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).

Question 11

If $A + B = 96$ and $A$ is half of $B$, then the value of $A$ will be

Accepted Answer

Concept & Logic This is a linear equation problem involving substitution. By expressing one variable in terms of the other based on the given ratio, you can solve for the unknown variable using a single-variable algebraic equation. Step-by-Step Solution • Given conditions: 1) A + B = 96 2) A is half of B, which translates to A = (B/2) or B = 2A • Substitute the value of B into the first equation: A + 2A = 96 • Combine the like terms: 3A = 96 • Solve for A: A = (96/3) A = 32 Exam Strategy & Shortcut Ratio Method: Since A is half of B, the ratio of A:B is 1:2. The total parts in the sum are 1 + 2 = 3 parts. We are given that the sum (3 parts) equals 96. The value of 1 part (which is A) is simply 96 \div 3 = 32. This avoids formal algebra and is extremely fast. Common Pitfall A frequent error is misreading "A is half of B" and mistakenly writing A = 2B. This leads to the incorrect calculation 3B = 96 \Rightarrow B = 32, and A = 64. Always double-check the relationship before setting up the equation. Final Answer Therefore, the correct answer is 32.

Question 12

$-76 	imes 33 + 221 = x$

Accepted Answer

Concept & Rule This question applies the BODMAS/PEMDAS rule, dictating that multiplication must be performed before addition. It also tests the handling of negative integers during arithmetic operations. Step-by-Step Solution • Given expression: -76 × 33 + 221 • According to BODMAS, perform the multiplication first: -76 × 33 • To make this easier, break 33 into (30 + 3): -76 × (30 + 3) = -2280 - 228 = -2508 • Substitute this back into the original expression: -2508 + 221 • Perform the addition (which acts as subtraction since the signs differ): -2508 + 221 = -2287 Exam Strategy & Shortcut Unit Digit Check: Multiply the unit digits of the first term: 6 × 3 = 18. The unit digit is an 8, but it carries a negative sign (so think of it as ending in -8). We are adding a number ending in 1. -8 + 1 = -7. The final answer must have a unit digit of 7 and be negative. Only -2287 fits this exact profile. Common Pitfall Students often ignore the negative sign during the multiplication step, calculating 2508 + 221 = 2729, or they accidentally add 221 to 33 first before multiplying, violating the BODMAS order of operations entirely. Final Answer Therefore, the correct answer is -2287.

Question 13

$4848 \div 24 	imes 11 - 222 = x$

Accepted Answer

Concept & Formula This simplification problem strictly follows the BODMAS rule (Brackets, Orders, Division, Multiplication, Addition, Subtraction). Operations must be executed left-to-right for Division and Multiplication in the sequence they appear if not bracketed, but Division takes precedence here. Step-by-Step Solution • Given expression: 4848 \div 24 × 11 - 222 • Step 1: Perform the Division first: (4848/24) = 202 • (You can quickly see 48 \div 24 = 2 and 48 \div 24 = 2, putting a zero in between gives 202). • Step 2: Substitute back and perform the Multiplication: 202 × 11 - 222 202 × 11 = 2222 • Step 3: Perform the final Subtraction: 2222 - 222 = 2000 Exam Strategy & Shortcut Multiplication by 11 Rule: When you reach 202 × 11, use the standard shortcut for multiplying by 11: write the outer digits, and insert the sums of adjacent digits in between. 2, (2+0), (0+2), 2 \Rightarrow 2222. Subtracting 222 from 2222 immediately yields 2000 visually without needing scratchpad math. Common Pitfall A major pitfall is executing the division incorrectly. Students often calculate 4848 \div 24 as 22 instead of 202 by forgetting to insert the zero placeholder when bringing down the next digit. This leads to entirely wrong subsequent calculations. Final Answer Therefore, the correct answer is 2000.

Question 14

$-224 + (-314) 	imes (-9) = x$

Accepted Answer

Concept & Rule This problem tests the order of operations alongside negative integer arithmetic rules. Multiplication must be performed prior to addition. Crucially, the product of two negative numbers is always a positive number. Step-by-Step Solution • Given expression: -224 + (-314) × (-9) • Step 1: According to BODMAS, evaluate the multiplication part first. (-314) × (-9) Since multiplying two negatives yields a positive: 314 × 9 = 2826 • Step 2: Substitute this result back into the original equation. -224 + 2826 • Step 3: Perform the addition. Since the signs are opposite, subtract the smaller absolute value from the larger one and keep the sign of the larger value. 2826 - 224 = 2602 Exam Strategy & Shortcut Sign and Unit Digit Elimination: You can solve this in seconds without full calculations. First, (-314) × (-9) will be a large positive number (around +2700). Subtracting 224 will still leave a large positive number. This instantly eliminates options (a) and (b). Next, look at the unit digits. The product 4 × 9 = 36, so it ends in a 6. The addition is essentially (number ending in 6) - (number ending in 4). 6 - 4 = 2. The final answer must be a positive number ending in 2. Option (d) 2602 is the only logical choice. Common Pitfall The most common trap is a sign error during multiplication. Students hastily write (-314) × (-9) = -2826, which leads to -224 - 2826 = -3050. Always double-check your integer sign rules before moving to the next step. Final Answer Therefore, the correct answer is 2602.

Question 15

$853 + x \div 17 = 1000$

Accepted Answer

Concept & Logic This is a fundamental algebraic equation. To isolate the unknown variable x, you must reverse the order of operations, peeling away the constants on the left side by moving them to the right side of the equals sign. Step-by-Step Solution • Given equation: 853 + (x/17) = 1000 • Step 1: Isolate the fraction containing x by subtracting 853 from both sides. (x/17) = 1000 - 853 (x/17) = 147 • Step 2: Solve for x by multiplying both sides by 17. x = 147 × 17 • Step 3: Perform the multiplication. You can break 17 into (10 + 7) for easier mental math. x = 147 × (10 + 7) x = 1470 + 1029 x = 2499 Exam Strategy & Shortcut Unit Digit Verification: Once you reach the step x = 147 × 17, you do not need to calculate the entire product. Simply multiply the unit digits of the two numbers: 7 × 7 = 49. The final product must have a unit digit of 9. Looking closely at the given options, only option (b) 2499 ends in a 9. You can confidently select this option and save precious seconds. Common Pitfall A very common error is misinterpreting the BODMAS rule and attempting to add 853 directly to x before dealing with the division, mistakenly writing ((853 + x)/17) = 1000. This leads to 853 + x = 17000, resulting in x = 16147, which is entirely incorrect (and cleverly placed as option d). Final Answer Therefore, the correct answer is 2499.

Question 16

$999 	imes 99 	imes 9 \div 99 \div 9 \div 3 = x$

Accepted Answer

Concept & Formula This question tests the rule of sequential division. When multiple divisions occur in sequence without brackets, they evaluate strictly from left to right. A faster algebraic method is to convert sequential divisions into a single multiplication in the denominator: a \div b \div c = (a/b × c) Step-by-Step Solution • Given expression: 999 × 99 × 9 \div 99 \div 9 \div 3 • Step 1: Group the multiplications in the numerator and move all the sequentially divided terms into the denominator as a product. (999 × 99 × 9/99 × 9 × 3) • Step 2: Cancel out the common terms in the numerator and denominator. Both contain a 99 and a 9. (999 × (99 × 9)/(99 × 9) × 3) This simplifies directly to: (999/3) • Step 3: Perform the final simple division. (999/3) = 333 Exam Strategy & Shortcut Visual Cancellation: Do not write anything down for this problem. Visually scan the equation from left to right. You see × 99 and later \div 99—they cancel out. You see × 9 and later \div 9—they cancel out. You are left strictly with 999 \div 3, which is instantly 333. This should take no more than 3 seconds to solve on an exam. Common Pitfall A common mistake is reading the expression right-to-left or grouping divisions incorrectly, such as treating \div 9 \div 3 as \div (9 \div 3) = \div 3, which completely alters the mathematical meaning and leads to an incorrect answer like 999. Final Answer Therefore, the correct answer is 333.

Question 17

If $a, b, c, \dots, x, y, z$ are $26$ natural numbers, then the value of $(x - a)(x - b)(x - c) \dots (x - y)(x - z)$ is

Accepted Answer

Concept & Logic This is a classic logical puzzle disguised as an algebra problem. It tests your ability to recognize patterns within a sequence rather than performing actual mathematical calculations. The key is identifying a specific term within the long product chain that forces the entire expression to evaluate to a constant. Step-by-Step Solution • Given expression: (x - a)(x - b)(x - c) \dots (x - y)(x - z) • Step 1: Understand the pattern of the sequence. The first term in each bracket is always the variable x. The second term iterates through the alphabet from a to z. • Step 2: Trace the sequence mentally. It goes (x - a), (x - b), (x - c), and so on. Since it iterates through the entire alphabet, it must eventually reach the letter x as the second term. • Step 3: Identify the critical bracket. At the 24th position in the sequence, the bracket will be: (x - x) • Step 4: Evaluate the critical bracket. (x - x) = 0 • Step 5: Apply the zero product property. Any number multiplied by 0 results in 0. (Product of previous terms) × 0 × (Product of remaining terms) = 0 Exam Strategy & Shortcut Look for the Zero Term: Whenever you see a long sequential product, especially one involving the alphabet or a long string of integers, immediately scan for a term that evaluates to zero. In alphanumeric sequences, look for self-subtraction like (x - x) or (n - n). The moment you realize (x - x) is part of the sequence, the answer is instantly 0 without any further thought. Common Pitfall The main pitfall is panic. Students see an infinite-looking algebra string with 26 unknown variables and assume it requires complex polynomial expansion or advanced theorems, wasting time on a problem designed to be solved in two seconds via logical deduction. Final Answer Therefore, the correct answer is 0.

Question 18

$$ \frac{18 	imes 14 - 6 	imes 8}{488 \div 4 - 20} = x $$

Accepted Answer

Concept & Logic: This expression requires a straightforward application of the BODMAS rule for both the numerator and the denominator independently. Remember the priority: Division comes before Multiplication, which comes before Subtraction. Step-by-Step Solution: • Step 1: Simplify the numerator. First, perform the multiplications: 18 × 14 = 252 6 × 8 = 48 Next, perform the subtraction: Numerator = 252 - 48 = 204 • Step 2: Simplify the denominator. First, perform the division: 488 \div 4 = 122 Next, perform the subtraction: Denominator = 122 - 20 = 102 • Step 3: Divide the simplified numerator by the simplified denominator. x = (204/102) x = 2 Exam Strategy & Shortcut: To calculate 18 × 14 quickly, you can use the base 10 multiplication trick or break it down algebraically: 18 × (10 + 4) = 180 + 72 = 252 . Once you see (204/102), you should immediately recognize that 102 × 2 = 204. You do not need to perform long division; simply look at the leading digits and unit digits (2 × 2 = 4). Common Pitfall: A major mistake is violating the order of operations in the denominator by subtracting before dividing (e.g., trying to do 4 - 20 first). Always scan for the division symbol and execute it before any addition or subtraction. Final Answer: Therefore, the correct answer is 2.

Question 19

$$ \frac{4+4 	imes 18 - 6 - 8}{123 	imes 6 - 146 	imes 5} = x $$

Accepted Answer

Concept & Logic: This problem requires careful execution of arithmetic operations according to the BODMAS rule, followed by decimal conversion. You must accurately multiply large numbers in the denominator and simplify the entire fraction before finding the exact decimal value. Step-by-Step Solution: • Step 1: Simplify the numerator. Expression: 4 + 4 × 18 - 6 - 8 First, perform the multiplication: 4 × 18 = 72. The expression becomes: 4 + 72 - 6 - 8. Now, process addition and subtraction from left to right: 76 - 6 - 8 = 70 - 8 = 62. Numerator = 62. • Step 2: Simplify the denominator. Expression: 123 × 6 - 146 × 5 Perform the multiplications: 123 × 6 = 738 146 × 5 = 730 Subtract the results: 738 - 730 = 8. Denominator = 8. • Step 3: Form the fraction and solve for the decimal. x = (62/8) Reduce the fraction by dividing by 2: x = (31/4) Convert to decimal: (31/4) = 7 (3/4) = 7.75. Exam Strategy & Shortcut: For 146 × 5, use the shortcut of multiplying by 10 and dividing by 2: 1460 \div 2 = 730. This is much faster than standard vertical multiplication. When you arrive at (62/8), look at the options. You know 8 × 7 = 56 and 8 × 8 = 64. Therefore, the answer must be between 7 and 8. The only option that fits this estimation is 7.75, eliminating the need to do the final long division manually. Common Pitfall: A standard arithmetic error occurs in the denominator calculation 123 × 6. If rushed, students might calculate it as 728 instead of 738, leading to a denominator of -2 and throwing off the entire result. Double-check single-digit multiplications carefully. Final Answer: Therefore, the correct answer is 7.75.

Question 20

$$4\frac{3}{7} - 1\frac{3}{14} = x + 2\frac{3}{28}$$

Accepted Answer

Concept & Strategy: This is a linear equation containing mixed fractions. The strategy is to isolate the variable x on one side of the equation, then separate the resulting expression into whole numbers and fractional parts to simplify efficiently. Notice that the denominators (7, 14, 28) are multiples of each other, making the common denominator obvious. Step-by-Step Solution: • Step 1: Isolate the variable x. Move 2(3/28) to the left side by subtracting it. x = 4(3/7) - 1(3/14) - 2(3/28) • Step 2: Split the expression into whole numbers and fractions. Whole numbers: (4 - 1 - 2) Fractions: ((3/7) - (3/14) - (3/28)) • Step 3: Evaluate the whole numbers. 4 - 1 - 2 = 1 • Step 4: Evaluate the fractions using the LCM of 28. Convert (3/7) to (12/28) Convert (3/14) to (6/28) The fraction expression becomes: (12/28) - (6/28) - (3/28) (12 - 6 - 3/28) = (3/28) • Step 5: Combine the results. Final Answer = 1 + (3/28) = 1(3/28) Exam Strategy & Shortcut: Since the denominators scale perfectly (7 × 2 = 14, 14 × 2 = 28), you can solve the fractions almost entirely in your head. Just think in terms of 28ths: 12 - 6 - 3 = 3. The whole numbers are trivial: 4 - 1 - 2 = 1. The answer assembles itself instantly. Common Pitfall: When shifting 2(3/28) across the equals sign, students sometimes forget to invert the sign of BOTH the whole number and the fraction. Subtracting a mixed number means subtracting the whole part AND subtracting the fractional part. Final Answer: Therefore, the correct answer is 1(3/28).

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