Question 1

Using exact trigonometric ratios, evaluate the value of sec 45° + tan 30° and express the result in simplest surd form.

Accepted Answer

Introduction / Context: This question checks your recall of exact trigonometric values for 30° and 45° and your ability to manipulate surds. You must compute sec 45° and tan 30°, then add them and simplify the result into a single compact expression involving square roots, as commonly required in exam style answers. Given Data / Assumptions: Expression: sec 45° + tan 30°. Angles are measured in degrees. Known exact values: sin 30°, cos 30°, sin 45°, cos 45°. We aim for a simplified surd expression. Concept / Approach: We use the definitions sec θ = 1 / cos θ and tan θ = sin θ / cos θ. For 45°, cos 45° = √2/2, so sec 45° is its reciprocal. For 30°, sin 30° = 1/2 and cos 30° = √3/2, so tan 30° = sin 30° / cos 30°. After substituting, we add the two resulting terms and then combine them into a single fraction with a common denominator, simplifying any surds that arise. Step-by-Step Solution: cos 45° = √2/2, so sec 45° = 1 / (√2/2) = 2/√2 = √2. For 30°, sin 30° = 1/2 and cos 30° = √3/2. Hence tan 30° = (1/2) / (√3/2) = 1/√3. Therefore sec 45° + tan 30° = √2 + 1/√3. To combine into a single fraction, express √2 with denominator √3. Write √2 = √2 · (√3/√3) = √6/√3. So √2 + 1/√3 = (√6/√3) + (1/√3) = (√6 + 1)/√3. Verification / Alternative check: Check numerically: √2 ≈ 1.414 and 1/√3 ≈ 0.577. Their sum is about 1.991. Now compute (√6 + 1)/√3: √6 ≈ 2.449, so numerator is roughly 3.449. Divide by √3 ≈ 1.732 to get about 1.991 again. The close match confirms that the simplified form (√6 + 1)/√3 is correct. Why Other Options Are Wrong: (1 + √3)/2 evaluates to roughly 1.366, which does not match 1.991. (√3 + 2)/√3 is about (1.732 + 2)/1.732 ≈ 2.155. The choice 5/√3 is about 2.887, which is far too large. The unsimplified form √2 + 1/√3 is equivalent numerically, but the question expects a single fraction. Only (√6 + 1)/√3 is both simplified and equal to the original expression. Common Pitfalls: Mixing up tan 30° with tan 60° is a common mistake, as is forgetting to rationalize or combine terms into a single fraction when needed. Some learners miscalculate sec 45° as 1/√2 instead of √2. Carefully using definitions and standard values for special angles helps avoid these errors. Final Answer: Thus, sec 45° + tan 30° simplifies exactly to (√6 + 1)/√3.

Question 2

In right triangle DEF, angle E is 90°. If m∠D = 45°, then what is the exact value of cosec F, where F is the remaining acute angle of the triangle?

Accepted Answer

Introduction / Context: This question involves basic properties of right triangles and trigonometric ratios of standard angles. When one acute angle of a right triangle is known, the other acute angle is determined, and we can use known sine values to compute cosecant. Here we work with a 45°–45°–90° triangle where both acute angles are equal. Given Data / Assumptions: Triangle DEF is right angled at E, so m∠E = 90°. m∠D = 45°. Angles of a triangle sum to 180°. cosec θ = 1 / sin θ. Concept / Approach: Since the triangle is right angled at E and one acute angle is 45°, the third angle must also be 45° because 90° + 45° + 45° = 180°. That means angles D and F are both 45°. The sine of 45° is √2/2, so the cosecant of 45° is its reciprocal, which simplifies to √2. We simply apply these standard ratios to find cosec F. Step-by-Step Solution: Sum of angles: m∠D + m∠E + m∠F = 180°. Given m∠E = 90° and m∠D = 45°. So m∠F = 180° − 90° − 45° = 45°. We need cosec F = cosec 45°. Recall sin 45° = √2/2. Hence cosec 45° = 1 / sin 45° = 1 / (√2/2) = 2/√2 = √2. Therefore cosec F = √2. Verification / Alternative check: Consider a 45°–45°–90° triangle with legs of length 1 and hypotenuse √2. For angle F, the sine is opposite/hypotenuse = 1/√2 = √2/2. The reciprocal of this value is √2, which matches the calculated cosec F. This geometric picture confirms the result derived from standard trigonometric values. Why Other Options Are Wrong: 1/2 and 1/√2 are values associated with sine or cosine of 30° or 45°, not with cosecant. 1/√3 corresponds to trigonometric ratios for 30° or 60°, not for 45°. The value 2 would be cosec 30° in a 30°–60°–90° triangle, not in a 45°–45°–90° triangle. Only √2 matches the exact value of cosec 45° and therefore cosec F. Common Pitfalls: Some learners mistakenly assume that if D is 45°, then F might be 30° or 60°, forgetting that E is already 90°. Others mix up the roles of sine and cosecant or fail to simplify 2/√2 to √2. Remember that the reciprocal of √2/2 is √2 and that in a right triangle the two acute angles must sum to 90°. Final Answer: Therefore, the value of cosec F is √2.

Question 3

If cos θ = 5/13 for an acute angle θ in a right triangle, then using the Pythagorean identity, what is the exact value of cosec θ?

Accepted Answer

Introduction / Context: This question links cosine and cosecant using the fundamental Pythagorean identity sin² θ + cos² θ = 1. Given cos θ for an acute angle, you can determine sin θ and then compute its reciprocal to obtain cosec θ. This is a standard exercise in basic trigonometry, especially in problems involving right triangles and exact ratios. Given Data / Assumptions: cos θ = 5/13. θ is an acute angle (0° < θ < 90°). sin² θ + cos² θ = 1. cosec θ = 1 / sin θ. Concept / Approach: From cos θ we can imagine a right triangle where the adjacent side is 5 units and the hypotenuse is 13 units. Using Pythagoras, the opposite side can be found as √(13² − 5²). This gives sin θ as opposite/hypotenuse and therefore cosec θ as hypotenuse/opposite. Alternatively, we can find sin² θ directly from the identity sin² θ = 1 − cos² θ and then take the positive square root because θ is acute. Step-by-Step Solution: Given cos θ = 5/13. Compute cos² θ = (5/13)² = 25/169. Use identity sin² θ + cos² θ = 1. So sin² θ = 1 − cos² θ = 1 − 25/169. Write 1 as 169/169: sin² θ = 169/169 − 25/169 = 144/169. Since θ is acute, sin θ = √(144/169) = 12/13. Now cosec θ = 1 / sin θ = 1 / (12/13) = 13/12. Verification / Alternative check: Visualize the triangle: adjacent side = 5, hypotenuse = 13. Then the opposite side is √(13² − 5²) = √(169 − 25) = √144 = 12. Thus sin θ = opposite/hypotenuse = 12/13, and cosec θ = hypotenuse/opposite = 13/12. This geometric interpretation perfectly matches the algebraic calculation and confirms the answer. Why Other Options Are Wrong: 5/12 and 12/13 represent cos/sin or sin/cos combinations but not cosec θ here. 12/5 is the reciprocal of 5/12 and does not correspond to any primary ratio for θ given cos θ = 5/13. The value 13/5 is much larger and would not match the reciprocal of sin θ. Only 13/12 is consistent with the derived value of cosec θ for this triangle. Common Pitfalls: Some learners may attempt to invert cos θ directly to get cosec θ, which is incorrect because cosec θ is the reciprocal of sin θ, not cos θ. Others might mistakenly choose the negative square root when solving for sin θ, forgetting that sin θ is positive in the first quadrant. Carefully applying the Pythagorean identity and the definitions avoids these errors. Final Answer: Hence, the value of cosec θ is 13/12.

Question 4

Solve the fractional linear equation (10x/3) + (5/2)(2 − x/3) = 7/2 and find the exact value of x.

Accepted Answer

Introduction / Context: This equation involves several fractional terms and a bracket, making it a good test of algebraic manipulation. You need to expand correctly, combine like terms, and then solve the resulting linear equation. Such problems are standard in aptitude tests under the topic of simplification or linear equations. Given Data / Assumptions: Equation: (10x/3) + (5/2)(2 − x/3) = 7/2. x is a real unknown. All denominators (3 and 2) are non zero. Standard rules for distribution and fraction arithmetic apply. Concept / Approach: The equation can be simplified by first expanding (5/2)(2 − x/3). Then, we add the resulting x terms together and move all terms to one side to isolate x. It can be helpful to clear denominators at some stage by multiplying both sides of the equation by a common multiple of 2 and 3, which is 6. This converts the equation into one with integer coefficients and reduces arithmetic errors. Step-by-Step Solution: Start from (10x/3) + (5/2)(2 − x/3) = 7/2. Expand the bracket: (5/2)(2 − x/3) = (5/2)·2 − (5/2)(x/3). This simplifies to 5 − (5x/6). So the equation becomes (10x/3) + 5 − (5x/6) = 7/2. Rewrite 10x/3 with denominator 6: 10x/3 = 20x/6. Thus 20x/6 − 5x/6 + 5 = 7/2. Combine x terms: (20x/6 − 5x/6) = 15x/6 = 5x/2. So we have 5x/2 + 5 = 7/2. Subtract 5 from both sides: 5x/2 = 7/2 − 5. Write 5 as 10/2: 7/2 − 10/2 = −3/2. Thus 5x/2 = −3/2. Multiply both sides by 2: 5x = −3, so x = −3/5. Verification / Alternative check: Substitute x = −3/5 into the original equation. Compute 10x/3 = 10(−3/5)/3 = (−6)/3 = −2. Next, 2 − x/3 becomes 2 − (−3/5)/3 = 2 + 3/15 = 2 + 1/5 = 11/5. Then (5/2)(2 − x/3) = (5/2)(11/5) = 11/2. The left side is −2 + 11/2 = −4/2 + 11/2 = 7/2, which equals the right side. This confirms that x = −3/5 is correct. Why Other Options Are Wrong: If you substitute 3/5, −5/3 or 5/3 into the equation, the left side does not simplify to 7/2. The option 0 gives a left side equal to 5, not 7/2. Only x = −3/5 satisfies the equation and balances both sides exactly. Common Pitfalls: Students may incorrectly expand (5/2)(2 − x/3) or mishandle the conversion of 10x/3 to a fraction with denominator 6. Some might forget to adjust both sides of the equation when clearing denominators or moving terms, causing sign errors. Following each algebraic step carefully and double checking arithmetic with fractions helps prevent such mistakes. Final Answer: Therefore, the solution to the equation is x = −3/5.

Question 5

If a − b = 2 and ab = 15 for real numbers a and b, then using the identity for the difference of cubes, what is the value of a³ − b³?

Accepted Answer

Introduction / Context: This problem uses the algebraic identity for the difference of cubes to relate a³ − b³ to the difference a − b and the product ab. By combining this identity with the square of the difference, we can find a³ − b³ without explicitly solving for a and b, which is a standard technique in algebra and aptitude exams. Given Data / Assumptions: a − b = 2. ab = 15. a and b are real numbers. We need to find a³ − b³. Concept / Approach: We recall the identity: a³ − b³ = (a − b)(a² + ab + b²). We know a − b directly, but not a² + ab + b². However, we can find a² + b² by using (a − b)² = a² − 2ab + b². Once we get a² + b², we add ab to obtain a² + ab + b², and then multiply by a − b to obtain a³ − b³. Step-by-Step Solution: Given a − b = 2 and ab = 15. First compute (a − b)² = a² − 2ab + b². Since a − b = 2, (a − b)² = 4. So a² − 2ab + b² = 4. Substitute ab = 15: a² − 2(15) + b² = 4. Thus a² + b² − 30 = 4, so a² + b² = 34. Now compute a² + ab + b² = (a² + b²) + ab = 34 + 15 = 49. Use the difference of cubes identity: a³ − b³ = (a − b)(a² + ab + b²). So a³ − b³ = 2 · 49 = 98. Verification / Alternative check: We can solve for a and b explicitly to confirm. Since a − b = 2 and ab = 15, they are roots of the quadratic t² − (a + b)t + ab = 0. We can find a + b from (a − b)² = a² − 2ab + b² and (a + b)² = a² + 2ab + b². Adding these gives 2(a² + b²) = (a − b)² + (a + b)². But it is easier to note that a + b must satisfy (a + b)² = (a − b)² + 4ab = 4 + 60 = 64, so a + b = 8 or −8. With ab = 15 and a − b = 2, we find a = 5 and b = 3. Then a³ − b³ = 125 − 27 = 98, confirming the result. Why Other Options Are Wrong: Values like 152, 112, 108 and 90 can arise from miscomputing a² + b² or from incorrectly applying the difference of cubes formula (for example using a² − ab + b²). None of these match the exact result verified both through identities and through explicit numbers. Only 98 satisfies all the given conditions. Common Pitfalls: Learners sometimes confuse the sum and difference of cubes identities or forget to include the ab term when computing a² + ab + b². Mistakes in squaring a − b or in substituting ab can also lead to the wrong result. Carefully following the identity and checking with a numerical example is a good way to avoid such errors. Final Answer: Therefore, the value of a³ − b³ is 98.

Question 6

The sum of a positive fraction and four times its reciprocal is 13/3. If the fraction lies between 1 and 2, what is the exact value of the fraction?

Accepted Answer

Introduction / Context: This question involves forming and solving a rational equation where a fraction and a multiple of its reciprocal sum to a given value. Because this leads to a quadratic equation in the fraction, we may obtain two algebraic solutions. The additional condition that the fraction lies between 1 and 2 is used to select the appropriate solution, which is a common technique in aptitude questions. Given Data / Assumptions: Let the fraction be x (x > 0). x + 4(1/x) = 13/3. The fraction lies strictly between 1 and 2. x ≠ 0 since its reciprocal is involved. Concept / Approach: We treat x as an unknown and set up the equation x + 4/x = 13/3. Multiplying through by x clears the denominator and results in a quadratic equation. After solving this quadratic, we examine both roots and use the given range for x (between 1 and 2) to pick the correct value. This approach highlights the connection between rational equations and quadratic equations. Step-by-Step Solution: Start from x + 4/x = 13/3. Multiply both sides by x: x² + 4 = (13/3)x. Multiply everything by 3 to clear the denominator: 3x² + 12 = 13x. Rearrange to standard quadratic form: 3x² − 13x + 12 = 0. Solve the quadratic 3x² − 13x + 12 = 0. Compute the discriminant Δ = (−13)² − 4·3·12 = 169 − 144 = 25. √Δ = 5, so x = [13 ± 5] / (2·3). Thus x₁ = (13 + 5)/6 = 18/6 = 3, and x₂ = (13 − 5)/6 = 8/6 = 4/3. Given that the fraction lies between 1 and 2, we discard x = 3. So the valid fraction is x = 4/3. Verification / Alternative check: Substitute x = 4/3 into the original condition. Then 1/x = 3/4 and 4(1/x) = 3. So x + 4(1/x) = 4/3 + 3 = 4/3 + 9/3 = 13/3, which matches the given sum exactly. Also, 4/3 is approximately 1.333, which is indeed between 1 and 2, so it satisfies the range condition. Why Other Options Are Wrong: 3/4 and 4/5 are less than 1 and violate the condition that the fraction lies between 1 and 2. 5/4 is between 1 and 2 but does not satisfy x + 4/x = 13/3 when substituted. 3/2 again fails the equation; its substitution gives a different sum. Only 4/3 both satisfies the rational equation and lies in the required interval. Common Pitfalls: A common error is to forget the range condition and accept both roots 3 and 4/3 as answers. Another mistake is miscalculating the discriminant or mishandling the algebra when clearing denominators. Always check each root in the original equation and apply any extra conditions given in the question. Final Answer: Therefore, the required fraction is 4/3.

Question 7

Using exact trigonometric values, evaluate the expression sin 30° − cosec 45° and simplify your answer in surd form.

Accepted Answer

Introduction / Context: This problem asks you to combine the sine of 30° with the cosecant of 45°, using exact trigonometric values. It is a straightforward exercise in recalling standard angle values, taking reciprocals for cosecant, and simplifying the resulting expression into a neat surd form. Given Data / Assumptions: Expression: sin 30° − cosec 45°. Angles are in degrees. Standard values: sin 30°, sin 45°. cosec θ = 1 / sin θ. Concept / Approach: We first recall sin 30° and sin 45°, then compute cosec 45° as the reciprocal of sin 45°. Substituting these exact values into the expression gives a difference of a simple fraction and a surd. We then combine them over a common denominator to express the result as a single fraction in simplest surd form. Step-by-Step Solution: Recall sin 30° = 1/2. Also sin 45° = √2/2. Then cosec 45° = 1 / sin 45° = 1 / (√2/2) = 2/√2 = √2. Now compute sin 30° − cosec 45° = 1/2 − √2. Write √2 with denominator 2: √2 = (2√2)/2. So 1/2 − √2 = 1/2 − (2√2)/2 = (1 − 2√2)/2. Verification / Alternative check: Check numerically for reassurance. sin 30° ≈ 0.5 and cosec 45° ≈ 1.414. Their difference is approximately 0.5 − 1.414 = −0.914. Now evaluate (1 − 2√2)/2: 2√2 ≈ 2(1.414) ≈ 2.828, so numerator is 1 − 2.828 ≈ −1.828. Dividing by 2 gives about −0.914, matching the numerical calculation and confirming the simplified form. Why Other Options Are Wrong: Expressions like (2√6 − 1)/√3 or (√2 − √3)/√6 involve different combinations of surds that evaluate to values with different magnitudes from −0.914. The option (1 − 2√3)/2 would be much more negative because 2√3 is larger than 2√2. The expression (1 − √2)/2 corresponds to subtracting only one √2 instead of two in the numerator and does not match the exact value we computed. Only (1 − 2√2)/2 equals 1/2 − √2 exactly. Common Pitfalls: Students sometimes confuse cosec 45° with sec 45° or forget to take the reciprocal of sin 45°. Another mistake is to combine 1/2 and √2 incorrectly, for example adding instead of subtracting or failing to place them over a common denominator. Carefully using the definitions and standard values and performing the subtraction with a common denominator eliminates these errors. Final Answer: Therefore, the exact value of sin 30° − cosec 45° is (1 − 2√2)/2.

Question 8

In trigonometry for an acute angle θ, you are given cos θ = 35/37. Using a right triangle model and the Pythagoras theorem, first determine the remaining side length and then calculate the exact value of cot θ from basic trigonometric ratios.

Accepted Answer

Introduction / Context: This aptitude question checks your understanding of basic trigonometric ratios and how to move from one ratio to another by using a right triangle. You are given the value of cos θ for an acute angle θ and asked to find cot θ. This is a very common pattern in exams, where one trigonometric ratio is given and another is required, without the need to find the actual angle in degrees or radians. Given Data / Assumptions: cos θ = 35/37 for an acute angle θ. Since θ is acute, all standard trigonometric ratios are positive. In a right triangle, cos θ = adjacent side / hypotenuse. cot θ is defined as adjacent side / opposite side. Pythagoras theorem holds: hypotenuse^2 = adjacent^2 + opposite^2. Concept / Approach: We interpret cos θ = 35/37 in geometric terms. Think of a right triangle where the side adjacent to θ has length 35 units and the hypotenuse has length 37 units. Once these two sides are known, the third side can be obtained from Pythagoras theorem. With all three side lengths available, any trigonometric ratio, including cot θ, can be computed as a simple ratio of sides. This approach avoids any need for calculators or angle tables. Step-by-Step Solution: From cos θ = 35/37, let adjacent side = 35 and hypotenuse = 37. Use Pythagoras theorem: hypotenuse^2 = adjacent^2 + opposite^2. Substitute the values: 37^2 = 35^2 + opposite^2. Compute squares: 1369 = 1225 + opposite^2, so opposite^2 = 1369 - 1225 = 144. Therefore opposite side = 12 (taking the positive root because θ is acute). Now cot θ = adjacent / opposite = 35 / 12 = 35/12. Verification / Alternative check: You can confirm the triple 12, 35, 37 as a valid Pythagorean triple because 12^2 + 35^2 = 144 + 1225 = 1369 = 37^2. Once this is verified, all ratios follow consistently. For example, sin θ would be 12/37 and tan θ would be 12/35, which agree with the usual relationships tan θ = 1 / cot θ and sin^2 θ + cos^2 θ = 1. Why Other Options Are Wrong: Option b (12/35) is actually tan θ, not cot θ. Option c (37/12) incorrectly uses the hypotenuse as numerator and mixes up the ratio definition. Option d (12/37) is sin θ. Option e (5/12) does not correspond to any main ratio in this triangle and likely comes from confusing 12^2 = 144 with 12 times 12 or 12 times some other number without correct reasoning. Common Pitfalls: A common mistake is to reverse the ratio and treat cos θ as hypotenuse over adjacent, which leads to incorrect side assignments. Another error is miscomputing 37^2 or 35^2, which then damages the calculation of the opposite side. Always compute the squares carefully and remember that cos θ is adjacent over hypotenuse, while cot θ is adjacent over opposite. Final Answer: The exact value of cot θ for cos θ = 35/37 with θ acute is 35/12.

Question 9

Solve the linear equation with fractional coefficients 5/2 - (6/5)(x - 15/2) = -x/5 by clearing denominators carefully and find the exact value of x from the given options.

Accepted Answer

Introduction / Context: This simplification question focuses on solving a linear equation that contains fractions and a bracketed term. Such equations are very common in aptitude tests and require careful handling of brackets and denominators. The goal is to isolate x by performing valid algebraic operations and then matching the result with one of the given answer choices. Given Data / Assumptions: The equation is 5/2 - (6/5)(x - 15/2) = -x/5. x is a real number. All fractions are to be treated exactly, without decimal approximation. We can clear denominators by multiplying through by the least common multiple of the denominators. Concept / Approach: The main idea is to first expand the bracket (x - 15/2) by multiplying it with 6/5, then simplify the left hand side. After that, it is convenient to clear all fractional denominators by multiplying the entire equation by a common multiple, such as 10. Finally, we collect like terms and solve the resulting simple linear equation for x. This step by step approach greatly reduces the chance of sign or arithmetic errors. Step-by-Step Solution: Start with 5/2 - (6/5)(x - 15/2) = -x/5. Expand (6/5)(x - 15/2) = (6/5)x - (6/5)*(15/2) = (6/5)x - 9. Substitute back: 5/2 - [(6/5)x - 9] = -x/5. Distribute the minus sign: 5/2 - (6/5)x + 9 = -x/5. Combine constants on the left: 5/2 + 9 = 5/2 + 18/2 = 23/2, so 23/2 - (6/5)x = -x/5. Multiply the entire equation by 10 to clear denominators: 10*(23/2) - 10*(6/5)x = 10*(-x/5). Compute: 10*(23/2) = 115, and 10*(6/5)x = 12x, and 10*(-x/5) = -2x, giving 115 - 12x = -2x. Bring x terms together: 115 - 12x + 2x = 0, so 115 - 10x = 0, hence -10x = -115 and x = 23/2. Verification / Alternative check: You can substitute x = 23/2 back into the original equation to verify. Compute x - 15/2 = 23/2 - 15/2 = 8/2 = 4, so (6/5)(x - 15/2) = (6/5)*4 = 24/5. Then 5/2 - 24/5 = (25/10 - 48/10) = -23/10. On the right side, -x/5 = -(23/2)/5 = -23/10. Both sides match, confirming the solution is correct. Why Other Options Are Wrong: Option b ( -23/2 ) results from losing a negative sign while rearranging. Options c and d (13/2 and -13/2) often appear if someone miscomputes constants when combining 5/2 and 9 or miscalculates 10*(23/2). Option e ( -11 ) suggests that denominators were not cleared properly or that the bracket was expanded incorrectly. Common Pitfalls: Common mistakes include forgetting to distribute the minus sign across both terms inside the bracket, incorrectly multiplying fractions like (6/5)*(15/2), and making sign errors when moving terms across the equals sign. Always expand carefully, convert all terms to a common denominator, and then simplify step by step to avoid these issues. Final Answer: The correct solution of the equation 5/2 - (6/5)(x - 15/2) = -x/5 is x = 23/2.

Question 10

Two numbers a and b satisfy a - b = 2 and ab = 24. Without finding a and b individually, use algebraic identities to compute the exact value of a^3 - b^3 and select the correct option.

Accepted Answer

Introduction / Context: This simplification question checks your ability to use algebraic identities cleverly so that you avoid explicitly solving a quadratic equation for a and b. Instead of finding the two numbers directly, you use the given information about their difference and product to compute a^3 - b^3 efficiently, which is a valuable skill in competitive exams where time is limited. Given Data / Assumptions: a - b = 2. ab = 24. a and b are real numbers. We need to find a^3 - b^3 exactly. We are encouraged to use standard algebraic identities rather than solving for a and b explicitly. Concept / Approach: The identity for the difference of cubes is a^3 - b^3 = (a - b)(a^2 + ab + b^2). However, we do not know a^2 + ab + b^2 directly. Instead of solving for a and b, we can express a^3 - b^3 in terms of a - b and ab using a rearranged identity: a^3 - b^3 = (a - b)^3 + 3ab(a - b). This version uses only a - b and ab, both of which are given. Once we plug in those values and compute carefully, we reach the final answer in a few steps. Step-by-Step Solution: Recall the identity: a^3 - b^3 = (a - b)^3 + 3ab(a - b). Given a - b = 2 and ab = 24, substitute into the formula. Compute (a - b)^3 = 2^3 = 8. Compute 3ab(a - b) = 3 * 24 * 2 = 144. Add the two parts: a^3 - b^3 = 8 + 144 = 152. Verification / Alternative check: As a verification, you may solve for a and b explicitly. From a - b = 2, we get a = b + 2. Substitute this into ab = 24 to obtain b(b + 2) = 24, which simplifies to b^2 + 2b - 24 = 0. Solving gives b = 4 or b = -6. Correspondingly, a = 6 or a = -4. For the pair (a, b) = (6, 4), a^3 - b^3 = 216 - 64 = 152. For the pair (-4, -6), we get (-4)^3 - (-6)^3 = -64 - (-216) = 152 again, confirming the identity based result. Why Other Options Are Wrong: Options b (140), c (124), and d (280) may arise from mistakes such as forgetting the 3 factor in 3ab(a - b), using ab(a - b) instead, or squaring instead of cubing. Option e (168) could come from miscomputing 3 * 24 * 2 as 160 or 168 through arithmetic errors. Only 152 matches correct use of the identity and accurate multiplication. Common Pitfalls: A frequent error is to start by computing a^2 + ab + b^2 separately, which can be more time consuming if done via separate identities. Another pitfall is misremembering the cube expansion or mixing up a^3 - b^3 with a^3 + b^3. Carefully recalling and applying the exact formula a^3 - b^3 = (a - b)^3 + 3ab(a - b) avoids these mistakes and leads directly to the solution. Final Answer: The exact value of a^3 - b^3, given a - b = 2 and ab = 24, is 152.

Question 11

The sum of twice a fraction and its reciprocal is 17/6. If the fraction, in lowest terms, has numerator 3, determine the fraction by forming and solving the appropriate equation.

Accepted Answer

Introduction / Context: This aptitude question involves fractions, reciprocals, and forming a linear equation from a word statement. The condition links twice a fraction and its reciprocal to a given rational number. An extra piece of information tells us that the fraction, in lowest terms, has numerator 3. You must translate the statement into algebra, solve for the fraction, and then match it with one of the answer options. Given Data / Assumptions: Let the fraction be f. The condition is 2f + 1/f = 17/6. The fraction is in lowest terms. The numerator of the reduced fraction is 3. The options list several candidate fractions; exactly one satisfies all conditions. Concept / Approach: First, express the fraction as f = 3/k, because the numerator is 3 in lowest terms and k is a positive integer not divisible by 3. Substitute this form into the equation 2f + 1/f = 17/6. This will give an equation in k. Solving this equation will yield the denominator k and therefore the exact fraction. Finally, check which option matches this fraction and verify that it satisfies the original condition. Step-by-Step Solution: Let the fraction be f = 3/k, where k is a positive integer and gcd(3, k) = 1. Substitute into 2f + 1/f = 17/6 to get 2*(3/k) + 1/(3/k) = 17/6. Simplify: 6/k + k/3 = 17/6. Multiply throughout by 6k to clear denominators: 6k*(6/k) + 6k*(k/3) = 6k*(17/6). This becomes 36 + 2k^2 = 17k. Rearrange to standard quadratic form: 2k^2 - 17k + 36 = 0. Solve the quadratic: the discriminant is 17^2 - 4*2*36 = 289 - 288 = 1. Thus k = (17 ± 1)/4, giving k = 18/4 = 4.5 or k = 16/4 = 4. Only k = 4 is an integer, so k = 4. Therefore f = 3/4. Verification / Alternative check: Check the fraction f = 3/4 directly in the original condition. Compute twice the fraction: 2*(3/4) = 3/2. The reciprocal is 4/3. Their sum is 3/2 + 4/3. Using a common denominator 6, this becomes 9/6 + 8/6 = 17/6, which matches the given condition exactly. None of the other options with numerator 3, such as 3/5, will produce a sum of 17/6 when used in the same way. Why Other Options Are Wrong: Option b (2/3) has numerator 2, not 3, so it violates the numerator condition. Options c (4/5) and d (5/4) also do not have numerator 3. Option e (3/5) has the correct numerator but fails the equation: 2*(3/5) + 5/3 does not simplify to 17/6. Therefore, these choices do not satisfy both the algebraic condition and the lowest terms numerator requirement. Common Pitfalls: A common mistake is to ignore the numerator condition and simply solve 2f + 1/f = 17/6 directly, which may lead to checking many fractions unnecessarily. Another pitfall is mishandling the algebra when clearing denominators, especially in the step that multiplies through by 6k. Carefully setting up the quadratic equation and verifying that k is an integer ensures you find the correct fraction efficiently. Final Answer: The fraction whose double plus its reciprocal equals 17/6 and whose numerator in lowest terms is 3 is 3/4.

Question 12

In coordinate geometry, find the reflection of the point (4, 7) in the horizontal line y = -1 by using the concept of equal perpendicular distances from the line, and choose the correct reflected coordinates.

Accepted Answer

Introduction / Context: Reflection of points in lines is a fundamental concept in coordinate geometry and is often tested in aptitude questions. Here, you are asked to reflect a point in a horizontal line, which is easier than a general line because only the y coordinate changes. Understanding how distances behave under reflection is the key to solving such problems quickly and accurately. Given Data / Assumptions: Original point P has coordinates (4, 7). The mirror line is the horizontal line y = -1. We want the coordinates of the reflected point P prime. The reflection is across a horizontal line, so the x coordinate remains unchanged. The vertical distance from the point to the line is equal on both sides after reflection. Concept / Approach: For reflection across a horizontal line y = k, the x coordinate of a point does not change, while its y coordinate moves so that the line lies exactly midway between the original point and its image. In other words, the line y = k is the perpendicular bisector of the segment joining the original point and its reflection. Therefore, the y coordinates of the original and the reflected point must be symmetric with respect to k, meaning their average is k. Step-by-Step Solution: The original point is P(4, 7) and the mirror line is y = -1. For a reflection in a horizontal line, the x coordinate stays the same, so x prime = 4. Let the reflected point be P prime with coordinates (4, y prime). The line y = -1 is the midpoint of the vertical segment between 7 and y prime. Therefore (7 + y prime)/2 = -1. Solve for y prime: 7 + y prime = -2, so y prime = -9. Thus the reflected point is (4, -9). Verification / Alternative check: You can check the vertical distances from both the original point and the reflected point to the line y = -1. From (4, 7) to y = -1, the distance is 7 - (-1) = 8 units upward. From (4, -9) to y = -1, the distance is -1 - (-9) = 8 units upward if you consider absolute distance. The line y = -1 lies exactly halfway between y = 7 and y = -9, confirming that the reflection is correct. Why Other Options Are Wrong: Option b ( -4, -9 ) changes both x and y coordinates, which is not correct for a reflection in a horizontal line. Options c and d change the x coordinate but do not maintain symmetry about y = -1. Option e (4, 1) maintains the x coordinate but places the point only 2 units above y = -1, whereas the original is 8 units above, so the distances are not equal. Common Pitfalls: A frequent error is to reflect across the x axis (y = 0) instead of the given line y = -1, which leads to incorrect y coordinates such as -7. Another pitfall is changing both coordinates when reflecting across a horizontal or vertical line, even though only the coordinate perpendicular to the line should change. Always identify whether the line is horizontal or vertical and adjust only the relevant coordinate accordingly. Final Answer: The reflection of the point (4, 7) in the line y = -1 is (4, -9).

Question 13

In trigonometry, if tan θ = 9/40 for an acute angle θ, model the situation with a right triangle, use the Pythagoras theorem to find the hypotenuse, and then determine the exact value of sec θ.

Accepted Answer

Introduction / Context: This question tests your ability to convert from one trigonometric ratio to another by using right triangle relationships. You are given tan θ for an acute angle θ and asked to find sec θ. Understanding how tangent relates to opposite and adjacent sides, and how secant relates to hypotenuse and adjacent sides, allows you to solve such questions quickly in aptitude and competitive exams. Given Data / Assumptions: tan θ = 9/40. θ is an acute angle, so all standard trigonometric ratios are positive. tan θ = opposite side / adjacent side. sec θ = hypotenuse / adjacent side. Pythagoras theorem: hypotenuse^2 = opposite^2 + adjacent^2. Concept / Approach: Interpret tan θ = 9/40 in terms of side lengths of a right triangle. Take the opposite side to θ as 9 units and the adjacent side as 40 units. Then, compute the hypotenuse using the Pythagoras theorem. Once all three sides are known, sec θ is simply the ratio of hypotenuse to adjacent side. This approach is far more efficient than trying to determine the angle in degrees or radians. Step-by-Step Solution: Let opposite side to θ be 9 and adjacent side be 40, so tan θ = 9/40. Apply Pythagoras theorem: hypotenuse^2 = 9^2 + 40^2. Compute squares: hypotenuse^2 = 81 + 1600 = 1681. Take the square root: hypotenuse = 41 since 41^2 = 1681. Now sec θ = hypotenuse / adjacent = 41 / 40. Verification / Alternative check: From the triangle, cos θ = adjacent / hypotenuse = 40/41. Since sec θ is the reciprocal of cos θ, sec θ = 1 / cos θ = 41/40. This confirms the direct computation. You may also note that 9, 40, and 41 form a well known Pythagorean triple, which further supports that the side lengths are consistent. Why Other Options Are Wrong: Option b (40/41) is cos θ rather than sec θ. Option c (41/9) incorrectly uses the hypotenuse over the opposite side and corresponds more closely to cosec θ if the roles of opposite and adjacent were mixed. Option d (9/41) is the sine of θ. Option e (40/9) is the cotangent of θ. None of these equals sec θ, which must be the reciprocal of cos θ in this right triangle model. Common Pitfalls: Learners sometimes confuse which side is opposite and which is adjacent when using tan θ, leading to wrong assignments of 9 and 40. Another common issue is forgetting that sec θ is 1/cos θ, not 1/tan θ. Carefully drawing a right triangle, labelling opposite and adjacent sides, and then applying Pythagoras theorem will usually prevent these mistakes. Final Answer: The exact value of sec θ when tan θ = 9/40 and θ is acute is 41/40.

Question 14

Using the algebraic identity (a - b)(a + b) = a^2 - b^2, compute the product 9997 × 10003 by treating the numbers as 10000 - 3 and 10000 + 3, and then choose the correct result.

Accepted Answer

Introduction / Context: This simplification question is designed to test your familiarity with special algebraic products, particularly the identity (a - b)(a + b) = a^2 - b^2. Instead of multiplying 9997 and 10003 directly, which is time consuming and error prone, you can exploit the fact that both numbers lie close to 10000 and are symmetrical around it. This is a classic mental math trick used in many quantitative aptitude exams. Given Data / Assumptions: We need to compute 9997 × 10003. Recognise that 9997 = 10000 - 3. Recognise that 10003 = 10000 + 3. The identity (a - b)(a + b) = a^2 - b^2 is valid for all real numbers a and b. No calculator is required if the identity is applied correctly. Concept / Approach: By setting a = 10000 and b = 3, the product 9997 × 10003 becomes (a - b)(a + b). According to the identity, this equals a^2 - b^2. Calculating a^2 is straightforward for powers of 10, and b^2 is a small integer. Subtracting b^2 from a^2 gives the final result in a few seconds, making this method both quick and reliable. Step-by-Step Solution: Write 9997 as 10000 - 3 and 10003 as 10000 + 3. Recognise the pattern (a - b)(a + b) with a = 10000 and b = 3. Apply the identity: (a - b)(a + b) = a^2 - b^2. Compute a^2 = 10000^2 = 100000000 (eight zeros after 1). Compute b^2 = 3^2 = 9. Subtract: 100000000 - 9 = 99999991. Verification / Alternative check: You can perform a rough check by approximating the numbers. Both 9997 and 10003 are close to 10000, so their product should be very close to 100000000. Since 9997 is slightly less than 10000 and 10003 is slightly more, the product will be slightly less than 100000000, which matches 99999991. You could also perform traditional multiplication digit by digit to confirm, but this is more tedious and prone to manual mistakes. Why Other Options Are Wrong: Options b, c, d, and e are numbers that look visually similar, but they correspond to subtracting 89, 1009, 889, or another incorrect quantity from 100000000 rather than 9. They may arise from miscalculating b^2 or from misplacing zeros in a^2. Only 99999991 fits the structure of 100000000 - 9 and respects the pattern from the identity. Common Pitfalls: A common error is to wrongly take a as 9997 and b as 3, leading to an incorrect (a - b)(a + b) representation. Another mistake is miscomputing 10000^2 by writing 10000000 instead of 100000000, which shifts all subsequent digits. Remember that 10^4 squared is 10^8, so there must be eight zeros. If you handle these details carefully, the identity method is very reliable. Final Answer: Using the identity (a - b)(a + b) = a^2 - b^2, the product 9997 × 10003 is 99999991.

Question 15

Solve the linear equation with a negative fractional coefficient, (-1/2)(x - 5) + 3 = -5/2, by simplifying step by step and find the exact value of x.

Accepted Answer

Introduction / Context: This question involves solving a simple linear equation that contains a bracket and a negative fractional coefficient. Such problems are routine in aptitude tests, and they are designed to check your ability to expand brackets correctly, handle negative signs, and work with fractions without making algebraic mistakes. The final goal is to isolate x and identify its value from the options. Given Data / Assumptions: The equation is (-1/2)(x - 5) + 3 = -5/2. x is a real number. All fractions should be treated exactly. We can clear fractions by multiplying through by a suitable common multiple if convenient. Concept / Approach: The key idea is to first expand the bracket (-1/2)(x - 5) and then combine like terms. Because the coefficient is negative, special care is needed with signs. Once the expression on the left hand side is simplified, we can move constants to one side and x terms to the other, or we can clear the denominator by multiplying the entire equation by 2. Either path yields the same solution for x when done carefully. Step-by-Step Solution: Start with (-1/2)(x - 5) + 3 = -5/2. Expand the bracket: (-1/2)(x - 5) = -(1/2)x + (5/2). Substitute this expansion: -(1/2)x + 5/2 + 3 = -5/2. Convert 3 to a fraction with denominator 2: 3 = 6/2, so 5/2 + 3 = 5/2 + 6/2 = 11/2. The equation becomes -(1/2)x + 11/2 = -5/2. Multiply the entire equation by 2 to clear denominators: -x + 11 = -5. Rearrange: -x = -5 - 11 = -16, therefore x = 16. Verification / Alternative check: Substitute x = 16 back into the original equation. Evaluate x - 5 = 11, so (-1/2)(x - 5) = (-1/2)*11 = -11/2. Then (-11/2) + 3 = -11/2 + 6/2 = -5/2, which matches the right hand side. This confirms that x = 16 satisfies the equation exactly. Why Other Options Are Wrong: Option b (4), option c (-4), and option d (-6) do not satisfy the equation when substituted back and result in unequal left and right sides. Option e (-16) comes from a common sign error when moving terms across the equals sign or misapplying the multiplication by 2 step, where -x + 11 = -5 is incorrectly converted to x + 11 = -5. Common Pitfalls: Typical mistakes include dropping the negative sign in front of 1/2, incorrectly expanding (-1/2)(x - 5) as -x/2 - 5/2, or failing to convert the integer 3 into a fraction with matching denominator when combining terms. Another error is to multiply only some terms by 2 when clearing denominators instead of multiplying both sides of the equation completely. Writing out each step carefully helps avoid these issues. Final Answer: The exact solution of the linear equation (-1/2)(x - 5) + 3 = -5/2 is x = 16.

Question 16

If two numbers satisfy a - b = 1 and ab = 6, use algebraic identities (without solving directly for a and b) to find the exact value of a^3 - b^3.

Accepted Answer

Introduction / Context: This question again focuses on the algebraic identity for the difference of cubes. Instead of finding the actual values of a and b from the given conditions, you are expected to use identities that express a^3 - b^3 in terms of a - b and ab. This is a powerful technique because it saves time and reduces the chance of algebraic errors that can occur when solving quadratics under exam pressure. Given Data / Assumptions: a - b = 1. ab = 6. a and b are real numbers. The task is to find a^3 - b^3. You should use identities rather than solving explicitly for a and b. Concept / Approach: The relevant identity is a^3 - b^3 = (a - b)^3 + 3ab(a - b). This identity is derived by expanding (a - b)^3 and then rearranging terms. Because the given information includes a - b and ab, this identity allows us to substitute directly and compute a^3 - b^3 in just a few steps. This approach is especially suited to quantitative aptitude questions where speed and accuracy are critical. Step-by-Step Solution: Use the identity: a^3 - b^3 = (a - b)^3 + 3ab(a - b). Substitute a - b = 1 and ab = 6. Compute (a - b)^3 = 1^3 = 1. Compute 3ab(a - b) = 3 * 6 * 1 = 18. Add them: a^3 - b^3 = 1 + 18 = 19. Verification / Alternative check: To verify, you can solve for a and b explicitly, although this is not necessary in an exam. From a - b = 1, we have a = b + 1. Substitute into ab = 6 to get b(b + 1) = 6, which simplifies to b^2 + b - 6 = 0. Solving gives b = 2 or b = -3. If b = 2, then a = 3, and a^3 - b^3 = 27 - 8 = 19. If b = -3, then a = -2, and a^3 - b^3 = (-8) - (-27) = 19. Both pairs confirm the identity based result. Why Other Options Are Wrong: Options b (21), c (23), d (25), and e (27) can arise from partial application of the identity or incorrect arithmetic. For example, using 3ab instead of 3ab(a - b), or miscomputing 3*6 as 16, will lead to such incorrect results. None of these values match both the identity calculation and the explicit check using actual values of a and b. Common Pitfalls: A common mistake is to use the sum of cubes identity a^3 + b^3 by accident, or to misremember the formula for the difference of cubes. Another error is to attempt to compute a^2 + ab + b^2 separately using additional identities, which increases the risk of mistakes. Staying with the compact identity a^3 - b^3 = (a - b)^3 + 3ab(a - b) and substituting carefully is the most reliable and efficient method. Final Answer: The exact value of a^3 - b^3, given a - b = 1 and ab = 6, is 19.

Question 17

Find the point where the line 2x - 3y = 6 intersects the y-axis by setting x = 0, and then choose the correct y-intercept coordinate.

Accepted Answer

Introduction / Context: This coordinate geometry question asks you to find the y-intercept of a line given in standard form. The y-intercept is the point where the line cuts the y-axis, which is a very common requirement in graph based aptitude problems. Understanding how to find intercepts quickly from equations of lines is an essential skill for both exams and real life applications involving graphs. Given Data / Assumptions: The equation of the line is 2x - 3y = 6. The y-axis corresponds to x = 0. The y-intercept is the point where the line crosses the y-axis. Coordinates are written in the form (x, y). Concept / Approach: To find where a line meets the y-axis, we simply set x = 0 in the equation, because all points on the y-axis have x coordinate equal to zero. Substituting x = 0 into the equation and solving for y gives the y coordinate of the intersection point. The resulting pair (0, y) is the y-intercept. This is a straightforward application of the definition of the axes in the Cartesian plane. Step-by-Step Solution: Start from the line equation: 2x - 3y = 6. For the y-axis, set x = 0. Substitute x = 0: 2(0) - 3y = 6, which simplifies to -3y = 6. Solve for y: y = 6 / (-3) = -2. Thus the line meets the y-axis at the point (0, -2). Verification / Alternative check: If you rewrite the equation in slope intercept form, you can verify the same result. From 2x - 3y = 6, isolate y to get -3y = -2x + 6 and then y = (2/3)x - 2. In this form, the constant term -2 represents the y-intercept directly, confirming that the line crosses the y-axis at (0, -2). This second method matches the result of the substitution method. Why Other Options Are Wrong: Option b (0, 2) has the correct x coordinate but the opposite sign for y, which would correspond to 2x - 3(2) = 2x - 6, not equal to 6 when x = 0. Options c and d, (-2, 0) and (2, 0), are x-intercepts rather than y-intercepts because they place the point on the x-axis. Option e (-3, 2) satisfies neither the y-axis condition nor the line equation correctly when substituted. Common Pitfalls: A common mistake is to mix up x and y intercepts and accidentally set y = 0 when asked for the y-intercept. Another pitfall is mishandling the algebraic signs when solving for y, especially when dividing by a negative coefficient. Carefully substituting x = 0 and then performing the division correctly avoids these issues. Final Answer: The line 2x - 3y = 6 cuts the y-axis at the point (0, -2).

Question 18

Evaluate the exact value of cot 45° + cosec 60° by recalling standard trigonometric values for special angles and simplifying the result into a single surd expression.

Accepted Answer

Introduction / Context: This trigonometry question checks your recall of exact values of trigonometric functions for special angles such as 30 degrees, 45 degrees, and 60 degrees. You are asked to compute cot 45 degrees plus cosec 60 degrees and express the answer as a simplified surd fraction. Such problems are common in aptitude tests because they reward memorisation of special angle values and basic algebraic manipulation. Given Data / Assumptions: Angle 45 degrees is a standard special angle. Angle 60 degrees is also a standard special angle. cot 45 degrees = 1. cosec θ = 1 / sin θ for any angle θ where sine is defined. sin 60 degrees = √3/2 for the principal non negative value. Concept / Approach: The strategy is to use the known exact values directly. First, evaluate cot 45 degrees. Then, find sin 60 degrees and invert it to get cosec 60 degrees. Finally, add the two values and simplify the result so that it matches one of the given options. No calculator is needed as long as the special angle values are memorised correctly. Step-by-Step Solution: Recall that tan 45 degrees = 1, so cot 45 degrees, which is 1 / tan 45 degrees, also equals 1. Recall sin 60 degrees = √3/2. Therefore cosec 60 degrees = 1 / sin 60 degrees = 1 / (√3/2) = 2/√3. Now compute the sum: cot 45 degrees + cosec 60 degrees = 1 + 2/√3. Express 1 with denominator √3: 1 = √3/√3. Add the fractions: √3/√3 + 2/√3 = (√3 + 2)/√3. Verification / Alternative check: You can perform a quick numerical check to confirm. Approximate √3 as about 1.732. Then cosec 60 degrees = 2/√3 is approximately 1.155, and cot 45 degrees is exactly 1. Their sum is roughly 2.155. Evaluating (√3 + 2)/√3 numerically yields (1.732 + 2)/1.732 ≈ 3.732/1.732 ≈ 2.155, which matches the earlier approximate sum and confirms the expression is correct. Why Other Options Are Wrong: Option b ( (√6 + 1)/√3 ) and option d ( (1 + √3)/2 ) represent different combinations of surds and fractions that do not match the exact sum of 1 and 2/√3. Option c ( 5/√3 ) would require the sum to be 5/√3 ≈ 2.886, which is larger than the actual value. Option e ( (2 + √3)/2 ) approximates to a value near 1.866 and therefore does not equal the computed sum. Common Pitfalls: A common mistake is to confuse sine and cosine values at 60 degrees, for example using sin 60 degrees = 1/2 and cos 60 degrees = √3/2 instead of the correct values. Another error is to forget that cosec is the reciprocal of sine, not cosine, leading to an incorrect 2/1 or 1/2. When simplifying the final expression, some learners also forget to put 1 over a common denominator before adding fractions, which leads to mismatched forms compared to the options. Final Answer: The exact simplified value of cot 45 degrees + cosec 60 degrees is (√3 + 2)/√3.

Question 19

In right triangle LMN, right angled at M, you are told that ∠N = 60°. Use the property that the acute angles are complementary to find the exact value of tan L and select the correct option.

Accepted Answer

Introduction / Context: This trigonometry question tests your understanding of complementary angles in a right triangle. When one angle in a right triangle is known, you can always find the other acute angle, since their sum must be 90 degrees. Using basic trigonometric values for 30 degrees and 60 degrees, you can then evaluate the required ratio, in this case tan L, without needing any complex computation. Given Data / Assumptions: Triangle LMN is right angled at M, so ∠M = 90 degrees. ∠N = 60 degrees. The sum of angles in a triangle is 180 degrees. tan θ is defined as opposite side / adjacent side for an acute angle θ in a right triangle. We work with standard exact values for 30 degrees and 60 degrees. Concept / Approach: In a right triangle, the two acute angles are complementary. That is, if one acute angle is 60 degrees, the other must be 30 degrees. Here, since ∠N = 60 degrees and ∠M = 90 degrees, angle L must be 30 degrees. Once we establish that L is 30 degrees, we can simply recall that tan 30 degrees has the exact value 1/√3, which is one of the standard special-angle trigonometric ratios. Step-by-Step Solution: Use the angle sum property: ∠L + ∠M + ∠N = 180 degrees. Substitute the known values: ∠L + 90 + 60 = 180, so ∠L + 150 = 180. Solve for ∠L: ∠L = 180 - 150 = 30 degrees. Now we need tan L, which is tan 30 degrees. Recall the standard value: tan 30 degrees = 1/√3. Verification / Alternative check: You can think of a standard 30–60–90 triangle, where the sides are in the ratio 1 : √3 : 2. For the 30 degree angle, opposite side is 1 and adjacent side is √3, so tan 30 degrees = 1/√3. For the 60 degree angle, the roles reverse and tan 60 degrees = √3. This confirms that the correct value for tan L (which is 30 degrees) is 1/√3. Why Other Options Are Wrong: Option b (1/2) is the value of sin 30 degrees, not tan 30 degrees. Option c (1/√2) is often associated with sin 45 degrees or cos 45 degrees, and does not apply here. Option d (2) is not a standard tangent value for 30 or 60 degrees. Option e (√3) is tan 60 degrees, which would be the tangent of angle N, not angle L. Common Pitfalls: Learners sometimes mix up which angle is 30 degrees and which is 60 degrees in the right triangle, especially when not drawing a diagram. Another common error is to recall tan 30 degrees as √3 or tan 60 degrees as 1/√3, effectively swapping them. Carefully identifying that L is the smaller acute angle and remembering the standard triangle with side ratio 1 : √3 : 2 helps avoid these mistakes. Final Answer: The correct value of tan L in the given right triangle is 1/√3.

Question 20

For an acute angle θ, if tan θ = 4/3, interpret this as a right triangle ratio, use Pythagoras theorem to find the hypotenuse, and then determine the exact value of sin θ.

Accepted Answer

Introduction / Context: This trigonometry question asks you to convert from one trigonometric ratio to another using a right triangle model. You are given tan θ and asked to find sin θ for an acute angle θ. Instead of working with approximate decimal values directly, you can interpret the ratio 4/3 as side lengths in a right triangle and then apply the Pythagoras theorem to find all sides and the required sine value. Given Data / Assumptions: tan θ = 4/3 for an acute angle θ. tan θ = opposite side / adjacent side. sin θ = opposite side / hypotenuse. Pythagoras theorem: hypotenuse^2 = opposite^2 + adjacent^2. θ is acute, so all basic trigonometric values are positive. Concept / Approach: Interpret tan θ = 4/3 by choosing a right triangle where the side opposite θ is 4 units and the side adjacent to θ is 3 units. This choice fits the given ratio and is convenient. Then apply Pythagoras theorem to calculate the hypotenuse. Once you have all three side lengths, compute sin θ as opposite divided by hypotenuse. Finally, convert the fraction to a decimal to compare with the options if needed. Step-by-Step Solution: Let the opposite side to θ be 4 and the adjacent side be 3, so tan θ = 4/3. Apply Pythagoras theorem: hypotenuse^2 = 4^2 + 3^2 = 16 + 9 = 25. So the hypotenuse is 5, since 5^2 = 25. Now sin θ = opposite / hypotenuse = 4 / 5. Convert 4/5 to a decimal: 4/5 = 0.8. Verification / Alternative check: In the well known 3–4–5 Pythagorean triangle, if the sides opposite and adjacent to the angle are 4 and 3 respectively, tan θ = 4/3, sin θ = 4/5, and cos θ = 3/5. All these satisfy the identity sin^2 θ + cos^2 θ = 1, since (4/5)^2 + (3/5)^2 = 16/25 + 9/25 = 25/25 = 1. This confirms that sin θ = 4/5, which is 0.8 in decimal form. Why Other Options Are Wrong: Option b (1.25) is greater than 1 and therefore cannot be a sine value, since sine values are bounded between -1 and 1. Option c (4/3) is tan θ itself, not sin θ. Option d (3/4) is the cosine of θ, which is adjacent over hypotenuse in this triangle. Option e (1/2) is often associated with sin 30 degrees but does not match the triangle representing tan θ = 4/3. Common Pitfalls: Learners sometimes mistakenly treat 4/3 as opposite/hypotenuse or adjacent/hypotenuse instead of opposite/adjacent, which leads to inconsistent side lengths. Another common error is to attempt to approximate values with a calculator rather than using the exact Pythagorean triple. Remembering the 3–4–5 triangle and the definitions of tan, sin, and cos makes these conversions much easier. Final Answer: The exact value of sin θ for tan θ = 4/3 is 4/5, which equals 0.8 in decimal form.

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