Question 1

1. Evaluate the value of: $$\frac{\sqrt{1296}}{12} + 25 \times 1.4 - 15$$

A) 20B) 23C) 25D) 28

Answer

✓ Correct Answer: Option B

Explanation: Solution: Step 1: Calculate $\sqrt{1296}$. Since $30^2 = 900$ and $40^2 = 1600$, and the unit digit is 6, $\sqrt{1296} = 36$. Step 2: Divide: 36 / 12 = 3. Step 3: Multiply: $25 \times 1.4 = 35$. Step 4: Combine: 3 + 35 - 15 = 38 - 15 = 23. Correct Answer: 23

Question 2

2. Find the value of: $37\frac{1}{2}%$ of 800 + 12.5% of 2400.

A) 450B) 500 600

Question 3

3. Solve for x: $x = \sqrt{6 + \sqrt{6 + \sqrt{6 + \dots \infty}}}$

A) 3B) 2C) 6D) Infinite

Answer

✓ Correct Answer: Option A

Explanation: Solution: Let $x = \sqrt{6 + x}$. Squaring both sides: $x^2 = 6 + x \Rightarrow x^2 - x - 6 = 0$. Factorizing: (x-3)(x+2) = 0. Since the value must be positive, x = 3. Correct Answer: 3

Question 4

4. Simplify: $$\frac{0.87 \times 0.87 \times 0.87 + 0.13 \times 0.13 \times 0.13}{0.87 \times 0.87 - 0.87 \times 0.13 + 0.13 \times 0.13}$$

A) 0.74B) 0.90C) 1.13D) 1.00

Answer

✓ Correct Answer: Option D

Explanation: Solution: This is in the form $\frac{a^3 + b^3}{a^2 - ab + b^2}$. We know $a^3 + b^3 = (a+b)(a^2 - ab + b^2)$. So, the expression simplifies to a + b. 0.87 + 0.13 = 1.00. Correct Answer: 1.00

Question 5

5. If $2^{x+3} \cdot 4^{2x-1} = 8^{x+1}$, then find the value of x.

A) 1/2B) 1C) 3/2D) 2

Answer

✓ Correct Answer: Option B

Explanation: Solution: Express all terms with base 2: $2^{x+3} \cdot (2^2)^{2x-1} = (2^3)^{x+1}$ $2^{x+3} \cdot 2^{4x-2} = 2^{3x+3}$ Equating exponents: (x+3) + (4x-2) = 3x+3 5x + 1 = 3x + 3 \Rightarrow 2x = 2 \Rightarrow x = 1. Correct Answer: 1

Question 6

6. Convert the following recurring decimal to a fraction: 0.1\overline{2}

A) 12/90B) 12/99C) 11/90D) 11/99

Answer

✓ Correct Answer: Option C

Explanation: Solution: Rule: (Full number - Non-repeating part) / (9 for each repeating digit, 0 for each non-repeating decimal digit). $0.1\overline{2} = \frac{12 - 1}{90} = \frac{11}{90}$. Correct Answer: 11/90

Question 7

7. Simplify: $\left(1 - \frac{1}{3}\right)\left(1 - \frac{1}{4}\right)\left(1 - \frac{1}{5}\right) \dots \left(1 - \frac{1}{n}\right)$

A) 1/nB) 2/nC) 2(n-1)/nD) (n+1)/n

Answer

✓ Correct Answer: Option B

Explanation: Solution: The series is: $\frac{2}{3} \times \frac{3}{4} \times \frac{4}{5} \times \dots \times \frac{n-1}{n}$. This is a telescoping product where numerators cancel with preceding denominators. The result is $\frac{First Numerator}{Last Denominator} = \frac{2}{n}$. Correct Answer: 2/n

Question 8

8. What is the value of $(48)^2 - (32)^2 + \sqrt{x} = 1300$? Solve for x.

A) 400B) 40000C) 200D) 20000

Answer

✓ Correct Answer: Option A

Explanation: Solution: Step 1: Use $a^2 - b^2 = (a-b)(a+b)$. $(48-32)(48+32) = 16 \times 80 = 1280$. Step 2: $1280 + \sqrt{x} = 1300$. Step 3: $\sqrt{x} = 20$. Step 4: $x = 20^2 = 400$. Correct Answer: 400

Question 9

9. Simplify: $\sqrt{10 + 2\sqrt{21}}$

A) $\sqrt{5} + \sqrt{2}$B) $\sqrt{8} + \sqrt{2}$C) $\sqrt{7} + \sqrt{3}$D) $\sqrt{6} + 2$

Answer

✓ Correct Answer: Option C

Explanation: Solution: We need to find two numbers whose sum is 10 and product is 21. The numbers are 7 and 3. $\sqrt{10 + 2\sqrt{21}} = \sqrt{(\sqrt{7})^2 + (\sqrt{3})^2 + 2\sqrt{7}\sqrt{3}} = \sqrt{(\sqrt{7} + \sqrt{3})^2} = \sqrt{7} + \sqrt{3}$. Correct Answer: $\sqrt{7} + \sqrt{3}$

Question 10

10. Find the value of $121 \div 1.1 + 100 \div 0.5 - 20 \times 1.5$.

A) 280B) 310C) 250D) 300

Answer

✓ Correct Answer: Option A

Explanation: Solution: Step 1: 121 / 1.1 = 110. Step 2: 100 / 0.5 = 200. Step 3: $20 \times 1.5 = 30$. Step 4: 110 + 200 - 30 = 280. Correct Answer: 280

Accepted Answer

✓ Correct Answer: Option C

Explanation: Solution: Step 1: Convert percentages to fractions. $37\frac{1}{2}% = \frac{3}{8}$ and $12.5% = \frac{1}{8}$. Step 2: Calculate: $(\frac{3}{8} \times 800) + (\frac{1}{8} \times 2400)$. Step 3: 300 + 300 = 600. Correct Answer: 600

Competitive Exams : Quantitative Aptitude - Simplification (Set 1)

1. Evaluate the value of: $$\frac{\sqrt{1296}}{12} + 25 \times 1.4 - 15$$

2. Find the value of: $37\frac{1}{2}%$ of 800 + 12.5% of 2400.

3. Solve for x: $x = \sqrt{6 + \sqrt{6 + \sqrt{6 + \dots \infty}}}$

4. Simplify: $$\frac{0.87 \times 0.87 \times 0.87 + 0.13 \times 0.13 \times 0.13}{0.87 \times 0.87 - 0.87 \times 0.13 + 0.13 \times 0.13}$$

5. If $2^{x+3} \cdot 4^{2x-1} = 8^{x+1}$, then find the value of x.

6. Convert the following recurring decimal to a fraction: 0.1\overline{2}

7. Simplify: $\left(1 - \frac{1}{3}\right)\left(1 - \frac{1}{4}\right)\left(1 - \frac{1}{5}\right) \dots \left(1 - \frac{1}{n}\right)$

8. What is the value of $(48)^2 - (32)^2 + \sqrt{x} = 1300$? Solve for x.

9. Simplify: $\sqrt{10 + 2\sqrt{21}}$

10. Find the value of $121 \div 1.1 + 100 \div 0.5 - 20 \times 1.5$.