Question 1

One-fourth of the sum of prime numbers, greater than 4 but less than 16, is the square of

Accepted Answer

Concept & Strategy: This is a multi-step arithmetic reasoning problem. It requires identifying prime numbers within a specific range, performing basic operations, and then finding the number whose square matches the result. Prime Number Definition: A number divisible only by 1 and itself. Step-by-Step Solution: - Step 1: Identify the prime numbers. - We need prime numbers greater than 4 and less than 16. - These prime numbers are: 5, 7, 11, and 13. - Step 2: Find their sum. - Sum = 5 + 7 + 11 + 13 = 36. - Step 3: Calculate one-fourth of the sum. - One-fourth of 36 = (36/4) = 9. - Step 4: Relate back to the question. - The result is 9. The question asks: "9 is the square of what number?" - Since 3^2 = 9, the required number is 3. Exam Strategy & Shortcut: Write down the primes quickly: 5, 7, 11, 13. Add them in pairs to be faster: (13+7) + (11+5) = 20 + 16 = 36. Divide by 4 to get 9. The square root of 9 is 3. This entire process should take less than 15 seconds. Common Pitfall: Including numbers that are not prime, such as 9 or 15, in the initial list. Remember that 9 is divisible by 3, and 15 is divisible by 3 and 5. Also, failing to read the final part of the prompt "is the square of" and looking for 9 in the options. Final Answer: Therefore, the correct answer is 3.

Question 2

The value of $\sqrt{10 + \sqrt{25 + \sqrt{108 + \sqrt{154 + \sqrt{225}}}}}$ is

Accepted Answer

Concept & Strategy: For nested square root problems, always begin evaluating from the innermost square root and work your way outwards step-by-step. Core logic: Simplify from inside out: \sqrt{a + \sqrt{b}} Step-by-Step Solution: - Step 1: Evaluate the innermost root: \sqrt{225} = 15. - Substitute this back: \sqrt{154 + 15}. - Step 2: Add and evaluate the next root: 154 + 15 = 169. - \sqrt{169} = 13. - Substitute this back: \sqrt{108 + 13}. - Step 3: Add and evaluate the next root: 108 + 13 = 121. - \sqrt{121} = 11. - Substitute this back: \sqrt{25 + 11}. - Step 4: Add and evaluate the next root: 25 + 11 = 36. - \sqrt{36} = 6. - Substitute this back into the final, outermost expression: \sqrt{10 + 6}. - Step 5: Final evaluation: 10 + 6 = 16. - \sqrt{16} = 4. Exam Strategy & Shortcut: Look at the outermost operation: \sqrt{10 + something}. Since the total expression evaluates to an integer (based on the options), the sum under the outermost root must be a perfect square greater than 10. The closest perfect square greater than 10 is 16. If the sum is 16, the root is 4. This skips the entire calculation. Common Pitfall: Trying to add the numbers while they are still under their respective square roots. You cannot add terms across different radical layers. You must resolve each radical completely before adding it to the next term. Final Answer: Therefore, the correct answer is 4.

Question 3

Evaluate $\sqrt{41 - \sqrt{21 + \sqrt{19 - \sqrt{9}}}}$ .

Accepted Answer

Concept & Strategy: Just like other nested radicals, this problem requires solving from the innermost square root outwards. Pay close attention to the alternating addition and subtraction signs. Core logic: Work from the innermost term outwards. Step-by-Step Solution: - Step 1: Evaluate the innermost root: \sqrt{9} = 3. - Substitute this into the expression: \sqrt{19 - 3}. - Step 2: Subtract and evaluate: 19 - 3 = 16. - \sqrt{16} = 4. - Substitute this back into the next layer: \sqrt{21 + 4}. - Step 3: Add and evaluate: 21 + 4 = 25. - \sqrt{25} = 5. - Substitute this into the outermost layer: \sqrt{41 - 5}. - Step 4: Final evaluation: 41 - 5 = 36. - \sqrt{36} = 6. Exam Strategy & Shortcut: Analyze the outermost layer: \sqrt{41 - something positive}. The result must be the square root of a perfect square strictly less than 41. The nearest perfect square less than 41 is 36. The square root of 36 is 6. Option (c) is 6, making it a highly confident guess without doing the inner math. Common Pitfall: Missing the alternating minus and plus signs. Carelessly adding 19 + 3 instead of subtracting, which completely throws off the chain of perfect squares. Always triple-check the operational signs in nested roots. Final Answer: Therefore, the correct answer is 6.

Question 4

$\sqrt{176 + \sqrt{2401}}$ is equal to

Accepted Answer

Concept & Strategy: Simplify the nested square root by finding the value of the inner radical first, adding it to the outer term, and then taking the square root of the sum. Core logic: Find inner root first: \sqrt{a + \sqrt{b}} Step-by-Step Solution: - Step 1: Evaluate the inner root, \sqrt{2401}. - The number ends in 1, so its root ends in 1 or 9. - Bounding ranges: 40^2 = 1600, 50^2 = 2500. - Since 2401 is very close to 2500, the root must be 49. - Step 2: Add to the outer term. - Substitute 49 back into the expression: 176 + 49. - 176 + 49 = 225. - Step 3: Evaluate the final root. - We are left with \sqrt{225}. - The square root of 225 is a standard known perfect square, which is 15. Exam Strategy & Shortcut: Look at the outer shell: \sqrt{176 + value}. The nearest perfect squares above 176 are 196 (root 14) and 225 (root 15). The inner value is \sqrt{2401}, which is approximately \sqrt{2500} = 50. Adding roughly 50 to 176 gives roughly 226. The perfect square near 226 is 225. Thus, the answer is \sqrt{225} = 15. Common Pitfall: Miscalculating the square root of 2401. Students might guess 41 or 51 if they don't carefully bound the number using 40^2 and 50^2. Final Answer: Therefore, the correct answer is 15.

Question 5

$\left(\sqrt{\frac{225}{729}} - \sqrt{\frac{25}{144}}\right) \div \sqrt{\frac{16}{81}} = x$

Accepted Answer

Concept & Formula: Simplify fractions containing perfect squares by evaluating the square root of the numerator and denominator separately before executing the arithmetic operations. Radical rule for division: \sqrt{(a/b)} = \frac{\sqrt{a}}{\sqrt{b}} Step-by-Step Solution: - Step 1: Evaluate each individual square root. - \sqrt{(225/729)} = (15/27) = (5/9) - \sqrt{(25/144)} = (5/12) - \sqrt{(16/81)} = (4/9) - Step 2: Simplify the expression inside the parentheses. - We have ((5/9) - (5/12)). - Find the LCM of 9 and 12, which is 36. - (5 × 4/36) - (5 × 3/36) = (20 - 15/36) = (5/36) - Step 3: Perform the division. - (5/36) \div (4/9) = (5/36) × (9/4) - (5/4 × 4) = (5/16) Exam Strategy & Shortcut: Simplify fractions at the very first step. Reducing (15/27) directly to (5/9) minimizes the LCM computation from 108 down to 36. When dividing fractions, instantly multiply by the reciprocal to keep your scratch work clean and swift. Common Pitfall: Forgetting to invert the divisor when changing from a division operator to a multiplication operator, which leads to computing (5/36) × (4/9) = (5/81) instead of the correct answer. Final Answer: Therefore, the correct answer is 5/16.

Question 6

$\sqrt{x} 	imes \sqrt{484} = 1034$

Accepted Answer

Concept & Strategy: Isolate the algebraic term \sqrt{x} by evaluating the known perfect square root term and dividing it through to the right-hand side. Core algebraic identity: If \sqrt{x} × a = b, then \sqrt{x} = (b/a) Step-by-Step Solution: - Step 1: Extract the known square root. - Recognize that \sqrt{484} = 22. - Step 2: Substitute and simplify. - \sqrt{x} × 22 = 1034 - \sqrt{x} = (1034/22) - Divide 1034 by 22: \sqrt{x} = 47 - Step 3: Square both sides to find x. - x = 47^2 - 47^2 = (50 - 3)^2 = 2500 - 300 + 9 = 2209 Exam Strategy & Shortcut: After finding \sqrt{x} = 47, determine 47^2 quickly by referencing 50^2 = 2500. Since 47 is just below 50, its square must be somewhat less than 2500. Furthermore, 7^2 = 49, so the final answer must end in 9. Both 2209 and 2401 fit this, but 2401 is 49^2, making 2209 the correct choice. Common Pitfall: Mistakenly taking the square root of 47 instead of squaring it, or halting calculations once you discover 47 because of an operational mixup. Always remember that removing a square root demands squaring the opposing balance. Final Answer: Therefore, the correct answer is 2209.

Question 7

$\sqrt{11881} 	imes \sqrt{x} = 10137$

Accepted Answer

Concept & Strategy: Extract the large integer root first using estimation limits, isolate \sqrt{x} via basic arithmetic division, and square the outcome. Approximation limits: 100^2 = 10000 \quad and \quad 110^2 = 12100 Step-by-Step Solution: - Step 1: Evaluate \sqrt{11881}. - 11881 lies perfectly between 10000 (100^2) and 12100 (110^2). - Since it ends in 1, the unit digit of its root is either 1 or 9. - Given that 11881 is much closer to 12100, we check 109. - Verification: 109^2 = (110 - 1)^2 = 12100 - 220 + 1 = 11881. Thus, \sqrt{11881} = 109. - Step 2: Isolate the unknown term. - 109 × \sqrt{x} = 10137 - \sqrt{x} = (10137/109) - Computing division: \sqrt{x} = 93 - Step 3: Compute x. - x = 93^2 - 93^2 = (100 - 7)^2 = 10000 - 1400 + 49 = 8649 Exam Strategy & Shortcut: Use last digit patterns for the division. In 109 × \sqrt{x} = 10137, the unit digit 9 multiplied by some number must result in a unit digit of 7. Since 9 × 3 = 27, \sqrt{x} must end in 3. Squaring a value ending in 3 (\dots3^2) guarantees the final value x must end in 9. This filters options to 8649 and 9409. Knowing 90^2 = 8100, 93^2 must be close to 8100, confirming 8649. Common Pitfall: Dividing 10137 by 109 incorrectly due to rushing. Take care with multi-digit divisions by rounding 109 to 110 mentally to check magnitude constraints. Final Answer: Therefore, the correct answer is 8649.

Question 8

In the equation $\frac{4050}{\sqrt{x}} = 450$, the value of $x$ is

Accepted Answer

Concept & Strategy: Rearrange the fractional statement to bring the variable radical expression to one side, simplify the integers, and square the remaining value. Algebraic property: If (a/\sqrt{x)} = b, then \sqrt{x} = (a/b) Step-by-Step Solution: - Step 1: Rearrange the algebraic fraction. - (4050/450) = \sqrt{x} - Step 2: Simplify the division balance. - Cancel out the common zeros: (405/45) = \sqrt{x} - Dividing 405 by 45: \sqrt{x} = 9 (since 45 × 10 = 450, then 450 - 45 = 405). - Step 3: Square both sides. - x = 9^2 = 81 Exam Strategy & Shortcut: Directly observe the ratio relationship. 450 goes into 4050 exactly 9 times. Therefore, \sqrt{x} equals 9. The square of 9 is 81. This can easily be checked in under 5 seconds. Common Pitfall: Selecting option (a) 9 by accident because of a lack of attention to the radical sign over the variable x. Always check whether the question requires x or \sqrt{x}. Final Answer: Therefore, the correct answer is 81.

Question 9

$\sqrt{\frac{16}{25}} 	imes \sqrt{\frac{x}{25}} 	imes \frac{16}{25} = \frac{256}{625}$

Accepted Answer

Concept & Strategy: Simplify algebraic fraction chains by isolating the radical element containing the variable and cancelling matching factors from both sides of the equation. Simplification rule: If a × \sqrt{x} = b, then \sqrt{x} = (b/a) Step-by-Step Solution: - Step 1: Simplify known square roots and separate variables. - \sqrt{(16/25)} = (4/5) - \sqrt{(x/25)} = \frac{\sqrt{x}}{5} - Step 2: Substitute expressions back into the primary equation. - (4/5) × \frac{\sqrt{x}}{5} × (16/25) = (256/625) - Combine constant multipliers on the left: (4 × 16/5 × 5 × 25) × \sqrt{x} = (64/625) × \sqrt{x} - Equation is now: (64/625) × \sqrt{x} = (256/625) - Step 3: Cancel denominators and solve for \sqrt{x}. - Multiply both sides by 625: 64 × \sqrt{x} = 256 - \sqrt{x} = (256/64) = 4 - Step 4: Square both sides to find x. - x = 4^2 = 16 Exam Strategy & Shortcut: Notice that the right hand side is (256/625) = ((16/25))^2. The expression on the left is (4/5) × \sqrt{(x/25)} × (16/25). Divide out (16/25) from both sides to get: (4/5) × \sqrt{(x/25)} = (16/25). We know (4/5) × (4/5) = (16/25), so \sqrt{(x/25)} = (4/5). This means (x/25) = (16/25), revealing directly that x = 16. Common Pitfall: Getting confused by the multiple repetitions of numbers like 16 and 25 throughout the problem structure. Work linearly and keep track of which components reside inside a radical and which do not. Final Answer: Therefore, the correct answer is 16.

Question 10

The square root of $(272^2 - 128^2)$ is

Accepted Answer

Concept & Formula: Avoid squaring large individual numbers directly. Instead, apply the difference of squares algebraic identity to simplify the internal expression. Difference of squares identity: a^2 - b^2 = (a - b)(a + b) Step-by-Step Solution: - Step 1: Set up the targeted expression. - We need to compute: \sqrt{272^2 - 128^2} - Step 2: Apply the algebraic identity. - Let a = 272 and b = 128. - 272^2 - 128^2 = (272 - 128)(272 + 128) - Step 3: Compute values within brackets. - 272 - 128 = 144 - 272 + 128 = 400 - So, the expression becomes \sqrt{144 × 400} - Step 4: Factor out the square roots. - \sqrt{144 × 400} = \sqrt{144} × \sqrt{400} - 12 × 20 = 240 Exam Strategy & Shortcut: Never square 272 and 128 manually. The difference of squares structure turns a messy subtraction problem into a product of two obvious perfect squares (\sqrt{144 × 400}). Taking roots gives 12 × 20 = 240 instantly. Common Pitfall: Mistakenly applying the distributive error \sqrt{a^2 - b^2} = a - b, which results in computing 272 - 128 = 144. Radical distribution over subtraction is completely invalid. Final Answer: Therefore, the correct answer is 240.

Question 11

If $y = 5$, then what is the value of $10y\sqrt{y^3 - y^2}$ ?

Accepted Answer

Concept & Strategy: Perform straight algebraic evaluation by replacing all instances of the variable y with its given value, 5. Factor or resolve internal values under the radical before final multiplication. Core step: Substitute y = 5 into the radical function. Step-by-Step Solution: - Step 1: Substitute y = 5 into the statement. - Expression: 10(5) × \sqrt{5^3 - 5^2} - Step 2: Simplify components outside and inside the radical. - Outside product: 10 × 5 = 50 - Inside powers: 5^3 = 125 and 5^2 = 25 - Subtraction under radical: 125 - 25 = 100 - Step 3: Resolve the remaining term. - Expression is now: 50 × \sqrt{100} - Since \sqrt{100} = 10, we evaluate: 50 × 10 = 500 Exam Strategy & Shortcut: Factor out y^2 inside the radical first to work with smaller values: \sqrt{y^3 - y^2} = \sqrt{y^2(y - 1)} = y\sqrt{y - 1}. Substitute y = 5 into this simplified form: 5\sqrt{5 - 1} = 5\sqrt{4} = 5 × 2 = 10. Multiply by the initial front term 10y = 50, yielding 50 × 10 = 500. Common Pitfall: Mistaking 5^3 as 15 or 5^2 as 10 due to rushing. Always write down exponential expansions carefully if you are prone to simple mental arithmetic slips. Final Answer: Therefore, the correct answer is 500.

Question 12

$\sqrt{110\frac{1}{4}} = x$

Accepted Answer

Concept & Strategy: Convert the mixed fraction into an improper fraction, take the square root of the numerator and denominator independently, and then convert back into decimal format. Fraction conversion identity: \sqrt{(a/b)} = \frac{\sqrt{a}}{\sqrt{b}} Step-by-Step Solution: - Step 1: Convert the mixed fraction to an improper fraction. - Multiply the integer by the denominator and add the numerator: 110 × 4 + 1 = 440 + 1 = 441. - The expression becomes: \sqrt{(441/4)} - Step 2: Extract the square roots. - Split the radical: \frac{\sqrt{441}}{\sqrt{4}} - We know \sqrt{441} = 21 and \sqrt{4} = 2. - Result: (21/2) - Step 3: Convert the fraction to a decimal value. - (21/2) = 10.5 Exam Strategy & Shortcut: Alternatively, express the value directly as a decimal from the start: \sqrt{110(1/4)} = \sqrt{110.25}. We know that 10^2 = 100 and 11^2 = 121, so the answer must be between 10 and 11. Since the number ends in .25, its square root must end in .5. This points straight to 10.5. Common Pitfall: Mistakenly picking option (a) 10.25 because the original question had a (1/4) fractional component, assuming that the root splits directly onto the component parts. Never split a radical across individual terms of an addition or mixed fraction format. Final Answer: Therefore, the correct answer is 10.5.

Question 13

$$\sqrt{\frac{25}{81} - \frac{1}{9}} = x$$

Accepted Answer

Concept & Formula: To simplify an expression under a square root involving fractions, find a common denominator to subtract the fractions first, then take the square root of the resulting fraction. \sqrt{(a/b)} = \frac{\sqrt{a}}{\sqrt{b}} Step-by-Step Solution: • Given: The expression \sqrt{(25/81) - (1/9)} • Calculation: Find the common denominator for 81 and 9, which is 81. Multiply the numerator and denominator of (1/9) by 9: (1/9) = (9/81) Subtract the fractions inside the radical: (25/81) - (9/81) = (16/81) Take the square root: \sqrt{(16/81)} = \frac{\sqrt{16}}{\sqrt{81}} = (4/9) Exam Strategy & Shortcut: Recognize perfect squares immediately. The numbers 25, 81, 9, 16 are all perfect squares. As soon as you compute the numerator subtraction 25 - 9 = 16, you know the answer must be \sqrt{(16/81)} = (4/9) without writing down intermediate steps. Common Pitfall: • Mistake: Distributing the square root over subtraction, i.e., mistakenly computing \sqrt{(25/81)} - \sqrt{(1/9)} = (5/9) - (1/3) = (2/9). • Correction: Remember that \sqrt{A - B} 
eq \sqrt{A} - \sqrt{B}. Always perform the subtraction inside the radical first. Final Answer: Therefore, the correct answer is 4/9.

Question 14

Which of the following is closest to $\sqrt{3}$?

Accepted Answer

Concept & Value: The square root of 3 (\sqrt{3}) is an irrational number with a well-known approximate value: \sqrt{3} \approx 1.73205 Step-by-Step Solution: • Given: Options representing approximations of \sqrt{3}. • Calculation: Convert all options to decimal form to compare with 1.73205: • (a) 1.69 • (b) (173/100) = 1.73 • (c) 1.75 • (d) (9/5) = 1.8 Calculate the absolute differences: • |1.73205 - 1.69| = 0.04205 • |1.73205 - 1.73| = 0.00205 • |1.73205 - 1.75| = 0.01795 • |1.73205 - 1.8| = 0.06795 The smallest difference is 0.00205, corresponding to 1.73. Exam Strategy & Shortcut: Memorizing basic radical values saves significant time. \sqrt{2} \approx 1.414 and \sqrt{3} \approx 1.732. Knowing \sqrt{3} \approx 1.732 lets you immediately match it to 1.73, which is precisely (173/100). Common Pitfall: • Mistake: Rounding up prematurely to 1.75 or confusing (9/5) (1.8) with a closer approximation. • Tip: Always write down the value of standard constants up to 3 decimal places to avoid proximity errors. Final Answer: Therefore, the correct answer is 173/100.

Question 15

How many two-digit numbers satisfy this property: The last digit (unit's digit) of the square of the two-digit number is 8?

Accepted Answer

Concept & Logic: Perfect squares can only end in specific digits. Let's evaluate the unit's digits of squares of all single digits: • 0^2 = 0, 1^2 = 1, 2^2 = 4, 3^2 = 9, 4^2 = 6 • 5^2 = 5, 6^2 = 6, 7^2 = 9, 8^2 = 4, 9^2 = 1 Thus, any perfect square must end in 0, 1, 4, 5, 6, or 9. A perfect square never ends in 2, 3, 7, or 8. Step-by-Step Solution: • Given property: The unit's digit of the square of a two-digit number is 8. • Deduction: Based on the core logic of perfect squares, no integer squared can yield a number ending in 8. Therefore, there are 0 two-digit numbers that satisfy this condition. This matches option (d) 'None of these'. Exam Strategy & Shortcut: Remember the 2-3-7-8 Rule: No perfect square ever ends in the digits 2, 3, 7, or 8. Since the question specifies a square ending in 8, you can instantly mark 'None of these' without performing any calculations. Common Pitfall: • Mistake: Confusing the rule and thinking that since 2^3 = 8, some square might end in 8, or listing numbers ending in 8 instead of checking their squares. • Tip: Keep a firm mental checklist of impossible unit digits for squares (2, 3, 7, 8) and cubes (0-9 are all possible). Final Answer: Therefore, the correct answer is None of these.

Question 16

What percentage of the numbers from 1 to 50 have squares that end in the digit 1?

Accepted Answer

Concept & Formula: A number's square ends in the digit 1 if and only if the number itself ends in either 1 or 9 (since 1^2 = 1 and 9^2 = 81). Percentage = ( \frac{Favorable Outcomes}{Total Outcomes} ) × 100 Step-by-Step Solution: • Given: Total count of numbers from 1 to 50 = 50. • Identification: Numbers between 1 and 50 ending in 1 or 9: • Ending in 1: 1, 11, 21, 31, 41 (5 numbers) • Ending in 9: 9, 19, 29, 39, 49 (5 numbers) Total favorable numbers = 5 + 5 = 10. • Calculation: Percentage = ( (10/50) ) × 100 = 20% Exam Strategy & Shortcut: In every block of 10 consecutive numbers (e.g., 1-10, 11-20), exactly 2 numbers end in 1 or 9. Since 2 out of 10 numbers satisfy the condition, the proportion is always (2/10) = 20%, regardless of whether the range goes up to 10, 20, 50, or 100. Common Pitfall: • Mistake: Forgetting to count the numbers ending in 9, or missing the boundary value like 49, leading to an incorrect count of 5 numbers (10%). • Tip: Always write out the unit possibilities (1^2=1 and 9^2=81) explicitly before counting. Final Answer: Therefore, the correct answer is 20.

Question 17

While solving a mathematical problem, Samidha squared a number and then subtracted 25 from it rather than the required i.e., first subtracting 25 from the number and then squaring it. But she got the right answer. What was the given number?

Accepted Answer

Concept & Formula: Translate the word problem into an algebraic equation. Let the given number be x. • Samidha's operation: x^2 - 25 • Required operation: (x - 25)^2 Set them equal to find x: x^2 - 25 = (x - 25)^2 Step-by-Step Solution: • Formulate the equation: x^2 - 25 = (x - 25)^2 • Expand the right side using (a-b)^2 = a^2 - 2ab + b^2: x^2 - 25 = x^2 - 50x + 625 • Simplify and solve: Subtract x^2 from both sides: -25 = -50x + 625 50x = 625 + 25 50x = 650 x = (650/50) = 13 Exam Strategy & Shortcut: Option Verification Method: Test the options directly into both scenarios. • Try Option (a) 13: • Scenario 1: 13^2 - 25 = 169 - 25 = 144 • Scenario 2: (13 - 25)^2 = (-12)^2 = 144 Both values match perfectly, confirming 13 is the correct answer within seconds. Common Pitfall: • Mistake: Thinking that since a negative number arises inside the square, i.e., (13-25) = -12, the method is invalid or results in "Cannot be determined". • Correction: Remember that the square of a negative number is always positive: (-12)^2 = 144. Final Answer: Therefore, the correct answer is 13.

Question 18

How many perfect squares lie between 120 and 300?

Accepted Answer

Concept & Logic: To find the number of perfect squares within a range, determine the integers whose squares fall strictly within the lower and upper bounds. Step-by-Step Solution: • Given range: Greater than 120 and less than 300. • Find lower limit: 10^2 = 100 \quad (too small) 11^2 = 121 \quad (greater than 120) So, the first perfect square is 11^2. • Find upper limit: 17^2 = 289 \quad (less than 300) 18^2 = 324 \quad (greater than 300) So, the last perfect square is 17^2. • Count the numbers: The integers are 11, 12, 13, 14, 15, 16, 17. Count = 17 - 11 + 1 = 7 Exam Strategy & Shortcut: Memorize squares up to 30. Knowing instantly that 11^2 = 121 and 17^2 = 289 allows you to find the count using the simple formula: Last Number - First Number + 1. 17 - 11 + 1 = 7 Common Pitfall: • Mistake: Off-by-one errors when subtracting limits (17 - 11 = 6) without adding 1 to include both endpoints. • Tip: Always use Upper - Lower + 1 when counting inclusive continuous integer sequences. Final Answer: Therefore, the correct answer is 7.

Question 19

The smallest natural number which is a perfect square and which ends in 3 identical digits lies between

Accepted Answer

Concept & Logic: Perfect squares have strict rules about their ending digits. A perfect square can only end in 0, 1, 4, 5, 6, or 9. Furthermore, if a perfect square ends in multiple identical digits: • It cannot end in 111, 555, 666, or 999 because of modular arithmetic constraints (e.g., an odd square must be a multiple of 8 plus 1). • It cannot end in an odd number of zeros (e.g., 000). • The only possible non-zero repeating identical triplet at the end of a perfect square is 444. Step-by-Step Solution: • Deduction: The square must end in 444. Since the number ends in 4, its square root must end in either 2 or 8 (since 2^2=4 and 8^2=64). • Calculation: Let's test squares ending in 2 or 8, starting from smaller numbers, to find one that ends in 444. • 12^2 = 144 (Ends in 44, not 444) • 22^2 = 484 • 28^2 = 784 • 32^2 = 1024 • 38^2 = 1444 We found it! 1444 is a perfect square (38^2) and it ends in three identical digits (444). • Conclusion: The number 1444 clearly falls into the range between 1000 and 2000. Exam Strategy & Shortcut: Knowing that 38^2 = 1444 is a tremendous shortcut for higher-level aptitude exams. 1444 is the only four-digit perfect square ending in three identical digits. Recognizing this instantly points you to the 1000-2000 bucket. Common Pitfall: • Mistake: Trying to guess numbers ending in 000 (like 1000 or 4000) and thinking they are perfect squares. • Correction: A perfect square must have an even number of trailing zeros. Three zeros is mathematically impossible for a perfect square. Final Answer: Therefore, the correct answer is 1000 and 2000.

Question 20

$$(\sqrt{2} + \frac{1}{\sqrt{2}})^2$$ is equal to

Accepted Answer

Concept & Formula: To expand a binomial expression squared, use the standard algebraic identity: (a + b)^2 = a^2 + 2ab + b^2 Step-by-Step Solution: • Given: The expression (\sqrt{2} + (1/\sqrt{2)})^2 • Calculation: Here, a = \sqrt{2} and b = (1/\sqrt{2)}. Apply the formula: (\sqrt{2})^2 + 2(\sqrt{2})((1/\sqrt{2)}) + ((1/\sqrt{2)})^2 Simplify each term individually: • (\sqrt{2})^2 = 2 • 2(\sqrt{2})((1/\sqrt{2)}) = 2 × 1 = 2 • ((1/\sqrt{2)})^2 = (1/2) Add them together: 2 + 2 + (1/2) = 4 + (1/2) = 4(1/2) Exam Strategy & Shortcut: Simplify the expression inside the parentheses first by finding a common denominator: \sqrt{2} + (1/\sqrt{2)} = (2 + 1/\sqrt{2)} = (3/\sqrt{2)} Now, simply square the result directly: ((3/\sqrt{2)})^2 = (9/2) = 4.5 = 4(1/2) Both methods are equally fast, pick the one that prevents careless errors. Common Pitfall: • Mistake: Distributing the square to both terms independently, computing (\sqrt{2})^2 + ((1/\sqrt{2)})^2 = 2 + 0.5 = 2.5. • Correction: Never forget the middle term (2ab) when expanding a squared binomial! (a+b)^2 
eq a^2 + b^2. Final Answer: Therefore, the correct answer is 4 1/2.

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