1A rectangular field is (2x + y) m long and (x + 2y) m wide. (a) What is its area? (b) What is its perimeter? (c) If a flower bed of length (x + 2y) m and breadth y m is made inside it, how much smaller is the bed than the whole field? (d) If x = 10 and y = 2, what are the two areas?
2Factorize: (a) 6x + 3 (b) x² + 4x (c) 12a + 3b (d) 12p² + 6q² (e) x + x³
3Factorize by grouping: (a) ax + bx + ay + by (b) x² + 3x + xy + 3y (c) 2a² + 5a - 6a - 15
4Factorize: (a) x² - 4 (b) a² - 4b² (c) 9x² - y² (d) 5x² - 20y² (e) zx² - zy²
5A square sheet of side x cm has a square of side 3 cm cut out of the middle. What is the area of the shaded part that is left?
6A square garden of side x m has a square pond of side 6 m in the middle. What is the area of the garden apart from the pond?
7A large circle of radius R cm has a smaller circle of radius r cm inside it. (a) What is the area of the shaded ring between them, given that the area of a circle is πr²? (b) Find that area when R = 5 cm and r = 3 cm.
8Fill in the missing term to make a perfect square: (a) x² + . . . + 16 (b) 4a² + . . . + y² (c) p² - . . . + 36 (d) 9a² - . . . + 16b²
9Factorize: (a) a² + 12a + 36 (b) y² + 14y + 49 (c) x² - 8x + 16 (d) 25y² - 60y + 36
10Factorize: (a) a² + 12a + 36 - b² (b) 9r² - s² - 6s - 9
11Find two numbers whose product is P and whose sum is S: (a) P = 2, S = 3 (b) P = 15, S = 8 (c) P = 45, S = 18 (d) P = 160, S = 28
12Find two numbers whose product is P and whose difference is D: (a) P = 8, D = 2 (b) P = 36, D = 5 (c) P = 96, D = 10
13Factorize: (a) x² + 5x + 4 (b) x² - 5x + 6 (c) b² + 13b + 42 (d) z² - 13z + 36
14Factorize: (a) x² + 4x - 21 (b) x² + x - 20 (c) b² - 7b - 120 (d) (a + b)² + 5(a + b) - 36
15Factorize: (a) 3x² + 5x + 2 (b) 7x² - 30x + 8 (c) 6b² - 4b - 10
16Find the highest common factor: (a) 4x²y and xy² (b) 25x²y³ and 15xy² (c) x² - 4 and 3x + 6 (d) x² - 9 and x² - 6x + 9
17Find the lowest common multiple: (a) 2x and 4 (b) 3x²y and 6xy² (c) 2a and 2a + 4 (d) x² + 4x + 4 and x² + 2x
18Use algebraic tiles to factorize x² - 4, x² + 8x + 16, x² - 10x + 21 and x² + 2x - 15, paste the arrangements on chart paper and present them to the class. What are the factors?
19Use algebraic tiles to factorize 2x² - x - 3.
20Why can x² + 9 not be written as (x + 3)(x - 3)? Give the reason.
21Factorize x⁴ - 16 completely.
22A rectangular room has an area of x² + 7x + 12 square metres. What could its length and breadth be?
23Two algebraic expressions have HCF (x + 3) and LCM (x + 3)(x - 2)(x + 4). One expression is x² + x - 6. Find the other.
24A student wrote the factorization of 6x² + 9x as 3(2x² + 3x). Is this answer complete? Correct it and say why.