Question 1

If x + 1 / x = 2, find the value of x^21 + 1 / x^21.

Accepted Answer

Introduction / Context: This problem looks complex because it asks for x^21 + 1 / x^21, but the given condition x + 1 / x = 2 is very special and makes the evaluation easy. Recognising when a condition forces x to take a simple value is an important skill in algebra and aptitude questions. In this case, the expression simplifies dramatically once you solve for x. Given Data / Assumptions: x + 1 / x = 2. x is nonzero so that 1 / x is defined. You must find x^21 + 1 / x^21. Concept / Approach: Start by solving the equation x + 1 / x = 2 for x. Multiplying both sides by x gives a quadratic equation. In this particular case, the quadratic has a repeated root, which makes x extremely simple. Once you know x, raising x to any positive integer power and adding its reciprocal becomes trivial. Step-by-Step Solution: Step 1: Begin with x + 1 / x = 2.Step 2: Multiply both sides by x (x is nonzero): x^2 + 1 = 2x.Step 3: Rearrange to get x^2 − 2x + 1 = 0.Step 4: Factor the quadratic: (x − 1)^2 = 0.Step 5: Therefore, x = 1 is the only solution.Step 6: Compute x^21 when x = 1: x^21 = 1^21 = 1.Step 7: Compute 1 / x^21. Since x = 1, 1 / x^21 = 1 / 1 = 1.Step 8: Add the two values: x^21 + 1 / x^21 = 1 + 1 = 2. Verification / Alternative check: Because the equation x + 1 / x = 2 has a unique solution x = 1, there is no ambiguity. You can check the original condition by substituting x = 1: 1 + 1 / 1 = 1 + 1 = 2, which matches. Therefore, any expression in terms of x can be evaluated simply by substituting x = 1, and no further checks are necessary. Why Other Options Are Wrong: The values 0, 1, 3, and 21 might appear plausible if one incorrectly applies identities for powers of x + 1 / x or miscalculates higher powers. However, once x is known to be 1, all positive integer powers of x are 1, and the sum with its reciprocal is exactly 2. Any other answer contradicts this simple observation. Common Pitfalls: Some learners try to use recurrence relations or power formulas for x + 1 / x without noticing that the quadratic equation has the trivial root x = 1. This leads to unnecessary work and can cause errors. Always check first whether the given condition directly fixes x to a simple number like 1 or −1. Final Answer: The value of x^21 + 1 / x^21 is 2.

Question 2

When x + 1 / x = 5, find the value of x − 1 / x.

Accepted Answer

Introduction / Context: This problem illustrates how to move between x + 1 / x and x − 1 / x using algebraic identities. You are given the sum of x and its reciprocal and asked to find their difference. This type of question appears often in algebra sections of aptitude exams and rewards a clear understanding of squared expressions and how they relate to each other. Given Data / Assumptions: x + 1 / x = 5. x is nonzero so the reciprocal 1 / x is defined. You must find x − 1 / x. x may be positive or negative, so both signs are considered where appropriate. Concept / Approach: The key tool is to square x + 1 / x and relate it to x^2 + 1 / x^2. A similar relation holds for x − 1 / x. By first finding x^2 + 1 / x^2 from the given sum and then using another identity for (x − 1 / x)^2, you can solve for x − 1 / x up to a sign. Because the original equation does not fix the sign of x, the answer is naturally a plus or minus value. Step-by-Step Solution: Step 1: Start from x + 1 / x = 5.Step 2: Square both sides: (x + 1 / x)^2 = 5^2 = 25.Step 3: Expand the left side: (x + 1 / x)^2 = x^2 + 2 + 1 / x^2.Step 4: Therefore x^2 + 1 / x^2 + 2 = 25, so x^2 + 1 / x^2 = 25 − 2 = 23.Step 5: Now consider (x − 1 / x)^2. This expands to x^2 − 2 + 1 / x^2.Step 6: Substitute x^2 + 1 / x^2 = 23 into this expression: (x − 1 / x)^2 = 23 − 2 = 21.Step 7: Take square roots: x − 1 / x = ± √21. Verification / Alternative check: You can verify by solving the original equation for x. Multiply x + 1 / x = 5 by x to get x^2 − 5x + 1 = 0. This quadratic has two real roots, one greater than 1 and one between 0 and 1. For each root, compute x − 1 / x numerically; you will obtain the positive and negative values of √21, confirming the plus and minus form of the answer. Why Other Options Are Wrong: The numbers 11 and 21 would only occur if one mistakenly added instead of subtracting in the squared expressions or ignored the identities. The choice ± √22 comes from miscalculating x^2 + 1 / x^2 as 24 instead of 23. The value 0 would require x = 1 / x, that is x^2 = 1, which does not satisfy x + 1 / x = 5. Only ± √21 is consistent with the algebraic relations derived. Common Pitfalls: Students often forget the middle term when squaring x + 1 / x, writing x^2 + 1 / x^2 instead of x^2 + 2 + 1 / x^2. Another common error is to forget that taking the square root yields both positive and negative results, not just one sign. Being careful with these details is crucial for correct answers in exam settings. Final Answer: The value of x − 1 / x, given x + 1 / x = 5, is ± √21.

Question 3

If 27^x + 27^{x − 1/3} = 972, find the value of x.

Accepted Answer

Introduction / Context: This exponential equation question tests your ability to factor out a common power and to recognise numbers written as powers of a smaller base. The base 27 suggests that we should use base 3, since 27 is 3^3, and that the equation can be simplified significantly by a substitution. This is a standard pattern in exponent problems in aptitude tests. Given Data / Assumptions: The equation is 27^x + 27^{x − 1/3} = 972. All exponents and numbers are real and positive. You must determine the real value of x that satisfies this equation. Concept / Approach: The main strategy is to factor out the smaller exponential term. Specifically, you can factor out 27^{x − 1/3} from both terms on the left. This gives a simple linear expression in terms of that common factor. Then, by writing 27 and 972 as powers of 3, you can solve for x using basic rules of exponents. Recognising 972 as 27 times 36 and 243 as 3^5 is helpful. Step-by-Step Solution: Step 1: Start with 27^x + 27^{x − 1/3} = 972.Step 2: Factor out 27^{x − 1/3} from the left side: 27^{x − 1/3}(27^{1/3} + 1) = 972.Step 3: Note that 27^{1/3} is the cube root of 27, which is 3. So the bracket becomes (3 + 1) = 4.Step 4: The equation simplifies to 27^{x − 1/3} * 4 = 972.Step 5: Divide both sides by 4: 27^{x − 1/3} = 972 / 4 = 243.Step 6: Express 27 and 243 as powers of 3. Since 27 = 3^3, we have 27^{x − 1/3} = (3^3)^{x − 1/3} = 3^{3x − 1}.Step 7: Also, 243 = 3^5.Step 8: Equate exponents: 3^{3x − 1} = 3^5 implies 3x − 1 = 5.Step 9: Solve for x: 3x = 6, so x = 2. Verification / Alternative check: Substitute x = 2 back into the original equation. Compute 27^2 = 729 and 27^{2 − 1/3} = 27^{5/3} = (27^{1/3})^5 = 3^5 = 243. Then 729 + 243 = 972, which matches the right side. This confirms that x = 2 is the correct solution. Why Other Options Are Wrong: The values 3, 4, 5, and 1 do not satisfy the original equation. If x were 3, 27^3 is already much larger than 972. If x were 1, the left side becomes 27 + 27^{2/3}, which is less than 972. Only x = 2 makes the sum exactly 972, consistent with both the algebraic solution and the numerical check. Common Pitfalls: Some students try to take logarithms immediately, which is unnecessary and can complicate the algebra. Others forget to factor out the common term correctly or miscalculate 972 / 4. Recognising patterns and working systematically with powers of 3 is the fastest and most reliable approach. Final Answer: The value of x that satisfies the equation is 2.

Question 4

The inradius of an equilateral triangle is 10 cm. Find the circumradius of the same triangle in centimetres.

Accepted Answer

Introduction / Context: This geometry question involves two important radii associated with an equilateral triangle: the inradius (radius of the inscribed circle) and the circumradius (radius of the circumscribed circle). You are given the inradius and asked to find the circumradius. For equilateral triangles, the relationship between these two radii is particularly simple, which makes the calculation very quick once you know the formula. Given Data / Assumptions: An equilateral triangle has inradius r = 10 cm. All sides of the triangle are equal and all internal angles are 60 degrees. You must find the circumradius R of this triangle. Standard formulas for equilateral triangles apply. Concept / Approach: For an equilateral triangle with side length a, the inradius r and circumradius R are given by r = (a√3) / 6 and R = (a√3) / 3. These formulas show that the circumradius is exactly double the inradius: R = 2r. Therefore, once the inradius is known, you can immediately find the circumradius without even computing the side length a explicitly. Step-by-Step Solution: Step 1: Recall the formulas for an equilateral triangle with side length a: r = (a√3) / 6 and R = (a√3) / 3.Step 2: Observe that R is twice r, because (a√3) / 3 = 2 * (a√3 / 6).Step 3: Therefore, R = 2r for any equilateral triangle.Step 4: Substitute the given value r = 10 cm into R = 2r.Step 5: Compute R = 2 * 10 = 20 cm.Step 6: Thus, the circumradius of the triangle is 20 cm. Verification / Alternative check: If you want an extra confirmation, you can first compute the side length a from the inradius formula. Since r = (a√3) / 6 and r = 10, we have 10 = (a√3) / 6, so a = 60 / √3 = 20√3. Then R = (a√3) / 3 = (20√3 * √3) / 3 = (20 * 3) / 3 = 20. This matches the direct relation R = 2r, confirming the answer. Why Other Options Are Wrong: The value 5 cm is half the inradius and has no basis in the geometry of an equilateral triangle. The values 10√3 and 20√3 correspond to misinterpreting the formulas or mixing up side length with radius. The value 30 cm would imply R = 3r, which is not the correct relationship. Only 20 cm satisfies the simple and exact relation R = 2r for equilateral triangles. Common Pitfalls: Many learners forget the specific formulas for r and R in an equilateral triangle and try to use general formulas for other triangles, which are more complicated. Remembering that R is always exactly twice r in this special case is a useful shortcut that can save time in exams. Final Answer: The circumradius of the equilateral triangle is 20 centimetres.

Question 5

If a = 73, b = 74 and c = 75, find the value of a^3 + b^3 + c^3 − 3abc.

Accepted Answer

Introduction / Context: This question asks you to evaluate a^3 + b^3 + c^3 − 3abc for three specific consecutive integers. Rather than expanding each cube directly, which would be long and error prone, you can use a standard identity that expresses this combination in terms of sums and products of a, b, and c. This approach is both faster and more reliable for exam purposes. Given Data / Assumptions: a = 73, b = 74, c = 75. You must find a^3 + b^3 + c^3 − 3abc. All numbers are integers, and the identity for cubes holds for real numbers. Concept / Approach: Use the identity a^3 + b^3 + c^3 − 3abc = (a + b + c)(a^2 + b^2 + c^2 − ab − bc − ca). Since the given numbers are consecutive, sums and products can be calculated systematically. Once you compute the sum a + b + c and the combination a^2 + b^2 + c^2 − ab − bc − ca, you simply multiply them to obtain the desired value. Step-by-Step Solution: Step 1: Compute the sum S1 = a + b + c = 73 + 74 + 75 = 222.Step 2: Compute the pairwise products ab, bc, and ca.Step 3: ab = 73 * 74 = 5402.Step 4: bc = 74 * 75 = 5550.Step 5: ca = 73 * 75 = 5475.Step 6: Sum these to get S2 = ab + bc + ca = 5402 + 5550 + 5475 = 16427.Step 7: Compute the squares a^2, b^2, and c^2.Step 8: a^2 = 73^2 = 5329, b^2 = 74^2 = 5476, c^2 = 75^2 = 5625.Step 9: Sum them to get S3 = a^2 + b^2 + c^2 = 5329 + 5476 + 5625 = 16430.Step 10: Compute S3 − S2 = 16430 − 16427 = 3.Step 11: Use the identity: a^3 + b^3 + c^3 − 3abc = S1 * (S3 − S2) = 222 * 3 = 666. Verification / Alternative check: You can confirm the pattern by noting that a, b, and c form an arithmetic progression with common difference 1 and that a + b + c is divisible by 3. In such cases, a^3 + b^3 + c^3 − 3abc is nonzero and equal to (a + b + c) times a small integer. Here, that small integer is 3, giving 222 * 3 = 666. Any attempt to expand the cubes directly will also yield the same result, though with more work. Why Other Options Are Wrong: The values 365, 444, 999, and 333 do not match the identity based calculation. For example, 999 would require S3 − S2 to be 4.5 instead of 3. Only 666 arises from the exact computations of the sums and squares as shown. Common Pitfalls: Errors often occur when computing large squares or products manually. Careless addition or multiplication can easily lead to a wrong S2 or S3. Using the identity keeps the structure clear, but you still need to be careful with arithmetic. Writing intermediate results clearly, as done above, helps avoid mistakes. Final Answer: The value of a^3 + b^3 + c^3 − 3abc is 666.

Question 6

If x^2 + 1 / x^2 = 31 / 9 and x > 0, find the value of x^3 + 1 / x^3.

Accepted Answer

Introduction / Context: This algebra problem shows how to move from information about x^2 + 1 / x^2 to a higher power expression x^3 + 1 / x^3. It uses standard identities that relate sums of powers of x and its reciprocal to lower degree expressions. This technique is often tested in aptitude exams to see whether you can use these identities systematically rather than trying to solve for x directly. Given Data / Assumptions: x^2 + 1 / x^2 = 31 / 9. x is positive, so x > 0. You are required to find x^3 + 1 / x^3. All expressions are real and defined (x is nonzero). Concept / Approach: The standard plan is to introduce t = x + 1 / x. There is a known relation (x + 1 / x)^2 = x^2 + 2 + 1 / x^2, which allows you to find t from the given x^2 + 1 / x^2. Once t is known, you use the identity x^3 + 1 / x^3 = (x + 1 / x)^3 − 3(x + 1 / x) to reach the target expression. This approach avoids solving a quartic equation and keeps the algebra manageable. Step-by-Step Solution: Step 1: Let t = x + 1 / x.Step 2: Recall that t^2 = x^2 + 2 + 1 / x^2.Step 3: You are given x^2 + 1 / x^2 = 31 / 9, so substitute into the relation: t^2 = 31 / 9 + 2.Step 4: Write 2 as 18 / 9. Then t^2 = 31 / 9 + 18 / 9 = 49 / 9.Step 5: Since x > 0, both x and 1 / x are positive, so t = x + 1 / x is positive. Therefore t = 7 / 3 (the positive square root of 49 / 9).Step 6: Use the identity x^3 + 1 / x^3 = t^3 − 3t.Step 7: Compute t^3 = (7 / 3)^3 = 343 / 27.Step 8: Compute 3t = 3 * 7 / 3 = 7.Step 9: Write 7 as a fraction with denominator 27: 7 = 189 / 27.Step 10: Therefore x^3 + 1 / x^3 = 343 / 27 − 189 / 27 = (343 − 189) / 27 = 154 / 27. Verification / Alternative check: You can verify the consistency by working backwards. If x^3 + 1 / x^3 = 154 / 27, and t = 7 / 3, then t^3 − 3t should reproduce 154 / 27, which it does. Additionally, squaring t gives back t^2 = 49 / 9, and subtracting 2 yields x^2 + 1 / x^2 = 31 / 9, confirming that all identities are used correctly. Why Other Options Are Wrong: The values 70/9, 349/27, 349/7, and 70/27 correspond to partial steps or arithmetic mistakes. For example, 343 / 27 without subtracting 3t, or subtracting 3t with the wrong denominator, leads to some of these incorrect options. Only 154 / 27 carefully follows the identity and the required fraction operations. Common Pitfalls: Students sometimes forget to include the plus 2 in the formula t^2 = x^2 + 2 + 1 / x^2, or they take the wrong sign for t after taking the square root. Others make fraction errors when converting 7 to 189 / 27. Keeping track of denominators and signs at each step is essential for an accurate result. Final Answer: The value of x^3 + 1 / x^3 is 154/27.

Question 7

In an algebraic simplification question, let x be a real number such that x − 1/x = 3. Using this relation, evaluate the expression (2x^4 + 3x^3 + 13x^2 − 3x + 2) ÷ (3x^4 + 3) and find its exact numerical value.

Accepted Answer

Introduction / Context:This aptitude question checks how well you can use an algebraic condition to simplify a complicated rational expression in terms of x. Instead of solving for x directly and getting messy values, we use the relation x − 1/x = 3 to reduce higher powers of x to something simple and then evaluate the full expression. This type of question is common in competitive exams to test algebraic manipulation, identities, and simplification skills. Given Data / Assumptions: x is a real number and x ≠ 0 because 1/x is defined. The relation x − 1/x = 3 holds. We need the value of (2x^4 + 3x^3 + 13x^2 − 3x + 2) ÷ (3x^4 + 3). No approximate decimal value is required, only an exact simplified fraction. Concept / Approach:The key idea is to use the relation x − 1/x = 3 to form an equation in x and then reduce higher powers of x. From x − 1/x = 3, we obtain a quadratic equation in x. Once we know that x satisfies this quadratic, we can replace x^2 or x^4 in the given expression using that relation. Another approach is to solve the quadratic for x and then substitute each root into the expression. Since both roots satisfy the same equation, the value of the expression will be the same for both roots, giving us a unique answer. Step-by-Step Solution:Step 1: Start from x − 1/x = 3.Step 2: Multiply both sides by x to remove the denominator: x^2 − 1 = 3x.Step 3: Rearrange to get a quadratic: x^2 − 3x − 1 = 0.Step 4: This shows x satisfies x^2 = 3x + 1. Hence x^4 = (x^2)^2 = (3x + 1)^2 = 9x^2 + 6x + 1.Step 5: Substitute x^4 into the numerator: 2x^4 + 3x^3 + 13x^2 − 3x + 2 = 2(9x^2 + 6x + 1) + 3x^3 + 13x^2 − 3x + 2.Step 6: Simplify the numerator: 18x^2 + 12x + 2 + 3x^3 + 13x^2 − 3x + 2 = 3x^3 + 31x^2 + 9x + 4.Step 7: Similarly, substitute x^4 into the denominator: 3x^4 + 3 = 3(9x^2 + 6x + 1) + 3 = 27x^2 + 18x + 6.Step 8: Now the expression is (3x^3 + 31x^2 + 9x + 4) ÷ (27x^2 + 18x + 6).Step 9: Instead of further algebraic reduction, use the quadratic x^2 − 3x − 1 = 0 to find numerical values for x and substitute.Step 10: The quadratic has two real roots. Substituting either root into the simplified fraction gives the same constant value, which is 4/3. Verification / Alternative check:An efficient verification method is to solve x^2 − 3x − 1 = 0 using the quadratic formula, compute x approximately, and then evaluate the original fraction numerically with a calculator. Doing this for both roots shows that the fraction is approximately 1.3333 in each case, which corresponds to 4/3 when written as an exact fraction. Since the value is independent of the chosen root, the simplified result is confirmed as 4/3. Why Other Options Are Wrong:Options 1/3 and 2/3 are much smaller than the evaluated expression, and substitution of any root yields a value significantly larger than these. Option 5/3 is approximately 1.6667, which does not match the actual value around 1.3333. The option 7/3 is even larger and clearly inconsistent. Only 4/3 matches the exact algebraic simplification and numerical verification. Common Pitfalls:Many learners try to expand and simplify the large polynomial expression directly without using the relation x − 1/x = 3, which leads to lengthy calculations and mistakes. Another common error is to assume that x has a single value and not check that the expression remains constant for all possible roots. Some students also square the relation incorrectly or mishandle denominators. Using the quadratic relation carefully and checking work avoids these issues. Final Answer:The exact value of the expression (2x^4 + 3x^3 + 13x^2 − 3x + 2) ÷ (3x^4 + 3) under the condition x − 1/x = 3 is 4/3.

Question 8

In trigonometry, simplify the expression sec^6 A − tan^6 A − 3 sec^2 A tan^2 A for an acute angle A and determine its exact numerical value.

Accepted Answer

Introduction / Context:This problem tests knowledge of trigonometric identities and algebraic factorisation. The expression sec^6 A − tan^6 A − 3 sec^2 A tan^2 A may look complicated, but it is built around the familiar identity sec^2 A − tan^2 A = 1. Recognising patterns similar to algebraic identities such as a^3 − b^3 and using basic trigonometric relations leads to a simple numerical result. These types of questions appear frequently in competitive exams under the simplification or trigonometry section. Given Data / Assumptions: A is an acute angle, so sec A and tan A are well defined. The expression to simplify is sec^6 A − tan^6 A − 3 sec^2 A tan^2 A. Standard trigonometric identities such as sec^2 A − tan^2 A = 1 and sec^2 A = 1 + tan^2 A can be used. The final answer is expected to be a simple number or a very simple trigonometric term. Concept / Approach:First, treat sec^2 A and tan^2 A as algebraic variables. Rewrite sec^6 A − tan^6 A as (sec^2 A)^3 − (tan^2 A)^3 and use the factorisation for a^3 − b^3. Then simplify the resulting terms using the identity sec^2 A − tan^2 A = 1. After this, combine the remaining terms with the − 3 sec^2 A tan^2 A that appears separately. The aim is to reach a compact form that can be evaluated using basic identities without substituting any specific angle. Step-by-Step Solution:Step 1: Let p = sec^2 A and q = tan^2 A for simplicity.Step 2: Then sec^6 A − tan^6 A becomes p^3 − q^3.Step 3: Use a^3 − b^3 = (a − b)(a^2 + ab + b^2). Hence p^3 − q^3 = (p − q)(p^2 + pq + q^2).Step 4: We know that p − q = sec^2 A − tan^2 A = 1.Step 5: Therefore sec^6 A − tan^6 A = 1 × (p^2 + pq + q^2) = p^2 + pq + q^2.Step 6: Now write the full expression: sec^6 A − tan^6 A − 3 sec^2 A tan^2 A = (p^2 + pq + q^2) − 3pq.Step 7: Combine terms: p^2 + pq + q^2 − 3pq = p^2 − 2pq + q^2.Step 8: Recognise another algebraic identity: p^2 − 2pq + q^2 = (p − q)^2.Step 9: Substitute back p − q = 1 to get (p − q)^2 = 1^2 = 1.Step 10: Therefore the whole expression simplifies to 1, independent of the actual value of A. Verification / Alternative check:To verify, pick a simple angle such as A = 45°. Then sec 45° = √2 and tan 45° = 1. Compute sec^2 A = 2 and tan^2 A = 1. Then sec^6 A − tan^6 A − 3 sec^2 A tan^2 A = 2^3 − 1^3 − 3(2)(1) = 8 − 1 − 6 = 1. This confirms our algebraic result. Any other convenient angle within the domain of the functions will give the same value. Why Other Options Are Wrong:The options −1 and 0 are incorrect because both algebraic simplification and numerical checking show the expression is strictly positive and equal to 1. The option sec A tan A depends on A, while our result is independent of the angle. The additional option sec^2 A − tan^2 A evaluates to 1, but this is the square root of our final squared expression, so it would match only by coincidence if misapplied. The correct and exact value is 1. Common Pitfalls:Common mistakes include forgetting the identity sec^2 A − tan^2 A = 1, or trying to expand everything numerically without recognising the a^3 − b^3 pattern. Some learners incorrectly apply sec^2 A = 1 + tan^2 A at the wrong stage, creating unnecessary complexity. Keeping the expression in terms of sec^2 A and tan^2 A and systematically using algebraic identities makes the problem straightforward. Final Answer:The simplified value of sec^6 A − tan^6 A − 3 sec^2 A tan^2 A is 1.

Question 9

For an acute angle A, simplify the trigonometric product (cosec A − sin A)(sec A − cos A)(tan A + cot A) and determine its exact value.

Accepted Answer

Introduction / Context:This question explores the interplay between reciprocal trigonometric functions and basic identities. The product (cosec A − sin A)(sec A − cos A)(tan A + cot A) looks complicated, but after rewriting all functions in terms of sine and cosine, many cancellations occur. Simplification questions like this are used in aptitude and competitive exams to test fluency with trigonometric identities and algebraic manipulation. Given Data / Assumptions: A is an acute angle, so sine, cosine, tangent, cotangent, secant and cosecant are all well defined and positive. The expression to simplify is (cosec A − sin A)(sec A − cos A)(tan A + cot A). We are allowed to use standard definitions cosec A = 1/sin A, sec A = 1/cos A, tan A = sin A/cos A, and cot A = cos A/sin A. The final answer should be a simple constant value. Concept / Approach:The main strategy is to express all trigonometric functions in terms of sin A and cos A. This converts the product into an algebraic expression in sin A and cos A, where we can factor and simplify. Often, terms like (1/sin A − sin A) and (1/cos A − cos A) give common factors. Together with tan A + cot A, which can be written as a single fraction, we can see cancellations that lead to a constant value. Step-by-Step Solution:Step 1: Rewrite cosec A − sin A as 1/sin A − sin A = (1 − sin^2 A)/sin A = cos^2 A/sin A.Step 2: Rewrite sec A − cos A as 1/cos A − cos A = (1 − cos^2 A)/cos A = sin^2 A/cos A.Step 3: Rewrite tan A + cot A as sin A/cos A + cos A/sin A = (sin^2 A + cos^2 A)/(sin A cos A) = 1/(sin A cos A), since sin^2 A + cos^2 A = 1.Step 4: Multiply the three factors: (cos^2 A/sin A) × (sin^2 A/cos A) × (1/(sin A cos A)).Step 5: Combine numerators and denominators: numerator = cos^2 A × sin^2 A × 1, denominator = sin A × cos A × sin A × cos A = sin^2 A cos^2 A.Step 6: Since numerator and denominator are both sin^2 A cos^2 A, the fraction simplifies to 1.Step 7: Therefore the entire product is equal to 1 for all acute angles A. Verification / Alternative check:Take a simple acute angle such as A = 45°. Then sin A = cos A = √2/2, tan A = 1, cot A = 1, sec A = √2 and cosec A = √2. Substitute into the original expression and compute numerically; the result comes out to 1, confirming our algebraic simplification. Any other acute angle will produce the same constant value when calculated correctly. Why Other Options Are Wrong:The options −1 and 0 are not possible because each individual factor is positive for an acute angle and the product cannot be negative or zero. The option 2 is also incorrect because our stepwise simplification leads precisely to 1, not a larger constant. The option involving cosec A sec A depends on A, but our final result is independent of the angle, so it cannot be correct. Common Pitfalls:Learners sometimes mis-handle the algebra when combining tan A and cot A, or they forget that sin^2 A + cos^2 A = 1. Another common mistake is failing to simplify 1/sin A − sin A and 1/cos A − cos A properly, leading to messy expressions that obscure the cancellations. Writing every function in terms of sin A and cos A and simplifying slowly prevents these errors. Final Answer:The simplified value of (cosec A − sin A)(sec A − cos A)(tan A + cot A) is 1.

Question 10

If A is an acute angle, simplify the trigonometric expression cos^4 A − sin^4 A and express your answer in terms of a standard trigonometric function of 2A.

Accepted Answer

Introduction / Context:This question checks familiarity with basic trigonometric identities and algebraic factorisation. The expression cos^4 A − sin^4 A resembles the difference of two fourth powers, which is naturally treated as a difference of squares. The goal is to connect the result to a well known double angle identity involving cos 2A. Such transformations are very common in trigonometry sections of aptitude and entrance exams. Given Data / Assumptions: A is an acute angle, though the identity will hold for any real angle. The expression to simplify is cos^4 A − sin^4 A. We may use standard identities such as cos^2 A + sin^2 A = 1 and cos 2A = cos^2 A − sin^2 A. The final answer must be expressed using a standard trigonometric function, preferably cos 2A. Concept / Approach:The main idea is to factor cos^4 A − sin^4 A as a difference of squares. We treat cos^4 A as (cos^2 A)^2 and sin^4 A as (sin^2 A)^2. The difference of squares formula a^2 − b^2 = (a − b)(a + b) can then be applied. Once we have a product of two simpler expressions, we can replace cos^2 A − sin^2 A by cos 2A and cos^2 A + sin^2 A by 1, leading to a very compact result. Step-by-Step Solution:Step 1: Rewrite cos^4 A as (cos^2 A)^2 and sin^4 A as (sin^2 A)^2.Step 2: Use the difference of squares identity: a^2 − b^2 = (a − b)(a + b).Step 3: Therefore cos^4 A − sin^4 A = (cos^2 A − sin^2 A)(cos^2 A + sin^2 A).Step 4: Use the Pythagorean identity cos^2 A + sin^2 A = 1.Step 5: This simplifies the expression to cos^2 A − sin^2 A.Step 6: Recognise that cos 2A = cos^2 A − sin^2 A.Step 7: Hence cos^4 A − sin^4 A = cos 2A. Verification / Alternative check:Choose an easy angle such as A = 30°. Then cos A = √3/2 and sin A = 1/2. Compute cos^4 30° − sin^4 30°. We have cos^2 30° = 3/4 and sin^2 30° = 1/4, so cos^4 30° = 9/16 and sin^4 30° = 1/16. The difference is 8/16 = 1/2. Now compute cos 2A = cos 60° = 1/2. Both results match, confirming our identity. Why Other Options Are Wrong:The option 0 would be correct only if cos^4 A and sin^4 A were equal for all A, which they are not. The option 2 cos 2A is twice the correct value and arises if one mistakenly multiplies by 2 when converting to double angle form. The option 1 ignores the dependence on A entirely. The option sin 2A is unrelated to cos^4 A − sin^4 A under the standard identities. Only cos 2A matches the correct algebraic simplification. Common Pitfalls:A frequent error is to try to convert cos^4 A and sin^4 A directly into double angle forms using power reduction formulas, which is unnecessarily long here. Some students also forget that cos^2 A + sin^2 A = 1 and therefore miss the simple factor that collapses the expression. Recognising the structure of a difference of squares and then applying standard identities is the most efficient route. Final Answer:The expression cos^4 A − sin^4 A simplifies to cos 2A.

Question 11

If a/(b + c) = b/(c + a) = c/(a + b) for nonzero real numbers a, b and c, determine which statement correctly describes the possible common value of each fraction.

Accepted Answer

Introduction / Context:This algebraic question examines your understanding of symmetric equations involving three variables and the consequences of setting several rational expressions equal to a common value. The condition a/(b + c) = b/(c + a) = c/(a + b) with nonzero real numbers a, b and c imposes strong restrictions on their relationships. The problem asks you to deduce what the common value of these fractions can be, which is a classic reasoning task in algebra and aptitude tests. Given Data / Assumptions: a, b and c are nonzero real numbers. a/(b + c) = b/(c + a) = c/(a + b) = k, say, where k is the common value. We are asked to identify which statement about the possible values of k is correct. No specific numeric values of a, b or c are given, so we must use algebraic reasoning. Concept / Approach:Let the common value be k. Then we can write a = k(b + c), b = k(c + a) and c = k(a + b). Adding these three equations gives a relation between the sum a + b + c and k. This leads to two main cases: either the sum a + b + c is zero or it is nonzero. Each case yields a different possible value of k. We then verify that these values actually work by choosing simple numerical examples for a, b and c. Step-by-Step Solution:Step 1: Let a/(b + c) = b/(c + a) = c/(a + b) = k.Step 2: Then a = k(b + c), b = k(c + a), and c = k(a + b).Step 3: Add the three equations: a + b + c = k[(b + c) + (c + a) + (a + b)].Step 4: The bracket simplifies to 2(a + b + c), so we get a + b + c = 2k(a + b + c).Step 5: Either a + b + c ≠ 0, giving 1 = 2k, or a + b + c = 0.Step 6: If a + b + c ≠ 0, then k = 1/2 and each fraction equals 1/2.Step 7: If a + b + c = 0, then from a = k(b + c) we have a = k(−a), which implies (1 + k)a = 0. Since a ≠ 0, we get k = −1.Step 8: Therefore the only possible common values of the fractions are 1/2 and −1. Verification / Alternative check:To confirm, choose an example where a = b = c. Let a = b = c = 2. Then a + b + c = 6 ≠ 0, and a/(b + c) = 2/(2 + 2) = 2/4 = 1/2, so k = 1/2 works. For the second case, choose numbers with sum zero, for example a = 1, b = 1 and c = −2. Then a/(b + c) = 1/(1 − 2) = −1, b/(c + a) = 1/(−2 + 1) = −1, and c/(a + b) = −2/(1 + 1) = −1. Here the common value is −1, confirming both possibilities. Why Other Options Are Wrong:The option stating 1 or −1 is incorrect because 1 is not possible under the given relations. Options that include 1 as a possibility, or only 1/2, ignore the valid case where a + b + c = 0. The suggestion that any real number is possible is far too broad and contradicts the strong algebraic constraints we derived. Only the choice indicating 1/2 or −1 matches all valid configurations. Common Pitfalls:One common mistake is to assume that a, b and c must be equal, leading to the conclusion that the common value is always 1/2. Another error is to perform incorrect algebra when adding the three equations, which can lead to a wrong relationship between k and a + b + c. It is also easy to forget that a + b + c could be zero, which opens the second case k = −1. Keeping track of both possibilities is essential. Final Answer:The common value of a/(b + c), b/(c + a), and c/(a + b) can be 1/2 or −1 depending on the specific values of a, b and c.

Question 12

If (4a + 7b)(4c − 7d) = (4a − 7b)(4c + 7d), which of the following relations between a, b, c and d is correct (assuming all are nonzero)?

Accepted Answer

Introduction / Context: This is an algebraic identity question connected with factorisation and symmetry. Given an equation involving products of linear expressions in a, b, c and d, you must deduce a proportional relation between these variables. Such problems train you to expand and compare coefficients, then interpret the resulting condition as a ratio relation. Given Data / Assumptions: (4a + 7b)(4c − 7d) = (4a − 7b)(4c + 7d). a, b, c and d are nonzero real numbers. We want to know which ratio relation is implied by this equality. Concept / Approach: We can either expand both sides and compare terms, or we can rearrange the equality to factor out common expressions. Expansion is straightforward and quickly leads to a single simple condition involving ad and bc, which can then be rewritten as a/b = c/d. Step-by-Step Solution: 1. Expand the left-hand side: (4a + 7b)(4c − 7d). 2. Left-hand side = 4a·4c + 4a·(−7d) + 7b·4c + 7b·(−7d). 3. This simplifies to 16ac − 28ad + 28bc − 49bd. 4. Expand the right-hand side: (4a − 7b)(4c + 7d). 5. Right-hand side = 4a·4c + 4a·7d + (−7b)·4c + (−7b)·7d. 6. This simplifies to 16ac + 28ad − 28bc − 49bd. 7. Set left-hand side equal to right-hand side: 16ac − 28ad + 28bc − 49bd = 16ac + 28ad − 28bc − 49bd. 8. Subtract 16ac and −49bd from both sides; they cancel. We get −28ad + 28bc = 28ad − 28bc. 9. Bring like terms to one side: −28ad − 28ad + 28bc + 28bc = 0, so −56ad + 56bc = 0. 10. Divide by 56 (nonzero): −ad + bc = 0, or bc = ad. 11. Since a, b, c, d are nonzero, divide both sides by bd: c/d = a/b. 12. Hence, a/b = c/d. Verification / Alternative check: You can pick simple values satisfying a/b = c/d, such as a = 2, b = 1, c = 4, d = 2. Substituting these into both sides of the original equation shows that the equality holds, which supports the derived relation. Why Other Options Are Wrong: a/d = c/b or a/b = d/c are different ratio relations and do not follow from bc = ad in general. None of these is incorrect because we have clearly derived a/b = c/d from the given equation. Common Pitfalls: Some students attempt to cancel terms prematurely or incorrectly, or they mis-expand one of the products, leading to sign errors. It is important to expand carefully and systematically, then simplify step by step. Also, one must avoid dividing by zero; that is why we assume a, b, c and d are nonzero when forming ratios. Final Answer: The correct relation implied by the equation is a/b = c/d.

Question 13

A 12 m long wire is cut into two parts. One part is bent into a circle and the other part is bent into a square that just encloses the circle (the circle is inscribed in the square). What is the radius of the circle, expressed in metres, in terms of…

Accepted Answer

Introduction / Context:This geometry problem combines the concepts of perimeter and circumference with the idea of an inscribed circle. A single wire is used to form both a circle and a square, and the circle just fits inside the square. We are asked to determine the radius of the circle in terms of π. Such questions test spatial reasoning and the ability to relate different geometric shapes formed from the same total length. Given Data / Assumptions: Total length of the wire is 12 metres. One part of the wire forms a circle of radius r metres. The other part forms a square that exactly encloses the circle (the circle is inscribed). The circle is tangent to all four sides of the square, so the diameter of the circle equals the side length of the square. We assume no wastage in bending the wire. Concept / Approach:When a circle is inscribed in a square, the diameter of the circle is equal to the side length of the square. If the radius is r, the diameter is 2r and the square has side 2r. The circumference of the circle uses part of the wire and equals 2πr. The perimeter of the square uses the rest of the wire and equals 4 times its side, that is 8r. The total wire length is therefore the sum 2πr + 8r, which must equal 12 metres. Solving this linear equation in r gives the required radius. Step-by-Step Solution:Step 1: Let r be the radius of the circle in metres.Step 2: The circumference of the circle formed by part of the wire is 2πr.Step 3: Since the circle is inscribed in the square, the side length of the square is equal to the diameter of the circle, which is 2r.Step 4: The perimeter of the square formed by the remaining wire is 4 × 2r = 8r.Step 5: The total length of the wire is 12 m, so 2πr + 8r = 12.Step 6: Factor out r: r(2π + 8) = 12.Step 7: Solve for r: r = 12/(2π + 8) = 12/[2(π + 4)].Step 8: Simplify the fraction: r = 6/(π + 4). Verification / Alternative check:To sense check the result, note that π is approximately 3.14, so π + 4 is roughly 7.14. Then r ≈ 6/7.14, which is a little less than 1 metre. This is reasonable, since the total wire length is 12 metres and both the circle and square must share that length. If r were much larger than 1, the combined perimeters would exceed 12 metres, which would be impossible. The expression 6/(π + 4) is therefore both algebraically correct and numerically sensible. Why Other Options Are Wrong:The option 12/(π + 4) corresponds to the side of the square or to twice the radius, not the radius itself. The option 3/(π + 4) is exactly half of the correct radius and results from mistakenly dividing by 4 instead of 2 at the final step. The expression 6/(π + 2√2) arises from confusing the geometry of a circle inside a square with the circle inside a different polygon. The last option 6/(π + 6) comes from an incorrect perimeter equation. Only 6/(π + 4) matches the correct derivation. Common Pitfalls:A typical error is to assume that the circumference of the circle equals the perimeter of the square or that they share equal wire lengths, which the question does not state. Another frequent mistake is misidentifying the relationship between the circle and square; for an inscribed circle, the diameter equals the side of the square, not the diagonal. Confusing this leads to wrong expressions in r. Carefully distinguishing diameter, side, and diagonal avoids these issues. Final Answer:The radius of the circle formed from the wire is 6/(π + 4) metres.

Question 14

Two equal circles intersect each other in such a way that the centre of each circle lies on the circumference of the other. If the length of their common chord is 10√3 cm, what is the diameter of each circle in centimetres?

Accepted Answer

Introduction / Context:This geometry question deals with two equal circles that intersect in a special way: the centre of each circle lies on the circumference of the other. This configuration is sometimes called a symmetric or lens shaped intersection. The problem provides the length of the common chord of the two circles and asks for the diameter of each circle. Solving it requires knowledge of basic circle geometry and right triangle relationships. Given Data / Assumptions: The two circles are equal, so both have the same radius r. Each circle passes through the centre of the other, so the distance between the centres is equal to r. The length of the common chord of intersection is 10√3 cm. We assume the usual Euclidean geometry properties of circles. We need to find the diameter, which is 2r. Concept / Approach:When two equal circles intersect, the line segment joining their centres is the perpendicular bisector of their common chord. In this special configuration, the distance between the centres is equal to the radius r. The common chord lies at equal distance from both centres. We can draw a right triangle whose hypotenuse is r (from centre to any end of the chord) and whose one leg is half of the chord length. Using the Pythagorean theorem, we solve for r. Step-by-Step Solution:Step 1: Let r be the radius of each circle.Step 2: The distance between the two centres is given to be equal to r, because each centre lies on the circumference of the other circle.Step 3: Let P and Q be the points of intersection of the two circles, and let AB be the common chord with endpoints A and B.Step 4: The line joining the centres is perpendicular to AB and bisects it.Step 5: The length of the chord AB is 10√3 cm, so half the chord is AB/2 = 5√3 cm.Step 6: Consider the right triangle formed by one centre, the midpoint of AB, and one endpoint of AB. The hypotenuse is r and one leg is half the chord, 5√3.Step 7: By the Pythagorean theorem, r^2 = (5√3)^2 + d^2, where d is the distance from the centre to the chord. But in this special configuration the distance between centres is r, and geometric symmetry shows that d = r/2.Step 8: Another direct approach is to use the standard relation for an equal intersecting circle configuration: chord length = r√3, so r√3 = 10√3.Step 9: Cancel √3 on both sides to get r = 10 cm.Step 10: Therefore the diameter is 2r = 20 cm. Verification / Alternative check:To verify, note that if r = 10, then the distance between centres is also 10. In a right triangle formed by the centre of one circle, the midpoint of the chord, and an endpoint of the chord, we have hypotenuse 10 and one leg 5√3. The remaining leg equals √(10^2 − (5√3)^2) = √(100 − 75) = √25 = 5. That distance represents how far the chord is from the centre, which is consistent in both circles. This confirms that r = 10 is geometrically valid and that the chord length 10√3 is correctly matched. Why Other Options Are Wrong:A diameter of 10 cm would correspond to a radius of 5 cm, which would make the chord shorter than 10√3 cm. Larger values like 30 cm or 25 cm give radii that produce significantly longer chords than the one specified. The 15 cm diameter also does not yield a chord of 10√3 cm in this configuration. Only a radius of 10 cm, resulting in a diameter of 20 cm, satisfies the geometric constraints of the intersecting circles and the given chord length. Common Pitfalls:A common mistake is to treat the distance between the centres as 2r instead of r when each centre lies on the circumference of the other circle. Another error is to forget that the chord is bisected by the line joining the centres and therefore half of the chord should be used in right triangle calculations. Careful interpretation of the configuration and correct use of Pythagoras prevent these errors. Final Answer:The diameter of each circle is 20 cm.

Question 15

Out of a total of n observations, the mean (average) of m observations is n and the mean of the remaining (n − m) observations is m. What is the mean of all n observations in terms of m and n?

Accepted Answer

Introduction / Context:This statistics question tests understanding of how means (averages) combine when we have subgroups within a data set. We are given the mean of two disjoint groups of observations and asked to find the overall mean of all observations. Instead of plugging in concrete numbers, we work algebraically with n and m. This type of question is common in aptitude exams and helps build intuition about weighted averages. Given Data / Assumptions: There are n total observations. m of these observations have mean n. The remaining n − m observations have mean m. All observations are real numbers, and n and m are positive integers with n ≥ m. We need to find the overall mean of all n observations. Concept / Approach:The overall mean of all observations is found by dividing the total sum of all observations by the total number of observations. When we know the means of subgroups, we can reconstruct the sums for each subgroup by multiplying each mean by the number of observations in that group. Then we add these sums to get the total sum, and finally divide by n to get the combined mean. This is a direct application of the definition of mean and the idea of weighted averages. Step-by-Step Solution:Step 1: For the first group of m observations with mean n, the total sum of these observations is S1 = m × n.Step 2: For the second group, there are n − m observations with mean m, so the total sum is S2 = (n − m) × m.Step 3: Compute S2 explicitly: S2 = mn − m^2.Step 4: The total sum of all n observations is S = S1 + S2 = mn + (mn − m^2) = 2mn − m^2.Step 5: The overall mean of all n observations is S divided by the total number of observations n.Step 6: Therefore overall mean = (2mn − m^2)/n.Step 7: Simplify the expression by dividing numerator and denominator: overall mean = 2m − m^2/n. Verification / Alternative check:Take a simple numeric example to verify the formula. Let n = 10 and m = 4, so there are 10 observations in total. The first 4 observations have mean 10, and the remaining 6 have mean 4. Then S1 = 4 × 10 = 40 and S2 = 6 × 4 = 24, giving total sum 64. The overall mean is 64/10 = 6.4. Now apply the derived expression: 2m − m^2/n = 2 × 4 − 16/10 = 8 − 1.6 = 6.4, which matches the direct computation. Why Other Options Are Wrong:The expression 2m − m/n ignores the squared term and does not correctly represent the weighting of the second group, so it gives wrong values in general. The option 2m is only correct in special contrived cases and fails for most choices of n and m. The option 2m − m simplifies to m, which clearly cannot represent the combined mean in general. The option m^2/n is far too small and does not involve the combined effect of both subgroup means. Only 2m − m^2/n matches the correct derivation from the definition of mean. Common Pitfalls:Many learners mistakenly average the two given means n and m directly, for example by computing (n + m)/2, which is incorrect because the groups may have different sizes. Others forget to multiply each mean by the corresponding number of observations, leading to an incorrect total sum. Remember that means from subgroups must be weighted by their group sizes when combining them. Final Answer:The mean of all n observations is 2m − m^2/n.

Question 16

If x and y are real numbers such that x + y = 5 and x^3 + y^3 = 35, what is the positive value of the difference between x and y, that is, |x − y|?

Accepted Answer

Introduction / Context:This algebra question uses symmetric expressions in two variables. We are given the sum x + y and the sum of cubes x^3 + y^3, and we are asked to find the absolute difference |x − y|. Rather than solving for x and y individually, we use identities that relate sums and products of numbers to their powers. Such problems are popular in aptitude exams because they reward knowledge of algebraic identities and efficient manipulation of expressions. Given Data / Assumptions: x and y are real numbers. x + y = 5. x^3 + y^3 = 35. We must find |x − y|, the positive difference between x and y. No additional information about ordering of x and y is required because we take the absolute value. Concept / Approach:We use the identity for the sum of cubes: x^3 + y^3 = (x + y)^3 − 3xy(x + y). The given values allow us to set up an equation involving xy. Once we know xy, we can use another identity that connects x − y with x + y and xy, namely (x − y)^2 = (x + y)^2 − 4xy. Taking the square root then yields |x − y|. This avoids solving a quadratic for x or y and keeps the work clean and quick. Step-by-Step Solution:Step 1: Start from the identity x^3 + y^3 = (x + y)^3 − 3xy(x + y).Step 2: Substitute x + y = 5 and x^3 + y^3 = 35.Step 3: The equation becomes 35 = 5^3 − 3xy × 5.Step 4: Compute 5^3 = 125, so 35 = 125 − 15xy.Step 5: Rearrange to solve for xy: 15xy = 125 − 35 = 90.Step 6: Therefore xy = 90/15 = 6.Step 7: Use the identity (x − y)^2 = (x + y)^2 − 4xy.Step 8: Substitute x + y = 5 and xy = 6: (x − y)^2 = 5^2 − 4 × 6 = 25 − 24 = 1.Step 9: Taking square roots, |x − y| = √1 = 1. Verification / Alternative check:We can find one explicit pair (x, y) that satisfies the conditions. Since x + y = 5 and xy = 6, the numbers are roots of t^2 − 5t + 6 = 0, which factors as (t − 2)(t − 3) = 0. So x and y must be 2 and 3 in some order. With x = 3 and y = 2, x^3 + y^3 = 27 + 8 = 35, which matches the given condition. The difference |x − y| = |3 − 2| = 1, confirming our result. Why Other Options Are Wrong:The option 0 would require x = y, but that would give x + y = 10 or x^3 + y^3 inconsistent with the data. The options 5 and 6 represent much larger differences and do not fit the derived relation (x − y)^2 = 1. The option 2 would correspond to a squared difference of 4, again inconsistent with the computation. Only 1 matches the algebraic relations obtained from the identities and the numeric verification. Common Pitfalls:Some learners try to guess x and y directly without using identities, which may work in simple cases but is not a reliable general strategy. Others misuse the formula for x^3 + y^3 or incorrectly expand (x + y)^3, leading to algebraic errors. Forgetting to use the absolute value and reporting x − y instead of |x − y| is another minor mistake. Sticking carefully to the identities and checking each step avoids these issues. Final Answer:The positive difference between x and y is |x − y| = 1.

Question 17

For an acute angle θ, if cosec^2 θ = 625/576, find the exact simplified value of the expression (sin θ − cos θ) ÷ (sin θ + cos θ).

Accepted Answer

Introduction / Context:This trigonometry question requires converting information about cosecant into values of sine and cosine, and then simplifying a ratio involving sin θ and cos θ. The given value of cosec^2 θ allows us to find sin θ exactly, and from there we can determine cos θ using the Pythagorean identity. Finally, substituting into (sin θ − cos θ)/(sin θ + cos θ) gives a rational number. Questions like this test precision and comfort with exact trigonometric values rather than approximate decimals. Given Data / Assumptions: θ is an acute angle, so sin θ and cos θ are positive. cosec^2 θ = 625/576. cosec θ = 1/sin θ by definition. We must find (sin θ − cos θ)/(sin θ + cos θ) in exact fractional form. Concept / Approach:First, use the relation between cosecant and sine: cosec^2 θ = 1/sin^2 θ. From this we recover sin^2 θ and then sin θ, taking the positive root because θ is acute. Next, apply the identity sin^2 θ + cos^2 θ = 1 to find cos θ. With exact values for both functions, we substitute into the expression (sin θ − cos θ)/(sin θ + cos θ) and simplify the resulting fraction. All steps are done symbolically to avoid rounding errors. Step-by-Step Solution:Step 1: From cosec^2 θ = 625/576, we have 1/sin^2 θ = 625/576.Step 2: Invert both sides to find sin^2 θ: sin^2 θ = 576/625.Step 3: Since θ is acute, sin θ is positive, so sin θ = √(576/625) = 24/25.Step 4: Use the identity sin^2 θ + cos^2 θ = 1 to find cos^2 θ.Step 5: cos^2 θ = 1 − sin^2 θ = 1 − 576/625 = (625 − 576)/625 = 49/625.Step 6: Again, θ is acute, so cos θ is positive, giving cos θ = √(49/625) = 7/25.Step 7: Now compute sin θ − cos θ: 24/25 − 7/25 = 17/25.Step 8: Compute sin θ + cos θ: 24/25 + 7/25 = 31/25.Step 9: Form the required ratio: (sin θ − cos θ)/(sin θ + cos θ) = (17/25)/(31/25) = 17/31. Verification / Alternative check:We can verify the result numerically by approximating sin θ = 24/25 ≈ 0.96 and cos θ = 7/25 ≈ 0.28. Then sin θ − cos θ ≈ 0.68, sin θ + cos θ ≈ 1.24, and their ratio is approximately 0.548. The fraction 17/31 is roughly 0.548 as well, so the numerical check aligns with the exact algebraic answer. This reinforces the correctness of the exact fraction 17/31. Why Other Options Are Wrong:The value 1 would require sin θ = cos θ, which is not consistent with sin θ = 24/25 and cos θ = 7/25. The option 31/17 is simply the reciprocal of the correct answer and arises if one accidentally flips numerator and denominator. The fraction 14/25 does not correspond to any ratio formed from 24/25 and 7/25 in this expression. The additional option 7/25 is equal to cos θ and does not represent the required ratio. Only 17/31 matches the correctly simplified value of the expression. Common Pitfalls:Frequent mistakes include forgetting that cosec θ is 1/sin θ and treating the given number directly as sin^2 θ instead of its reciprocal, or taking the negative square root even though θ is acute. Another issue is mismanaging fractions when simplifying (sin θ − cos θ)/(sin θ + cos θ), for example by not cancelling the common denominator 25 correctly. Careful fraction operations and attention to the sign and quadrants of trigonometric functions avoid these errors. Final Answer:The simplified value of (sin θ − cos θ)/(sin θ + cos θ) is 17/31.

Question 18

For distinct real numbers x, y and z, evaluate the expression ((x − y)^3 + (y − z)^3 + (z − x)^3) ÷ (3(x − y)(y − z)(z − x)).

Accepted Answer

Introduction / Context:This algebraic identity problem involves symmetric expressions in three variables x, y and z. The numerator is a sum of three cubes of pairwise differences, while the denominator is three times the product of those same differences. At first glance, it may look complicated, but it actually relates to a well known identity involving a^3 + b^3 + c^3 − 3abc. The question checks understanding of identities and ability to apply them in a nonstandard form. Given Data / Assumptions: x, y and z are real numbers, all distinct so that denominators are nonzero. The numerator is (x − y)^3 + (y − z)^3 + (z − x)^3. The denominator is 3(x − y)(y − z)(z − x). We need to simplify the entire fraction to a single, simple value. Concept / Approach:The expression resembles the identity a^3 + b^3 + c^3 − 3abc = (a + b + c)[(a − b)^2 + (b − c)^2 + (c − a)^2]/2. In this problem, we can set a = x − y, b = y − z and c = z − x. Notice that a + b + c equals zero in this case, which considerably simplifies the identity. Once we understand that the sum of cubes equals 3abc whenever a + b + c = 0, the fraction simplifies immediately. Step-by-Step Solution:Step 1: Let a = x − y, b = y − z and c = z − x.Step 2: Observe that a + b + c = (x − y) + (y − z) + (z − x) = 0.Step 3: Recall the identity a^3 + b^3 + c^3 − 3abc = (a + b + c)[(a − b)^2 + (b − c)^2 + (c − a)^2]/2.Step 4: When a + b + c = 0, the right side becomes zero, so a^3 + b^3 + c^3 − 3abc = 0.Step 5: Therefore a^3 + b^3 + c^3 = 3abc.Step 6: Substitute back a = x − y, b = y − z and c = z − x to get (x − y)^3 + (y − z)^3 + (z − x)^3 = 3(x − y)(y − z)(z − x).Step 7: Hence the given fraction becomes [3(x − y)(y − z)(z − x)] ÷ [3(x − y)(y − z)(z − x)].Step 8: The numerator and denominator are identical nonzero expressions, so the fraction simplifies to 1. Verification / Alternative check:To verify numerically, pick distinct values such as x = 1, y = 2 and z = 4. Compute the differences: x − y = −1, y − z = −2 and z − x = 3. Then the numerator is (−1)^3 + (−2)^3 + 3^3 = −1 − 8 + 27 = 18. The denominator is 3(−1)(−2)(3) = 3 × 6 = 18. The fraction is 18/18 = 1, confirming the result for this specific choice of x, y and z. Since the derivation used an identity, it holds for all distinct real numbers x, y and z. Why Other Options Are Wrong:The option 0 would require the numerator to vanish while the denominator stays nonzero, which contradicts the proven identity. The value 1/3 or 3 could only appear if the numerator were a constant multiple of the denominator, which it is not in this case after using the correct identity. The option −1 would require a sign mismatch between numerator and denominator, which does not occur. Only 1 is consistent with both algebraic reasoning and numerical verification. Common Pitfalls:Students sometimes attempt to expand each cube fully and then factor the resulting polynomial, which is tedious and error prone. Another frequent mistake is misremembering the a^3 + b^3 + c^3 − 3abc identity or failing to notice that a + b + c equals zero for these particular differences. Recognising this special condition immediately simplifies the problem and avoids long calculations. Final Answer:The expression ((x − y)^3 + (y − z)^3 + (z − x)^3) ÷ (3(x − y)(y − z)(z − x)) simplifies to 1.

Question 19

Let p and q be the roots of the quadratic equation x^2 − 15x + r = 0, and suppose that the difference between the roots is p − q = 1. What is the value of the constant r?

Accepted Answer

Introduction / Context:This problem uses standard properties of quadratic equations, in particular the relationships between the coefficients of a quadratic and the sum and product of its roots. The equation x^2 − 15x + r = 0 has roots p and q. We are told that p − q = 1 and must find r. Instead of solving the quadratic explicitly for p and q, we use simple identities involving p + q, pq and the square of the difference between the roots. Such questions appear frequently in competitive exams to test algebraic reasoning. Given Data / Assumptions: The quadratic equation is x^2 − 15x + r = 0. Its roots are p and q. p − q = 1. Coefficients are real, and the usual relationships between roots and coefficients hold. We must determine the real value of r. Concept / Approach:For a quadratic equation x^2 − Sx + P = 0, the sum of the roots is S and the product is P. Here, p + q = 15 and pq = r. We are given the difference p − q = 1. There is a standard identity relating the difference of roots to the sum and product: (p − q)^2 = (p + q)^2 − 4pq. Substituting known values for p + q and p − q gives a simple equation in pq = r, from which we can solve for r directly. Step-by-Step Solution:Step 1: From the quadratic x^2 − 15x + r = 0, identify that p + q = 15 and pq = r.Step 2: We are given that p − q = 1.Step 3: Use the identity (p − q)^2 = (p + q)^2 − 4pq.Step 4: Substitute p − q = 1, p + q = 15 and pq = r.Step 5: The equation becomes 1^2 = 15^2 − 4r.Step 6: Compute 15^2 = 225, so 1 = 225 − 4r.Step 7: Rearrange to solve for r: 4r = 225 − 1 = 224.Step 8: Therefore r = 224/4 = 56. Verification / Alternative check:To verify, consider the quadratic x^2 − 15x + 56 = 0. Factor this equation: 56 factors as 7 × 8, and 7 + 8 = 15, so x^2 − 15x + 56 = (x − 7)(x − 8). Hence the roots are p = 7 and q = 8 (or 8 and 7). Their difference is |p − q| = 1, as required. This confirms that r = 56 satisfies all conditions given in the problem. Why Other Options Are Wrong:The values 55, 60, 64 and 65 do not produce a quadratic with roots differing by 1. For example, if r were 60, the quadratic x^2 − 15x + 60 = 0 factors as (x − 5)(x − 12), whose roots differ by 7, not 1. Similar checks show discrepancies for the other options. Only r = 56 leads to factors (x − 7)(x − 8), which produce the required unit difference between roots. Common Pitfalls:Common mistakes include trying to solve the quadratic using the quadratic formula for each candidate r or guessing roots instead of using the root sum and product relations. Another error is misapplying the identity for (p − q)^2 or forgetting to square the given difference correctly. Using the standard relationships p + q = 15 and pq = r together with (p − q)^2 = (p + q)^2 − 4pq is the most reliable and efficient approach. Final Answer:The required constant in the quadratic equation is r = 56.

Question 20

In trigonometry, let θ be an acute angle with 0 < θ < π/2 such that the relation 3 * tan θ = cot θ holds exactly. Using this condition, find the exact value of θ expressed in radians.

Accepted Answer

Introduction / Context: This aptitude question tests understanding of basic trigonometric functions and how tangent and cotangent are related as reciprocals. By giving a simple equation involving tan θ and cot θ for an acute angle, the problem asks you to determine the exact radian measure of θ. Such questions are common in competitive exams because they check both algebraic manipulation and familiarity with standard special angles. Given Data / Assumptions: θ is an acute angle, so 0 < θ < π/2. The trigonometric relation 3 * tan θ = cot θ is given. cot θ is the reciprocal of tan θ for all angles where these functions are defined. We need to find θ in exact radian form, not as a decimal approximation. Concept / Approach: The key concept is that cot θ = 1 / tan θ. Substituting this into the given equation transforms it into an algebraic equation in tan θ alone. Solving this equation gives the value of tan θ. Once tan θ is known, we match it with standard tangent values of special angles in the first quadrant, such as π/6, π/4, and π/3, to determine θ. Since θ is acute, we only consider positive tangent values. Step-by-Step Solution: 1) Start from the given equation: 3 * tan θ = cot θ. 2) Replace cot θ with 1 / tan θ to obtain 3 * tan θ = 1 / tan θ. 3) Multiply both sides by tan θ (tan θ is non zero for an acute angle except at 0, which is excluded): 3 * tan^2 θ = 1. 4) Solve for tan^2 θ: tan^2 θ = 1 / 3. 5) Take the positive square root because θ is acute and tangent is positive in the first quadrant: tan θ = 1 / √3. 6) Recall that tan(π/6) = 1 / √3, tan(π/4) = 1, and tan(π/3) = √3. 7) Therefore θ must be π/6, since this is the only acute angle in the standard set with tan θ = 1 / √3. Verification / Alternative check: We can substitute θ = π/6 directly into the original equation. First compute tan(π/6) = 1 / √3. Then cot(π/6) = √3. The left side becomes 3 * tan θ = 3 * (1 / √3) = 3 / √3 = √3. The right side is cot θ = √3. Both sides are equal, so the equation is satisfied. No other standard acute angle produces tan θ = 1 / √3, which confirms that θ = π/6 is the unique solution within 0 < θ < π/2. Why Other Options Are Wrong: Option b (π/4) would require tan θ = 1, which does not satisfy 3 * tan θ = cot θ. Option c (π/3) gives tan θ = √3, leading to 3 * √3 on the left and 1 / √3 on the right, which do not match. Option d (π/2) is not valid because tan θ and cot θ are not both defined in a useful way at π/2. Option e (π/12) does not have the correct tangent value. Only π/6 makes the given identity true. Common Pitfalls: A frequent mistake is to forget that cot θ is 1 / tan θ and instead treat cot θ as an independent variable. Another common issue is taking both positive and negative square roots without considering the restriction that θ is acute, which forces tan θ to be positive. Some learners also guess from the options without checking the equation carefully. Systematically substituting cot θ = 1 / tan θ and solving the resulting quadratic in tan θ avoids these errors. Final Answer: The acute angle θ that satisfies 3 * tan θ = cot θ is π/6 radians.

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