Practice MCQ Questions and Answer on Simplification

21.

The value of $$\frac{{{x^2} - {{\left( {y - z} \right)}^2}}}{{{{\left( {x + z} \right)}^2} - {y^2}}}{\text{ + }}$$   $$\frac{{{y^2} - {{\left( {x - z} \right)}^2}}}{{{{\left( {x + y} \right)}^2} - {z^2}}} + $$   $$\frac{{{z^2} - {{\left( {x - y} \right)}^2}}}{{{{\left( {y + z} \right)}^2} - {x^2}}}$$   is = ?

• (A) -1
• (B) 0
• (C) 1
• (D) None of these

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 $$\eqalign{ & {\text{Given xepression ,}} \cr & \frac{{\left( {x + y - z} \right)\left( {x - y + z} \right)}}{{\left( {x + y + z} \right)\left( {x + z - y} \right)}} + \frac{{\left( {y + x - z} \right)\left( {y - x + z} \right)}}{{\left( {x + y + z} \right)\left( {x + y - z} \right)}} + \frac{{\left( {z + x - y} \right)\left( {z - x + y} \right)}}{{\left( {y + z + x} \right)\left( {y + z - x} \right)}} \cr & = \frac{{\left( {x + y - z} \right)}}{{\left( {x + y + z} \right)}} + \frac{{\left( {y - x + z} \right)}}{{\left( {x + y + z} \right)}} + \frac{{\left( {x - y + z} \right)}}{{\left( {x + y + z} \right)}} \cr & = \frac{{\left( {x + y - z} \right) + \left( {y - x + z} \right) + \left( {x - y + z} \right)}}{{\left( {x + y + z} \right)}} \cr & = \frac{{x + y + z}}{{x + y + z}} \cr & = 1 \cr} $$

22.

Simplify : $$\root 3 \of { - 2197} \,\times $$  $$\root 3 \of { - 125} \,\,\div $$   $$\root 3 \of {\frac{{27}}{{512}}} $$  = ?

• (A) $$\frac{{492}}{7}$$
• (B) $$\frac{{520}}{3}$$
• (C) $$\frac{{554}}{7}$$
• (D) $$\frac{{571}}{5}$$

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 $$\,\,\,\,\,\,\root 3 \of { - 2197} \times \root 3 \of { - 125} \div \root 3 \of {\frac{{27}}{{512}}} \,$$   $$ = \root 3 \of {13 \times 13 \times 13 \times - 1} \,\, \times $$     $$\sqrt {5 \times 5 \times 5 \times - 1} \,\,\div $$     $$\root 3 \of {\frac{{27}}{{512}}} $$ $$\eqalign{ & = \left( {13 \times 5\root 3 \of { - 1 \times - 1} } \right) \div \frac{3}{8} \cr & = 13 \times 5 \times \frac{8}{3} \cr & = \frac{{520}}{3} \cr} $$

23.

$$4\frac{4}{5} \div 6\frac{2}{5} = ?$$

• (A) $$\frac{3}{4}$$
• (B) $$\frac{5}{7}$$
• (C) $$\frac{5}{8}$$
• (D) $$\frac{7}{11}$$

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 $$\eqalign{ & {\text{Given expression}} \cr & \frac{{24}}{5} \div \frac{{32}}{5} \cr & = \frac{{24}}{5} \times \frac{5}{{32}} \cr & = \frac{3}{4} \cr} $$

24.

If 56 × 75 × 60 × 84 × 210 = 2 × 3 × 5 × 7, then what is the value of $$\left[ {\frac{{\left( {{\text{p}} + {\text{q}}} \right)}}{{\text{s}}}} \right] + {\text{r}}?$$pqrs

• (A) 6
• (B) 8
• (C) 12
• (D) 10

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 56 × 75 × 60 × 84 × 210 = 2p × 3q × 5r × 7s 7 × 23 × 52 × 3 × 22 × 3 × 5 × 22 × 3 × 7 × 7 × 3 × 2 × 5 = 2p × 3q × 5r × 7s 28 × 34 × 54 × 73 = 2p × 3q × 5r × 7s ∴ p = 8 q = 4 r = 4 s = 3 $$\eqalign{ & \therefore \left[ {\frac{{\left( {{\text{p}} + {\text{q}}} \right)}}{{\text{s}}}} \right] + {\text{r}} \cr & = \left[ {\frac{{\left( {8 + 4} \right)}}{3}} \right] + 4 \cr & = \frac{{12}}{3} + 4 \cr & = 4 + 4 \cr & = 8 \cr} $$

25.

Simplify : $$1 - \left[ {1 - \left\{ {1 - \left( {1 - \overline {1 - 1} } \right)} \right\}} \right]$$

• (A) 0
• (B) 1
• (C) 2
• (D) 3

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 Given Expression = 1 - [ 1 - { 1 - ( 1 - 0 ) } ] = 1 - [ 1 - ( 1 - 1 ) ] = 1 - [ 1 - 0 ] = 0

26.

$$\frac{{{{\left( {469 + 174} \right)}^2} - {{\left( {469 - 174} \right)}^2}}}{{469 \times 174}} $$      = ?

• (A) 2
• (B) 4
• (C) 295
• (D) 643

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 $$\eqalign{ & {\text{Given expression,}} \cr & \frac{{{{\left( {a + b} \right)}^2} - {{\left( {a - b} \right)}^2}}}{{a b}} \cr & = \frac{{4ab}}{{ab}} \cr & = 4\,\,\,\,\,\left\{ {{\text{where }}a = 469,b = 174} \right\} \cr} $$

27.

The value (1001)³ is = ?

• (A) 1003003001
• (B) 100303001
• (C) 100300301
• (D) 103003001

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 (1001)3 =1001 × 1001 × 1001 =1002001 × 1001 =1003003001

28.

$$\frac{{38 \times 38 \times 38 + 34 \times 34 \times 34 + 28 \times 28 \times 28 - 38 \times 34 \times 84}}{{38 \times 38 + 34 \times 34 + 28 \times 28 - 38 \times 34 - 34 \times 28 - 38 \times 28}}$$            is equal to = ?

• (A) 24
• (B) 32
• (C) 44
• (D) 100

Show Answer

 $$\eqalign{ & {\text{Given experssion ,}} \cr & \frac{{{a^3} + {b^3} + {c^3} - 3abc}}{{{a^2} + {b^2} + {c^2} - ab - bc - ca}} \cr & \left( {Let,\,38 = a,\,34 = b,\,28 = c} \right) \cr & = a + b + c \cr & = 38 + 34 + 28 \cr & = 100 \cr} $$

29.

$$853 + ? \div 17 = 1000$$

• (A) 2482
• (B) 2499
• (C) 2516
• (D) 16147

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 $$\eqalign{ & {\text{Let,}} \cr & {\text{853 + }}x \div 17 = 1000 \cr & {\text{Then,}} \cr & {\text{853 + }}\frac{x}{{17}} = 1000 \cr & \Leftrightarrow \frac{x}{{17}} = 1000 - 853 \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 147 \cr & \Leftrightarrow x = 147 \times 17 \cr & \,\,\,\,\,\,\,\,\,\,\,\,\, = 2499 \cr} $$

30.

If x + y + z = 0, then x³ + y³ + z³ + 3xyz is equal to = ?

• (A) 0
• (B) 6xyz
• (C) 12xyz
• (D) xyz

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 As we know a3 + b3 + c3 + 3abc = (a2 + b2 + c2 - ab - bc - ca)(a + b + c) When, a + b + c = 0 Then a3 + b3 + c3 - 3abc = 0 When, x + y + z = 0 ⇒ x3 + y3 + z3 = 3xyz ⇒ x3 + y3 + z3 + 3xyz = 3xyz + 3xyz ⇒ x3 + y3 + z3 + 3xyz = 6xyz