Question 1

Evaluate a mixed fraction inside a square root: Compute √(110 1/4).

Accepted Answer

Introduction / Context: This question checks recognizing perfect decimal squares in mixed fractions. Given Data / Assumptions: Inside the root we have 110 1/4 = 110.25. Concept / Approach: Recall 10.5^2 = (10 + 0.5)^2 = 100 + 10 + 0.25 = 110.25. Step-by-Step Solution: Convert mixed number: 110 1/4 = 110.25.Recognize 10.5^2 = 110.25.Therefore, √(110 1/4) = 10.5. Verification / Alternative check: Square 10.5 directly to confirm: 110.25. Why Other Options Are Wrong: 19.5, 11.5, 10.25, 9.5 do not square to 110.25. Common Pitfalls: Misconverting the mixed number, or confusing 10.25^2 with 10.5^2. Final Answer: 10.5

Question 2

Solve a square-root proportion: If √(x/169) = 18/13, find the value of x.

Accepted Answer

Introduction / Context: This is a straightforward equation involving square roots and rational numbers. Given Data / Assumptions: √(x/169) = 18/13. Concept / Approach: Square both sides to eliminate the root, then solve for x. Step-by-Step Solution: (x/169) = (18/13)^2 = 324/169.Multiply both sides by 169: x = 324. Verification / Alternative check: Substitute back: √(324/169) = 18/13, true. Why Other Options Are Wrong: Other values do not satisfy the equality when substituted. Common Pitfalls: Forgetting to square both numerator and denominator when squaring a fraction. Final Answer: 324

Question 3

Sixth root interpretation: Evaluate the sixth root of 0.000729.

Accepted Answer

Introduction / Context: The notation ⁶√(number) means the sixth root. Recognizing perfect powers helps evaluate roots quickly. Given Data / Assumptions: Compute ⁶√(0.000729). Concept / Approach: Write 0.000729 as 729 × 10^-6. Since 729 = 3^6, taking the sixth root is straightforward. Step-by-Step Solution: 0.000729 = 729 × 10^-6.⁶√(729 × 10^-6) = ⁶√(729) × ⁶√(10^-6) = 3 × 10^-1 = 0.3. Verification / Alternative check: (0.3)^6 = (3 × 10^-1)^6 = 3^6 × 10^-6 = 729 × 10^-6 = 0.000729. Why Other Options Are Wrong: 0.027, 0.03, 0.09, 0.9 are the square, cube, or other root results but not the sixth root. Common Pitfalls: Misreading 6√ as 6 × √; here it denotes the sixth root, not 6 times the square root. Final Answer: 0.3

Question 4

Evaluate a multi-step numeric expression: Compute S = √(16 + (9.5 × 0.0085 × 18.9) / (0.021 × 0.0017 × 1.9)).

Accepted Answer

Introduction / Context: This evaluates a nested expression mixing decimals and a square root. Careful order of operations and fraction handling are key. Given Data / Assumptions: S = √(16 + (9.5 × 0.0085 × 18.9) / (0.021 × 0.0017 × 1.9)). Concept / Approach: Compute the numerator and denominator separately, simplify the fraction, add 16, then take the square root. Step-by-Step Solution: Numerator: 9.5 × 0.0085 × 18.9 = 1.526175.Denominator: 0.021 × 0.0017 × 1.9 = 0.00006783.Fraction = 1.526175 / 0.00006783 = 22500 exactly (since the decimals reduce to 22500).Inside root: 16 + 22500 = 22516.S = √22516 ≈ 150.05, which rounds to 150 among the given choices. Verification / Alternative check: Quick check: since √22500 = 150 and our value is √22516, the answer must be just slightly above 150, so 150 is the nearest listed value. Why Other Options Are Wrong: 154, 158, 160, 169 are far from the precise √22516 ≈ 150.05. Common Pitfalls: Losing precision in decimal multiplication/division or misapplying square root to the fraction before summing with 16. Final Answer: 150

Question 5

What is the cube root of 0.000216?

Accepted Answer

Introduction / Context: This is a basic question on cube roots of decimal numbers. It asks for the cube root of 0.000216, which can be handled either by converting the decimal to a fraction or by using known cube values and decimal place rules. Given Data / Assumptions: - We need ∛0.000216. - The given options are 0.6, 0.06, 0.77 and 0.87. Concept / Approach: We recall that 216 is a perfect cube, since 6³ = 6 * 6 * 6 = 216. Then we use the idea that moving the decimal point in the original number corresponds to adjusting the decimal point in its cube root. Expressing 0.000216 in scientific notation helps in understanding how the cube root acts on powers of 10. Step-by-Step Solution: Step 1: Write 0.000216 as 216 / 10⁶, because 0.000216 = 216 divided by 1,000,000.Step 2: Recognize that 216 = 6³.Step 3: Also note that 10⁶ = (10²)³, since (10²)³ = 10⁶.Step 4: Therefore, 0.000216 = 216 / 10⁶ = 6³ / (10²)³ = (6 / 10²)³.Step 5: This shows that the cube root of 0.000216 is 6 / 10² = 6 / 100 = 0.06. Verification / Alternative check: Compute 0.06³ directly: 0.06 * 0.06 = 0.0036, and 0.0036 * 0.06 = 0.000216. This confirms that 0.06 is indeed the cube root of 0.000216. Why Other Options Are Wrong: 0.6³ equals 0.216, which is 1000 times larger than 0.000216. The options 0.77 and 0.87 are not simple fractions of known cubes and their cubes would be well above 0.000216. Therefore they do not match the required cube root. Common Pitfalls: A common mistake is to treat decimal cube roots as if they were square roots and move the decimal point incorrectly. Another error is ignoring the exponent on 10 when splitting the number into mantissa and power of 10. Using scientific notation or writing the decimal as a fraction helps avoid these errors. Final Answer: The cube root of 0.000216 is 0.06.

Question 6

If x = 7 − 4√3, then what is the value of x + 1/x?

Accepted Answer

Introduction / Context: This algebra question involves surds and reciprocals. We are given x = 7 − 4√3 and must compute x + 1/x. The problem tests skill in rationalizing expressions with square roots and recognizing patterns that lead to simplification. Given Data / Assumptions: - x = 7 − 4√3. - We need the exact value of x + 1/x. - All operations are in real numbers and √3 is the positive square root of 3. Concept / Approach: When x is of the form a − b√c, its reciprocal 1/x can often be found by multiplying numerator and denominator by the conjugate a + b√c. Then x + 1/x frequently simplifies to a rational number because the surd parts cancel. We use the fact that (a − b√c)(a + b√c) = a² − b²c. Step-by-Step Solution: Step 1: Start with x = 7 − 4√3.Step 2: To find 1/x, multiply numerator and denominator by the conjugate 7 + 4√3.Step 3: So 1/x = (7 + 4√3) / [(7 − 4√3)(7 + 4√3)].Step 4: Compute the denominator using a² − b²c: 7² − (4² * 3) = 49 − 16 * 3 = 49 − 48 = 1.Step 5: Therefore, 1/x = 7 + 4√3.Step 6: Now compute x + 1/x = (7 − 4√3) + (7 + 4√3).Step 7: The surd terms cancel: −4√3 + 4√3 = 0, leaving 7 + 7 = 14. Verification / Alternative check: We can compute x numerically. Since √3 is approximately 1.732, x ≈ 7 − 4 * 1.732 ≈ 7 − 6.928 ≈ 0.072. Then 1/x ≈ 1 / 0.072 ≈ 13.888. Their sum is approximately 13.96, which is close to 14, confirming the exact result with minor rounding error. Why Other Options Are Wrong: The options involving surds, such as 3√3, 8√3 or 14 + 8√3, would indicate that the surd term survived the simplification. Because of the conjugate property, the surd part cancels when we add x and 1/x, leaving a pure rational result. Thus those options do not match the correct algebraic simplification. Common Pitfalls: Students may forget to use the conjugate properly or may incorrectly compute a² − b²c in the denominator, leading to wrong values for 1/x. Another mistake is to try to add x and 1/x before simplifying 1/x, which makes the algebra more complicated and error prone. Final Answer: The value of x + 1/x is 14.

Question 7

If √0.00000676 = 0.0026, then what is the value of √6760000?

Accepted Answer

Introduction / Context: This question relates two square roots whose numbers share the same digits but differ by their decimal placement. We are told the square root of a small decimal number and asked to find the square root of a large number constructed from the same digits. The problem tests understanding of how square roots behave with powers of 10. Given Data / Assumptions: - √0.00000676 = 0.0026 is given. - We need √6760000. - All numbers are positive real numbers. Concept / Approach: We use scientific notation to understand the relation between the two numbers. Multiplying a number by a power of 10 shifts the decimal point. The square root of 10ⁿ is 10^(n/2). If two numbers differ by a factor of 10ᵏ, then their square roots differ by a factor of 10^(k/2). We use the given square root to deduce the required one by scaling appropriately. Step-by-Step Solution: Step 1: Rewrite 0.00000676 in scientific form: 0.00000676 = 6.76 × 10⁻⁶.Step 2: Then √0.00000676 = √(6.76 × 10⁻⁶) = √6.76 × 10⁻³, which is given as 0.0026 = 2.6 × 10⁻³. So √6.76 = 2.6.Step 3: Now consider 6760000. Write 6760000 = 6.76 × 10⁶ (the same 6.76 with 10⁶ instead of 10⁻⁶).Step 4: Then √6760000 = √(6.76 × 10⁶) = √6.76 × 10³.Step 5: Since √6.76 = 2.6, we get √6760000 = 2.6 × 10³ = 2600. Verification / Alternative check: We can square 2600 to verify. 2600² = (26 × 100)² = 26² × 100² = 676 × 10⁴ = 6,760,000 which matches 6760000, confirming that 2600 is the correct square root. Why Other Options Are Wrong: 2.6 and 26 are far too small; their squares are 6.76 and 676, respectively, which are nowhere near 6760000. The option 260 has square 67600, still smaller than the target by a factor of 100. Only 2600 gives the correct square when multiplied by itself. Common Pitfalls: A common error is to miscount decimal places when converting between forms like 0.00000676 and 6760000. Another mistake is to forget that when a number is multiplied by 10⁶, its square root is multiplied by 10³, not by 10⁶. Carefully handling the exponents of 10 avoids these issues. Final Answer: The value of √6760000 is 2600.

Question 8

A group of students decide to collect as many paise from each member as the number of members in the group. If the total collection amounts to Rs 59.29, how many students are there in the group?

Accepted Answer

Introduction / Context: This is an application of basic arithmetic and the concept of square numbers. The collection rule is that each student contributes a number of paise equal to the total number of students in the group. The question asks us to find the group size from the total collection given in rupees. Given Data / Assumptions: - Each student contributes paise equal to the number of students in the group. - Total collection is Rs 59.29. - 1 rupee = 100 paise. Concept / Approach: If there are n students, each contributes n paise. So the total collection in paise is n × n = n². We first convert the rupee amount into paise, then recognize that this total must be a perfect square. Taking the square root gives the number of students. Step-by-Step Solution: Step 1: Let the number of students be n.Step 2: Each student contributes n paise, so total collection in paise is n × n = n².Step 3: Convert Rs 59.29 to paise. Since 1 rupee = 100 paise, Rs 59.29 = 59.29 × 100 = 5929 paise.Step 4: Therefore, n² = 5929.Step 5: Find n by taking the square root: n = √5929.Step 6: Check: 77² = 77 × 77 = 5929, so n = 77. Verification / Alternative check: You can quickly square 57, 67 and 87 to see that none of these give 5929. Only 77² equals 5929. This confirms that the number of students is 77. Why Other Options Are Wrong: 57² = 3249, 67² = 4489 and 87² = 7569. None match the total collection in paise (5929). Therefore those options cannot represent the correct group size under the given contribution rule. Common Pitfalls: Students sometimes forget to convert rupees into paise and mistakenly set n² equal to 59.29 instead of 5929. Also, miscalculating squares of two digit numbers may lead to wrong conclusions. Using mental math shortcuts or quick multiplication helps verify squares accurately. Final Answer: The number of students in the group is 77.

Question 9

If 89² is added to the square of a number and the result is 16202, what is one twenty-sixth (1/26) of that original number?

Accepted Answer

Introduction / Context: This is an algebraic problem involving squares and fractions. We are told that when 89² is added to the square of an unknown number, the sum is 16202. We need to determine one twenty-sixth of that unknown number. The question tests equation formation and manipulation with square numbers. Given Data / Assumptions: - Let the unknown number be n. - n² + 89² = 16202. - We need n / 26 (one twenty-sixth of n). Concept / Approach: We first compute 89², subtract it from 16202 to find n², and then take the positive square root to get n. Once n is known, we divide it by 26 to get n / 26, which should match one of the options. Step-by-Step Solution: Step 1: Let the required number be n. Then n² + 89² = 16202.Step 2: Compute 89² = 89 × 89 = 7921.Step 3: Substitute into the equation: n² + 7921 = 16202.Step 4: Subtract 7921 from both sides: n² = 16202 − 7921.Step 5: Calculate the difference: 16202 − 7921 = 8281.Step 6: So n² = 8281. Take the square root: n = √8281.Step 7: Recognize that 91² = 8281, so n = 91.Step 8: We are asked for n / 26, so compute 91 / 26.Step 9: 91 divided by 26 = 3.5. Verification / Alternative check: Check the original equation: 91² = 8281, 89² = 7921, and 8281 + 7921 = 16202. This confirms that n = 91. Dividing 91 by 26 gives 3.5, which matches the derived result and one of the answer choices. Why Other Options Are Wrong: Values like 5.65, 2.7 and 6.66 would correspond to different values of n and therefore would not satisfy the original equation n² + 89² = 16202. Testing them by multiplying by 26 and then squaring shows that none produce n² = 8281. Common Pitfalls: Errors often occur in computing 89² or in subtracting 7921 from 16202. Some students also misidentify the square root of 8281. Knowing that 90² = 8100 and 95² = 9025 helps estimate that 8281 lies between them, leading to the correct value n = 91 by checking 91² explicitly. Final Answer: One twenty-sixth of the number is 3.5.

Question 10

Given that √18225 = 135, find the value of √182.25 + √1.8225 + √0.018225 + √0.00018225.

Accepted Answer

Introduction / Context: This question involves square roots of several decimal numbers that are scaled versions of 18225. We are given that √18225 = 135 and asked to evaluate a sum of related square roots. The problem tests recognition of how square roots behave under scaling and decimal shifts. Given Data / Assumptions: - √18225 = 135 is known. - We must compute S = √182.25 + √1.8225 + √0.018225 + √0.00018225. - All quantities are positive real numbers. Concept / Approach: Each decimal can be written as 18225 multiplied by a power of 10. Since √(k × 18225) = √k × √18225, and √18225 is given, we can express each term in the sum S in terms of 135 and appropriate powers of 10. Then we add the results to find S. Step-by-Step Solution: Step 1: Note that 182.25 = 18225 / 100 = 18225 × 10⁻².Step 2: So √182.25 = √(18225 × 10⁻²) = √18225 × √10⁻² = 135 × 10⁻¹ = 13.5.Step 3: Next, 1.8225 = 18225 / 10000 = 18225 × 10⁻⁴, so √1.8225 = 135 × 10⁻² = 1.35.Step 4: For 0.018225, write 0.018225 = 18225 / 10⁶ = 18225 × 10⁻⁶, hence √0.018225 = 135 × 10⁻³ = 0.135.Step 5: For 0.00018225, we have 0.00018225 = 18225 / 10⁸ = 18225 × 10⁻⁸, so √0.00018225 = 135 × 10⁻⁴ = 0.0135.Step 6: Now compute the sum S = 13.5 + 1.35 + 0.135 + 0.0135.Step 7: Add them stepwise: 13.5 + 1.35 = 14.85; 14.85 + 0.135 = 14.985; 14.985 + 0.0135 = 14.9985. Verification / Alternative check: We can check each square root using a calculator. For example, √182.25 ≈ 13.5, √1.8225 ≈ 1.35, √0.018225 ≈ 0.135 and √0.00018225 ≈ 0.0135. Adding these approximate values gives 14.9985, matching the exact calculation. Why Other Options Are Wrong: The options 1.49985, 149.985 and 1499.85 correspond to shifting the decimal point incorrectly by one or more places. These arise if the power of 10 under the square root is mishandled, for example treating √10⁻² as 10⁻² instead of 10⁻¹. Common Pitfalls: Errors often occur when converting decimals to powers of 10 or when taking square roots of these powers. Remember that √10ⁿ = 10^(n/2). For example, √10⁻⁴ = 10⁻², not 10⁻⁴. Misplacing the decimal in intermediate steps can easily lead to answers that differ by factors of 10. Final Answer: The value of √182.25 + √1.8225 + √0.018225 + √0.00018225 is 14.9985.

Question 11

Find the value of 1.52 × 0.0225 (give your answer as a decimal number).

Accepted Answer

Introduction / Context: This question checks your comfort with decimal multiplication, which is very common in bank exams, SSC exams, and other aptitude tests. Handling place values correctly while multiplying decimals is important because a small mistake with zeros and decimal points can completely change the answer. Given Data / Assumptions: We need to evaluate the numerical product 1.52 * 0.0225. Both numbers are positive decimal numbers. The answer should be written as a decimal value, without any rounding instruction beyond normal precision. Concept / Approach: To multiply decimals, ignore the decimal points first and multiply the numbers as whole numbers. Then count the total number of decimal places in the factors. Finally, place the decimal point in the product so that it has the same total number of decimal places. This systematic approach prevents confusion and makes calculations cleaner in aptitude exams. Step-by-Step Solution: Step 1: Rewrite the expression as 1.52 * 0.0225. Step 2: Count decimal places: 1.52 has 2 decimal places, and 0.0225 has 4 decimal places, for a total of 2 + 4 = 6 decimal places. Step 3: Remove decimal points and multiply 152 * 225 as whole numbers. Step 4: Compute 152 * 225 = 34200. Step 5: Now place the decimal point so that the product has 6 decimal places: 34200 becomes 0.034200. Step 6: Remove the unnecessary trailing zeros. So the final value is 0.0342. Verification / Alternative check: We know that 0.0225 is a small number, slightly more than 2 percent. Multiplying 1.52 by a little more than 0.02 should give a result slightly more than 1.52 * 0.02 = 0.0304. Our obtained answer 0.0342 is only a bit larger than 0.0304, which is perfectly reasonable and confirms that the magnitude is correct. Why Other Options Are Wrong: Option 0.342 is ten times larger than the correct value, which would happen if you shifted the decimal point only five places instead of six. Option 0.00342 is one tenth of the correct answer, which would come from shifting the decimal point seven places. Option 0.375 and option 0.0375 are unrelated to the correct multiplication and likely come from guessing without checking decimal places carefully. Common Pitfalls: Students often ignore the total number of decimal places and just write the whole number product as the final answer, which is completely wrong. Another frequent error is to miscount the decimal digits in one of the factors. Always count decimal places in each factor, add them, multiply as whole numbers, and then shift the decimal point exactly that many places from the right in the product. Final Answer: Therefore, the correct value of 1.52 × 0.0225 is 0.0342.

Question 12

Find the value of 0.0100 + 0.0064 (express your answer in decimal form).

Accepted Answer

Introduction / Context: This problem tests basic decimal addition, a foundational skill in quantitative aptitude. Being able to add small decimal numbers accurately is essential in topics such as simple interest, discounts, and data interpretation, where values often appear in decimal form. Given Data / Assumptions: We must compute 0.0100 + 0.0064. Both numbers are written to four decimal places, so proper alignment of decimal points is important. The answer should be presented as a decimal number. Concept / Approach: For decimal addition, we align the numbers by the decimal point, add digit by digit from right to left, and carry over when needed. Writing trailing zeros does not change a number, but it helps keep place values clear. Here, each number has four decimal places, which makes the addition straightforward. Step-by-Step Solution: Step 1: Write the numbers one below the other: 0.0100 and 0.0064. Step 2: Add the digits in the ten thousandths place: 0 + 4 = 4. Step 3: Add the digits in the thousandths place: 0 + 6 = 6. Step 4: Add the digits in the hundredths place: 1 + 0 = 1. Step 5: Add the digits in the tenths place: 0 + 0 = 0. Step 6: Combine these to get 0.0164 as the final sum. Verification / Alternative check: We can quickly check reasonableness by observing that both numbers are close to 0.01. Their sum should be slightly more than 0.015. The computed result is 0.0164, which is indeed a little larger than 0.015, so the answer fits our expectation and confirms that the addition is correct. Why Other Options Are Wrong: Option 0.164 is ten times larger than the correct answer and comes from shifting the decimal point one place too far to the right. Option 0.00164 is ten times smaller than the correct value and results from shifting the decimal in the opposite direction. Options 0.024 and 0.026 are arbitrary values that do not match the precise sum obtained through correct decimal addition. Common Pitfalls: A very common mistake is to ignore place values and just add the digits without aligning the decimal points. Another issue is dropping or misplacing zeros, which changes the value entirely. Always ensure that tenths align with tenths, hundredths with hundredths, and so on before performing the addition. Final Answer: Thus, the correct value of 0.0100 + 0.0064 is 0.0164.

Question 13

If a custom operation is defined as x × y = x + y + x*y, what is the value of 6 × 24?

Accepted Answer

Introduction / Context: This problem introduces a custom binary operation on numbers. Such questions frequently appear in aptitude and reasoning sections to test whether the candidate reads definitions carefully instead of assuming the usual meaning of mathematical symbols like multiplication. You must apply the given definition exactly as stated. Given Data / Assumptions: The operation × is not ordinary multiplication. It is defined by the rule: x × y = x + y + x*y. We need to compute 6 × 24 using this special rule. Concept / Approach: Whenever a new operation is defined, always replace the symbol with the given formula. Here, for any numbers x and y, we must calculate x + y + x*y. That means we first find the product x*y, then add x and y to this product. We never treat × as normal multiplication in this context, even though the same symbol is used. Step-by-Step Solution: Step 1: Identify x and y in the expression 6 × 24. Here x = 6 and y = 24. Step 2: Use the definition x × y = x + y + x*y. Step 3: Compute the product x*y = 6*24 = 144. Step 4: Compute x + y = 6 + 24 = 30. Step 5: Add these results: 144 + 30 = 174. Step 6: Therefore, 6 × 24 under the given operation equals 174. Verification / Alternative check: We can quickly check by recomputing: 6*24 = 144 and 6 + 24 = 30. Adding them again gives 174, confirming that our arithmetic is consistent. There is no other interpretation for the custom operation given, so 174 is the unique correct value for this question. Why Other Options Are Wrong: Option 150 might come from adding 6 and 24 to the normal product 120 instead of 144. Option 186 could come from a mistaken product like 6*27 instead of 6*24. Option 210 may be the usual product 6*35 or a random guess that ignores the definition. Option 144 corresponds only to the ordinary product 6*24 and does not include x + y, so it does not satisfy the rule x × y = x + y + x*y. Common Pitfalls: A very frequent mistake in such questions is to interpret the symbol × as normal multiplication without using the new definition at all. Another pitfall is to partly apply the rule, for example adding only one of the numbers to the product instead of both x and y. Always rewrite the custom operation explicitly in terms of standard operations to avoid confusion. Final Answer: Therefore, using the custom rule x × y = x + y + x*y, the value of 6 × 24 is 174.

Question 14

First find the cube root of 729, then square that result. What final value do you obtain?

Accepted Answer

Introduction / Context: This question connects cube roots and squares, two very basic but important operations in algebra. Problems like this are common in school level mathematics and competitive exams to test how well you understand powers and roots of numbers, especially perfect cubes and perfect squares. Given Data / Assumptions: We are given the number 729. We must first find its cube root. Then we must square the cube root obtained in the first step. No approximation is needed because 729 is a perfect cube. Concept / Approach: The cube root of a number N is the value r such that r^3 = N. After finding the cube root, squaring that result means calculating r^2. For perfect powers, it helps to remember common cubes and squares: for example 3^3 = 27, 5^3 = 125, 9^3 = 729, and 9^2 = 81. Recognizing these patterns makes such questions extremely fast to solve. Step-by-Step Solution: Step 1: Observe that 729 can be written as 9^3, because 9 * 9 * 9 = 729. Step 2: Therefore, the cube root of 729 is 9. Step 3: Next, we must square this cube root. So we compute 9^2. Step 4: Calculate 9^2 = 9 * 9 = 81. Step 5: Hence, after taking the cube root of 729 and then squaring the result, we obtain 81. Verification / Alternative check: We can check in reverse. If the final answer is 81, its square root is 9. Cubing 9 gives 729, which is exactly the original number. This confirms that we have correctly identified 9 as the cube root of 729 and 81 as its square, so the overall two step process is consistent. Why Other Options Are Wrong: Option 9 is only the cube root of 729, not the final squared value. Option 27 could appear if someone mistakenly thought that 3 is the cube root and then computed 3^3, but 3^3 is 27, not 729. Option 144 is 12^2 and does not arise from any correct operation here. Option 36 might come from squaring 6, but 6 is not related to 729 as a cube root. Only 81 matches the described two step procedure. Common Pitfalls: A common mistake is to confuse square roots and cube roots, for example trying to find a number whose square is 729 instead of whose cube is 729. Another error is to stop after finding the cube root and forget to square it as the second step. Always read multi step instructions fully and complete each operation in order. Final Answer: Thus, the cube root of 729 is 9 and squaring this gives 81 as the final result.

Question 15

What is the value of 729 + 72.9 + 7.29 (correct to two decimal places)?

Accepted Answer

Introduction / Context: This question checks your skill with adding whole numbers and decimal numbers together. Such mixed sums are very common in data interpretation, profit and loss, and general arithmetic sections of aptitude tests, so it is important to keep track of place values accurately. Given Data / Assumptions: We must compute the sum 729 + 72.9 + 7.29. All numbers are positive. The answer is required correct to two decimal places. Concept / Approach: When adding whole numbers and decimal numbers, it is best to align them by their decimal point. The whole number can be written with a decimal point and trailing zeros, like 729.00. Then we add column wise, starting from the smallest place value on the right. Finally, we round the result to the required number of decimal places if necessary. Step-by-Step Solution: Step 1: Write the numbers as 729.00, 72.90, and 7.29 to align the decimals. Step 2: Add the hundredths place: 0 + 0 + 9 = 9. Step 3: Add the tenths place: 0 + 9 + 2 = 11, write 1 in the tenths place and carry 1 to the units place. Step 4: Add the units place: 9 (from 729) + 2 (from 72) + 7 (from 7) + carried 1 = 19, write 9 in the units place and carry 1 to the tens place. Step 5: Add the tens place: 2 (from 729) + 7 (from 72) + carried 1 = 10, write 0 in tens and carry 1 to the hundreds place. Step 6: Add the hundreds place: 7 (from 729) + carried 1 = 8. Step 7: The final result is 809.19, which is already correct to two decimal places. Verification / Alternative check: We can group the terms in a simpler way: 729 + 72.9 = 801.9. Then 801.9 + 7.29 = 809.19. This quick mental verification confirms that our detailed column addition is correct. The consistency between both methods increases confidence in the final answer. Why Other Options Are Wrong: Option 809.29 arises from misadding the tenths or hundredths place. Option 802.19 suggests an error in adding the tens digit of 72.9. Option 729.19 ignores adding 72.9 and 7.29 properly. Option 809.09 results from mishandling the hundredths place when adding 0.90 and 0.29. Only 809.19 matches both systematic calculation and mental checking. Common Pitfalls: Typical mistakes include not aligning decimal points, forgetting to carry from the tenths to the units place, or dropping decimal digits when converting whole numbers like 729 into 729.00. Careful alignment and stepwise addition prevent these errors and are essential for speed and accuracy in competitive exams. Final Answer: Therefore, the value of 729 + 72.9 + 7.29 is 809.19 when written to two decimal places.

Question 16

What is the value of 361 + 3.61 + 0.0361 (correct to four decimal places)?

Accepted Answer

Introduction / Context: This question again focuses on decimal addition, but now with more digits after the decimal point. Being comfortable with four decimal places is useful in questions related to percentages, interest rates, and scientific style calculations where precision is important. Given Data / Assumptions: We must evaluate 361 + 3.61 + 0.0361. The result must be expressed correct to four decimal places. All values are positive real numbers. Concept / Approach: The key idea is to line up decimal points and then add each column of digits from right to left. Writing each number with the same number of decimal places simplifies the process. Here, the smallest place value is ten thousandths, so we convert each number into a form with four decimal places before adding. Step-by-Step Solution: Step 1: Rewrite 361 as 361.0000, 3.61 as 3.6100, and 0.0361 remains 0.0361. Step 2: Add the ten thousandths place: 0 + 0 + 1 = 1. Step 3: Add the thousandths place: 0 + 0 + 6 = 6. Step 4: Add the hundredths place: 0 + 1 + 3 = 4. Step 5: Add the tenths place: 0 + 6 + 0 = 6. Step 6: Add the units place: 1 (from 361) + 3 (from 3.6100) + 0 = 4. Step 7: Add the tens and hundreds of the whole part: 6 tens and 3 hundreds simply give 60 and 300, so the total whole number part is 364. Step 8: Combining the whole and decimal parts, we obtain 364.6461. Verification / Alternative check: We can first add the whole numbers and then the decimals separately. Whole parts: 361 + 3 = 364. Decimal parts: 0.61 + 0.0361 = 0.6461. Adding these gives 364 + 0.6461 = 364.6461. This matches the earlier detailed calculation, confirming that the answer is consistent. Why Other Options Are Wrong: Option 364.6614 rearranges the decimal digits incorrectly and does not match proper column addition. Option 364.6416 also represents a misplacement of digits when summing the thousandths and ten thousandths places. Option 361.6461 comes from forgetting to add the 3 in 3.61 to the whole part. Option 374.6461 mistakenly increases the hundreds digit to 7 instead of 3 plus 0, leading to an inflated total. Common Pitfalls: Many learners rush through decimal addition and mix up the order of digits, especially when more than two decimal places are involved. Another common issue is treating 3.61 as 3.610 or 3.601 accidentally. Carefully rewriting all numbers with the same number of decimal digits before adding is a reliable way to avoid such mistakes. Final Answer: Hence, the value of 361 + 3.61 + 0.0361 is 364.6461 when written to four decimal places.

Question 17

If a is positive and a^2 + 1/a^2 = 7, then what is the value of a^3 + 1/a^3 ?

Accepted Answer

Introduction / Context: This question tests your understanding of algebraic identities involving expressions of the form a + 1/a, a^2 + 1/a^2, and a^3 + 1/a^3. Such identities are very popular in competitive exams because they allow you to avoid solving for a directly and instead use smart manipulation of formulas. Given Data / Assumptions: a is a positive real number. We are given that a^2 + 1/a^2 = 7. We must find the value of a^3 + 1/a^3. Concept / Approach: The standard approach is to introduce a new variable t = a + 1/a. Then we use algebraic identities to relate t, a^2 + 1/a^2, and a^3 + 1/a^3. The key identities are: a^2 + 1/a^2 = t^2 - 2 a^3 + 1/a^3 = t^3 - 3t Once we find t from the first identity, we substitute in the second identity to get the required value. Step-by-Step Solution: Step 1: Let t = a + 1/a. Step 2: Use the identity a^2 + 1/a^2 = t^2 - 2. Step 3: We are given a^2 + 1/a^2 = 7, so t^2 - 2 = 7. Step 4: Solve for t^2: t^2 = 7 + 2 = 9. Step 5: Since a is positive, a + 1/a is also positive, therefore t = 3 (and not -3). Step 6: Now use the identity a^3 + 1/a^3 = t^3 - 3t. Step 7: Substitute t = 3 to get a^3 + 1/a^3 = 3^3 - 3*3. Step 8: Compute 3^3 = 27 and 3*3 = 9, so a^3 + 1/a^3 = 27 - 9 = 18. Verification / Alternative check: We can cross check by finding a explicitly. From t = a + 1/a = 3, we have a^2 - 3a + 1 = 0. Solving this quadratic gives two positive roots, but both will yield the same value of a^2 + 1/a^2 and a^3 + 1/a^3. Substituting either root into a^3 + 1/a^3 confirms that the expression is indeed 18, which matches our identity based method. Why Other Options Are Wrong: Option 9 corresponds to t^2, not a^3 + 1/a^3. Option 27 is t^3, which ignores the subtraction of 3t in the identity. Option 0 would suggest a^3 + 1/a^3 cancels out completely, which is not supported by the given condition. Option 7 simply repeats the value of a^2 + 1/a^2 and does not apply the correct identity to move from squares to cubes. Common Pitfalls: A frequent error is to assume t = ±3 without using the information that a is positive. Another mistake is to try to solve directly for a^3 and 1/a^3 separately, which is unnecessary and time consuming. Some students also confuse the identities and write a^3 + 1/a^3 = t^3 - 2t, which is incorrect. Remember the correct forms and apply them step by step. Final Answer: Therefore, when a is positive and a^2 + 1/a^2 = 7, the value of a^3 + 1/a^3 is 18.

Question 18

When the polynomial 4x^4 + 10x^3 - 20x^2 + 90 is divided by (x + 2), what is the remainder?

Accepted Answer

Introduction / Context: This question tests the remainder theorem for polynomials, which is a very common topic in aptitude and algebra exams. Instead of performing full long division, we can quickly find the remainder when a polynomial in x is divided by a linear factor of the form x - a or x + a by evaluating the polynomial at a suitable value of x. Given Data / Assumptions: Polynomial: P(x) = 4x^4 + 10x^3 - 20x^2 + 90 Divisor: x + 2 We assume x is a real number and standard polynomial division rules apply. Concept / Approach: The remainder theorem states that when a polynomial P(x) is divided by (x - a), the remainder is P(a). For a divisor of the form (x + 2), we can rewrite it as (x - (-2)), so the remainder will be P(-2). This avoids long division and gives a fast and accurate result. Step-by-Step Solution: Step 1: Identify a such that x + 2 = x - (-2). So a = -2.Step 2: Write P(x) = 4x^4 + 10x^3 - 20x^2 + 90.Step 3: Compute P(-2) = 4(-2)^4 + 10(-2)^3 - 20(-2)^2 + 90.Step 4: Evaluate powers: (-2)^4 = 16, (-2)^3 = -8, (-2)^2 = 4.Step 5: Substitute: P(-2) = 4*16 + 10*(-8) - 20*4 + 90.Step 6: Simplify term by term: 4*16 = 64, 10*(-8) = -80, -20*4 = -80.Step 7: Add all terms: 64 - 80 - 80 + 90 = (64 - 80) = -16, then -16 - 80 = -96, and -96 + 90 = -6.Step 8: Therefore, the remainder when P(x) is divided by (x + 2) is -6. Verification / Alternative check: We can verify by performing synthetic division with -2 as the root. Using the coefficients 4, 10, -20, 0, 90 and applying synthetic division with -2 will produce a final remainder of -6. This confirms that our direct substitution using the remainder theorem is correct. Why Other Options Are Wrong: 0: This would mean x + 2 is an exact factor of the polynomial, which is not true because P(-2) is -6, not 0. 45: This value is not supported by any correct evaluation method for P(-2). 90: This is simply the constant term of the polynomial, not the remainder after division by (x + 2). Common Pitfalls: Students sometimes plug in x = 2 instead of x = -2 because they forget that x + 2 corresponds to x - (-2). Another common error is incorrect evaluation of powers of negative numbers, especially confusing (-2)^3 with -8 and (-2)^4 with 16. Careless arithmetic in combining the terms can also lead to wrong answers. Final Answer: The remainder when 4x^4 + 10x^3 - 20x^2 + 90 is divided by (x + 2) is -6.

Question 19

If r and s are real numbers such that r^3 + s^3 = 0 and r + s = 6, what is the value of 1/r + 1/s?

Accepted Answer

Introduction / Context: This problem combines symmetric expressions in algebra, specifically sums and cubes of two real numbers r and s. Questions like this often appear in aptitude tests to check comfort with algebraic identities and the ability to express one symmetric combination in terms of others. Given Data / Assumptions: r and s are real numbers. r^3 + s^3 = 0. r + s = 6. We need to compute 1/r + 1/s. We assume r and s are nonzero so that 1/r and 1/s are defined. Concept / Approach: We use the identity for the sum of cubes: r^3 + s^3 = (r + s)^3 - 3rs(r + s). This connects r^3 + s^3 with r + s and rs. Once we find rs, we can compute 1/r + 1/s using the identity 1/r + 1/s = (r + s) / (rs). Step-by-Step Solution: Step 1: Start from r^3 + s^3 = 0.Step 2: Apply the identity r^3 + s^3 = (r + s)^3 - 3rs(r + s).Step 3: Substitute r + s = 6 into the identity.Step 4: Compute (r + s)^3 = 6^3 = 216.Step 5: So r^3 + s^3 = 216 - 3rs * 6 = 216 - 18rs.Step 6: Given r^3 + s^3 = 0, set 216 - 18rs = 0.Step 7: Solve for rs: 18rs = 216, so rs = 216 / 18 = 12.Step 8: Now compute 1/r + 1/s = (r + s) / (rs) = 6 / 12 = 1/2 = 0.5. Verification / Alternative check: We can construct explicit values for r and s. Suppose r + s = 6 and rs = 12, then r and s are roots of t^2 - 6t + 12 = 0. These roots are complex conjugates, but the expression 1/r + 1/s only depends on the symmetric data r + s and rs, so the result 0.5 is still valid and consistent with the algebraic relationships. Why Other Options Are Wrong: 0: This would imply r + s = 0 or rs is infinite, which contradicts r + s = 6. 1: This would require rs = 6, but we found rs = 12. 2: This would require rs = 3, again inconsistent with the derived value rs = 12. Common Pitfalls: A common mistake is to try to solve directly for r and s instead of using symmetric identities, which may become messy. Another frequent error is misapplying the sum of cubes formula or forgetting the factor 3rs(r + s), leading to an incorrect value for rs and hence for 1/r + 1/s. Final Answer: The required value of 1/r + 1/s is 0.5.

Question 20

Which of the following numbers is the exact square root of 14161?

Accepted Answer

Introduction / Context: This problem checks the ability to identify the square root of a given perfect square without using a calculator. Such questions are standard in aptitude exams, where speed and estimation are very important. Given Data / Assumptions: We are given the number 14161. Options are 119, 121, 131, and 141. We assume that one of these numbers is the exact positive square root of 14161. Concept / Approach: To find the square root of 14161, we can estimate based on nearby squares. Notice that 120^2 = 14400, which is slightly larger than 14161. So the square root must be slightly less than 120. The most reasonable option near 120 is 119, so we explicitly compute 119^2 to check if it equals 14161. Step-by-Step Solution: Step 1: Observe that 120^2 = 14400, which is greater than 14161.Step 2: This suggests that the square root must be less than 120.Step 3: Among the options, 119 is closest to 120 from below.Step 4: Compute 119^2. We can use (120 - 1)^2 = 120^2 - 2*120*1 + 1^2.Step 5: 120^2 = 14400, 2*120*1 = 240, and 1^2 = 1.Step 6: So 119^2 = 14400 - 240 + 1 = 14400 - 239 = 14161.Step 7: Since 119^2 equals 14161 exactly, the square root of 14161 is 119. Verification / Alternative check: We can also check another option to see that it fails. For example, 121^2 = (120 + 1)^2 = 14400 + 240 + 1 = 14641, which is larger than 14161. This confirms that 121 is not the correct square root. Why Other Options Are Wrong: 121: Its square is 14641, which is greater than 14161. 131: Since it is larger than 120, its square will be significantly larger than 14400, clearly exceeding 14161. 141: This is also far larger than 120, and 141^2 would be well above the target number. Common Pitfalls: Some students may guess based purely on the last digit of the number or may choose 121 because they know it is 11^2 multiplied by 11, but that pattern does not match 14161. Careful estimation around 120 and exact computation avoids these mistakes. Final Answer: The exact square root of 14161 is 119.

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