DAV Class 7 Maths Chapter 6 Worksheet 4

Solution:

\begin{align*} [(x + 3)(x^2 + 2x - 1)] \\ \end{align*}

\begin{align*} &= [(x + 3)(x^2 + 2x - 1)] \\ &= [x(x^2 + 2x - 1) + 3(x^2 + 2x - 1)] \\ &= x^3 + 2x^2 - x + 3x^2 + 6x - 3 \\ &= x^3 + 2x^2 + 3x^2 -x + 6x - 3 \\ &= x^3 + 5x^2 + 5x - 3 \end{align*}

Solution:

\begin{align*} [(7y - 2)(5y^2 - 3y + 2)] \\ \end{align*}

\begin{align*} &= [(7y - 2)(5y^2 - 3y + 2)] \\ &= [7y(5y^2 - 3y + 2) - 2(5y^2 - 3y + 2)] \\ &= [7y \times 5y^2 + 7y \times (-3y) + 7y \times 2 - 2 \times 5y^2 + 2 \times 3y - 2 \times 2] \\ &= [35y^3 - 21y^2 + 14y - 10y^2 + 6y - 4] \\ &= 35y^3 + (-21y^2 - 10y^2) + (14y + 6y) - 4 \\ &= 35y^3 - 31y^2 + 20y - 4 \end{align*}

Solution:

\begin{align*} [(p^3 + 3p + q)(9p + 2q)] \\ \end{align*}

\begin{align*} &= [(p^3 + 3p + q)(9p + 2q)] \\ &= [p^3(9p + 2q) + 3p(9p + 2q) + q(9p + 2q)] \\ &= [p^3 \times 9p + p^3 \times 2q + 3p \times 9p + 3p \times 2q + q \times 9p + q \times 2q] \\ &= 9p^4 + 2p^3q + 27p^2 + 6pq + 9pq + 2q^2 \\ &= 9p^4 + 2p^3q + 27p^2 + 15pq + 2q^2 \end{align*}

Solution:

\begin{align*} [(-2x^2 + xy - y^2)(3x + 4y)] \\ \end{align*}

\begin{align*} &= [(-2x^2 + xy - y^2)(3x + 4y)] \\ &= [-2x^2(3x + 4y) + xy(3x + 4y) - y^2(3x + 4y)] \\ &= -2x^2 \times 3x - 2x^2 \times 4y + xy \times 3x + xy \times 4y - y^2 \times 3x - y^2 \times 4y \\ &= -6x^3 - 8x^2y + 3x^2y + 4xy^2 - 3xy^2 - 4y^3 \\ &= -6x^3 - 5x^2y + xy^2 - 4y^3 \end{align*}

Solution:

\begin{align*} \left[\left(\frac{2}{5} a + \frac{1}{7}b\right) (3a + 4b - 2)\right] \\ \end{align*}

\begin{align*} &= \left[\left(\frac{2}{5} a + \frac{1}{7}b\right) (3a + 4b - 2)\right] \\ &= \left[\frac{2}{5} a (3a + 4b - 2) + \frac{1}{7}b (3a + 4b - 2)\right] \\ &= \left[\frac{2}{5} a \times 3a + \frac{2}{5} a \times 4b - \frac{2}{5} a \times 2 + \frac{1}{7}b \times 3a + \frac{1}{7}b \times 4b - \frac{1}{7}b \times 2\right] \\ &= \frac{6}{5} a^2 + \frac{8}{5} ab - \frac{4}{5} a + \frac{3}{7} ab + \frac{4}{7} b^2 - \frac{2}{7}b\\ &= \frac{6}{5} a^2 + \left(\frac{8}{5} + \frac{3}{7}\right) ab + \frac{4}{7} b^2 - \frac{4}{5} a - \frac{2}{7}b \\ &= \frac{6}{5} a^2 + \left(\frac{56}{35} + \frac{15}{35}\right) ab + \frac{4}{7} b^2 - \frac{4}{5} a - \frac{2}{7}b \\ &= \frac{6}{5} a^2 + \frac{4}{7} b^2 + \frac{71}{35} ab - \frac{4}{5} a - \frac{2}{7}b \end{align*}

Solution:

\begin{align*} (0.1a - 0.2c) (a + c + ac) \\ \end{align*}

\begin{align*} &= (0.1a - 0.2c) (a + c + ac) \\ &= 0.1a(a + c + ac) - 0.2c(a + c + ac) \\ &= 0.1a^2 + 0.1ac + 0.1a^2c - 0.2ac - 0.2c^2 - 0.2ac^2 \\ &= 0.1a^2 + (0.1 - 0.2)ac + 0.1a^2c - 0.2c^2 - 0.2ac^2 \\ &= 0.1a^2 - 0.1ac + 0.1a^2c - 0.2c^2 - 0.2ac^2 \end{align*}

Solution:

\begin{align*} (x^2 - 4xy + y^2) \times (x - 2y) \\ \text{Values: } & x = 3, y = 2 \\ \end{align*}

\begin{align*} &=(x^2 - 4xy + y^2) \times (x - 2y) \\ &= x^2 \times (x - 2y) - 4xy \times (x - 2y) + y^2 \times (x - 2y) \\ &= x^3 - 2x^2y - 4x^2y + 8xy^2 + xy^2 - 2y^3 \\ &= x^3 - 6x^2y + 9xy^2 - 2y^3 \\ \\ & \text{Verification with } x = 3 \text{ and } y = 2 \\ & \text{L.H.S} \\ &=(x^2 - 4xy + y^2) \times (x - 2y) \\ &=(3^2 - 4 \times 3 \times 2 + 2^2) \times (3 - 2 \times 2) \\ &=(9 - 24 + 4) \times (3 - 4) \\ &=-11 \times -1 \\ &= 11 \\ \\ & \text{R.H.S} \\ &= x^3 - 6x^2y + 9xy^2 - 2y^3 \\ &= 3^3 - 6 \times 3^2 \times 2 + 9 \times 3 \times 2^2 - 2 \times 2^3 \\ &= 27 - 108 + 108 - 16 \\ &= 11 \\ \\ & \text{L.H.S = R.H.S} \\ & \text{Hence Verified} \\ \end{align*}

Solution:

\begin{align*} (7x^2y - 3z^2) \times (x + y + z) \\ \text{Values: } & x =1, y = 1, z = -1 \\ \end{align*}

\begin{align*} &=(7x^2y - 3z^2) \times (x + y + z) \\ &= 7x^2y \times (x + y + z) - 3z^2 \times (x + y + z) \\ &= 7x^3y + 7x^2y^2 + 7x^2yz - 3xz^2 - 3yz^2 - 3z^3 \\ \\ & \text{Verification with } x = 1, y = 1, \text{ and } z = -1 \\ & \text{L.H.S} \\ &=(7x^2y - 3z^2) \times (x + y + z) \\ &=(7(1)^2(1) - 3(-1)^2) \times (1 + 1 - 1) \\ &=(7 - 3) \times 1 \\ &= 4 \\ \\ & \text{R.H.S} \\ &= 7x^3y + 7x^2y^2 + 7x^2yz - 3xz^2 - 3yz^2 - 3z^3 \\ &= 7(1)^3(1) + 7(1)^2(1)^2 + 7(1)^2(1)(-1) - 3(1)(-1)^2 - 3(1)(-1)^2 - 3(-1)^3 \\ &= 7 + 7 - 7 - 3 - 3 + 3 \\ &= 7 - 3 \\ &= 4 \\ \\ & \text{L.H.S = R.H.S} \\ & \text{Hence Verified} \\ \end{align*}

Solution:

\begin{align*} (4a^2 - 6ab + 9b^2) \times (2a + 3b) \\ \text{Values: } a = 2, b = 1 \\ \end{align*}

\begin{align*} &=(4a^2 - 6ab + 9b^2) \times (2a + 3b) \\ &= 4a^2 \times (2a + 3b) - 6ab \times (2a + 3b) + 9b^2 \times (2a + 3b) \\ &= 8a^3 + \cancel{12a^2b} - \cancel{12a^2b} - \cancel{18ab^2} + \cancel{18ab^2} + 27b^3 \\ &= 8a^3 + 27b^3 \\ \\ & \text{Verification with } a = 2 \text{ and } b = 1 \\ & \text{L.H.S} \\ &=(4a^2 - 6ab + 9b^2) \times (2a + 3b) \\ &=(4(2)^2 - 6(2)(1) + 9(1)^2) \times (2(2) + 3(1)) \\ &=(16 - 12 + 9) \times (4 + 3) \\ &= 13 \times 7 \\ &= 91 \\ \\ & \text{R.H.S} \\ &= 8a^3 + 27b^3 \\ &= 8(2)^3 + 27(1)^3 \\ &= 8 \times 8 + 27 \\ &= 64 + 27 \\ &= 91 \\ \\ & \text{L.H.S = R.H.S} \\ & \text{Hence Verified} \\ \end{align*}

Solution:

\begin{align*} (m^2 - 10m + 25) \times (m - 5) \\ \text{Values: } m = -2 \\ \end{align*}

\begin{align*} &=(m^2 - 10m + 25) \times (m - 5) \\ &= m^2 \times (m - 5) - 10m \times (m - 5) + 25 \times (m - 5) \\ &= m^3 - 5m^2 - 10m^2 + 50m + 25m - 125 \\ &= m^3 - 15m^2 + 75m - 125 \\ \\ & \text{Verification with } m = -2 \\ & \text{L.H.S} \\ &=(m^2 - 10m + 25) \times (m - 5) \\ &=((-2)^2 - 10(-2) + 25) \times (-2 - 5) \\ &=(4 + 20 + 25) \times (-7) \\ &= 49 \times -7 \\ &= -343 \\ \\ & \text{R.H.S} \\ &= m^3 - 15m^2 + 75m - 125 \\ &= (-2)^3 - 15(-2)^2 + 75(-2) - 125 \\ &= -8 - 60 - 150 - 125 \\ &= -343 \\ \\ & \text{L.H.S = R.H.S} \\ & \text{Hence Verified} \\ \end{align*}

Solution:

\begin{align*} (p^2 + q^2 + r^2) \times (pq + qr) \\ \text{Values: } p = 2, q = -3, r = 1 \\ \end{align*}

\begin{align*} &=(p^2 + q^2 + r^2) \times (pq + qr) \\ &= p^2 \times (pq + qr) + q^2 \times (pq + qr) + r^2 \times (pq + qr) \\ &= p^3q + p^2qr + pq^3 + q^3r + pqr^2 + qr^3 \\ \\ & \text{Verification with } p = 2, q = -3, \text{ and } r = 1 \\ & \text{L.H.S} \\ &=(p^2 + q^2 + r^2) \times (pq + qr) \\ &=((2)^2 + (-3)^2 + (1)^2) \times (2 \times -3 + 1 \times -3) \\ &=(4 + 9 + 1) \times (-6 - 3) \\ &=14 \times -9 \\ &=-126 \\ \\ & \text{R.H.S} \\ &= p^3q + p^2qr + pq^3 + q^3r + pqr^2 + qr^3 \\ &= 2^3(-3) + 2^2(-3)(1) + 2(-3)^3 + (-3)^3(1) + (2)(-3)(1)^2 + (-3)(1)^3 \\ &= -24 - 12 - 54 - 27 - 6 - 3 \\ &= -126 \\ \\ & \text{L.H.S = R.H.S} \\ & \text{Hence Verified} \\ \end{align*}

Solution:

\begin{align*} \left(\frac{5}{4} x^2 - \frac{3}{2} xy\right) \times (x + y + y^2) \\ \text{Values: } x = 2, y = 2 \\ \end{align*}

\begin{align*} &=\left(\frac{5}{4} x^2 - \frac{3}{2} xy\right) \times (x + y + y^2) \\ &= \frac{5}{4} x^2 \times (x + y + y^2) - \frac{3}{2} xy \times (x + y + y^2) \\ &= \frac{5}{4} x^3 + \frac{5}{4} x^2y + \frac{5}{4} x^2y^2 - \frac{3}{2} x^2y - \frac{3}{2} xy^2 - \frac{3}{2} xy^3 \\ \\ &= \frac{5}{4} x^3 + \frac{5}{4} x^2y - \frac{3}{2} x^2y + \frac{5}{4} x^2y^2 - \frac{3}{2} xy^2 - \frac{3}{2} xy^3 \\ \\ &= \frac{5}{4} x^3 - \frac{1}{4} x^2y + \frac{5}{4} x^2y^2 - \frac{3}{2} xy^2 - \frac{3}{2} xy^3 \\ \\ & \text{Verification with } x = 2, y = 2 \\ & \text{L.H.S} \\ &=\left(\frac{5}{4} x^2 - \frac{3}{2} xy\right) \times (x + y + y^2) \\ &=\left(\frac{5}{4} (2)^2 - \frac{3}{2} (2)(2)\right) \times (2 + 2 + 2^2) \\ &=\left(\frac{5}{4} \cdot 4 - \frac{3}{2} \cdot 4\right) \times (2 + 2 + 4) \\ &=\left(5 - 6\right) \times 8 \\ &=-1 \times 8 \\ &=-8 \\ \\ & \text{R.H.S} \\ &= \frac{5}{4} x^3 - \frac{1}{4} x^2y + \frac{5}{4} x^2y^2 - \frac{3}{2} xy^2 - \frac{3}{2} xy^3 \\ &= \frac{5}{4} (2)^3 - \frac{1}{4} (2)^2(2) + \frac{5}{4} (2)^2(2)^2 - \frac{3}{2} (2)(2)^2 - \frac{3}{2} (2)(2)^3 \\ &= \frac{5}{4} \times 8 - \frac{1}{4} \times 4 \times 2 + \frac{5}{4} \times 4 \times 4 - \frac{3}{2} \times 2 \times 4 - \frac{3}{2} \times 2 \times 8 \\ &= 10 - 2 + 20 - 12 - 24 \\ &= 28 - 36 \\ &= -8 \\ \\ & \text{L.H.S = R.H.S} \\ & \text{Hence Verified} \\ \end{align*}