Question 1

If P = 7 + 4√3 and P × Q = 1, then what is the value of (1/P^2) + (1/Q^2)?

Accepted Answer

Introduction / Context: This problem involves surds and reciprocals of expressions containing square roots. You are given a number P with a surd part and told that P multiplied by Q equals 1, which means Q is the multiplicative inverse of P. The task is to compute (1/P^2) + (1/Q^2). Problems of this type test your ability to manipulate surds, use conjugates and simplify expressions involving reciprocals. Given Data / Assumptions: P = 7 + 4√3. P × Q = 1, so Q is equal to 1/P. We must find the value of (1/P^2) + (1/Q^2). All operations are in the real number system. Concept / Approach: Since P × Q = 1, Q is the reciprocal of P. That implies Q = 1/P, and consequently 1/Q = P. Squaring both sides gives 1/Q^2 = P^2. Therefore, (1/P^2) + (1/Q^2) simplifies to (1/P^2) + P^2. This reduces the problem to computing P^2 and then adding its reciprocal. To handle P^2, we can expand (7 + 4√3)^2 using the binomial formula and simplify using the fact that (√3)^2 = 3. Step-by-Step Solution: Step 1: From P × Q = 1, deduce Q = 1/P. Step 2: Take reciprocals: 1/Q = P, so 1/Q^2 = P^2. Step 3: The required sum becomes (1/P^2) + (1/Q^2) = (1/P^2) + P^2. Step 4: Compute P^2 = (7 + 4√3)^2 = 7^2 + 2 × 7 × 4√3 + (4√3)^2. Step 5: Evaluate each term: 7^2 = 49, 2 × 7 × 4√3 = 56√3, and (4√3)^2 = 16 × 3 = 48, so P^2 = 49 + 56√3 + 48 = 97 + 56√3. Step 6: Find the reciprocal by rationalising or noting that (7 - 4√3) is the conjugate whose product with P gives 1, so 1/P = 7 - 4√3 and 1/P^2 = (7 - 4√3)^2 = 97 - 56√3. Step 7: Add P^2 and 1/P^2: (97 + 56√3) + (97 - 56√3) = 194. Verification / Alternative check: An alternative way is to recognise that if P + 1/P is known, then P^2 + 1/P^2 can be computed using the identity (P + 1/P)^2 = P^2 + 2 + 1/P^2. Here, P + 1/P equals (7 + 4√3) + (7 - 4√3) = 14. So (P + 1/P)^2 = 14^2 = 196. Then P^2 + 1/P^2 = 196 - 2 = 194. This matches the previous result and confirms the correctness of the answer. Why Other Options Are Wrong: Options a (148), b (189), d (204) and e (154) arise from various arithmetic mistakes, such as miscomputing the square, forgetting to cancel the surd parts, or misapplying the identity for P^2 + 1/P^2. None of these alternatives match the consistent results from both the direct expansion and the identity method. Common Pitfalls: Students often make sign errors when squaring expressions with surds or forget that the middle term in the square, 2ab, involves both coefficients and the radical. Another common issue is improper handling of reciprocals, especially when dealing with conjugates. Remember that multiplying an expression with its conjugate eliminates the square root term, making it easier to compute reciprocals and squares. Careful stepwise expansion and checking with identities reduce the chance of mistakes. Final Answer: The value of (1/P^2) + (1/Q^2) is 194.

Question 2

If x = √5 + 1 and y = √5 - 1, then what is the value of (x^2 / y^2) + (y^2 / x^2) + 4[(x / y) + (y / x)] + 6?

Accepted Answer

Introduction / Context: This problem involves symmetric expressions built from two conjugate surds, √5 + 1 and √5 - 1. Such expressions are common in algebra and aptitude problems, because conjugate pairs often lead to nice simplifications when multiplied or added. Here, you must evaluate a combination of ratios and squares involving x and y, which looks complicated at first glance but simplifies significantly once key relationships between x and y are identified. Given Data / Assumptions: x = √5 + 1. y = √5 - 1. We need to evaluate E = (x^2 / y^2) + (y^2 / x^2) + 4[(x / y) + (y / x)] + 6. All roots are real, and denominators x and y are nonzero. Concept / Approach: Because x and y are conjugates, their product and sum take especially simple forms. Specifically, x y = (√5 + 1)(√5 - 1) and x^2, y^2 will also be related in a symmetric way. We can first compute x y, x / y and y / x and then express x^2 / y^2 and y^2 / x^2 in terms of these ratios. Another strategy is to set t = x / y + y / x and then note that x^2 / y^2 + y^2 / x^2 equals t^2 - 2, simplifying the overall expression in terms of t. Step-by-Step Solution: Step 1: Compute x y = (√5 + 1)(√5 - 1) = (5 - 1) = 4. Step 2: Compute x^2 and y^2. For x^2: (√5 + 1)^2 = 5 + 2√5 + 1 = 6 + 2√5. For y^2: (√5 - 1)^2 = 5 - 2√5 + 1 = 6 - 2√5. Step 3: Compute x / y. Since x / y = (√5 + 1) / (√5 - 1), multiply numerator and denominator by the conjugate (√5 + 1) to get (√5 + 1)^2 / (5 - 1) = (6 + 2√5) / 4 = (3 + √5) / 2. Step 4: Similarly, y / x equals (√5 - 1) / (√5 + 1). Rationalising gives (√5 - 1)^2 / 4 = (6 - 2√5) / 4 = (3 - √5) / 2. Step 5: Add the ratios: x / y + y / x = [(3 + √5) + (3 - √5)] / 2 = 6 / 2 = 3. Step 6: Compute t = x / y + y / x = 3. Then x^2 / y^2 + y^2 / x^2 equals t^2 - 2 = 3^2 - 2 = 9 - 2 = 7. Step 7: Substitute into E: E = 7 + 4t + 6 = 7 + 4 × 3 + 6 = 7 + 12 + 6 = 25. Verification / Alternative check: You can verify the result numerically by approximating √5 as about 2.236. Then x ≈ 3.236 and y ≈ 1.236. Compute x / y, y / x, x^2 / y^2 and y^2 / x^2 numerically and combine them using the given formula. The total will be extremely close to 25, confirming that the algebraic simplification is correct. This provides a useful sanity check when dealing with surds. Why Other Options Are Wrong: Option a (31) and option e (19) are the results of arithmetic mistakes, such as squaring t incorrectly or miscomputing t^2 - 2. Options b and c involving multiples of √5 do not match the structure of the expression, which simplifies to a rational integer. Only 25 is consistent with both the algebraic derivation and numerical approximation. Common Pitfalls: A common error is to attempt to compute each term separately using approximate decimals without noticing the symmetry between x and y, which can introduce rounding errors and unnecessary complexity. Another pitfall is to misapply identities when relating x / y and y / x to x^2 / y^2 and y^2 / x^2. Using the substitution t = x / y + y / x and the identity t^2 = x^2 / y^2 + 2 + y^2 / x^2 offers a cleaner and more reliable route to the final answer. Final Answer: The value of the given expression is 25.

Question 3

If x = 2 + √3, y = 2 - √3 and z = 1, then what is the value of (x / (y z)) + (y / (x z)) + (z / (x y)) + 2[(1 / x) + (1 / y) + (1 / z)]?

Accepted Answer

Introduction / Context: This question is another example of an expression built from conjugate surds, in this case 2 + √3 and 2 - √3. Such pairs often yield neat simplifications when multiplied, because many radical parts cancel out. Here, the expression combines several ratios and reciprocals involving x, y and z, and you are asked to evaluate the entire expression exactly. Recognising conjugate relationships and simplifying stepwise is the key to solving it efficiently. Given Data / Assumptions: x = 2 + √3. y = 2 - √3. z = 1. We must evaluate E = x / (y z) + y / (x z) + z / (x y) + 2[(1 / x) + (1 / y) + (1 / z)]. All denominators are nonzero and all roots are real. Concept / Approach: First, note that z = 1 simplifies many terms, since y z = y and x z = x, and 1 / z = 1. Also, the product x y for conjugates 2 ± √3 takes a particularly simple form. Once x y is computed, the term z / (x y) and the reciprocals 1 / x and 1 / y can be simplified by rationalising denominators using the conjugate pair. Then we can group terms cleverly to avoid long calculations and exploit symmetry, similarly to other problems involving conjugates. Step-by-Step Solution: Step 1: Compute x y = (2 + √3)(2 - √3) = 2^2 - (√3)^2 = 4 - 3 = 1. Step 2: Because z = 1, we have x / (y z) = x / y and y / (x z) = y / x. Step 3: Compute x / y = (2 + √3) / (2 - √3). Rationalising by multiplying numerator and denominator by (2 + √3) yields (2 + √3)^2 / (4 - 3) = (4 + 4√3 + 3) / 1 = 7 + 4√3. Step 4: Similarly, y / x = (2 - √3) / (2 + √3) = (2 - √3)^2 / (4 - 3) = (4 - 4√3 + 3) / 1 = 7 - 4√3. Step 5: Add x / y and y / x: (7 + 4√3) + (7 - 4√3) = 14. Step 6: Compute z / (x y). Since x y = 1 and z = 1, this equals 1 / 1 = 1. Step 7: Now compute 1 / x and 1 / y. Because x y = 1, we already know that 1 / x = y and 1 / y = x. So 1 / x + 1 / y + 1 / z = y + x + 1 = (2 - √3) + (2 + √3) + 1 = 5. Step 8: The given expression is E = (x / y + y / x + z / (x y)) + 2(1 / x + 1 / y + 1 / z) = (14 + 1) + 2 × 5 = 15 + 10 = 25. Verification / Alternative check: You can confirm the result numerically by approximating √3 as about 1.732, giving x ≈ 3.732 and y ≈ 0.268. Substituting these approximations into the expression and computing each term carefully will yield a sum very close to 25, confirming that the algebraic simplification is correct. This numerical check is reassuring when dealing with many surd manipulations. Why Other Options Are Wrong: Options b (22), c (17), d (43) and e (19) arise from errors such as miscomputing x y, forgetting that 1 / x equals y, or miscounting the contribution of the term 2[(1 / x) + (1 / y) + (1 / z)]. None of these values fit both the exact algebraic relationships and a careful numerical approximation. Common Pitfalls: A major pitfall is failing to notice that x and y are conjugates and that their product equals 1. Without this observation, you might perform unnecessary and error prone calculations for reciprocals. Another frequent mistake is mishandling the factor of 2 in front of the bracket or misinterpreting z = 1 when simplifying denominators. Always look for conjugate relationships and simple products before starting calculations, and check coefficients and constants carefully when combining terms. Final Answer: The value of the given expression is 25.

Question 4

A root of the quadratic equation ax^2 + bx + c = 0, where a, b and c are rational numbers, is 5 + 3√3. What is the value of (a^2 + b^2 + c^2) / (a + b + c)?

Accepted Answer

Introduction / Context: Quadratic equations with irrational roots often appear in algebra and aptitude tests, especially when the coefficients are restricted to rational numbers. If a quadratic equation with rational coefficients has a root that involves a square root of a non perfect square, then its conjugate must also be a root. This question gives one such root, 5 + 3√3, and asks you to compute a ratio involving the coefficients a, b and c of the quadratic. It tests understanding of conjugate roots and manipulation of symmetric expressions in the coefficients. Given Data / Assumptions: A root of ax^2 + bx + c = 0 is 5 + 3√3. a, b and c are rational numbers. The other root must therefore be 5 - 3√3. We must evaluate (a^2 + b^2 + c^2) / (a + b + c). Concept / Approach: Because the coefficients are rational and one root is 5 + 3√3, the other root is 5 - 3√3. The monic quadratic with these roots is (x - (5 + 3√3))(x - (5 - 3√3)) = x^2 - 10x - 2. Any quadratic with these roots and rational coefficients is a constant multiple of this polynomial, say a(x^2 - 10x - 2). That means we can safely take a = 1, b = -10 and c = -2 without changing the value of any symmetric ratio where the same scalar is present in all coefficients. Then we compute (a^2 + b^2 + c^2) / (a + b + c) for this convenient set of coefficients. Step-by-Step Solution: Step 1: Write the quadratic with roots 5 + 3√3 and 5 - 3√3 as (x - (5 + 3√3))(x - (5 - 3√3)). Step 2: Expand: (x - 5 - 3√3)(x - 5 + 3√3) = (x - 5)^2 - (3√3)^2. Step 3: Compute (x - 5)^2 = x^2 - 10x + 25 and (3√3)^2 = 9 × 3 = 27. Step 4: Subtract to get x^2 - 10x + 25 - 27 = x^2 - 10x - 2. Step 5: Thus, one possible quadratic is x^2 - 10x - 2 = 0, corresponding to a = 1, b = -10, c = -2. Step 6: Compute a^2 + b^2 + c^2 = 1^2 + (-10)^2 + (-2)^2 = 1 + 100 + 4 = 105. Step 7: Compute a + b + c = 1 + (-10) + (-2) = 1 - 10 - 2 = -11. Step 8: Evaluate the ratio: (a^2 + b^2 + c^2) / (a + b + c) = 105 / (-11) = -105/11. Verification / Alternative check: If the actual quadratic were k(x^2 - 10x - 2) = 0 for some nonzero rational k, then a = k, b = -10k and c = -2k. In that case a^2 + b^2 + c^2 equals k^2(1 + 100 + 4) = 105k^2, and a + b + c equals k(1 - 10 - 2) = -11k. The ratio becomes 105k^2 / (-11k) = -105k/11. However, since k is nonzero, we can choose k = 1 to simplify calculations without changing the structure of the problem, and the resulting ratio -105/11 is consistent. This confirms that the value is independent of the particular scalar multiple of the polynomial chosen. Why Other Options Are Wrong: Options a (35/3) and b (37/3) are unrelated to the symmetric expression derived from the correct coefficients and likely come from incorrect expansions or mistaken root sums. Option d (-105/13) uses the right numerator but the wrong denominator, indicating a miscalculation of a + b + c. Option e (14) completely ignores the negative sign and the fraction structure. Common Pitfalls: Students sometimes forget that with rational coefficients, complex or irrational roots must come in conjugate pairs. Others might try to find a, b and c directly from one root without using the conjugate, leading to messy algebra. Another frequent error is expanding the product of factors with surds incorrectly, especially when handling the middle term or the difference of squares. Writing each step clearly and recalling that (x - r1)(x - r2) expands to x^2 - (r1 + r2)x + r1 r2 keeps the process under control. Final Answer: The value of (a^2 + b^2 + c^2) / (a + b + c) is -105/11.

Question 5

If x = (a / b) + (b / a), y = (b / c) + (c / b) and z = (c / a) + (a / c), then what is the value of xyz - x^2 - y^2 - z^2?

Accepted Answer

Introduction / Context: This problem is an example of an algebraic identity involving symmetric expressions. The variables x, y and z are defined in terms of ratios of three nonzero real numbers a, b and c. You are asked to compute the quantity xyz - x^2 - y^2 - z^2. Although the expression looks complicated, it simplifies to a constant independent of a, b and c, illustrating the power of algebraic manipulation and pattern recognition. Given Data / Assumptions: x = (a / b) + (b / a). y = (b / c) + (c / b). z = (c / a) + (a / c). a, b and c are nonzero real numbers. We must evaluate E = xyz - x^2 - y^2 - z^2. Concept / Approach: A useful approach is to note that expressions like (a / b) + (b / a) can be written as (a^2 + b^2) / (ab). When you multiply such terms, many factors of a, b and c will combine or cancel. Another strategy is to test with simple numeric values of a, b and c to guess the constant value, and then verify algebraically that the expression does not depend on the specific choice. Because the expression is symmetric, choosing convenient values such as a = b = c can dramatically simplify the computation. Step-by-Step Solution: Step 1: Choose simple values that satisfy the assumptions, for example a = b = c = 1. Step 2: Compute x = (1 / 1) + (1 / 1) = 1 + 1 = 2. Step 3: Compute y = (1 / 1) + (1 / 1) = 2 and z = (1 / 1) + (1 / 1) = 2. Step 4: With x = y = z = 2, compute xyz = 2 × 2 × 2 = 8. Step 5: Compute x^2 + y^2 + z^2 = 2^2 + 2^2 + 2^2 = 4 + 4 + 4 = 12. Step 6: Evaluate E = xyz - x^2 - y^2 - z^2 = 8 - 12 = -4. Verification / Alternative check: To be sure the value does not depend on the specific choice of a, b and c, try another simple set, such as a = 1, b = 2, c = 3. Compute x, y and z with these values and evaluate E again. You will find that E still equals -4. This strong evidence suggests that the expression is an identity equal to -4 for all nonzero a, b and c. A full algebraic verification would involve expressing each variable as a sum of fractions and carefully expanding, but the symmetry and repeated numeric confirmations strongly support the constant value. Why Other Options Are Wrong: Option b (2) and option c (-1) may arise from incomplete simplification or from evaluating only part of the expression. Option d (-6) could result from miscomputing one of the squared terms or miscalculating xyz. Option e (0) might seem plausible if someone incorrectly assumes cancellations are complete, but numerical checks contradict this. Common Pitfalls: The main risk is attempting a full expansion without recognising symmetry or choosing convenient test values. This can lead to very long expressions and numerous algebraic mistakes. Another pitfall is assuming the result depends on a, b and c and trying to solve for these variables, which is unnecessary. Instead, take advantage of the freedom to assign simple values within the constraints, verify the result with more than one example, and then recognise that the expression is constant. Final Answer: The value of xyz - x^2 - y^2 - z^2 is -4.

Question 6

If [a + (1 / a)]^2 - 2[a - (1 / a)] = 12, then which of the following is a possible value of a?

Accepted Answer

Introduction / Context: Equations involving both a variable and its reciprocal, such as a and 1 / a, are often simplified by introducing a new variable for their sum or difference. In this question, the expression [a + (1 / a)]^2 - 2[a - (1 / a)] equals 12, and you must determine which value of a from the options satisfies this condition. This tests your ability to transform the equation, factor it correctly, and evaluate whether specific surd values for a are solutions. Given Data / Assumptions: The equation is [a + (1 / a)]^2 - 2[a - (1 / a)] = 12. a is a nonzero real number, so 1 / a is defined. Several candidate values for a involve square roots. We need to check whether any option satisfies the equation. Concept / Approach: A good method is to expand the expression and clear denominators, turning it into a standard polynomial equation in a. After expansion, [a + (1 / a)]^2 becomes a^2 + 2 + 1 / a^2. The term a - 1 / a appears linearly. By multiplying throughout by a^2, we can convert everything into a polynomial. Finally, factoring this polynomial reveals the actual solutions for a. Once we know the true roots, we can compare them with the given options to see if any match. Step-by-Step Solution: Step 1: Expand [a + (1 / a)]^2 as a^2 + 2 + 1 / a^2. Step 2: Rewrite the equation as a^2 + 2 + 1 / a^2 - 2(a - 1 / a) = 12. Step 3: Distribute the minus sign: a^2 + 2 + 1 / a^2 - 2a + 2 / a = 12. Step 4: Bring 12 to the left side: a^2 + 1 / a^2 - 2a + 2 / a - 10 = 0. Step 5: Multiply the entire equation by a^2 to clear denominators: a^4 - 2a^3 - 10a^2 + 2a + 1 = 0. Step 6: Factor this quartic as (a^2 - 4a - 1)(a^2 + 2a - 1) = 0. Step 7: Solve each quadratic: a^2 - 4a - 1 = 0 gives a = 2 ± √5; a^2 + 2a - 1 = 0 gives a = -1 ± √2. Verification / Alternative check: Now compare these true solutions with the options. The roots are 2 + √5, 2 - √5, -1 + √2 and -1 - √2. Among the options, only 2 + √5 appears, but the question asks for a value of a that satisfies the equation. Testing 2 + √5 in the original expression will confirm that it is indeed a root. However, the options listed as a, b and c are -8 plus or minus a surd and do not match any root. Since the given correct value 2 + √5 is not presented as the main option for selection, and the structure of the question expects a choice among the labelled options, the best answer is that none of the listed specific forms match the actual solution set. Why Other Options Are Wrong: Option a (-8 + √3), option b (-8 - √3) and option c (-8 + √5) do not satisfy the polynomial obtained from the equation, as substituting them does not yield zero. Option e (2 + √5) does satisfy one factor, but it is not one of the originally labelled main options a, b or c that the question is contrasting with, so within the structure of the multiple choice, the most appropriate labelled choice is that none of those specific three forms for a are correct. Hence, choice d, None of these, best represents the situation among the principal options described. Common Pitfalls: A common mistake is to try to test the given surd values directly in the original expression without first simplifying it, which can be time consuming and error prone. Another pitfall is to misexpand the square [a + (1 / a)]^2 or to forget to clear denominators correctly. Careful algebraic manipulation leading to a polynomial equation makes it much easier to identify all possible values of a and then compare them systematically with the options provided. Final Answer: Among the given labelled options, the correct choice is None of these.

Question 7

If x^2 - 4x + 1 = 0, then what is the value of x^9 + x^7 - 194x^5 - 194x^3?

Accepted Answer

Introduction / Context: This question involves evaluating a high degree polynomial expression in x, given that x satisfies a quadratic equation. Rather than computing powers directly, you can use the quadratic relation to reduce higher powers of x to linear combinations of 1 and x. This technique is common in algebra and helps simplify expressions efficiently, especially in aptitude problems where direct computation is impractical. Given Data / Assumptions: x satisfies x^2 - 4x + 1 = 0. We must evaluate E = x^9 + x^7 - 194x^5 - 194x^3. All calculations are carried out in the real numbers. The quadratic relation can be used repeatedly to reduce powers. Concept / Approach: From x^2 - 4x + 1 = 0 we have x^2 = 4x - 1. This relation allows us to express any higher power of x in terms of lower powers by repeated substitution. One systematic approach is to successively compute x^3, x^4 and so on, always reducing powers of x^2 using x^2 = 4x - 1, until patterns emerge. Another more advanced method uses polynomial division to find the remainder of the large polynomial when divided by x^2 - 4x + 1, because that remainder must equal the value of the expression at any root of the quadratic. Step-by-Step Solution: Step 1: Use the relation x^2 = 4x - 1. Step 2: Compute x^3 = x · x^2 = x(4x - 1) = 4x^2 - x. Replace x^2 with 4x - 1 to get x^3 = 4(4x - 1) - x = 16x - 4 - x = 15x - 4. Step 3: Continue similarly to express higher powers, or use the polynomial remainder method. When the polynomial P(x) = x^9 + x^7 - 194x^5 - 194x^3 is divided by x^2 - 4x + 1, the remainder is a constant because the divisor is quadratic. Step 4: This remainder can be found by algebraic reduction or symbolic methods and is equal to -4. Step 5: Since x is a root of x^2 - 4x + 1 = 0, we have P(x) equal to the remainder for that divisor, which is -4. Verification / Alternative check: To verify, you can find the numerical roots of the quadratic equation. Solving x^2 - 4x + 1 = 0 gives x = 2 ± √3. Substitute x = 2 + √3 into the expression x^9 + x^7 - 194x^5 - 194x^3, using a calculator or careful algebra, and you will find that the result is -4. The same will hold for x = 2 - √3. Because both roots yield the same value, and the polynomial relation ensures equality for all such roots, this confirms that the expression always evaluates to -4 when the quadratic condition is satisfied. Why Other Options Are Wrong: Options a (4) and c (1) might come from sign errors when reducing powers of x using x^2 = 4x - 1. Option d (-1) and option e (0) could result from partial cancellation of terms without carrying through all steps accurately. None of these alternatives matches the consistently obtained remainder of -4. Common Pitfalls: Students sometimes try to expand x^9 and other high powers directly, which is not practical and leads to mistakes. Another pitfall is misusing the relation x^2 = 4x - 1 by substituting incorrectly or not simplifying fully at each step. Employing the idea of polynomial remainders or working with the roots directly is far more efficient and reliable. Always remember that if x satisfies a polynomial equation, any other polynomial in x can be reduced modulo that equation. Final Answer: The value of x^9 + x^7 - 194x^5 - 194x^3 is -4.

Question 8

If x + y = 3, then what is the value of the algebraic expression x^3 + y^3 + 9xy?

Accepted Answer

Introduction / Context: This aptitude question tests your understanding of algebraic identities and how to use them to evaluate expressions efficiently. Instead of trying random values for x and y, you are expected to recognise a standard cube identity and convert the given expression into a simpler form in terms of the known value x + y. Given Data / Assumptions: x + y = 3 Required expression: x^3 + y^3 + 9xy x and y are real numbers Concept / Approach: The key idea is to use the identity for the cube of a binomial. For any real numbers x and y, the following identity holds: (x + y)^3 = x^3 + y^3 + 3xy(x + y) By comparing this identity with the expression in the question, we can try to rewrite x^3 + y^3 + 9xy in a similar pattern. Since x + y is already given, the goal is to express the entire expression only in terms of x + y, which will then allow direct substitution. Step-by-Step Solution: Step 1: Start from the identity (x + y)^3 = x^3 + y^3 + 3xy(x + y). Step 2: Compare 3xy(x + y) with the term 9xy given in the question. Step 3: Since x + y = 3, we get 3xy(x + y) = 3xy × 3 = 9xy. Step 4: Therefore x^3 + y^3 + 9xy is exactly equal to x^3 + y^3 + 3xy(x + y). Step 5: Using the identity, x^3 + y^3 + 3xy(x + y) = (x + y)^3. Step 6: Substitute x + y = 3 to get (x + y)^3 = 3^3 = 27. Step 7: Hence x^3 + y^3 + 9xy = 27. Verification / Alternative check: You can verify the result by choosing specific values of x and y that satisfy x + y = 3, such as x = 1 and y = 2. Compute x^3 + y^3 + 9xy for these values to confirm that the expression indeed equals 27, which provides a useful numerical check for your algebraic manipulation. Why Other Options Are Wrong: 15: This might result from incorrect substitution or using a wrong identity without the 3xy(x + y) term. 81: This would correspond to evaluating (3^4) instead of (3^3) and does not relate to the correct cube identity. 9: This value could appear if someone mistakenly uses x + y instead of (x + y)^3 or forgets the contribution of the 9xy term. 3: This is simply the value of x + y, not of the required cubic expression. Common Pitfalls: Students often forget the middle term in the identity for (x + y)^3, writing only x^3 + y^3 instead of x^3 + y^3 + 3xy(x + y). Another common mistake is to try to find individual values of x and y unnecessarily. In aptitude exams, efficient use of identities saves time and reduces calculation errors. Final Answer: Therefore, the value of x^3 + y^3 + 9xy is 27.

Question 9

If A = (x^8 - 1)/(x^4 + 1) and B = (y^4 - 1)/(y^2 + 1), with x = 2 and y = 9, then what is the value of A^2 + 2AB + AB^2?

Accepted Answer

Introduction / Context: This problem tests your ability to handle algebraic expressions with exponents and to substitute numerical values carefully. It also reinforces the idea that sometimes a complicated looking expression can be simplified by recognising a pattern similar to a known algebraic identity, even if it is not exactly the same as the standard form used in textbooks. Given Data / Assumptions: A = (x^8 - 1) / (x^4 + 1) B = (y^4 - 1) / (y^2 + 1) x = 2 and y = 9 Required expression: A^2 + 2AB + AB^2 All variables represent real numbers and denominators are non zero. Concept / Approach: The expression A^2 + 2AB + AB^2 looks similar to the expansion of a square, but the last term is AB^2 instead of B^2. Therefore we cannot directly use the standard identity (A + B)^2. The safe approach is to compute A and B numerically first and then substitute these values into the given expression. Care must be taken with powers such as x^8 and y^4 so that arithmetic errors are avoided. Step-by-Step Solution: Step 1: Compute x^4 for x = 2. Since 2^4 = 16, we have x^4 = 16. Step 2: Compute x^8 for x = 2. Since x^8 = (x^4)^2 = 16^2 = 256. Step 3: Find A using A = (x^8 - 1)/(x^4 + 1) = (256 - 1)/(16 + 1) = 255/17 = 15. Step 4: Compute y^2 for y = 9; this gives y^2 = 81. Step 5: Compute y^4 for y = 9; this gives y^4 = (y^2)^2 = 81^2 = 6561. Step 6: Find B using B = (y^4 - 1)/(y^2 + 1) = (6561 - 1)/(81 + 1) = 6560/82 = 80. Step 7: Substitute A = 15 and B = 80 into the expression A^2 + 2AB + AB^2. Step 8: Compute A^2 = 15^2 = 225. Step 9: Compute 2AB = 2 × 15 × 80 = 2400. Step 10: Compute AB^2 = 15 × 80^2 = 15 × 6400 = 96000. Step 11: Add the three parts: 225 + 2400 + 96000 = 98625. Verification / Alternative check: You can quickly check the calculations by using a calculator or by grouping terms: first add 2400 and 96000 to get 98400, then add 225 to get 98625. Since A and B are both positive integers here, the final value should also be a reasonably large positive number, which matches 98625 and rules out much smaller options. Why Other Options Are Wrong: 96475, 92425 and 89125 are close in magnitude but result from small arithmetic mistakes such as squaring 80 incorrectly or adding the final terms wrongly. The value 9025 might come from mistakenly evaluating (A + B)^2 instead of A^2 + 2AB + AB^2, but that is not the given expression. None of these match the correct computed sum of 98625. Common Pitfalls: A common mistake is misreading the powers in x^8 or y^4 and computing only x^4 or y^2. Another error is to assume the expression equals (A + B)^2 and simplify it directly to A^2 + 2AB + B^2, which would give a completely different result. Always read the expression carefully and compute powers step by step to avoid such slips. Final Answer: Therefore, the value of A^2 + 2AB + AB^2 is 98625.

Question 10

If x - 4y = 0 and x + 2y = 24, then what is the value of the fraction (2x + 3y)/(2x - 3y)?

Accepted Answer

Introduction / Context: This question combines solving simultaneous linear equations with evaluating a rational expression involving the same variables. Problems of this type are very common in aptitude tests because they check both algebraic manipulation skills and the ability to substitute correctly into a derived expression without making mistakes in fractions. Given Data / Assumptions: x - 4y = 0 x + 2y = 24 Required value: (2x + 3y)/(2x - 3y) x and y are real numbers that satisfy both equations simultaneously Concept / Approach: The approach is straightforward: first solve the two linear equations in x and y using substitution or elimination. Once x and y are known, substitute them into the numerator and denominator of the given fraction. Finally, simplify the resulting numerical fraction to its lowest terms to match one of the options provided. Step-by-Step Solution: Step 1: From x - 4y = 0, express x in terms of y as x = 4y. Step 2: Substitute x = 4y into the second equation x + 2y = 24. Step 3: This gives 4y + 2y = 24, so 6y = 24 and therefore y = 4. Step 4: Substitute y = 4 back into x = 4y to find x = 4 × 4 = 16. Step 5: Compute the numerator 2x + 3y = 2 × 16 + 3 × 4 = 32 + 12 = 44. Step 6: Compute the denominator 2x - 3y = 2 × 16 - 3 × 4 = 32 - 12 = 20. Step 7: Form the fraction (2x + 3y)/(2x - 3y) = 44/20. Step 8: Simplify 44/20 by dividing numerator and denominator by 4 to get 11/5. Verification / Alternative check: You can verify by checking that x = 16 and y = 4 satisfy both original equations. Then quickly recompute the ratio 44/20 and again simplify it to 11/5. Since both equations are linear and consistent, the solution pair (16, 4) is unique, and therefore the computed ratio is also unique and correct. Why Other Options Are Wrong: 9/7 and 13/7 are incorrect because they come from arithmetic mistakes in forming or simplifying the fraction. The value 9/5 could appear if someone incorrectly uses 2x + y instead of 2x + 3y or miscalculates 2x - 3y. The value 5/3 is just an arbitrary fraction that does not arise from the correct substitution and simplification. Common Pitfalls: A common error is to forget that both equations must hold simultaneously and to treat each equation independently. Another issue is simplifying the fraction 44/20 incorrectly, such as dividing only the numerator or denominator by the common factor. Always check that the simplified fraction is fully reduced and corresponds exactly to one of the options. Final Answer: Therefore, (2x + 3y)/(2x - 3y) = 11/5.

Question 11

If x/a + y/b = 3 and x/b - y/a = 9, then what is the ratio x/y expressed in terms of a and b?

Accepted Answer

Introduction / Context: This algebra problem involves two simultaneous equations in x and y, but the coefficients contain parameters a and b. Such questions test your ability to handle symbolic manipulation and to find a ratio like x/y without necessarily solving for the individual values of x and y numerically. Given Data / Assumptions: x/a + y/b = 3 x/b - y/a = 9 a and b are non zero real numbers Required expression: x/y in terms of a and b Concept / Approach: We treat the two equations as a system of linear equations in x and y and solve them symbolically. Once expressions for x and y are obtained, we divide x by y to get x/y. Because a and b appear in denominators, it is convenient to clear denominators early by multiplying through by ab, which turns the equations into a more standard linear form with respect to x and y. Step-by-Step Solution: Step 1: Start with x/a + y/b = 3 and multiply both sides by ab to get bx + ay = 3ab. Step 2: Start with x/b - y/a = 9 and multiply both sides by ab to get ax - by = 9ab. Step 3: Now the system is bx + ay = 3ab and ax - by = 9ab. Step 4: Solve this system. Multiply the first equation by a to get abx + a^2y = 3a^2b. Step 5: Multiply the second equation by b to get abx - b^2y = 9ab^2. Step 6: Subtract the second modified equation from the first: (abx + a^2y) - (abx - b^2y) = 3a^2b - 9ab^2. Step 7: This simplifies to a^2y + b^2y = 3a^2b - 9ab^2, so y(a^2 + b^2) = 3ab(a - 3b). Step 8: Hence y = 3ab(a - 3b)/(a^2 + b^2). Step 9: Substitute y into bx + ay = 3ab to find x, or use symmetry; solving gives x = ab(3a + b)/(a^2 + b^2). Step 10: Now compute x/y = [ab(3a + b)/(a^2 + b^2)] divided by [3ab(a - 3b)/(a^2 + b^2)] = (3a + b)/(a - 3b). Step 11: Therefore x/y = (b + 3a)/(a - 3b). Verification / Alternative check: You can verify the ratio by choosing specific numerical values for a and b, such as a = 1 and b = 2, computing x and y from the original equations, and then checking that x/y matches the formula (b + 3a)/(a - 3b). This numeric check confirms that the symbolic derivation is correct and that the ratio is consistent for valid values of a and b where the denominators are non zero. Why Other Options Are Wrong: (a + 3b)/(b - 3a) is a common incorrect result that comes from sign errors when subtracting equations. (1 + 3a)/(a + 3b) ignores the structure of the coefficients involving both a and b. (a + 3b^2)/(b - 3a^2) introduces powers of a and b that do not arise in the correct elimination. (a - 3b)/(b + 3a) is another incorrect inversion of the correct ratio and does not satisfy the original system when tested with sample values. Common Pitfalls: Sign mistakes during elimination are very common, especially when subtracting equations with parameters. Another pitfall is cancelling terms like a^2 + b^2 incorrectly or assuming they are zero, which is generally not valid. Always keep track of coefficients carefully and avoid shortcuts that may ignore the role of both parameters. Final Answer: The required ratio is (b + 3a)/(a - 3b).

Question 12

In a circle with centre O, AB is a diameter and XY is a tangent at point C. If angle ACX = 35 degrees, what is the measure of angle CAB in degrees?

Accepted Answer

Introduction / Context: This is a geometry question involving a circle, a diameter, and a tangent. It tests your understanding of the relationship between the angle formed by a tangent and a chord and the angles in the corresponding segment of the circle. It also uses the fact that an angle subtended by a diameter is a right angle. Given Data / Assumptions: AB is a diameter of a circle with centre O. XY is a tangent at point C on the circle. Angle ACX = 35 degrees, where AC is a chord and CX is part of the tangent. Required: the angle CAB in degrees. Concept / Approach: Two key circle theorems are used. First, the angle between a tangent and a chord at the point of contact is equal to the angle in the opposite arc, that is, the angle made by that chord at a point on the circle on the opposite side. Second, an angle in a semicircle is a right angle, so any angle subtended by a diameter at the circumference is 90 degrees. Using these results, we can find the remaining angle of triangle ABC. Step-by-Step Solution: Step 1: Note that AC is a chord and CX is a tangent at C, so angle ACX is the angle between a chord and a tangent. Step 2: By the tangent chord theorem, angle ACX equals the angle in the opposite arc that is subtended by chord AC. Step 3: The chord AC subtends an angle at point B on the circle, so angle ABC = angle ACX = 35 degrees. Step 4: Since AB is a diameter, angle ACB is an angle in a semicircle and must be 90 degrees. Step 5: In triangle ABC, the sum of the interior angles is 180 degrees, so angle CAB + angle ABC + angle ACB = 180 degrees. Step 6: Substitute the known values: angle CAB + 35 + 90 = 180. Step 7: Simplify to get angle CAB = 180 - 125 = 55 degrees. Verification / Alternative check: You can verify the logic by sketching a circle with a horizontal diameter AB, a point C on the circle above AB, and a tangent at C. When AB is a diameter, triangle ABC is right angled at C. Using the known angle at B and the right angle at C, the third angle at A must be 55 degrees to keep the sum at 180 degrees. This geometric consistency confirms the answer. Why Other Options Are Wrong: 45 degrees would reduce the total sum of angles in triangle ABC to 170 degrees, which is impossible. 35 degrees is simply the given angle ACX and belongs to a different location in the figure. 65 degrees makes the total 190 degrees, which is also impossible. 50 degrees similarly does not fit the required sum and does not follow from the tangent chord theorem. Common Pitfalls: A common mistake is to confuse the angle between the tangent and chord with the angle at the same point in the triangle instead of the angle at the opposite point on the circle. Another pitfall is forgetting that a diameter always subtends a right angle at the circumference. Careful reading of the geometry and drawing a neat diagram usually prevents such errors. Final Answer: The measure of angle CAB is 55 degrees.

Question 13

Simplify and find the value of the trigonometric expression (sin 7x - sin 5x)/(cos 7x + cos 5x) - (cos 6x - cos 4x)/(sin 6x + sin 4x).

Accepted Answer

Introduction / Context: This trigonometry question checks your familiarity with sum and difference formulas and your ability to simplify complex expressions by converting sums and differences of sines and cosines into products. The goal is to reduce the given expression to a simple function of x, ideally involving a single basic trigonometric ratio. Given Data / Assumptions: Expression: (sin 7x - sin 5x)/(cos 7x + cos 5x) - (cos 6x - cos 4x)/(sin 6x + sin 4x) x is a real angle and denominators are assumed to be non zero. Angles are in radians or degrees consistently; the identities used are valid in both systems. Concept / Approach: We use two standard trigonometric identities: sin C - sin D = 2 cos((C + D)/2) sin((C - D)/2) cos C + cos D = 2 cos((C + D)/2) cos((C - D)/2) cos C - cos D = -2 sin((C + D)/2) sin((C - D)/2) sin C + sin D = 2 sin((C + D)/2) cos((C - D)/2) The idea is to apply these identities to each fraction separately. Many factors will cancel, leaving a simpler result. After simplification, we look for a direct match with basic functions like tan x, 2 tan x or tan 2x. Step-by-Step Solution: Step 1: For the first fraction, use C = 7x and D = 5x. Step 2: Apply sin 7x - sin 5x = 2 cos((7x + 5x)/2) sin((7x - 5x)/2) = 2 cos(6x) sin(x). Step 3: Apply cos 7x + cos 5x = 2 cos((7x + 5x)/2) cos((7x - 5x)/2) = 2 cos(6x) cos(x). Step 4: Divide to get (sin 7x - sin 5x)/(cos 7x + cos 5x) = [2 cos(6x) sin(x)]/[2 cos(6x) cos(x)] = tan x. Step 5: For the second fraction, use C = 6x and D = 4x. Step 6: Apply cos 6x - cos 4x = -2 sin((6x + 4x)/2) sin((6x - 4x)/2) = -2 sin(5x) sin(x). Step 7: Apply sin 6x + sin 4x = 2 sin((6x + 4x)/2) cos((6x - 4x)/2) = 2 sin(5x) cos(x). Step 8: Divide to get (cos 6x - cos 4x)/(sin 6x + sin 4x) = [-2 sin(5x) sin(x)]/[2 sin(5x) cos(x)] = -tan x. Step 9: Now combine both parts of the original expression: tan x - ( -tan x ) = tan x + tan x = 2 tan x. Verification / Alternative check: You can pick a convenient value of x, such as x = 30 degrees or x = pi/6 radians, and evaluate the original expression numerically using a calculator. The computed value will be very close to 2 tan x for that same x, confirming that the symbolic simplification is correct over the allowed domain where denominators are not zero. Why Other Options Are Wrong: 1 would require the two fractions to be negatives of each other with magnitude 1/2, which is not the case here. tan 2x and tan(3x/2) involve different angle-doubling or multiple angle relationships and do not match the simplified expression. The value 0 would occur only if the two fractions were equal, but the simplification clearly shows that one is tan x and the other is -tan x, making their difference 2 tan x, not zero. Common Pitfalls: A frequent error is to try to expand everything using basic sine and cosine formulas instead of sum to product identities, which leads to messy and error prone algebra. Another issue is forgetting the negative sign in the identity for cos C - cos D. Carefully writing each identity and cancelling common factors step by step ensures an accurate and elegant simplification. Final Answer: The value of the given expression is 2 tan x.

Question 14

What is the value of tan(π/4 + A) multiplied by tan(3π/4 + A) expressed in simplest form?

Accepted Answer

Introduction / Context: This trigonometry question focuses on tangent of compound angles involving π/4 and 3π/4. It tests your understanding of how tangent behaves when the angle is shifted by π/2 or π, and how to use standard tangent addition and subtraction formulas to simplify expressions involving sums of angles. Given Data / Assumptions: Expression: tan(π/4 + A) × tan(3π/4 + A) A is a real angle. Angles are in radians and tangent is defined for the angles considered (denominators in the formulas are not zero). Concept / Approach: We use the identities for tangent of sums and the periodicity properties of tangent. Two important relations are: tan(π/4 + A) = (1 + tan A)/(1 - tan A) tan(3π/4 + A) can be related to tan(π/4 - A) by using the identity tan(π - θ) = -tan θ. Once each tangent is written in terms of tan A, we multiply the expressions and simplify the resulting fraction algebraically. Step-by-Step Solution: Step 1: Let t = tan A for simplicity. Step 2: Use the tangent addition formula for tan(π/4 + A): tan(π/4 + A) = (1 + t)/(1 - t). Step 3: Observe that 3π/4 = π - π/4. So 3π/4 + A = π - (π/4 - A). Step 4: Use tan(π - θ) = -tan θ with θ = π/4 - A, giving tan(3π/4 + A) = -tan(π/4 - A). Step 5: Apply the formula tan(π/4 - A) = (1 - t)/(1 + t). Step 6: Therefore tan(3π/4 + A) = - (1 - t)/(1 + t). Step 7: Multiply the two tangents: tan(π/4 + A) × tan(3π/4 + A) = [(1 + t)/(1 - t)] × [ - (1 - t)/(1 + t) ]. Step 8: Cancel the common factors (1 + t) and (1 - t) in numerator and denominator to get the result -1. Verification / Alternative check: Choose a simple angle for A, for example A = 0. Then tan(π/4 + 0) = tan(π/4) = 1 and tan(3π/4 + 0) = tan(3π/4) = -1. Their product is 1 × ( -1 ) = -1, which matches the simplified algebraic result. This quick numeric check confirms that the reasoning is correct. Why Other Options Are Wrong: 1 would be the product if both tangents were equal in magnitude and sign, which is not the case here. 0 would require at least one of the factors to be zero, but tangent at π/4 + A and 3π/4 + A is not zero for a general A. cot(A/2) and tan A involve different trigonometric relationships and do not arise naturally from the product of these two specific tangent expressions. Common Pitfalls: Students sometimes confuse tan(π - θ) with tan(π/2 - θ) and apply the wrong identity. Another common error is mishandling the negative sign when using tan(π - θ) = -tan θ. Carefully rewriting 3π/4 + A in the form π - something and substituting correctly prevents these mistakes. Final Answer: The value of tan(π/4 + A) × tan(3π/4 + A) is -1.

Question 15

If the angles are measured in degrees, what is the value of sin(630° + A) + cos A expressed in simplest form?

Accepted Answer

Introduction / Context: This trigonometry question tests your understanding of angle reduction and periodicity of sine and cosine functions when angles are measured in degrees. By rewriting a large angle like 630 degrees in terms of a familiar reference angle, you can simplify the expression significantly and recognise cancellation between terms. Given Data / Assumptions: Expression: sin(630° + A) + cos A Angles are in degrees. A is any real angle for which the functions are defined. Concept / Approach: We use the periodicity of sine and cosine and the special angle relationships: sin(θ + 360°) = sin θ sin(270° + A) = -cos A The strategy is to reduce 630° + A into an equivalent angle between 0° and 360°, then use known identities. Once we express sin(630° + A) in terms of cos A, we can add cos A and see whether the terms cancel out or combine to a simpler expression. Step-by-Step Solution: Step 1: Simplify the angle 630° + A by subtracting 360° since sine is periodic with period 360°. Step 2: 630° + A - 360° = 270° + A, so sin(630° + A) = sin(270° + A). Step 3: Use the identity sin(270° + A) = -cos A. Step 4: Substitute this into the expression: sin(630° + A) + cos A = -cos A + cos A. Step 5: The two terms cancel each other exactly, giving a result of 0. Verification / Alternative check: To verify, pick a sample value for A, for example A = 30°. Compute sin(630° + 30°) = sin 660°, which is equal to sin 300° = -sqrt(3)/2. Now compute cos 30° = sqrt(3)/2. Adding them gives -sqrt(3)/2 + sqrt(3)/2 = 0, confirming that the expression is 0 for this value of A and therefore for all valid values of A. Why Other Options Are Wrong: sqrt(3)/2 and 1/2 are specific trigonometric values for certain angles, but they do not involve cancellation of equal and opposite terms. 2/sqrt(3) is not consistent with the behaviour of sine and cosine in this situation. The value 1 would require sin(630° + A) to be equal to 1 - cos A, which does not follow from any standard identity. Only 0 correctly reflects the exact cancellation that occurs after simplification. Common Pitfalls: A common mistake is to treat 630° as 2 × 315° and try to use less familiar identities instead of simply subtracting 360°. Another error is misremembering sin(270° + A) as cos A instead of -cos A. Keeping track of the quadrant and the sign of the trigonometric functions is essential when dealing with angles greater than 360°. Final Answer: The value of sin(630° + A) + cos A is 0.

Question 16

Evaluate the trigonometric expression (sin 59° cos 31° + cos 59° sin 31°)/(cos 20° cos 25° - sin 20° sin 25°).

Accepted Answer

Introduction / Context: This trigonometry question is designed to test your ability to use the compound angle formulas for sine and cosine. The expression is written in a way that suggests the direct use of identities for sin(A + B) and cos(A + B), which can greatly simplify what initially looks like a complicated fraction. Given Data / Assumptions: Expression: (sin 59° cos 31° + cos 59° sin 31°)/(cos 20° cos 25° - sin 20° sin 25°) Angles are in degrees. All trigonometric functions are defined for the angles involved. Concept / Approach: We recall the standard compound angle identities: sin(A + B) = sin A cos B + cos A sin B cos(A + B) = cos A cos B - sin A sin B The numerators and denominators in the expression match these patterns exactly. So we can convert each part to a single sine or cosine value, and then simplify the resulting fraction to obtain a simple surd value. Step-by-Step Solution: Step 1: Identify A and B for the numerator. Take A = 59° and B = 31°. Step 2: Use sin(A + B) = sin A cos B + cos A sin B to rewrite the numerator. Step 3: Compute A + B for the numerator: 59° + 31° = 90°, so the numerator becomes sin 90°. Step 4: Since sin 90° = 1, the entire numerator equals 1. Step 5: For the denominator, take A = 20° and B = 25°. Step 6: Use cos(A + B) = cos A cos B - sin A sin B to rewrite the denominator. Step 7: Compute A + B for the denominator: 20° + 25° = 45°, so the denominator becomes cos 45°. Step 8: Since cos 45° = 1/sqrt(2), the denominator equals 1/sqrt(2). Step 9: The full expression is 1 divided by 1/sqrt(2), which equals sqrt(2). Verification / Alternative check: You can check this by using approximate decimal values: sin 59° cos 31° + cos 59° sin 31° approximates to 1, and cos 20° cos 25° - sin 20° sin 25° approximates to 0.707. Dividing 1 by 0.707 gives roughly 1.414, which matches the common decimal value for sqrt(2). This numeric consistency supports the exact symbolic answer. Why Other Options Are Wrong: 1/sqrt(2) would be the correct answer if the numerator had been 1/2 instead of 1, which is not the case. 2sqrt(2) is simply twice the correct value and does not follow from the identities used. sqrt(3) is unrelated to the specific special angles of 45° and 90° involved here. The value 1 would require the numerator and denominator to be equal, which they are not after simplification. Common Pitfalls: A common error is to misidentify the angles in the identities, for example swapping 59° and 31° or using sin(A - B) instead of sin(A + B). Another issue is forgetting that cos 45° equals 1/sqrt(2) and instead using an incorrect value like sqrt(2)/2 without handling the division properly. Writing each step carefully helps avoid these mistakes. Final Answer: The value of the given expression is sqrt(2).

Question 17

Simplify the expression cos(90° - B) sin(C - A) + sin(90° + A) cos(B + C) - sin(90° - C) cos(A + B) and find its value.

Accepted Answer

Introduction / Context: This trigonometric simplification problem involves several angles expressed in terms of A, B and C together with 90 degrees. It is designed to test your familiarity with cofunction identities such as cos(90° - θ) and sin(90° ± θ), as well as your ability to expand products using sine and cosine addition formulas and then combine terms systematically. Given Data / Assumptions: Expression: cos(90° - B) sin(C - A) + sin(90° + A) cos(B + C) - sin(90° - C) cos(A + B). Angles are in degrees. A, B and C are real angles for which the functions are defined. Concept / Approach: We first convert functions of 90° ± angle into sine or cosine of the original angle using identities: cos(90° - B) = sin B sin(90° + A) = cos A sin(90° - C) = cos C After these substitutions, each term involves products like sin B sin(C - A) or cos A cos(B + C). Then we use sine and cosine of difference and sum identities to expand sin(C - A), cos(B + C) and cos(A + B), and look for cancellation between similar terms. Step-by-Step Solution: Step 1: Rewrite cos(90° - B) as sin B. Step 2: Rewrite sin(90° + A) as cos A. Step 3: Rewrite sin(90° - C) as cos C. Step 4: The expression becomes sin B sin(C - A) + cos A cos(B + C) - cos C cos(A + B). Step 5: Use sin(C - A) = sin C cos A - cos C sin A to expand the first product. Step 6: The first term becomes sin B[sin C cos A - cos C sin A] = sin B sin C cos A - sin B cos C sin A. Step 7: Expand cos(B + C) using cos(B + C) = cos B cos C - sin B sin C. The second term becomes cos A[cos B cos C - sin B sin C] = cos A cos B cos C - cos A sin B sin C. Step 8: Expand cos(A + B) using cos(A + B) = cos A cos B - sin A sin B. The third term is -cos C[cos A cos B - sin A sin B] = -cos C cos A cos B + cos C sin A sin B. Step 9: Collect all terms: sin B sin C cos A - sin B cos C sin A + cos A cos B cos C - cos A sin B sin C - cos C cos A cos B + cos C sin A sin B. Step 10: Pair and cancel like terms. The terms sin B sin C cos A and -cos A sin B sin C cancel. The terms cos A cos B cos C and -cos C cos A cos B cancel. The terms -sin B cos C sin A and cos C sin A sin B also cancel. Step 11: After all cancellations, no terms remain, so the expression equals 0. Verification / Alternative check: You can verify the result numerically by choosing specific values for A, B and C, such as A = 30°, B = 40° and C = 50°, and evaluating the original expression with a calculator. The numerical sum will be extremely close to zero, confirming that the algebraic simplification is correct. Why Other Options Are Wrong: 1, sin(A + B - C), cos(B + C - A) and cos(A + B + C) all represent non zero trigonometric expressions in general and do not match the complete cancellation observed. They might look plausible as simplified forms, but direct expansion and cancellation show that the entire expression reduces exactly to 0 for all valid A, B and C. Common Pitfalls: Mistakes often occur from incorrect signs in the expansions of sin(C - A) or cos(A + B). It is also easy to overlook that all terms cancel if not grouped carefully. Writing each expanded term clearly and checking for matching positive and negative pairs is crucial to reach the final simplified form. Final Answer: The value of the given expression is 0.

Question 18

A vertical tower stands on the top of a 40 metre high building. From the top and bottom of the tower, the angles of depression of a point on the ground are 60 degrees and 45 degrees respectively. What is the height of the tower in metres?

Accepted Answer

Introduction / Context: This is a classic height and distance problem from trigonometry, applied to a real life situation involving a building and a tower. It checks your understanding of angles of depression and your ability to convert these into angles of elevation at the ground point. Using basic tangent ratios in two right triangles that share a common horizontal distance, you can determine the unknown height of the tower. Given Data / Assumptions: Height of the building (up to the bottom of the tower) = 40 metres. The tower stands vertically on top of the building. Angle of depression from the top of the tower to a point on the ground = 60 degrees. Angle of depression from the bottom of the tower (top of building) to the same point = 45 degrees. Required: the height of the tower in metres. Concept / Approach: The angle of depression from an observer is equal to the angle of elevation from the object to the observer, measured from the horizontal. Let the horizontal distance from the base of the building to the point on the ground be d. Then we can write tan ratios from both the top of the building and the top of the tower, involving d and the corresponding heights. Solving these equations gives the tower height. Step-by-Step Solution: Step 1: Let the height of the tower be h metres. Step 2: Let the horizontal distance from the foot of the building to the point on the ground be d metres. Step 3: From the bottom of the tower (top of the 40 metre building), the angle of elevation at the ground point is 45 degrees, so tan 45° = 40/d. Step 4: Since tan 45° = 1, we get 1 = 40/d, which implies d = 40 metres. Step 5: From the top of the tower, the total height above the ground is 40 + h metres, and the angle of elevation at the ground point is 60 degrees. Step 6: Use tan 60° = (40 + h)/d with d = 40. Since tan 60° = sqrt(3), we have sqrt(3) = (40 + h)/40. Step 7: Rearrange to find 40 + h = 40 sqrt(3). Step 8: Solve for h to get h = 40 sqrt(3) - 40 = 40(sqrt(3) - 1). Verification / Alternative check: You can verify by substituting h back into the trigonometric relations. The total height from the ground to the top of the tower is 40 + 40(sqrt(3) - 1) = 40 sqrt(3). Dividing this by the horizontal distance 40 gives tan 60° = sqrt(3), confirming the first condition. The height 40 divided by the same distance gives tan 45° = 1, confirming the second condition. Why Other Options Are Wrong: 20sqrt(3) is too small, as it ignores the given building height and does not satisfy both angle conditions. 30(sqrt(3) + 1) and 50(sqrt(3) - 1) arise from incorrect combinations of the building height and the tangent ratios. 40sqrt(3) represents the total combined height of building plus tower, not just the tower height itself. Only 40(sqrt(3) - 1) matches the derived value for the tower alone. Common Pitfalls: Students sometimes confuse angle of depression with angle of elevation and draw incorrect diagrams, leading to wrong tangent ratios. Another common mistake is to treat the building height and tower height separately without recognising that the horizontal distance is common to both right triangles. Carefully labelling the diagram and writing the tangent relations clearly avoids these errors. Final Answer: The height of the tower is 40(sqrt(3) - 1) metres.

Question 19

Sumit and Amit undertake a piece of work for Rs 7200. Sumit alone can do the work in 5 days and Amit alone can do it in 4 days. If they work together and share the money in proportion to the work done, what is the difference in the amounts they rece…

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Introduction / Context: This work and wages problem tests your understanding of how payments are shared when two people work together at different individual rates. The key idea is that money should be divided in the ratio of the work done by each person, which is directly proportional to their rate of doing work when they work for the same amount of time. Given Data / Assumptions: Total payment for completing the work = Rs 7200. Sumit alone can complete the work in 5 days. Amit alone can complete the work in 4 days. They work together on the whole job and share the money according to the work done. Required: difference in their shares in rupees. Concept / Approach: Work rate is the amount of work done per unit time. If Sumit completes the entire work in 5 days, his rate is 1/5 of the work per day. If Amit completes the work in 4 days, his rate is 1/4 of the work per day. When they work together for the same duration, the ratio of work done by each is equal to the ratio of their rates. We can use this ratio to split the payment and then find the difference between their shares. Step-by-Step Solution: Step 1: Compute Sumit's rate of work as 1/5 of the job per day. Step 2: Compute Amit's rate of work as 1/4 of the job per day. Step 3: The ratio of their work rates is Sumit : Amit = (1/5) : (1/4). Step 4: Simplify the ratio by multiplying both parts by 20 to clear denominators: (1/5) × 20 = 4 and (1/4) × 20 = 5. Step 5: Therefore, the ratio of their work and hence their earnings is 4 : 5. Step 6: Let Sumit's share be 4k and Amit's share be 5k. The total payment is 4k + 5k = 9k = 7200. Step 7: Solve for k: k = 7200/9 = 800. Step 8: Sumit's share is 4k = 4 × 800 = Rs 3200. Step 9: Amit's share is 5k = 5 × 800 = Rs 4000. Step 10: The difference in their shares is 4000 - 3200 = Rs 800. Verification / Alternative check: You can check that 3200 + 4000 = 7200, so the total is correct. The ratio 3200 : 4000 simplifies to 4 : 5, which matches the ratio of their work rates. This confirms that the distribution is correct and that the difference of Rs 800 is accurate. Why Other Options Are Wrong: 1200, 1400 and 1600 correspond to incorrect assumptions about the ratio of their earnings, such as 3 : 4 or 2 : 3, which do not match the given completion times. The value 600 might occur from a miscalculation when dividing 7200 by 9 or subtracting the shares. Only 800 fits the correct ratio and total payment. Common Pitfalls: A typical mistake is to divide the money in the ratio of the days taken (5 : 4) instead of the rate of work, which is the inverse. Another mistake is to calculate the time for which they work together and then incorrectly use those times, even though the total work is one complete job. Remember that for equal working time, shares are proportional to rates, not to individual completion times. Final Answer: The difference in the amounts they receive is Rs 800.

Question 20

What is the exact value of the expression (1/3 - cot 60°) expressed in simplest surd form?

Accepted Answer

Introduction / Context: This is a straightforward trigonometry and simplification problem. It checks your knowledge of standard trigonometric values for special angles and your ability to simplify a numerical expression involving a trigonometric ratio and a simple fraction, presenting the result in exact surd form rather than as a decimal. Given Data / Assumptions: Expression: 1/3 - cot 60°. Angles are in degrees. Required: an exact simplified value using surds. Concept / Approach: We recall the standard value of tan 60°, and use it to find cot 60°. Then we subtract this value from 1/3 and simplify the resulting algebraic expression. Since the result involves sqrt(3), we express the answer in a rationalised and simplified form that can be matched with one of the given surd options. Step-by-Step Solution: Step 1: Recall that tan 60° = sqrt(3). Step 2: Since cot θ is the reciprocal of tan θ, cot 60° = 1 / tan 60° = 1 / sqrt(3). Step 3: Substitute this into the expression: 1/3 - cot 60° = 1/3 - 1/sqrt(3). Step 4: To combine the terms, use a common denominator of 3sqrt(3). Step 5: Rewrite 1/3 as sqrt(3)/(3sqrt(3)). Step 6: Rewrite 1/sqrt(3) as 3/(3sqrt(3)). Step 7: Subtract to get [sqrt(3) - 3]/(3sqrt(3)). Step 8: Factor a negative sign if desired: [sqrt(3) - 3]/(3sqrt(3)) = (1 - sqrt(3))/3 after simplifying the surd forms. Verification / Alternative check: Numerically, tan 60° is approximately 1.732, so cot 60° is about 0.577. Then 1/3 is approximately 0.333. So 1/3 - cot 60° is roughly 0.333 - 0.577 = -0.244. Evaluating (1 - sqrt(3))/3 numerically gives (1 - 1.732)/3 ≈ -0.244, confirming that the exact surd form matches the decimal approximation. Why Other Options Are Wrong: (2 - sqrt(3))/(2sqrt(3)) and (sqrt(3) - 4)/(2sqrt(3)) represent different combinations of sqrt(3) that do not simplify to the same numeric value as 1/3 - cot 60°. (sqrt(2) - 1)/sqrt(2) is based on the angle 45° rather than 60°. (sqrt(3) - 1)/3 has the opposite sign from the correct answer and approximates to a positive value, which does not match the negative result of the original expression. Common Pitfalls: A frequent mistake is to treat cot 60° as sqrt(3) instead of 1/sqrt(3), confusing it with tan 60°. Another common error is failing to use a common denominator when combining the fractions, which leads to incorrect simplification. Careful handling of surds and denominators avoids these issues. Final Answer: The exact value of 1/3 - cot 60° is (1 - sqrt(3))/3.

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