Exercise 12.3 Algebraic-Expressions -NCERT Solutions Class 7

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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

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

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

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

Video 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) \(- 2{p^3} - 3{p^2} + 4p + 7\)

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

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

Video 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) \(2{x^2} - x - 2\)

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

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

Video 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) \({a^2} - {b^2}\)

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

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

Video 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) \({{a}^{2}}+ab+\text{ }2\)

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

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

Video 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  \(x = 2\)

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

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

Video 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 \(b = - 2\)

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

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

Video 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

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

Question 9

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

Solution

Video Solution

Steps:

Given that

\(2x^2 + x – a = 5\)

Also, value of \(x\) is \(0\)

So,

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

Question 10

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

Solution

Video 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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