 4n
✅ Answer: b) n²
Explanation: Sum of first k odd numbers = k². Here k = n−1 → (n−1)². But options don’t match? Wait, they say first (n−1) odd numbers sum = (n−1)². But given options: n² is for first n odds. This seems mismatch. Likely answer: n² if they meant first n−1 odds sum = (n−1)² not in options, so maybe misprint.

Q7: How many inches in 1 ft?
a) 142 b) 145 c) 144 d) 146
✅ Answer: None match — actually 1 ft = 12 inches. But maybe they meant in cm? 1 ft = 30.48 cm ≈ 30.5 cm. Not matching. Possibly misprint.

Q8: If 3/5 = y/35, y = ?
a) 15 b) 21 c) 25 d) 30
✅ Answer: b) 21
Explanation: y = (3/5)×35 = 21.

Q9: Equation aⁿ + bⁿ = cⁿ has no solution in positive integers when:
a) n=2 b) n>2 c) n<2 d) n>0
✅ Answer: b) n>2
Explanation: Fermat’s Last Theorem: no integer solutions for n>2.

Q10: Sum of first 10 natural numbers?
a) 45 b) 50 c) 55 d) 60
✅ Answer: c) 55
Explanation: Sum = n(n+1)/2 = 10×11/2 = 55.

Q11: Number divisible by 2 and 3?
a) 15 b) 16 c) 18 d) 21
✅ Answer: c) 18
Explanation: Divisible by 6: 18/6=3.

Q12: a(b+c) = ab + ac → property?
a) associative b) closure c) distributive d) all
✅ Answer: c) distributive
Explanation: Distributive property of multiplication over addition.

Q13: Cost of 5 pencils = ₹30, cost of 7 pencils?
a) 6 b) 12 c) 14 d) 21
✅ Answer: c) 42 but not in options — recalc: Cost per pencil = 30/5=6, so 7 pencils = 42. Options wrong? They have 6,12,14,21. Possibly misprint.

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Assertion-Reason

Q14:
A: Product of two natural numbers always greater than both.
R: Multiplication always increases value.
✅ Answer: (d) A false, R true
Explanation: If one number is 1, product equals the other. So A false. R: multiplication by number >1 increases, but not always.

Q15:
A: 5/10 = 1/2
R: Fraction unchanged if numerator/denominator divided by same nonzero number.
✅ Answer: (a) Both true, R explains A
Explanation: 5/10 ÷ 5/5 = 1/2.

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Section B – Short Answers (2 marks)

Q16: Diagonal of square with side 5 cm.
✅ Answer: Diagonal = side√2 = 5√2 cm ≈ 7.07 cm.

Q17: Digital root of 8-digit number is 5. Digital root of number + 10?
✅ Answer: Adding 10 adds 1 to digital root sum → new digital root = 5+1 = 6 (since 10 ≡ 1 mod 9).

Q18: Compute 96² using identity.
✅ Answer: 96² = (100−4)² = 10000 − 800 + 16 = 9216.
OR Multiply 3874 × 101 = 3874 × (100+1) = 387400 + 3874 = 391,274.

Q19: If 3a7b8 divisible by 44, find possible a,b.
✅ Explanation: Divisible by 44 means divisible by 4 and 11.

• Divisible by 4: last two digits b8 divisible by 4 → b=0,2,4,6,8.

• Divisible by 11: (3+7+8) − (a+b) = multiple of 11 → 18 − (a+b) = 0 or 11 or… Try values.

Let’s solve: 18 − (a+b) = 0 ⇒ a+b=18 (max digit 9+9=18).
If 18 − (a+b) = 11 ⇒ a+b=7.
If 18 − (a+b) = −11 ⇒ a+b=29 (not possible).
So a+b=18 or 7. With b even from step 1.
Test b=0 → a=18 or 7 (a digit so ≤9) → a=7 works for a+b=7.
List pairs:
(a,b) = (7,0), (5,2), (3,4), (1,6), (8,8) for sum=7? Wait sum=7 cases:
(7,0),(5,2),(3,4),(1,6) yes.
Also sum=18: (9,9) but b=9 not even → no.
So possible: (7,0),(5,2),(3,4),(1,6).

Q20: TV bought ₹21,000, depreciates 5% yearly → value after 2 years?
✅ Answer: After 1 year: 21000×0.95 = ₹19,950.
After 2 years: 19950×0.95 = ₹18,952.5.
OR Profit/loss: SP of 3 pencils = CP of 5 pencils → Let CP per pencil = ₹1, CP of 5 = ₹5 = SP of 3 → SP per pencil = 5/3 ≈ 1.67 → Profit % = (0.67/1)×100 = 67%.

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Section C – Short Answers (3 marks)

Q21: Scooter: 120 km in 3h → speed 40 km/h.
Train: 120 km in 2h → speed 60 km/h.
Ratio of speeds = 40:60 = 2:3.
OR Parking: cycles=115, scooters=75, bikes=45, total=235.
Ratio cycles:total = 115:235 = 23:47.

Q22: Isosceles right triangle hypotenuse = √72 → other sides?
✅ Answer: side√2 = √72 = 6√2 → side = 6. So sides 6,6.

Q23: Manju: ₹12,500 at 12% p.a. simple interest for 3 years.
Anju: same at 10% p.a. compound annually.
Who pays more interest and by how much?
✅ Answer:
Manju SI = 12500×0.12×3 = ₹4500.
Anju CI: A = 12500×(1.1)³ = 12500×1.331 = ₹16,637.5 → Interest = 16637.5−12500=4137.5.
Manju pays more by 4500−4137.5=₹362.5.

Q24(a): Simplify (am + bn)² − (am − bn)².
✅ Answer: = (a²m²+2abmn+b²n²) − (a²m²−2abmn+b²n²) = 4abmn.

(b): If a+b=8, a−b=4 → find a²+b².
✅ Answer: Add: 2a=12 → a=6, b=2.
a²+b² = 36+4 = 40.

Q25: Given ΔXDC in figure: AB=5 cm, AD=4 cm. Find area ΔXDC.
Need diagram. Likely rectangle ABCD, X on BC, triangle base DC=5, height=4 → Area = ½×5×4 = 10 cm².
OR Trapezium: parallel sides 6 in, 14 in, distance 10 in → Area = ½×10×(6+14) = 5×20 = 100 in².

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Section D – Long Answers (4 marks)

Q26: Side of rhombus diagonals 24 cm and 70 cm.
✅ Answer: Side = √[(24/2)² + (70/2)²] = √(12²+35²) = √(144+1225) = √1369 = 37 cm.
OR Equilateral triangle side 6 m → height = 3√3 m, area = ½×6×3√3 = 9√3 m² ≈ 15.588 m².

Q27: Number remainder 3 when ÷7, another remainder 5 when ÷7. Remainder of sum ÷7?
✅ Answer: Sum remainder = (3+5) mod 7 = 8 mod 7 = 1.
OR Newspapers: English=216, Hindi=312−216=96.
Ratio English:Hindi = 216:96 = 9:4.
Ratio Hindi:Total = 96:312 = 8:26 = 4:13.

Q28: Population reduces 10% per decade. Current 1250 → after 4 decades?
✅ Answer: After each decade = 90% of previous. After 4 decades = 1250×(0.9)⁴ = 1250×0.6561 ≈ 820.

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Section E – Case-Based (4 marks)

Q29: Hexagonal cake side 4 cm, 10 friends, equal pieces. Quantity each? Perimeter?
✅ Answer:
Area of hexagon = (3√3/2)×side² = (3√3/2)×16 = 24√3 cm² ≈ 41.57 cm².
Each friend gets ≈ 4.157 cm².
Perimeter = 6×4 = 24 cm.

Q30: Deepa earns ₹150,000/month.
Spends:

• Food: 1/10 = ₹15,000.

• Shopping: 1/4 = ₹37,500.
  Remaining = 150,000 − (15,000+37,500) = ₹97,500.
  Education: 1/5 of remaining = ₹19,500.
  Savings = 97,500 − 19,500 = ₹78,000.

(a) Food = ₹15,000.
(b) Education = ₹19,500.
(c) Savings = ₹78,000.

Class 8 SESSION ENDING EXAMINATION (2026) SAMPLE PAPER-2 WITH ANSWERS

 Class 8 SESSION ENDING EXAMINATION (2026)  SAMPLE PAPER-2 WITH ANSWERS

 SAMPLE Paper 2

General Instructions: All questions compulsory. Sections A-F.

SECTION A (1 mark each)

1. Which statement of proportion is true:
   a) 4:7::12:21 b) 8:3::24:6 c) 2:3::5:7 d) 7:12::12:7

2. 1 Hectare =
   a) 10,000 m b) 10,000 sq m c) 1000 sq m d) 1 km

3. 9/20 in % is:
   a) 45% b) 90% c) 9% d) 80%

4. Nandini has 25 marbles, out of which 15 are white. What % is white:
   a) 15% b) 25% c) 100% d) 60%

5. Smallest multiple of 9 with no odd digits is:
   a) 288 b) 504 c) 902 d) 348

6. A number that has a remainder of 3 when divided by 5 is:
   a) 15 b) 9 c) 13 d) 2

7. If area of rectangle is 44 and breadth is 4, what is the length:
   a) 8 b) 11 c) 48 d) 40

8. Area of right triangle with perpendicular sides 3 and 4 is:
   a) 12 b) 7 c) 24 d) 6

9. Fill appropriate sign: 50/10 ___ 50%

10. 5w² + 6w =
    a) 11w² b) 11w c) 30w³ d) 5w²+6w

11. Area of isosceles triangle SUB is 48 sq m. What is area of triangle SEB? (diagram needed)

12. Two more than square a number:
    a) 2+s b) s²+2 c) (s+2)² d) 2s²

13. HCF of 72 and 96 is:
    a) 32 b) 24 c) 72 d) 2

Assertion-Reason Questions (14-15):
14. Assertion: (a−b)² = (b−a)² for any integers a,b
Reason: Square of any integer is positive
15. Assertion: Population of tigers increased by 5% last decade
Reason: Population now is 0.05m

SECTION B (2 marks each)
16. Sum of 4 consecutive numbers is 34. Find them.
17. What will be digital root of 9a + 36b + 13?
18. Express 100 as difference of two squares.
19. Maximum marks=75, need 80% for A grade. How much to score?
20. Is 10% of a day longer than 1% of a week? Justify.

SECTION C (3 marks each)
21. Anagh mixes 600 mL orange juice with 900 mL apple juice. Write ratio in simplest form.
22. Solve cryptarithm: MEOW (diagram needed)
23. In parallelogram, AM=12 cm, CN=8 cm, DC=10 cm. Find AD.
24. Samson bought car for ₹4,40,000 after 15% discount. Find original price.
25. Find side lengths of rhombus whose diagonals are 24 cm and 70 cm.

SECTION D (4 marks each)
26. Right triangle has sides 5 cm and 12 cm. Find hypotenuse by drawing and then verify by Pythagoras theorem.
27. Compute using identity: a) 43×45 using (a+b)(a−b) b) Earth travels 940 million km/year → km/week.

SECTION E (Case-Based)
29. Giridhar borrows ₹12,500 at 12% p.a. simple interest for 3 years; Raghu borrows same at 10% p.a. compounded annually.
a) Who pays more interest?
b) Difference in amount paid after 3 years?
c) Who incurs loss?

30. Varun's toy looks like two trapeziums + rectangle in middle (diagram).
    a) Height of trapezium
    b) Total area of toy
    c) Length and breadth of rectangle part

SAMPLE Paper 2 – Complete Solutions with Explanations

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Section A – MCQs with Explanations

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Q1: Which proportion is true?
a) 4:7 :: 12:21 b) 8:3 :: 24:6 c) 2:3 :: 5:7 d) 7:12 :: 12:7
✅ Answer: a) 4:7 :: 12:21
Explanation: Check cross products: 4×21 = 84, 7×12 = 84 → Equal → Proportional.

Q2: 1 Hectare =
a) 10,000 m b) 10,000 sq m c) 1000 sq m d) 1 km
✅ Answer: b) 10,000 sq m
Explanation: 1 hectare = 10,000 m².

Q3: 9/20 in % is
a) 45% b) 90% c) 9% d) 80%
✅ Answer: a) 45%
Explanation: 9/20 = 0.45 = 45%.

Q4: Nandini has 25 marbles, 15 white. % white?
a) 15% b) 25% c) 100% d) 60%
✅ Answer: d) 60%
Explanation: (15/25)×100 = 60%.

Q5: Smallest multiple of 9 with no odd digits is
a) 288 b) 504 c) 902 d) 348
✅ Answer: a) 288
Explanation: 288 digits: 2,8,8 (all even). Multiples of 9: sum of digits divisible by 9. 2+8+8=18 ✔. Check others: 504 has 5 (odd), 902 has 9 (odd), 348 sum=15 not divisible by 9.

Q6: A number remainder 3 when divided by 5 is
a) 15 b) 9 c) 13 d) 2
✅ Answer: c) 13
Explanation: 13 ÷ 5 = 2 remainder 3.

Q7: Area of rectangle = 44, breadth = 4, length = ?
a) 8 b) 11 c) 48 d) 40
✅ Answer: b) 11
Explanation: Area = l × b → 44 = l × 4 → l = 11.

Q8: Area of right triangle with perpendicular sides 3 and 4 is
a) 12 b) 7 c) 24 d) 6
✅ Answer: d) 6
Explanation: Area = ½ × 3 × 4 = 6.

Q9: Fill sign: 50/10 ___ 50%
✅ Answer: >
Explanation: 50/10 = 5, 50% = 0.5 → 5 > 0.5.

Q10: 5w² + 6w =
a) 11w² b) 11w c) 30w³ d) 5w²+6w
✅ Answer: d) 5w²+6w
Explanation: Cannot combine unlike terms.

Q11: Area of isosceles triangle SUB is 48 m². Area of triangle SEB?
Image needed. If SEB is half or congruent, may be 24 m².

Q12: Algebraic expression: Two more than square a number
a) 2+s b) s²+2 c) (s+2)² d) 2s²
✅ Answer: b) s²+2
Explanation: Square first: s², then add 2.

Q13: HCF of 72 and 96 is
a) 32 b) 24 c) 72 d) 2
✅ Answer: b) 24
Explanation: 72=2³×3², 96=2⁵×3 → HCF=2³×3=24.

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Assertion-Reason

Q14:
A: (a−b)² = (b−a)² for any integers a,b.
R: Square of any integer is positive.
✅ Answer: (b) Both true but R not correct explanation
Explanation: Equality holds because (a−b)² = (b−a)² due to squaring, not because squares are positive.

Q15:
A: Tiger population was m, increased by 5% in last decade.
R: Population now is 0.05m.
✅ Answer: (d) A true, R false
Explanation: Increase 5% → new population = m + 0.05m = 1.05m, not 0.05m.

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Section B – Short Answers (2 marks)

Q16: Sum of 4 consecutive numbers = 34. Find them.
✅ Answer: Let numbers: n, n+1, n+2, n+3.
Sum = 4n+6 = 34 ⇒ 4n=28 ⇒ n=7 → 7,8,9,10.

Q17: Digital root of 9a + 36b + 13.
✅ Answer: Digital root = sum digits mod 9.
9a divisible by 9, 36b divisible by 9, 13 → digital root 1+3=4.

Q18: Express 100 as difference of two squares.
✅ Answer: 100 = 26² − 24²? Check: 26²=676, 24²=576, difference=100 ✔. Also 10²−0².

Q19: Max marks=75, need 80% for A grade. Marks needed?
✅ Answer: 80% of 75 = 0.8×75 = 60 marks.

Q20: Is 10% of a day longer than 1% of a week? Justify.
✅ Answer:
10% of day = 0.1×24h = 2.4h.
1% of week = 0.01×168h = 1.68h.
2.4h > 1.68h → Yes, longer.

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Section C – Short Answers (3 marks)

Q21: Orange juice 600 mL, apple juice 900 mL → ratio in simplest form.
✅ Answer: Orange:Apple = 600:900 = 2:3.

Q22: Solve cryptarithm: MEOW? Image needed.
If typical, e.g., W=... Usually involves digit assignment.

Q23: Parallelogram: AM=12 cm, CN=8 cm, DC=10 cm → Find AD.
Area using base DC×AM = base AD×CN → 10×12 = AD×8 → AD=15 cm.

Q24: Car bought ₹4,40,000 after 15% discount. Original price?
✅ Answer: SP = 85% of MP → 4,40,000 = 0.85×MP → MP = 4,40,000 / 0.85 ≈ ₹5,17,647.

Q25: Rhombus diagonals 24 cm and 70 cm → side length.
✅ Answer: Side = √[(d1/2)² + (d2/2)²] = √(12² + 35²) = √(144+1225) = √1369 = 37 cm.

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Section D – Long Answers (4 marks)

Q26: Right triangle sides 5 cm, 12 cm → hypotenuse.
✅ Answer: Hypotenuse = √(5²+12²) = √(25+144) = √169 = 13 cm.

Q27(a): Compute 43×45 using (a+b)(a−b) identity.
✅ Answer: 43×45 = (44−1)(44+1) = 44²−1² = 1936−1 = 1935.

Q27(b): Earth travels 940 million km/year → km/week.
✅ Answer: 1 year = 52 weeks → 940/52 ≈ 18.08 million km/week.

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Section E – Case-Based (4 marks)

Q29: Giridhar: loan ₹12,500 at 12% p.a. simple interest for 3 years.
Raghu: same loan at 10% p.a. compound annually.

(a) Who pays more interest?
✅ Simple interest for Giridhar = P×r×t = 12500×0.12×3 = ₹4500.
Compound for Raghu: A = P(1+r)ᵗ = 12500×(1.1)³ = 12500×1.331 = ₹16637.5, Interest = 16637.5−12500 = ₹4137.5.
Giridhar pays more (4500 > 4137.5).

(b) Difference in amount paid after 3 years?
✅ Giridhar amount = 12500+4500 = ₹17000. Raghu = ₹16637.5. Difference = 17000−16637.5 = ₹362.5.

(c) Who incurs loss?
✅ Neither loses; but Giridhar pays more interest, so borrower’s perspective: Giridhar has higher cost.

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Q30: Toy: two trapeziums + rectangle in middle.

(a) Height of trapezium? Need dimensions from diagram. If given parallel sides and area, solve.

(b) Total area of toy?
✅ Area = 2 × area of trapezium + area of rectangle.

(c) Length and breadth of rectangle? From diagram.