Question 1

If $x = \frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} - \sqrt{3}}$ and $y = \frac{\sqrt{5} - \sqrt{3}}{\sqrt{5} + \sqrt{3}}$, then $(x + y)$ equals

Accepted Answer

Concept & Formula: The core insight is recognizing that x and y are reciprocal conjugates. Instead of rationalizing x and y separately, we can add them directly using a common denominator and a standard algebraic identity. \frac{\sqrt{a} + \sqrt{b}}{\sqrt{a} - \sqrt{b}} + \frac{\sqrt{a} - \sqrt{b}}{\sqrt{a} + \sqrt{b}} = (2(a + b)/a - b) Step-by-Step Solution: Given: x = \frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} - \sqrt{3}} and y = \frac{\sqrt{5} - \sqrt{3}}{\sqrt{5} + \sqrt{3}} We need to find x + y: x + y = \frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} - \sqrt{3}} + \frac{\sqrt{5} - \sqrt{3}}{\sqrt{5} + \sqrt{3}} Find the common denominator by multiplying the two denominators: Denominator = (\sqrt{5} - \sqrt{3})(\sqrt{5} + \sqrt{3}) Using the identity (a-b)(a+b) = a^2 - b^2: Denominator = (\sqrt{5})^2 - (\sqrt{3})^2 = 5 - 3 = 2 Now, cross-multiply to find the numerator: Numerator = (\sqrt{5} + \sqrt{3})^2 + (\sqrt{5} - \sqrt{3})^2 Using the algebraic expansion (a+b)^2 + (a-b)^2 = 2(a^2 + b^2): Numerator = 2((\sqrt{5})^2 + (\sqrt{3})^2) Numerator = 2(5 + 3) = 2(8) = 16 Substitute the numerator and denominator back into the fraction: x + y = (16/2) = 8 Exam Strategy & Shortcut: Memorize the standard result for the sum of inverse conjugate surds: If x = \frac{\sqrt{a} + \sqrt{b}}{\sqrt{a} - \sqrt{b}} and y = \frac{\sqrt{a} - \sqrt{b}}{\sqrt{a} + \sqrt{b}}, then x + y = (2(a + b)/a - b). Here, a = 5 and b = 3. x + y = (2(5 + 3)/5 - 3) = (16/2) = 8. This completely bypasses intermediate calculations and solves the problem instantly without writing anything down. Common Pitfall: A frequent mistake is attempting to rationalize x and y individually first. While it yields the correct answer (x = 4 + \sqrt{15} and y = 4 - \sqrt{15}, so x+y=8), it requires double the calculation effort and time compared to using the direct identity. Final Answer: Therefore, the correct answer is 8.

Question 2

$\frac{12}{3 + \sqrt{5} + 2\sqrt{2}}$ is equal to

Accepted Answer

Concept & Formula: When a denominator contains three surd terms, you must rationalize it by strategically grouping the terms into two parts to create a binomial structure, and then multiplying by its conjugate. (a+b)(a-b) = a^2 - b^2 Step-by-Step Solution: Given: The algebraic fraction (12/3 + \sqrt{5) + 2\sqrt{2}} Calculation / Deduction: • Group the denominator into a single binomial structure to prepare for rationalization: ((3 + \sqrt{5}) + 2\sqrt{2}) • Rationalize by multiplying the numerator and the denominator by the conjugate ((3 + \sqrt{5}) - 2\sqrt{2}): Denominator = (3 + \sqrt{5})^2 - (2\sqrt{2})^2 • Expand the first term of the denominator using (a+b)^2 = a^2 + 2ab + b^2: Denominator = (9 + 5 + 6\sqrt{5}) - 8 = 14 + 6\sqrt{5} - 8 = 6 + 6\sqrt{5} = 6(1 + \sqrt{5}) • Update the entire fraction with the simplified denominator: Fraction = \frac{12(3 + \sqrt{5} - 2\sqrt{2})}{6(1 + \sqrt{5})} = \frac{2(3 + \sqrt{5} - 2\sqrt{2})}{1 + \sqrt{5}} • The denominator still contains a surd, so rationalize again by multiplying the top and bottom by (\sqrt{5} - 1): New Denominator = (\sqrt{5})^2 - (1)^2 = 5 - 1 = 4 • Multiply the numerator expressions out carefully: Numerator = 2(3 + \sqrt{5} - 2\sqrt{2})(\sqrt{5} - 1) = 2(3\sqrt{5} - 3 + 5 - \sqrt{5} - 2\sqrt{10} + 2\sqrt{2}) = 2(2 + 2\sqrt{5} + 2\sqrt{2} - 2\sqrt{10}) Factor out the common 2: Numerator = 4(1 + \sqrt{5} + \sqrt{2} - \sqrt{10}) • Divide the final numerator by the new denominator (4): Result = \frac{4(1 + \sqrt{5} + \sqrt{2} - \sqrt{10})}{4} = 1 + \sqrt{5} + \sqrt{2} - \sqrt{10} Exam Strategy & Shortcut: For three-term rationalization, always group the two terms whose squares sum closest to the square of the third term. Here, 3^2 + (\sqrt{5})^2 = 9 + 5 = 14, and (2\sqrt{2})^2 = 8. This keeps the resulting coefficients manageable. Keep constants factored out (like keeping the 12 outside the brackets) until the very end to maximize cancellation opportunities. Common Pitfall: Students often distribute the numerator's constant immediately instead of looking for structural cancellations. Distributing the 12 into the numerator right away makes the subsequent multiplication by (\sqrt{5} - 1) much harder and severely increases the likelihood of simple arithmetic errors. Final Answer: Therefore, the correct answer is 1 + \sqrt{5} + \sqrt{2} - \sqrt{10}.

Question 3

If $a = \frac{\sqrt{5} + 1}{\sqrt{5} - 1}$ and $b = \frac{\sqrt{5} - 1}{\sqrt{5} + 1}$, the value of $\left(\frac{a^2 + ab + b^2}{a^2 - ab + b^2}\right)$ is

Accepted Answer

Concept & Logic: This requires simplifying complex algebraic expressions by evaluating the sum and product of reciprocal surd fractions. By rearranging the numerator and denominator, we can express the entire fraction in terms of (a^2 + b^2) and ab. Step-by-Step Solution: • Given: a = \frac{\sqrt{5} + 1}{\sqrt{5} - 1} and b = \frac{\sqrt{5} - 1}{\sqrt{5} + 1} • Notice that b is the reciprocal of a. Therefore, the product ab = 1. • Now, rationalize a by multiplying the numerator and denominator by (\sqrt{5} + 1): • a = \frac{(\sqrt{5} + 1)^2}{(\sqrt{5} - 1)(\sqrt{5} + 1)} = \frac{5 + 1 + 2\sqrt{5}}{5 - 1} = \frac{6 + 2\sqrt{5}}{4} = \frac{3 + \sqrt{5}}{2} • Similarly, rationalizing b gives the conjugate: • b = \frac{3 - \sqrt{5}}{2} • Find the sum (a + b): • a + b = \frac{3 + \sqrt{5} + 3 - \sqrt{5}}{2} = (6/2) = 3 • Find the sum of their squares using the identity a^2 + b^2 = (a+b)^2 - 2ab: • a^2 + b^2 = (3)^2 - 2(1) = 9 - 2 = 7 • Now, substitute these values into the target expression: • Target Expression: (a^2 + ab + b^2/a^2 - ab + b^2) • Rearrange to group terms: ((a^2 + b^2) + ab/(a^2 + b^2) - ab) • Substitute (a^2 + b^2) = 7 and ab = 1: • (7 + 1/7 - 1) = (8/6) • Simplify the fraction: (4/3) Exam Strategy & Shortcut: Use the direct sum formula for reciprocal surds of the form a = \frac{\sqrt{x} + \sqrt{y}}{\sqrt{x} - \sqrt{y}}. Note that 1 can be written as \sqrt{1}. The sum a + b = 2 \cdot (x + y/x - y). Here, x = 5 and y = 1. a + b = 2 \cdot (5 + 1/5 - 1) = 2 \cdot (6/4) = 3. Since ab = 1, we instantly know a^2 + b^2 = 3^2 - 2 = 7. Plug directly into (7 + 1/7 - 1) = (8/6) = (4/3). This solves the problem in under 30 seconds. Common Pitfall: A common mistake is incorrectly expanding the terms in the target expression or failing to recognize that a and b are reciprocals. If you miss that ab = 1 immediately, you will waste valuable time multiplying complex fractions. Final Answer: Therefore, the correct answer is 4/3.

Question 4

$\sqrt{2 + \sqrt{2 + \sqrt{2 + \cdots}}}$ is equal to

Accepted Answer

Concept & Logic: This evaluates an infinite repeating nested square root. When the repeating number can be expressed as the product of two consecutive integers, a rapid logical deduction can be made instead of full algebraic calculation. The core substitution logic remains: y = \sqrt{a + y} Step-by-Step Solution: • Let the entire expression be x. • x = \sqrt{2 + \sqrt{2 + \sqrt{2 + \cdots}}} • Because the nested roots continue infinitely, the part under the first square root after "2 +" is exactly equal to our original definition of x. • Replace the infinite tail with x: • x = \sqrt{2 + x} • Square both sides of the equation: • x^2 = 2 + x • Bring all terms to one side to form a quadratic equation: • x^2 - x - 2 = 0 • Factor the quadratic equation. We need two numbers that multiply to -2 and add to -1. Those numbers are -2 and 1. • (x - 2)(x + 1) = 0 • This gives two possible solutions: x = 2 or x = -1. • Since a square root function (and a sum of positive terms) must yield a positive result, x cannot be negative. • We discard -1. Thus, x = 2. Exam Strategy & Shortcut: Whenever you face an infinite series in the form \sqrt{a + \sqrt{a + \sqrt{a + \cdots}}}, immediately check if a can be factored into two consecutive integers (n × (n+1)). If it is a "+" series, the answer is always the larger integer (n+1). If it is a "-" series, the answer is the smaller integer (n). Here, 2 = 1 × 2. Since it's a "+" series, the answer is the larger factor, which is 2. Zero calculations required. Common Pitfall: The most frequent error is choosing the smaller factor (1) instead of the larger factor (2). Remember: addition sequences yield the larger consecutive factor, while subtraction sequences yield the smaller one. Final Answer: Therefore, the correct answer is 2.

Question 5

One-fourth of a herd of camels was seen in the forest. Twice the square root of the herd had gone to mountains and the remaining 15 camels were seen on the bank of a river. Find the total number of camels.

Accepted Answer

Concept & Logic: This word problem is solved by translating the given conditions into an algebraic equation. Since the problem involves the square root of the total number of camels, it will naturally form a quadratic equation when formalized. Step-by-Step Solution: • Given: Let the total number of camels in the herd be x. • Number of camels in the forest = (x/4) • Number of camels gone to the mountains = 2\sqrt{x} • Number of remaining camels on the bank = 15 • The sum of these three groups must equal the total number of camels: • (x/4) + 2\sqrt{x} + 15 = x • Rearrange the terms to group the x variables on one side: • x - (x/4) - 2\sqrt{x} = 15 • (3x/4) - 2\sqrt{x} = 15 • Multiply the entire equation by 4 to clear the fraction: • 3x - 8\sqrt{x} = 60 • Let y = \sqrt{x}. This implies y^2 = x. Substitute this into the equation to form a standard quadratic: • 3y^2 - 8y - 60 = 0 • Factor the quadratic equation. We need two numbers that multiply to (3 × -60) = -180 and add to -8. Those numbers are -18 and 10. • 3y^2 - 18y + 10y - 60 = 0 • 3y(y - 6) + 10(y - 6) = 0 • (3y + 10)(y - 6) = 0 • This gives y = -(10/3) or y = 6. • Since y represents a square root of a physical count of camels, it must be positive. Thus, y = 6. • Substitute back to find x: \sqrt{x} = 6, so x = 36. Exam Strategy & Shortcut: Option Elimination is the fastest method here. The problem states "Twice the square root of the herd...". This strongly implies that the total number of camels must be a perfect square integer (since you can't have a fractional number of live camels). Look at the options: (a) 32 - Not a perfect square. (b) 34 - Not a perfect square. (c) 35 - Not a perfect square. (d) 36 - Perfect square! (\sqrt{36} = 6) You can solve this question in exactly 2 seconds without writing a single equation. Common Pitfall: The most common mistake is setting up the equation correctly but struggling to factor the resulting quadratic with surds. Students often panic when they see 3x - 8\sqrt{x} = 60. Using substitution (y = \sqrt{x}) simplifies it, but relying on option elimination bypasses the risk of arithmetic errors entirely. Final Answer: Therefore, the correct answer is 36.

Question 6

The number of trees in each row of a garden is equal to the total number of rows in the garden. After 111 trees have been uprooted in a storm, there remain 10914 trees in the garden. The number of rows of trees in the garden is

Accepted Answer

Concept & Logic: This problem is a two-step application of algebraic modeling and square roots. We must first account for the trees lost in the storm before calculating the grid dimensions. Step-by-Step Solution: • Let the initial number of rows in the garden be x. • Since the number of trees in each row equals the number of rows, the initial number of trees per row is also x. • Original total number of trees = x \cdot x = x^2. • Given: 111 trees were uprooted, leaving 10914 trees remaining. • We can set up the equation for the original total: • Original Total = Remaining Trees + Uprooted Trees • x^2 = 10914 + 111 • x^2 = 11025 • Now, solve for x by finding the square root: • x = \sqrt{11025} • Any number ending in 25 has a square root ending in 5. • Ignore the last two digits (25) and look at the remaining part: 110. • We need to find two consecutive integers whose product is 110. Those integers are 10 and 11 (since 10 × 11 = 110). • Take the smaller integer (10) and append the 5. • Thus, x = 105. Exam Strategy & Shortcut: Master the shortcut for squaring numbers that end in 5. The pattern for (n5)^2 is to compute n × (n+1) and append 25. When you reach x^2 = 11025, you can immediately see the 25 suffix. The prefix 110 is clearly 10 × 11. This instantly tells you the root is 105. No manual calculation or division is required. Common Pitfall: A common error is setting up the equation backwards (e.g., x^2 - 10914 = 111), which works mathematically, but some students mistakenly subtract 111 from 10914 instead of adding it, finding x^2 = 10803 which is not a perfect square and leads to panic. Always ensure your total represents the intact original state. Final Answer: Therefore, the correct answer is 105.

Question 7

1250 oranges were distributed among a group of girls of a class. Each girl got twice as many oranges as the number of girls in that group. The number of girls in the group was

Accepted Answer

Concept & Logic: This is a straightforward algebra problem where the quantity distributed per person depends directly on the total number of people. It requires setting up a basic quadratic equation and solving for the positive root. Step-by-Step Solution: • Let the total number of girls in the group be x. • Given: Each girl received twice as many oranges as the total number of girls. • Therefore, oranges received by one girl = 2x. • The total number of oranges distributed is the product of the number of girls and the oranges per girl: • Total Oranges = (Number of girls) × (Oranges per girl) • Total Oranges = x \cdot 2x • 2x^2 = 1250 • Divide both sides by 2 to isolate x^2: • x^2 = 625 • Take the square root of both sides to find x: • x = \sqrt{625} • We know that 25 × 25 = 625. • x = 25. Exam Strategy & Shortcut: You can solve this rapidly using the options. The total oranges (1250) must be equal to 2x^2. Divide the total by 2 in your head to get 625. Now, you just need to identify which option is the square root of 625. 50^2 = 2500 (Too big) 45^2 > 40^2 = 1600 (Too big) 25^2 = 625. (Perfect match). Common Pitfall: The most common trap is forgetting to include the multiplier "twice as many". If a student misreads it as "as many oranges as the number of girls", they will set up x^2 = 1250, which doesn't have an integer square root, causing confusion and wasted time. Always pay close attention to proportional keywords like "twice" or "half". Final Answer: Therefore, the correct answer is 25.

Question 8

A mobile company offered to pay the Indian Cricket Team as much money per run scored by the side as the total number it gets in a one-dayer against Australia.   Which one of the following cannot be the total amount to be spent by the company in this…

Accepted Answer

Concept & Logic The company pays an amount equal to the number of runs multiplied by the number of runs. Thus, the total amount spent must be a perfect square. The question is asking us to identify the single option that is not a perfect square. Step-by-Step Solution • Given: Total amount = (Runs Scored) × (Runs Scored) = R^2. • We need to inspect the unit digits of the given options to rule out non-perfect squares. • A fundamental rule of mathematics: A perfect square can never end in 2, 3, 7, or 8. • Let us check the last digit of each given option: - Option (a) 21,904 ends in 4 (possible perfect square). - Option (b) 56,169 ends in 9 (possible perfect square). - Option (c) 1,01,761 ends in 1 (possible perfect square). - Option (d) 1,21,108 ends in 8 (impossible). • Since no integer squared ends in 8, 1,21,108 mathematically cannot be the total amount spent. Exam Strategy & Shortcut Always check the unit digits first when asked if a number is a perfect square! Perfect squares only end in 0, 1, 4, 5, 6, or 9. Option (d) ends in 8, making it instantly incorrect as a perfect square. Absolutely no calculation is required. Common Pitfall Wasting valuable exam time trying to calculate the exact square root of each large 5- or 6-digit number. Recognizing unit digit rules saves massive amounts of time on these types of questions. Final Answer Therefore, the correct answer is 1,21,108.

Question 9

Value of $\sqrt{.01} 	imes \sqrt[3]{.008} - .02$ is

Accepted Answer

Concept & Logic Evaluate the square root and cube root of the decimal numbers individually, paying strict attention to the number of decimal places for each radical. After evaluating, apply BODMAS (Multiplication before Subtraction) to find the final numerical value. Step-by-Step Solution • Given: \sqrt{0.01} × \sqrt[3]{0.008} - 0.02 • Step 1: Evaluate the square root of 0.01. The number 0.01 has 2 decimal places. The square root will have 2 \div 2 = 1 decimal place. Since \sqrt{1} = 1, the result is 0.1. • Step 2: Evaluate the cube root of 0.008. The number 0.008 has 3 decimal places. The cube root will have 3 \div 3 = 1 decimal place. Since \sqrt[3]{8} = 2, the result is 0.2. • Step 3: Substitute back and simplify using BODMAS. 0.1 × 0.2 - 0.02 • Perform the multiplication first. When multiplying 0.1 and 0.2, multiply the numbers (1 × 2 = 2) and add the decimal places (1 + 1 = 2). 0.1 × 0.2 = 0.02 • Perform the final subtraction: 0.02 - 0.02 = 0 Exam Strategy & Shortcut Quickly convert the decimals in your head to fractions to avoid decimal errors: \sqrt{1/100} = 1/10. \sqrt[3]{8/1000} = 2/10. Multiply them: (1/10) × (2/10) = 2/100 = 0.02. Subtracting 0.02 from 0.02 is instantly 0. Fraction conversion often prevents silly decimal placement mistakes. Common Pitfall Miscalculating the product of 0.1 × 0.2. Many students incorrectly write 0.2 instead of 0.02, leading them to calculate 0.2 - 0.02 = 0.18, which isn't an option but causes panic and wasted time checking steps. Final Answer Therefore, the correct answer is 0.

Question 10

A rationalising factor of $(\sqrt[3]{9} - \sqrt[3]{3} + 1)$ is

Accepted Answer

Concept & Formula To rationalise an expression containing cube roots, we use the standard algebraic identities for the sum or difference of cubes. a^3 + b^3 = (a + b)(a^2 - ab + b^2) We need to multiply the given irrational expression by a factor that transforms it into a rational integer (like a^3 + b^3). Step-by-Step Solution • Given expression: \sqrt[3]{9} - \sqrt[3]{3} + 1 • Rewrite the expression to identify the algebraic structure. Note that 9 = 3^2. (\sqrt[3]{3})^2 - \sqrt[3]{3}(1) + (1)^2 • Let a = \sqrt[3]{3} and b = 1. The expression exactly matches the quadratic factor of the sum of cubes formula: (a^2 - ab + b^2) • To rationalize it (meaning, to get rid of the cube roots by turning the whole thing into a^3 + b^3), we must multiply it by the missing factor: (a + b) • Substitute a and b back into the missing factor: (\sqrt[3]{3} + 1) • If we multiply them, the result is (\sqrt[3]{3})^3 + 1^3 = 3 + 1 = 4, which is perfectly rational. Exam Strategy & Shortcut Recognize the pattern a^2 - ab + b^2. The presence of the mixed minus and plus signs (+, -, +) always pairs with the factor (a + b) to create a sum of cubes (a^3 + b^3). Here, the base elements are clearly \sqrt[3]{3} and 1. Therefore, the rationalizing multiplier is simply their sum: \sqrt[3]{3} + 1. Common Pitfall Confusing the sum of cubes with the difference of cubes. If a student sees the minus sign and instinctively chooses (\sqrt[3]{3} - 1), the resulting multiplication would not clear the radicals properly, as the signs in the trinomial would be mismatched. Final Answer Therefore, the correct answer is \sqrt[3]{3} + 1.

Question 11

Which smallest number must be added to 710 so that the sum is a perfect cube?

Accepted Answer

Concept & Logic The core logic is to find the nearest perfect cube that is strictly greater than the given number. Step-by-Step Solution • Identify the closest perfect cubes around 710. • We know that 8^3 = 512 and 9^3 = 729. • Since 729 > 710, the next perfect cube immediately after 710 is 729. • To find the smallest number to add, subtract the given number from the perfect cube: 729 - 710 = 19. Exam Strategy & Shortcut Memorize perfect cubes up to 15. Recognizing 9^3 = 729 instantly gives the difference without any manual calculation. Common Pitfall A common mistake is finding the nearest perfect square instead of a cube, or subtracting instead of adding. Always read carefully whether the question asks to "add" or "subtract". Final Answer Therefore, the correct answer is 19.

Question 12

$\sqrt{24 \div 0.5 + 1} + \sqrt{18 \div 0.6 + 6} = $ $x$

Accepted Answer

Concept & Logic This problem tests your ability to apply the BODMAS rule (specifically handling division with decimals) inside radical expressions. You must completely simplify the expressions under each square root before adding them together. \sqrt{a} + \sqrt{b} 
eq \sqrt{a+b} Step-by-Step Solution • Given: \sqrt{24 \div 0.5 + 1} + \sqrt{18 \div 0.6 + 6} = x • Calculation: First, evaluate the expression inside the first square root: 24 \div 0.5 = 24 \div (1/2) = 24 × 2 = 48 48 + 1 = 49 So, the first term is \sqrt{49} = 7. • Next, evaluate the expression inside the second square root: 18 \div 0.6 = (180/6) = 30 30 + 6 = 36 So, the second term is \sqrt{36} = 6. • Finally, add the simplified terms: 7 + 6 = 13 Exam Strategy & Shortcut Division by a decimal like 0.5 is exactly the same as multiplying by 2. Division by 0.6 is the same as multiplying by (10/6). Recognizing these fraction equivalents allows you to do the math mentally: 24 × 2 = 48, and 18 × (10/6) = 30. This skips the need for long division. Common Pitfall A frequent mistake is adding the numbers inside the roots before performing the division, or incorrectly evaluating 18 \div 0.6 as 3 instead of 30 due to decimal misplacement. Always align your decimal points carefully! Final Answer Therefore, the correct answer is 13.

Question 13

$\sqrt{x} + 14 = \sqrt{2601}$

Accepted Answer

Concept & Logic To solve this equation, you must first evaluate the perfect square root on the right side of the equation. Once calculated, use basic algebra to isolate \sqrt{x} and then square both sides to find x. If \sqrt{x} = y, then x = y^2 Step-by-Step Solution • Given: \sqrt{x} + 14 = \sqrt{2601} • Calculation: First, determine the square root of 2601. We know 50^2 = 2500. Since 2601 ends in 1, its square root must end in 1 or 9. The number 2601 is very close to 2500, so the root is 51. \sqrt{2601} = 51 • Substitute this back into the equation: \sqrt{x} + 14 = 51 • Isolate \sqrt{x} by subtracting 14 from both sides: \sqrt{x} = 51 - 14 \sqrt{x} = 37 • Square both sides to solve for x: x = 37^2 x = 1369 Exam Strategy & Shortcut Memorize squares up to 40. Knowing 37^2 = 1369 instantly saves you from having to do manual multiplication during the final step. To quickly verify 37^2, you can use the (a-b)^2 method: (40-3)^2 = 1600 - 240 + 9 = 1369. Common Pitfall A frequent mistake is stopping at \sqrt{x} = 37 and looking for 37 in the options (if it were present), or incorrectly doubling 37 instead of squaring it. Always remember to apply the inverse operation (squaring) to remove the radical. Final Answer Therefore, the correct answer is 1369.

Question 14

If $a = \frac{\sqrt{3}}{2}$, then   $\sqrt{1+a} + \sqrt{1-a} = $ $x$

Accepted Answer

Concept & Formula Instead of plugging in the complex fraction directly under nested radicals, it is much easier to square the entire target expression. This takes advantage of the (x+y)^2 identity to eliminate the outer square roots and simplify the inner terms via the difference of squares. (x+y)^2 = x^2 + y^2 + 2xy (1+a)(1-a) = 1 - a^2 Step-by-Step Solution • Given: a = \frac{\sqrt{3}}{2} and we need to evaluate y = \sqrt{1+a} + \sqrt{1-a} • Calculation: Square the entire expression we want to evaluate: y^2 = (\sqrt{1+a} + \sqrt{1-a})^2 y^2 = (\sqrt{1+a})^2 + (\sqrt{1-a})^2 + 2(\sqrt{1+a})(\sqrt{1-a}) y^2 = (1+a) + (1-a) + 2\sqrt{(1+a)(1-a)} • Simplify the terms: y^2 = 2 + 2\sqrt{1 - a^2} • Now, substitute the given value of a: Since a = \frac{\sqrt{3}}{2}, then a^2 = (3/4) • Plug a^2 into our simplified equation: y^2 = 2 + 2\sqrt{1 - (3/4)} y^2 = 2 + 2\sqrt{(1/4)} y^2 = 2 + 2((1/2)) y^2 = 2 + 1 y^2 = 3 • Solve for y (the original expression): y = \sqrt{3} (Taking the positive root since the sum of positive square roots must be positive). Exam Strategy & Shortcut Whenever you see a pattern like \sqrt{1+a} \pm \sqrt{1-a}, immediately square the expression. It elegantly collapses the a terms and creates a simple 1-a^2 under the remaining radical. This is a standard competitive exam trick that bypasses nested radical denesting algorithms entirely. Common Pitfall Attempting to evaluate \sqrt{1 + \frac{\sqrt{3}}{2}} directly requires finding a perfect square in the form of (x+y)^2 hidden under the root (e.g., multiplying top and bottom by 2 to get \frac{\sqrt{4+2\sqrt{3}}}{\sqrt{2}}). While mathematically possible, it is incredibly time-consuming and prone to arithmetic errors. Final Answer Therefore, the correct answer is \sqrt{3}.

Question 15

What is   $\frac{5+\sqrt{10}}{5\sqrt{5}-2\sqrt{20}-\sqrt{32}+\sqrt{50}}$   equal to

Accepted Answer

Concept & Formula To simplify complex radical fractions, you must break down every surd into its simplest form by extracting perfect squares. Once the denominator is simplified, look for common factors to factor out in both the numerator and denominator to cancel terms. \sqrt{a^2b} = a\sqrt{b} Step-by-Step Solution • Given: \frac{5+\sqrt{10}}{5\sqrt{5}-2\sqrt{20}-\sqrt{32}+\sqrt{50}} • Calculation: First, simplify each term in the denominator independently: 5\sqrt{5} (Already simplified) 2\sqrt{20} = 2\sqrt{4 × 5} = 2(2\sqrt{5}) = 4\sqrt{5} \sqrt{32} = \sqrt{16 × 2} = 4\sqrt{2} \sqrt{50} = \sqrt{25 × 2} = 5\sqrt{2} • Reconstruct and simplify the denominator: = 5\sqrt{5} - 4\sqrt{5} - 4\sqrt{2} + 5\sqrt{2} Group like terms: = (5 - 4)\sqrt{5} + (-4 + 5)\sqrt{2} = \sqrt{5} + \sqrt{2} • The full expression is now: \frac{5+\sqrt{10}}{\sqrt{5}+\sqrt{2}} • Now, look at the numerator. Factor out \sqrt{5}: Notice that 5 = \sqrt{25} = \sqrt{5} × \sqrt{5} and \sqrt{10} = \sqrt{5} × \sqrt{2}. 5 + \sqrt{10} = \sqrt{5}(\sqrt{5} + \sqrt{2}) • Substitute this back into the fraction: = \frac{\sqrt{5}(\sqrt{5} + \sqrt{2})}{\sqrt{5}+\sqrt{2}} • Cancel the common binomial term (\sqrt{5} + \sqrt{2}) from the top and bottom: = \sqrt{5} Exam Strategy & Shortcut Don't immediately jump to rationalizing the denominator. Rationalizing \sqrt{5} + \sqrt{2} by multiplying by its conjugate \sqrt{5} - \sqrt{2} works, but it takes an extra step. In exam settings, numerators are almost always designed to factor cleanly into a multiple of the simplified denominator. Always check for a common factor first! Common Pitfall A common mistake is incorrectly simplifying the surds (e.g., extracting the wrong perfect square) which prevents the denominator from collapsing into a simple binomial. Always double-check your prime factorizations for numbers like 20, 32, and 50. Final Answer Therefore, the correct answer is \sqrt{5}.

Square Root and Cube Root Questions

Questions

Practice smarter