3 Mark Question

Question 13 Marks

Expand :

(7m – 3n – 4k)²

Answer

(7m – 3n – 4k)² = (7m)² + (- 3n)² + (- 4k)² + 2(7m) (- 3n) + 2 (- 3n) (- 4k) + 2 (7m) (- 4k)
… [(a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ac]
= 49m² + 9n² + 16k² – 42mn + 24nk – 56km

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Question 23 Marks

Expand :

(3x + 4y – 5p)²

Answer

(3x + 4y – 5p)² = (3x)² + (4y)² + (- 5p)² + 2(3x) (4y) + 2(4y) (- 5p) + 2(3x) (- 5p)
… [(a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ac]
= 9x + 16y² + 25p² + 24xy – 40py – 30px

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Question 33 Marks

Expand :

(m + 2n + 3r)²

Answer

(m + 2n + 3r)² = (m)² + (2n)² + (3r)² + 2(m) (2n) + 2(2n) (3r) + 2(m) (3r)
… [(a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ac]
= m² + 4n² + 9r² + 4mn + 12nr + 6mr

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Question 43 Marks

Expand :

(2p + q + 5)²

Answer

(2p + q + 5)² = (2p)² + (q)² + (5)² + 2(2p) (q) + 2(q) (5) + 2(2p) (5)
… [(a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ac]
= 4p² + q² + 25 + 4pq + 10q + 20p

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Question 53 Marks

Expand: $\left(\frac{x}{3}-\frac{3}{x}\right)^3$

Answer

Here, $\mathrm{a}=\frac{x}{3}$ and $\mathrm{b}=\frac{3}{x}$
$\begin{aligned}
\left(\frac{x}{3}-\frac{3}{x}\right)^3= & \left(\frac{x}{3}\right)^3-3\left(\frac{x}{3}\right)^2\left(\frac{3}{x}\right) \\
& +3\left(\frac{x}{3}\right)\left(\frac{3}{x}\right)^2-\left(\frac{3}{x}\right)^3 \\
& \quad \ldots\left[(a-b)^3=a^3-3 a^2 b+3 a b^2-b^3\right] \\
& \frac{x^3}{27}-3\left(\frac{x}{3}\right)\left(\frac{x}{3}\right)\left(\frac{3}{x}\right)+3\left(\frac{x}{3}\right)\left(\frac{3}{x}\right)\left(\frac{3}{x}\right)-\frac{27}{x^3} \\
= & \frac{x^3}{27}-3\left(\frac{x}{3}\right)+3\left(\frac{3}{x}\right)-\frac{27}{x^3} \\
= & \frac{x^3}{27}-x+\frac{9}{x}-\frac{27}{x^3}
\end{aligned}$

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Question 63 Marks

Expand: $\left(1-\frac{1}{a}\right)^3$

Answer

Here, $A=1$ and $B=\frac{1}{a}$
$\begin{aligned}
\left(1-\frac{1}{a}\right)^3= & (1)^3-3(1)^2\left(\frac{1}{a}\right)+3(1)\left(\frac{1}{a}\right)^2-\left(\frac{1}{a}\right)^3 \\
& \ldots\left[(a-b)^3=a^3-3 a^2 b+3 a b^2-b^3\right] \\
= & \mathbf{1}-\frac{3}{a}+\frac{3}{a^2}-\frac{1}{a^3}
\end{aligned}$

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Question 73 Marks

Expand: $\left(2 p-\frac{1}{2 p}\right)^3$

Answer

$\begin{aligned} & \text { Here, } a=2 p \text { and } b=\frac{1}{2 p} \\ & \left(2 p-\frac{1}{2 p}\right)^3 \\ & =(2 p)^3-3(2 p)^2\left(\frac{1}{2 p}\right)+3(2 p)\left(\frac{1}{2 p}\right)^2-\left(\frac{1}{2 p}\right)^3 \\ & \quad \ldots\left[(a-b)^3=a^3-3 a^2 b+3 a b^2-b^3\right] \\ & =8 p^3-3(2 p)(2 p)\left(\frac{1}{2 p}\right)+3(2 p)\left(\frac{1}{2 p}\right)\left(\frac{1}{2 p}\right)-\frac{1}{8 p^3} \\ & =8 p^3-3(2 p)+3\left(\frac{1}{2 p}\right)-\frac{1}{8 p^3} \\ & \mathbf{8 p}^3-\mathbf{6 p}+\frac{\mathbf{3}}{2 p}-\frac{1}{8 p^3}\end{aligned}$

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Question 83 Marks

Expand: $(198)^3$

Answer

(198)³ = (200 – 2)³
Here, a = 200 and b = 2
(198)³ = (200)³ – 3(200)²(2) + 3(200)(2)² – (2)³
… [(a – b)³ = a³ – 3a²b + 3ab² – b³]
= 8000000 – 3(40000)(2) + 3(200)(4) – 8
= 8000000 – 240000 + 2400 – 8
= 7762392

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Question 93 Marks

Expand: $(58)^3$

Answer

(58)³ = (60 – 2)³
Here, a = 60 and b = 2
(58)³ = (60)³ – 3(60)²(2) + 3(60)(2)² – (2)³
… [(a – b)³ = a³ – 3a²b + 3ab² – b³]
= 216000 – 3(3600)(2) + 3(60)(4) – 8
= 216000 – 21600 + 720 – 8
=195112

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Question 103 Marks

Expand: $(7 x-9 y)^3$

Answer

Here, a = 7x and b = 9y
(7x – 9y)³
= (7x)³ – 3(7x)² (9y) + 3 (7x)(9y)² – (9y)³
…[(a – b)³ = a³ – 3a²b + 3ab² – b³]
= 343x³ – 3(49x²)(9y) + 3(7x)(81y²) – 729y³
= 343x³ – 1323x²y + 1701xy² – 729y³

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Question 113 Marks

Expand: $(4-p)^3$

Answer

Here, a = 4 and b = p
(4 – p)³ = (4)³ – 3(4)²(p) + 3(4)(p)² – (p)³
… [(a – b)³ = a³ – 3a²b + 3ab² – b³]
= 64 – 3(16)(p) + 3(4)(p²) – p³
= 64 – 48p + 12p² – p³

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Question 123 Marks

Expand: $(2 m-5)^3$

Answer

Here, a = 2m and b = 5
(2m – 5)³
= (2m)³ – 3(2m)² (5) + 3(2m) (5)² – (5)³
… [(a – b)³ = a³ – 3a²b + 3ab² – b³]
= 8m³ – 3(4m²)(5) + 3(2m)(25) – 125
= 8m³ – 60m² + 150m – 125

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Question 133 Marks

Expand : $\left(\frac{5 x}{y}+\frac{y}{5 x}\right)^3$

Answer

\text { Here, } \mathrm{a}=\frac{5 x}{y} \text { and } \mathrm{b}=\frac{y}{5 x}

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Question 143 Marks

Expand : $\left(2 m+\frac{1}{5}\right)^3$

Answer

$\begin{aligned} & \text { Here, } a=2 m \text { and } b=\frac{1}{5} \\ & \left(2 m+\frac{1}{5}\right)^3 \\ & =(2 m)^3+3(2 m)^2\left(\frac{1}{5}\right)+3(2 m)\left(\frac{1}{5}\right)^2+\left(\frac{1}{5}\right)^3 \\ & \quad \ldots\left[\because(a+b)^3=a^3+3 a^2 b+3 a b^2+b^3\right] \\ & =8 m^3+3\left(4 m^2\right)\left(\frac{1}{5}\right)+3(2 m)\left(\frac{1}{25}\right)+\frac{1}{125} \\ & =8 m^3+\frac{12 m^2}{5}+\frac{6 m}{25}+\frac{1}{125}\end{aligned}$

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Question 153 Marks

Expand : $\left(x+\frac{1}{x}\right)^3$

Answer

Here, $a=x$ and $b=\frac{1}{x}$
$
\begin{aligned}
& \left(x+\frac{1}{x}\right)^3 \\
& =(x)^3+3(x)^2\left(\frac{1}{x}\right)+3(x)\left(\frac{1}{x}\right)^2+\left(\frac{1}{x}\right)^3 \\
& \quad \ldots\left[\because(\mathrm{a}+\mathrm{b})^3=\mathrm{a}^3+3 \mathrm{a}^2 \mathrm{~b}+3 \mathrm{ab}^2+\mathrm{b}^3\right] \\
& =x^3+3 x+3 x\left(\frac{1}{x^2}\right)+\frac{1}{x^3} \\
& =x^3+3 x+\frac{3}{x}+\frac{1}{x^3}
\end{aligned}
$

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Question 163 Marks

Expand : $(101)^3$

Answer

(101)³ = (100 + 1)³
Here, a = 100 and b = 1
(101)³
= (100)³ + 3(100)²(1) + 3(100)(1)² + (1)³
…[∵ (a + b)³ = a³ + 3a²b + 3ab² + b³]
= 1000000 + 3(10000) + 3(100) (1) + 1
= 1000000 + 30000 + 300 + 1
= 1030301

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Question 173 Marks

Expand : $(52)^3$

Answer

(52)³ = (50 + 3)³
Here, a = 50 and b = 2
(52)³ = (50)³ + 3(50)² (2) + 3(50)(2)² + (2)³
…[∵ (a + b)³ = a³ + 3a²b + 3ab² + b³]
= 125000 + 3(2500)(2) + 3(50)(4) + 8
= 125000 + 15000 + 600 + 8
=140608

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Question 183 Marks

Expand : $(7 x+m)^3$

Answer

Here, a = 7 and b = m
(7 + m)³ = (7)³ + 3(7)²(m) + 3(7)(m)² + (m)³
…[∵ (a + b)³ = a³ + 3a²b + 3ab² + b³]
= 343 + 3(49)(m) + 3(7)(m²) + m³
= 343 + 147m + 21m² + m³

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Question 193 Marks

Expand : $(7 x+8 y)^3$

Answer

Here, a = 7x and b = 8y
(7x + 8y)³
= (7x)³ + 3(7x)² (8y) + 3(7x) (8y)² + (8y)³
…[∵ (a + b)³ = a³ + 3a²b + 3ab² + b³]
= 343x³ + 3(49x²)(8y) + 3(7x)(64y²) + 512y³
= 343x³ + 1176x²y + 1344xy² + 512y³

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Question 203 Marks

Expand : $(k+4)^3$

Answer

Here, a = k and b = 4
(k + 4)³ = (k)³ + 3(k)² (4) + 3(k)(4)² + (4)³
…[∵ (a + b)³ = a³ + 3a²b + 3ab² + b³]
= k³ + 12k² + 3(k)(16) + 64
= k³ + 12k² + 48k + 64

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Question 213 Marks

Expand :$\left(\frac{1}{y}+4\right)\left(\frac{1}{y}-9\right)$

Answer

$\begin{aligned} & \text {}\left(\frac{1}{y}+4\right)\left(\frac{1}{y}-9\right) \\ & =\left(\frac{1}{y}\right)^2+(4-9) \frac{1}{y}+4 \times(-9) \\ & \quad \ldots\left[\because(x+a)(x+\mathrm{b})=x^2+(a+b) x+a b\right] \\ & =\frac{1}{y^2}-\frac{5}{y}-36\end{aligned}$

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Question 223 Marks

Expand :$\left(x+\frac{1}{x}\right)\left(x-\frac{1}{x}\right)$

Answer

$\begin{aligned} & \text {}\left(x+\frac{1}{x}\right)\left(x-\frac{1}{x}\right) \\ & =x^2+\left(\frac{1}{x}-\frac{1}{x}\right) x+\frac{1}{x} \times\left(-\frac{1}{x}\right) \\ & \quad \ldots\left[\because(x+\mathrm{a})(x+\mathrm{b})=x^2+(\mathrm{a}+\mathrm{b}) x+\mathrm{ab}\right] \\ & =x^2+0 \times x-\frac{1}{x^2} \\ & =x^2-\frac{1}{x^2}\end{aligned}$

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Question 233 Marks

Expand :$\left(m+\frac{2}{3}\right)\left(m-\frac{7}{3}\right)$

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Question 243 Marks

Expand :$(9 x-5 t)(9 x+3 t)$

Answer

(9x – 5t) (9x + 3t)
= (9x)² + [(-5t) + 3t] 9x + (-5t) × 3t
…[∵ (x + a) (x + b) = x² + (a + b)x + ab]
= 81x² + (-2t) × 9x – 15t²
= 81x² – 18xt – 15t²

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Question 253 Marks

Expand :$(3 x+4 y)(3 x+5 y)$

Answer

(3x + 4y) (3x + 5y)
= (3x)² + (4y + 5y) 3x + 4y × 5y
…[∵ (x + a) (x + b) = x² + (a + b)x + ab]
= 9x² + 9y × 3x + 20y²
= 9x² + 27xy + 20y²

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Question 263 Marks

Expand :$(13+x)(13-x)$

Answer

(13 + x) (13 – x)
= (13)² + (x – x) 13 + x × (-x)
…[∵ (x + a) (x + b) = x² + (a + b)x + ab]
= 169 + 0 × 13 – x²
= 169 – x²

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Question 273 Marks

Expand :$(p+8)(p-3)$

Answer

(p + 8) (p – 3)
= p² + (8 – 3) p + 8 x (-3)
…[∵ (x + a) (x + b) = x² + (a + b)x + ab]
= p² + 5p – 24

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Question 283 Marks

Expand :$(m-4)(m+6)$

Answer

(m – 4)(m + 6)
= m² + (- 4 + 6) m + (-4) × 6
…[∵ (x + a) (x + b) = x² + (a + b)x + ab]
= m² + 2m – 24

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Question 293 Marks

Expand :$(a+2)(a-1)$

Answer

(a + 2)(a – 1)
= a² + (2 – 1) a + 2 × (-1)
..[∵ (x + A) (x + B) = x² + (A + B)x + AB]
= a² + a – 2

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