DAV Class 7 Maths Chapter 6 Worksheet 5

by

1. Express the following as a product of its any two factors (in four different ways).

a. 12x2y

Solution:

\begin{align*} 12x^2y \\ \end{align*}

\begin{align*} 1. & \quad 4x^2y \times 3 \\ 2. & \quad 2x^2 \times 6y \\ 3. & \quad 3xy \times 4x \\ 4. & \quad 2x \times 6xy \\ \end{align*}

b. 18ab2

Solution:

\begin{align*} 18ab^2 \\ \end{align*}

\begin{align*} 1. & \quad 9a \times 2b^2 \\ 2. & \quad 18a \times b^2 \\ 3. & \quad 3ab \times 6b \\ 4. & \quad 6a \times 3b^2 \\ \end{align*}

c. 24c2b

Solution:

\begin{align*} 24c^2b \\ \end{align*}

\begin{align*} 1. & \quad 2c \times 12cb \\ 2. & \quad 3cb \times 8c \\ 3. & \quad 4b \times 6c^2 \\ 4. & \quad 24c \times cb \\ \end{align*}

2. Find the H.C.F of the following monomials

a. 2a5 and 12a2

Solution:

\begin{align*} 2a^5 \text{ and } 12 a^2 \\ \end{align*}

\begin{align*} \text{Factors of 2:} & \quad 1, \boxed{2} \\ \text{Factors of 12:} & \quad 1, \boxed{2}, 3, 4, 6, 12 \\ \text{HCF of 2 and 12:} & \quad 2 \\ \\ \text{Highest common power of a :}& \quad a^2 \\ \\ \text{HCF = }& \quad 2a^2 \\ \\ \end{align*}

b. 9x3y and 18x2y3

Solution:

\begin{align*} 9x^3y \text{ and } 18x^2y^3 \\ \end{align*}

\begin{align*} \text{Factors of 9:} & \quad 1, 3, \boxed{9} \\ \text{Factors of 18:} & \quad 1, 3, 6, \boxed{9}, 18 \\ \text{HCF of 9 and 18:} & \quad 9 \\ \\ \text{Highest common power of \( x \):}& \quad x^2 \\ \text{Highest common power of \( y \):}& \quad y \\ \\ \text{HCF = }& \quad 9x^2y \\ \\ \end{align*}

c. a2b3 and a3b2

Solution:

\begin{align*} a^2b^3 \text{ and } a^3b^2 \\ \end{align*}

\begin{align*} \text{Highest common power of \( a \):}& \quad a^2 \\ \text{Highest common power of \( b \):}& \quad b^2 \\ \\ \text{HCF = }& \quad a^2b^2 \\ \end{align*}

d. 15a3, -45a2, 150a

Solution:

\begin{align*} 15a^3 , -45a^2 , 150a \\ \end{align*}

\begin{align*} \text{Factors of 15:} & \quad 1, 3, 5, \boxed{15} \\ \text{Factors of -45:} & \quad 1, 3, 5, 9, \boxed{15},45 \\ \text{Factors of 150:} & \quad 1, 2, 3, 5, 6, 10, \boxed{15}, 25, 30, 50, 75, 150 \\ \text{HCF of 15, -45, and 150:} & \quad 15 \\ \\ \text{Highest common power of \( a \):}& \quad a^1 \\ \\ \text{HCF = }& \quad 15a \\ \end{align*}

e. 2x3y2, 10x2y3, 14xy

Solution:

\begin{align*} 2x^3y^2 , 10x^2y^3 , 14xy \\ \end{align*}

\begin{align*} \text{Factors of 2:} & \quad 1, \boxed{2} \\ \text{Factors of 10:} & \quad 1, \boxed{2}, 5, 10 \\ \text{Factors of 14:} & \quad 1, \boxed{2}, 7, 14 \\ \text{HCF of 2, 10, and 14:} & \quad 2 \\ \\ \text{Highest common power of } x: & \quad x^1 \\ \text{Highest common power of } y: & \quad y^1 \\ \\ \text{HCF = } & \quad 2xy \\ \end{align*}

f. x3y2, -8y2

Solution:

\begin{align*} x^3y^2, -8y^2 \\ \end{align*}

\begin{align*} \text{HCF of 1 and 8:} & \quad 1 \text{ (since 1 is the only common factor)} \\ \\ \text{Highest common power of } x: & \quad x^0 \\ \text{Highest common power of } y: & \quad y^2 \\ \\ \text{HCF = } & \quad y^2 \\ \end{align*}

3. Find the H.C.F of the terms and factorise

a. 5y - 15y2

Solution:

\begin{align*} 5y - 15y^2 \\ \end{align*}

\begin{align*} \text{HCF of 5 and -15 :} & \quad 5 \\ \text{Highest common power } (y): & \quad y \\ \\ \text{HCF = } & \quad 5y \\ \\ 5y - 15y^2 &= (5y \times 1) - (5y \times 3y) \\ & = 5y(1 - 3y) \\ \\ \text{Answer:} \quad & = 5y(1 - 3y) \end{align*}

b. 16m - 4m2

Solution:

\begin{align*} 16m - 4m^2 \\ \end{align*}

\begin{align*} \text{HCF of 16 and -4:} & \quad 4 \\ \text{Highest common power of } m: & \quad m \\ \\ \text{HCF = } & \quad 4m \\ \\ 16m - 4m^2 &= (4m \times 4) - (4m \times m) \\ &= 4m(4 - m) \\ \\ \text{Answer:} \quad & = 4m(4 - m) \end{align*}

c. 8x3y2 + 8x3

Solution:

\begin{align*} 8x^3y^2 + 8x^3 \\ \end{align*}

\begin{align*} \text{HCF of 8 and 8:} & \quad 8 \\ \text{Highest common power of } x: & \quad x^3 \\ \\ \text{HCF = } & \quad 8x^3 \\ \\ 8x^3y^2 + 8x^3 &= (8x^3 \times y^2)+ (8x^3 \times 1) \\ & = 8x^3(y^2 + 1)\\ \\ \text{Answer:} \quad & = 8x^3(y^2 + 1) \\ \end{align*}

d. 20x3 - 40x2 + 80x

Solution:

\begin{align*} 20x^3 - 40x^2 + 80x \\ \end{align*}

\begin{align*} \text{HCF of 20, -40, and 80:} & \quad 20 \\ \text{Highest common power of } x: & \quad x \\ \\ \text{HCF = } & \quad 20x \\ \\ 20x^3 - 40x^2 + 80x &= (20x \times x^2) - (20x \times 2x) + (20x \times 4) \\ & = 20x(x^2 - 2x + 4) \\ \\ \text{Answer: } \quad & = 20x(x^2 - 2x + 4) \end{align*}

e. x4y - 3x2y2 - 6xy3

Solution:

\begin{align*} x^4y - 3x^2y^2 - 6xy^3 \\ \end{align*}

\begin{align*} \text{HCF of 1, -3, and -6: } & \quad 1 \\ \text{Highest common power of } x: & \quad x \\ \text{Highest common power of } y: & \quad y \\ \\ \text{HCF = } & \quad xy \\ \\ x^4y - 3x^2y^2 - 6xy^3 &= (xy \times x^3) - (xy \times 3xy) - (xy \times 6y^2) \\ & = xy(x^3 - 3xy - 6y^2) \\ \\ \text{Answer: } \quad & = xy(x^3 - 3xy - 6y^2) \end{align*}

f. 8x2y2- 16xy3 + 24xy

Solution:

\begin{align*} 8x^2y^2 - 16xy^3 + 24xy \\ \end{align*}

\begin{align*} \text{HCF of 8, -16, and 24:} & \quad 8 \\ \text{Highest common power of } x: & \quad x \\ \text{Highest common power of } y: & \quad y \\ \\ \text{HCF = } & \quad 8xy \\ \\ 8x^2y^2 - 16xy^3 + 24xy &= 8xy \times xy - 8xy \times 2y^2 + 8xy \times 3 \\ & = 8xy(xy - 2y^2 + 3) \\ \\ \text{Answer: } \quad & = 8xy(xy - 2y^2 + 3) \end{align*}