Question 1

If sin P + cosec P = 2 for an angle P (in degrees), what is the value of sin 7P + cosec 7P?

Accepted Answer

Introduction / Context: This question explores the relationship between a trigonometric function and its reciprocal. You are given that sin P + cosec P = 2 and asked to find sin 7P + cosec 7P. The key idea is to use an inequality for sin P + 1/sin P and identify when equality occurs, which pins down the value of sin P and hence P itself. Given Data / Assumptions: sin P + cosec P = 2. cosec P is defined, so sin P ≠ 0. Angles are measured in degrees. Concept / Approach: For any nonzero real number s, the expression s + 1/s satisfies s + 1/s ≥ 2 if s is positive, with equality only when s = 1. Here s = sin P. Thus the equation sin P + 1/sin P = 2 implies sin P = 1 and P is a special angle. Once we know P, we can compute 7P and then find sin 7P and cosec 7P directly from the unit circle. Step-by-Step Solution: Let s = sin P. Given s + 1/s = 2. Multiply both sides by s: s^2 + 1 = 2s. Rearrange: s^2 − 2s + 1 = 0. This factorises as (s − 1)^2 = 0. Thus s = 1, so sin P = 1. In degrees, sin P = 1 when P = 90° + 360°k, where k is any integer. Take P = 90° for simplicity. Then 7P = 7 × 90° = 630°. Reduce 630° modulo 360°: 630° − 360° = 270°. So sin 7P = sin 270° = −1. Then cosec 7P = 1/sin 7P = 1/(−1) = −1. Therefore sin 7P + cosec 7P = −1 + (−1) = −2. Verification / Alternative check: Using the general solution P = 90° + 360°k, 7P = 630° + 2520°k. Subtract multiples of 360° to get 270° + 360°k, for any integer k. The sine of angles of the form 270° + 360°k is always −1, so sin 7P + cosec 7P is always −2, independent of k. Why Other Options Are Wrong: Values 2, 3, 1 and 0 would require sin 7P or its reciprocal to be positive or zero, which contradicts the fact that sin 7P is always −1 for the set of permissible P. Option 2 corresponds to sin 7P = 1, which would only happen if 7P were of the form 90° + 360°k, not satisfied here. Common Pitfalls: A common error is to forget that sin P + 1/sin P ≥ 2 only for positive sin P, or to mis-handle the reduction of 630° to a standard angle. Another mistake is to assume P is 30° or 60° by guesswork, which does not satisfy the original equation. Always solve the algebraic condition on sin P first. Final Answer: The value of sin 7P + cosec 7P is −2.

Question 2

If cos x · cos y + sin x · sin y = −1, what is the value of cos x + cos y?

Accepted Answer

Introduction / Context: This trigonometric identity question uses a well known formula for the cosine of a difference. You are given cos x · cos y + sin x · sin y and asked to compute cos x + cos y. Such questions test how well you know compound angle formulas and how to use them to deduce relationships between trigonometric expressions. Given Data / Assumptions: cos x · cos y + sin x · sin y = −1. x and y are real angles. We need cos x + cos y. Concept / Approach: The identity cos(x − y) = cos x · cos y + sin x · sin y directly matches the left side. Once we know cos(x − y) = −1, we know that x − y must be an odd multiple of π (or 180°) and can then use the sum formula for cos x + cos y in terms of cos((x + y)/2) and cos((x − y)/2). That leads quickly to a product in which one factor is zero. Step-by-Step Solution: Recall the identity: cos(x − y) = cos x · cos y + sin x · sin y. Given cos x · cos y + sin x · sin y = −1, we have cos(x − y) = −1. The cosine of an angle equals −1 when the angle is an odd multiple of π (or 180°). Thus x − y = (2k + 1)π for some integer k. Now use the sum formula for cosines: cos x + cos y = 2 cos((x + y)/2) cos((x − y)/2). We know x − y = (2k + 1)π, so (x − y)/2 = (2k + 1)π/2. Cosine of an odd multiple of π/2 is zero. Therefore cos((x − y)/2) = 0. So cos x + cos y = 2 cos((x + y)/2) · 0 = 0. Verification / Alternative check: Choose a simple pair of angles to test the result. Let y = 0 and x = π, then cos x · cos y + sin x · sin y = cos π·cos 0 + sin π·sin 0 = (−1)·1 + 0·0 = −1, which matches the given condition. Now cos x + cos y = cos π + cos 0 = −1 + 1 = 0, confirming the value. Why Other Options Are Wrong: Values such as ±2 would require both cosines to be ±1 simultaneously, which is impossible if cos(x − y) is −1. A value 1 or −1 would correspond to partial cancellation rather than total cancellation. The identity and sample check both show that the sum must be exactly zero. Common Pitfalls: A frequent mistake is to misremember the identity and use cos(x + y) instead of cos(x − y). Another issue is forgetting to use the product-to-sum formula for cos x + cos y, which makes it harder to see that a factor becomes zero. Keeping a clear list of standard identities helps avoid such confusion. Final Answer: The value of cos x + cos y is 0.

Question 3

Simplify the expression 2(sin^6θ + cos^6θ) − 3(sin^4θ + cos^4θ) + 1 and determine its constant value for all real θ.

Accepted Answer

Introduction / Context: This problem is another example of simplifying trigonometric expressions involving higher powers of sine and cosine. The expression 2(sin^6θ + cos^6θ) − 3(sin^4θ + cos^4θ) + 1 may look complicated, but with the standard identity sin^2θ + cos^2θ = 1 and some algebraic identities for sums of powers, it reduces to a constant independent of θ. Given Data / Assumptions: Expression: 2(sin^6θ + cos^6θ) − 3(sin^4θ + cos^4θ) + 1. θ is any real angle. Basic identity: sin^2θ + cos^2θ = 1. Concept / Approach: As in similar problems, we set s = sin^2θ and c = cos^2θ. We then express sin^6θ + cos^6θ and sin^4θ + cos^4θ in terms of s and c, and further in terms of the product sc. After substitution, the expression becomes a linear combination of 1 and sc, and the sc terms cancel out, leaving a simple constant value. Step-by-Step Solution: Let s = sin^2θ and c = cos^2θ. Then s + c = 1. Compute sin^6θ + cos^6θ = s^3 + c^3. Use a^3 + b^3 = (a + b)^3 − 3ab(a + b). Thus s^3 + c^3 = (s + c)^3 − 3sc(s + c) = 1^3 − 3sc·1 = 1 − 3sc. Next compute sin^4θ + cos^4θ = s^2 + c^2. Use a^2 + b^2 = (a + b)^2 − 2ab. Thus s^2 + c^2 = (s + c)^2 − 2sc = 1^2 − 2sc = 1 − 2sc. Now substitute into the original expression: 2(sin^6θ + cos^6θ) − 3(sin^4θ + cos^4θ) + 1 = 2(1 − 3sc) − 3(1 − 2sc) + 1. Expand each part: 2 − 6sc − 3 + 6sc + 1. Combine like terms: (2 − 3 + 1) + (−6sc + 6sc) = 0 + 0 = 0. Verification / Alternative check: Check with a few values of θ. For θ = 0°, sin θ = 0 and cos θ = 1, so the expression becomes 2·1 − 3·1 + 1 = 0. For θ = 45°, sin θ = cos θ = √2/2, so sin^2θ = cos^2θ = 1/2 and both sin^6θ and cos^6θ equal 1/8. Substitution again yields zero, confirming the identity. Why Other Options Are Wrong: Values −1, 1 or 2 would not hold for all θ. For example, at θ = 0° the expression equal to 0 contradicts these options. The value −2 is also impossible because the expression can be shown to be zero for many different angles. Common Pitfalls: A common error is misinterpreting sin^6θ as sin 6θ. Another pitfall is misusing the algebraic identities for sums of squares and cubes, leading to extra terms or incorrect coefficients. By carefully substituting s = sin^2θ, c = cos^2θ and using s + c = 1, the computation becomes straightforward. Final Answer: The simplified value of the expression is 0 for all real θ.

Question 4

In trigonometry, if cos θ is given by the expression (x^2 − y^2) / (x^2 + y^2) for real numbers x and y, then what is the value of cot θ expressed purely in terms of x and y?

Accepted Answer

Introduction / Context: This question tests a standard trigonometric parametrisation where cosine is written in terms of two real parameters x and y. The form cos θ = (x^2 − y^2) / (x^2 + y^2) is closely related to the tangent half angle formulas and to expressing sine and cosine using rational functions. The goal is to convert this information into cot θ, which is the ratio of cosine to sine, written only in terms of x and y. Given Data / Assumptions: cos θ = (x^2 − y^2) / (x^2 + y^2). θ is an acute angle so sine and cosine are positive. x and y are real numbers, and x^2 + y^2 ≠ 0. We need cot θ, which is cos θ / sin θ, expressed using x and y only. Concept / Approach: The key idea is to first compute sin θ from the identity sin^2 θ + cos^2 θ = 1. Once we obtain sin θ in terms of x and y, we can form cot θ = cos θ / sin θ. Because cos θ already has denominator x^2 + y^2, we will square it, subtract from 1, and then take the square root to get sin θ. Simplification will produce a clean expression involving the product xy and the difference x^2 − y^2. Step-by-Step Solution: 1. Start from cos θ = (x^2 − y^2) / (x^2 + y^2). 2. Compute cos^2 θ = (x^2 − y^2)^2 / (x^2 + y^2)^2. 3. Use sin^2 θ = 1 − cos^2 θ. 4. Substitute: sin^2 θ = 1 − (x^2 − y^2)^2 / (x^2 + y^2)^2. 5. Write 1 as (x^2 + y^2)^2 / (x^2 + y^2)^2. 6. Then sin^2 θ = [(x^2 + y^2)^2 − (x^2 − y^2)^2] / (x^2 + y^2)^2. 7. Expand numerators: (x^2 + y^2)^2 = x^4 + 2x^2 y^2 + y^4, (x^2 − y^2)^2 = x^4 − 2x^2 y^2 + y^4. 8. Subtract: (x^4 + 2x^2 y^2 + y^4) − (x^4 − 2x^2 y^2 + y^4) = 4x^2 y^2. 9. So sin^2 θ = 4x^2 y^2 / (x^2 + y^2)^2. 10. For an acute θ and real x, y, take the positive root: sin θ = 2xy / (x^2 + y^2). 11. Now form cot θ = cos θ / sin θ. 12. Substitute cos θ and sin θ: cot θ = [(x^2 − y^2) / (x^2 + y^2)] / [2xy / (x^2 + y^2)]. 13. Cancel x^2 + y^2 in numerator and denominator. 14. This gives cot θ = (x^2 − y^2) / (2xy). Verification / Alternative check: We can cross check using the tangent half angle substitution t = tan(θ / 2). In that parametrisation, cos θ = (1 − t^2) / (1 + t^2) and sin θ = 2t / (1 + t^2). If we identify t = y / x, then cos θ becomes (x^2 − y^2) / (x^2 + y^2) and sin θ becomes 2xy / (x^2 + y^2). This matches our derived expressions and confirms that cot θ = (x^2 − y^2) / (2xy). Why Other Options Are Wrong: Option 2: 2xy / (x^2 − y^2) is equal to tan θ, not cot θ, because it is sin θ / cos θ. Option 3: (x^2 + y^2) / (2xy) is the reciprocal of sin θ, which would correspond to cosec θ, not cot θ. Option 4: (x^2 + y^2) / (x^2 − y^2) is the reciprocal of cos θ, that is sec θ, not cot θ. Common Pitfalls: A common mistake is to forget to square the denominator when squaring cos θ, which leads to an incorrect expression for sin^2 θ. Another frequent error is mixing up tan θ and cot θ when forming cos θ / sin θ versus sin θ / cos θ. Students may also incorrectly cancel terms before properly simplifying the fraction, especially with (x^2 + y^2) terms. Careful algebra at each step avoids these errors. Final Answer: Therefore, the correct value of cot θ in terms of x and y is (x^2 − y^2) / (2xy).

Question 5

If x = cosec θ − sin θ and y = sec θ − cos θ for an acute angle θ, then which of the following relations between x and y is true?

Accepted Answer

Introduction / Context: This problem explores special transformations of trigonometric functions. The expressions x = cosec θ − sin θ and y = sec θ − cos θ often appear in algebraic manipulation questions where the aim is to derive a compact identity relating x and y. The task is to work out an exact relation involving x^2 and y^2, then match it with one of the given options. Given Data / Assumptions: x = cosec θ − sin θ. y = sec θ − cos θ. θ is acute, so sin θ and cos θ are positive. We must find which of the given algebraic relations between x and y is identically true. Concept / Approach: The strategy is to rewrite x and y in terms of sin θ and cos θ only, remove reciprocal functions, and then compute x^2, y^2, and x^2 y^2. Known algebraic simplifications show that such expressions often collapse nicely due to the identity sin^2 θ + cos^2 θ = 1. After simplification we look for a compact expression that equals 1, and match that with the most suitable option. Step-by-Step Solution: 1. Start with x = cosec θ − sin θ = 1 / sin θ − sin θ. 2. Combine terms: x = (1 − sin^2 θ) / sin θ = cos^2 θ / sin θ. 3. Similarly, y = sec θ − cos θ = 1 / cos θ − cos θ. 4. Combine y: y = (1 − cos^2 θ) / cos θ = sin^2 θ / cos θ. 5. Compute x^2: x^2 = cos^4 θ / sin^2 θ. 6. Compute y^2: y^2 = sin^4 θ / cos^2 θ. 7. Find x^2 y^2: x^2 y^2 = (cos^4 θ / sin^2 θ) * (sin^4 θ / cos^2 θ) = sin^2 θ cos^2 θ. 8. Now compute x^2 + y^2: x^2 + y^2 = cos^4 θ / sin^2 θ + sin^4 θ / cos^2 θ. 9. Bring to a common denominator sin^2 θ cos^2 θ to combine, but instead of fully expanding we evaluate x^2 y^2 (x^2 + y^2 + 3). 10. Substitute x^2 y^2 = sin^2 θ cos^2 θ into x^2 y^2 (x^2 + y^2 + 3). 11. Compute x^2 + y^2 in terms of sin θ and cos θ and simplify; after algebraic manipulation it reduces so that x^2 y^2 (x^2 + y^2 + 3) equals 1. 12. Therefore the identity that always holds is x^2 y^2 (x^2 + y^2 + 3) = 1. Verification / Alternative check: We can verify numerically with a convenient acute angle, for example θ = 45 degrees. Then sin θ = cos θ = √2 / 2. This gives x = cosec θ − sin θ = (√2) − (√2 / 2) = √2 / 2, and y = sec θ − cos θ = √2 − √2 / 2 = √2 / 2, so x = y. Then x^2 = y^2 = 1 / 2, and x^2 y^2 = 1 / 4. Plugging into x^2 y^2 (x^2 + y^2 + 3) gives (1 / 4) * (1 / 2 + 1 / 2 + 3) = (1 / 4) * 4 = 1, which confirms option 2 is correct. Why Other Options Are Wrong: Option 1: x^2 + y^2 + 3 = 1 would give x^2 + y^2 = −2, which is impossible for real x and y since squares are non negative. Option 3: x^2 (x^2 + y^2 − 5) = 1 is not satisfied for θ = 45 degrees because this would give (1 / 2) * (1 − 5) = −2, which is not equal to 1. Option 4: y^2 (x^2 + y^2 − 5) = 1 fails by the same reasoning as option 3 since x^2 = y^2 in the test case. Common Pitfalls: Many learners forget to square both the numerator and denominator when finding x^2 and y^2. Another pitfall is attempting to simplify x^2 + y^2 directly without considering a more guided expression such as x^2 y^2 (x^2 + y^2 + 3). It is also easy to mismanage signs when expanding powers of sine and cosine. Careful substitution and patient algebra avoid inconsistencies. Final Answer: The correct relation between x and y is x^2 y^2 (x^2 + y^2 + 3) = 1.

Question 6

A hydrogen filled balloon is ascending vertically at a constant speed of 18 km/h while being drifted horizontally by the wind. From a fixed point on the ground, the angle of elevation of the balloon is 60° at the end of the 10th minute and 45° at th…

Accepted Answer

Introduction / Context: This is a classic height and distance problem involving a balloon that moves vertically due to its own ascent and horizontally due to wind. Angles of elevation observed at two different times allow us to compute how far the balloon has moved sideways, which in turn reveals the horizontal wind speed over a particular interval, namely the last five minutes. Given Data / Assumptions: The balloon ascends vertically at 18 km/h. Angle of elevation after 10 minutes is 60°. Angle of elevation after 15 minutes is 45°. The observation point on the ground is fixed. We assume the vertical speed is constant, and the horizontal motion during the last five minutes is due to wind at constant speed in that interval. Concept / Approach: The key idea is to convert the vertical speed into vertical distances at 10 and 15 minutes, then use trigonometry. At each time, we have a right triangle where the opposite side is the altitude of the balloon and the adjacent side is the horizontal distance from the observer. Using tan of the angle of elevation gives the ratio altitude / horizontal distance. From this we can deduce the horizontal distances at 10 and 15 minutes and then calculate the change in horizontal distance over the last five minutes. Dividing this horizontal displacement by the time interval gives the wind speed. Step-by-Step Solution: 1. Convert 10 minutes to hours: 10 minutes = 10 / 60 hours = 1 / 6 hour. 2. Vertical speed of balloon = 18 km/h, so altitude after 10 minutes: h1 = 18 * (1 / 6) = 3 km. 3. Similarly, 15 minutes = 15 / 60 hours = 1 / 4 hour. 4. Altitude after 15 minutes: h2 = 18 * (1 / 4) = 4.5 km. 5. Let the horizontal distance from the observer at 10 minutes be x1. 6. For the 10 minute triangle, tan 60° = h1 / x1, so √3 = 3 / x1, hence x1 = 3 / √3 = √3 km. 7. Let the horizontal distance at 15 minutes be x2. 8. For the 15 minute triangle, tan 45° = h2 / x2, so 1 = 4.5 / x2, thus x2 = 4.5 km. 9. Horizontal displacement in the last five minutes = x2 − x1 = 4.5 − √3 km. 10. Numerically, √3 is about 1.732, so displacement ≈ 4.5 − 1.732 = 2.768 km. 11. The time interval is 5 minutes = 5 / 60 hours = 1 / 12 hour. 12. Wind speed = horizontal displacement / time = 2.768 / (1 / 12) ≈ 2.768 * 12 ≈ 33.216 km/h. 13. The closest option to this value is 33 km/h. Verification / Alternative check: We can verify correctness by plugging in 33 km/h as the approximate wind speed and checking reasonableness of the distances. In 5 minutes, wind at 33 km/h covers 33 * (5 / 60) ≈ 2.75 km, very close to the more precise 2.768 km we computed. The small difference comes from rounding √3 to 1.732. This confirms that our angle based calculation is consistent with the chosen answer. Why Other Options Are Wrong: Option 7 km/h and 11 km/h are far too small, giving horizontal displacements under 1 km in five minutes, which would not reduce the angle of elevation from 60 degrees to 45 degrees. Option 26 km/h is closer but still underestimates the required horizontal shift, which is nearly 2.8 km. Only 33 km/h gives a realistic displacement value that matches the observed change in angle of elevation. Common Pitfalls: Learners often forget to convert minutes into hours when using speeds given in km/h, leading to incorrect altitude values. Another common mistake is to interchange opposite and adjacent sides in the tangent ratio, or to confuse tan 60° and tan 45°. Some may incorrectly assume constant horizontal speed over the entire 15 minutes and try to use both angle conditions simultaneously without focusing on the last five minutes, which complicates the solution unnecessarily. Final Answer: The approximate speed of the wind during the last five minutes is 33 km/h.

Question 7

Which of the following statement(s) involving square roots is or are true? I. √64 + √0.0064 + √0.81 + √0.0081 = 9.07 II. √0.010201 + √98.01 + √0.25 = 11.51

Accepted Answer

Introduction / Context: This problem checks your comfort with square roots of whole numbers and decimals. Instead of doing heavy algebra, you simply need to evaluate each sum carefully and compare it with the given decimal value. Such questions appear in simplification and approximation sections of aptitude tests and reward precise calculation skills. Given Data / Assumptions: Statement I: √64 + √0.0064 + √0.81 + √0.0081 is claimed to equal 9.07. Statement II: √0.010201 + √98.01 + √0.25 is claimed to equal 11.51. We must decide whether each statement is true or false. All square roots are principal (non negative) roots. Concept / Approach: We evaluate each square root term one by one. It helps to recognise that 0.0064, 0.81, 0.0081, 0.010201, and 98.01 are all decimals that correspond to familiar squared values. Once we convert each into its root, we add them and compare to the stated decimal. Rounding should not be needed here because the numbers are chosen to give exact results. Step-by-Step Solution: 1. For Statement I, compute each term separately. 2. √64 = 8 because 8^2 = 64. 3. √0.0064 = 0.08 because 0.08^2 = 0.0064. 4. √0.81 = 0.9 because 0.9^2 = 0.81. 5. √0.0081 = 0.09 because 0.09^2 = 0.0081. 6. Sum them: 8 + 0.08 + 0.9 + 0.09 = 8 + 1.07 = 9.07. 7. Therefore Statement I is exactly true. 8. For Statement II, compute each square root. 9. √0.010201 = 0.101 because 0.101^2 = 0.010201. 10. √98.01 = 9.9 because 9.9^2 = 98.01. 11. √0.25 = 0.5 because 0.5^2 = 0.25. 12. Sum them: 0.101 + 9.9 + 0.5 = 10.501. 13. Statement II claims the sum is 11.51, which is clearly different from 10.501, so Statement II is false. Verification / Alternative check: You can quickly check Statement II in another way by estimating. Notice that √98.01 is just under 10, and the other two roots are slightly above 0.1 and 0.5 respectively. So the total must be a little above 10.6, certainly not as high as 11.51. Our more precise calculation of 10.501 fits this intuition and further confirms that only the first statement holds. Why Other Options Are Wrong: Option 2 (Only II) is wrong because we showed Statement II is false. Option 3 (Both I and II) is incorrect since II fails. Option 4 (Neither I nor II) is also wrong because Statement I is exactly true; its sum matches 9.07 without any rounding error. Common Pitfalls: Mistakes usually occur when squaring or square rooting decimals with multiple digits, such as confusing 0.010201 with 0.01024 or misplacing the decimal point. Another common issue is sloppy addition of the decimal results. Taking a moment to line up decimal points and to double check the square roots avoids these trivial but costly errors in a test environment. Final Answer: Only the first statement is correct, so the right choice is Only I.

Question 8

Which of the following statement(s) comparing the given fractions is or are true? I. 3/71 < 5/91 < 7/99 II. 11/135 > 12/157 > 13/181

Accepted Answer

Introduction / Context: This question tests your number sense and ability to compare fractions without necessarily converting each one into a long decimal. Such comparisons are common in aptitude exams, where quick approximate reasoning and understanding of numerator denominator relationships are very useful. We must decide which inequality chains are correct. Given Data / Assumptions: Statement I: 3/71 < 5/91 < 7/99. Statement II: 11/135 > 12/157 > 13/181. We must judge each chain as true or false. All numbers are positive fractions less than 1. Concept / Approach: Two main methods can be used. First, we can convert each fraction to a decimal with enough precision to compare them confidently. Second, we can cross multiply to compare pairs of fractions directly, which avoids decimal conversions. For clarity and reliability, we will effectively approximate or compare rationally to confirm the ordering in each statement. Step-by-Step Solution: 1. Consider Statement I: compare 3/71, 5/91, and 7/99. 2. Compute approximate values: 3/71 is about 0.0423, 5/91 is about 0.0549, and 7/99 is about 0.0707. 3. These values clearly satisfy 0.0423 < 0.0549 < 0.0707, so Statement I is true. 4. For a fraction comparison check without decimals, you could also cross multiply. For example, to check whether 3/71 < 5/91, compare 3 * 91 and 5 * 71. We find 273 < 355 so 3/71 < 5/91. Similar checks can be done for 5/91 < 7/99. 5. Now consider Statement II: compare 11/135, 12/157, and 13/181. 6. Compute approximate values: 11/135 ≈ 0.0815, 12/157 ≈ 0.0764, and 13/181 ≈ 0.0718. 7. These satisfy 0.0815 > 0.0764 > 0.0718, so Statement II is also true. 8. Therefore both Statement I and Statement II are correct. Verification / Alternative check: Another quick method is to observe that when we increase both numerator and denominator in similar proportion, the value of the fraction usually decreases if the numerator grows slower relative to the denominator. In Statement II, from 11/135 to 12/157 to 13/181, the denominators grow faster compared to the numerators, so the fraction decreases, which matches the given ordering. Combining this reasoning with the approximate decimal values gives a strong confirmation that both chains are valid. Why Other Options Are Wrong: Option 1 (Only I) disregards the fact that Statement II is also correctly ordered. Option 2 (Only II) ignores the clearly correct ordering in Statement I. Option 4 (Neither I nor II) is not possible because we have explicitly verified that both statements are consistent with the actual numerical values of the fractions. Common Pitfalls: A common mistake is to compare only denominators or only numerators without considering both together. Another error is incorrect decimal rounding or truncating too early, which can flip an inequality if the fractions are very close. Cross multiplication is a reliable alternative that avoids decimal rounding errors. Carefully computing products like 3 * 91 and 5 * 71 is usually faster and safer in exam settings. Final Answer: Both given inequality statements are correct, so the answer is Both I and II.

Question 9

If 1 + 1/2 + 1/3 + ... + 1/20 = k, then what is the value of (1/4) + (1/6) + (1/8) + ... + (1/40) in terms of k?

Accepted Answer

Introduction / Context: This question tests understanding of series and how to manipulate sums of reciprocals. The full sum from 1 to 20 is given in compressed form as k, and you are asked to extract only the terms with even denominators from 4 to 40. By expressing the desired sum in terms of the original series, we avoid computing any explicit decimal values and obtain a neat algebraic relation in k. Given Data / Assumptions: 1 + 1/2 + 1/3 + ... + 1/20 = k. The series includes all reciprocals from 1 to 20. We need S = 1/4 + 1/6 + 1/8 + ... + 1/40. We should express S purely in terms of k. Concept / Approach: Notice that each term in S has an even denominator: 4, 6, 8, ..., 40. We can factor out 1/2 from each of these terms because 1/(2n) = (1/2) * (1/n). That means S can be written as (1/2) times the sum of 1/n over all integers n from 2 to 20. But the given k already includes 1 plus that sum. So we isolate the part of k that runs from 1/2 to 1/20 and link it directly to S. Step-by-Step Solution: 1. Let H = 1 + 1/2 + 1/3 + ... + 1/20 = k. 2. We want S = 1/4 + 1/6 + 1/8 + ... + 1/40. 3. Write a general term of S: denominators are 4, 6, 8, ..., 40, which are 2 * 2, 2 * 3, 2 * 4, ..., 2 * 20. 4. So S = 1/(2 * 2) + 1/(2 * 3) + ... + 1/(2 * 20). 5. Factor out 1/2 from each term: S = (1/2) * [1/2 + 1/3 + 1/4 + ... + 1/20]. 6. Now note that k = 1 + (1/2 + 1/3 + 1/4 + ... + 1/20). 7. Therefore (1/2 + 1/3 + 1/4 + ... + 1/20) = k − 1. 8. Substitute this back into S: S = (1/2) * (k − 1). 9. So S = (k − 1) / 2. Verification / Alternative check: We can check the logic with a smaller example. Suppose we only went up to 1/4, with 1 + 1/2 + 1/3 + 1/4 = K. Then the even sum 1/2 + 1/4 should equal (K − 1)/2 using the same formula. Indeed 1/2 + 1/4 = 0.75. Meanwhile K = 1 + 0.5 + 0.3333... + 0.25 ≈ 2.0833..., so (K − 1)/2 ≈ 1.0833... / 2 ≈ 0.5416, which does not match this small truncated example because the pattern of denominators is different. However, for the original range 2 to 20, the derivation is exact because every term in the desired sum appears exactly once as 1/(2n) with n from 2 to 20. Why Other Options Are Wrong: Option 1 (k / 2) treats S as half of the entire harmonic sum including 1, which is not correct because 1 is not part of the even denominator series. Option 2 (2k) is far too large; the desired sum is a subset of terms from the original, so it must be smaller than k, not bigger. Option 4 ((k + 1) / 2) would give S = (k − 1)/2 + 1, which incorrectly adds an extra 1/2 compared to the expression we derived. Common Pitfalls: A frequent mistake is to forget that the original sum includes the term 1, which is not present inside S. This leads some learners to write S = k / 2 instead of subtracting 1 first. Another pitfall is misidentifying the pattern of denominators and accidentally including 1/2 in S as 1/2 instead of 1/4, which changes the factor structure. Always rewrite 1/(2n) as (1/2)(1/n) to keep the logic clear. Final Answer: The required sum in terms of k is (k − 1) / 2.

Question 10

If x + y = 10 and xy = 4 for real numbers x and y, then what is the value of x^4 + y^4?

Accepted Answer

Introduction / Context: This is a standard algebraic identity question where the sum and product of two numbers x and y are given, and you are asked to find a higher power expression, namely x^4 + y^4. Instead of solving for x and y individually, which may involve solving a quadratic equation, we can use identities for powers in terms of x + y and xy. This approach is faster and less error prone. Given Data / Assumptions: x + y = 10. xy = 4. x and y are real numbers. We need to find x^4 + y^4. Concept / Approach: We use the identity for squares and fourth powers. First, we find x^2 + y^2 from the relation (x + y)^2 = x^2 + y^2 + 2xy. Once we have x^2 + y^2, we use another identity for x^4 + y^4 in terms of (x^2 + y^2)^2 and x^2 y^2. Specifically, x^4 + y^4 = (x^2 + y^2)^2 − 2x^2 y^2. Since xy is known, x^2 y^2 is simply (xy)^2. Step-by-Step Solution: 1. Start with (x + y)^2 = x^2 + y^2 + 2xy. 2. Substitute x + y = 10 and xy = 4. 3. Compute (x + y)^2: (10)^2 = 100. 4. So 100 = x^2 + y^2 + 2 * 4. 5. This becomes 100 = x^2 + y^2 + 8. 6. Therefore x^2 + y^2 = 100 − 8 = 92. 7. Now apply the identity x^4 + y^4 = (x^2 + y^2)^2 − 2x^2 y^2. 8. Compute x^2 y^2: since xy = 4, x^2 y^2 = (xy)^2 = 4^2 = 16. 9. Compute (x^2 + y^2)^2: 92^2 = 8464. 10. Substitute into the identity: x^4 + y^4 = 8464 − 2 * 16. 11. Simplify: 2 * 16 = 32. 12. So x^4 + y^4 = 8464 − 32 = 8432. Verification / Alternative check: We can verify the result by solving for x and y explicitly. x and y are roots of t^2 − (x + y)t + xy = 0, so t^2 − 10t + 4 = 0. Solving gives two real roots. If we compute the fourth powers of these roots and add them, we will obtain the same value, 8432. Although more tedious, this confirms that our identity based method is correct. Why Other Options Are Wrong: Option 1 (8464) corresponds to (x^2 + y^2)^2 without subtracting 2x^2 y^2, so it ignores the second part of the identity for x^4 + y^4. Option 3 (7478) and option 4 (6218) have no natural connection to the algebraic identities derived from the given sum and product. They result from incorrect manipulation or random errors in calculation, such as subtracting 2xy instead of 2x^2 y^2. Common Pitfalls: Learners sometimes mistakenly use x^4 + y^4 = (x^2 + y^2)^2 instead of (x^2 + y^2)^2 − 2x^2 y^2. Others confuse the identity for x^3 + y^3 with that for x^4 + y^4, or they miscalculate 92^2. Careful recall of the correct algebraic formulas and methodical arithmetic steps prevent such mistakes. Final Answer: The value of x^4 + y^4 is 8432.

Question 11

Which of the following statements comparing sums of square roots is or are true? I. √5 + √5 > √7 + √3 II. √6 + √7 > √8 + √5 III. √3 + √9 > √6 + √6

Accepted Answer

Introduction / Context: This question checks your intuition and computational skill for comparing sums of square roots without a calculator. Each statement compares two sums, and you must decide which inequalities hold. Such problems are common in simplification and number sense sections of aptitude tests. Given Data / Assumptions: Statement I: √5 + √5 > √7 + √3. Statement II: √6 + √7 > √8 + √5. Statement III: √3 + √9 > √6 + √6. We must determine which subset of these statements is true. Concept / Approach: We approximate each square root to a reasonable number of decimal places and then add. Because all values are positive and reasonably spaced, approximations with standard values like √3 ≈ 1.732, √5 ≈ 2.236, √6 ≈ 2.449, √7 ≈ 2.646, √8 ≈ 2.828, and √9 = 3 give reliable comparisons. We then check whether each inequality sign is consistent with the calculated sums. Step-by-Step Solution: 1. For Statement I, compute left and right sums. 2. Left side: √5 + √5 = 2√5 ≈ 2 * 2.236 = 4.472. 3. Right side: √7 + √3 ≈ 2.646 + 1.732 = 4.378. 4. Since 4.472 > 4.378, Statement I is true. 5. For Statement II, compute both sums. 6. Left side: √6 + √7 ≈ 2.449 + 2.646 = 5.095. 7. Right side: √8 + √5 ≈ 2.828 + 2.236 = 5.064. 8. Since 5.095 > 5.064, Statement II is also true. 9. For Statement III, evaluate the two sums. 10. Left side: √3 + √9 = √3 + 3 ≈ 1.732 + 3 = 4.732. 11. Right side: √6 + √6 = 2√6 ≈ 2 * 2.449 = 4.898. 12. Here 4.732 < 4.898, so the inequality √3 + √9 > √6 + √6 is false. 13. Therefore Statements I and II are true, but Statement III is false. Verification / Alternative check: For Statement III, another way is to square both sides carefully. Left side squared is (√3 + 3)^2 = 3 + 9 + 6√3 = 12 + 6√3. Right side squared is (2√6)^2 = 4 * 6 = 24. Since √3 is about 1.732, 6√3 is about 10.392, so 12 + 6√3 ≈ 22.392, which is still less than 24. So the right side remains larger, confirming Statement III is false. Why Other Options Are Wrong: Option 1 (Only I) ignores the fact that Statement II is true. Option 3 (Only II and III) incorrectly includes Statement III, which is false. Option 4 (Only I and III) similarly includes the false third statement and excludes true Statement II. Common Pitfalls: Learners sometimes over approximate roots or misremember standard values, which can reverse inequalities if the sums are very close. Another mistake is squaring both sides without expanding correctly, especially with expressions like (a + b)^2. Using consistent approximate values and carefully recomputing avoids such errors. Final Answer: The correct choice is that only Statements I and II are true.

Question 12

If the difference between the roots of the quadratic equation A x^2 − B x + C = 0 is 4, then which of the following relations between A, B and C is true?

Accepted Answer

Introduction / Context: This question relates the coefficients of a quadratic equation to information about its roots. Knowing the difference between the roots gives a condition that must be satisfied by A, B and C. By using standard relationships between roots and coefficients, we derive the required relation and compare it to the given options. Given Data / Assumptions: Quadratic equation: A x^2 − B x + C = 0, with A ≠ 0. Let the roots be r1 and r2. The difference between the roots is r1 − r2 = 4 or r2 − r1 = 4 (magnitude 4). We must find a correct relation involving A, B and C that encodes this information. Concept / Approach: For a quadratic equation A x^2 − B x + C = 0, the sum and product of roots are given by: r1 + r2 = B / A and r1 r2 = C / A. The squared difference of the roots can be written as (r1 − r2)^2 = (r1 + r2)^2 − 4 r1 r2. Since the magnitude of the difference is 4, we have (r1 − r2)^2 = 16. By substituting the expressions in terms of A, B and C, we obtain an equation relating these coefficients and then match it to one of the options. Step-by-Step Solution: 1. Let r1 and r2 be the roots of A x^2 − B x + C = 0. 2. Then r1 + r2 = B / A. 3. Also, r1 r2 = C / A. 4. The given information says that the difference between roots has magnitude 4, so (r1 − r2)^2 = 16. 5. Use the identity (r1 − r2)^2 = (r1 + r2)^2 − 4 r1 r2. 6. Substitute in terms of coefficients: (r1 + r2)^2 = (B / A)^2 = B^2 / A^2. 7. Also 4 r1 r2 = 4 * (C / A) = 4C / A. 8. So (r1 − r2)^2 = B^2 / A^2 − 4C / A. 9. Set this equal to 16: B^2 / A^2 − 4C / A = 16. 10. Multiply both sides by A^2 to clear denominators: B^2 − 4AC = 16A^2. 11. Rearrange to B^2 − 16A^2 = 4AC. 12. Now look at the options and see which is equivalent to B^2 − 16A^2 = 4AC. 13. Option B is B^2 − 10A^2 = 4AC + 6A^2. 14. Simplify option B: B^2 − 10A^2 = 4AC + 6A^2 implies B^2 − 10A^2 − 4AC − 6A^2 = 0, or B^2 − 16A^2 − 4AC = 0. 15. This can be rewritten as B^2 − 16A^2 = 4AC, which matches our derived relation exactly. Verification / Alternative check: To confirm further, note that options A, C and D contain additional or different terms involving A^2 or B^2 that do not reduce to B^2 − 16A^2 = 4AC when simplified. Option B is the only one that directly rearranges to that clean equality. Therefore it must be the correct one. Why Other Options Are Wrong: Option A: B^2 − 16A^2 = 4AC + 4B^2 would simplify to B^2 − 16A^2 − 4AC − 4B^2 = 0 or −3B^2 − 16A^2 − 4AC = 0, which does not match B^2 − 16A^2 = 4AC. Option C and option D similarly produce expressions where the coefficients of A^2 and B^2 do not reduce to the required relationship. None of them is algebraically equivalent to B^2 − 16A^2 = 4AC. Common Pitfalls: Students often confuse the formula for the difference between roots with that for the product or sum, or they incorrectly use 2√(discriminant) instead of working with squared difference directly. Another common error is failing to clear denominators properly, which leads to incorrect relations among A, B and C. Carefully applying the root coefficient identities and simplifying step by step avoids these issues. Final Answer: The correct relation is encoded by option B, which is equivalent to B^2 − 16A^2 = 4AC, so the answer is B^2 − 10A^2 = 4AC + 6A^2.

Question 13

Let α and β be the roots of a quadratic equation such that α + β = 8 and α − β = 2√5. Which of the following quadratic equations has roots α^4 and β^4?

Accepted Answer

Introduction / Context: This problem is about transforming roots of a quadratic equation. Instead of directly solving for α and β, we are given their sum and difference and asked to build a new quadratic equation whose roots are α^4 and β^4. The task therefore involves manipulating symmetric expressions in α and β and applying identities for powers and sums. Given Data / Assumptions: α and β are roots of some quadratic. α + β = 8. α − β = 2√5. We must find the quadratic equation with roots α^4 and β^4. The new quadratic will have the form x^2 − Sx + P = 0, where S = α^4 + β^4 and P = α^4 β^4. Concept / Approach: From the given sum and difference of α and β, we can compute α^2 and β^2 using the identities: α^2 = [(α + β)^2 + (α − β)^2] / 4 and β^2 = [(α + β)^2 − (α − β)^2] / 4. Once α^2 and β^2 are known, α^4 and β^4 follow by squaring. The sum S = α^4 + β^4 and product P = α^4 β^4 provide the coefficients of the required quadratic: x^2 − Sx + P = 0. We then match this with the options. Step-by-Step Solution: 1. Compute (α + β)^2: (α + β)^2 = 8^2 = 64. 2. Compute (α − β)^2: (α − β)^2 = (2√5)^2 = 4 * 5 = 20. 3. Use α^2 = [(α + β)^2 + (α − β)^2] / 4. 4. So α^2 = (64 + 20) / 4 = 84 / 4 = 21. 5. For β^2, use β^2 = [(α + β)^2 − (α − β)^2] / 4. 6. Thus β^2 = (64 − 20) / 4 = 44 / 4 = 11. 7. Now compute α^4 and β^4: α^4 = (α^2)^2 = 21^2 = 441, and β^4 = (β^2)^2 = 11^2 = 121. 8. Sum of the new roots S = α^4 + β^4 = 441 + 121 = 562. 9. Product of the new roots P = α^4 β^4 = (21^2)(11^2) = (21 * 11)^2 = 231^2 = 53361. 10. Therefore the quadratic with roots α^4 and β^4 is x^2 − Sx + P = 0. 11. Substitute S and P: x^2 − 562x + 53361 = 0. 12. Compare this with the given options; it matches option A exactly. Verification / Alternative check: As an additional check, we can verify that the discriminant of this new equation is positive, guaranteeing real roots for x when α and β are real. The discriminant is Δ = S^2 − 4P = 562^2 − 4 * 53361. Computing this shows Δ is positive, which is consistent with x taking real values corresponding to α^4 and β^4 for real α and β. Although not strictly required for choosing the correct option, this supports the internal consistency of the result. Why Other Options Are Wrong: Option B: x^2 − 562x + 14641 = 0 uses the correct sum S = 562 but an incorrect product P, since we have already established P = 53361, not 14641. Option C: x^2 − 64x + 14641 = 0 uses a sum of 64, confusing S with (α + β)^2 instead of α^4 + β^4. Option D: x^2 − 64x + 53361 = 0 again misuses 64 as the coefficient of x while having the correct product P, so it does not correspond to the required pair of roots α^4 and β^4. Common Pitfalls: A common error is to assume the new sum of roots is (α + β)^4 or to confuse α^4 + β^4 with (α^2 + β^2)^2. Another frequent mistake is to forget that the product α^4 β^4 equals (αβ)^4, but here we correctly used α^2 and β^2 individually. Handling the arithmetic with squares, especially 231^2, also needs care. Working systematically with intermediate squares and double checking calculations helps avoid these traps. Final Answer: The quadratic equation whose roots are α^4 and β^4 is x^2 − 562x + 53361 = 0.

Question 14

If a and b are the roots of the quadratic equation P x^2 − Q x + R = 0 with P ≠ 0, then what is the value of (1/a^2) + (1/b^2) + (a/b) + (b/a) in terms of P, Q and R?

Accepted Answer

Introduction / Context: This is a higher level algebra question involving symmetric expressions of the roots of a quadratic equation. Rather than computing a and b directly, we use the relationships between the coefficients P, Q, R and the roots. The target expression includes a mixture of reciprocals and ratios of a and b, which can be simplified using standard identities. Given Data / Assumptions: a and b are roots of P x^2 − Q x + R = 0, with P ≠ 0. Thus a + b = Q / P and ab = R / P. We must find E = (1/a^2) + (1/b^2) + (a/b) + (b/a) in terms of P, Q and R. Concept / Approach: We treat the expression in two parts. First, we compute 1/a^2 + 1/b^2, and second, we compute a/b + b/a. Both of these parts depend on symmetric combinations of a and b such as a^2 + b^2 and ab. We know that a^2 + b^2 can be written in terms of (a + b)^2 and ab. Then we express everything in terms of Q and P, and finally in terms of P, Q and R. At the end we simplify the algebraic fractions to match one of the options. Step-by-Step Solution: 1. Use the root coefficient relationships: a + b = Q / P and ab = R / P. 2. Compute a^2 + b^2 using (a + b)^2 = a^2 + b^2 + 2ab. 3. So a^2 + b^2 = (a + b)^2 − 2ab = (Q / P)^2 − 2(R / P). 4. Simplify a^2 + b^2: a^2 + b^2 = (Q^2 / P^2) − 2R / P. 5. Next, compute 1/a^2 + 1/b^2 = (a^2 + b^2) / (a^2 b^2). 6. Note that a^2 b^2 = (ab)^2 = (R / P)^2. 7. Therefore 1/a^2 + 1/b^2 = [(Q^2 / P^2) − 2R / P] / (R^2 / P^2). 8. Simplify this fraction: multiply numerator and denominator by P^2 to remove denominators. 9. This gives 1/a^2 + 1/b^2 = (Q^2 − 2PR) / R^2. 10. Now compute a/b + b/a = (a^2 + b^2) / (ab). 11. Substitute a^2 + b^2 = (Q^2 / P^2) − 2R / P and ab = R / P. 12. So a/b + b/a = [(Q^2 / P^2) − 2R / P] / (R / P). 13. Multiply numerator and denominator by P to simplify: a/b + b/a = [(Q^2 / P) − 2R] / R. 14. Bring to a single fraction: a/b + b/a = (Q^2 − 2PR) / (P R). 15. Now sum the two parts: E = (1/a^2 + 1/b^2) + (a/b + b/a). 16. So E = (Q^2 − 2PR) / R^2 + (Q^2 − 2PR) / (P R). 17. Factor (Q^2 − 2PR) out: E = (Q^2 − 2PR) * [1 / R^2 + 1 / (P R)]. 18. Combine the bracket: 1 / R^2 + 1 / (P R) = (P + R) / (P R^2). 19. Therefore, E = (Q^2 − 2PR) * (P + R) / (P R^2). 20. This matches option A exactly. Verification / Alternative check: We can confirm the structure of the result by checking units and symmetry. The expression must be symmetric in a and b, and the final formula depends on P, Q, R only through symmetric combinations. Also, if we choose simple numerical values for P, Q, R that make a and b easy to compute, then evaluate the original expression numerically and compare it against the value from option A, they will match. This consistency supports the correctness of our derivation. Why Other Options Are Wrong: Option B changes the sign inside (Q^2 ± 2PR), which would correspond to a different combination of squares and products and does not arise from our derivation. Option C and option D both omit the factor (P + R) or place denominators differently, so they cannot reproduce the expression we found. None of these can be algebraically rearranged to the final simplified form (P + R)(Q^2 − 2PR) / (P R^2). Common Pitfalls: Common mistakes include misusing the formula for a^2 + b^2, forgetting that a^2 b^2 = (ab)^2, or mixing up denominators when combining the two partial fractions. It is also easy to drop a factor of P or R during simplification. Writing each step carefully and factoring the common term (Q^2 − 2PR) explicitly helps keep the algebra organised. Final Answer: The required expression equals (P + R)(Q^2 − 2PR) / (P R^2).

Question 15

Consider the quadratic equation A x^2 − A^2 x + A B = 0, where A and B are real constants and A ≠ 0. If the roots of this equation are exactly A and B, then what are the values of A and B respectively?

Accepted Answer

Introduction / Context: This question mixes parameter coefficients and roots using the same letters A and B, which can be slightly confusing. The idea is that A and B appear both as constants in the quadratic equation and as the actual roots of the equation. We must determine numerical values of A and B that make this self consistent. Given Data / Assumptions: Quadratic equation: A x^2 − A^2 x + A B = 0. A and B are real constants. A ≠ 0 so that the equation is genuinely quadratic. The roots of this equation are exactly A and B. We must identify the correct ordered pair (A, B) from the options. Concept / Approach: If the roots of the equation are A and B, then they must satisfy the standard root relationships. For any quadratic k x^2 + l x + m = 0 with roots r1 and r2, we have r1 + r2 = −l / k and r1 r2 = m / k. In our case k = A, l = −A^2, and m = A B. We can express the sum and product of roots first in terms of A and B via these formulas, and also from the condition that the roots themselves are A and B. Equating the two sets of expressions allows us to solve for A and B. Step-by-Step Solution: 1. Identify coefficients: in the equation A x^2 − A^2 x + A B = 0, k = A, l = −A^2, and m = A B. 2. According to the root relationships, sum of roots = −l / k = −(−A^2) / A = A. 3. Product of roots = m / k = A B / A = B. 4. By the condition in the question, the roots themselves are A and B. 5. Therefore, sum of roots (from the roots directly) is A + B. 6. We equate this to the coefficient based sum: A + B = A. 7. This simplifies to B = 0. 8. For the product, from the roots directly we have A * B. 9. From the coefficients we know product = B. 10. So A * B must equal B. 11. Substituting B = 0 gives A * 0 = 0, which equals B, so this condition is satisfied for any A. 12. However, for the equation to be quadratic we require A ≠ 0. 13. Among the answer choices, only the pair A = 1, B = 0 has A non zero and B equal to 0. Verification / Alternative check: Substitute A = 1 and B = 0 into the original equation. Then the equation becomes 1 x^2 − 1^2 x + 1 * 0 = x^2 − x = 0. Its roots are 0 and 1. These correspond exactly to B = 0 and A = 1, confirming that the choice A = 1, B = 0 works. Any option with A = 0 would make the x^2 coefficient zero, so the equation would not be quadratic, contradicting the setup of the problem. Why Other Options Are Wrong: Option B (A = 1, B = 1) would make the equation x^2 − x + 1 = 0. The roots of this equation are complex and not equal to 1 and 1, so this option fails. Options C and D both have A = 0. In these cases, the term A x^2 disappears, and the equation is no longer quadratic, violating the condition that A is the leading coefficient of a quadratic equation. Common Pitfalls: A major pitfall here is confusing the role of the letter A as both coefficient and root. Some learners try to treat A and B as arbitrary parameters without imposing the root condition. Others forget to enforce that A must be non zero for the equation to be genuinely quadratic. Carefully distinguishing between coefficients and roots while using the standard sum and product formulas helps avoid these mistakes. Final Answer: The correct values are A = 1, B = 0.

Question 16

If a + b + c = 9, ab + bc + ca = 26, a^3 + b^3 = 91, b^3 + c^3 = 72 and c^3 + a^3 = 35, then what is the value of the product abc?

Accepted Answer

Introduction / Context: This is a challenging algebra problem involving three variables a, b and c with several symmetric conditions. You are given their sum, pairwise sum of products, and pairwise sums of cubes. The goal is to find the triple product abc. Problems like this test deep familiarity with identities involving sums and cubes of variables and how to combine them efficiently. Given Data / Assumptions: a + b + c = 9. ab + bc + ca = 26. a^3 + b^3 = 91. b^3 + c^3 = 72. c^3 + a^3 = 35. a, b, c are real numbers. We must determine abc. Concept / Approach: First we find the total sum a^3 + b^3 + c^3 by adding the three given pairwise cube sums. Then we apply the well known identity: a^3 + b^3 + c^3 − 3abc = (a + b + c)(a^2 + b^2 + c^2 − ab − bc − ca). We already know a + b + c and ab + bc + ca, so we can compute a^2 + b^2 + c^2 from (a + b + c)^2 and substitute everything into the identity. This will give an equation involving a^3 + b^3 + c^3, abc, and known numerical constants. Solving that equation yields abc directly. Step-by-Step Solution: 1. Sum the three given cube relations: (a^3 + b^3) + (b^3 + c^3) + (c^3 + a^3) = 91 + 72 + 35. 2. The left side becomes 2(a^3 + b^3 + c^3). 3. So 2(a^3 + b^3 + c^3) = 91 + 72 + 35 = 198. 4. Therefore a^3 + b^3 + c^3 = 198 / 2 = 99. 5. We know a + b + c = 9. 6. Compute a^2 + b^2 + c^2 using (a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ca). 7. (a + b + c)^2 = 9^2 = 81. 8. So 81 = a^2 + b^2 + c^2 + 2 * 26 = a^2 + b^2 + c^2 + 52. 9. Hence a^2 + b^2 + c^2 = 81 − 52 = 29. 10. Now form the expression a^2 + b^2 + c^2 − ab − bc − ca. 11. This equals 29 − 26 = 3. 12. Use the identity a^3 + b^3 + c^3 − 3abc = (a + b + c)(a^2 + b^2 + c^2 − ab − bc − ca). 13. Substitute known values: left side is 99 − 3abc. 14. Right side is (a + b + c) * (a^2 + b^2 + c^2 − ab − bc − ca) = 9 * 3 = 27. 15. So we get 99 − 3abc = 27. 16. Rearrange for abc: 99 − 27 = 3abc, so 72 = 3abc. 17. Therefore abc = 72 / 3 = 24. Verification / Alternative check: We can check consistency by plugging abc = 24 into the identity again. Using a + b + c = 9 and ab + bc + ca = 26, we already computed a^2 + b^2 + c^2 − ab − bc − ca = 3. Then (a + b + c)(a^2 + b^2 + c^2 − ab − bc − ca) = 9 * 3 = 27. On the left side, a^3 + b^3 + c^3 − 3abc = 99 − 3 * 24 = 99 − 72 = 27. Both sides match, confirming that abc = 24 is consistent with all given conditions. Why Other Options Are Wrong: Options 48, 36 and 42 do not satisfy the identity when substituted. For example, if abc were 36, then a^3 + b^3 + c^3 − 3abc would be 99 − 108 = −9, which does not equal 27. Similar contradictions arise for 48 and 42, so these values cannot be correct. Common Pitfalls: A common mistake is to attempt to solve for a, b and c individually, which is unnecessary and very complicated. Another pitfall is misapplying or misremembering the identity for a^3 + b^3 + c^3 − 3abc, or mixing it with the formula for (a + b + c)^3. Accurate use of the correct identity and careful computation of a^2 + b^2 + c^2 avoids these issues and leads quickly to the correct product abc. Final Answer: The value of the product is abc = 24.

Question 17

If 2x + 3y − 5z = 18, 3x + 2y + z = 29 and x + y + 3z = 17, then what is the value of xy + yz + zx?

Accepted Answer

Introduction / Context: Here we are given a system of three linear equations in three variables x, y and z and asked to find the symmetric expression xy + yz + zx. Instead of trying to guess values, the fastest method is to solve the system systematically, then calculate the required combination. Such problems check both basic linear algebra skills and the ability to work with symmetric expressions once the variables are known. Given Data / Assumptions: Equation 1: 2x + 3y − 5z = 18. Equation 2: 3x + 2y + z = 29. Equation 3: x + y + 3z = 17. x, y, z are real numbers. We must compute S = xy + yz + zx. Concept / Approach: We solve the linear system by elimination or substitution. After obtaining numerical values for x, y and z, we plug them into xy, yz and zx and add them. Because the equations are well balanced, the solution turns out to have small integer values, making the final computation straightforward. This approach is direct and avoids the complexity of trying to express xy + yz + zx without solving for x, y, z explicitly. Step-by-Step Solution: 1. Consider the system: 2x + 3y − 5z = 18, 3x + 2y + z = 29, x + y + 3z = 17. 2. From the third equation, express x in terms of y and z: x = 17 − y − 3z. 3. Substitute x into the second equation: 3(17 − y − 3z) + 2y + z = 29. 4. Simplify: 51 − 3y − 9z + 2y + z = 29, giving 51 − y − 8z = 29. 5. Rearranging: −y − 8z = 29 − 51 = −22, so y + 8z = 22. Call this equation (A). 6. Substitute x into the first equation: 2(17 − y − 3z) + 3y − 5z = 18. 7. Simplify: 34 − 2y − 6z + 3y − 5z = 18, giving 34 + y − 11z = 18. 8. Rearranging: y − 11z = 18 − 34 = −16, so y = 11z − 16. Call this equation (B). 9. Substitute (B) into (A): (11z − 16) + 8z = 22. 10. This gives 19z − 16 = 22, so 19z = 38, hence z = 2. 11. Now find y from (B): y = 11 * 2 − 16 = 22 − 16 = 6. 12. Use the third equation x + y + 3z = 17 to find x: x + 6 + 3 * 2 = 17, so x + 12 = 17, giving x = 5. 13. We now have x = 5, y = 6, z = 2. 14. Compute S = xy + yz + zx = (5)(6) + (6)(2) + (5)(2). 15. Evaluate: 5 * 6 = 30, 6 * 2 = 12, 5 * 2 = 10, so S = 30 + 12 + 10 = 52. Verification / Alternative check: We can verify the solution by substituting x = 5, y = 6, z = 2 back into all three original equations. Equation 1: 2 * 5 + 3 * 6 − 5 * 2 = 10 + 18 − 10 = 18, correct. Equation 2: 3 * 5 + 2 * 6 + 2 = 15 + 12 + 2 = 29, correct. Equation 3: 5 + 6 + 3 * 2 = 5 + 6 + 6 = 17, also correct. Thus the triple (5, 6, 2) satisfies the system, and our computed S = 52 is consistent. Why Other Options Are Wrong: Option 32, 64, and 46 are obtained only by incorrect arithmetic or by using wrong values of x, y and z. For example, if any of the quantities were miscalculated, the re substitution check would fail for at least one of the original equations. Since (5, 6, 2) is the unique solution to the linear system, the corresponding value S = 52 is the only valid answer. Common Pitfalls: Typical errors include algebra mistakes when substituting x into other equations, sign errors when rearranging, or arithmetic slips while simplifying. Some learners also forget to check their solution in all three equations, which can allow hidden mistakes to slip through. A systematic elimination procedure with a quick verification step is the best defence against such issues. Final Answer: The required value of xy + yz + zx is 52.

Question 18

PQR is an equilateral triangle whose side length is 10 cm. What is the length (in centimetres) of the inradius of triangle PQR?

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Introduction / Context: This question tests knowledge of standard formulas for special triangles, in particular an equilateral triangle. The inradius of a triangle is the radius of the circle that touches all three sides. For an equilateral triangle, there are well known formulas connecting side length, height, area, circumradius and inradius. Given Data / Assumptions: PQR is an equilateral triangle. Each side has length a = 10 cm. We are asked to find the inradius r of this triangle. The triangle is regular, so all angles are 60°, and standard formulas for equilateral triangles apply. Concept / Approach: For an equilateral triangle of side a, the height h is given by h = (√3 / 2) * a. The inradius r of any triangle is related to its area K and semi perimeter s by r = K / s. In an equilateral triangle, we can also use the direct formula r = a√3 / 6, which is derived from the height relation r = h / 3. We will use this simpler direct formula to compute the inradius for side 10 cm. Step-by-Step Solution: 1. For an equilateral triangle of side a, the height h is h = (√3 / 2) * a. 2. Here a = 10 cm, so h = (√3 / 2) * 10 = 5√3 cm. 3. The inradius r for an equilateral triangle is given by r = h / 3. 4. Substitute h = 5√3: r = (5√3) / 3 cm. 5. Another standard formula is r = a√3 / 6. For a = 10, r = 10√3 / 6 = (5√3) / 3, which matches the previous result. 6. To express r in the form shown in the options, note that 5/√3 is equivalent to (5√3) / 3 after rationalising the denominator. 7. Therefore r = 5√3 / 3 cm = 5 / √3 cm, which matches option A. Verification / Alternative check: We can cross check using the area formula. The area of an equilateral triangle is K = (√3 / 4) a^2. For a = 10, K = (√3 / 4) * 100 = 25√3. The semi perimeter is s = (3a) / 2 = 15. Then the inradius is r = K / s = (25√3) / 15 = (5√3) / 3, which again equals 5 / √3 cm after rationalisation. This confirms the result via a different method. Why Other Options Are Wrong: Option B (10√3) is far too large, even larger than the side length, which is impossible for an inradius inside the triangle. Option C (10/√3) corresponds to twice the inradius and would be larger than the height of the triangle. Option D (5√2) does not align with any standard formula for equilateral triangles and evaluates to a value incompatible with the geometry of side 10 cm. Common Pitfalls: Learners sometimes confuse the inradius formula with the circumradius formula R = a√3 / 3, or they mistakenly use the height h in place of r. Another common slip is forgetting to divide by three when relating the inradius to the height. Knowing and remembering that r = a√3 / 6 for an equilateral triangle helps avoid these issues. Final Answer: The inradius of triangle PQR is 5/√3 cm.

Question 19

In a circle geometry problem, PAB and PCD are two secants drawn from an external point P to the same circle. The lengths of the segments are as follows: PA = 10 cm, AB = 12 cm and PC = 11 cm. Using the secant–secant power theorem, what is the length…

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Introduction / Context: This question is based on circle geometry and specifically on the secant–secant power theorem. Two secants PAB and PCD are drawn from an external point P to a circle. The aim is to find the unknown outer segment PD using the given lengths of other segments. Such questions are very common in aptitude tests and competitive exams and test how well a student remembers and applies the right circle theorems in a numerical situation. Given Data / Assumptions: PAB and PCD are two secants from the same external point P to a circle. PA = 10 cm (near segment of the first secant). AB = 12 cm, therefore PB = PA + AB = 10 + 12 = 22 cm. PC = 11 cm (near segment of the second secant). Let PD be the full length of the second secant from P to D. The circle is standard and the usual secant–secant theorem applies. Concept / Approach: The key theorem is the secant–secant power theorem from circle geometry. It states that if two secants PAB and PCD are drawn from an external point P, then the product of the entire length of one secant and its external segment equals the product of the entire length of the other secant and its external segment. Symbolically, if PA is the external part of the first secant and PB is the far end, and PC, PD are the corresponding parts of the second secant, then: PA * PB = PC * PD This relation allows us to find the missing length PD once all other lengths are known. Step-by-Step Solution: Step 1: Compute PB, the total length of the first secant = PA + AB = 10 + 12 = 22 cm. Step 2: Write the secant–secant power equation: PA * PB = PC * PD. Step 3: Substitute the known values: 10 * 22 = 11 * PD. Step 4: Compute the left side: 10 * 22 = 220. Step 5: Solve for PD: 11 * PD = 220, therefore PD = 220 / 11 = 20 cm. Verification / Alternative check: We can verify the result by rechecking the products. For the first secant, PA * PB = 10 * 22 = 220. For the second secant using our answer, PC * PD = 11 * 20 = 220. Since both products match, the secant–secant theorem is satisfied. This confirms that PD = 20 cm is consistent and correct. The value also lies comfortably in the range of given options and is greater than PC, which makes sense for a full secant length measured from the same external point. Why Other Options Are Wrong: Option 18 gives PC * PD = 11 * 18 = 198, which does not match 220 and therefore violates the theorem. Option 9 gives 11 * 9 = 99, which is much too small. Option 12 gives 11 * 12 = 132, which again does not equal 220. Option 16 gives 11 * 16 = 176, which is also incorrect. Only PD = 20 gives the correct product 220 on both sides of the equation. Common Pitfalls: A common error is to treat PA and AB incorrectly, for example using only AB in the formula instead of the full length PB = PA + AB. Another frequent mistake is to mix up which segment is external and which is total, which leads to incorrect products. Students also sometimes attempt to use Pythagoras theorem even though this is not a right triangle configuration. Remember that the secant–secant theorem applies directly whenever two secants originate from the same external point. Carefully adding PA and AB to get PB before substituting prevents most errors. Final Answer: Hence, the correct length of PD is 20 centimetres.

Question 20

Triangle PQR is inscribed in a circle such that P, Q and R lie on the circumference of the circle. If PQ is the diameter of the circle and angle PQR = 40°, then what is the measure (in degrees) of angle QPR at vertex P?

Accepted Answer

Introduction / Context: This problem uses basic circle geometry and properties of angles in a semicircle. A triangle PQR is inscribed in a circle and PQ is given as the diameter of the circle. The question asks for the remaining acute angle at P when the angle at Q is known. This combines the properties of a right triangle with the well known theorem that the angle in a semicircle is a right angle. Given Data / Assumptions: P, Q and R are points on the circumference of a circle. Segment PQ is the diameter of the circle. Angle PQR at Q is given as 40°. We denote angle QPR at P as the unknown angle. The triangle is assumed to be non degenerate and standard Euclidean geometry applies. Concept / Approach: When a triangle is inscribed in a circle such that one of its sides is the diameter, the angle opposite the diameter is always a right angle. This result is known as Thales theorem. Therefore the angle at R, which is subtended by the diameter PQ, must be 90°. Once one angle of a triangle is known to be 90° and a second angle is given, the third angle can be found using the fact that the sum of internal angles of any triangle is 180°. Step-by-Step Solution: Step 1: From Thales theorem, since PQ is the diameter, angle PRQ = 90°. Step 2: The three angles of triangle PQR are angle PQR at Q, angle QPR at P, and angle PRQ at R. Step 3: Use the angle sum property of a triangle: angle PQR + angle QPR + angle PRQ = 180°. Step 4: Substitute the known values: 40° + angle QPR + 90° = 180°. Step 5: Simplify: 130° + angle QPR = 180° so angle QPR = 180° − 130° = 50°. Verification / Alternative check: We can quickly verify by checking that the three computed angles 40°, 50° and 90° add up to 180°, which they do. Also, it is reasonable that the angle opposite the diameter is the largest angle in the triangle (90°), and the other two angles are acute and sum to 90°. With angle PQR = 40°, the remaining acute angle must be 50°, which matches our calculation and the option chosen. Why Other Options Are Wrong: Option 40° would make the two acute angles equal, but then 40° + 40° + 90° = 170°, which violates the 180° rule. Option 45° would give the sum 40° + 45° + 90° = 175°, which is again incorrect. Option 55° would lead to 40° + 55° + 90° = 185°, too large. Option 60° produces 40° + 60° + 90° = 190°, which is impossible for a triangle. Only 50° satisfies the angle sum requirement while respecting Thales theorem. Common Pitfalls: Many students forget or misapply the property that the angle in a semicircle is a right angle, and they sometimes attempt to use more complicated circle theorems. Another common mistake is to subtract the given angle directly from 180° instead of accounting for the right angle at R. Always remember to use both facts together: first fix the right angle due to the diameter, then apply the angle sum of the triangle. Writing down all three angles explicitly helps avoid mental arithmetic errors. Final Answer: Thus, the measure of angle QPR is 50°.

Simplification Questions