Question 1

If 3x + 5y + 7z = 49 and 9x + 8y + 21z = 126, then what is the numerical value of y in this system of linear equations?

Accepted Answer

Introduction / Context: In this aptitude problem we are given two linear equations in the three variables x, y and z, and we are asked to determine only the value of y. The question checks whether you can use elimination in a focused way, without trying to find the entire solution set of the system. Given Data / Assumptions: - 3x + 5y + 7z = 49 - 9x + 8y + 21z = 126 - x, y and z are real numbers and we only need the value of y. Concept / Approach: The key idea is to eliminate x and z simultaneously. If we can create a situation where the x and z terms cancel when we combine the equations, the remaining equation will involve only y, which can then be solved directly. Step-by-Step Solution: Step 1: Multiply the first equation by 3 so that the coefficient of x matches the second equation: 3(3x + 5y + 7z) = 3 * 49.Step 2: This gives 9x + 15y + 21z = 147.Step 3: Now subtract the original second equation (9x + 8y + 21z = 126) from this new equation.Step 4: (9x + 15y + 21z) − (9x + 8y + 21z) = 147 − 126.Step 5: The x and z terms cancel, leaving 7y = 21.Step 6: Divide both sides by 7 to get y = 21 / 7 = 3. Verification / Alternative check: We can quickly check consistency. If y = 3, then the first equation becomes 3x + 5(3) + 7z = 49, so 3x + 15 + 7z = 49, or 3x + 7z = 34. The second equation becomes 9x + 8(3) + 21z = 126, so 9x + 24 + 21z = 126, or 9x + 21z = 102. These two equations in x and z are consistent, which confirms that y = 3 is valid. Why Other Options Are Wrong: Values like y = 2, 4 or 5 do not satisfy the relationship between the two equations after elimination. Substituting any of these into both equations will lead to inconsistent equations for x and z, so they cannot be correct. Common Pitfalls: A common mistake is to think that because there are three variables and only two equations, no single variable can be determined. However, sometimes a specific combination of coefficients makes one variable uniquely determined. Another error is performing elimination incorrectly, such as multiplying or subtracting equations wrongly, which leads to incorrect values of y. Final Answer: The correct value of y is 3.

Question 2

The cost of 4 pens, 6 notebooks and 9 files is Rs 305, and the cost of 3 pens, 4 notebooks and 2 files is Rs 145. Based on this information, what is the total cost in rupees of buying 5 pens, 8 notebooks and 16 files together?

Accepted Answer

Introduction / Context: This is a linear equations word problem about the cost of pens, notebooks and files. Instead of asking for the individual prices of each item, the question directly asks for the cost of a particular combination. This allows us to use linear combination of equations without fully solving for every variable. Given Data / Assumptions: - Cost of 4 pens, 6 notebooks and 9 files is Rs 305. - Cost of 3 pens, 4 notebooks and 2 files is Rs 145. - We want the cost of 5 pens, 8 notebooks and 16 files. - All prices are in rupees and are assumed to be constant per item. Concept / Approach: Let the cost of one pen be p rupees, one notebook be n rupees and one file be f rupees. We translate the statements into linear equations in p, n and f. Then we try to express the target combination (5p + 8n + 16f) as a linear combination of the given equations, which is more efficient than solving for p, n and f individually. Step-by-Step Solution: Step 1: From the first sentence we get 4p + 6n + 9f = 305.Step 2: From the second sentence we get 3p + 4n + 2f = 145.Step 3: We want S = 5p + 8n + 16f. Try to write S as a(4p + 6n + 9f) + b(3p + 4n + 2f).Step 4: This gives a(4, 6, 9) + b(3, 4, 2) = (5, 8, 16).Step 5: Solve 4a + 3b = 5 and 6a + 4b = 8. Subtracting gives a + b = 1, so b = 1 − a.Step 6: Substitute into 4a + 3(1 − a) = 5, which gives a = 2 and hence b = −1.Step 7: Therefore S = 2(305) − 1(145) = 610 − 145 = 465. Verification / Alternative check: If desired, we could solve for p, n and f explicitly and then compute 5p + 8n + 16f, but the linear combination method is faster and less error prone. The final total of Rs 465 is consistent with both original equations when back-substituted through the derived coefficients a = 2 and b = −1. Why Other Options Are Wrong: Values like 415 or 440 arise from incorrect algebra such as choosing wrong coefficients a and b, or making arithmetic mistakes while subtracting or adding the equations. The option “Cannot be determined” is incorrect because the specific combination we need lies in the span of the two given combinations, so it is uniquely determined even though we have three unknown prices. Common Pitfalls: Students often think that three unknowns always require three equations. However, when we only need a specific linear combination, two equations may be enough. Another common pitfall is to attempt full substitution, which is longer and increases chances of arithmetic errors, instead of directly expressing the target combination as a linear combination of the given ones. Final Answer: The total cost of 5 pens, 8 notebooks and 16 files is Rs 465.

Question 3

In triangle ABC, which is right angled at A (∠BAC = 90°) and has ∠ACB = 60°, what is the ratio of the circumradius of the triangle to the side AB?

Accepted Answer

Introduction / Context: This geometry question involves a right angled triangle and its circumradius. The triangle ABC is right angled at A and another angle is given as 60 degrees. We are asked to find the ratio of the circumradius to one specific side, AB. The problem tests knowledge of right triangle properties, 30-60-90 triangles and the relation between the hypotenuse and circumradius. Given Data / Assumptions: - Triangle ABC is right angled at A, so ∠BAC = 90°. - ∠ACB = 60°, so the remaining angle ∠ABC is 30°. - We need the ratio circumradius R : side AB. - All sides are positive and the triangle is non-degenerate. Concept / Approach: For a right angled triangle, the circumcenter lies at the midpoint of the hypotenuse and the circumradius R is half of the hypotenuse. Also, a triangle with angles 30°, 60° and 90° has well known side ratios. By identifying which side corresponds to which angle, we can express R and AB in terms of a common scale factor and then take their ratio. Step-by-Step Solution: Step 1: Since ∠BAC = 90°, side BC is the hypotenuse.Step 2: ∠ACB = 60° means that side AB is opposite 60°, and ∠ABC = 30° means that side AC is opposite 30°.Step 3: In a standard 30°-60°-90° triangle, if the side opposite 30° is a, then the side opposite 60° is a√3 and the hypotenuse is 2a.Step 4: Matching this to our triangle, AC (opposite 30°) = a, AB (opposite 60°) = a√3, and BC (hypotenuse) = 2a.Step 5: For a right angled triangle, circumradius R = hypotenuse / 2 = BC / 2 = (2a) / 2 = a.Step 6: Therefore, R : AB = a : (a√3) = 1 : √3. Verification / Alternative check: You can assign a specific value to a, for example a = 1. Then AC = 1, AB = √3 and BC = 2. The circumradius is BC / 2 = 1. So the ratio R : AB = 1 : √3, which confirms our symbolic result. Why Other Options Are Wrong: The ratio 1 : 2 would correspond to R being half of AB, which is incorrect because the hypotenuse, not AB, is directly related to R. Ratios like 2 : √3 or 2 : 3 come from confusing which side corresponds to which angle or mixing up the standard 30°-60°-90° side ratios. Common Pitfalls: Students sometimes misidentify the hypotenuse or pair the wrong side with the wrong angle, which leads to wrong ratios. Another frequent mistake is assuming R is half of any side instead of specifically half of the hypotenuse in a right angled triangle. Remember always to map each angle to its opposite side before applying known ratios. Final Answer: The required ratio of circumradius to side AB is 1 : √3.

Question 4

For real angles x and y, what is the simplified value of the expression [(sin x + sin y)(sin x − sin y)] / [(cos x + cos y)(cos y − cos x)]?

Accepted Answer

Introduction / Context: This problem tests simplification of a trigonometric expression using basic identities. The expression involves sums and differences of sines and cosines in product form, and the goal is to simplify it to a constant, if possible, without assuming any special values of x or y. Given Data / Assumptions: - Expression: [(sin x + sin y)(sin x − sin y)] / [(cos x + cos y)(cos y − cos x)]. - x and y are real angles for which all trigonometric functions involved are defined. Concept / Approach: We recognize that (a + b)(a − b) = a² − b². The numerator is a difference of squares in sine terms, and the denominator is a difference of squares in cosine terms, except for the sign hidden in (cos y − cos x). By rewriting both numerator and denominator using this pattern and Pythagorean identities, we can see whether the expression reduces to a simple constant. Step-by-Step Solution: Step 1: Simplify the numerator: (sin x + sin y)(sin x − sin y) = sin²x − sin²y.Step 2: Simplify the denominator: (cos x + cos y)(cos y − cos x) = cos y² − cos x² = cos²y − cos²x.Step 3: Use the identity sin²θ = 1 − cos²θ.Step 4: Compute sin²x − sin²y = (1 − cos²x) − (1 − cos²y) = −cos²x + cos²y = cos²y − cos²x.Step 5: Notice that the numerator is exactly cos²y − cos²x, which matches the denominator.Step 6: Therefore, the expression equals (cos²y − cos²x) / (cos²y − cos²x) = 1, as long as the denominator is not zero. Verification / Alternative check: You can verify with sample values, for example x = 30° and y = 60°. Substituting these into the expression and simplifying numerically will give a value extremely close to 1, confirming the algebraic simplification. Why Other Options Are Wrong: The values 0, −1 and 2 do not match the algebraic simplification. Getting 0 often comes from mistakenly cancelling terms incorrectly. Obtaining −1 or 2 typically results from sign errors in handling cos y − cos x or misuse of the Pythagorean identity. Common Pitfalls: A common mistake is to forget that (cos y − cos x) = −(cos x − cos y) and mishandle the sign, which changes the result. Another pitfall is mixing up sin²θ + cos²θ = 1 with incorrect variants like sin²θ − cos²θ = 1, which is not true. Always apply algebraic identities carefully and match patterns like a² − b² correctly. Final Answer: The simplified value of the expression is 1.

Question 5

What is the simplified value of the trigonometric expression [(tan 5θ + tan 3θ) / (4 cos 4θ (tan 5θ − tan 3θ))]?

Accepted Answer

Introduction / Context: This question involves a nontrivial trigonometric expression containing tan 5θ, tan 3θ and cos 4θ. The goal is to simplify it to a basic trigonometric function. It tests understanding of tangent sum and difference identities and how they relate to sine and cosine expressions. Given Data / Assumptions: - Expression: (tan 5θ + tan 3θ) / (4 cos 4θ (tan 5θ − tan 3θ)). - θ is an angle for which all functions tan 5θ, tan 3θ and cos 4θ are defined and nonzero as required. Concept / Approach: We use the identities for tangent of sum and difference expressed in terms of sine and cosine. Specifically, we rewrite tan 5θ and tan 3θ as sin / cos and then form the ratio (tan 5θ + tan 3θ) / (tan 5θ − tan 3θ). This ratio can be converted into a ratio of sines, which simplifies using double angle and multiple angle formulas. Finally, we simplify with the extra factor 4 cos 4θ in the denominator. Step-by-Step Solution: Step 1: Write tan 5θ = sin 5θ / cos 5θ and tan 3θ = sin 3θ / cos 3θ.Step 2: Compute tan 5θ + tan 3θ = (sin 5θ cos 3θ + cos 5θ sin 3θ) / (cos 5θ cos 3θ) = sin(5θ + 3θ) / (cos 5θ cos 3θ) = sin 8θ / (cos 5θ cos 3θ).Step 3: Similarly, tan 5θ − tan 3θ = (sin 5θ cos 3θ − cos 5θ sin 3θ) / (cos 5θ cos 3θ) = sin(5θ − 3θ) / (cos 5θ cos 3θ) = sin 2θ / (cos 5θ cos 3θ).Step 4: Therefore, (tan 5θ + tan 3θ) / (tan 5θ − tan 3θ) = sin 8θ / sin 2θ.Step 5: Now the whole expression becomes [sin 8θ / sin 2θ] / [4 cos 4θ] = sin 8θ / (4 sin 2θ cos 4θ).Step 6: Use the identity sin 8θ = 2 sin 4θ cos 4θ and sin 4θ = 2 sin 2θ cos 2θ, so sin 8θ = 4 sin 2θ cos 2θ cos 4θ.Step 7: Substitute to get sin 8θ / (4 sin 2θ cos 4θ) = [4 sin 2θ cos 2θ cos 4θ] / [4 sin 2θ cos 4θ] = cos 2θ. Verification / Alternative check: Pick a simple angle, for example θ = 10°, and evaluate the original expression numerically with a calculator. Then compute cos 2θ at the same θ. Both values will match closely, confirming that the simplification is correct. Why Other Options Are Wrong: The expression does not simplify to sin 2θ, tan 4θ or cot 2θ. These alternatives usually arise from incomplete simplification, skipping the step that uses sin 8θ and the factor 4 cos 4θ, or mixing up formulas for tan of sum and difference. Common Pitfalls: Students sometimes try to apply tan(A + B) formulas directly in a complicated way instead of first converting to sine and cosine. It is also easy to lose track of the factor 4 cos 4θ in the denominator, which is essential to canceling terms and arriving at cos 2θ. Careful step-by-step rewriting avoids these issues. Final Answer: The simplified value of the given expression is cos 2θ.

Question 6

What is the value of the expression (4/3)cot²(π/6) + 3cos² 150° − 4cosec² 45° + 8sin(π/2)?

Accepted Answer

Introduction / Context: This question asks you to evaluate a mixed trigonometric expression involving cotangent, cosine, cosecant and sine, at specific standard angles. It tests recall of exact trigonometric values for special angles such as 30°, 45°, 90° and 150°, and the ability to handle squares of these functions accurately. Given Data / Assumptions: - Expression: (4/3)cot²(π/6) + 3cos² 150° − 4cosec² 45° + 8sin(π/2). - π/6 corresponds to 30° and π/2 corresponds to 90°. - All angles are in the standard sense used in trigonometry, and we use their well known exact values. Concept / Approach: We evaluate each trigonometric term separately using special angle values, then substitute into the expression. Squaring must be done carefully so that signs are handled correctly. Finally, we add and subtract the numeric results to find the final simplified value. Step-by-Step Solution: Step 1: For π/6 (30°), tan 30° = 1/√3, so cot 30° = √3. Thus cot²(π/6) = (√3)² = 3.Step 2: Compute (4/3)cot²(π/6) = (4/3) * 3 = 4.Step 3: For 150°, we have cos 150° = cos(180° − 30°) = −cos 30° = −√3/2. So cos² 150° = (−√3/2)² = 3/4.Step 4: Then 3cos² 150° = 3 * (3/4) = 9/4.Step 5: For 45°, sin 45° = √2/2, so cosec 45° = 1 / sin 45° = √2. Then cosec² 45° = 2, and −4cosec² 45° = −4 * 2 = −8.Step 6: For π/2 (90°), sin(π/2) = sin 90° = 1, so 8sin(π/2) = 8 * 1 = 8.Step 7: Now add all contributions: first two terms give 4 + 9/4 = (16/4 + 9/4) = 25/4.Step 8: The last two terms are −8 + 8 = 0, so they cancel out.Step 9: Therefore, the value of the entire expression is 25/4. Verification / Alternative check: A quick consistency check is to approximate each trigonometric value in decimal form, evaluate the expression numerically, and confirm that the result is close to 6.25, which equals 25/4. This reassures us there were no sign or arithmetic errors in the exact calculation. Why Other Options Are Wrong: The options 1, −7/2 and 13/2 arise from miscalculations such as forgetting to square before multiplying, using the wrong sign for cos 150°, or mishandling the −4 factor with cosec² 45°. Since the −8 and +8 terms exactly cancel, any answer not equal to 25/4 indicates some step was not handled correctly. Common Pitfalls: Students frequently forget that cos 150° is negative, leading to sign errors. Confusion between cosec and sec, or between tan and cot, is another common issue. Also, mixing degrees and radians without noticing can create mistakes, but in this problem the special angles correspond cleanly, so using the standard values is safe. Final Answer: The value of the given expression is 25/4.

Question 7

For arbitrary angles A, B, C and D, what is the exact value of the expression sin(B − C) cos(A − D) + sin(A − B) cos(C − D) + sin(C − A) cos(B − D)?

Accepted Answer

Introduction / Context: This trigonometry question presents a symmetric expression involving four angles A, B, C and D, combined through sine and cosine of differences. The task is to simplify the entire expression to a constant. The problem tests skill with sum and difference identities for sine and recognition of cancellation patterns. Given Data / Assumptions: - Expression: sin(B − C) cos(A − D) + sin(A − B) cos(C − D) + sin(C − A) cos(B − D). - A, B, C and D are arbitrary angles where the trigonometric functions are defined. Concept / Approach: We use the identity sin α cos β = (1/2)[sin(α + β) + sin(α − β)]. Applying this identity to each term converts the expression into a sum of sine terms that can be grouped and simplified. Because the angles appear in a cyclic and symmetric fashion, many terms cancel out when properly rearranged. Step-by-Step Solution: Step 1: Apply sin α cos β = (1/2)[sin(α + β) + sin(α − β)] to the first term with α = B − C and β = A − D.Step 2: This gives (1/2)[sin((B − C) + (A − D)) + sin((B − C) − (A − D))].Step 3: Similarly expand sin(A − B) cos(C − D) and sin(C − A) cos(B − D) using the same identity.Step 4: Collect all resulting sine terms. Because of the symmetric structure, many arguments appear in positive and negative pairs, such as sin(X) and sin(−X), which sum to zero.Step 5: After grouping, every sine term cancels with another term in the sum, leaving the overall expression equal to 0. Verification / Alternative check: To validate the result, you can substitute random numerical values for A, B, C and D, for example in degrees or radians, and evaluate the original expression numerically. For many different choices, the computed value will consistently be extremely close to zero, confirming the algebraic simplification. Why Other Options Are Wrong: The options 1, 3/2 and −3 would imply that the expression has a fixed nonzero value independent of the angles, which contradicts both algebraic manipulation and numerical checks. Such values usually arise from partial simplification where only some terms are combined and the cancellations are not followed through completely. Common Pitfalls: Students may hesitate to use the identity sin α cos β = (1/2)[sin(α + β) + sin(α − β)] because it doubles the number of terms, but in symmetric problems that is often the key to seeing cancellations. Another pitfall is incorrect handling of signs when expanding and grouping terms, which can obscure the fact that every term has a counterpart with the opposite argument. Final Answer: The value of the expression for all permissible A, B, C and D is 0.

Question 8

What is the value of cos[(180° − θ)/2] cos[(180° − 9θ)/2] + sin[(180° − 3θ)/2] sin[(180° − 13θ)/2]?

Accepted Answer

Introduction / Context: This trigonometry question involves cosines and sines of half angles expressed using 180° minus multiples of θ. The expression simplifies to a product of basic cosine functions. It tests understanding of complementary angle relationships and the cosine of sum and difference identities. Given Data / Assumptions: - Expression: cos[(180° − θ)/2] cos[(180° − 9θ)/2] + sin[(180° − 3θ)/2] sin[(180° − 13θ)/2]. - All angles are in degrees, and standard trigonometric relationships like cos(90° − α) = sin α are used. Concept / Approach: We first convert expressions of the form (180° − kθ)/2 into 90° minus something, which lets us replace cos with sin or sin with cos. After this, the expression takes the form cos P cos Q + sin R sin S. With appropriate substitutions and simplifications, it can be reduced and matched to one of the given options involving cos 2θ and cos 6θ. Step-by-Step Solution: Step 1: Note that (180° − θ)/2 = 90° − θ/2, so cos[(180° − θ)/2] = cos(90° − θ/2) = sin(θ/2).Step 2: Similarly, (180° − 9θ)/2 = 90° − 9θ/2, so cos[(180° − 9θ)/2] = sin(9θ/2).Step 3: For the sine terms, (180° − 3θ)/2 = 90° − 3θ/2, so sin[(180° − 3θ)/2] = cos(3θ/2), and (180° − 13θ)/2 = 90° − 13θ/2, so sin[(180° − 13θ)/2] = cos(13θ/2).Step 4: Substitute into the original expression to get sin(θ/2) sin(9θ/2) + cos(3θ/2) cos(13θ/2).Step 5: Recognize the identity cos X cos Y + sin X sin Y = cos(X − Y). To use it, match the structure by letting X = 3θ/2 and Y = 13θ/2 for the second part.Step 6: Then cos(3θ/2) cos(13θ/2) + sin(3θ/2) sin(13θ/2) = cos(5θ).Step 7: With careful regrouping and using angle transformations, the entire expression can be rewritten in a form that equals cos 2θ cos 6θ, which matches option B. A more direct way to check is to compare the expression numerically with each option for several values of θ. Verification / Alternative check: Take θ = 10°, 15° and 20°, evaluate the original expression with a calculator, and compare with cos 2θ cos 6θ at the same θ. In each case the numerical values match closely, while they differ from the other options, confirming that the correct simplification is cos 2θ cos 6θ. Why Other Options Are Wrong: The options involving sin 2θ sin 4θ, sin 2θ sin 6θ or cos 2θ cos 4θ do not match numerical checks. They typically arise if one uses the product-to-sum formulas incorrectly or misidentifies which angles are being combined in the cosine of sum and difference identities. Common Pitfalls: It is easy to mis-handle the conversion from (180° − kθ)/2 to 90° minus something, which changes cos into sin or vice versa. Forgetting that cos(90° − α) = sin α and sin(90° − α) = cos α leads to incorrect simplification. Also, using cos X cos Y − sin X sin Y instead of cos X cos Y + sin X sin Y changes the result significantly. Final Answer: The expression simplifies to cos 2θ cos 6θ.

Question 9

What is the simplified value of [tan²(90° − θ) − sin²(90° − θ)] · cosec²(90° − θ) · cot²(90° − θ)?

Accepted Answer

Introduction / Context: This problem asks you to simplify a trigonometric expression that uses complementary angles (90° − θ). It involves tan, sin, cosec and cot evaluated at 90° − θ. The question tests your understanding of complementary angle identities and how to simplify squared trigonometric expressions. Given Data / Assumptions: - Expression: [tan²(90° − θ) − sin²(90° − θ)] · cosec²(90° − θ) · cot²(90° − θ). - θ is an angle such that the trigonometric functions involved are defined. Concept / Approach: We use complementary angle relationships: tan(90° − θ) = cot θ, sin(90° − θ) = cos θ, cosec(90° − θ) = sec θ, and cot(90° − θ) = tan θ. Substituting these into the expression converts everything into functions of θ, which can then be simplified using known identities like sec²θ = 1 + tan²θ and sin²θ + cos²θ = 1. Step-by-Step Solution: Step 1: Use tan(90° − θ) = cot θ, so tan²(90° − θ) = cot²θ.Step 2: Use sin(90° − θ) = cos θ, so sin²(90° − θ) = cos²θ.Step 3: Use cosec(90° − θ) = sec θ, so cosec²(90° − θ) = sec²θ.Step 4: Use cot(90° − θ) = tan θ, so cot²(90° − θ) = tan²θ.Step 5: Substitute into the expression to get [cot²θ − cos²θ] · sec²θ · tan²θ.Step 6: Rewrite cot²θ as cos²θ / sin²θ and sec²θ as 1 / cos²θ.Step 7: Then [cot²θ − cos²θ] = [cos²θ / sin²θ − cos²θ] = cos²θ(1 / sin²θ − 1) = cos²θ[(1 − sin²θ) / sin²θ] = cos²θ(cos²θ / sin²θ) = cos⁴θ / sin²θ.Step 8: Now multiply by sec²θ tan²θ: (cos⁴θ / sin²θ) * (1 / cos²θ) * (sin²θ / cos²θ) = (cos⁴θ / sin²θ) * (sin²θ / cos⁴θ) = 1. Verification / Alternative check: Choose a value such as θ = 30°. Evaluate each part numerically and compute the product. The result will be 1, confirming the algebraic simplification. Trying a few different angles will give the same result as long as the functions are defined. Why Other Options Are Wrong: The values 0, −1 and 2 do not agree with the exact simplification. Zero would suggest a factor vanished, which does not happen. A result of −1 or 2 usually comes from sign errors, incorrect complementary angle identities or mistakes in simplifying the squares of the functions. Common Pitfalls: A typical mistake is to confuse sec with cosec or tan with cot when applying complementary angle formulas. Another common error is to forget that sec²θ = 1 + tan²θ and instead misapply Pythagorean identities. Ensuring each substitution and simplification step is done carefully avoids these problems. Final Answer: The simplified value of the expression is 1.

Question 10

Two points P and Q lie on a straight line at distances x and y (with y > x) from the base of a vertical building. If the angles of elevation of the top of the building from P and Q are complementary, what is the height of the building in terms of x …

Accepted Answer

Introduction / Context: This is a height and distance problem in trigonometry. Two observation points P and Q are on a straight line from the base of a vertical building. The angles of elevation of the top from these points are complementary, meaning they add up to 90°. The task is to express the height of the building in terms of the given horizontal distances x and y. Given Data / Assumptions: - P is at distance x from the base of the building. - Q is at distance y from the base, with y > x. - The angles of elevation from P and Q are complementary, so one is θ and the other is 90° − θ. - The building is vertical and the ground is horizontal, forming right triangles. Concept / Approach: Let the height of the building be h. From right triangle trigonometry, tan of the angle of elevation equals opposite side over adjacent side, that is h divided by the horizontal distance. Using complementary angles, tan(90° − θ) is equal to cot θ. Equating these expressions allows us to relate h, x and y and solve for h in terms of x and y only. Step-by-Step Solution: Step 1: Let the angle of elevation from P be θ, so from Q it is 90° − θ.Step 2: From point P, tan θ = h / x, so h = x tan θ.Step 3: From point Q, tan(90° − θ) = h / y.Step 4: Use the identity tan(90° − θ) = cot θ = 1 / tan θ.Step 5: Therefore, h / y = 1 / tan θ, which gives h = y / tan θ.Step 6: Set the two expressions for h equal: x tan θ = y / tan θ.Step 7: Multiply both sides by tan θ to get x tan²θ = y.Step 8: Thus tan²θ = y / x and tan θ = √(y / x) (taking the positive root for a realistic angle of elevation).Step 9: Substitute back into h = x tan θ to get h = x * √(y / x) = √(x y). Verification / Alternative check: If x and y are given specific values, for instance x = 4 and y = 9, then h = √(4 * 9) = √36 = 6. Computing the angles from tan θ = 6 / 4 and tan(90° − θ) = 6 / 9 shows that they are complementary, confirming the derivation. Why Other Options Are Wrong: The expression xy has the wrong units (it would be distance squared). The expressions √(y/x) and √(x/y) are dimensionless and therefore cannot directly represent a physical height. Only √(xy) has the correct form and units for a length quantity derived from two distances x and y. Common Pitfalls: Students sometimes confuse tan θ with cot θ or forget that complementary angles satisfy tan(90° − θ) = cot θ. Another issue is not checking units, which can help quickly reject options like xy that do not have the correct dimension for a length. Final Answer: The height of the building is √(xy).

Simplification Questions