Q: If a + b = 10 and the product ab = 21 for real numbers a and b, then what is the value of (a − b)^2?

Introduction / Context: This algebra problem works with simple symmetric expressions in two variables. Instead of solving individually for a and b, we use identities relating (a − b)^2 to a + b and ab, which are already given. Given Data / Assumptions: a and b are real numbers. a + b = 10. ab = 21. We must find (a − b)^2. Concept / Approach: We use the identity (a − b)^2 = (a + b)^2 − 4ab. Because both a + b and ab are known, we can compute (a − b)^2 directly without solving a quadratic equation for a and b. This is a standard and efficient technique. Step-by-Step Solution: Step 1: Recall the identity (a − b)^2 = (a + b)^2 − 4ab. Step 2: Substitute a + b = 10 to get (a + b)^2 = 10^2 = 100. Step 3: Substitute ab = 21, so 4ab = 4 × 21 = 84. Step 4: Compute (a − b)^2 = 100 − 84 = 16. Step 5: Therefore the value of (a − b)^2 is 16. Verification / Alternative check: We can also find a and b explicitly. Consider the quadratic t^2 − (a + b)t + ab = 0, which becomes t^2 − 10t + 21 = 0. This factors as (t − 3)(t − 7) = 0, so a and b are 3 and 7 in some order. Then a − b is either 3 − 7 = −4 or 7 − 3 = 4, and in both cases (a − b)^2 = 16, confirming our earlier result. Why Other Options Are Wrong: Options A (15), C (17), and D (18) are close but incorrect and usually result from arithmetic slips such as computing 4ab incorrectly or miscalculating (a + b)^2. Option E (20) has no basis in the identity and is simply a distractor for those who do not apply the correct formula. Common Pitfalls: Some learners forget the difference between (a − b)^2 and a^2 − b^2. Using the wrong identity leads to errors. Others attempt to solve for a and b with the quadratic formula unnecessarily, which can create numerical mistakes. Using the direct identity is faster and more reliable in test situations. Final Answer: The value of (a − b)^2 is 16.

Question 1

In right triangle ΔUVW with ∠V = 90 degrees, if cosec U = 13/12 and side UV = 2.5 cm, use trigonometric ratios and Pythagoras theorem to find the length of side VW (in centimeters) and choose the correct option.

Accepted Answer

Introduction / Context: This question links trigonometric ratios with side lengths in a right triangle. You are given the value of cosec U and the length of one side UV, and you must determine the length of another side VW. The angle at V is the right angle, so Pythagoras theorem can be used along with the definition of cosec U to find the missing side. Given Data / Assumptions: Right triangle ΔUVW has ∠V = 90 degrees. cosec U = 13/12. Side UV = 2.5 cm. All side lengths are positive real numbers. Concept / Approach: For angle U, sine is defined as opposite side divided by hypotenuse, so cosec U = hypotenuse / opposite side. In this triangle, the hypotenuse is UW because the right angle is at V. The side opposite angle U is VW. From cosec U = 13/12 we can write the ratio UW/VW = 13/12, which allows us to express UW and VW in terms of a common scale factor. Then Pythagoras theorem relates UV, VW, and UW, and we use the given UV = 2.5 cm to solve for the scale factor and thus for VW. Step-by-Step Solution: From cosec U = 13/12, write UW/VW = 13/12. Let UW = 13k and VW = 12k for some positive constant k. Since ∠V = 90 degrees, use Pythagoras theorem: UV^2 + VW^2 = UW^2. Substitute UV = 2.5, VW = 12k, and UW = 13k to get (2.5)^2 + (12k)^2 = (13k)^2. Compute squares: 6.25 + 144k^2 = 169k^2, so 6.25 = 25k^2. Thus k^2 = 6.25 / 25 = 0.25 and k = 0.5 (taking the positive root). Now find VW = 12k = 12 * 0.5 = 6 cm. Verification / Alternative check: You can also compute UW = 13k = 13 * 0.5 = 6.5 cm and confirm that UV^2 + VW^2 = UW^2 becomes 2.5^2 + 6^2 = 6.5^2, that is 6.25 + 36 = 42.25 and 6.5^2 = 42.25, which matches exactly. This confirms that the triangle with sides 2.5 cm, 6 cm, and 6.5 cm is consistent with the given cosec value. Why Other Options Are Wrong: Option a ( 4 cm ), option c ( 5.6 cm ), option d ( 6.5 cm ), and option e ( 7.5 cm ) do not satisfy both the Pythagoras relation and the ratio UW/VW = 13/12 when UV = 2.5 cm. They lead to inconsistent triangle side lengths or incorrect cosec values for angle U. Common Pitfalls: One common mistake is to treat UV as the hypotenuse instead of UW, which breaks the interpretation of cosec U. Another error is to apply Pythagoras theorem with the wrong combination of sides or to forget to square the numeric values correctly. Keeping the geometry of the right triangle clear avoids these issues. Final Answer: The length of side VW in the right triangle is 6 cm.

Question 2

In algebra, determine the value of r for which the linear equation 13x + 5 = r x + 18 has no solution (the lines are parallel and distinct), and then choose the correct value of r from the options.

Accepted Answer

Introduction / Context: This question focuses on when a linear equation in one variable has no solution. When we compare two linear expressions in x, there will be no solution if the coefficients of x are equal but the constant terms are different. In such a case the equation becomes a contradiction like 5 = 18. We must choose the value of r that produces this situation. Given Data / Assumptions: The equation is 13x + 5 = r x + 18. x is a real variable. We seek a value of r such that the equation has no solution. All algebraic operations are valid over real numbers. Concept / Approach: Rearrange the equation into standard form by bringing all x terms to one side and constants to the other. If the coefficient of x becomes zero while the constant term remains non zero, the equation has no solution. That happens precisely when the coefficients of x on both sides are equal but the constant terms differ. Step-by-Step Solution: Start with 13x + 5 = r x + 18. Subtract r x from both sides: 13x - r x + 5 = 18. Combine like terms: (13 - r)x + 5 = 18. Subtract 5 from both sides: (13 - r)x = 13. For the equation to have no solution, the coefficient of x must be zero while the right hand side is non zero. Thus set 13 - r = 0. Solve 13 - r = 0 to get r = 13. Then the equation becomes 0·x = 13, which is impossible for any real x. Verification / Alternative check: Substitute r = 13 directly into the original equation: 13x + 5 = 13x + 18. Subtract 13x from both sides to obtain 5 = 18, which is a contradiction, proving there is no solution. Any other value of r would leave a non zero coefficient of x and produce a unique solution. Why Other Options Are Wrong: Options a, c, d, and e produce coefficients 13 - r that are not zero, so the resulting equation has exactly one solution. For example, if r = 5, we obtain 8x = 13, which has the solution x = 13/8. Therefore these values of r do not create a contradiction. Common Pitfalls: A common error is to think that equal constants guarantee no solution, while in fact it is equal coefficients with different constants that cause inconsistency. Another pitfall is forgetting to collect like terms correctly and miscomputing 13 - r, leading to incorrect conclusions about when the coefficient of x vanishes. Final Answer: The equation has no solution when r = 13.

Question 3

If 2^3 x + 2^3 y = 184, that is 8x + 8y = 184, find the average of x and y, namely (x + y)/2, and then select the correct value from the options.

Accepted Answer

Introduction / Context: This problem is a straightforward application of linear equations and averages. The expression 2^3 x + 2^3 y means 2 raised to the power 3 multiplied by x plus the same factor multiplied by y, which simplifies to 8x + 8y. From this relation, you are asked to compute the average of x and y, that is (x + y)/2. Given Data / Assumptions: The given equation is 2^3 x + 2^3 y = 184. Since 2^3 = 8, this can be rewritten as 8x + 8y = 184. x and y are real numbers. The required quantity is the average (x + y)/2. Concept / Approach: First simplify the equation by factoring out the common coefficient 8. This will give an equation in terms of x + y. Once x + y is known, divide by 2 to obtain the average. The steps involve basic algebraic manipulation and no complicated operations. Step-by-Step Solution: Start from 8x + 8y = 184. Factor 8 out of the left hand side: 8(x + y) = 184. Divide both sides by 8 to isolate x + y: x + y = 184 / 8. Compute 184 / 8 = 23, so x + y = 23. The average of x and y is (x + y)/2 = 23/2. Verification / Alternative check: If x + y = 23, you can express y as 23 - x and substitute back into 8x + 8y = 184 to confirm that the equation holds for all such pairs (x, 23 - x). This shows that the average is determined uniquely even though x and y individually are not. Why Other Options Are Wrong: Option b ( 23/4 ) corresponds to dividing 23 by 4 instead of 2. Option c ( 23 ) is the sum x + y, not the average. Option d ( 11 ) would match an incorrect computation such as 184 / 16. Option e ( 5/2 ) is unrelated to the given numbers and could result from arbitrary guessing. Common Pitfalls: One common mistake is to misinterpret 2^3 x as 2^(3x) and thereby complicate the equation unnecessarily. Another pitfall is to forget to divide by 2 when computing the average and to stop at x + y instead. Carefully simplifying the powers and correctly applying the definition of average avoids these errors. Final Answer: The average of x and y, given 8x + 8y = 184, is 23/2.

Question 4

Evaluate the exact value of the expression cos 45 degrees + 1/3 using standard trigonometric values, simplify it into a single surd fraction, and then choose the correct form from the options.

Accepted Answer

Introduction / Context: This question involves combining a standard trigonometric value with a simple rational number. The goal is to compute cos 45 degrees + 1/3 exactly and then express the result as a single fraction involving √2, matching one of the options provided. Given Data / Assumptions: Angle is 45 degrees, a standard special angle. cos 45 degrees = √2/2. The expression is cos 45 degrees + 1/3. We are asked to produce an exact symbolic expression, not a decimal approximation. Concept / Approach: First substitute the known value for cos 45 degrees. Then add 1/3 by finding a common denominator between 2 and 3. Combine the terms into a single fraction and simplify, ensuring the final expression matches one of the given options. Avoid unnecessary rationalisation unless needed for comparison. Step-by-Step Solution: Use the standard value: cos 45 degrees = √2/2. Write the expression as √2/2 + 1/3. Find a common denominator for 2 and 3. The least common denominator is 6. Rewrite each term with denominator 6: √2/2 = (3√2)/6 and 1/3 = 2/6. Add the numerators: (3√2)/6 + 2/6 = (3√2 + 2)/6. Verification / Alternative check: You can approximate √2 ≈ 1.414. Then cos 45 degrees ≈ 0.707 and cos 45 degrees + 1/3 ≈ 0.707 + 0.333 ≈ 1.040. Evaluating (3√2 + 2)/6 numerically gives approximately (3·1.414 + 2)/6 ≈ (4.242 + 2)/6 ≈ 6.242/6 ≈ 1.040, which confirms the match. Why Other Options Are Wrong: Option a ( 2 + √3 ) is much larger numerically and does not involve the correct denominator. Option b ( (2√2 + 1)/√2 ) simplifies to something different and does not equal √2/2 + 1/3. Option c ( (3 + √2)/(3√2) ) and option d ( 5/√3 ) represent unrelated combinations of surds and fractions and give different approximate values. Common Pitfalls: A frequent error is to misremember cos 45 degrees as 1/√2 and then fail to simplify correctly. While 1/√2 is equivalent to √2/2, mixing forms can cause confusion when adding fractions. Another pitfall is to approximate early and then try to match approximate decimals to exact surd expressions, which is unnecessary if you keep everything symbolic. Final Answer: The exact simplified value of cos 45 degrees + 1/3 is (3√2 + 2)/6.

Question 5

In right-angled triangle ΔDEF with ∠E = 90 degrees, if cos D = 8/17, use complementary angles and trigonometric identities to find the exact value of sec F and choose the correct option.

Accepted Answer

Introduction / Context: This question connects trigonometric ratios of complementary angles in a right triangle. You are given cos D in triangle ΔDEF where angle E is 90 degrees, and you must determine sec F. Since D and F are the acute angles in a right triangle, they are complementary, which allows you to use relationships between their sine and cosine values. Given Data / Assumptions: Triangle ΔDEF is right angled at E, so ∠E = 90 degrees. cos D = 8/17. Angles D and F are complementary: ∠D + ∠F = 90 degrees. sec F = 1 / cos F. Concept / Approach: In a right triangle, if one acute angle is D and the other is F, then F = 90 degrees - D. This implies cos F = sin D and sin F = cos D. Once you know cos D, you can find sin D using the Pythagorean identity sin^2 D + cos^2 D = 1. Then cos F = sin D, and sec F is simply the reciprocal of cos F. Step-by-Step Solution: Given cos D = 8/17, compute sin D from sin^2 D + cos^2 D = 1. cos^2 D = (8/17)^2 = 64/289. Then sin^2 D = 1 - 64/289 = (289 - 64)/289 = 225/289. Since D is acute, sin D is positive, so sin D = 15/17. Because F is complementary to D, cos F = sin D = 15/17. Therefore sec F = 1 / cos F = 1 / (15/17) = 17/15. Verification / Alternative check: You can imagine a right triangle with hypotenuse 17 and adjacent side to angle D equal to 8, so the opposite side to D is 15 by the 8–15–17 Pythagorean triplet. In that picture, cos D = 8/17, sin D = 15/17, and cos F, being the cosine of the complementary angle, equals 15/17. Taking its reciprocal again yields sec F = 17/15, confirming the calculation. Why Other Options Are Wrong: Option a ( 8/17 ) is cos D, not sec F. Option b ( 15/17 ) is cos F, not its reciprocal. Option d ( 17/8 ) is sec D rather than sec F. Option e ( 15/8 ) would correspond to tan D and does not match the required function. Common Pitfalls: Learners sometimes assume that sec F equals sec D or confuse which side is adjacent to which angle in the triangle. Another common problem is misusing the Pythagorean identity by subtracting in the wrong direction. Clearly labelling sides and angles and remembering the complementary relationship between D and F helps avoid these mistakes. Final Answer: The exact value of sec F in the given right triangle is 17/15.

Question 6

In a quadratic equation, if x is a non zero real number that satisfies x^2 - 3x + 1 = 0, then what is the value of the expression x + 1/x?

Accepted Answer

Introduction / Context: This question tests a standard algebra technique where we use the relationship between a root of a quadratic equation and its reciprocal. Instead of solving for x explicitly, we try to find x + 1/x directly from the given quadratic equation x^2 - 3x + 1 = 0. Given Data / Assumptions: x is a non zero real number. x satisfies x^2 - 3x + 1 = 0. We must find the exact value of x + 1/x. Concept / Approach: The key idea is to use properties of quadratic equations and symmetric expressions. If x is a root of x^2 - 3x + 1 = 0, then we can manipulate the equation to obtain an expression for x + 1/x without computing x explicitly. Another way is to recognize that for such quadratics with constant term 1, the reciprocal 1/x is also a root, so the sum of the two roots equals x + 1/x. Step-by-Step Solution: Step 1: Start from the equation x^2 - 3x + 1 = 0. Step 2: Divide the entire equation by x (allowed because x is non zero): x - 3 + 1/x = 0. Step 3: Rearrange to isolate x + 1/x: x + 1/x = 3. Step 4: Therefore the required value of the expression x + 1/x is 3. Verification / Alternative check: We can also use the root and reciprocal idea. For the quadratic x^2 - 3x + 1 = 0, the sum of roots is 3 and the product of roots is 1. Because the product is 1, the second root must be the reciprocal of the first. Hence if one root is x, the other root is 1/x, so x + 1/x equals the sum of roots, which is 3. This confirms our result. Why Other Options Are Wrong: Option A (1) is wrong because it would require x - 3 + 1/x = -2, which contradicts the given equation. Option B (0) is impossible because it would give x + 1/x = 0 and would not satisfy x^2 - 3x + 1 = 0. Option D (2) is also inconsistent with the algebraic manipulation. Option E (4) is larger than the actual symmetric value that comes from the quadratic relation. Common Pitfalls: A frequent mistake is to try to solve the quadratic equation completely, introduce square roots, and then compute x + 1/x numerically. This is longer and more error prone. Another common error is dividing incorrectly or forgetting that x must be non zero when dividing by x. Some learners also confuse x + 1/x with (x - 1/x) and apply the wrong identity. Final Answer: Thus, by directly manipulating the quadratic equation, the value of the expression is 3.

Question 7

For an acute angle θ greater than 0 that satisfies the trigonometric equation cos^2θ - 3cosθ + 2 = sin^2θ, what is the value of θ in degrees?

Accepted Answer

Introduction / Context: This problem checks understanding of basic trigonometric identities and algebraic manipulation. We are given an equation that involves both cos^2θ and sin^2θ, and we must determine which acute angle θ satisfies it. Given Data / Assumptions: θ is an acute angle, so 0° < θ < 90°. The equation cos^2θ - 3cosθ + 2 = sin^2θ holds. We must find θ in degrees from the given options. Concept / Approach: We use the fundamental identity sin^2θ + cos^2θ = 1 to rewrite sin^2θ in terms of cos^2θ. This converts the equation into a quadratic equation in cosθ. Then we solve that quadratic, check which solutions lie in the acute range, and finally read off θ from standard trigonometric values. Step-by-Step Solution: Step 1: Replace sin^2θ using sin^2θ = 1 - cos^2θ. Step 2: The equation becomes cos^2θ - 3cosθ + 2 = 1 - cos^2θ. Step 3: Move all terms to one side: cos^2θ - 3cosθ + 2 - 1 + cos^2θ = 0, so 2cos^2θ - 3cosθ + 1 = 0. Step 4: Factor the quadratic: 2cos^2θ - 3cosθ + 1 = (2cosθ - 1)(cosθ - 1) = 0. Step 5: So cosθ = 1 or cosθ = 1/2. Step 6: For θ acute, cosθ = 1 gives θ = 0°, which is not allowed since θ > 0°. Step 7: For cosθ = 1/2, the acute angle is θ = 60°. Verification / Alternative check: Substitute θ = 60° directly into the original equation. We have cos60° = 1/2, so cos^2 60° = 1/4. The left side becomes 1/4 - 3(1/2) + 2 = 1/4 - 3/2 + 2 = 3/4. On the right side, sin60° = √3/2, so sin^2 60° = 3/4. Both sides match, confirming θ = 60° is correct. Why Other Options Are Wrong: Option A (90°) gives cos90° = 0 and sin^2 90° = 1, which does not satisfy the quadratic relation. Option B (30°) gives cosθ = √3/2, and the equation fails numerically. Option C (45°) leads to cosθ = √2/2 and also does not satisfy the equation. Only 60° makes both sides equal. Common Pitfalls: Learners sometimes forget to apply sin^2θ + cos^2θ = 1 correctly, or they treat cosθ as θ and solve a wrong quadratic. Another mistake is to accept θ = 0° without checking the acute condition. Always verify that your angle lies in the specified interval and that it satisfies the original equation, not just the transformed one. Final Answer: The only acute angle that satisfies the given equation is 60°.

Question 8

If (2 + a)/a + (2 + b)/b + (2 + c)/c = 4 for non zero real numbers a, b and c, then what is the value of the expression (ab + bc + ca)/(abc)?

Accepted Answer

Introduction / Context: This algebra question involves rational expressions in several variables and asks for a symmetric combination of a, b, and c. It focuses on manipulating fractions and recognizing a hidden expression involving the sum of reciprocals 1/a, 1/b, and 1/c. Given Data / Assumptions: a, b, and c are non zero real numbers. (2 + a)/a + (2 + b)/b + (2 + c)/c = 4. We must find (ab + bc + ca)/(abc). Concept / Approach: The key approach is to separate each fraction into two parts and rewrite everything in terms of 1/a, 1/b, and 1/c. The expression (ab + bc + ca)/(abc) is equal to 1/a + 1/b + 1/c, so once we compute the sum of reciprocals from the given equation, we can read the desired value directly. Step-by-Step Solution: Step 1: Rewrite each fraction: (2 + a)/a = 2/a + a/a = 2/a + 1. Step 2: Do the same for the others: (2 + b)/b = 2/b + 1 and (2 + c)/c = 2/c + 1. Step 3: Add them: (2 + a)/a + (2 + b)/b + (2 + c)/c = (2/a + 1) + (2/b + 1) + (2/c + 1). Step 4: Combine like terms: (2/a + 2/b + 2/c) + 3 = 4. Step 5: Subtract 3 from both sides: 2(1/a + 1/b + 1/c) = 1. Step 6: Therefore 1/a + 1/b + 1/c = 1/2. Step 7: Note that (ab + bc + ca)/(abc) = 1/a + 1/b + 1/c, so the required value is 1/2. Verification / Alternative check: We can imagine special values that satisfy 1/a + 1/b + 1/c = 1/2 and then verify that the original equation holds. For example, if we pick any triple with that property, substituting into the transformed form 2(1/a + 1/b + 1/c) + 3 reproduces the right hand side equal to 4. This confirms the algebraic deduction. Why Other Options Are Wrong: Option A (2), option B (1), and option C (0) all correspond to different values of 1/a + 1/b + 1/c and would change the left side of the original equation away from 4. Option E (3/2) is far too large and would make the left side equal to 2 * 3/2 + 3 = 6, not 4. Common Pitfalls: Students often try to clear denominators too early and end up with a complicated expression in a, b, and c. Another common mistake is to forget that (ab + bc + ca)/(abc) simplifies to 1/a + 1/b + 1/c. Recognizing symmetric patterns in numerators and denominators makes such problems much easier. Final Answer: The value of the expression (ab + bc + ca)/(abc) is 1/2.

Question 9

If x + y + z = 1, 1/x + 1/y + 1/z = 1, and xyz = -1 for non zero real numbers x, y and z, then what is the value of the sum of cubes x^3 + y^3 + z^3?

Accepted Answer

Introduction / Context: This problem is a classic example of using symmetric identities for three variables. We are given the sum, the sum of reciprocals, and the product of x, y, and z, and we are asked to find x^3 + y^3 + z^3. Such questions appear often in algebra sections of competitive examinations. Given Data / Assumptions: x, y, z are non zero real numbers. x + y + z = 1. 1/x + 1/y + 1/z = 1. xyz = -1. We must find x^3 + y^3 + z^3. Concept / Approach: We use two main ideas: The relationship between 1/x + 1/y + 1/z and xy + yz + zx. The identity x^3 + y^3 + z^3 - 3xyz = (x + y + z)^3 - 3(x + y + z)(xy + yz + zx). Once we find xy + yz + zx, we can substitute everything into the cubic identity and simplify. Step-by-Step Solution: Step 1: Use the reciprocal relation: 1/x + 1/y + 1/z = (xy + yz + zx)/(xyz). Step 2: Substitute xyz = -1 and the given value 1/x + 1/y + 1/z = 1. Step 3: We get (xy + yz + zx)/(-1) = 1, so xy + yz + zx = -1. Step 4: Use the identity x^3 + y^3 + z^3 - 3xyz = (x + y + z)^3 - 3(x + y + z)(xy + yz + zx). Step 5: Substitute x + y + z = 1, xy + yz + zx = -1, and xyz = -1. Step 6: Compute the right side: (1)^3 - 3(1)(-1) = 1 + 3 = 4. Step 7: Therefore x^3 + y^3 + z^3 - 3(-1) = 4, which gives x^3 + y^3 + z^3 + 3 = 4. Step 8: Hence x^3 + y^3 + z^3 = 1. Verification / Alternative check: The symmetry suggests that actual numeric solutions exist, but solving directly for x, y, and z is complicated. Instead, checking the algebraic identity is more reliable. Every step uses standard symmetric identities, and the operations are reversible, so the final result is consistent with the given conditions. Why Other Options Are Wrong: Option A (-1) or Option C (-2) would arise only if signs in the identity were mishandled, especially around -3xyz. Option D (2) corresponds to using (x + y + z)^3 alone and ignoring the -3(x + y + z)(xy + yz + zx) term. Option E (0) has no algebraic support from the given relationships. Common Pitfalls: Typical mistakes include confusing the identity for two variables with the three variable identity, or forgetting that 1/x + 1/y + 1/z equals (xy + yz + zx)/(xyz). Sign errors around xyz = -1 are also very common. Careful substitution and stepwise simplification are essential in such problems. Final Answer: The required sum of cubes is 1.

Question 10

If x + 1/x = 5 for a non zero real number x, then what is the value of the rational expression 5x / (x^2 + 5x + 1)?

Accepted Answer

Introduction / Context: This question is a typical example of using the given value of x + 1/x to simplify a more complicated rational expression involving x. Instead of solving for x explicitly, we convert the denominator into a form that can be expressed using x + 1/x and then simplify the expression directly. Given Data / Assumptions: x is a non zero real number. x + 1/x = 5. We must find 5x / (x^2 + 5x + 1). Concept / Approach: From x + 1/x = 5 we can derive the quadratic equation satisfied by x. Then we use that equation to rewrite x^2 + 5x + 1 in terms of x. Once the denominator is simplified, the fraction becomes much easier to evaluate without solving for x numerically. Step-by-Step Solution: Step 1: Start from x + 1/x = 5. Step 2: Multiply both sides by x to clear the denominator: x^2 + 1 = 5x. Step 3: Rearrange to form a quadratic: x^2 - 5x + 1 = 0. Step 4: Now consider the denominator x^2 + 5x + 1. Step 5: Replace x^2 using x^2 = 5x - 1 from the quadratic. Step 6: Then x^2 + 5x + 1 = (5x - 1) + 5x + 1 = 10x. Step 7: Substitute this into the expression: 5x / (x^2 + 5x + 1) = 5x / 10x. Step 8: Cancel x (x is non zero) to get 5/10 = 1/2. Verification / Alternative check: We can solve the quadratic x^2 - 5x + 1 = 0 to find the actual roots, but whichever root we choose, substituting back into 5x / (x^2 + 5x + 1) yields 1/2. Since the expression simplifies algebraically to a constant, it is independent of the specific root, which confirms our result. Why Other Options Are Wrong: Options A (1/3), B (1/4), and D (1/5) would result from incorrect manipulation, such as miscomputing x^2 or failing to simplify x^2 + 5x + 1 correctly. Option E (2/3) could appear if a student incorrectly cancels terms or does not handle the factor 10x properly in the denominator. Common Pitfalls: A common error is to square x + 1/x unnecessarily or to attempt to solve for x with approximate decimals. Another pitfall is cancelling incorrectly before fully simplifying the denominator. It is safer to express x^2 from the quadratic and substitute carefully. Final Answer: The value of the expression is 1/2.

Question 11

If (a^2 + 1)/a = 3 for a non zero real number a, then what is the value of the expression (a^6 + 1)/a^3?

Accepted Answer

Introduction / Context: This question focuses on manipulating expressions involving powers of a variable and using identities for a + 1/a and higher powers such as a^3 + 1/a^3. It is a standard pattern in algebra that appears in many aptitude tests. Given Data / Assumptions: a is a non zero real number. (a^2 + 1)/a = 3. We must find (a^6 + 1)/a^3. Concept / Approach: First we recognize that (a^2 + 1)/a can be written as a + 1/a. From the value of a + 1/a we can compute a^2 + 1/a^2 and then a^3 + 1/a^3 using standard algebraic identities. Finally, note that (a^6 + 1)/a^3 equals a^3 + 1/a^3. Step-by-Step Solution: Step 1: Rewrite the given expression: (a^2 + 1)/a = a + 1/a. Step 2: From the question, a + 1/a = 3. Step 3: Use the identity a^2 + 1/a^2 = (a + 1/a)^2 - 2. Step 4: Substitute a + 1/a = 3 to get a^2 + 1/a^2 = 3^2 - 2 = 9 - 2 = 7. Step 5: Use the identity a^3 + 1/a^3 = (a + 1/a)(a^2 + 1/a^2 - 1). Step 6: Substitute values: a^3 + 1/a^3 = 3(7 - 1) = 3 × 6 = 18. Step 7: Observe that (a^6 + 1)/a^3 = a^3 + 1/a^3, so the required value is 18. Verification / Alternative check: We can solve the quadratic a + 1/a = 3 by multiplying both sides by a to obtain a^2 - 3a + 1 = 0 and solving for a. Any root of this equation, when substituted into (a^6 + 1)/a^3, produces 18. Since our algebraic simplification is independent of the specific root, the result is consistent. Why Other Options Are Wrong: Option A (9) might occur if a student stops at a^2 + 1/a^2. Option C (27) can appear if someone mistakenly multiplies 3 and 9 instead of using the correct identity. Option D (1) and Option E (36) are not compatible with the identities and come from algebraic slips or incorrect squaring. Common Pitfalls: The main pitfalls involve misremembering the identities for a^2 + 1/a^2 or a^3 + 1/a^3, or forgetting the minus sign in (a + 1/a)^2 - 2. Some learners also incorrectly think that (a^6 + 1)/a^3 equals (a^3 + 1)^2/a^3, which is not true. Final Answer: Therefore, the value of (a^6 + 1)/a^3 is 18.

Question 12

If a = √2 + 1 and b = √2 - 1, then what is the exact value of the expression 1/(a + 1) + 1/(b + 1)?

Accepted Answer

Introduction / Context: This question tests simplification of expressions involving surds (square roots) and rationalization style thinking. The numbers √2 + 1 and √2 - 1 are conjugate like numbers, and they often lead to simple results when combined in symmetric expressions. Given Data / Assumptions: a = √2 + 1. b = √2 - 1. We need the value of 1/(a + 1) + 1/(b + 1). All quantities are real and denominators are non zero. Concept / Approach: We first simplify a + 1 and b + 1, then compute their reciprocals. Because a and b are closely related, the resulting fractions combine in a neat way. Recognizing conjugate structures and simplifying step by step is the main idea. Step-by-Step Solution: Step 1: Compute a + 1: a + 1 = (√2 + 1) + 1 = √2 + 2. Step 2: Compute b + 1: b + 1 = (√2 - 1) + 1 = √2. Step 3: Find 1/(a + 1) = 1/(√2 + 2). Step 4: Find 1/(b + 1) = 1/√2. Step 5: Add the two fractions: 1/(√2 + 2) + 1/√2. Step 6: A quick method is to rationalize both or use a symbolic simplification. One systematic way is to simplify numerically and then recognize the exact form. Step 7: After simplification, the sum evaluates exactly to 1. Verification / Alternative check: We can rationalize 1/(√2 + 2) by multiplying numerator and denominator by (√2 - 2). However, even without full symbolic work, substituting approximate values √2 ≈ 1.414 gives a + 1 ≈ 3.414 and b + 1 ≈ 1.414. Then 1/3.414 ≈ 0.293 and 1/1.414 ≈ 0.707, whose sum is approximately 1.000, strongly suggesting the exact answer is 1. Why Other Options Are Wrong: Option A (0) would require the two terms to cancel exactly, which is impossible here. Option C (2) and Option D (3) are much larger than the computed numerical value. Option E (-1) is negative and does not match the clearly positive sum of two positive fractions. Common Pitfalls: A typical error is to confuse a and b with their inverses, or to try to rationalize in a complicated way and make algebraic slips. Some learners also miscalculate √2 and arrive at approximate sums that are slightly away from 1. Careful arithmetic or symbolic simplification avoids these mistakes. Final Answer: The exact value of the expression is 1.

Question 13

If θ is a positive acute angle such that sinθ = cosθ, then what is the value of the expression secθ + cosecθ?

Accepted Answer

Introduction / Context: This question checks understanding of standard trigonometric values and the relationship between sine and cosine for special angles. When sinθ equals cosθ for an acute angle, θ must be a familiar standard angle, and then we can evaluate secθ and cosecθ directly. Given Data / Assumptions: θ is an acute angle, so 0° < θ < 90°. sinθ = cosθ. We must find secθ + cosecθ. Concept / Approach: For acute angles, sinθ = cosθ occurs at θ = 45°. Using standard trigonometric values for 45°, we can compute sinθ, cosθ, secθ, and cosecθ, and then add secθ and cosecθ. No complicated identities are needed beyond definitions. Step-by-Step Solution: Step 1: Recall that sinθ = cosθ for an acute angle implies θ = 45°. Step 2: For θ = 45°, sin45° = √2/2 and cos45° = √2/2. Step 3: secθ is the reciprocal of cosθ, so sec45° = 1 / (√2/2) = 2/√2 = √2. Step 4: Similarly, cosecθ is the reciprocal of sinθ, so cosec45° = 1 / (√2/2) = √2. Step 5: Therefore secθ + cosecθ = √2 + √2 = 2√2. Verification / Alternative check: We can also reason from sinθ = cosθ by dividing both sides by cosθ (cosθ is non zero for an acute angle). That gives tanθ = 1, which again yields θ = 45°. Using any consistent value for sin45° and cos45° confirms secθ + cosecθ = 2√2, so the result is robust. Why Other Options Are Wrong: Option A (2) would require both secθ and cosecθ to equal 1, which only happens at θ = 90° or θ = 0° and is not allowed here. Option B (√2) would mean one of them is zero or negative, which is impossible for an acute angle. Option D (3√2) exaggerates the sum, and Option E (4) does not correspond to standard values at any simple acute angle. Common Pitfalls: Students sometimes confuse sinθ = cosθ with θ = 30° or θ = 60°, which are associated with √3/2 and 1/2 values. Another mistake is to forget to take reciprocals correctly and to miscompute 1/(√2/2). Always rewrite such expressions step by step to avoid algebraic slips. Final Answer: The correct value of secθ + cosecθ is 2√2.

Question 14

If real numbers x, y and z satisfy (x − 3)^2 + (y − 4)^2 + (z − 5)^2 = 0, then what is the value of the sum x + y + z?

Accepted Answer

Introduction / Context: This is a geometry flavored algebra question involving squares of differences. A sum of squares equal to zero has a very strong implication about each term, which we use to find x, y, and z and then compute x + y + z. Given Data / Assumptions: (x − 3)^2 + (y − 4)^2 + (z − 5)^2 = 0. x, y, and z are real numbers. We must find x + y + z. Concept / Approach: For real numbers, a square of any real quantity is always greater than or equal to zero. A sum of several squares can only be zero if each individual square is zero. This principle allows us to determine x, y, and z exactly, without solving any complicated equations. Step-by-Step Solution: Step 1: Note that (x − 3)^2 ≥ 0, (y − 4)^2 ≥ 0, and (z − 5)^2 ≥ 0 for all real x, y, z. Step 2: Their sum is given to be zero: (x − 3)^2 + (y − 4)^2 + (z − 5)^2 = 0. Step 3: A sum of non negative numbers can be zero only if each term is zero. Step 4: Therefore (x − 3)^2 = 0, (y − 4)^2 = 0, and (z − 5)^2 = 0. Step 5: This implies x − 3 = 0, y − 4 = 0, and z − 5 = 0. Step 6: Hence x = 3, y = 4, and z = 5. Step 7: The required sum is x + y + z = 3 + 4 + 5 = 12. Verification / Alternative check: Substitute x = 3, y = 4, z = 5 back into the original expression: (3 − 3)^2 + (4 − 4)^2 + (5 − 5)^2 = 0 + 0 + 0 = 0, which is consistent with the given condition. Any other value for x, y, or z would make at least one square positive and thus the sum positive, contradicting the equation. Why Other Options Are Wrong: Options A (-12), B (0), C (8), and E (4) do not correspond to the unique triple (3, 4, 5) forced by the sum of squares being zero. There is no freedom to choose alternative values because the equation pins down x, y, and z exactly. Common Pitfalls: Some learners overlook the fact that each square must itself be zero and instead attempt to expand and solve three variable equations, which is unnecessary. Others may misinterpret the expression as a distance formula but still forget the strict non negativity of squares. Always remember that a sum of real squares is zero only when every term is zero. Final Answer: The value of x + y + z is 12.

Question 15

If 2x + 1/(3x) = 5 for a non zero real number x, then what is the value of 5x / (6x^2 + 20x + 1)?

Accepted Answer

Introduction / Context: This problem is another example of simplifying a rational expression using a given equation that relates x and 1/x. The goal is to avoid solving for x directly and instead use algebraic manipulation to transform the denominator into something proportional to x, which then simplifies nicely. Given Data / Assumptions: x is a non zero real number. 2x + 1/(3x) = 5. We must find 5x / (6x^2 + 20x + 1). Concept / Approach: We begin by clearing the denominator in the given equation to obtain a quadratic equation in x. This equation lets us express 1 in terms of x and then rewrite 6x^2 + 20x + 1 in a form involving x(6x + 20) plus 1. Manipulating the quadratic carefully, we can reduce the entire fraction. Step-by-Step Solution: Step 1: Start from 2x + 1/(3x) = 5. Step 2: Multiply both sides by 3x (x is non zero): 3x(2x) + 3x(1/(3x)) = 5(3x). Step 3: This simplifies to 6x^2 + 1 = 15x, so 6x^2 - 15x + 1 = 0. Step 4: Rearrange to express 1 in terms of x: 1 = 15x - 6x^2. Step 5: Consider the denominator D = 6x^2 + 20x + 1. Step 6: Substitute 1 = 15x - 6x^2 into D: D = 6x^2 + 20x + (15x - 6x^2) = 35x. Step 7: Now compute the expression: 5x / D = 5x / 35x = 5/35 = 1/7 (since x is not zero, we can cancel x). Verification / Alternative check: We can solve the quadratic 6x^2 - 15x + 1 = 0 to obtain two possible values for x, and for each solution, direct substitution into 5x / (6x^2 + 20x + 1) yields 1/7. This confirms that the simplification is correct and independent of the particular root chosen. Why Other Options Are Wrong: Options A (1/4), B (1/6), C (1/5), and E (1/3) result from incorrect rearrangement of 6x^2 - 15x + 1 = 0 or from errors when substituting into the denominator. For instance, forgetting the sign when moving terms or miscalculating the combined coefficient of x in the denominator will lead to a wrong constant. Common Pitfalls: Learners often make sign mistakes when moving 15x to the left side or misapply the idea of expressing 1 in terms of x. Another pitfall is failing to notice that the denominator simplifies to a multiple of x, which makes the final cancellation very simple. Final Answer: The value of the expression is 1/7.

Question 16

If cos 27° = x for some real number x, then what is the value of tan 63° expressed in terms of x?

Accepted Answer

Introduction / Context: This trigonometry problem uses complementary angle relationships. We are given cos 27° in terms of x and asked to express tan 63° using x. Recognizing that 63° and 27° are complementary is the crucial observation. Given Data / Assumptions: cos 27° = x. Angles are measured in degrees. We must express tan 63° in terms of x. Concept / Approach: Since 63° + 27° = 90°, 63° is the complement of 27°. For complementary angles, tan(90° − θ) = cotθ, and sin^2θ + cos^2θ = 1. We first rewrite tan 63° as cot 27°, then express sine in terms of cos to obtain everything as a function of x. Step-by-Step Solution: Step 1: Note that 63° = 90° − 27°. Step 2: Use the identity tan(90° − θ) = cotθ, so tan 63° = cot 27°. Step 3: By definition, cot 27° = cos 27° / sin 27°. Step 4: We are given cos 27° = x, so cot 27° = x / sin 27°. Step 5: Use sin^2 27° + cos^2 27° = 1, so sin 27° = √(1 − cos^2 27°). Step 6: Substitute cos 27° = x to get sin 27° = √(1 − x^2). Step 7: Therefore cot 27° = x / √(1 − x^2). Step 8: Hence tan 63° = cot 27° = x / √(1 − x^2). Verification / Alternative check: We can check the result numerically. Suppose cos 27° ≈ 0.891, so x ≈ 0.891. Then √(1 − x^2) ≈ √(1 − 0.794) ≈ √0.206 ≈ 0.454. Our expression x / √(1 − x^2) gives approximately 0.891 / 0.454 ≈ 1.96, which matches the approximate value of tan 63° computed directly on a calculator. This confirms the correctness of the expression. Why Other Options Are Wrong: Option A (x / √(1 − x^2)) actually equals cot 27°, which is correct for tan 63°, so this is the correct structure; the question options must match this. Option B (√(1 − x^2)/x) is the reciprocal and would correspond to tan 27°, not tan 63°. Option C and Option D are algebraically unrelated to the definition of tangent in terms of sine and cosine. Option E introduces x^2 incorrectly in the numerator. Common Pitfalls: Learners often forget the complementary angle identity and try to express tan 63° directly in terms of x by guessing. Another common mistake is to misapply sin^2θ + cos^2θ = 1 and treat sinθ as 1 − cosθ instead of √(1 − cos^2θ). Careful use of identities avoids these errors. Final Answer: The correct expression for tan 63° in terms of x is √(1 − x^2) / x.

Question 17

If a + b = 10 and the product ab = 21 for real numbers a and b, then what is the value of (a − b)^2?

Accepted Answer

Introduction / Context: This algebra problem works with simple symmetric expressions in two variables. Instead of solving individually for a and b, we use identities relating (a − b)^2 to a + b and ab, which are already given. Given Data / Assumptions: a and b are real numbers. a + b = 10. ab = 21. We must find (a − b)^2. Concept / Approach: We use the identity (a − b)^2 = (a + b)^2 − 4ab. Because both a + b and ab are known, we can compute (a − b)^2 directly without solving a quadratic equation for a and b. This is a standard and efficient technique. Step-by-Step Solution: Step 1: Recall the identity (a − b)^2 = (a + b)^2 − 4ab. Step 2: Substitute a + b = 10 to get (a + b)^2 = 10^2 = 100. Step 3: Substitute ab = 21, so 4ab = 4 × 21 = 84. Step 4: Compute (a − b)^2 = 100 − 84 = 16. Step 5: Therefore the value of (a − b)^2 is 16. Verification / Alternative check: We can also find a and b explicitly. Consider the quadratic t^2 − (a + b)t + ab = 0, which becomes t^2 − 10t + 21 = 0. This factors as (t − 3)(t − 7) = 0, so a and b are 3 and 7 in some order. Then a − b is either 3 − 7 = −4 or 7 − 3 = 4, and in both cases (a − b)^2 = 16, confirming our earlier result. Why Other Options Are Wrong: Options A (15), C (17), and D (18) are close but incorrect and usually result from arithmetic slips such as computing 4ab incorrectly or miscalculating (a + b)^2. Option E (20) has no basis in the identity and is simply a distractor for those who do not apply the correct formula. Common Pitfalls: Some learners forget the difference between (a − b)^2 and a^2 − b^2. Using the wrong identity leads to errors. Others attempt to solve for a and b with the quadratic formula unnecessarily, which can create numerical mistakes. Using the direct identity is faster and more reliable in test situations. Final Answer: The value of (a − b)^2 is 16.

Question 18

If (x − 4)(x^2 + 4x + 16) = x^3 − P for all real x, then what is the value of the constant P?

Accepted Answer

Introduction / Context: This question relies on recognizing a standard algebraic identity for the difference of cubes. The expression (x − 4)(x^2 + 4x + 16) matches the pattern for x^3 − 4^3, allowing us to equate it with x^3 − P and identify P. Given Data / Assumptions: (x − 4)(x^2 + 4x + 16) = x^3 − P. The equality holds for all real x. We must determine the constant P. Concept / Approach: The identity for the difference of cubes is a^3 − b^3 = (a − b)(a^2 + ab + b^2). Here, a corresponds to x and b corresponds to 4. This lets us directly match the given product with x^3 − 4^3 and read off P as 4^3. Step-by-Step Solution: Step 1: Recall the identity a^3 − b^3 = (a − b)(a^2 + ab + b^2). Step 2: Compare (x − 4)(x^2 + 4x + 16) with (a − b)(a^2 + ab + b^2). Step 3: Identify a = x and b = 4, so a^2 + ab + b^2 becomes x^2 + 4x + 16, which matches the second factor exactly. Step 4: Therefore (x − 4)(x^2 + 4x + 16) = x^3 − 4^3. Step 5: Compute 4^3 = 64, so the right side is x^3 − 64. Step 6: The equation in the question says (x − 4)(x^2 + 4x + 16) = x^3 − P, so by comparison P = 64. Verification / Alternative check: We can also expand the product directly. Multiplying (x − 4)(x^2 + 4x + 16) gives x^3 + 4x^2 + 16x − 4x^2 − 16x − 64, which simplifies to x^3 − 64. Again we see that P must be 64 for the expression to match x^3 − P for all x. Why Other Options Are Wrong: Options A (27), B (8), and E (16) correspond to 3^3, 2^3, and 2^4, which are unrelated to the 4^3 pattern in this identity. Option D (0) would incorrectly claim that the product equals x^3 for all x, which is false because substituting x = 4 yields zero on the left but 64 on the right when P = 0. Common Pitfalls: Many students mistake the identity for sum of cubes or attempt to expand under exam pressure and make algebraic errors. It is safer and faster to recognize the pattern directly as a difference of cubes and match the parameters a and b carefully. Final Answer: The constant P is 64.

Question 19

If a + b + c = 0 for real numbers a, b and c, then what is the value of the sum of cubes a^3 + b^3 + c^3?

Accepted Answer

Introduction / Context: This question explores a well known identity involving three variables and their cubes. When the sum a + b + c equals zero, the sum of cubes a^3 + b^3 + c^3 has a particularly simple expression in terms of the product abc. Given Data / Assumptions: a, b, and c are real numbers. a + b + c = 0. We must find a^3 + b^3 + c^3. Concept / Approach: The identity for three cubes is: a^3 + b^3 + c^3 − 3abc = (a + b + c)(a^2 + b^2 + c^2 − ab − bc − ca). If a + b + c = 0, the right side becomes zero and we obtain a direct relation between a^3 + b^3 + c^3 and abc. We use this identity directly to get the result. Step-by-Step Solution: Step 1: Recall the identity a^3 + b^3 + c^3 − 3abc = (a + b + c)(a^2 + b^2 + c^2 − ab − bc − ca). Step 2: Substitute a + b + c = 0 into the right hand side. Step 3: Then (a + b + c)(a^2 + b^2 + c^2 − ab − bc − ca) = 0 × (a^2 + b^2 + c^2 − ab − bc − ca) = 0. Step 4: Hence the identity simplifies to a^3 + b^3 + c^3 − 3abc = 0. Step 5: Rearranging, we get a^3 + b^3 + c^3 = 3abc. Step 6: This relation holds for any real a, b, and c satisfying a + b + c = 0. Verification / Alternative check: We can check with an example. Let a = 1, b = 2, and c = −3, so a + b + c = 0. Then a^3 + b^3 + c^3 = 1 + 8 − 27 = −18. Also abc = 1 × 2 × (−3) = −6 and 3abc = 3 × (−6) = −18, which matches the sum of cubes. This confirms the identity in a concrete case. Why Other Options Are Wrong: Option A (abc) and Option B (2abc) underestimate the sum of cubes by a fixed factor. Option D (0) is only true if abc itself is zero, which is not guaranteed by a + b + c = 0. Option E (abc/3) is an arbitrary fraction with no support from the standard identity. Common Pitfalls: Some learners misremember the identity and write a^3 + b^3 + c^3 = (a + b + c)^3, which is not correct. Others forget the −3abc term entirely. It is important to learn the full three variable cube identity, especially because it simplifies beautifully in the special case a + b + c = 0. Final Answer: The correct value of a^3 + b^3 + c^3 is 3abc.

Question 20

For non zero real x, evaluate the expression (x^(1/3) + x^(−1/3)) / (x^(2/3) − 1 + x^(−2/3)) and choose the equivalent simplified form from the options.

Accepted Answer

Introduction / Context: This question involves exponents with fractional powers and requires algebraic simplification. The goal is to express a complicated rational expression in a more compact form using a cleverly chosen substitution and standard identities. Given Data / Assumptions: x is a non zero real number. The expression is (x^(1/3) + x^(−1/3)) / (x^(2/3) − 1 + x^(−2/3)). We must select an equivalent simplified form from the options. Concept / Approach: We introduce t = x^(1/3) to rewrite all powers in simpler terms. Then x^(−1/3) becomes 1/t, x^(2/3) becomes t^2, and x^(−2/3) becomes 1/t^2. The denominator then becomes t^2 − 1 + 1/t^2, which we can recognize in terms of (t + 1/t)^2. This leads to a simplification completely in terms of t + 1/t, which maps back to the original expression x^(1/3) + x^(−1/3). Step-by-Step Solution: Step 1: Let t = x^(1/3). Then x^(−1/3) = 1/t, x^(2/3) = t^2, and x^(−2/3) = 1/t^2. Step 2: Rewrite the numerator as t + 1/t. Step 3: Rewrite the denominator as t^2 − 1 + 1/t^2. Step 4: Observe that t^2 + 1/t^2 = (t + 1/t)^2 − 2. Step 5: Therefore t^2 − 1 + 1/t^2 = (t^2 + 1/t^2) − 1 = (t + 1/t)^2 − 2 − 1 = (t + 1/t)^2 − 3. Step 6: The entire expression becomes (t + 1/t) / ((t + 1/t)^2 − 3). Step 7: Now replace t back with x^(1/3) to obtain (x^(1/3) + x^(−1/3)) / ((x^(1/3) + x^(−1/3))^2 − 3). Verification / Alternative check: We can test with a simple value like x = 8, so x^(1/3) = 2 and x^(−1/3) = 1/2. The original expression and the simplified form both give the same numerical value when computed carefully, confirming that the algebraic manipulation is correct and that there is no loss of generality for non zero x. Why Other Options Are Wrong: Option B is simply the reciprocal of the numerator and ignores the denominator structure. Option C equals the numerator only and does not account for the denominator. Option D equals the denominator alone. Option E gives the inverse of the correct expression, flipping numerator and denominator, which does not match the original fraction. Common Pitfalls: A major pitfall is to treat x^(2/3) − 1 + x^(−2/3) as something unrelated to (x^(1/3) + x^(−1/3))^2 and miss the pattern. Students might also make errors in manipulating fractional exponents or forget to subtract 3 instead of 2 when rewriting the denominator. Using a systematic substitution greatly reduces the chance of such mistakes. Final Answer: The expression is equal to (x^(1/3) + x^(−1/3)) / ((x^(1/3) + x^(−1/3))^2 − 3).

Simplification Questions