Question 1

In a circle with centre O, let AB be a diameter and P be a point on the circumference. If the central angle POA is 120 degrees with vertex at O, what is the measure in degrees of angle PBO?

Accepted Answer

Introduction / Context: This geometry question involves angles in a circle with a diameter and a central angle. It tests understanding of the relationships between central angles, points on the circle and angles formed at other points on the circle. Such questions train you to visualise circle geometry and use symmetry effectively. Given Data / Assumptions: - AB is a diameter of the circle with centre O. - P is a point on the circumference of the circle, different from A and B. - The central angle POA measures 120 degrees at the centre O. - We need to find the measure of angle PBO, which is the angle at point B between segments BP and BO. Concept / Approach: One clean approach is to use coordinate geometry. Place the circle with centre at the origin, take the radius as 1, and put A and B conveniently on the x axis. Then interpret the angle POA = 120 degrees as giving the direction of OP. After that, compute the angle at B using vectors or geometric reasoning. The result is independent of the specific radius choice. Step-by-Step Solution: Place the circle with centre O at (0, 0) and radius 1. Let A be at (1, 0) and B at (-1, 0), so AB is a diameter. The angle POA = 120 degrees means that OP makes an angle of 120 degrees with OA, which lies along the positive x axis. Thus, the coordinates of P are (cos 120°, sin 120°) = (-1/2, √3/2). Now consider angle PBO at B formed by vectors BP and BO. Vector BP = P - B = (-1/2 + 1, √3/2 - 0) = (1/2, √3/2). Vector BO = O - B = (0 + 1, 0 - 0) = (1, 0). The angle between these two vectors is 60 degrees because BP lies at 60 degrees to the positive x axis while BO points along the positive x axis. Verification / Alternative check: A purely geometric argument also works. Since AB is a diameter, triangle AOB is a straight line at the centre. Rotating OP 120 degrees from OA places P symmetrically in such a way that triangle BOP forms a 60 degree angle at B due to the known angles in equilateral type configurations. Either method yields 60 degrees. Why Other Options Are Wrong: Angles 30, 40, 45 and 50 degrees do not match the strict symmetry of this configuration. They usually come from misinterpreting which angle is 120 degrees or confusing central angles with inscribed angles that subtend the same arc. Common Pitfalls: A frequent mistake is to assume that angle PBO equals half of 120 degrees directly, which is not true here because PBO is not an inscribed angle subtending the same arc as POA. Another pitfall is misplacing the points around the circle mentally and mixing up which angle is at the centre versus at the circumference. Final Answer: The measure of angle PBO is 60°.

Question 2

Suppose a non zero real number P satisfies the relation P^2 + 1/P^2 = 47. Using standard algebraic identities, what is the value of the sum P + 1/P?

Accepted Answer

Introduction / Context: This is a classic algebra identity question where you are given P^2 + 1/P^2 and asked to find P + 1/P. The objective is to recognise and apply the correct identity rather than attempting to solve a quadratic equation for P and then back substitute, which would be slower. Given Data / Assumptions: - P is a non zero real number, so division by P is allowed. - P^2 + 1/P^2 = 47. - We must find the value of P + 1/P. Concept / Approach: The key identity relating these expressions is (P + 1/P)^2 = P^2 + 2 + 1/P^2. Knowing P^2 + 1/P^2 allows you to compute (P + 1/P)^2 directly. After that, you take a square root, remembering that both positive and negative roots are possible in general, but aptitude questions typically expect the positive value unless otherwise specified. Step-by-Step Solution: Start from the identity (P + 1/P)^2 = P^2 + 2 + 1/P^2. Substitute the given value P^2 + 1/P^2 = 47. Then (P + 1/P)^2 = 47 + 2 = 49. So P + 1/P is a real number whose square is 49. Therefore P + 1/P = 7 or P + 1/P = -7. Among the provided options, only 7 appears, so the intended answer is 7. Verification / Alternative check: If P + 1/P = 7, then P^2 + 1/P^2 is (P + 1/P)^2 - 2 = 7^2 - 2 = 49 - 2 = 47, which matches the given value. This confirms that choosing 7 is consistent with the original relation. Why Other Options Are Wrong: If you try P + 1/P = 5, then P^2 + 1/P^2 = 25 - 2 = 23, not 47. For 6 we get 36 - 2 = 34, for 8 we get 64 - 2 = 62, and for 4 we get 16 - 2 = 14. None of these match the required 47, so these options are incorrect. Common Pitfalls: Students sometimes mistakenly use P^2 + 1/P^2 = (P + 1/P)^2, forgetting the extra 2 term, which would lead to the wrong value. Others try to solve P^2 + 1/P^2 = 47 as a quadratic in P^2, which is unnecessary and complicated. Recognising the simple square identity is the most efficient approach. Final Answer: The value of the sum is 7.

Question 3

For a real angle x, consider the trigonometric expression tan^2 x + cot^2 x. What is the least possible value that this expression can take?

Accepted Answer

Introduction / Context: This trigonometry question asks for the minimum possible value of tan^2 x + cot^2 x. It tests your ability to convert the problem into a simpler algebraic form and then apply basic inequality ideas, such as the AM GM inequality, rather than trying random angle values. Given Data / Assumptions: - x is a real angle for which both tan x and cot x are defined, so tan x is non zero. - We need the least possible value of tan^2 x + cot^2 x. Concept / Approach: Let t = tan^2 x. Then cot^2 x = 1 / tan^2 x = 1 / t, provided t is positive. The expression becomes t + 1 / t for t > 0. The well known inequality t + 1 / t >= 2 for all positive real t gives the minimum value directly, which is attained at t = 1, corresponding to tan^2 x = 1. Step-by-Step Solution: Set t = tan^2 x. Then t > 0 because a square is always non negative and tan x is non zero. Then cot^2 x = 1 / tan^2 x = 1 / t. So tan^2 x + cot^2 x = t + 1 / t. For any positive real t, we have the inequality t + 1 / t >= 2, with equality only when t = 1. Thus the minimum possible value of t + 1 / t is 2. This minimum occurs when tan^2 x = 1, which means tan x = 1 or tan x = -1. So there are many angles where the minimum is attained. Verification / Alternative check: Take x = 45 degrees, where tan x = 1. Then tan^2 x = 1 and cot^2 x = 1. The sum is 1 + 1 = 2. If you try another angle, such as x = 30 degrees, tan^2 30 is 1 / 3 and cot^2 30 is 3, giving 1 / 3 + 3 = 10 / 3 which is larger than 2. This confirms that 2 is indeed the least value. Why Other Options Are Wrong: Values such as 0 or 1 are impossible because both tan^2 x and cot^2 x are non negative and cannot both be zero. A value of 3 or 4 is achievable but not minimal according to the inequality t + 1 / t >= 2. Common Pitfalls: Some learners mistakenly try to differentiate t + 1 / t or plug in arbitrary angles instead of using the simple inequality. Others misidentify the minimum as 0 or 1 by incorrectly assuming that one of the terms can be zero, which is not allowed because tan x cannot be zero if cot x is defined. Always check domain conditions carefully. Final Answer: The least possible value of tan^2 x + cot^2 x is 2.

Question 4

Given the equation (x - 5)^2 + (y - 2)^2 + (z - 9)^2 = 0 in three real variables x, y and z, what is the value of the linear expression x + y - z?

Accepted Answer

Introduction / Context: This problem uses a simple but powerful idea from algebra and geometry. A sum of non negative square terms equals zero only when each individual square term is zero. The question then asks for a simple linear combination of x, y and z once you identify their exact values. Given Data / Assumptions: - (x - 5)^2 + (y - 2)^2 + (z - 9)^2 = 0. - x, y and z are real numbers. - We must find x + y - z. Concept / Approach: Each term (x - 5)^2, (y - 2)^2 and (z - 9)^2 is a square of a real number, hence each is greater than or equal to zero. A sum of such squares can be zero only if every term is individually zero. This observation quickly gives the values of x, y and z without any expansion. Step-by-Step Solution: Consider the expression S = (x - 5)^2 + (y - 2)^2 + (z - 9)^2. Each square term is non negative because a square of any real number is at least zero. The sum S is given to be zero. The only way a sum of non negative numbers can equal zero is if each number is itself zero. Therefore, (x - 5)^2 = 0, (y - 2)^2 = 0 and (z - 9)^2 = 0. Solving these, we get x - 5 = 0 so x = 5, y - 2 = 0 so y = 2, and z - 9 = 0 so z = 9. Now compute x + y - z = 5 + 2 - 9 = -2. Verification / Alternative check: Substituting x = 5, y = 2 and z = 9 back into the original expression gives (5 - 5)^2 + (2 - 2)^2 + (9 - 9)^2 = 0 + 0 + 0 = 0, which matches the condition. The computed value x + y - z = -2 is therefore consistent with the equation. Why Other Options Are Wrong: Values like 16, -1, 12 or 0 would require at least one of x, y or z to differ from 5, 2 or 9. In that case, at least one squared term would be positive and the sum could not be zero. So those options contradict the fundamental property of sums of squares. Common Pitfalls: A typical mistake is to expand all squares and attempt to solve a multivariable equation, which is unnecessary. Some learners also overlook that the condition forces a unique point in three dimensional space. Remember that a sum of squares equal to zero is a strong statement that each square is zero. Final Answer: The value of x + y - z is -2.

Question 5

If a real number x satisfies x + 1/x = 3, use power identities to find the exact value of x^8 + 1/x^8.

Accepted Answer

Introduction / Context: This question is a typical example of using recursive identities to compute higher powers of x and 1/x when their sum is known. Instead of trying to find x explicitly, you build up x^2 + 1/x^2, x^4 + 1/x^4 and finally x^8 + 1/x^8 using simple algebraic relations. Given Data / Assumptions: - x is a real number and x is non zero. - x + 1/x = 3. - We must find x^8 + 1/x^8. Concept / Approach: The method is stepwise. First compute x^2 + 1/x^2 from (x + 1/x)^2. Then compute x^4 + 1/x^4 from (x^2 + 1/x^2)^2. Finally, compute x^8 + 1/x^8 from (x^4 + 1/x^4)^2. Each step uses the identity (u + v)^2 = u^2 + 2uv + v^2 with u and v chosen appropriately. Step-by-Step Solution: Start with x + 1/x = 3. Square both sides: (x + 1/x)^2 = x^2 + 2 + 1/x^2 = 3^2 = 9. So x^2 + 1/x^2 = 9 - 2 = 7. Next, square x^2 + 1/x^2: (x^2 + 1/x^2)^2 = x^4 + 2 + 1/x^4. We have 7^2 = 49 = x^4 + 2 + 1/x^4. Thus x^4 + 1/x^4 = 49 - 2 = 47. Now square x^4 + 1/x^4: (x^4 + 1/x^4)^2 = x^8 + 2 + 1/x^8. We have 47^2 = 2209 = x^8 + 2 + 1/x^8. Therefore x^8 + 1/x^8 = 2209 - 2 = 2207. Verification / Alternative check: You could in principle solve x + 1/x = 3 as a quadratic equation x^2 - 3x + 1 = 0 and find the two possible values of x. Substituting either root into x^8 + 1/x^8 would give the same numeric result 2207. However, that route is longer than the identity based method used above. Why Other Options Are Wrong: Values 2201, 2203, 2211 and 2213 are close to the correct answer but differ by a few units. They are typical of arithmetic errors where the constant term 2 is added or subtracted incorrectly at one of the squaring steps. Only 2207 is consistent with the chain of identities and with any exact substitution using the quadratic roots. Common Pitfalls: Common mistakes include forgetting to subtract 2 when computing x^2 + 1/x^2 from (x + 1/x)^2, or using 4 instead of 2 when dealing with x^4 + 1/x^4 and x^8 + 1/x^8. Keeping track of the constant 2 at each squaring step is crucial for arriving at the correct answer. Final Answer: The required value of x^8 + 1/x^8 is 2207.

Question 6

Two acute angles alpha and beta satisfy alpha + beta = 90 degrees and the ratio alpha : beta = 2 : 1. What is the simplified ratio cos alpha : cos beta?

Accepted Answer

Introduction / Context: This trigonometry question links angle ratios with complementary angles. You must convert the given angle ratio into actual angle measures, then use standard cosine values to find the ratio cos alpha : cos beta. It tests both basic angle manipulation and knowledge of special angles. Given Data / Assumptions: - alpha and beta are acute angles. - alpha + beta = 90 degrees, so they are complementary. - The ratio alpha : beta = 2 : 1. - We must find cos alpha : cos beta. Concept / Approach: From the ratio alpha : beta = 2 : 1 and the fact that alpha + beta = 90 degrees, we can treat alpha as 2k and beta as k for some positive k. Solving 2k + k = 90 gives their exact measures. Then we use well known cosine values for 60 degrees and 30 degrees and form the desired ratio. Step-by-Step Solution: Let alpha = 2k and beta = k for some positive k. Given alpha + beta = 90 degrees, we have 2k + k = 90, so 3k = 90. Thus k = 30 degrees. So alpha = 2k = 60 degrees and beta = k = 30 degrees. Now compute cos alpha and cos beta using special angle values. cos 60 degrees = 1 / 2. cos 30 degrees = √3 / 2. Therefore cos alpha : cos beta = (1 / 2) : (√3 / 2). Dividing both terms by 1 / 2, we get the simplified ratio 1 : √3. Verification / Alternative check: You can verify complementary relations by noting that cos 60 degrees equals sin 30 degrees and cos 30 degrees equals sin 60 degrees. Substituting alpha and beta into these relationships still leads to the same ratio, confirming the calculation. Why Other Options Are Wrong: Ratios like 1 : 3, 1 : √2, 1 : 2 or 2 : 1 come from incorrect angle values or from forgetting to simplify the ratio by cancelling the common factor 1 / 2. None of these match the exact cosine values for 60 and 30 degrees. Common Pitfalls: A common mistake is to misinterpret the ratio alpha : beta = 2 : 1 as alpha = 2 degrees and beta = 1 degree or to assign angles that do not sum to 90 degrees. Another pitfall is to invert the ratio and compute cos beta : cos alpha instead. Always check that your angles satisfy both the ratio and the sum condition. Final Answer: The required ratio is 1:√3.

Question 7

Let x = 999, y = 1000 and z = 1001. Using the standard identity for x^3 + y^3 + z^3 - 3xyz, find the numerical value of x^3 + y^3 + z^3 - 3xyz.

Accepted Answer

Introduction / Context: This algebra problem uses a well known identity for the sum of cubes of three numbers minus three times their product. When the three numbers form an arithmetic progression, especially of the form n - 1, n and n + 1, the expression simplifies beautifully. Recognising this pattern saves a lot of computation. Given Data / Assumptions: - x = 999. - y = 1000. - z = 1001. - We must find x^3 + y^3 + z^3 - 3xyz. Concept / Approach: The key identity is x^3 + y^3 + z^3 - 3xyz = (x + y + z)(x^2 + y^2 + z^2 - xy - yz - zx). For numbers in arithmetic progression, such as n - 1, n and n + 1, this expression simplifies further. In fact, you can show that it equals 9n. Here, n is 1000, so we expect a multiple of 9000. Step-by-Step Solution: First compute x + y + z = 999 + 1000 + 1001 = 3000. Using the identity, x^3 + y^3 + z^3 - 3xyz = (x + y + z)(x^2 + y^2 + z^2 - xy - yz - zx). Recognise that x, y and z are consecutive integers: x = n - 1, y = n, z = n + 1 with n = 1000. For this pattern, it can be shown that x^2 + y^2 + z^2 - xy - yz - zx = 3n. Thus, x^3 + y^3 + z^3 - 3xyz = (x + y + z)(3n) = 3000 * 3 = 9000. Verification / Alternative check: If you wish, you can confirm by direct computation using a calculator: 999^3 + 1000^3 + 1001^3 is a large number, but subtracting 3 * 999 * 1000 * 1001 yields 9000. This matches the identity based derivation and confirms the result. Why Other Options Are Wrong: Values like 1000, 1, 9 or 3000 do not match the identity. For example, 1000 is just one of the numbers, 1 and 9 are much too small compared with the magnitudes of the cubes, and 3000 ignores the additional factor of 3n in the identity. Only 9000 is consistent with a correct application of the formula. Common Pitfalls: A common error is to forget to subtract 3xyz and instead compute only x^3 + y^3 + z^3. Another pitfall is to misapply the identity or incorrectly assume that the expression always factors as (x + y + z)^3, which is not true. Careful use of the correct identity avoids these mistakes. Final Answer: The value of x^3 + y^3 + z^3 - 3xyz is 9000.

Question 8

In a circle of radius 13 cm, a chord lies at a perpendicular distance of 12 cm from the centre. Using the Pythagorean theorem, what is the length of the chord in centimetres?

Accepted Answer

Introduction / Context: This geometry question deals with chords in a circle and their distance from the centre. It primarily tests whether you can apply the Pythagorean theorem in the right triangle formed by the radius, the perpendicular from the centre to the chord and half the chord length. Given Data / Assumptions: - The circle has radius 13 cm. - A chord is at a perpendicular distance of 12 cm from the centre. - We must find the length of the chord in centimetres. Concept / Approach: When you drop a perpendicular from the centre of a circle to a chord, it bisects the chord. This gives a right triangle where the radius is the hypotenuse, the distance from the centre to the chord is one leg, and half the chord length is the other leg. The Pythagorean theorem then connects these three quantities. Step-by-Step Solution: Let the circle have centre O and the chord be AB. Drop a perpendicular from O to chord AB, meeting it at point M. Then OM is perpendicular to AB and OM = 12 cm. The radius OA is 13 cm, and triangle OMA is a right triangle at M. Let AM be half the length of the chord AB. Then AO^2 = OM^2 + AM^2 by the Pythagorean theorem. So 13^2 = 12^2 + AM^2. Compute: 169 = 144 + AM^2, hence AM^2 = 169 - 144 = 25. Therefore AM = 5 cm. Since the perpendicular from the centre bisects the chord, AB = 2 * AM = 2 * 5 = 10 cm. Verification / Alternative check: You can also think of a circle with radius 13 forming a 5 12 13 right triangle from the well known Pythagorean triplet. Recognising that 5, 12 and 13 form such a triangle immediately suggests that the half chord is 5 cm, leading again to a chord length of 10 cm. Why Other Options Are Wrong: A chord of length 5 cm would correspond to half chord 2.5 cm, which does not satisfy 13^2 = 12^2 + 2.5^2. Similarly, chord lengths 6, 8 or 12 cm produce half chords that do not fit the Pythagorean relation with legs 12 and radius 13. Only 10 cm is consistent with the given distances. Common Pitfalls: Students sometimes mistakenly treat the distance from the centre to the chord as the chord length itself or forget that the perpendicular bisects the chord. Others misapply Pythagoras by writing 12^2 = 13^2 + AM^2, which is incorrect. Always identify the hypotenuse correctly as the radius in this setup. Final Answer: The length of the chord is 10 cm.

Question 9

If θ is a positive acute angle and 7 cos²θ + 3 sin²θ = 4, then what is the exact value of θ in degrees?

Accepted Answer

Introduction / Context: This trigonometry question checks how well you can use the Pythagorean identity sin²θ + cos²θ = 1 and manipulate equations involving sin²θ and cos²θ. The key idea is to convert the expression into a single trigonometric function, solve for its value, and then identify the corresponding acute angle θ in degrees. Given Data / Assumptions: θ is a positive acute angle (0° < θ < 90°). The equation 7 cos²θ + 3 sin²θ = 4 holds. Standard trigonometric identity sin²θ + cos²θ = 1 can be used. All trigonometric values are in degrees for this question. Concept / Approach: The main idea is to rewrite the equation so that it involves only one trigonometric function. Since sin²θ and cos²θ are linked by sin²θ = 1 − cos²θ, substituting this into the expression will convert everything into cos²θ. Then we can solve a simple algebraic equation in cos²θ, obtain cos θ, and match it with a standard angle whose cosine is known exactly. Step-by-Step Solution: Start with 7 cos²θ + 3 sin²θ = 4. Use sin²θ = 1 − cos²θ. So 7 cos²θ + 3(1 − cos²θ) = 4. Expand: 7 cos²θ + 3 − 3 cos²θ = 4. Combine like terms: (7 − 3) cos²θ + 3 = 4. Therefore 4 cos²θ + 3 = 4. So 4 cos²θ = 1. Hence cos²θ = 1/4. For an acute angle, cos θ is positive, so cos θ = 1/2. The acute angle with cos θ = 1/2 is θ = 60°. Verification / Alternative check: Substitute θ = 60° back into the original equation. We have cos 60° = 1/2, so cos² 60° = 1/4. Also sin 60° = √3/2, so sin² 60° = 3/4. Then 7 cos² 60° + 3 sin² 60° = 7(1/4) + 3(3/4) = 7/4 + 9/4 = 16/4 = 4, which exactly matches the right side of the equation, confirming that θ = 60° is correct. Why Other Options Are Wrong: 30° gives cos² 30° = 3/4 and sin² 30° = 1/4, which leads to 7(3/4) + 3(1/4) = 24/4 = 6, not 4. 45° makes cos² 45° = sin² 45° = 1/2, giving 7/2 + 3/2 = 5, not 4. 90° is not even valid here because cos 90° = 0 which leads to 3, not 4. The option 75° does not produce the required balance either. Common Pitfalls: Common mistakes include forgetting to use sin²θ = 1 − cos²θ, mishandling the algebra when combining terms, or taking cos θ = −1/2 even though the angle is specified as acute. Remember that for acute angles, sine and cosine are both positive, so the positive square root must be chosen for cos θ in this context. Final Answer: Therefore, the required acute angle is 60°.

Question 10

If x = 3^(1/3) − 3^(−1/3), where 3^(1/3) denotes the real cube root of 3, then what is the exact value of the expression 3x³ + 9x?

Accepted Answer

Introduction / Context: This question involves simplifying an expression built from cube roots of 3. It checks your ability to use algebraic identities with terms like t − 1/t and then substitute back into a polynomial expression, here 3x³ + 9x. Such manipulations are common in aptitude and algebra questions involving surds and roots. Given Data / Assumptions: x = 3^(1/3) − 3^(−1/3). All roots are real; let 3^(1/3) be the real cube root of 3. The expression to evaluate is 3x³ + 9x. Basic algebraic expansion rules can be applied safely. Concept / Approach: The key technique is to introduce a simpler symbol for the cube root of 3. Let t = 3^(1/3). Then 3^(−1/3) = 1/t. This allows us to rewrite x as t − 1/t. We then expand (t − 1/t)³ carefully, group like terms, and express x³ in terms of x itself. Finally, we compute 3x³ + 9x using this relation, which removes the surds completely and leaves a rational number. Step-by-Step Solution: Let t = 3^(1/3). Then 3^(−1/3) = 1/t. So x = t − 1/t. Compute x³ = (t − 1/t)³. Expand: (t − 1/t)³ = t³ − 3t(1/t)(t − 1/t) − 1/t³. More directly: (t − 1/t)³ = t³ − 3t + 3(1/t) − 1/t³. But t³ = 3 and 1/t³ = 1/3. So x³ = 3 − 3t + 3/t − 1/3. Combine 3 − 1/3 = 8/3. Notice that t − 1/t = x, so −3t + 3/t = −3(t − 1/t) = −3x. Thus x³ = 8/3 − 3x. Multiply by 3: 3x³ = 8 − 9x. Therefore 3x³ + 9x = 8 − 9x + 9x = 8. Verification / Alternative check: As a rough numerical check, approximate t ≈ 3^(1/3) ≈ 1.442. Then 1/t ≈ 0.693, so x ≈ 1.442 − 0.693 ≈ 0.749. Compute 3x³ + 9x ≈ 3(0.749³) + 9(0.749). This is approximately 3(0.42) + 6.741 ≈ 1.26 + 6.74 ≈ 8.00, which confirms the exact result of 8. Why Other Options Are Wrong: 9, 16 and 27 are common distractors that might arise from incorrectly cubing 3^(1/3) or from ignoring the 3^(−1/3) term. Zero would require 3x³ and 9x to cancel exactly, which does not happen given the algebraic relation derived. Only 8 is consistent with both algebraic derivation and numerical verification. Common Pitfalls: Typical mistakes include expanding (t − 1/t)³ incorrectly, forgetting that 3^(−1/3) = 1/t, or trying to compute everything numerically without recognizing the pattern t − 1/t. Another error is to treat t³ as 1 instead of 3. Always expand step by step and simplify using the definitions of t and 1/t carefully. Final Answer: Hence, the exact value of 3x³ + 9x is 8.

Question 11

If xy(x + y) = m for real numbers x and y, then what is the simplified value of x³ + y³ + 3m in terms of x and y?

Accepted Answer

Introduction / Context: This algebra question focuses on using standard identities for the sum of cubes. It checks whether you recognize how to rewrite x³ + y³ in terms of x + y and xy, and how to incorporate the given expression xy(x + y) = m into the identity. Mastering such identities is very helpful in simplification problems in aptitude and competitive exams. Given Data / Assumptions: xy(x + y) = m. x and y are real numbers. We need x³ + y³ + 3m expressed in simplest form. Standard algebraic identities can be used. Concept / Approach: We recall the identity for the sum of cubes of two numbers: x³ + y³ = (x + y)³ − 3xy(x + y). If we then consider x³ + y³ + 3xy(x + y), the −3xy(x + y) and +3xy(x + y) cancel, leaving just (x + y)³. Since xy(x + y) is given as m, the term 3m is exactly 3xy(x + y). This observation allows a very quick simplification of the whole expression. Step-by-Step Solution: Start with the known identity: x³ + y³ = (x + y)³ − 3xy(x + y). We are asked to evaluate x³ + y³ + 3m. Given that m = xy(x + y), we have 3m = 3xy(x + y). Substitute the identity into the expression: x³ + y³ + 3m = [(x + y)³ − 3xy(x + y)] + 3xy(x + y). The terms −3xy(x + y) and +3xy(x + y) cancel out. So we get x³ + y³ + 3m = (x + y)³. Verification / Alternative check: Take a simple numerical example to validate. Let x = 1 and y = 2. Then x + y = 3 and xy = 2. Here m = xy(x + y) = 2·3 = 6. Now x³ + y³ + 3m = 1³ + 2³ + 3·6 = 1 + 8 + 18 = 27. On the other hand, (x + y)³ = 3³ = 27. Both sides are equal, confirming the correctness of the simplification for at least one concrete pair of values. Why Other Options Are Wrong: m³ would imply cubing xy(x + y), which is not supported by the identity. m(x + y) is dimensionally inconsistent with x³ + y³. The option (x³ + y³)³ is far larger power-wise and does not reduce to our expression. The plain x³ + y³ ignores the +3m term completely. Only (x + y)³ matches the identity-based simplification. Common Pitfalls: Students sometimes confuse the identity for x³ + y³ with that for x³ − y³, or they misremember the factor 3xy(x + y). Another common mistake is to attempt expansion from scratch instead of using the identity, which is longer and prone to error. Recognizing standard patterns like x³ + y³ + 3xy(x + y) saves time and reduces algebraic mistakes. Final Answer: Therefore, the simplified value of x³ + y³ + 3m is (x + y)³.

Question 12

Without assuming specific values of θ and φ, simplify the trigonometric expression (sin θ + sin φ)/(cos θ + cos φ) + (cos θ − cos φ)/(sin θ − sin φ) to its exact value.

Accepted Answer

Introduction / Context: This question tests your understanding of trigonometric identities, especially the sum-to-product and difference-to-product formulas for sine and cosine. The goal is to simplify a seemingly complicated expression involving sums and differences of angles without substituting any particular numerical values for θ and φ. Given Data / Assumptions: Expression: (sin θ + sin φ)/(cos θ + cos φ) + (cos θ − cos φ)/(sin θ − sin φ). θ and φ are angles for which the denominators are non zero. Standard trigonometric identities can be applied (sum and difference formulas). We seek a simplified exact value, not a numerical approximation. Concept / Approach: We use the sum-to-product identities: sin θ + sin φ = 2 sin((θ + φ)/2) cos((θ − φ)/2), cos θ + cos φ = 2 cos((θ + φ)/2) cos((θ − φ)/2), cos θ − cos φ = −2 sin((θ + φ)/2) sin((θ − φ)/2), sin θ − sin φ = 2 cos((θ + φ)/2) sin((θ − φ)/2). By substituting these into the two fractions, most factors cancel, leaving expressions in tan((θ + φ)/2). The two terms then combine in a simple way. Step-by-Step Solution: Let A = (θ + φ)/2 and B = (θ − φ)/2 for convenience. Then sin θ + sin φ = 2 sin A cos B. cos θ + cos φ = 2 cos A cos B. So the first fraction becomes (2 sin A cos B)/(2 cos A cos B) = tan A. Next, cos θ − cos φ = −2 sin A sin B. And sin θ − sin φ = 2 cos A sin B. So the second fraction becomes (−2 sin A sin B)/(2 cos A sin B) = −tan A. Now add the two results: tan A + (−tan A) = 0. Therefore the entire expression simplifies to 0. Verification / Alternative check: You can pick specific angles to confirm the result, provided denominators are non zero. For example, take θ = 60° and φ = 30°. Direct substitution into the original expression using known values of sine and cosine yields a numerical result very close to zero (up to rounding), which supports the algebraic simplification. However, the identity based derivation is exact and holds for all valid θ and φ. Why Other Options Are Wrong: The value 1 would require the two fractions to combine constructively rather than cancel. A value like 2 or 1/2 would suggest a fixed non zero multiple of some basic ratio, which is not supported once the identities are applied. The option −1 would require tan A + (−tan A) to equal −1, which is impossible as they cancel exactly. Only 0 matches the simplification. Common Pitfalls: Learners might try to expand sine and cosine of sums directly, which is messy and error prone, instead of using sum-to-product formulas. Another pitfall is incorrect signs in formulas like cos θ − cos φ, which has a leading negative sign in its identity. Cancelling common factors incorrectly can also lead to wrong non zero answers. Carefully applying the standard identities avoids these issues. Final Answer: Thus, the expression simplifies to the exact value 0.

Question 13

If (3a + 4b)/(3c + 4d) = (3a − 4b)/(3c − 4d) for real numbers a, b, c and d (with denominators non zero), then which of the following relations between a, b, c and d must hold?

Accepted Answer

Introduction / Context: This algebra question explores what happens when two rational expressions involving the same four variables are equal. By cross multiplying and comparing coefficients, we can extract a necessary relation among a, b, c and d. Such questions test the ability to handle symmetrical expressions and factor out common terms effectively. Given Data / Assumptions: (3a + 4b)/(3c + 4d) = (3a − 4b)/(3c − 4d). 3c + 4d ≠ 0 and 3c − 4d ≠ 0 so the fractions are defined. a, b, c and d are real numbers. We need to deduce a relation among a, b, c and d that must always hold when the equality is true. Concept / Approach: The standard approach is to cross multiply the two fractions and simplify the resulting equality. This yields a polynomial equation in a, b, c and d. By expanding both sides and then moving all terms to one side, we can factor or group the terms to reveal the underlying product relation. Often, expressions of this type simplify nicely to something like ad = bc, ab = cd or similar symmetric conditions. Step-by-Step Solution: Given (3a + 4b)/(3c + 4d) = (3a − 4b)/(3c − 4d). Cross multiply: (3a + 4b)(3c − 4d) = (3a − 4b)(3c + 4d). Expand the left side: (3a + 4b)(3c − 4d) = 9ac − 12ad + 12bc − 16bd. Expand the right side: (3a − 4b)(3c + 4d) = 9ac + 12ad − 12bc − 16bd. Set them equal and subtract the right side from the left side: (9ac − 12ad + 12bc − 16bd) − (9ac + 12ad − 12bc − 16bd) = 0. The 9ac and −16bd terms cancel out. We are left with −12ad + 12bc − 12ad + 12bc = 0. This simplifies to −24ad + 24bc = 0. Divide by 24: −ad + bc = 0, so bc = ad. Verification / Alternative check: As a check, choose numbers satisfying ad = bc, for example a = 2, b = 3, c = 4, d = 6 (since 2·6 = 3·4 = 12). Then 3a + 4b = 6 + 12 = 18 and 3c + 4d = 12 + 24 = 36, giving (3a + 4b)/(3c + 4d) = 18/36 = 1/2. On the other side, 3a − 4b = 6 − 12 = −6 and 3c − 4d = 12 − 24 = −12, so (3a − 4b)/(3c − 4d) = (−6)/(−12) = 1/2. The fractions are indeed equal, illustrating that ad = bc works. Why Other Options Are Wrong: ab = cd, ac = bd, and a = b = c ≠ d would not, in general, guarantee equality of the two fractions when checked with arbitrary numeric examples. The condition a + d = b + c is also unrelated to the cross multiplied expression and fails for many sets of values. Only ad = bc emerges directly from algebraic simplification of the given equality. Common Pitfalls: Students sometimes expand incorrectly or forget to move all terms to one side, missing the chance to factor. Another pitfall is cancelling non common terms incorrectly or dividing by expressions that might be zero. Carefully expand, combine like terms and factor out constants before dividing to avoid losing valid solutions. Final Answer: The necessary relation is ad = bc.

Question 14

If cot θ = 4 for an acute angle θ, then what is the value of the expression (5 sin θ + 3 cos θ) / (5 sin θ − 3 cos θ)?

Accepted Answer

Introduction / Context: This trigonometry problem combines a given value of cot θ with an expression involving both sin θ and cos θ. The goal is to convert everything into one trigonometric ratio, use the given information, and then simplify the resulting fraction. Handling such expressions is a common skill tested in aptitude exams. Given Data / Assumptions: cot θ = 4 for an acute angle θ (0° < θ < 90°). Expression to evaluate: (5 sin θ + 3 cos θ) / (5 sin θ − 3 cos θ). Since θ is acute, sin θ > 0 and cos θ > 0. Standard trigonometric relationships and right triangle interpretation may be used. Concept / Approach: We know that cot θ = cos θ / sin θ. If cot θ = 4, we can imagine a right triangle where the adjacent side is 4k and the opposite side is k, for some positive k. Then sin θ and cos θ can be expressed in terms of k and the hypotenuse. An even quicker method is to use the identity sin θ = 1 / √(1 + cot² θ) and cos θ = cot θ · sin θ, which directly give numerical values for sin θ and cos θ. Substituting these into the expression allows us to simplify the ratio fully. Step-by-Step Solution: Given cot θ = 4 = cos θ / sin θ. Compute sin θ using sin θ = 1 / √(1 + cot² θ). cot² θ = 16, so sin θ = 1 / √(1 + 16) = 1 / √17. Then cos θ = cot θ · sin θ = 4 · (1 / √17) = 4 / √17. Compute the numerator: 5 sin θ + 3 cos θ = 5(1 / √17) + 3(4 / √17). This is (5 + 12) / √17 = 17 / √17. Compute the denominator: 5 sin θ − 3 cos θ = 5(1 / √17) − 3(4 / √17). This is (5 − 12) / √17 = −7 / √17. Therefore, (5 sin θ + 3 cos θ) / (5 sin θ − 3 cos θ) = (17 / √17) / (−7 / √17) = −17/7. Verification / Alternative check: We can approximate numerically: sin θ = 1 / √17 ≈ 0.2425 and cos θ = 4 / √17 ≈ 0.9701. The numerator is about 5(0.2425) + 3(0.9701) ≈ 1.2125 + 2.9103 ≈ 4.1228. The denominator is about 5(0.2425) − 3(0.9701) ≈ 1.2125 − 2.9103 ≈ −1.6978. The ratio 4.1228 / (−1.6978) ≈ −2.428 which matches −17/7 ≈ −2.4286, confirming the exact value. Why Other Options Are Wrong: The options 1/9, 1/3, 3 and 9 are all positive, whereas the true value is negative because the denominator turns out to be negative while the numerator is positive. None of those values are close to −17/7 in magnitude either. Only −17/7 matches the result obtained from both exact and approximate calculations. Common Pitfalls: Common mistakes include mixing up cot θ with tan θ, miscomputing sin θ from cot θ, or ignoring the sign of the denominator. Some learners might incorrectly assume the answer must be positive because θ is acute, forgetting that the linear combination in the denominator can still be negative. Carefully computing sin θ and cos θ from cot θ and substituting them avoids these errors. Final Answer: Thus, the value of (5 sin θ + 3 cos θ) / (5 sin θ − 3 cos θ) is −17/7.

Question 15

Solve the linear equation (−3/2) + (2/3)(3x + 9) = x/2 and find the exact value of x.

Accepted Answer

Introduction / Context: This is a standard linear equation with fractional coefficients. It tests your ability to expand brackets, distribute fractions correctly, and then collect like terms to isolate the variable x. Such questions are very common in aptitude tests under the simplification or algebra section. Given Data / Assumptions: Equation: (−3/2) + (2/3)(3x + 9) = x/2. x is a real number. All algebraic operations follow the usual rules of arithmetic. No denominator is zero in the process. Concept / Approach: The equation can be simplified step by step. First expand the product (2/3)(3x + 9) using distribution. This will give a linear expression in x plus a constant term. Then move all x terms to one side and constants to the other. Because the equation is linear, you will end up with a single solution for x. It is often easier to clear fractions early, but even without that, systematic manipulation works well. Step-by-Step Solution: Start from (−3/2) + (2/3)(3x + 9) = x/2. Expand (2/3)(3x + 9): it equals (2/3)·3x + (2/3)·9 = 2x + 6. So the equation becomes (−3/2) + 2x + 6 = x/2. Combine the constant terms on the left: 6 − 3/2 = (12/2 − 3/2) = 9/2. Thus 2x + 9/2 = x/2. Multiply every term by 2 to clear the denominator: 4x + 9 = x. Subtract x from both sides: 3x + 9 = 0. So 3x = −9 and x = −9/3 = −3. Verification / Alternative check: Substitute x = −3 back into the original equation. Then 3x + 9 = 3(−3) + 9 = −9 + 9 = 0, so (2/3)(3x + 9) becomes (2/3)·0 = 0. The left side is then (−3/2) + 0 = −3/2. The right side is x/2 = (−3)/2 = −3/2. Both sides match, confirming that x = −3 satisfies the equation exactly. Why Other Options Are Wrong: If x = −9, then x/2 = −9/2, but the left side becomes much more negative, so it does not balance. Positive values like 11 or 9 make the right side positive, while the left side evaluates to a different number completely. Zero would give −3/2 on the left and 0 on the right. Only x = −3 satisfies the equality. Common Pitfalls: A typical error is distributing (2/3) incorrectly or mishandling the sign of the −3/2 term. Some learners forget to multiply all terms when clearing denominators, causing inconsistencies. Others accidentally move terms across the equality sign without changing their signs. Working slowly and checking each arithmetic step helps avoid such mistakes. Final Answer: Therefore, the solution of the equation is x = −3.

Question 16

Using standard trigonometric ratios, evaluate the exact value of sec 30° + tan 60° and express the result in simplest surd form.

Accepted Answer

Introduction / Context: This problem tests your knowledge of exact trigonometric ratios for special angles, particularly 30° and 60°. It asks you to use those values to evaluate an expression involving secant and tangent, and then simplify it into a compact surd form commonly used in exam answers. Given Data / Assumptions: Expression: sec 30° + tan 60°. Angles are measured in degrees. We use standard exact values for sin, cos and tan at 30° and 60°. No calculators are required; algebraic simplification with surds is enough. Concept / Approach: We recall that sec θ = 1 / cos θ and tan θ = sin θ / cos θ. For 30° and 60°, the cosine and sine values are well known: cos 30° = √3/2, sin 30° = 1/2, sin 60° = √3/2 and cos 60° = 1/2. Using these, we compute sec 30° and tan 60° exactly, then add them and simplify the result into a single fraction with a rational numerator and a surd denominator. Step-by-Step Solution: cos 30° = √3/2, so sec 30° = 1 / cos 30° = 1 / (√3/2) = 2/√3. For 60°, sin 60° = √3/2 and cos 60° = 1/2. So tan 60° = sin 60° / cos 60° = (√3/2) / (1/2) = √3. Now compute sec 30° + tan 60° = 2/√3 + √3. Write √3 as 3/√3 (since 3/√3 = √3). Then 2/√3 + √3 = 2/√3 + 3/√3 = 5/√3. Verification / Alternative check: We can approximate numerically to confirm. cos 30° ≈ 0.866 so sec 30° ≈ 1.155. tan 60° ≈ 1.732. Their sum is about 2.887. Now 5/√3 ≈ 5 / 1.732 ≈ 2.887 as well. The close numerical match confirms that the simplified expression 5/√3 is correct. Why Other Options Are Wrong: (√6 + 1)/√3 and (√3 + 2)/√3 have different numerators and do not evaluate to approximately 2.887. (1 + √3)/2 is around 1.366, which is far from the true value. The option 2√3 is approximately 3.464, again not matching the computed sum. Only 5/√3 agrees with the exact algebraic simplification and the numerical check. Common Pitfalls: Students sometimes confuse sec 30° with sec 60° or mistakenly use tan 30° instead of tan 60°. Another frequent error is to skip the step of rewriting √3 as 3/√3, which can make it harder to combine terms. Carefully recall the exact values for special angles and handle surds step by step to avoid mistakes. Final Answer: Thus, sec 30° + tan 60° has the exact value 5/√3.

Question 17

In triangle PQR, angle Q is 90°. If m∠R = 30°, then using angle sum in a triangle, what is the exact value of cot P?

Accepted Answer

Introduction / Context: This question uses basic properties of a right triangle and standard trigonometric ratios. Given one acute angle in a right triangle, you can determine the other acute angle and then compute the cotangent of that angle using known values from the 30°–60°–90° triangle. It is a straightforward but important concept in trigonometry and geometry. Given Data / Assumptions: Triangle PQR is right angled at Q, so m∠Q = 90°. m∠R = 30°. The sum of interior angles of a triangle is 180°. Cotangent is defined as cot θ = cos θ / sin θ or adjacent/opposite in a right triangle. Concept / Approach: First, use the angle sum property of a triangle: the three interior angles add to 180°. With m∠Q = 90° and m∠R = 30°, we can find m∠P. Once we know that P corresponds to an angle of 60°, we recall the exact trigonometric values for a 30°–60°–90° triangle. The cotangent of 60° is the reciprocal of tan 60°, and standard tables give tan 60° = √3, so cot 60° = 1/√3. Step-by-Step Solution: m∠P + m∠Q + m∠R = 180° for any triangle. Given m∠Q = 90° and m∠R = 30°. So m∠P = 180° − 90° − 30° = 60°. We need cot P = cot 60°. Recall that tan 60° = √3. Hence cot 60° = 1 / tan 60° = 1/√3. Therefore cot P = 1/√3. Verification / Alternative check: In a 30°–60°–90° triangle, side lengths are in the ratio 1 : √3 : 2, with the smallest side opposite 30°, the largest side opposite 90° and the medium side opposite 60°. For angle P = 60°, cot P = adjacent/opposite = (side opposite 30°)/(side opposite 60°) = 1/√3, confirming the value obtained from the tan-cot relationship. Why Other Options Are Wrong: 1/2 and 1/√2 are values associated with cos 60° or sin 45°, not with cot 60°. The value 2 is cot 30°, not cot 60°. √3 is tan 60°, not its reciprocal. Only 1/√3 matches the standard exact value of cot 60° and therefore cot P in this triangle. Common Pitfalls: A common mistake is to assign 30° to P and 60° to R, reversing the angles, which leads to choosing cot 30° = √3 instead of cot 60°. Another error is mixing up tan and cot values for these standard angles. Carefully using the angle sum rule and recalling the exact trigonometric ratios prevents these errors. Final Answer: Thus, cot P in the given triangle is 1/√3.

Question 18

If sec θ = 5/3 for an acute angle θ in a right triangle, then what is the value of cosec θ?

Accepted Answer

Introduction / Context: This trigonometry question links secant and cosecant through sine and cosine. Given sec θ, you can determine cos θ, then use the Pythagorean identity to find sin θ, and finally obtain cosec θ as the reciprocal of sin θ. It is a standard exercise in relating different trigonometric ratios for an acute angle. Given Data / Assumptions: sec θ = 5/3. θ is an acute angle (0° < θ < 90°). sec θ = 1 / cos θ and cosec θ = 1 / sin θ. We use sin² θ + cos² θ = 1. Concept / Approach: From sec θ = 5/3 we immediately get cos θ = 3/5. Then we use the Pythagorean identity sin² θ + cos² θ = 1 to compute sin θ. Because θ is acute, sin θ will be positive. Once sin θ is known, cosec θ is simply the reciprocal of sin θ. We can also interpret this geometrically in a right triangle for additional intuition. Step-by-Step Solution: Given sec θ = 5/3. Recall sec θ = 1 / cos θ, so cos θ = 3/5. Use identity sin² θ + cos² θ = 1. So sin² θ = 1 − cos² θ = 1 − (3/5)². (3/5)² = 9/25, so sin² θ = 1 − 9/25 = 16/25. Since θ is acute, sin θ is positive, so sin θ = 4/5. Now cosec θ = 1 / sin θ = 1 / (4/5) = 5/4. As a decimal, 5/4 = 1.25. Verification / Alternative check: Visualize a right triangle with hypotenuse 5 units and adjacent side 3 units. Then cos θ = 3/5, matching the given sec θ = 5/3. The opposite side by Pythagoras is √(5² − 3²) = √(25 − 9) = √16 = 4, so sin θ = 4/5. The reciprocal of 4/5 is 5/4, confirming the value of cosec θ. As a decimal, this is 1.25, which aligns with the correct option given. Why Other Options Are Wrong: 0.8 is 4/5, the sine itself rather than its reciprocal. 4/3 and 3/4 correspond to other ratios not linked to the computed sine in this context. The decimal 0.8 and the fraction 3/4 are both less than 1, while cosec θ must be greater than or equal to 1 for acute angles. Only 1.25 (which equals 5/4) matches the derived value of cosec θ. Common Pitfalls: Learners often confuse sec θ with cos θ or mistakenly take sec θ as cos θ directly. Another mistake is to forget that for acute angles, we always take the positive square root when evaluating sin² θ. A few may incorrectly invert cos θ instead of sin θ for cosec θ. Following the identity and definitions step by step avoids these pitfalls. Final Answer: Thus, the value of cosec θ is 1.25 (that is, 5/4).

Question 19

Solve the linear equation −3[1 − (x/2)] + 5x/3 = 1/6 and determine the exact value of x.

Accepted Answer

Introduction / Context: This equation combines brackets and fractional coefficients, which makes it a good test of basic algebra skills. You must distribute the negative sign correctly, handle the fractions carefully, and then collect like terms to solve for x. Such questions appear frequently in the simplification and algebra segments of aptitude exams. Given Data / Assumptions: Equation: −3[1 − (x/2)] + 5x/3 = 1/6. x is a real number. Standard arithmetic and distributive laws apply. Fractions are defined (no division by zero occurs). Concept / Approach: The essential steps are to expand the bracket −3[1 − (x/2)], simplify the expression on the left, and then isolate x. One effective strategy is to clear denominators by multiplying the entire equation by the least common multiple of 2, 3 and 6. This converts the equation into one involving integers only, which is easier to solve without arithmetic errors. Step-by-Step Solution: Start with −3[1 − (x/2)] + 5x/3 = 1/6. Expand the bracket: 1 − (x/2). Multiply by −3: −3[1 − (x/2)] = −3 + (3x/2). So the equation becomes −3 + (3x/2) + 5x/3 = 1/6. Now multiply the entire equation by 6 (LCM of 2, 3 and 6): 6(−3) + 6(3x/2) + 6(5x/3) = 6(1/6). This gives −18 + 9x + 10x = 1. Combine like terms: −18 + 19x = 1. So 19x = 19, and x = 19/19 = 1. Verification / Alternative check: Substitute x = 1 back into the original equation. Inside the bracket we get 1 − (1/2) = 1/2. Thus −3[1 − (x/2)] = −3(1/2) = −3/2. Next, 5x/3 = 5(1)/3 = 5/3. So the left side becomes −3/2 + 5/3. With denominator 6, that is −9/6 + 10/6 = 1/6, which is exactly the right side. This confirms that x = 1 is the correct solution. Why Other Options Are Wrong: If x = 2 or x = −2, the left side evaluates to values different from 1/6. Similarly, x = −1 or x = 0 do not satisfy the original equation when substituted. Only x = 1 makes the entire equation balance with the right side of 1/6. Common Pitfalls: Common errors include distributing the negative sign incorrectly inside the bracket, forgetting to multiply every term when clearing denominators, or mishandling the addition of fractions. Another pitfall is to cancel terms prematurely, which can change the structure of the equation. Keeping track of each step with care helps prevent these mistakes. Final Answer: Therefore, the solution to the equation is x = 1.

Question 20

If a + b = 3 and ab = −4 for real numbers a and b, then using the identity for the sum of cubes, what is the value of a³ + b³?

Accepted Answer

Introduction / Context: This algebra problem uses the standard identity relating a³ + b³ to the sum and product of a and b. It shows how you can avoid solving for a and b individually by manipulating symmetric expressions instead. This technique is useful in many competitive exam problems involving sums and products of roots. Given Data / Assumptions: a + b = 3. ab = −4. a and b are real numbers. The goal is to find a³ + b³. Concept / Approach: We use the identity: a³ + b³ = (a + b)³ − 3ab(a + b). Both a + b and ab are given, so we substitute directly into this formula. This avoids the need to find a and b explicitly through solving a quadratic. This method is efficient and reduces the chance of algebraic mistakes. Step-by-Step Solution: Recall the identity: a³ + b³ = (a + b)³ − 3ab(a + b). Given a + b = 3 and ab = −4. Compute (a + b)³ = 3³ = 27. Compute 3ab(a + b) = 3(−4)(3) = −36. Substitute into the formula: a³ + b³ = 27 − [−36]. So a³ + b³ = 27 + 36 = 63. Verification / Alternative check: We can confirm by constructing the quadratic equation whose roots are a and b. Since a + b = 3 and ab = −4, they are roots of t² − 3t − 4 = 0. This factors as (t − 4)(t + 1) = 0, so a and b are 4 and −1 in some order. Then a³ + b³ = 4³ + (−1)³ = 64 − 1 = 63, which matches the identity based computation perfectly. Why Other Options Are Wrong: 36 and 12 could appear if one incorrectly computes (a + b)² or forgets the factor 3 in 3ab(a + b). The value −15 might appear from incorrectly adding 27 and −36. The value 27 corresponds to (a + b)³ alone and ignores the product term. Only 63 is consistent with both the identity and the explicit substitution check using roots 4 and −1. Common Pitfalls: Common errors include using the difference of cubes formula instead of the sum of cubes, misplacing the minus sign in −3ab(a + b), or mistakenly computing (a + b)³. Some learners also try to find a and b directly without realizing that the identity gives a much quicker route. Keeping the correct identity in mind simplifies the problem greatly. Final Answer: Thus, the value of a³ + b³ is 63.

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