Question 1

$$\sqrt{\frac{0.081 	imes 0.324 	imes 4.624}{1.5625 	imes 0.0289 	imes 72.9 	imes 64}}$$ 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 2

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 3

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 4

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 5

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 6

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 7

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 8

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 9

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 10

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 11

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 12

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 13

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.

Question 14

The least perfect square number divisible by $3$, $4$, $5$, $6$ and $8$ is

Accepted Answer

Concept & Strategy: To find a perfect square divisible by multiple numbers, first find the Least Common Multiple (LCM) of those numbers. Then, prime factorize the LCM and multiply it by any prime factors needed to make all exponents even. Step-by-Step Solution: Find the LCM of 3, 4, 5, 6, and 8. List their prime factorizations: 3 = 3^1 4 = 2^2 5 = 5^1 6 = 2^1 × 3^1 8 = 2^3 The LCM takes the highest power of each prime present: LCM = 2^3 × 3^1 × 5^1 LCM = 8 × 3 × 5 = 120. Now, analyze the prime factorization of the LCM (2^3 × 3^1 × 5^1) for odd powers. The power of 2 is 3 (odd). The power of 3 is 1 (odd). The power of 5 is 1 (odd). To make it a perfect square, we need to multiply by one more of each prime factor to make their powers even (2^4, 3^2, 5^2). Multiplier needed = 2 × 3 × 5 = 30. Multiply the LCM by this required multiplier: Least Perfect Square = 120 × 30 = 3600. Exam Strategy & Shortcut: Instead of calculating the LCM fully, look at the options. The number must be a perfect square. Options (b) 1200 and (c) 2500 are not divisible by 3, so eliminate them. Option (a) 900 is a perfect square and divisible by 3, 4, 5, 6, but it is NOT divisible by 8 (900 / 8 = 112.5). This leaves only 3600 by process of elimination. Common Pitfall: A common error is stopping once the LCM (120) is found and selecting the nearest multiple from the options without ensuring the final result is a perfect square. Final Answer: Therefore, the correct answer is 3600.

Question 15

The greatest four-digit perfect square number is

Accepted Answer

Concept & Formula: To find the greatest n-digit perfect square, we identify the largest number whose square is less than or equal to the largest n-digit number. The greatest 4-digit number is 9999. We must find the integer root closest to it. Step-by-Step Solution: • Given: We need the greatest 4-digit perfect square. • Calculation / Deduction: • The largest 4-digit number is 9999. • Let's approximate its square root. We know that 100^2 = 10000, which is a 5-digit number. • The integer just below 100 is 99. • Therefore, the greatest 4-digit perfect square must be 99^2. • Calculate 99^2 = (100 - 1)^2. • Applying the formula (a-b)^2 = a^2 - 2ab + b^2: 99^2 = 10000 - 200 + 1 99^2 = 9801 Exam Strategy & Shortcut: Memorize the squares of numbers up to at least 30, and know the baseline powers of 10 (like 100^2 = 10000). Since 100^2 is the smallest 5-digit square, the largest 4-digit square is immediately (100-1)^2 = 9801. You can solve this in 2 seconds without pen and paper. Common Pitfall: Students often try to find the square root of 9999 using the long division method, which consumes valuable time. Avoid long division when simple algebraic identities or baseline approximations can bridge the gap much faster. Final Answer: Therefore, the correct answer is 9801.

Question 16

The sum of 18 consecutive natural numbers is a perfect square. What is the smallest possible value of this sum?

Accepted Answer

Concept & Formula: The sum of an arithmetic progression (AP) is calculated using the formula: S_n = (n/2) [2a + (n-1)d] For consecutive natural numbers, the common difference d = 1. The sum must then be evaluated to find what conditions make it a perfect square. Step-by-Step Solution: • Given: Number of terms n = 18. The sum S_{18} is a perfect square. We need the smallest possible value. • Calculation / Deduction: • Let the 18 consecutive natural numbers start from a. (Where a \ge 1) • Using the AP sum formula: S_{18} = (18/2) [2a + (18-1)(1)] S_{18} = 9 [2a + 17] • For S_{18} to be a perfect square, the expression 9(2a + 17) must be a perfect square. • Since 9 is already a perfect square (3^2), the factor (2a + 17) must also be a perfect square. • Because a is a natural number (a \ge 1), the minimum value of (2a + 17) is 2(1) + 17 = 19. • We need the smallest perfect square greater than or equal to 19. • The perfect squares are 1, 4, 9, 16, 25, 36 \dots • The smallest perfect square greater than 19 is 25. • So, we set 2a + 17 = 25. 2a = 8 \Rightarrow a = 4 • The condition holds. Now calculate the sum: S_{18} = 9 × 25 = 225 Exam Strategy & Shortcut: You don't need to calculate a. Once you realize the sum is 9 × (an odd number greater than 17), you know the odd number must be a perfect square. The first perfect square after 17 is 25. The minimum sum is simply 9 × 25 = 225. Common Pitfall: A common error is forgetting that a must be a natural number (a \ge 1), which sets a lower limit on 2a+17. Students might just look for *any* perfect square multiplier, but checking the boundary condition (2a+17 \ge 19) is crucial to finding the true minimum sum. Final Answer: Therefore, the correct answer is 225.

Question 17

$\sqrt{2 + \sqrt{3}} \cdot \sqrt{2 + \sqrt{2 + \sqrt{3}}} \cdot \sqrt{2 + \sqrt{2 + \sqrt{2 + \sqrt{3}}}} \cdot \sqrt{2 - \sqrt{2 + \sqrt{2 + \sqrt{3}}}}$ is equal to

Accepted Answer

Concept & Formula: This problem uses the algebraic identity for the difference of two squares applied continuously from right to left under nested square roots. (a+b)(a-b) = a^2 - b^2 When numbers are multiplied under square roots of the same degree, we can combine them: \sqrt{x} \cdot \sqrt{y} = \sqrt{x \cdot y} Step-by-Step Solution: • Calculation / Deduction: • Focus on the last two terms on the right: \sqrt{2 + \sqrt{2 + \sqrt{2 + \sqrt{3}}}} \cdot \sqrt{2 - \sqrt{2 + \sqrt{2 + \sqrt{3}}}} • Combine them under a single square root. Let a = 2 and b = \sqrt{2 + \sqrt{2 + \sqrt{3}}}. • This forms \sqrt{(a+b)(a-b)} = \sqrt{a^2 - b^2}. = \sqrt{2^2 - (\sqrt{2 + \sqrt{2 + \sqrt{3}}})^2} = \sqrt{4 - (2 + \sqrt{2 + \sqrt{3}})} = \sqrt{2 - \sqrt{2 + \sqrt{3}}} • Now, multiply this result by the second term in the original expression: \sqrt{2 + \sqrt{2 + \sqrt{3}}} \cdot \sqrt{2 - \sqrt{2 + \sqrt{3}}} • Apply the same logic. Let a = 2 and b = \sqrt{2 + \sqrt{3}}. = \sqrt{2^2 - (\sqrt{2 + \sqrt{3}})^2} = \sqrt{4 - (2 + \sqrt{3})} = \sqrt{2 - \sqrt{3}} • Finally, multiply this result by the first term in the original expression: \sqrt{2 + \sqrt{3}} \cdot \sqrt{2 - \sqrt{3}} • Let a = 2 and b = \sqrt{3}. = \sqrt{2^2 - (\sqrt{3})^2} = \sqrt{4 - 3} = \sqrt{1} = 1 Exam Strategy & Shortcut: For infinitely or finitely nested square roots of the form (2 + \sqrt{\dots}) \cdot (2 - \sqrt{\dots}), the pattern collapses rapidly inward. Each step eliminates one layer of the root. Recognize the (x+y)(x-y) pattern immediately and collapse it backwards mentally to save writing time. Common Pitfall: Getting intimidated by the length of the expression and attempting to calculate inner roots first. Always look for structural algebraic patterns (like difference of squares) before attempting brute-force calculation. Final Answer: Therefore, the correct answer is 1.

Question 18

The expression $1 - \frac{1}{1 + \sqrt{3}} + \frac{1}{1 - \sqrt{3}}$ equals

Accepted Answer

Concept & Formula: This problem requires combining fractions that have conjugate binomials in their denominators. Instead of rationalizing each fraction separately, it is much faster to find a common denominator for the fractional parts, leveraging the difference of squares formula: (a + b)(a - b) = a^2 - b^2 Step-by-Step Solution: • Calculation / Deduction: • Let's isolate and simplify the fractional portion of the expression first to keep things clean. - (1/1 + \sqrt{3)} + (1/1 - \sqrt{3)} • The common denominator for these two fractions is (1 + \sqrt{3})(1 - \sqrt{3}). • Combine the numerators over this common denominator: = \frac{-(1 - \sqrt{3}) + (1 + \sqrt{3})}{(1 + \sqrt{3})(1 - \sqrt{3})} • Simplify the numerator by distributing the negative sign carefully: = \frac{-1 + \sqrt{3} + 1 + \sqrt{3}}{1^2 - (\sqrt{3})^2} = \frac{2\sqrt{3}}{1 - 3} = \frac{2\sqrt{3}}{-2} = -\sqrt{3} • Now, substitute this simplified result back into the original full expression: 1 + (-\sqrt{3}) = 1 - \sqrt{3} Exam Strategy & Shortcut: Whenever you see a pair of fractions taking the form (x/a+b) \pm (y/a-b), immediately cross-multiply to form a single fraction. The denominator instantly becomes a clean integer via difference of squares (a^2 - b^2), and the numerators will usually feature satisfying cancellations of integers or surds. Do not waste time rationalizing them as two separate, isolated problems. Common Pitfall: Many students lose track of the leading negative sign on the first fraction: - (1/1 + \sqrt{3)}. When finding the common denominator, they might forget to distribute that negative to *both* terms in (1 - \sqrt{3}), mistakenly writing -1 - \sqrt{3} instead of -1 + \sqrt{3} in the combined numerator. Always use parentheses when carrying a negative sign across a combined fraction. Final Answer: Therefore, the correct answer is 1 - \sqrt{3}.

Question 19

Given that $\sqrt{3} = 1.732$, the value of   $$ \frac{3 + \sqrt{6}}{5\sqrt{3} - 2\sqrt{12} - \sqrt{32} + \sqrt{50}} $$   is

Accepted Answer

Concept & Formula: The core strategy is simplifying compound surds by extracting perfect square factors from inside the square roots. Use the property: \sqrt{a^2 × b} = a\sqrt{b} After breaking down all terms into their simplest radical forms, combine like terms to simplify the fraction. Step-by-Step Solution: Let's simplify the numerator and denominator separately. Numerator: 3 + \sqrt{6} We can extract a common factor of \sqrt{3} because 3 = (\sqrt{3})^2 and \sqrt{6} = \sqrt{3}\sqrt{2}: Numerator = \sqrt{3}(\sqrt{3} + \sqrt{2}) Denominator: 5\sqrt{3} - 2\sqrt{12} - \sqrt{32} + \sqrt{50} Simplify each term by factoring out perfect squares: \sqrt{12} = \sqrt{4 × 3} = 2\sqrt{3} \sqrt{32} = \sqrt{16 × 2} = 4\sqrt{2} \sqrt{50} = \sqrt{25 × 2} = 5\sqrt{2} Substitute these back into the denominator: = 5\sqrt{3} - 2(2\sqrt{3}) - 4\sqrt{2} + 5\sqrt{2} = 5\sqrt{3} - 4\sqrt{3} - 4\sqrt{2} + 5\sqrt{2} Group the like terms (the \sqrt{3} terms together and the \sqrt{2} terms together): = (5 - 4)\sqrt{3} + (5 - 4)\sqrt{2} = \sqrt{3} + \sqrt{2} Recombine Fraction: Expression = \frac{\sqrt{3}(\sqrt{3} + \sqrt{2})}{\sqrt{3} + \sqrt{2}} The term (\sqrt{3} + \sqrt{2}) cancels out from the top and bottom: Expression = \sqrt{3} Substitute the given value \sqrt{3} = 1.732. Exam Strategy & Shortcut: In competitive exams, complicated-looking surd fractions almost always simplify down to a single radical term. Once you spot that the denominator simplifies to \sqrt{3} + \sqrt{2}, look at the numerator 3 + \sqrt{6}. Recognize instantly that \sqrt{3} is a multiplier that maps the denominator to the numerator. Cancel them mentally to arrive at \sqrt{3} directly. Common Pitfall: A major trap is attempting to plug in 1.732 for \sqrt{3} and approximating other roots like \sqrt{2} at the very beginning. This creates a nightmare of decimal arithmetic that wastes time and guarantees calculation errors. Always simplify variables and radicals fully before substituting numeric values. Final Answer: Therefore, the correct answer is 1.732.

Question 20

$\left(\frac{2 + \sqrt{3}}{2 - \sqrt{3}} + \frac{2 - \sqrt{3}}{2 + \sqrt{3}} + \frac{\sqrt{3} - 1}{\sqrt{3} + 1}\right)$ simplifies to

Accepted Answer

Concept & Formula: This problem requires grouping terms smartly. The first two fractions are conjugates of each other. Summing them using a direct algebraic identity is much faster than rationalizing them individually. For conjugate fractions: (a + b/a - b) + (a - b/a + b) = (2(a^2 + b^2)/a^2 - b^2) Step-by-Step Solution: Let's divide the expression into two parts to simplify calculation. Part 1: The First Two Fractions Let E_1 = \frac{2 + \sqrt{3}}{2 - \sqrt{3}} + \frac{2 - \sqrt{3}}{2 + \sqrt{3}} Using the identity above where a = 2 and b = \sqrt{3}: E_1 = \frac{2((2)^2 + (\sqrt{3})^2)}{(2)^2 - (\sqrt{3})^2} E_1 = (2(4 + 3)/4 - 3) E_1 = (2(7)/1) = 14 Part 2: The Third Fraction Let E_2 = \frac{\sqrt{3} - 1}{\sqrt{3} + 1} Rationalize the denominator by multiplying top and bottom by (\sqrt{3} - 1): E_2 = \frac{(\sqrt{3} - 1)^2}{(\sqrt{3} + 1)(\sqrt{3} - 1)} Expand the numerator and simplify the denominator: E_2 = \frac{3 + 1 - 2\sqrt{3}}{3 - 1} E_2 = \frac{4 - 2\sqrt{3}}{2} Divide both terms in the numerator by 2: E_2 = 2 - \sqrt{3} Total Expression: Total = E_1 + E_2 = 14 + (2 - \sqrt{3}) = 16 - \sqrt{3} Exam Strategy & Shortcut: Memorize the structure \frac{x + \sqrt{y}}{x - \sqrt{y}} + \frac{x - \sqrt{y}}{x + \sqrt{y}}. It always results in an integer if x and y are integers. For the third term, knowing that \frac{\sqrt{3} - 1}{\sqrt{3} + 1} = 2 - \sqrt{3} is a standard result (often used in Trigonometry for 	an 15^\circ) allows you to bypass the rationalization step entirely, turning this into a 15-second visual question. Common Pitfall: Students often try to find a single massive common denominator for all three fractions at once. This leads to extremely heavy algebraic multiplication, making it highly probable to drop a sign or miscalculate a coefficient. Always solve conjugate pairs first. Final Answer: Therefore, the correct answer is 16 - \sqrt{3}.

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