RS Aggarwal solution class 8 chapter 6 Operations on Algebraic Expressions Exercise 6E

Exercise 6E

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Question 1:

Tick (✓) the correct answer:
The sum of (6a + 4b − c + 3), (2b − 3c + 4), (11b − 7a + 2c − 1) and (2c − 5a − 6) is
(a) (4a − 6b + 2)
(b) (−3a + 14b − 3c + 2)
(d) (−6a + 17b)
(d) (−6a + 6b + c −4)

Answer 1:

(c) (−6a + 17b)

Page-95

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Question 2:

Tick (✓) the correct answer:
(3q + 7p² − 2r³ + 4) − (4p² − 2q + 7r³ − 3) = ?
(a) (p² + 2q + 5r³ + 1)
(b) (11p² + q + 5r³ + 1)
(c) (−3p² − 5q + 9r³ − 7)
(d) (3p² + 5q − 9r³ +7)

Answer 2:

(d) (3p² + 5q − 9r³ +7)

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Question 3:

Tick (✓) the correct answer:
(x + 5)(x − 3) = ?
(a) x² + 5x − 15
(b) x² − 3x − 15
(c) x² + 2x + 15
(d) x² + 2x − 15

Answer 3:

(d) x² + 2x − 15

$$\begin{gathered} \left(x+5\right)\left(x-3\right) \\ \Rightarrow \left(x\right)\left(x-3\right)+\left(5\right)\left(x-3\right) \\ \Rightarrow x^2-3x +5x -15 \\ \Rightarrow x^2+2x-15 \end{gathered}$$

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Question 4:

Tick (✓) the correct answer:
(2x + 3)(3x − 1) = ?
(a) (6x² + 8x − 3)
(b) (6x² + 7x − 3)
(c) 6x² − 7x − 3
(d) (6x² − 7x + 3)

Answer 4:

(b) (6x² + 7x − 3)

$$\begin{gathered} \left(2x+3\right)\left(3x-1\right) \\ \Rightarrow \left(2x\right)\left(3x-1\right)+\left(3\right)\left(3x-1\right) \\ \Rightarrow 6x^2-2x +9x-3 \\ \Rightarrow 6x^2+7x-3 \end{gathered}$$

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Question 5:

Tick (✓) the correct answer:
(x + 4)(x + 4) = ?
(a) (x² + 16)
(b) (x² + 4x + 16)
(c) (x² + 8x + 16)
(d) (x² + 16x)

Answer 5:

(c) (x² + 8x + 16)

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Question 6:

Tick (✓) the correct answer:
(x − 6)(x − 6) = ?
(a) (x² − 36)
(b) (x² + 36)
(c) (x² − 6x + 36)
(d) (x² − 12x + 36)

Answer 6:

(d) (x² − 12x + 36)

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Question 7:

Tick (✓) the correct answer:
(2x + 5)(2x − 5) = ?
(a) (4x² + 25)
(b) (4x² − 25)
(c) (4x² − 10x + 25)
(d) (4x² + 10x − 25)

Answer 7:

(b) (4x² − 25)

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Question 8:

Tick (✓) the correct answer:
8a²b³ ÷ (−2ab) = ?
(a) 4ab²
(b) 4a²b
(c) −4ab²
(d) −4a²b

Answer 8:

(c) −4ab²

$$\begin{gathered} 8a^2{b}^{3 }÷ \left(-2ab\right) \\ \Rightarrow \left(\frac{8}{-2}\right)\left(a^{2-1}\right)\left(b^{3-1}\right) \\ \Rightarrow -4a{b}^2 \end{gathered}$$

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Question 9:

Tick (✓) the correct answer:
(2x² + 3x + 1) ÷ (x + 1) = ?
(a) (x + 1)
(b) (2x + 1)
(c) (x + 3)
(d) (2x + 3)

Answer 9:

(b) (2x + 1)

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Question 10:

Tick (✓) the correct answer:
(x² − 4x + 4) ÷ (x − 2) = ?
(a) (x − 2)
(b) (x + 2)
(c) (2 − x)
(d) (2 + x + x²)

Answer 10:

(a) (x − 2)

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Question 11:

Tick (✓) the correct answer:
(a + 1)(a − 1)(a² + 1) = ?
(a) (a⁴ − 2a² − 1)
(b) (a⁴ − a² − 1)
(c) (a⁴ − 1)
(d) (a⁴ + 1)

Answer 11:

(c) (a⁴ − 1)

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Question 12:

Tick (✓) the correct answer:
$\left(\frac{1}{x}+\frac{1}{y}\right)\left(\frac{1}{x}-\frac{1}{y}\right)=?$
(a) $\left(\frac{1}{x^2}-\frac{1}{y^2}\right)$
(b) $\left(\frac{1}{x^2}+\frac{1}{y^2}\right)$
(c) $\left(\frac{1}{x^2}+\frac{1}{y^2}-\frac{1}{xy}\right)$
(d) $\left(\frac{1}{x^2}-\frac{1}{y^2}+\frac{1}{xy}\right)$

Answer 12:

a) $\left(\frac{1}{x^2}-\frac{1}{y^2}\right)$

$$\begin{gathered} \left(\frac{1}{x}+\frac{1}{y}\right)\left(\frac{1}{x}-\frac{1}{y}\right) \\ \Rightarrow According to the formula \left(a+b\right)\left(a-b\right)={\left(a\right)}^2-{\left(b\right)}^2: \\ \Rightarrow \left(\frac{1}{x^2}-\frac{1}{y^2}\right) \end{gathered}$$(1x2−1y2)

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Question 13:

Tick (✓) the correct answer:
If $\left(x+\frac{1}{x}\right)=5$, then $\left(x^2+\frac{1}{x^2}\right)=?$
(a) 25
(b) 27
(c) 23
(d) $25\frac{1}{25}$

Answer 13:

(c) 23

$$\begin{gathered} \left(x+\frac{1}{x}\right)= 5 \\ \Rightarrow Squaring both the sides: \\ \Rightarrow {\left(x+\frac{1}{x}\right)}^2= {\left(5\right)}^2 \\ \Rightarrow \left(x^2+\frac{1}{x^2}+2\left(x\right)\left(\frac{1}{x}\right)\right)= 25 \\ \Rightarrow \left(x^2+\frac{1}{x^2}\right)+2 =25 \\ \Rightarrow \left(x^2+\frac{1}{x^2}\right)=25-2 \\ \Rightarrow \left(x^2+\frac{1}{x^2}\right)= 23 \end{gathered}$$

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Question 14:

Tick (✓) the correct answer:
If $\left(x-\frac{1}{x}\right)=6$, then $\left(x^2+\frac{1}{x^2}\right)=?$
(a) 36
(b) 38
(c) 32
(d) $36\frac{1}{36}$

Answer 14:

(b) 38

$$\begin{gathered} \left(x-\frac{1}{x}\right)= 6 \\ \Rightarrow Squaring both the sides: \\ \Rightarrow {\left(x-\frac{1}{x}\right)}^2= {\left(6\right)}^2 \\ \Rightarrow \left(x^2+\frac{1}{x^2}-2\left(x\right)\left(\frac{1}{x}\right)\right)= 36 \\ \Rightarrow \left(x^2+\frac{1}{x^2}\right)-2 =36 \\ \Rightarrow \left(x^2+\frac{1}{x^2}\right)=36+2 \\ \Rightarrow \left(x^2+\frac{1}{x^2}\right)= 38 \end{gathered}$$

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Question 15:

Tick (✓) the correct answer:
(82)² − (18)² = ?
(a) 8218
(b) 6418
(c) 6400
(d) 7204

Answer 15:

(c) 6400

$$\begin{gathered} {\left(82\right)}^2-{\left(18\right)}^2 \\ =\left(82 + 18\right)\left(82 - 18\right) \\ =\left(100\right)\left(64\right) \\ =6400 \end{gathered}$$                           [using the identity (a-b)(a+b)=a² -b²]

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Question 16:

Tick (✓) the correct answer:
(197 × 203) = ?
(a) 39991
(b) 39999
(c) 40009
(d) 40001

Answer 16:

(a) 39991

$$\begin{gathered} \left(197\right)\times \left(203\right) \\ \Rightarrow \left(200-3\right)\left(200+3\right) \\ \Rightarrow {\left(200\right)}^2-{\left(3\right)}^2 \\ \Rightarrow 40000-9 \\ \Rightarrow 39991 \end{gathered}$$                   [using the identity (a+b) (a-b) = a²  -b²]

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Question 17:

Tick (✓) the correct answer:
If (a + b) = 12 and ab = 14, then (a² + b²) = ?
(a) 172
(b) 116
(c) 165
(d) 126

Answer 17:

(b) 116

$$\begin{gathered} \left(a+b\right) =12 \\ \Rightarrow Squaring both the sides: \\ \Rightarrow {\left(a+b\right)}^2 = {\left(12\right)}^2 \\ \Rightarrow \left(a^2+ b^2 +2ab\right) = 144 \\ \Rightarrow \left(a^2+ b^2\right) = 144-2ab \\ \Rightarrow \left(a^2+ b^2\right) = 144 -2\left(14\right) \\ \Rightarrow \left(a^2+ b^2\right) = 144 -28 \\ \Rightarrow \left(a^2+ b^2\right) = 116 \end{gathered}$$

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Question 18:

Tick (✓) the correct answer:
If (a − b) = 7 and ab = 9, then (a² + b²) = ?
(a) 67
(b) 31
(c) 40
(d) 58

Answer 18:

(a) 67

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Question 19:

Tick (✓) the correct answer:
If x = 10, then the value of (4x² + 20x + 25) = ?
(a) 256
(b) 425
(c) 625
(d) 575

Answer 19:

(c) 625

$$\begin{gathered} \left(4x^2+20x+25\right) \\ \Rightarrow {\left(2x\right)}^2+2\left(2x\right)\left(5\right)+{\left(5\right)}^2 \\ \Rightarrow {\left(2x + 5\right)}^2 \\ \Rightarrow {\left(2\left(10\right)+5\right)}^2 \\ \Rightarrow {\left(20+5\right)}^2 \\ \Rightarrow {\left(25\right)}^2 \\ \Rightarrow 625 \end{gathered}$$