Question 1

$$\sqrt{\frac{0.204 	imes 42}{0.07 	imes 3.4}}$$ is equal to

Accepted Answer

Concept & Formula: To simplify expressions involving decimals inside a square root, count the total number of decimal places in the numerator and denominator to eliminate the decimals. Step-by-Step Solution: Count the decimal places: • Numerator has 3 decimal places (from 0.204). • Denominator has 3 decimal places (2 from 0.07 and 1 from 3.4). Since the number of decimal places is equal, we can remove the decimal points entirely: \sqrt{(204 × 42/7 × 34)} Simplify the fraction: • Divide 42 by 7: (42/7) = 6 • Divide 204 by 34: (204/34) = 6 Substitute these back into the expression: \sqrt{6 × 6} = \sqrt{36} = 6 Exam Strategy & Shortcut: Count decimal places immediately. Since they balance out (3 on top, 3 on bottom), treat it instantly as an integer problem: \sqrt{(200 × 40/7 × 30)} \approx \sqrt{(8000/210)} \approx \sqrt{40} \approx 6. Looking at the options, 6 is the only reasonable integer answer. Common Pitfall: Students often miscount the decimal places or make errors while canceling out numbers, leading to an incorrect power of 10 such as 0.6 or 0.06. Always verify that the decimal shifts in the numerator and denominator balance out perfectly. Final Answer: Therefore, the correct answer is 6.

Question 2

$$\sqrt{\frac{0.081 	imes 0.324 	imes 4.624}{1.5625 	imes 0.0289 	imes 72.9 	imes 64}}$$ is equal to

Accepted Answer

Concept & Formula: Eliminate the decimal points by counting the total number of decimal places in the numerator and the denominator. Once converted to integers, simplify the perfect squares. Step-by-Step Solution: Count the total decimal places: • Numerator: 3 (from 0.081) + 3 (from 0.324) + 3 (from 4.624) = 9 decimal places. • Denominator: 4 (from 1.5625) + 4 (from 0.0289) + 1 (from 72.9) = 9 decimal places. Since the decimal places balance out, remove the decimals entirely: \sqrt{(81 × 324 × 4624/15625 × 289 × 729 × 64)} Find the square roots of the individual perfect squares: • \sqrt{81} = 9 • \sqrt{324} = 18 • \sqrt{4624} = 68 • \sqrt{15625} = 125 • \sqrt{289} = 17 • \sqrt{729} = 27 • \sqrt{64} = 8 Substitute the square roots back into the expression: (9 × 18 × 68/125 × 17 × 27 × 8) Simplify the expression by cancellation: • (9/27) = (1/3) • (18/3) = 6 • (68/17) = 4 • (4/8) = (1/2) • (6/2) = 3 This leaves us with: (3/125) = (3 × 8/125 × 8) = (24/1000) = 0.024 Exam Strategy & Shortcut: Instead of fully calculating, look at the denominator's dominant term \sqrt{15625} = 125. Any fraction with 125 in the denominator converted to a decimal will be a multiple of 0.008. Since 3/125 = 24/1000 = 0.024, you can quickly match the option ending in 24 with the correct decimal placement. Common Pitfall: Misidentifying the square root of 4624 or 729 can lead to wrong values. Memorizing squares up to 30 and practicing base-method estimation for larger squares helps prevent these calculation errors. Final Answer: Therefore, the correct answer is 0.024.

Question 3

$$\sqrt{\frac{9.5 	imes .085}{.0017 	imes .19}}$$ equals

Accepted Answer

Concept & Formula: Equalize the number of decimal places in the numerator and the denominator by multiplying by powers of 10, then simplify the remaining expression under the radical. Step-by-Step Solution: Count the decimal places: • Numerator: 1 (from 9.5) + 3 (from 0.085) = 4 decimal places. • Denominator: 4 (from 0.0017) + 2 (from 0.19) = 6 decimal places. To remove the decimals, multiply the numerator by 10^2 (or 100) to balance the 6 decimal places of the denominator: \sqrt{(95 × 85 × 100/17 × 19)} Now simplify the integers: • (95/19) = 5 • (85/17) = 5 Substitute these back into the radical: \sqrt{5 × 5 × 100} = \sqrt{25 × 100} = 5 × 10 = 50 Exam Strategy & Shortcut: Notice that 95/19 = 5 and 85/17 = 5. The core numbers give 5 × 5 = 25, whose square root is 5. The choices are all variants of 5 shifted by decimals. The denominator has 2 more decimal places than the numerator, which shifts a factor of 100 to the numerator inside the root, translating to a factor of 10 outside the root: 5 × 10 = 50. Common Pitfall: Forgetting that the extra decimal places are in the denominator can lead to dividing by 10 instead of multiplying, resulting in an incorrect answer of 0.5 or 5. Always remember: more decimals in the denominator means a larger value when simplified. Final Answer: Therefore, the correct answer is 50.

Question 4

The value of $$\sqrt{\frac{(0.03)^2 + (0.21)^2 + (0.065)^2}{(0.003)^2 + (0.021)^2 + (0.0065)^2}}$$ is

Accepted Answer

Concept & Formula: Express the terms in the denominator as a scaled version of the terms in the numerator by factoring out a common multiple of 10. Step-by-Step Solution: Observe the relationship between the numbers in the numerator and the denominator: • 0.003 = (0.03/10) • 0.021 = (0.21/10) • 0.0065 = (0.065/10) Substitute these relationships into the denominator: Denominator = ((0.03/10))^2 + ((0.21/10))^2 + ((0.065/10))^2 Factor out (1/10^2) from the denominator: Denominator = (1/10^2) [ (0.03)^2 + (0.21)^2 + (0.065)^2 ] Now substitute this back into the original square root expression: \sqrt{((0.03)^2 + (0.21)^2 + (0.065)^2/\frac{1){10^2} [ (0.03)^2 + (0.21)^2 + (0.065)^2 ]}} Cancel out the common expression in both the numerator and denominator: \sqrt{(1/\frac{1){10^2}}} = \sqrt{10^2} = 10 Exam Strategy & Shortcut: Every term in the denominator is exactly (1/10)th of its corresponding term in the numerator. Since the terms are squared, the denominator is (1/10^2) = (1/100)th of the numerator. Taking the square root of the reciprocal gives \sqrt{100} = 10 instantly. Common Pitfall: Students frequently choose 10^2 (or 100) by forgetting to take the final square root at the end of the simplification process. Always check the outermost operation! Final Answer: Therefore, the correct answer is 10.

Question 5

The square root of $$(7 + 3\sqrt{5})(7 - 3\sqrt{5})$$ is

Accepted Answer

Concept & Formula: Use the algebraic identity for the difference of squares: (a + b)(a - b) = a^2 - b^2 Step-by-Step Solution: Let a = 7 and b = 3\sqrt{5}. Apply the identity to simplify the product inside the square root: (7 + 3\sqrt{5})(7 - 3\sqrt{5}) = 7^2 - (3\sqrt{5})^2 Calculate the squared values: • 7^2 = 49 • (3\sqrt{5})^2 = 3^2 × 5 = 9 × 5 = 45 Subtract the values: 49 - 45 = 4 The question asks for the square root of this product: \sqrt{4} = 2 Exam Strategy & Shortcut: Recognize the difference of squares instantly: 49 - 9 × 5 = 49 - 45 = 4. The square root of 4 is 2. This entire computation can be done mentally within 5 seconds. Common Pitfall: A common mistake is forgetting to take the square root at the end, leading to selecting option (c) 4 instead of the actual square root, which is 2. Final Answer: Therefore, the correct answer is 2.

Question 6

The square root of $$\frac{(0.75)^3}{1 - 0.75} + [0.75 + (0.75)^2 + 1]$$ is

Accepted Answer

Concept & Formula: Use the algebraic identity for the difference of two cubes to simplify the fraction: a^3 - b^3 = (a - b)(a^2 + ab + b^2) (a^3/1 - a) + (a^2 + a + 1) = (a^3 + (1 - a)(a^2 + a + 1)/1 - a) Step-by-Step Solution: Let a = 0.75. The expression under the square root can be rewritten as: (a^3/1 - a) + (a^2 + a + 1) Find a common denominator to combine the terms: (a^3 + (1 - a)(1 + a + a^2)/1 - a) Expand the numerator using the identity (1 - a)(1 + a + a^2) = 1^3 - a^3: (a^3 + (1 - a^3)/1 - a) = (1/1 - a) Substitute a = 0.75 back into the simplified expression: (1/1 - 0.75) = (1/0.25) = (1/\frac{1){4}} = 4 The question asks for the square root of this entire value: \sqrt{4} = 2 Exam Strategy & Shortcut: Recognize the structure: (a^3/1-a) + a^2 + a + 1. Combining them forms a telescoping algebraic fraction that reduces cleanly to (1/1-a). With a = 0.75, 1-a = 0.25 = 1/4. The reciprocal is 4, and its square root is 2. Common Pitfall: Students often waste valuable time converting 0.75 to 3/4 and manually cubing the decimals. Recognizing the algebraic identity simplifies the entire problem to a single step. Final Answer: Therefore, the correct answer is 2.

Question 7

If $3a = 4b = 6c$ and $a + b + c = 27\sqrt{29}$, then $\sqrt{a^2 + b^2 + c^2}$ is

Accepted Answer

Concept & Formula: Use ratio proportions to express variables a, b, and c in terms of a single common constant k, then substitute them into the given sum to solve for k. Step-by-Step Solution: Let the given ratio equation be equal to a constant k: 3a = 4b = 6c = k Express a, b, and c in terms of k: a = (k/3), \quad b = (k/4), \quad c = (k/6) Substitute these values into the given sum equation a + b + c = 27\sqrt{29}: (k/3) + (k/4) + (k/6) = 27\sqrt{29} Find the common denominator for the fractions, which is 12: (4k + 3k + 2k/12) = 27\sqrt{29} (9k/12) = 27\sqrt{29} (3k/4) = 27\sqrt{29} \implies k = \frac{27\sqrt{29} × 4}{3} = 36\sqrt{29} Now, compute the values of a, b, and c: a = \frac{36\sqrt{29}}{3} = 12\sqrt{29} b = \frac{36\sqrt{29}}{4} = 9\sqrt{29} c = \frac{36\sqrt{29}}{6} = 6\sqrt{29} Substitute these values into the required expression \sqrt{a^2 + b^2 + c^2}: \sqrt{(12\sqrt{29})^2 + (9\sqrt{29})^2 + (6\sqrt{29})^2} \sqrt{29 × (12^2 + 9^2 + 6^2)} \sqrt{29 × (144 + 81 + 36)} = \sqrt{29 × 261} Factorize 261 to check for perfect squares: 261 = 29 × 9. \sqrt{29 × 29 × 9} = 29 × 3 = 87 Exam Strategy & Shortcut: Notice that a : b : c = (1/3) : (1/4) : (1/6) = 4 : 3 : 2. Let a=4x, b=3x, c=2x. Then 9x = 27\sqrt{29} \implies x = 3\sqrt{29}. The value required is \sqrt{(4x)^2 + (3x)^2 + (2x)^2} = \sqrt{29x^2} = x\sqrt{29}. Substitute x: (3\sqrt{29})\sqrt{29} = 3 × 29 = 87. Common Pitfall: Avoid calculating out the massive numbers like 12^2 × 29. Keep the common radical term \sqrt{29} factored outside the addition until the final step to keep calculations light. Final Answer: Therefore, the correct answer is 87.

Question 8

Which one of the following numbers has rational square root?

Accepted Answer

Concept & Strategy: For a decimal number to have a rational square root, when converted to a fraction of the form p/q, both the numerator and the denominator (after simplification or scaling to a power of 100) must be perfect squares. Step-by-Step Solution: Convert each decimal option into a fraction to inspect its square root: • Option (a) 0.4 = (4/10). Taking the square root gives (2/\sqrt{10)}, which is irrational. • Option (b) 0.09 = (9/100). Taking the square root gives (3/10) = 0.3, which is a rational number. • Option (c) 0.9 = (9/10). Taking the square root gives (3/\sqrt{10)}, which is irrational. • Option (d) 0.025 = (25/1000) = (1/40). Taking the square root gives (1/\sqrt{40)}, which is irrational. Exam Strategy & Shortcut: Count the decimal places! For a base-10 decimal to be a perfect square, it MUST have an even number of decimal places (e.g., 2, 4, 6). • 0.4, 0.9, and 0.025 have odd numbers of decimal places (1, 1, and 3 respectively). They cannot be perfect squares. • 0.09 has 2 decimal places and 9 is a perfect square. Thus, it is the only viable candidate. Common Pitfall: Students often see 4, 9, and 25 and mistakenly assume all these options yield rational roots. Always check the decimal place count; an odd number of decimal places always results in an irrational square root involving \sqrt{10}. Final Answer: Therefore, the correct answer is 0.09.

Question 9

The value of $$\sqrt{0.4}$$ is

Accepted Answer

Concept & Formula: When finding the square root of a decimal with an odd number of decimal places, convert it to a fraction with a denominator that is a perfect square (like 100 or 10000) to accurately estimate the numerator's root. Step-by-Step Solution: Convert the decimal 0.4 into a fraction: \sqrt{0.4} = \sqrt{(4/10)} Multiply the numerator and denominator by 10 to make the denominator a perfect square: \sqrt{(40/100)} = \frac{\sqrt{40}}{\sqrt{100}} = \frac{\sqrt{40}}{10} Estimate the value of \sqrt{40}: We know that 6^2 = 36 and 7^2 = 49. Since 40 is closer to 36, \sqrt{40} \approx 6.32. Substitute the estimation back into the fraction: (6.32/10) = 0.632 Rounding to two decimal places, we get 0.63. Exam Strategy & Shortcut: Never assume \sqrt{0.4} = 0.2. Instead, immediately shift the decimal point two places to the right to think in terms of out-of-100: \sqrt{0.4} = \sqrt{40%}. The square root of 40 is slightly more than 6, so the decimal answer must be slightly more than 0.6. Option (d) 0.63 matches instantly. Common Pitfall: The most common mistake is incorrectly evaluating \sqrt{0.4} as 0.2 by ignoring decimal place parity. Remember that (0.2)^2 = 0.04, not 0.4. Final Answer: Therefore, the correct answer is 0.63.

Question 10

The value of $$\sqrt{0.064}$$ is

Accepted Answer

Concept & Formula: Just like the previous problems, the number 0.064 has 3 decimal places (an odd number). Despite containing the perfect square 64, it will not yield a rational root. Scale to an even number of decimal places. Step-by-Step Solution: Express as a fraction: \sqrt{0.064} = \sqrt{(64/1000)} Multiply the top and bottom by 10: \sqrt{(640/10000)} = \frac{\sqrt{640}}{100} Estimate \sqrt{640}: We know 25^2 = 625 and 26^2 = 676. Since 640 is exactly between these bounds, \sqrt{640} \approx 25.29. Divide by 100: (25.29/100) = 0.2529 Rounding to three decimal places gives 0.252. Exam Strategy & Shortcut: Apply the percentage trick: \sqrt{0.064} is the same as \sqrt{6.4 / 100} = \sqrt{6.4} / 10. The value of \sqrt{6.4} lies between \sqrt{4} (2) and \sqrt{9} (3). Specifically, it is around 2.5. Thus, the final answer must be around 2.5 / 10 = 0.25. Option (c) is the only mathematical possibility. Common Pitfall: Seeing "64" triggers the urge to select an option containing "8", like 0.08 or 0.8. However, (0.8)^2 = 0.64 and (0.08)^2 = 0.0064. Neither equals 0.064. Count the decimal places! Final Answer: Therefore, the correct answer is 0.252.

Question 11

The value of $$\sqrt{\frac{0.16}{0.4}}$$ is

Accepted Answer

Concept & Formula: Simplify the fraction inside the square root first before attempting to extract the root. Eliminate decimals in the numerator and denominator by multiplying by the appropriate power of 10. Step-by-Step Solution: Write the expression: \sqrt{(0.16/0.4)} Multiply the numerator and the denominator by 100 to clear the decimals: \sqrt{(16/40)} Simplify the fraction by dividing both terms by 8: \sqrt{(2/5)} Convert back to a decimal to evaluate: \sqrt{0.4} Evaluate \sqrt{0.4} (as seen in earlier logic): \sqrt{0.4} = \sqrt{(40/100)} = \frac{\sqrt{40}}{10} Since \sqrt{40} \approx 6.32, we get: (6.32/10) = 0.632 Rounding gives 0.63. Exam Strategy & Shortcut: Instantly simplify the internal fraction: 0.16 / 0.4 = 1.6 / 4 = 0.4. The problem then reduces simply to \sqrt{0.4}. Knowing that 0.4 = 40% and \sqrt{40%} is slightly more than 60% (because 6^2 = 36), the answer must be around 0.63. Common Pitfall: Students sometimes take the square roots of the numerator and denominator separately *before* simplifying: \sqrt{0.16} = 0.4, but then they get stuck on \sqrt{0.4} in the denominator, making the calculation unnecessarily complicated. Always simplify the fraction first! Final Answer: Therefore, the correct answer is 0.63.

Question 12

The value of $$\frac{1 + \sqrt{0.01}}{1 - \sqrt{0.1}}$$ is close to

Accepted Answer

Concept & Formula: Evaluate the individual square roots first. Identify which roots are rational (perfect squares with even decimal places) and which require approximation (odd decimal places). Step-by-Step Solution: Evaluate the square root in the numerator: \sqrt{0.01} has two decimal places and 1 is a perfect square, so it is rational. \sqrt{0.01} = 0.1 Evaluate the square root in the denominator: \sqrt{0.1} has one decimal place (odd), so it is irrational. \sqrt{0.1} = \sqrt{(10/100)} = \frac{\sqrt{10}}{10} Knowing \sqrt{10} \approx 3.162: (3.162/10) = 0.3162 Substitute these values back into the main expression: (1 + 0.1/1 - 0.3162) (1.1/0.6838) Perform the final division: (1.1/0.6838) \approx 1.608 The value is closest to 1.6. Exam Strategy & Shortcut: Memorize \sqrt{10} \approx 3.16. This makes \sqrt{0.1} = 0.316. The expression becomes (1.1/1 - 0.316) = (1.1/0.684). Notice that 0.684 × 1.5 = 1.026, and 0.684 × 1.6 = 1.0944. Since 1.0944 is remarkably close to 1.1, the quotient is roughly 1.6. Common Pitfall: Assuming \sqrt{0.1} = 0.01 or some other erroneous value due to misinterpreting decimal rules. Always remember that \sqrt{0.1} is equivalent to \sqrt{10}/10. Final Answer: Therefore, the correct answer is 1.6.

Question 13

The square root of $535.9225$ is

Accepted Answer

Concept & Strategy: To find the square root of a large decimal, use perfect squares to establish a bounding range, then use unit digit logic to pinpoint the exact answer without heavy calculation. Step-by-Step Solution: Determine the integer bounds for the number 535.9225. We know that 23^2 = 529 and 24^2 = 576. Since 535 falls between these two, the square root must be strictly between 23 and 24. This immediately eliminates 24.15 and 28.25. Next, look at the last digit of the decimal 535.9225, which is 5. This means the square root must end in a 5. Both remaining options, 23.15 and 23.45, end in 5. Notice that 535 is much closer to 529 (which is 23^2) than it is to 576. Therefore, the square root will be on the lower end of the 23 range. We can verify 23.15 by squaring it, or by recognizing that 23.45 would yield a number closer to the midpoint. Exam Strategy & Shortcut: Use the range approximation method. 23^2 = 529. The number 535 is only slightly larger than 529. An increase of 0.15 makes sense, whereas an increase of 0.45 (nearly half) would push the square much closer to 550+. Therefore, 23.15 is the only logical choice. Common Pitfall: A common mistake is attempting the long division square root method on 535.9225, which consumes far too much time under exam conditions and is prone to arithmetic errors. Final Answer: Therefore, the correct answer is 23.15.

Question 14

If $\sqrt{5} = 2.236$, then the value of $\frac{1}{\sqrt{5}}$ is

Accepted Answer

Concept & Strategy: When dividing by an irrational number like a square root, always rationalize the denominator first. This transforms complex division into simple multiplication or division by an integer. (a/\sqrt{b)} = \frac{a × \sqrt{b}}{b} Step-by-Step Solution: Given: \sqrt{5} = 2.236 We need to evaluate the expression: (1/\sqrt{5)} Rationalize the denominator by multiplying both the numerator and the denominator by \sqrt{5}: (1/\sqrt{5)} × \frac{\sqrt{5}}{\sqrt{5}} = \frac{\sqrt{5}}{5} Substitute the given value for \sqrt{5}: (2.236/5) Perform the simple division: 2.236 \div 5 = 0.4472 Rounding to three decimal places to match the options, we get 0.447. Exam Strategy & Shortcut: Instead of dividing by 5, you can multiply the numerator by 2 and divide by 10. 2.236 × 2 = 4.472. Dividing by 10 moves the decimal point one place to the left, yielding 0.4472. This is much faster mental math. Common Pitfall: The most common trap is attempting to divide 1 directly by the decimal 2.236 (i.e., 1 / 2.236). This leads to a messy, time-consuming calculation that easily results in errors. Final Answer: Therefore, the correct answer is .447.

Question 15

If $\sqrt{24} = 4.899$, the value of $\sqrt{\frac{8}{3}}$ is

Accepted Answer

Concept & Formula: To utilize a given square root value, manipulate the fraction under the radical by multiplying the numerator and denominator by the same number to create the known value. \sqrt{(a/b)} = \sqrt{(a × k/b × k)} Step-by-Step Solution: Given: \sqrt{24} = 4.899 We need to find the value of \sqrt{(8/3)}. Notice that the numerator 8 can be turned into 24 if multiplied by 3. Multiply both the numerator and the denominator by 3: \sqrt{(8 × 3/3 × 3)} Simplify the fraction inside the square root: \sqrt{(24/9)} Apply the square root to the numerator and denominator separately: \frac{\sqrt{24}}{\sqrt{9}} This simplifies to: \frac{\sqrt{24}}{3} Substitute the given value for \sqrt{24}: (4.899/3) Perform the division: 4.899 \div 3 = 1.633 Exam Strategy & Shortcut: Look for relationships immediately. 8 and 24 are related by a factor of 3. Multiply the fraction by 3/3 mentally to get 24/9. The square root of 9 is 3, so you just need to divide the given value by 3. 4.8 / 3 = 1.6. Only option (c) matches. Common Pitfall: Students often try to divide 8 by 3 to get 2.666... and then attempt to find the square root of 2.666... manually. This ignores the given information and guarantees a difficult calculation. Final Answer: Therefore, the correct answer is 1.633.

Question 16

If $\sqrt{6} = 2.449$, then the value of $\frac{3\sqrt{2}}{2\sqrt{3}}$ is

Accepted Answer

Concept & Formula: Simplify surds and fractions by rationalizing the denominator or by cancelling common radical factors between the numerator and the denominator. \frac{\sqrt{a}}{\sqrt{b}} = \sqrt{(a/b)} Step-by-Step Solution: Given: \sqrt{6} = 2.449 We need to evaluate the expression: \frac{3\sqrt{2}}{2\sqrt{3}} Let's simplify the fraction by splitting the integers from the radicals: (3/2) × \frac{\sqrt{2}}{\sqrt{3}} To eliminate the square root in the denominator, rationalize it by multiplying the top and bottom by \sqrt{3}: \frac{3 × \sqrt{2} × \sqrt{3}}{2 × \sqrt{3} × \sqrt{3}} Combine the radicals in the numerator (\sqrt{2} × \sqrt{3} = \sqrt{6}) and simplify the denominator (\sqrt{3} × \sqrt{3} = 3): \frac{3 × \sqrt{6}}{2 × 3} The 3 in the numerator and the 3 in the denominator cancel each other out: \frac{\sqrt{6}}{2} Substitute the given value for \sqrt{6}: (2.449/2) Perform the division: 2.449 \div 2 = 1.2245 Exam Strategy & Shortcut: Recognize that 3 is simply (\sqrt{3})^2. So, (3/\sqrt{3)} instantly simplifies to \sqrt{3}. The expression becomes \frac{\sqrt{3} × \sqrt{2}}{2}, which is \frac{\sqrt{6}}{2}. Divide 2.449 in half to get 1.2245. This takes fewer than ten seconds. Common Pitfall: Substituting approximate values for \sqrt{2} (1.414) and \sqrt{3} (1.732) directly into the initial expression creates a messy calculation (3 × 1.414 / 2 × 1.732) and amplifies rounding errors. Final Answer: Therefore, the correct answer is 1.2245.

Question 17

If $\sqrt{5} = 2.236$, then the value of $\frac{\sqrt{5}}{2} - \frac{10}{\sqrt{5}} + \sqrt{125}$ is equal to

Accepted Answer

Concept & Strategy: When confronted with an expression containing multiple surds of the same base, simplify and rationalize every term so they all contain a common radical. This allows for straightforward addition and subtraction. Step-by-Step Solution: Given: \sqrt{5} = 2.236 We need to evaluate the expression: \frac{\sqrt{5}}{2} - (10/\sqrt{5)} + \sqrt{125} Simplify each term individually to have \sqrt{5} as a common factor. Term 1: \frac{\sqrt{5}}{2} remains as is (or can be written as 0.5\sqrt{5}). Term 2: Rationalize (10/\sqrt{5)} by multiplying top and bottom by \sqrt{5}: \frac{10\sqrt{5}}{5} = 2\sqrt{5} Term 3: Simplify \sqrt{125} by extracting the largest perfect square: \sqrt{25 × 5} = 5\sqrt{5} Substitute these simplified terms back into the original expression: 0.5\sqrt{5} - 2\sqrt{5} + 5\sqrt{5} Combine the coefficients: (0.5 - 2 + 5)\sqrt{5} = 3.5\sqrt{5} Now, substitute the given value for \sqrt{5}: 3.5 × 2.236 Calculate the final product: 3.5 × 2.236 = 7.826 Exam Strategy & Shortcut: Factor out the \sqrt{5} early on. The expression is \sqrt{5} × ((1/2) - (10/5) + \sqrt{25}). This simplifies instantly to \sqrt{5} × (0.5 - 2 + 5) = 3.5 × \sqrt{5}. To multiply 3.5 quickly, multiply 2.236 by 3 (6.708) and add half of 2.236 (1.118). 6.708 + 1.118 = 7.826. Common Pitfall: Failing to simplify \sqrt{125} properly or trying to evaluate 10 \div 2.236 manually. Always clean up the algebraic roots before plugging in the decimal values. Final Answer: Therefore, the correct answer is 7.826.

Question 18

If $2 * 3 = \sqrt{13}$ and $3 * 4 = 5$, then the value of $5 * 12$ is

Accepted Answer

Concept & Logic: Observe the relationship between the inputs and outputs to decode the hidden mathematical operation represented by the asterisk. a * b = \sqrt{a^2 + b^2} Step-by-Step Solution: Let's test our hypothesis on the first given equation. 2 * 3 = \sqrt{2^2 + 3^2} = \sqrt{4 + 9} = \sqrt{13}. This matches the given condition perfectly. Now, verify the pattern with the second equation. 3 * 4 = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5. This also holds true. Finally, apply this decoded logic to the target expression. 5 * 12 = \sqrt{5^2 + 12^2} Calculate the squares inside the root: = \sqrt{25 + 144} = \sqrt{169} Evaluate the square root: = 13 Exam Strategy & Shortcut: Recognize standard Pythagorean triplets immediately. The numbers (3, 4, 5) and (5, 12, 13) are classic sides of right-angled triangles. As soon as you see 3 * 4 = 5, you should suspect the Pythagorean distance formula. Applying this instinctively gives 5, 12 \rightarrow 13 without calculating the large squares manually. Common Pitfall: Trying to force standard arithmetic operations (like direct addition or multiplication) on the numbers and getting stuck when they don't yield the given roots. Always consider squares and roots for aptitude operator puzzles. Final Answer: Therefore, the correct answer is 13.

Question 19

If $1537*$ is a perfect square, then the digit which replaces $*$ is

Accepted Answer

Concept & Strategy: To find a missing digit in a perfect square, estimate the root by finding the nearest known perfect squares, then square numbers in that specific narrow range to find the exact match. Step-by-Step Solution: The number falls in the 15300 range. Let's establish a base estimate. We know 100^2 = 10000 and 120^2 = 14400. The target number 1537* is larger than 14400, so the square root must be greater than 120. Let's test 125^2. 125^2 = 15625. The target 1537* is smaller than 15625. Thus, the root must lie precisely between 120 and 125. Test the integers in this narrow window: 123^2 = 15129 (Too small) 124^2 = 15376 (This matches the 1537 prefix perfectly!) Therefore, the complete number is 15376, which means the missing final digit is 6. Exam Strategy & Shortcut: Look at the options: 2, 4, 5, 6. A perfect square can NEVER end in 2, 3, 7, or 8. This instantly eliminates option (a). If a perfect square ends in 5, the preceding digit MUST be 2 (since squares ending in 5 always end in 25). Here, the preceding digit is 7, so option (c) is eliminated. We are left with 4 and 6. Check 124^2 since 120^2 is 14400. 124 × 124 = 15376. Common Pitfall: Blindly testing all squares from 100 upwards or trying to extract a square root of an incomplete number using long division. This wastes precious time. Always use bounding squares to narrow the search radius immediately. Final Answer: Therefore, the correct answer is 6.

Question 20

The smallest perfect square that is divisible by $7!$ is

Accepted Answer

Concept & Formula: A perfect square must have even powers for all of its prime factors. To find the smallest perfect square divisible by a number, break that number into its prime factorization and multiply by the exact missing prime factors needed to make all exponents even. N = p_1^{e_1} × p_2^{e_2} \dots Step-by-Step Solution: First, expand 7! (7 factorial): 7! = 7 × 6 × 5 × 4 × 3 × 2 × 1 Break it down into its fundamental prime factors: 7! = 7 × (2 × 3) × 5 × (2^2) × 3 × 2 Combine the prime factors by adding their exponents: 2^{(1 + 2 + 1)} × 3^{(1 + 1)} × 5^1 × 7^1 = 2^4 × 3^2 × 5^1 × 7^1 Check the exponents for parity (even/odd). The powers of 2 (4) and 3 (2) are already even. The powers of 5 (1) and 7 (1) are odd. To make the entire expression a perfect square, we must multiply by 5^1 and 7^1 so their powers become 2 (an even number). Multiplier needed = 5 × 7 = 35. Calculate the final perfect square: We know 7! = 5040. Smallest Perfect Square = 5040 × 35. 5040 × 35 = 176400. Exam Strategy & Shortcut: 7! contains the prime numbers 2, 3, 5, 7. Count the highest powers conceptually: 2 appears in 2, 4, 6 (total power 4, even). 3 appears in 3, 6 (total power 2, even). 5 appears once (odd). 7 appears once (odd). You immediately know you must multiply 7! by 35. Multiplying 5040 × 35 quickly gives 176400. Common Pitfall: Forgetting to break composite numbers like 4 and 6 down into their prime components when finding the factorization. This leads to incorrect power counts and a wrong final multiplier. Final Answer: Therefore, the correct answer is 176400.

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