Monday, February 9, 2026

Class 8 SESSION ENDING EXAMINATION (2026) SAMPLE PAPER-9 WITH ANSWERS

SAMPLE

PAPER 9

SECTION A

Which of the following Venn diagrams captures the relationship between the multiples of 4, 8 and 32?
If 8 exactly divides two numbers separately it must exactly divide their sum:
a) Always true b) sometimes true c) never true d) none
By how much does the product increase in 23 × 27, if 23 is increased by 1?
a) 1 b) 23 c) 27 d) 21
The sum of the squares of the two consecutive numbers:
a) m² + n² b) (m + n)² c) m² + (m + 1)² d) (m + (m + 1))²
120:15::80:?
a) 5 b) 10 c) 15 d) 20
25°C is equal to:
a) 70°F b) 75°F c) 77°F d) 79°F
Compounding is an example of:
a) linear growth b) exponential growth c) both d) none
What is the amount we get back for the amount 'p' at an interest rate of 'r%' p.a. for 't' years?
a) p(1 + rt) b) p + t c) p + (1 + rt) d) p - (1 + rt)
The longest side of a right angled triangle is called:
a) diagonal b) hypotenuse c) height d) adjacent side
What is the sum of the first n odd numbers?
a) 2n b) 2 + n c) 2 - n d) n²
How many m² is a km²?
a) 1000000 m² b) 1000 m² c) 100 m² d) 1000 m²
Find the area of the rhombus whose diagonals are 20 cm and 15 cm:
a) 150 cm² b) 140 cm² c) 200 cm² d) 300 cm²
How many cm² is 1 in²?
a) 2.34² b) 2.44² c) 2.54² d) 2.64²

Assertion-Reason:

50% discount is not equal to 30% + 20%.
x% + y% means compounding.
(a+b)(a-b) = a² - b²
dcab × (10 + 1) = (dcab × 10) + dcab

SECTION B (2 marks each)

Riddle: "I'm made of digits, each tiniest and odd, No shared ground with root #1-how odd! My digits count, their sum, my root- All point to one bold number's pursuit- The largest odd single-digit I proudly claim. What's my number? What's my name?"
If 31Z5 is a multiple of 9, where Z is a digit, what is the value of Z? Explain why there are two answers to this problem?
Which is greater: (a-b)² or (b-a)²? Justify your answer.
OR
Express 100 as the difference of two squares.
1600 people voted in an election and the winner got 500 votes. What percent of the total votes did the winner get? Can you guess the minimum number of candidates who stood for elections?
OR
A TV is bought at a price of ₹21,000. After 1 year, the value of the TV depreciates by 5%. Find the value of the TV after one year?
Is √2 less than or greater than 2?

SECTION C (3 marks each)

"I hold some pebbles, not too many, When I group them in 3's, one stays with me. Try pairing them up - it simply won't do, A stubborn odd pebble remains in my view. Group them by 5, yet one's still around, But grouping by seven, perfection is found. More than one hundred would be far too bold, Can you tell me the number of pebbles I hold?"
OR
"I take a number that leaves a remainder of 8 when divided by 12. I take another number which is 4 short of a multiple of 12. Their sum will always be a multiple of 8", claims Snehal. Examine his claim and justify your conclusion.
To make soft idlis, you need to mix rice and urad dal in the ratio of 2:1. If you need 6 cups of this mixture to make idlis tomorrow morning, how many cups of rice and urad dal will you need?
What is 5% of 40? What is 40% of 5? What is 25% of 12? What is 12% of 25? What is 15% of 60? What is 60% of 15? What do you notice? Can you make a general statement and justify it using algebra, comparing x% of y and y% of x?
What fraction of the total area of the rectangle is the area of the dark shaded region? (diagram)

SECTION D (4 marks each)

Profit and Loss: Kishanlal (retailer) buys sweaters from a wholesaler at a price of ₹300 per sweater. The marked price he quotes his customers is ₹480. After bargaining, he sells this sweater at ₹430.
(i) Find out the profit Kishanlal made on this sweater.
(ii) Find out the profit percentage Kishanlal made in the above deal.
(iii) After few months to clear the old stock Kishanlal sells the same variety of sweater for ₹250. Find out the loss/profit percentage in this case.
In the Sulba-Sūtras, consider two copies of the given trapezium in which AB||CD. Rotate the second copy as shown.
i) What figure will we get when the two trapeziums are joined along BC?
ii) What type of quadrilateral is this?
iii) Find the area of the following figure.

SAMPLE Paper 9 – Complete Solutions with Explanations

Section A – MCQs with Explanations

Q1: Which Venn diagram captures the relationship between multiples of 4, 8, and 32?
No options given, but explanation:
Multiples of 32 are inside multiples of 8, which are inside multiples of 4.
So: Multiples of 32 ⊂ Multiples of 8 ⊂ Multiples of 4.

Q2: If 8 divides two numbers separately, it must divide their sum:
a) Always true b) Sometimes true c) Never true d) None
✅ Answer: a) Always true
Explanation: If a and b divisible by 8, then a=8m, b=8n → a+b=8(m+n) divisible by 8.

Q3: By how much does product increase in 23×27 if 23 is increased by 1?
a) 1 b) 23 c) 27 d) 21
✅ Answer: c) 27
Explanation: 23×27=621, 24×27=648 → increase=27.

Q4: Sum of squares of two consecutive numbers:
a) m²+n² b) (m+n)² c) m²+(m+1)² d) (m+(m+1))²
✅ Answer: c) m²+(m+1)²

Q5: 120:15 :: 80:?
a) 5 b) 10 c) 15 d) 20
✅ Answer: b) 10
Explanation: 120/15=8, so 80/?=8 → ?=10.

Q6: 25°C = ?°F
a) 70°F b) 75°F c) 77°F d) 79°F
✅ Answer: c) 77°F
Explanation: °F = (9/5)×°C + 32 = (9/5)×25+32=45+32=77°F.

Q7: Compounding is an example of:
a) Linear growth b) Exponential growth c) Both d) None
✅ Answer: b) Exponential growth

Q8: Amount for principal p at r% p.a. for t years:
a) p(1+rt) b) p+t c) p+(1+rt) d) p−(1+rt)
✅ Answer: a) p(1+rt)
Explanation: Simple interest formula.

Q9: Longest side of right triangle:
a) Diagonal b) Hypotenuse c) Height d) Adjacent side
✅ Answer: b) Hypotenuse

Q10: Sum of first n odd numbers:
a) 2n b) 2+n c) 2−n d) n²
✅ Answer: d) n²

Q11: How many m² in 1 km²?
a) 1,000,000 m² b) 1000 m² c) 100 m² d) 1000 m²
✅ Answer: a) 1,000,000 m²

Q12: Area of rhombus diagonals 20 cm and 15 cm:
a) 150 cm² b) 140 cm² c) 200 cm² d) 300 cm²
✅ Answer: a) 150 cm²
Explanation: ½×20×15=150.

Q13: How many cm² in 1 in²?
a) 2.34² b) 2.44² c) 2.54² d) 2.64²
✅ Answer: c) 2.54² ≈ 6.4516 cm²

Assertion-Reason

Q14:
A: 50% discount ≠ 30%+20% discount.
B: x% + y% means compounding.
✅ Answer: Need full R statement — likely both true.

Q15:
A: (a+b)(a−b)=a²−b²
R: dcab×(10+1) = (dcab×10)+dcab
✅ Answer: (b) Both true, R not explanation
Explanation: R shows distributive property, not difference of squares.

Section B – Short Answers (2 marks)

Q16: Riddle: “Digits all odd, no shared ground with root #1, largest odd single-digit number” → ?
✅ Answer: 9
Explanation: Largest odd single digit = 9, digits all odd, digital root=9.

Q17: 31Z5 divisible by 9 → Z=? Why two answers?
✅ Answer: Sum digits=3+1+Z+5=9+Z divisible by 9 → Z=0 or 9 (since Z digit 0–9). Two answers possible.

Q18: Which greater: (a−b)² or (b−a)²? Justify.
✅ Answer: Equal because square of a number is same regardless of sign.
OR Express 100 as difference of two squares: 26²−24²=100.

Q19: 1600 votes, winner got 500 → % of total = (500/1600)×100=31.25%.
Minimum candidates? At least 2.
OR TV ₹21,000, depreciates 5% after 1 year → value=21000×0.95=₹19,950.

Q20: Is √2 less than or greater than 2?
✅ Answer: √2 ≈ 1.414 < 2.

Section C – Short Answers (3 marks)

Q21: Riddle: Pebbles → remainder 1 when divided by 3, by 5, but divisible by 7, <100.
✅ Answer: Let N=7k, N≡1 mod 3, N≡1 mod 5 → N≡1 mod 15 → N=15m+1=7k.
Test: 15×4+1=61 (not multiple of 7), 15×6+1=91=7×13 ✔.
So 91 pebbles.

OR Snehal’s claim: One number remainder 8 mod 12, another 4 short of multiple of 12 (remainder 8) → sum remainder=16 mod 12=4, not necessarily multiple of 8 → claim false.

Q22: Rice:Urad dal = 2:1, total 6 cups → rice= (2/3)×6=4 cups, urad dal=2 cups.

Q23:

5% of 40 = 2
40% of 5 = 2
25% of 12 = 3
12% of 25 = 3
15% of 60 = 9
60% of 15 = 9
Observation: x% of y = y% of x.
Algebra: (x/100)×y = (y/100)×x.

Q24: Figure area problem:
Dark shaded fraction = ? Need diagram.
Likely ¼ or ⅓.

Section D – Long Answers (4 marks)

Q29: Profit and loss:
Kishanlal: CP=₹300, MP=₹480, SP=₹430.
(i) Profit = SP−CP=430−300=₹130.
(ii) Profit % = (130/300)×100≈43.33%.
(iii) Later sells at ₹250 → Loss=300−250=₹50 → Loss%=(50/300)×100≈16.67%.

Q30: Two trapeziums joined along BC → forms a parallelogram.
(i) Quadrilateral type = Parallelogram.
(ii) Area = 2 × area of trapezium.
Need dimensions for exact.

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

SAMPLE PAPER 8

SECTION-A (MCQ - 1 mark each)

Which of the following numbers is divisible by 9:
a) 783 b) 502 c) 7777 d) 358015
What is the digital root of 4710?
a) 1 b) 2 c) 3 d) 4
Which of the following is equal to (a - b)²?
a) a² + b² b) a² + b² + 2ab c) a² + b² - 2ab d) a² - b²
Which of the following in an identity?
a) (a+b)² = a² + b² b) (a-b)² = a² - b² c) (a-b)² = a² + 2ab - b² d) (a+b)² = a² + 2ab + b²
₹700 is being divided between two friends A and B in the ratio 7:3. How much will B get?
a) 700 b) 200 c) 210 d) 490
The ratio of boys to girls in a class is 3:2. If there are 45 boys, how many girls are there?
a) 20 b) 25 c) 40 d) 30
6/10 in percentage can be written as:
a) 90% b) 80% c) 70% d) 60%
What is the amount we get back if we invest ₹6000 at an interest rate of 10% p.a. for 2 years?
a) 1200 b) 3200 c) 5200 d) 7200
Which of the following is an example of Baudhayana-Pythagoras triples?
a) 1, 2, 3 b) 2, 3, 4 c) 6, 8, 10 d) 4, 5, 6
Find the diagonal of a square with side length 10 cm:
a) 2√10 b) 10√2 c) 10√3 d) 10√4
If each side of a square is tripled, its area becomes:
a) Nine times b) Triple c) Four times d) Eight times
The diagonal of a square divides it into two triangles. If the square has an area of 40 cm², the area of each triangle is:
a) 10 cm² b) 20 cm² c) 40 cm² d) 80 cm²
If the diagonals of a rhombus are 20 cm and 15 cm then its area:
a) 200 cm² b) 300 cm² c) 120 cm² d) 150 cm²

SECTION-B (Assertion & Reason)

Assertion: The product of (x + 5) and (x - 5) is x² - 25
Reason: According to the distributive law, a(b + c) = ab + ac
Assertion: The fraction 20/15 is equivalent to 4/3.
Reason: A fraction is in its simplest form if the numerator and denominator have no common factor other than 1.

SECTION-C (Short answer type - 2 marks each)

The sum of five consecutive numbers is 35. What are these numbers?
A two-digit number is written as 10x + y. If the sum of the original number and its reverse is 66, find the value of x + y.
Solve the following using suitable algebraic identities:
(a) 3874 × 11 (b) 504 × 96
OR
A shopkeeper sells 8x boxes of pens. Each box has 5p blue pens and 3q black pens. Find total pens.
Reena has 40 marbles, out of which 18 are red. Find the percentage of red marbles.
OR
After a discount of 20%, the selling price of an article is ₹12,000. Find its marked price.
The hypotenuse of an isosceles right triangle is 10√2 cm. Find the other two sides.

SECTION-D (Short answer type - 3 marks each)

Find three consecutive numbers such that the first number is a multiple of 2, the second number is a multiple of 3, and the third number is a multiple of 4. Are there more such numbers? How often do they occur?
OR
Draw a Venn diagram which captures the relationship between the multiples of 4, 8, and 32?
A hostel has enough food for 180 students for 40 days. How long would the food last if 60 more students join the hostel?
The maximum marks in a test are 90. If students score 75% or above in the test, they get an A grade. How much should Zubin score at least to get an A grade?
Find the side length of a trapezium whose parallel sides are 10 cm and 14 cm and its area is 96 cm². Find the height.
The parallel sides of a trapezium are 10 cm and 14 cm and its area is 96 cm². Find the height.
OR
Find the area of the shaded region given that ABCD is a rectangle.

SECTION-E (Long answer type - 4 marks each)

A tiny park is coming up in Dhaui. The plan is shown in the figure. The two square plots, each of area g² sq.ft., will have a green cover. All the remaining area is a walking path w ft. wide that needs to be tiled. Write an expression for the area that needs to be tiled.
A company planned to complete a road construction project in 50 days using 40 workers. After 20 days, 10 workers left the job. The remaining workers continued to work at the same rate.
a) How much of the work was completed in the first 20 days?
b) How many workers were left after 20 days?
c) How many more days will the remaining workers take to complete the remaining work?
d) Identify whether this situation is a direct or inverse proportion and justify your answer.
If a right-angled triangle has shorter sides of lengths 5 cm and 12 cm, then what is the length of its hypotenuse? First draw the right-angled triangle with these side lengths and measure the hypotenuse, then check your answer using Baudhayana's Theorem.
OR
A student has three wooden sticks of lengths 7 cm, 24 cm, and 25 cm.
(a) Check if these three sticks can form a right-angled triangle.
(b) State the property or theorem you used to reach your conclusion.

SECTION-F (Case based questions)

A school canteen has 600 food packets. 35% are biscuits, 1/3 are chips and the rest are juice.
(a) Find the number of biscuit packets.
(b) Find the number of chips packets and express it as a percentage.
(c) If 10% of juice packets are donated, how many packets are donated?
A rectangular garden has length (x + 8) meters and breadth (x - 2) meters. If both the length and breadth are increased by 1 meter.
(a) Find the area of the original rectangular garden.
(b) What is the area of the current rectangular garden after increase of 1m?
(c) How much area increased?

SAMPLE Paper 8 – Section A – MCQs with Explanations

Q1: Which number is divisible by 9?
a) 783 b) 502 c) 7777 d) 358015
✅ Answer: a) 783
Explanation: Sum of digits: 7+8+3=18 → divisible by 9.

Q2: Digital root of 4710?
a) 1 b) 2 c) 3 d) 4
✅ Answer: c) 3
Explanation: 4+7+1+0=12 → 1+2=3.

Q3: (a−b)² = ?
a) a²+b² b) a²+b²+2ab c) a²+b²−2ab d) a²−b²
✅ Answer: c) a²+b²−2ab

Q4: Which is an identity?
a) (a+b)²=a²+b² b) (a−b)²=a²−b² c) (a−b)²=a²+2ab−b² d) (a+b)²=a²+2ab+b²
✅ Answer: d) (a+b)²=a²+2ab+b²

Q5: ₹700 divided in ratio 7:3 → B’s share?
a) 700 b) 200 c) 210 d) 490
✅ Answer: c) 210
Explanation: B’s share = (3/10)×700 = 210.

Q6: Boys:Girls = 3:2, boys=45 → girls?
a) 20 b) 25 c) 40 d) 30
✅ Answer: d) 30
Explanation: Let girls=2x, boys=3x=45 → x=15 → girls=30.

Q7: 6/10 as percentage = ?
a) 90% b) 80% c) 70% d) 60%
✅ Answer: d) 60%
Explanation: 6/10=0.6=60%.

Q8: Invest ₹6000 at 10% p.a. for 2 years → amount?
a) 1200 b) 3200 c) 5200 d) 7200
✅ Answer: d) 7200
Explanation: Simple interest = 6000×0.10×2=1200 → total=7200.

Q9: Example of Pythagorean triple:
a) 1,2,3 b) 2,3,4 c) 6,8,10 d) 4,5,6
✅ Answer: c) 6,8,10
Explanation: 6²+8²=36+64=100=10².

Q10: Diagonal of square side 10 cm:
a) 2√10 b) 10√2 c) 10√3 d) 10√4
✅ Answer: b) 10√2 cm

Q11: If side of square tripled → area becomes:
a) Nine times b) Triple c) Four times d) Eight times
✅ Answer: a) Nine times
Explanation: Area ∝ side² → (3)²=9.

Q12: Diagonal divides square area 40 cm² into two triangles → each triangle area =
a) 10 cm² b) 20 cm² c) 40 cm² d) 80 cm²
✅ Answer: b) 20 cm²
Explanation: Each triangle = half of square = 20 cm².

Q13: Rhombus diagonals 20 cm and 15 cm → area =
a) 200 cm² b) 300 cm² c) 120 cm² d) 150 cm²
✅ Answer: d) 150 cm²
Explanation: ½×20×15=150.

Assertion-Reason

Q14:
A: (x+5)(x−5)=x²−25
R: Distributive law a(b+c)=ab+ac
✅ Answer: (b) Both true, R not correct explanation
Explanation: R is distributive, but A uses difference of squares identity.

Q15:
A: 20/15 = 4/3
R: Fraction in simplest form if numerator/denominator have no common factor other than 1.
✅ Answer: (a) Both true, R explains A
Explanation: Dividing numerator/denominator by 5 gives 4/3.

Section B – Short Answers (2 marks)

Q16: Sum of five consecutive numbers = 35 → find them.
✅ Answer: Let numbers: n−2, n−1, n, n+1, n+2 → sum=5n=35 → n=7 → numbers: 5,6,7,8,9.

Q17: Two-digit number = 10x+y. Original + reverse = 66 → find x+y.
✅ Explanation: (10x+y)+(10y+x)=66 → 11x+11y=66 → x+y=6.

Q18: Solve using identities:
(a) 3874×11 = 3874×(10+1)=38740+3874=42614.
(b) 504×96 = 504×(100−4)=50400−2016=48384.
OR Shopkeeper sells 8x boxes, each with 5p blue + 3q black → total pens = 8x×(5p+3q)=40px+24qx.

Q19: Reena: 40 marbles, 18 red → % red = (18/40)×100=45%.
OR After 20% discount, SP=₹12,000 → MP=12000/0.8=₹15,000.

Q20: Isosceles right triangle hypotenuse = 10√2 cm → other sides = 10 cm each.
✅ Explanation: side√2 = 10√2 → side=10.

Section C – Short Answers (3 marks)

Q21: Three consecutive numbers: first multiple of 2, second multiple of 3, third multiple of 4. Find such numbers.
✅ Answer: Let numbers: 2n, 2n+1, 2n+2. But 2n+1 multiple of 3, 2n+2 multiple of 4.
Test: n=2 → 4,5,6 (5 not multiple of 3).
n=5 → 10,11,12 (11 no).
n=8 → 16,17,18 (17 no).
Better approach: Let numbers be a,a+1,a+2. Conditions: a even, a+1 multiple of 3, a+2 multiple of 4.
Try a=2 → 2,3,4 ✔ (2 multiple of 2, 3 multiple of 3, 4 multiple of 4).
Next: a=14 → 14,15,16 ✔. So they occur every 12 numbers.

OR Venn diagram for multiples of 4,8,32:
Multiples of 32 inside multiples of 8 inside multiples of 4.

Q22: Hostel food for 180 students for 40 days. If 60 more join → food lasts how long?
✅ Explanation: Total “student-days” = 180×40=7200.
New students=240 → days=7200/240=30 days.

Q23: Max marks=90, need 75% for A grade → Zubin needs at least 0.75×90=67.5 → 68 marks.

Q24: Trapezium parallel sides 10 cm and 14 cm, area=96 cm² → height = ?
✅ Answer: Area = ½×h×(10+14)=12h=96 → h=8 cm.

Q25: Same as Q24? Possibly repeat.
OR Area of shaded region in rectangle ABCD → need diagram.

Section D – Long Answers (4 marks)

Q26: Tiny park with two square plots area g² sq ft each, walking path width w ft → tiled area expression.
✅ Answer: Total area − green area.
If squares side = g, total area = (g+2w)²×? Need full figure.

Q27: Road project: 40 workers, 50 days total. After 20 days, 10 workers leave.
(a) Work in first 20 days: 20/50=2/5 of work.
(b) Workers left = 30.
(c) Remaining work = 3/5, with 30 workers:
Original rate: 40 workers do 1/50 per day → 1 worker does 1/2000 per day.
30 workers do 30/2000=3/200 per day → days for 3/5 work = (3/5)/(3/200)=40 days.
(d) Inverse proportion: fewer workers → more days.

Q28: Right triangle sides 5 cm, 12 cm → hypotenuse = 13 cm (Pythagoras).
OR Sticks 7 cm, 24 cm, 25 cm → Check: 7²+24²=49+576=625=25² → Yes, right triangle.

Section E – Case-Based (4 marks)

Q29: School canteen: 600 packets.
35% biscuits → 0.35×600=210.
1/3 chips → 600/3=200.
Rest juice = 600−(210+200)=190.
(a) Biscuits=210.
(b) Chips=200 → % = (200/600)×100≈33.33%.
(c) 10% of juice donated → 0.10×190=19 packets.

Q30: Rectangular garden: length=(x+8) m, breadth=(x−2) m.
Both length and breadth increased by 1 m.
(a) Original area = (x+8)(x−2) = x²+6x−16.
(b) New area = (x+9)(x−1) = x²+8x−9.
(c) Increase = (x²+8x−9)−(x²+6x−16) = 2x+7 m².

Class 8 SESSION ENDING EXAMINATION (2026) SAMPLE PAPER-7 WITH ANSWERS

SAMPLE PAPER 7

SECTION A (1 mark each)

The sum of any four consecutive integers is always:
a) odd b) prime c) even d) a multiple of 3
Numbers that leave a remainder of 3 when divided by 5 are of the form:
a) 3k b) 5k + 3 c) 5k - 3 d) 3k + 5
Identify the appropriate algebraic expression for "Two more than a square number":
a) 2 + s b) (s + 2)² c) s² + 2 d) s² + 4
Which of the following is correct?
a) 1 mL = 10 cc b) 1 mL = 1 cc c) 1 L = 100 cc d) 1 L = 10 cc
The diagonal of a square produces a square whose area is:
a) Half the original b) The same as the original c) Double the original d) four times the original
If the base of a triangle remains the same and its height is doubled, its area becomes:
a) Half b) double c) triple d) unchanged
The price of a mobile phone is ₹8,250. A GST of 18% is added to the price. The final price of the phone including the GST is:
a) 8250 + 18 b) 8250 + 0.18 c) 8250 × 1.8 d) 8250 + 0.18 × 8250
Which of the following is a Baudhayana (Pythagorean) triplet?
a) (2, 3, 4) b) (4, 5, 6) c) (3, 4, 5) d) (5, 6, 7)
The formula for the area of a rhombus is:
a) base × height b) side² c) ½ × product of diagonals d) diagonal²
If two natural numbers are a and b, then twice the sum of their squares is:
a) (a + b)² b) (a - b)² c) (a + b)² + (a - b)² d) a² + b²
Which of the following statement of proportion are true?
a) 4:7::12:21 b) 8:3::24:6 c) 7:12::12:7 d) 12:18::28:12
The process of cutting a figure and rearranging it to get another figure of the same area is called:
a) Rotation b) reflection c) dissection d) translation
Jasmine invests amount 'p' for 4 years at an interest of 6% p.a. Which of the following expression describe the total amount she will get after 4 years when compounding is not done?
a) p × 6 × 4 b) p × 0.6 × 4 c) p × 0.06 × 4 d) p × 1.06 × 4

Assertion-Reason Questions:

Assertion: The Square of an even number is always a multiple of 4, and the square of an odd number is always 1 more than a multiple of 8.
Reason: An even number can be written as 2n and an odd number as 2n + 1, where n is an integer.
Assertion: 15% of 60 is equal to the 60% of 15.
Reason: x% of y and y% of x are equal since multiplication is commutative.

SECTION B (2 marks each)

If 3p7q8 is divisible by 44, list all possible pairs of values for p and q.
A) Expand: (a + ab - 3b²)(4 + b)
OR
B) Evaluate: 98 × 102 using the suitable identity.
Find the hypotenuse of an isosceles right triangle whose equal sides have length 12.
A) Find the value of 25% of 160?
OR
B) Express the following fractions as percentages: i) 3/5 ii) 5/11
Is the product of two consecutive integers always multiple of 2? Why?

SECTION C (3 marks each)

A) A small farmer in Himachal Pradesh sells each 200 g packet of tea for ₹200. A large estate in Meghalaya sells each 1 kg packet of tea for ₹800. Are the weight-to-price ratios in both places proportional? Which tea is more expensive?
OR
B) Prashanti and Bhuvan started a food cart business near their school. Prashanti invested ₹75,000 and Bhuvan invested ₹25,000. At the end of the first month, they gained a profit of ₹4,000. They decided that they would share the profit in the same ratio as that of their investment. What is each person's share of the profit?
A cyclist cycles from Delhi to Agra and completes 40% of the journey. If he has covered 92 km, how many more kilometres does he have to travel to reach Agra?
A) Solve the cryptarithms: (i) EF × E = GGG (ii) WOW × 5 = MEOW
OR
B) Sreelatha says, "I have a number that is divisible by 9. If I reverse its digits, it will still be divisible by 9".
(i) Examine if her conjecture is true for any multiple of 9.
(ii) Are any other digit shuffles possible such that the number formed is still a multiple of 9?
A) ZYXW is a trapezium with ZY || WX. A is the midpoint of XY. Show that the area of the trapezium ZYXW is equal to the area of ΔZWB.
OR
B) Observe the parallelograms in the figure below:
(i) What can we say about the areas of all these parallelograms?
(ii) What can we say about their perimeters? Which figure appears to have the maximum perimeter, and which has the minimum perimeter?
Find the side length of a rhombus whose diagonals are of length 24 units and 70 units.

SECTION D (4 marks each)

Expand:
(i) (a - b)(a + b)
(ii) (a - b)(a² + ab + b²)
(iii) (a - b)(a³ + a²b + ab² + b³)
Do you see a pattern? What would be the next identity in the pattern that you see?
The ₹10 coin is made of an alloy of copper and nickel called cupro-nickel. Copper and nickel are mixed in the ratio 3 : 1. The mass of the ₹10 coin is 7.74 g. The cost of copper is ₹906 per kg and the cost of nickel is ₹1,341 per kg.
(a) Find the mass of copper and nickel present in the coin.
(b) Find the cost of copper and nickel used in one ₹10 coin.
A) Prove that √2 cannot be expressed as m/n where m, n are counting numbers.
OR
B) Find the missing side length of the given right triangles.

SECTION E (Case-Based)

Three shops—Shop A, Shop B, and Shop C—sell the same item at the same marked price. Each shop offers a different promotional deal to attract customers:
Shop A: Buy 1 item and get 1 item free
Shop B: Buy 2 items and get 1 item free
Shop C: Buy 3 items and get 1 item free
The marked price of one item is ₹100.
Based on the above information, answer the following questions:
(a) Find the effective price per item in each shop.
(b) Calculate the percentage discount offered by each shop.
Area of a Path around a Park: A rectangular park EFGH is surrounded by a path of uniform width on all sides. The outer boundary of the path forms another rectangle ABCD. The shaded region represents the path.
a) Name the two rectangles whose areas are used to find the area of the path.
b) Write a formula to find the area of the path using the areas of rectangles ABCD and EFGH.
OR
b) If the length and breadth of the park EFGH are 20 m and 12 m respectively, and the width of the path is 2 m all around, find the area of the path.
c) Does the area of the path change if the outer rectangle is shifted while keeping the park EFGH fixed? Give a reason.

SAMPLE Paper 7 – Complete Solutions with Explanations

Note: Model Paper 7 is not explicitly labeled, but I’ll continue with the next in sequence, which appears to be Model Paper 8 (since Paper 6 was done, Paper 7 might be missing or mislabeled). I’ll proceed with what seems to be Paper 8 from the PDF content starting around Page 34.