Chapter 4 · Model TestAttempt first – solutions on the next page

Model Test Paper മാതൃകാ ചോദ്യപേപ്പർ

Set a timer for 1½ hours, keep the textbook closed, and write your answers in a notebook. Then check them on the solved paper, which also shows how marks are given. Where a question says "using an identity", write the identity and the working – not just the answer.

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Kerala State Syllabus · Standard IX · Mathematics Part 1

MATHEMATICS – Chapter 4: Multiplication Identities (ഗുണനസമവാക്യങ്ങൾ)

Time: 1½ hoursMaximum marks: 40
Instructions: Read each question carefully. Show your working – marks are given for correct steps even when the final answer is wrong. Where a proof is asked for, use a general form (such as 2n + 1 or 10m + n) and an identity; checking a few examples alone is not a proof. Section A: 8 × 1 = 8, Section B: 5 × 2 = 10, Section C: 4 × 3 = 12, Section D: 2 × 5 = 10.
SECTION A – Answer all questions8 × 1 = 8 marks
1.
(x + y)(u + v) = .
(1)
2.
Using an identity, 21 × 31 =
(a) 641(b) 651(c) 661(d) 630
(1)
3.
If xy = 50 and x + y = 15, then (x + 1)(y + 1) = .
(1)
4.
19 × 21 =
(a) 389(b) 399(c) 401(d) 419
(1)
5.
The general form of a natural number that leaves remainder 2 on division by 3 is .
(1)
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)
7.
(x − 1)(y − 1) = xy − + 1.
(1)
8.
The product of two odd numbers is always
(a) even(b) odd(c) a multiple of 3(d) a perfect square
(1)
SECTION B – Answer all questions5 × 2 = 10 marks
9.
Compute 32 × 52 using the identity (x + y)(u + v) = xu + xv + yu + yv. Show the four products.
(2)
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)
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)
12.
Compute 38 × 49 by writing each number as a difference and using the identity for (x − y)(u − v).
(2)
13.
Compute mentally, showing the identity used:
  1. 6½ × 8½
  2. 9½ × 19½
(2)
SECTION C – Answer all questions4 × 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)
15.
Four dates on a calendar form a 2 × 2 square, as in the figure.
  1. Taking the first date as x, write the other three dates.
  2. Prove that the difference of the two diagonal products is always 7.
  3. What would the difference be in a number grid with 5 columns (1 to 5 in the first row, 6 to 10 in the second, and so on)?
(3)
16.
For two natural numbers x and y, (x + 1)(y + 1) = 1056 and (x − 1)(y − 1) = 930.
  1. Find xy and x + y.
  2. Find x − y.
  3. Find the two numbers.
(3)
17.
A rectangle has perimeter 50 cm and area 150 sq cm. Each side is increased by 2 cm, as in the figure. Find the area of the new rectangle without finding the sides.
(3)
SECTION D – Answer all questions2 × 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.
  1. Find (x + 1)(y − 1).
  2. Find (x − 1)(y + 1).
  3. Find x + y, and hence the two numbers.
(5)
19.
The figure shows part of the multiplication table (1 to 9).
  1. 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.
  2. In a 3 × 3 square of the table, by how much do the diagonal sums of the four corner numbers differ? Justify.
  3. 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 of numbers.
(5)

— End of the paper —

📝 After the test

Check your answers on the next page. Give yourself marks honestly using the marking hints there. Anything below 28/40 means the chapter needs one more round – the solved paper tells you exactly which page to revisit for each question.

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