Exercise 12.3 Algebraic-Expressions -NCERT Solutions Class 7

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Chapter 12 Ex.12.3 Question 1

If $$m = 2$$, find the value of:

(i)  $$m = 2$$

(ii) $$3m - 5$$

(iii) $${\rm{ }}9{\rm{ }}-{\rm{ }}5m$$

(iv) $$3{m^2} - 2m - 7$$

(v) \begin{align} \frac{{5m}}{2} - 4\end{align}

Solution

What is Known?

Value of $$m$$.

What is unknown?

Value of the given expressions.

Reasoning:

This is based on concept of putting given value of variable and then performing the arithmetic operation as given in the question.

Steps:

Value of $$m$$ is given as $$2$$.

(i)  $$m = 2$$

\begin{align}& {\rm{ = 2}} - {\rm{2}}\\{\rm{ Ans }} &\,{\rm{ = }}\,{\rm{0 }}\end{align}

(ii) $$3m - 5$$

\begin{align}& = 3 \times 2 - \left( 5 \right)\\& = 6 - 5\\{\rm{ Ans }} &= 1\end{align}

(iii) $${\rm{ }}9{\rm{ }}-{\rm{ }}5m$$

\begin{align}& = 9 - \left( {5 \times 2} \right)\\& = 9 - 10\\{\rm{ Ans }} &= - 1\end{align}

(iv) $$3{m^2} - 2m - 7$$

\begin{align}& = 3{{\left( 2 \right)}^2} - \left( {2 \times 2} \right) - 7\\& = 3 \times 2 \times 2 - \left( 4 \right) - 7\\& = 12 - 4 - 7\\{\rm{ Ans }} & = 1\end{align}

(v) \begin{align} \frac{{5m}}{2} - 4\end{align}

\begin{align}& = \frac{{5 \times 2}}{2} - 4\\& = \frac{{10}}{2} - 4\\& = 5 - 4\\{\rm{ Ans }} & = 1\end{align}

Chapter 12 Ex.12.3 Question 2

If $$p = \,– 2$$, find the value of:

(i) $$4p + 7$$

(ii) $$- 3{p^2} + 4p + 7$$

(iii) $$- 2{p^3} - 3{p^2} + 4p + 7$$

Solution

What is known?

Value of $$p$$.

What is unknown?

Value of the given expressions.

Reasoning:

This is based on concept of putting given value of variable and then performing the arithmetic operation as given in the question.

Steps:

Value of $$p$$ is given as $$- 2$$

(i) $$4p + 7$$

\begin{align} &=\! 4 \!\times\!\! -\! 2 \!+\! \left( 7 \right)\\ &= \!-\! 8 \!+\! 7\\&{\rm{ Ans }} = \!-\! 1\end{align}

(ii) $$- 3{p^2} + 4p + 7$$

\begin{align}&= \!-\! 3 \!\times\! {{\left( { - 2} \right)}^2} \!+\!4 \!\times\! \left( { - 2} \right) \!+\! 7\\ &= \! \left( { - 3 \!\times\! \!- 2 \!\times\! \!- 2} \right) \!+\! \left( { - 8} \right) \!+\! 7\\ &= \!-\! 12 \!-\! 8 \!+\! 7\\&{\rm{ Ans }} = \!-\! 13\end{align}

(iii) $$- 2{p^3} - 3{p^2} + 4p + 7$$

\begin{align} &=\!-\!2{{\left( -2 \right)}^{3}}\!-\!3{{\left( -2 \right)}^{2}}\!+\!4\left( -2 \right)\!+\!7 \\ &=\!-\!2 \!\times\! \!-2\!\times\! \!-2\!\times\!\! -2\left( 3\!\times\! \!-2\!\times\! \!-2 \right)\!+\!\left( 4\!\times\! \!-2 \right)\!+\!7 \\ &=\!16\left( 12 \right)\!+\!\left( -8 \right)\!+\!7 \\ &\text{Ans} =\!3 \end{align}

Chapter 12 Ex.12.3 Question 3

Find the value of the following expressions, when $$x = \,–1$$

(i) $$2x - 7$$

(ii) $$- x + 2$$

(iii) $${x^2} + 2x + 1$$

(iv) $$2{x^2} - x - 2$$

Solution

What is Known?

Value of $$x$$

What is unknown?

Value of the given expressions.

Reasoning:

This is based on concept of putting given value of variable and then performing the arithmetic operation as given in the question.

Steps:

Value of $$x$$ is given as $$–1$$

(i) $$2x - 7$$

\begin{align}& = 2 \times - 1 - (7)\\& = - 2 - 7\\{\rm{ Ans }} & = - 9\end{align}

(ii) $$- x + 2$$

\begin{align}& = - \left( { - 1} \right) + 2\\& = 1 + 2\\{\rm{Ans}} & = 3\end{align}

(iii) $${x^2} + 2x + 1$$

\begin{align}& = {\left( { - 1} \right)^2} + \left( {2 \times - 1} \right) + 1\\& = - 1 \times - 1 + \left( { - 2} \right) + 1\\& = 1 - 2 + 1\\{\rm{ Ans}} & = 0\end{align}

(iv) $$2{x^2} - x - 2$$

\begin{align}&= 2{{\left( { - 1} \right)}^2} - \left( { - 1} \right) - 2\\&= 2 \times - 1 \times - 1 + 1 - 2\\&= 2 + 1 - 2\\{\rm{ Ans }} &= 1\end{align}

Chapter 12 Ex.12.3 Question 4

If $$a = 2$$, $$b =\, – 2$$, find the value of:

(i) $${a^2} + {b^2}$$

(ii) $${a^2} + ab + {b^2}$$

(iii) $${a^2} - {b^2}$$

Solution

What is Known?

Value of $$a$$ and $$b$$

What is unknown?

Value of the given expressions.

Reasoning:

This is based on concept of putting given value of variable and then performing the arithmetic operation as given in the question.

Steps:

Value of $$a$$ is given as $$2$$ and $$b$$ is $$-2$$

(i) $${a^2} + {b^2}$$

\begin{align}& = {2^2} + {\left( { - 2} \right)^2}\\& = \left( {2 \times 2} \right) + \left( { - 2 \times - 2} \right)\\& = 4 + 4\\{\rm{Ans}} & = 8\,\end{align}

(ii) $${a^2} + ab + {b^2}$$

\begin{align} & = {2^2} + \left\{ {\left( 2 \right) \times \left( { - 2} \right)} \right\} + {\left( { - 2} \right)^2}\\& = 4 + \left( { - 4} \right) + 4\\& = 4 - 4 + 4\\{\rm{Ans}} & = 4\,\end{align}

(iii) $${a^2} - {b^2}$$

\begin{align}& = {2^2} - {\left( { - 2} \right)^2}\\& = 4 - 4\\{\rm{ Ans}} & = 0\end{align}

Chapter 12 Ex.12.3 Question 5

When $$a = 0$$, $$b = \,– 1$$, find the value of the given expressions:

(i) $$2a+\text{ }2b$$

(ii) $$2{{a}^{2}}+{{b}^{2}}+\text{ }1$$

(iii) $$2{{a}^{2}}b+\text{ }2a{{b}^{2}}+ab$$

(iv) $${{a}^{2}}+ab+\text{ }2$$

Solution

What is Known?

Value of $$a$$ and $$b$$

What is unknown?

Value of the given expressions.

Reasoning:

This is based on concept of putting given value of variable and then performing the arithmetic operation as given in the question.

Steps:

Value of $$a$$ is given as $$0$$ and $$b$$ is $$–1$$

(i) $$2a+\text{ }2b$$

\begin{align}& = \left( {2 \times 0} \right) + \left( {2 \times - 1} \right)\\& = 0 + \left( { - 2} \right)\\{\rm{ Ans }} & = - 2\end{align}

(ii) $$2{{a}^{2}}+{{b}^{2}}+\text{ }1$$

\begin{align}& = \left( {2 \times {0^2}} \right) + {\left( { - 1} \right)^2} + 1\\ & = 0 + 1 + 1\\& = 2\end{align}

(iii) $$2{{a}^{2}}b+\text{ }2a{{b}^{2}}+ab$$

\begin{align} &= \left[ \begin{array}{l}\,\,2 \times 0 \times 0 \times - 1 + \\\left( {2 \times 0 \times - {1^2}} \right) + 0 \times - 1\end{array} \right]\\ &= 0 + 0 + 0\\{\rm{Ans}} &= 0\end{align}

(iv) $${{a}^{2}}+ab+\text{ }2$$

\begin{align}& = {0^2} + 0 \times - 1 + 2\\& = 0 + 0 + 2\\{\rm{ Ans }} & = 2\end{align}

Chapter 12 Ex.12.3 Question 6

Simplify the expressions and find the value if $$x$$ is equal to $$2$$

(i) $$x + 7 + 4\left( {x - 5} \right)$$

(ii) $$3\left( {x + 2} \right) + 5x - 7$$

(iii) $$6x + 5\left( {x - 2} \right)$$

(iv) $$4\left( {2x - 1} \right) + 3x + 11$$

Solution

What is Known?

Value of $$x$$

What is unknown?

Value of the given expressions.

Reasoning:

This is based on concept of simplification of like terms and then putting given value of variable and then performing the arithmetic operation as given in the question. Value of $$x$$ is given as $$2$$

Steps:

(i) $$x + 7 + 4\left( {x - 5} \right)$$

\begin{align}& =x+7+4x-20 \\ & =5x-13\end{align}

Now putting value of $$x=2$$

\begin{align}&5x-13 \\ & =\left( 5\times 2 \right)-13 \\ & =10-13 \\ \text{ Ans} & =-3 \\ \end{align}

(ii) $$3\left( {x + 2} \right) + 5x - 7$$

\begin{align}&=3x+6+5x-7 \\&=8x-1\end{align}

Now putting value of $$x=2$$

\begin{align}&=\left( 8\times 2 \right)-1 \\&=16-1 \\\text{ Ans} &=15 \\\end{align}

(iii) $$6x + 5\left( {x - 2} \right)$$

\begin{align}& = 6x + 5x - 10\\& = 11x - 10\end{align}

Now putting value of $$x = 2$$

\begin{align}& = \left( {11 \times 2} \right) - 10\\\text{ Ans} & = 12\end{align}

(iv) $$4\left( {2x - 1} \right) + 3x + 11$$

\begin{align}& = 8x - 4 + 3x + 11\\& = 11x + 7\end{align}

Now putting value of  $$x = 2$$

\begin{align}&= (11 \times 2) + 7\\ & = 22 + 7\\\text{ Ans} & = 29\end{align}

Chapter 12 Ex.12.3 Question 7

Simplify these expressions and find their values if $$x = 3$$, $$a = \,– 1$$, $$b =\, – 2$$.

(i) $$3x - 5 - x + 9$$

(ii) $$2 - 8x + 4x + 4$$

(iii) $$3a + 5 - 8a + 1$$

(iv) $$10 - 3b - 4 - 5b$$

(v) $$2a - 2b - 4 - 5 + a$$

Solution

What is Known?

Value of $$x$$, $$a$$ and $$b$$

What is unknown?

Value of the given expressions.

Reasoning:

This is based on concept of simplification of like terms and then putting given value of variable and then performing the arithmetic operation as given in the question.

Steps:

Value of $$x$$ is given as $$3$$, $$a$$ as $$-1$$ and $$b$$ is $$-2$$

(i) $$3x - 5 - x + 9$$

\begin{align}& = 2x + 4\end{align}

Now putting value of $$x = 3$$

\begin{align}& = \left( {2 \times 3} \right) + 4\\& = 6 + 4\\{\text{ Ans }} & = 10\end{align}

(ii) $$2 - 8x + 4x + 4$$

\begin{align}& = - 4x + 6\end{align}

Now putting value of $$x = 3$$

\begin{align}&= \left( { - 4 \times 3} \right) + 6\\& = - 12 + 6\\{\text{ Ans }} & = - 6\end{align}

(iii) $$3a + 5 - 8a + 1$$

\begin{align}= - 5a + 6\end{align}

Now putting value of $$a = - 1$$

\begin{align}&= \left( { - 5\,\, \times - 1} \right) + 6\\ & = 5 + 6\\{\text{ Ans }} & = 11\end{align}

(iv) $$10 - 3b - 4 - 5b$$

\begin{align}& = - 8b + 6\end{align}

Now putting value of  $$b=- 2$$

\begin{align}& = ( - 8 \times - 2) + 6\\& = 16 + 6\\{\text{ Ans }} & = 22\end{align}

(v) $$2a - 2b - 4 - 5 + a$$

\begin{align}& = 3a - 2b - 9\end{align}

Now putting value of $$a = - 1$$ and $$b = - 2$$

\begin{align} &\left( {3 \times - 1} \right) - \left( {2 \times - 2} \right) - 9\\& = - 3 - ( - 4) - 9\\ & = - 3 + 4 - 9\\{\text{Ans}} & = - 8\,\end{align}

Chapter 12 Ex.12.3 Question 8

(i) If $$z = 10$$, find the value of $$z^3 – 3(z – 10)$$.

(ii) If $$p = \,– 10$$, find the value of $$p^2 – 2p\, – 100$$

Solution

Steps:

First simplify the expression

\begin{align}={{z}^{3}}-3z+30\end{align}

Now putting value of $$z=10$$

\begin{align} &={{\left( 10 \right)}^{3}}-\left( 3\times 10 \right)+30 \\&=1000-30+30 \\\text{ Ans } &=1000 \\\end{align}

(ii) If $$p = \,– 10$$, find the value of $$p^2 – 2p \,– 100$$

Put value of $$p = -10$$ to solve the expression

\begin{align}&= {\left( { - 10} \right)^2} - \left( {2 \times - 10} \right) - 100\\&= 100 + 20 - 100\\{\text{Ans}} &= 20\end{align}

Chapter 12 Ex.12.3 Question 9

What should be the value of $$a$$ if the value of $$2x^2 + x – a$$ equals to $$5$$, when $$x = 0$$?

Solution

Steps:

Given that

$$2x^2 + x – a = 5$$

Also, value of $$x$$ is $$0$$

So,

\begin{align}2 \times {0^2} + 0 - a &= 5\\ 0- a &= 5\\ - a &= 5\\{\rm{ Ans: }}\; a &= - 5\end{align}

Chapter 12 Ex.12.3 Question 10

Simplify the expression and find its value when $$a = 5$$ and $$b = \,– 3$$.

Solution

Steps:

\begin{align}&2\left( {{a^2} + ab} \right){\rm{ }} + {\rm{ }}3{\rm{ }}-ab\\&= 2{a^2} + 2ab + 3 - ab\\&= 2{a^2} + ab + 3\\&= (2 \times 5 \times 5) + (5 \times - 3) + 3\\&= 50 - 15 + 3\\&= 38\end{align}

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