1State whether each sentence is true or false: (a) adding \( x \) and 3 gives \( 3x \); (b) adding \( 2a \) and \( 4a^2 \) gives \( 6a^2 \); (c) subtracting \( 2x \) from \( 5x \) gives \( 3x \); (d) in \( x - 2 = 8 \) the value of \( x \) is 10; (e) if \( y = 1 \) then \( y^2 + 1 \) is 3; (f) if \( x = 2 \) and \( y = 3 \) then \( x^2 + y^2 \) is 3.
2Say whether each letter stands for a variable or a constant: (a) \( x \) is the number of books you have; (b) \( y \) is the total number of teachers in your school; (c) \( z \) is the odd number between 2 and 4; (d) the value of \( a \) is 7; (e) \( b \) stands for the counting numbers between 1 and 9; (f) \( c \) is the sum of 1 and 2.
3Write an algebraic expression for each, and say how many terms it has: (a) Ram had \( x \) exercise books and his sister gave him 5 more; (b) Resma gave 3 of her \( y \) pens to a friend; (c) twice as many sticks were added to \( x \) sticks; (d) of \( z \) students, \( y \) were absent; (e) a friend gave Hari one more biscuit to add to his \( a \) biscuits.
4Find the sum: (a) \( 3s \) and \( 7s \); (b) \( 2xy \) and \( 5xy \); (c) \( 6a^2 \) and \( 8a^2 \); (d) \( a \), \( 2a \) and \( 3a \); (e) \( 3x^2y \), \( 4x^2y \) and \( 5xy^2 \); (f) \( m^2n \), \( 2mn^2 \) and \( 6mn^2 \); (g) \( (2a + 3) \) and \( (3a + 5) \); (h) \( (3x^2 + 4x + 5) \) and \( (5x^2 + 6x + 7) \).
5Find the difference: (a) \( (2x + 2y) \) from \( (3x + 4y) \); (b) \( (a + b) \) from \( (6a + 4b) \); (c) \( (2x + 3y + 4z) \) from \( (6x + 7y + 8z) \); (d) \( (a - 2b - 3c) \) from \( (7a - 5b - 7c) \); (e) \( (x^2 - xy + 2y^2) \) from \( (2x^2 - xy + y^2) \).
6Simplify: (a) \( 4a + 6a - 9a \); (b) \( 5x - 6x + 3x \); (c) \( 6m^2 + 3m^2 - 9m^2 \); (d) \( (x^2 + xy + y^2) - (x^2 - xy + y^2) \); (e) \( (5x^2 + xy + y^2) + (3x^2 + 2xy + 4y^2) \); (f) \( (2a - 3b + 7c) - (2a + 3b + 7c) \); (g) \( (a + 2b + 3c) - (3a + 4b + 5c) \); (h) \( (6x^3 - 2x^2y - y^3) + (4x^3 + x^2y + 3y^3) \).
7Find the total length of each line: (a) AB = \( x \) cm and BC = \( 2x \) cm; (b) AB = \( x \) cm and BC = \( (x + 5) \) cm; (c) AB = \( x \) cm, BC = \( 2x \) cm and CD = \( 3x \) cm; (d) AB = \( x \) cm, BC = \( 2x \) cm and CD = \( (x + 6) \) cm.
8Fill in the blanks: (a) the product of \( x \) and 1 is ....... ; (b) \( 2a \) multiplied by 0 gives ....... ; (c) the product of \( 3x \) and 4 is ....... ; (d) \( 5y \) multiplied by ....... gives \( 15y \); (e) \( 3y(2x - 2) = \) ....... ; (f) if \( p = 2 \) then \( p^2 - 1 \) is ....... ; (g) if \( a = 2 \) and \( b = 3 \) then \( a(a + b) \) is ....... .
9Multiply: (a) \( 0 \times a \); (b) \( 1 \times b \); (c) \( 2a \times 9b \); (d) \( a \times b \times c \); (e) \( 8b \times 9c \); (f) \( 2p \times 3q \times 4r \); (g) \( 2a \times 5a \); (h) \( 4x \times 7x \); (i) \( 3l \times 4m \times 5n \); (j) \( 2x \times 4y \times 5y \); (k) \( 5p \times 6p \times 7q \); (l) \( \frac{2x}{5} \times 20y \times \frac{z}{4} \).
10Multiply: (a) \( a \times (b + c) \); (b) \( 2x \times (3y + 4z) \); (c) \( 2p \times (q + 3) \); (d) \( 3l \times (2l + 3m) \); (e) \( 3a \times (5a + 7b) \); (f) \( 2x \times (3x + 4y) \); (g) \( 5x \times (4x - 5y) \).
11A rectangle has length \( 3x \) and breadth \( 2x \). (a) What is its area? (b) What is its perimeter? (c) If \( x = 7 \) cm, what are the area and the perimeter?
12The area (A) of a rectangle is length (l) × breadth (b). Find the area of a rectangle whose sides are: (a) \( 2a \) cm and \( 2b \) cm; (b) \( 5a \) cm and \( 6b \) cm; (c) \( (b + c) \) cm and \( 2a \) cm; (d) \( (5x + 6y) \) cm and \( 7x \) cm; (e) \( (a + 2b) \) cm and \( 3a \) cm; (f) \( (4y + 7x) \) cm and \( 9y \) cm.
13Taking \( a = 1 \), \( b = 2 \), \( c = 3 \), \( x = 4 \) and \( y = 5 \), find the actual area of each of those rectangles.
14The volume (V) of a cuboid is length (l) × breadth (b) × height (h). Find the volume when the three measurements are: (a) \( 2a \), \( a \), \( a \) cm; (b) \( 3c \), \( b \), \( 2a \) cm; (c) \( 3x \), \( 2y \), \( 4z \) cm; (d) \( 5a \), \( 2b \), \( 3x \) cm; (e) \( 7l \), \( 3m \), \( 4n \) cm.
15If \( a = -2 \), \( b = 3 \) and \( c = 4 \), find the value of: (a) \( a + b \); (b) \( a - b + c \); (c) \( 2a + 3b + 4c \); (d) \( 5a - 3c + 7b \); (e) \( 3a \times 4b \times 5c \); (f) \( a^2 + 2abc + b^2 \); (g) \( b^2 + 2bc - ac^2 \).
16Divide: (a) \( ab \div a \); (b) \( 3x^2y \div y \); (c) \( 20x^3y^2 \div 4x^2y \); (d) \( 12x^3y^3z^2 \div 3x^3y^2z \); (e) \( 36l^7m^3n^2 \div 4l^6m^2n \); (f) \( 100p^4q^7r^6 \div 25p^3q^4r^5 \).
17Divide: (a) \( (xy + xz) \div x \); (b) \( (x^2 - 2xy) \div x \); (c) \( (a^2bc + abc^2) \div abc \); (d) \( (5l^2m - 15lm^2) \div 5lm \); (e) \( (14p^3q^2 + 49p^2q^3) \div 7p^2q^2 \); (f) \( (36ax^3y^3 - 18bx^2y^2) \div 9x^2y^2 \); (g) \( (40u^3v^2 - 24u^2v^3) \div 8u^2v^2 \).
18(a) A rectangular plot has an area of \( 9x^2y^3 \) square metres. What could its length and breadth be? (b) A rectangular garden has an area of \( 32a^2b^2 \) square metres. What could its length and breadth be?
19Find the missing side of each rectangle: (a) area \( 9x^2y \) cm² with one side \( 3xy \) cm; (b) area \( 25a^2b^2c \) cm² with one side \( 5abc \) cm; (c) area \( 24p^4q^3r^2 \) cm² with one side \( 8p^3q^2r \) cm; (d) area \( (4u^2v - 6uv^2) \) cm² with one side \( 2uv \) cm.
20A rectangle ABCD has length \( (2x + y) \) and breadth \( 2x \). (a) What is its perimeter? (b) What is its area? (c) The diagonal DB divides the rectangle into two equal parts, so what is the area of triangle ABD? (d) How much must be taken off the length to make it a square? (e) What is the area of that square?
21Are \( 5p^2q \) and \( 5pq^2 \) like terms or unlike terms? Give the reason.
22In a shop one exercise book costs Rs. \( x \) and one pen costs Rs. \( y \). Sita buys 4 exercise books and 3 pens. (a) Write an algebraic expression for what she pays. (b) If \( x = 40 \) and \( y = 15 \), how much does she pay?
23Work out \( (12a^2b + 18ab^2) \div 6ab \) and check that the answer is right.
24A student wrote \( 7m - 3n - (2m - 5n) \) as \( 5m - 8n \). Where is the mistake, and what is the correct answer?