Model Test Paper – Solved ഉത്തരങ്ങൾ
Every question of the model paper, with a full solution, an animated figure wherever a picture or a derivation helps, and a marking hint so you can score your own paper honestly.
📖 How marks are usually given
1-mark questions: the correct answer only. 2- and 3-mark questions: about one mark for writing the right identity, the rest for the working and the final answer. 5-mark questions: marks for each part as shown. In a proof, using a general form (2n + 1, x, x + 1, 10m + n, …) earns a mark by itself; examples alone earn nothing. A correct method with an arithmetic slip usually loses only one mark.
SECTION A8 × 1 = 8 marks
1.
(x + y)(u + v) = ______.
(1)Answer: xu + xv + yu + yv
Every term of the first sum times every term of the second – four products.Review: Page 5 · Multiplying sums
2.
Using an identity, 21 × 31 = (a) 641 (b) 651 (c) 661 (d) 630
(1)Answer: (b) 651
(20 + 1)(30 + 1) = 600 + 20 + 30 + 1 = 651.Review: Page 5 · (x + 1)(y + 1)
3.
If xy = 50 and x + y = 15, then (x + 1)(y + 1) = ______.
(1)Answer: 66
(x + 1)(y + 1) = xy + x + y + 1 = 50 + 15 + 1 = 66. (The numbers are 5 and 10: 6 × 11 = 66.)Review: Page 5 · The "next numbers" rule
4.
19 × 21 = (a) 389 (b) 399 (c) 401 (d) 419
(1)Answer: (b) 399
(20 − 1)(20 + 1) = 400 + 20 − 20 − 1 = 399 (the Class 8 identity (x − y)(x + y) = x² − y²).Review: Page 3 · Class 8 identities
5.
The general form of a natural number that leaves remainder 2 on division by 3 is ______.
(1)Answer: 3n + 2
Multiples of 3 are 3n; two more than a multiple of 3 is 3n + 2 (n = 0, 1, 2, … gives 2, 5, 8, 11, …).Review: Page 2 · General forms
6.
In any 2 × 2 square of dates on a calendar, the difference of the two diagonal products is (a) 1 (b) 5 (c) 7 (d) 8
(1)Answer: (c) 7
(x + 1)(x + 7) − x(x + 8) = x² + 8x + 7 − x² − 8x = 7.Review: Page 7 · Calendar squares
7.
(x − 1)(y − 1) = xy − ______ + 1.
(1)Answer: (x + y)
(x − 1)(y − 1) = xy − x − y + 1 = xy − (x + y) + 1: the product of the previous numbers is product − sum + 1.Review: Page 9 · Multiplying differences
8.
The product of two odd numbers is always (a) even (b) odd (c) a multiple of 3 (d) a perfect square
(1)Answer: (b) odd
(2m + 1)(2n + 1) = 4mn + 2m + 2n + 1 = 2(2mn + m + n) + 1 – one more than an even number.SECTION B5 × 2 = 10 marks
9.
Compute 32 × 52 using the identity (x + y)(u + v) = xu + xv + yu + yv. Show the four products.
(2)🎬 Solution
Answer: 1664
32 × 52 = (30 + 2)(50 + 2) = 30 × 50 + 30 × 2 + 2 × 50 + 2 × 2 = 1500 + 60 + 100 + 4 = 1664.Marks: 1 for splitting into (30 + 2)(50 + 2) and writing the four products, 1 for 1664.
Review: Page 5 · Multiplying sums
10.
The product of two natural numbers is 600 and their sum is 50. Find the product of the two numbers that come just after them.
(2)Answer: 651
(x + 1)(y + 1) = xy + (x + y) + 1 = 600 + 50 + 1 = 651. (The numbers are 20 and 30; 21 × 31 = 651 ✔.)Marks: 1 for the identity (x + 1)(y + 1) = xy + x + y + 1, 1 for 651.
Review: Page 6 · Exercise 1, Q2
11.
The product of two odd numbers is 195 and their sum is 28. Find the product of the odd numbers that come just after each of them.
(2)Answer: 255
The next odd numbers are x + 2 and y + 2: (x + 2)(y + 2) = xy + 2(x + y) + 4 = 195 + 56 + 4 = 255. (The numbers are 13 and 15; 15 × 17 = 255 ✔.)Marks: 1 for using x + 2 and y + 2 with the identity xy + 2(x + y) + 4, 1 for 255. (Using x + 1 loses both marks – the next odd number is 2 more.)
Review: Page 6 · Exercise 1, Q3
12.
Compute 38 × 49 by writing each number as a difference and using the identity for (x − y)(u − v).
(2)🎬 Solution
Answer: 1862
38 × 49 = (40 − 2)(50 − 1) = 40 × 50 − 40 × 1 − 2 × 50 + 2 × 1 = 2000 − 40 − 100 + 2 = 1862.Marks: 1 for (40 − 2)(50 − 1) with the four signed products, 1 for 1862. A sign error in the last term (− 2 instead of + 2) loses 1 mark.
Review: Page 9 · Multiplying differences
13.
Compute mentally, showing the identity used: (i) 6½ × 8½ (ii) 9½ × 19½
(2)Answer: (i) 55¼ (ii) 185¼
(i) (x + ½)(y + ½) = xy + ½(x + y) + ¼: 48 + ½(14) + ¼ = 48 + 7 + ¼ = 55¼.(ii) (10 − ½)(20 − ½) = 200 − 5 − 10 + ¼ = 185¼.
Marks: 1 for each part (identity + answer). Decimals 55.25 and 185.25 are also accepted.
Review: Page 5 · Halves · Page 9 · Practice products
SECTION C4 × 3 = 12 marks
14.
Numbers of the form 4n + 1 (such as 5, 9, 13, …) leave remainder 1 on division by 4. Prove that the product of any two such numbers is also of the form 4n + 1.
(3)Proof
Take two such numbers 4m + 1 and 4n + 1 (m, n = 0, 1, 2, …).(4m + 1)(4n + 1) = 16mn + 4m + 4n + 1 = 4(4mn + m + n) + 1.
4mn + m + n is a whole number, so the product is 4 × (a whole number) + 1: it leaves remainder 1 on division by 4. Check: 5 × 9 = 45 = 4 × 11 + 1 ✔.
Marks: 1 for writing the numbers as 4m + 1 and 4n + 1 (two different letters!), 1 for expanding with the identity, 1 for regrouping as 4(…) + 1 and the conclusion.
15.
Four dates on a calendar form a 2 × 2 square. (i) Taking the first date as x, write the other three dates. (ii) Prove that the difference of the two diagonal products is always 7. (iii) What would the difference be in a number grid with 5 columns?
(3)🎬 Solution
Answer: (i) x + 1, x + 7, x + 8 (ii) 7 (iii) 5
(i) The date to the right is x + 1; a week later is x + 7, and x + 8.(ii) x(x + 8) = x² + 8x and (x + 1)(x + 7) = x² + 7x + x + 7 = x² + 8x + 7. Difference = 7, whatever x is.
(iii) With 5 columns the square is x, x + 1, x + 5, x + 6: (x + 1)(x + 5) − x(x + 6) = x² + 6x + 5 − x² − 6x = 5.
Marks: 1 for the four dates, 1 for the two products and the difference 7, 1 for 5 (with the products or a clear reason).
16.
For two natural numbers x and y, (x + 1)(y + 1) = 1056 and (x − 1)(y − 1) = 930. (i) Find xy and x + y. (ii) Find x − y. (iii) Find the two numbers.
(3)🎬 Solution
Answer: xy = 992, x + y = 63, x − y = 1; the numbers are 32 and 31
(i) xy + (x + y) + 1 = 1056 and xy − (x + y) + 1 = 930. Adding: 2xy + 2 = 1986 ⇒ xy = 992. Subtracting: 2(x + y) = 126 ⇒ x + y = 63.(ii) (x − y)² = (x + y)² − 4xy = 3969 − 3968 = 1 ⇒ x − y = 1.
(iii) x = ½(63 + 1) = 32, y = ½(63 − 1) = 31. Check: 33 × 32 = 1056 ✔, 31 × 30 = 930 ✔.
Marks: 1 for xy = 992 and x + y = 63 (adding and subtracting the identities), 1 for x − y = 1 via (x − y)² = (x + y)² − 4xy, 1 for 32 and 31.
17.
A rectangle has perimeter 50 cm and area 150 sq cm. Each side is increased by 2 cm. Find the area of the new rectangle without finding the sides.
(3)🎬 Solution
Answer: 204 sq cm
Let the sides be x and y: 2(x + y) = 50 ⇒ x + y = 25, and xy = 150.New area = (x + 2)(y + 2) = xy + 2x + 2y + 4 = xy + 2(x + y) + 4 = 150 + 50 + 4 = 204 sq cm. (The sides are 15 and 10; 17 × 12 = 204 ✔.)
Marks: 1 for x + y = 25 (from the perimeter), 1 for the identity (x + 2)(y + 2) = xy + 2(x + y) + 4, 1 for 204 sq cm.
SECTION D2 × 5 = 10 marks
18.
The product of two natural numbers is 925 and their difference is 12. Let x be the larger number and y the smaller. (i) Find (x + 1)(y − 1). (ii) Find (x − 1)(y + 1). (iii) Find x + y, and hence the two numbers.
(5)🎬 Solution
Answer: (i) 912 (ii) 936 (iii) x + y = 62; the numbers are 37 and 25
(i) (x + 1)(y − 1) = xy − x + y − 1 = xy − (x − y) − 1 = 925 − 12 − 1 = 912.(ii) (x − 1)(y + 1) = xy + x − y − 1 = xy + (x − y) − 1 = 925 + 12 − 1 = 936.
(iii) (x + y)² = (x − y)² + 4xy = 144 + 3700 = 3844 = 62² ⇒ x + y = 62. Then x = ½(62 + 12) = 37, y = ½(62 − 12) = 25. Check: 37 × 25 = 925 ✔, 38 × 24 = 912 ✔, 36 × 26 = 936 ✔.
Marks: 1 for each identity with its value in (i) and (ii) (2 marks), 1 for (x + y)² = (x − y)² + 4xy = 3844, 1 for x + y = 62, 1 for 37 and 25.
19.
The figure shows part of the multiplication table (1 to 9). (i) Take the 2 × 2 square whose top-left number is mn (row m, column n). Write the other three numbers and prove that the two diagonal sums differ by 1. (ii) In a 3 × 3 square of the table, by how much do the diagonal sums of the four corner numbers differ? Justify. (iii) 12 × 63 = 21 × 36. Writing the numbers as 10m + n and 10p + q, explain why reversing the digits of both numbers keeps the product exactly when mp = nq, and give one more such pair.
(5)🎬 Solution
Answer: (i) differ by 1 (ii) by 4 (iii) mp = nq; e.g. 13 × 62 = 31 × 26 = 806
(i) Row m holds m, 2m, 3m, …; so the square is mn, m(n + 1) = mn + m (to the right), (m + 1)n = mn + n (below) and (m + 1)(n + 1) = mn + m + n + 1. Diagonal sums: mn + (mn + m + n + 1) = 2mn + m + n + 1 and (mn + m) + (mn + n) = 2mn + m + n. Difference 1.(ii) Corners mn, m(n + 2) = mn + 2m, (m + 2)n = mn + 2n, (m + 2)(n + 2) = mn + 2m + 2n + 4. Sums 2mn + 2m + 2n + 4 and 2mn + 2m + 2n: difference 4. (Example: 35 + 63 = 98, 45 + 49 = 94.)
(iii) (10m + n)(10p + q) = 100mp + 10mq + 10np + nq and (10n + m)(10q + p) = 100nq + 10np + 10mq + mp. The middle terms are the same, so the products are equal exactly when 100mp + nq = 100nq + mp, i.e. 99mp = 99nq, i.e. mp = nq. For 12 and 63: 1 × 6 = 2 × 3 ✔. Another pair: 13 and 62 (1 × 6 = 3 × 2): 13 × 62 = 806 = 31 × 26.
Marks: 1 for the four cells mn, mn + m, mn + n, mn + m + n + 1, 1 for the two sums and the difference 1, 1 for the 3 × 3 case (difference 4 with reason), 1 for the expansion showing 99(mp − nq), 1 for a correct new pair.
Review: Page 7 · Curiosities · Page 8 · Exercise 3, Q2
Marks summary
| Section | Questions | Marks | What it tests |
|---|---|---|---|
| A | 1 – 8 | 8 | The identities by heart, next/previous products, general forms, calendar fact, odd × odd |
| B | 9 – 13 | 10 | Mental products with sums and differences, next numbers, next odd numbers, halves |
| C | 14 – 17 | 12 | A remainder proof, the calendar proof, finding numbers from neighbours' products, a rectangle enlarged |
| D | 18 – 19 | 10 | One-up-one-down products with the full "find the numbers" recipe; multiplication-table and digit-reversal proofs |
| Total | 40 |
⏱️ Time management (1½ hours = 90 minutes)
- Section A: about 10 minutes – these are one-line recalls; do not get stuck, move on and come back.
- Section B: about 20 minutes (4 minutes a question). Always write the identity before the arithmetic – it earns the first mark even if the arithmetic slips.
- Section C: about 30 minutes. In proofs, start with the general form (4m + 1, x, x + 1, x + 7 …) – that line alone is worth a mark.
- Section D: about 25 minutes. Follow the recipe: add, subtract, bridge identity, halve. Check your numbers by multiplying back.
- Keep the last 5 minutes to check signs in every (x − y)(u − v) expansion and to make sure every blank is filled.
🎉 Finished the chapter?
Scored 32+? Excellent. Between 24 and 31? Revisit the pages named under the questions you lost marks on. Below 24? Go back to page 5 and work through the three section pages again with the animations – then retake the test in a week.