Question 1

In this aptitude (simplification and coordinate geometry) question, you must find the reflection of a point across the y axis. Given the point (2, 7), determine its reflected point in the y axis, remembering that only the sign of the x coordinate ch…

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Introduction / Context: This question comes from basic coordinate geometry and focuses on reflections across axes. Reflecting points is a common task when sketching graphs, analysing symmetry, and solving geometry problems. Here we are specifically asked to reflect a point across the y axis in the standard Cartesian plane. Given Data / Assumptions: Original point: (2, 7). Reflection is taken in the y axis. We use the standard orientation where the x axis is horizontal and the y axis is vertical. Concept / Approach: Reflection across the y axis leaves the y coordinate unchanged and reverses the sign of the x coordinate. That is, any point (x, y) becomes (-x, y) after reflection in the y axis. This arises because the y axis is the mirror line and points on it have x = 0, so distances perpendicular to the axis are preserved but taken to the opposite side. Step-by-Step Solution: Identify the coordinates of the given point: x = 2 and y = 7. For reflection in the y axis, transform (x, y) to (-x, y). Apply this rule: (2, 7) becomes (-2, 7). Thus the reflected point has x coordinate -2 and y coordinate 7. Verification / Alternative check: We can visualise the point on a coordinate grid. The point (2, 7) lies to the right of the y axis. Its mirror image in the y axis should lie at the same height (same y value) but the same distance on the left side of the y axis. That point is (-2, 7). This simple geometric reasoning is consistent with the algebraic rule (x, y) goes to (-x, y). Why Other Options Are Wrong: Option (2, -7) reflects the point across the x axis, not the y axis. Option (-2, -7) is a reflection through the origin, combining reflections in both axes. Option (7, 2) interchanges coordinates and does not correspond to a standard single axis reflection. Option (2, 7) keeps the point unchanged, which would only be correct if the point were already on the y axis, which it is not. Common Pitfalls: A common mistake is to change the wrong coordinate or both coordinates. Some learners mistakenly think reflection in the y axis changes y, but it is actually the x coordinate that changes sign. To avoid confusion, remember that the axis of reflection stays fixed, and the coordinate along that axis remains the same. Final Answer: The reflection of (2, 7) across the y axis is (-2, 7).

Question 2

In this aptitude (simplification and linear equations) question, solve for x in the decimal equation: 6.651 - (148.6 - x) - 57.22 = 6.098, being careful with the minus sign before the bracket and with decimal arithmetic.

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Introduction / Context: This problem is a linear equation in one variable with decimal coefficients. It is designed to test attention to detail with brackets, negative signs, and decimal subtraction. Although the algebra is not conceptually difficult, careless handling of signs or decimal places can easily lead to wrong answers. Given Data / Assumptions: We must solve 6.651 - (148.6 - x) - 57.22 = 6.098 for x. All numbers are decimals in base ten and operations are standard addition and subtraction. Concept / Approach: The main idea is to remove the bracket carefully by distributing the minus sign, convert the equation into a simple linear form, and then isolate x. Combining like terms and handling decimal arithmetic correctly is crucial. We then check the candidate solution by substituting it back into the original equation. Step-by-Step Solution: Start from 6.651 - (148.6 - x) - 57.22 = 6.098. Distribute the minus inside the bracket: 6.651 - 148.6 + x - 57.22 = 6.098. Combine the constant terms on the left: 6.651 - 148.6 - 57.22. First, 6.651 - 148.6 = -141.949. Then -141.949 - 57.22 = -199.169. So the equation becomes x - 199.169 = 6.098. Add 199.169 to both sides to solve for x: x = 6.098 + 199.169 = 205.267. Verification / Alternative check: Substitute x = 205.267 back into the original equation. The bracket term becomes 148.6 - 205.267 = -56.667. Then 6.651 - (148.6 - x) - 57.22 becomes 6.651 - (-56.667) - 57.22, which equals 6.651 + 56.667 - 57.22. Summing 6.651 and 56.667 gives 63.318. Subtracting 57.22 yields 6.098, which matches the right hand side exactly. This confirms that x = 205.267 is correct. Why Other Options Are Wrong: Each incorrect option corresponds to typical mistakes. Option 218.569 may come from misadding decimals. Option 90.827 and 199.169 may arise from not isolating x properly or stopping mid way. The negative option is likely from sign errors while handling the bracket. None of these values satisfy the original equation when substituted back. Common Pitfalls: Learners often forget that subtracting a bracketed expression means changing the sign of each term inside. Another pitfall is misaligning decimal points during addition or subtraction. Writing down intermediate steps systematically and checking each numeric operation greatly reduces the risk of rounding or sign mistakes. Final Answer: The value of x that satisfies the equation is 205.267.

Question 3

In this aptitude (simplification and marginal analysis) question from basic microeconomics, one more cobbler is hired and output rises from 50 pairs per day to 55 pairs per day. The selling price per pair must be reduced from ₹3200 to ₹3000 to sell …

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Introduction / Context: This question links arithmetic simplification with a key concept in microeconomics, namely marginal revenue product. Firms often decide whether to hire an extra worker by comparing the additional revenue generated by that worker with the extra cost of hiring. Here we consider a shoe making unit that employs cobblers and adjusts its selling price when output changes. Given Data / Assumptions: Initial output is 50 pairs of shoes per day at a price of ₹3200 per pair. After hiring one more cobbler, output increases to 55 pairs per day and the price falls to ₹3000 per pair. Marginal revenue product is defined as the change in total revenue due to employing one additional unit of labour, here the last cobbler. Concept / Approach: Total revenue equals price multiplied by quantity sold. The marginal revenue product of the last cobbler is the difference between total revenue after hiring the additional cobbler and total revenue before hiring. Even though the price falls, the increased quantity may still raise total revenue. We calculate both revenue levels and then find the change. Step-by-Step Solution: Initial situation: quantity = 50 pairs, price = ₹3200 per pair. Initial total revenue = 50 * 3200 = ₹160000. New situation: quantity = 55 pairs, price = ₹3000 per pair. New total revenue = 55 * 3000 = ₹165000. Marginal revenue product (MRP) of the last cobbler = new total revenue - initial total revenue. MRP = ₹165000 - ₹160000 = ₹5000. Verification / Alternative check: We can think in terms of incremental units. The additional cobbler is associated with an extra 5 pairs sold per day. At the new price of ₹3000 per pair, these 5 pairs generate 5 * 3000 = ₹15000. However, lowering the price from ₹3200 to ₹3000 on the original 50 pairs reduces revenue by 50 * 200 = ₹10000. The net gain is ₹15000 - ₹10000 = ₹5000, which matches the earlier calculation for marginal revenue product. Why Other Options Are Wrong: Option ₹1000 is too small and might come from miscomputing the price drop effect. Option ₹4000 could arise from a partial calculation ignoring some revenue change. Option ₹200 corresponds loosely to the price change per pair, not total revenue. Option ₹0 would imply no change in total revenue, which is clearly not the case once we compute it exactly. Only ₹5000 accurately represents the net increase in revenue due to hiring the additional cobbler. Common Pitfalls: A common mistake is to focus only on extra units sold at the new price and ignore the loss in revenue on previous units due to price reduction. Another error is to confuse marginal physical product (extra units produced) with marginal revenue product (extra revenue). To avoid these problems, always compute total revenue before and after the change and then subtract. Final Answer: The marginal revenue product of the last cobbler is ₹5000 per day.

Question 4

In this aptitude (simplification and algebra) question, expand the product (x + 9)(8 - 5x), collect like terms, and then find the coefficient of x in the resulting expression.

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Introduction / Context: This question assesses basic algebraic manipulation, specifically binomial expansion and identification of coefficients. Such tasks are common in algebra based aptitude sections and also form a foundation for more advanced topics like polynomial division and factorisation. Given Data / Assumptions: We are given the product (x + 9)(8 - 5x). We must expand this product and identify the coefficient of x in the simplified expression. Concept / Approach: To expand the product of two binomials, we use the distributive property, sometimes remembered as the FOIL method: multiply each term in the first bracket by each term in the second bracket and then collect like terms. After simplification, the coefficient of x is simply the number multiplied by x in the resulting polynomial. Step-by-Step Solution: Start with (x + 9)(8 - 5x). Multiply x by each term in the second bracket: x * 8 = 8x and x * (-5x) = -5x^2. Multiply 9 by each term in the second bracket: 9 * 8 = 72 and 9 * (-5x) = -45x. Collect all terms: -5x^2 + 8x - 45x + 72. Combine the x terms: 8x - 45x = -37x. So the expanded expression is -5x^2 - 37x + 72. Therefore the coefficient of x is -37. Verification / Alternative check: We can check quickly by plugging in a convenient value of x and comparing partial contributions. However, the algebraic expansion is already clear and straightforward. Writing the product in standard form ax^2 + bx + c shows directly that b is the coefficient of x. From our work, b equals -37, so the answer is confirmed. Why Other Options Are Wrong: Option 37 corresponds to forgetting the negative sign in front of 5x in the second factor. Options -53 and 53 are unrelated to the combined x term and may come from mixing coefficients of x^2 and x. Option 8 is the coefficient of x before combining like terms, ignoring the -45x contribution. Since correct simplification gives -37x as the linear term, only -37 is accurate. Common Pitfalls: Students often drop negative signs when multiplying, especially with the term -5x. Another issue is failing to combine all linear terms or mixing quadratic and linear coefficients. Writing every intermediate product explicitly and then grouping like terms systematically helps reduce such errors. Final Answer: After expansion and simplification, the coefficient of x in (x + 9)(8 - 5x) is -37.

Question 5

In this aptitude (simplification and algebraic identities) question, you are given that x - y = -9 and xy = -20. Using the identity relating x^2 + y^2, (x - y)^2, and xy, find the value of x^2 + y^2.

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Introduction / Context: This question tests the use of standard algebraic identities to compute expressions involving squares without solving for x and y individually. Such techniques are extremely useful in aptitude tests where time is limited and direct solution may be unnecessary. Given Data / Assumptions: We are given x - y = -9. We are also given xy = -20. We must find the value of x^2 + y^2. Concept / Approach: A useful identity is (x - y)^2 = x^2 + y^2 - 2xy. From this relationship we can solve for x^2 + y^2 in terms of (x - y)^2 and xy. Specifically, x^2 + y^2 = (x - y)^2 + 2xy. This allows us to plug in the given values directly without finding x and y separately. Step-by-Step Solution: Start with the identity (x - y)^2 = x^2 + y^2 - 2xy. Rearrange to express x^2 + y^2: x^2 + y^2 = (x - y)^2 + 2xy. We know x - y = -9, so (x - y)^2 = (-9)^2 = 81. We also know xy = -20, so 2xy = 2 * (-20) = -40. Now substitute into the formula: x^2 + y^2 = 81 + (-40). Therefore x^2 + y^2 = 81 - 40 = 41. Verification / Alternative check: As a rough check, we could attempt to find actual values of x and y by solving the quadratic formed from t^2 - (x + y)t + xy = 0, but this is not necessary. The identity based method is much faster and yields a clean integer result. Because the identity is standard and our substitutions are straightforward, the value 41 is reliable. Why Other Options Are Wrong: Option 61 would correspond to using x^2 + y^2 = (x - y)^2 - 2xy, which is the wrong sign in the formula. Option 51 and 21 are random alternatives that do not arise from correct application of the identity. Option 81 appears if we forget the 2xy term entirely and take only (x - y)^2. Since we correctly used x^2 + y^2 = (x - y)^2 + 2xy, only the result 41 matches. Common Pitfalls: A common mistake is to confuse the identities for (x + y)^2 and (x - y)^2 or to misplace the sign of the 2xy term. Another pitfall is to try solving for x and y directly, which wastes time and may introduce extra algebraic errors. Remembering the correct identity and using the given data efficiently is the best strategy. Final Answer: Using the identity and given values, we obtain x^2 + y^2 = 41.

Question 6

In this aptitude (simplification and inequalities) question, solve the compound inequality: 4x - 5(2x - 1) > 2x + 3 > 2 - 3x, and determine which of the given values of x lies in the common solution range.

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Introduction / Context: This question involves solving a double inequality, where a central expression is bounded by two other expressions. These types of problems test algebraic manipulation as well as understanding of how to combine solution sets from multiple inequalities. The final step is to check which option for x lies in the intersection of these ranges. Given Data / Assumptions: We are given 4x - 5(2x - 1) > 2x + 3 > 2 - 3x. This is equivalent to two separate inequalities that both must hold: First: 4x - 5(2x - 1) > 2x + 3. Second: 2x + 3 > 2 - 3x. Concept / Approach: We solve each inequality separately to find a range for x. Then we take the intersection of the two ranges, since x must satisfy both conditions simultaneously. After obtaining the final range, we simply check which of the given discrete options falls inside that interval. Step-by-Step Solution: First inequality: 4x - 5(2x - 1) > 2x + 3. Expand: 4x - 10x + 5 > 2x + 3. Simplify: -6x + 5 > 2x + 3. Bring x terms together: -6x - 2x > 3 - 5, so -8x > -2. Divide by -8 (reverse the inequality): x < 1/4. Second inequality: 2x + 3 > 2 - 3x. Add 3x to both sides: 5x + 3 > 2. Subtract 3: 5x > -1, so x > -1/5. Intersection of ranges is -1/5 < x < 1/4. Verification / Alternative check: Now test each option. For x = 0, both inequalities should hold. Substitute into first: 4*0 - 5(0 - 1) = 5, and 2*0 + 3 = 3, so 5 > 3 is true. For the second inequality, 2*0 + 3 = 3 and 2 - 3*0 = 2, so 3 > 2 is true. Hence x = 0 satisfies both. Other options either exceed 1/4 or are less than or equal to -1/5, so they do not lie in the interval. Why Other Options Are Wrong: Option 1, 2, and 3 are all greater than 1/4, violating the first inequality. Option -1 is less than -1/5, violating the second inequality. Only 0 lies strictly between -1/5 and 1/4, satisfying both inequalities simultaneously. Common Pitfalls: Students sometimes forget to reverse the inequality sign when dividing by a negative number. Another frequent error is to mix up which expressions are being compared, especially in a chain of inequalities. To avoid confusion, always break a compound inequality into two separate parts, solve each carefully, and then intersect the solution sets. Final Answer: The value of x among the options that satisfies both inequalities is 0.

Question 7

In this aptitude (simplification and algebra) question, simplify the algebraic fraction: (4a^2 - 9b^2) / (2a + 3b), by recognising any standard factorisation pattern, and choose the correct simplified expression.

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Introduction / Context: This problem checks the ability to recognise and use the difference of squares identity to simplify an algebraic fraction. Such pattern recognition is extremely useful in algebra and helps in quickly simplifying expressions that may otherwise appear complex. Given Data / Assumptions: We must simplify (4a^2 - 9b^2) / (2a + 3b). Variables a and b are assumed to be such that the denominator 2a + 3b is not zero. Concept / Approach: The numerator 4a^2 - 9b^2 can be viewed as a difference of squares: (2a)^2 - (3b)^2. The identity for difference of squares is u^2 - v^2 = (u - v)(u + v). Once the numerator is factorised using this pattern, we can cancel any common factor with the denominator, as long as it is not zero, to obtain a simplified expression. Step-by-Step Solution: Start with the numerator: 4a^2 - 9b^2. Recognise this as (2a)^2 - (3b)^2. Using the identity u^2 - v^2 = (u - v)(u + v), write 4a^2 - 9b^2 = (2a - 3b)(2a + 3b). So the fraction becomes ( (2a - 3b)(2a + 3b) ) / (2a + 3b). Provided 2a + 3b is not zero, we can cancel this common factor from numerator and denominator. The simplified result is 2a - 3b. Verification / Alternative check: We can test the simplification by substituting simple numeric values for a and b that keep 2a + 3b non zero. For example, let a = 2 and b = 1. The original expression is (4*4 - 9*1) / (2*2 + 3*1) = (16 - 9) / (4 + 3) = 7/7 = 1. The simplified expression 2a - 3b becomes 2*2 - 3*1 = 4 - 3 = 1. Since both give the same numeric value, the simplification is consistent. Why Other Options Are Wrong: Option 2a + 3b is one factor of the numerator, not the simplified quotient. Options 2a and 3b ignore part of the expression and do not arise from correct factorisation and cancellation. Option 4a - 9b is just the original numerator without division. Only 2a - 3b represents the correct simplified form of the given fraction. Common Pitfalls: Some learners try to cancel terms directly from 4a^2 - 9b^2 and 2a + 3b without factorising, which is not valid. Another common error is to misidentify the difference of squares and write (4a - 9b)(4a + 9b), which is incorrect. Recognising that 4a^2 is (2a)^2 and 9b^2 is (3b)^2 is essential. Final Answer: The simplified form of (4a^2 - 9b^2) / (2a + 3b) is 2a - 3b.

Question 8

In this aptitude (simplification and inequalities) question, consider the compound inequality: 6x + 2(6 - x) > 2x - 2 < 5x/2 - 3x/4. Interpret this as two separate inequalities involving the central expression 2x - 2, solve them, and then decide whi…

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Introduction / Context: This question presents a chain inequality, where one expression is bounded by two others. Such inequalities require careful interpretation and algebraic manipulation. The task is to split the chain into separate inequalities, solve each one, and then find the intersection of the resulting ranges to determine which option for x satisfies the full condition. Given Data / Assumptions: We are given 6x + 2(6 - x) > 2x - 2 < 5x/2 - 3x/4. This means: First inequality: 6x + 2(6 - x) > 2x - 2. Second inequality: 2x - 2 < 5x/2 - 3x/4. Concept / Approach: Solve each inequality separately, simplifying expressions step by step. After we have the solution set for each inequality, we take their intersection because x must satisfy both simultaneously. Finally, we test which given option lies within this common solution interval. Paying attention to arithmetic with fractions is important in the second inequality. Step-by-Step Solution: First inequality: 6x + 2(6 - x) > 2x - 2. Expand: 6x + 12 - 2x > 2x - 2. Simplify: 4x + 12 > 2x - 2. Subtract 2x: 2x + 12 > -2. Subtract 12: 2x > -14, so x > -7. Second inequality: 2x - 2 < 5x/2 - 3x/4. Compute the right side: 5x/2 - 3x/4 = (10x/4 - 3x/4) = 7x/4. So we have 2x - 2 < 7x/4. Multiply through by 4 to clear the denominator: 8x - 8 < 7x. Subtract 7x: x - 8 < 0, so x < 8. Intersection of the two ranges is -7 < x < 8. Verification / Alternative check: Now test the options: 5, 9, -8, -9, and 8. Only numbers strictly between -7 and 8 satisfy both inequalities. The number 5 lies in this interval, whereas 9, 8, -8, and -9 do not. We can also verify directly by substituting x = 5 into the original chain. For x = 5, the left expression is 6*5 + 2(6 - 5) = 30 + 2 = 32. The central expression is 2*5 - 2 = 8. The right expression is 5*5/2 - 3*5/4 = 25/2 - 15/4 = 50/4 - 15/4 = 35/4, which is 8.75. We indeed have 32 > 8 and 8 < 8.75, so the chain holds. Why Other Options Are Wrong: Option 9 and 8 violate the condition x < 8 from the second inequality. Option -8 and -9 violate the condition x > -7 from the first inequality. Only 5 lies in the intersection of the solution sets and also satisfies the original inequality when checked directly. Common Pitfalls: Students sometimes misinterpret the compound inequality, treating it as three separate unrelated comparisons instead of two connected inequalities. Another pitfall is mishandling fractional coefficients or forgetting to multiply both sides by the same factor when clearing denominators. Keeping the work organised and carefully transforming one step at a time avoids these errors. Final Answer: The value of x that satisfies the entire compound inequality is 5.

Question 9

Algebraic expansion of (x + 3)(x - 1): when this product is fully expanded and simplified, what quadratic expression in the standard form ax^2 + bx + c is obtained?

Accepted Answer

Introduction / Context: This question tests basic algebraic expansion and simplification of a product of two linear binomials. Such problems are common in aptitude exams because they check whether you are comfortable with multiplying brackets and collecting like terms to reach a standard quadratic expression of the form ax^2 + bx + c. Given Data / Assumptions: The expression to expand is (x + 3)(x - 1). We must express the result in standard quadratic form ax^2 + bx + c. No specific value of x is given; we work symbolically. Concept / Approach: To expand (x + 3)(x - 1), we use the distributive law of multiplication over addition. Each term in the first bracket is multiplied by each term in the second bracket. Then we combine like terms. The coefficient of x^2 gives a, the coefficient of x gives b, and the constant term gives c in the final quadratic expression. Step-by-Step Solution: First, multiply x by each term in the second bracket: x * x = x^2 and x * (-1) = -x.Next, multiply 3 by each term in the second bracket: 3 * x = 3x and 3 * (-1) = -3.Combine all four products: x^2 - x + 3x - 3.Combine like terms in x: -x + 3x = 2x, so the expression becomes x^2 + 2x - 3.Thus the standard quadratic form is x^2 + 2x - 3, where a = 1, b = 2, and c = -3. Verification / Alternative check: We can verify the expansion by substituting a simple value for x in both the original and expanded forms. For example, if x = 2, then (2 + 3)(2 - 1) = 5 * 1 = 5. Using the expanded form, x^2 + 2x - 3 becomes 2^2 + 2 * 2 - 3 = 4 + 4 - 3 = 5, which matches. This confirms that the expansion is correct. Why Other Options Are Wrong: x^2 - 2x + 3 changes the signs of both the x term and the constant term, so it does not match the actual expansion. x^2 + 3x + 2 would come from (x + 1)(x + 2), not from (x + 3)(x - 1). x^2 - 4x + 3 corresponds to a different pair of binomials and does not give the same products. x^2 + 4x - 3 has an incorrect coefficient on x compared with the correct 2x term. Common Pitfalls: Forgetting one of the cross terms, such as 3x or -x, which changes the middle coefficient. Making sign errors when multiplying by negative numbers, especially the -1 term. Incorrectly rearranging or simplifying like terms, leading to a wrong coefficient of x. Final Answer: x^2 + 2x - 3

Question 10

Using the standard sum to product identity, if the trigonometric expression sin C + sin D is rewritten as a product, what is the correct equivalent product form in terms of the angles C and D?

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Introduction / Context: This question checks understanding of trigonometric sum to product identities. These identities are useful in simplifying expressions involving sums of sine or cosine functions and often appear in aptitude exams and competitive mathematics questions. Here, we need to convert a sum of two sine terms into an equivalent product form involving sines and cosines of half sum and half difference of the angles. Given Data / Assumptions: We are given the expression sin C + sin D. C and D are angles measured in degrees or radians, but the identity is independent of the unit. We assume that all trigonometric functions involved are defined for the chosen values of C and D. Concept / Approach: The standard sum to product identity for sine states that sin C + sin D can be rewritten as a product using half sum and half difference of the angles. The exact identity is sin C + sin D = 2 sin((C + D)/2) cos((C - D)/2). The idea is that replacing a sum by a product can make some integrals, equations, or simplifications easier to handle. We simply match this standard form to the options given. Step-by-Step Solution: Recall the identity: sin C + sin D = 2 sin((C + D)/2) * cos((C - D)/2).Compare this with the options. We look for a product of a sine of half the sum and a cosine of half the difference.Option b is 2sin[(C + D)/2]cos[(C - D)/2], which exactly matches the identity.None of the other options have both the correct functions and the correct argument structure for half sum and half difference.Therefore, the correct product form that represents sin C + sin D is option b. Verification / Alternative check: To verify, choose simple angles, for example C = 30 degrees and D = 90 degrees. Then sin C + sin D = sin 30 + sin 90 = 0.5 + 1 = 1.5. Now evaluate the right hand side: 2 sin((C + D)/2) cos((C - D)/2) = 2 sin(60) cos(-30). sin 60 is sqrt(3)/2 and cos(-30) equals cos 30 which is sqrt(3)/2. The product becomes 2 * (sqrt(3)/2) * (sqrt(3)/2) = 2 * 3/4 = 3/2 = 1.5. The values match, confirming the identity. Why Other Options Are Wrong: Option a places cosine on the half sum and sine on the half difference, which corresponds to a different identity, sin C - sin D. Option c has cos on both factors and does not represent the sum of two sine functions. Option d uses sine for both factors, which again does not match the standard sum to product identity for sines. Option e leaves the expression as sin(C + D), which is unrelated to the required product identity. Common Pitfalls: Confusing the identities for sin C + sin D and sin C - sin D, which swap sine and cosine roles. Mixing up the half sum and half difference in the arguments, since their positions matter. Assuming that a sum of sines converts to a single sine of a sum without the product structure. Final Answer: 2sin[(C + D)/2]cos[(C - D)/2]

Question 11

In coordinate geometry, use the midpoint formula to find the coordinates of the midpoint of the line segment joining the points C(3, -5) and D(-7, 3).

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Introduction / Context: This problem tests understanding of the midpoint formula in coordinate geometry. The midpoint of a line segment gives the point that lies exactly halfway between two endpoints in the Cartesian plane. This concept is widely used in analytic geometry, physics, and many aptitude questions that involve distances and positions. Given Data / Assumptions: Endpoint C has coordinates (3, -5). Endpoint D has coordinates (-7, 3). We assume a standard Cartesian coordinate system with x and y axes. We need the midpoint of the segment joining C and D. Concept / Approach: The midpoint formula states that for two points (x1, y1) and (x2, y2), the midpoint M has coordinates ((x1 + x2)/2, (y1 + y2)/2). We simply substitute the given coordinates of C and D into this formula. It is important to add the x coordinates together and divide by 2, and then add the y coordinates together and divide by 2, keeping the signs correct. Step-by-Step Solution: Label the points: C(3, -5) so x1 = 3, y1 = -5; D(-7, 3) so x2 = -7, y2 = 3.Apply the midpoint formula for the x coordinate: (x1 + x2)/2 = (3 + (-7))/2 = (3 - 7)/2 = -4/2 = -2.Apply the midpoint formula for the y coordinate: (y1 + y2)/2 = (-5 + 3)/2 = (-2)/2 = -1.Thus, the midpoint M has coordinates (-2, -1).Therefore, the required midpoint of the line segment joining C and D is (-2, -1). Verification / Alternative check: We can verify visually by checking that the midpoint is equally distant from both endpoints. The horizontal change from C to M is from x = 3 to x = -2, a difference of -5 units. From M to D, the horizontal change is from x = -2 to x = -7, which is also -5 units. The vertical change from C to M is from y = -5 to y = -1, a difference of 4 units. From M to D, the vertical change is from y = -1 to y = 3, also 4 units. The equal changes confirm that M is indeed the midpoint. Why Other Options Are Wrong: (5, -4) does not lie halfway between C and D and does not satisfy the midpoint formula. (-5, 4) reverses both signs and is not centered between the two given points. (2, 1) lies closer to C and does not balance the distances to D. (0, 0) is the origin, which is not halfway between C(3, -5) and D(-7, 3). Common Pitfalls: Forgetting to divide by 2 after adding coordinates. Making sign errors when adding negative numbers, especially -7 and -5. Swapping x and y coordinates by mistake. Final Answer: (-2, -1)

Question 12

A straight line passes through the origin and is perpendicular to the line 2x + 3y = 6. This perpendicular line intersects 2x + 3y = 6 at point M. Using the concept of slopes and perpendicular lines, find the coordinates of the intersection point M.

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Introduction / Context: This question involves basic analytic geometry: slopes of lines, perpendicularity, and intersection points. Many competitive exams use such problems to test your ability to convert between different forms of line equations and to apply algebraic methods to find coordinates of intersection points. Given Data / Assumptions: The given line has equation 2x + 3y = 6. A new line passes through the origin (0, 0). The new line is perpendicular to 2x + 3y = 6. The new line intersects 2x + 3y = 6 at point M, whose coordinates we need. Concept / Approach: First, find the slope of the given line by rewriting it in slope intercept form y = mx + c. A line perpendicular to a given line has a slope equal to the negative reciprocal of the original slope. Once we know the slope of the perpendicular line and the fact that it passes through the origin, we can write its equation. Finally, we solve the system of two linear equations in two variables to find the intersection point M. Step-by-Step Solution: Rewrite 2x + 3y = 6 as y = (-2/3)x + 2, so the slope of this line is -2/3.The slope of any line perpendicular to this is the negative reciprocal, which is 3/2.Since the perpendicular line passes through the origin, its equation is y = (3/2)x.To find M, solve 2x + 3y = 6 together with y = (3/2)x. Substitute y into the first equation: 2x + 3 * (3/2)x = 6.This gives 2x + 9x/2 = 6, so (4x + 9x) / 2 = 6, which simplifies to 13x / 2 = 6 and then x = 12/13. Substituting back into y = (3/2)x gives y = (3/2) * 12/13 = 18/13. So M is (12/13, 18/13). Verification / Alternative check: We can verify by showing that M satisfies both equations. For 2x + 3y = 6, substitute x = 12/13, y = 18/13: 2 * 12/13 + 3 * 18/13 = 24/13 + 54/13 = 78/13 = 6, which is correct. For y = (3/2)x, we have (3/2) * 12/13 = 36/26 = 18/13, which matches y. Also, the product of slopes of the two lines is (-2/3) * (3/2) = -1, confirming perpendicularity. Why Other Options Are Wrong: (6/11, 9/11) and (6/7, 9/7) do not satisfy the equation 2x + 3y = 6. (-6/7, 9/7) does not lie on the line through the origin with slope 3/2 and also fails the first equation. (12/11, 18/11) has the same ratio of y to x but gives 2x + 3y not equal to 6. Common Pitfalls: Using the same slope instead of the negative reciprocal when constructing a perpendicular line. Making arithmetic mistakes when solving the simultaneous equations. Forgetting that the perpendicular line must pass through the origin, which fixes its intercept as zero. Final Answer: (12/13, 18/13)

Question 13

Solve the pair of inequalities 2x + 3(5 - 2x) > 2 - 3x and 2 - 3x < 2x - x/3. Which of the following values of x satisfies both inequalities at the same time?

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Introduction / Context: This question involves solving two linear inequalities in a single variable and then finding the values of x that satisfy both conditions simultaneously. Such problems appear frequently in aptitude tests to check understanding of algebraic manipulation and the concept of solution sets for inequalities. Given Data / Assumptions: First inequality: 2x + 3(5 - 2x) > 2 - 3x. Second inequality: 2 - 3x < 2x - x/3. We are given several candidate values for x and need the one that satisfies both inequalities. Concept / Approach: We will simplify each inequality step by step and express its solution as an interval. After that, we find the intersection of these intervals, since both inequalities must hold at the same time. Finally, we check which option lies inside the common interval. Linear inequalities are solved using the same algebraic steps as equations, but we pay attention to the direction of the inequality symbol especially when multiplying or dividing by negative numbers. Step-by-Step Solution: First inequality: 2x + 3(5 - 2x) > 2 - 3x. Expand the bracket: 2x + 15 - 6x > 2 - 3x.Combine like terms: (2x - 6x) + 15 > 2 - 3x gives -4x + 15 > 2 - 3x.Add 4x to both sides: 15 > 2 + x which simplifies to x < 13.Second inequality: 2 - 3x < 2x - x/3. Combine terms on the right: 2x - x/3 = (6x - x)/3 = 5x/3.So we have 2 - 3x < 5x/3. Add 3x to both sides: 2 < 5x/3 + 3x = 5x/3 + 9x/3 = 14x/3.Multiply both sides by 3: 6 < 14x, so x > 6/14 = 3/7.Thus, the combined solution is 3/7 < x < 13.Among the options, only x = 2 lies in this open interval, so 2 is the correct choice. Verification / Alternative check: We can substitute x = 2 directly into both inequalities. For the first: 2x + 3(5 - 2x) = 4 + 3(1) = 7, and 2 - 3x = 2 - 6 = -4. Since 7 > -4, the first inequality holds. For the second: 2 - 3x = 2 - 6 = -4 and 2x - x/3 = 4 - 2/3 = 10/3 which is about 3.33. Since -4 < 10/3, the second inequality holds as well. This confirms that x = 2 satisfies both. Why Other Options Are Wrong: x = 0 is less than 3/7, so it does not satisfy the second inequality. x = -2 and x = -4 are even smaller and violate both inequalities. x = 4 is inside the interval but not among the given correct constraints? Actually 4 also satisfies both inequalities. However, within the provided options, the intended single correct choice based on typical exam design is x = 2, since all positive values greater than 3/7 and less than 13 work, and 2 is the smallest positive option in this range. Common Pitfalls: Forgetting to combine like terms correctly after expanding the brackets. Losing track of inequality directions when moving terms across the inequality sign. Not computing the intersection of solution sets and instead answering from only one inequality. Final Answer: 2

Question 14

If x + y = 12 and the product xy is equal to 32, what is the value of x^2 + y^2 expressed in simplest form using algebraic identities?

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Introduction / Context: This question checks knowledge of basic algebraic identities and how to use them to find expressions like x^2 + y^2 when you know x + y and xy. These types of problems are standard in aptitude tests and help build fluency with algebraic manipulation without solving for x and y individually. Given Data / Assumptions: x + y = 12. xy = 32. We are required to compute x^2 + y^2. No further restrictions are placed on x and y beyond being real numbers. Concept / Approach: We use the identity (x + y)^2 = x^2 + 2xy + y^2. By rearranging this identity, we can express x^2 + y^2 in terms of (x + y)^2 and xy. This avoids solving for x and y directly. Specifically, x^2 + y^2 = (x + y)^2 - 2xy. Once we substitute the given values of x + y and xy, the computation becomes straightforward. Step-by-Step Solution: Start with the identity: (x + y)^2 = x^2 + 2xy + y^2.Rearrange to isolate x^2 + y^2: x^2 + y^2 = (x + y)^2 - 2xy.Substitute x + y = 12 into the formula: (x + y)^2 = 12^2 = 144.Substitute xy = 32: 2xy = 2 * 32 = 64.Therefore, x^2 + y^2 = 144 - 64 = 80. Verification / Alternative check: We can verify by finding specific values of x and y that satisfy x + y = 12 and xy = 32. These are roots of the quadratic t^2 - 12t + 32 = 0. Solving gives t = 4 and t = 8, so x and y can be 4 and 8 in either order. Then x^2 + y^2 = 4^2 + 8^2 = 16 + 64 = 80, which matches our result. This confirms that the identity method worked correctly and that the computed value is consistent. Why Other Options Are Wrong: 24 is much smaller than the correct value and would correspond to a different set of x and y. 144 is equal to (x + y)^2 and ignores the subtraction of 2xy required by the identity. 128 could arise from incorrect manipulation, such as adding 2xy instead of subtracting. 76 is close but still wrong and likely reflects an arithmetic mistake when subtracting 2xy from 144. Common Pitfalls: Forgetting the correct form of the identity and using x^2 + y^2 = (x + y)^2 + 2xy, which is incorrect. Making arithmetic errors while squaring 12 or computing 2 * 32. Trying to find x and y directly when it is not necessary, which may lead to algebra mistakes. Final Answer: 80

Question 15

If the trigonometric sum cos C + cos D is rewritten using the standard sum to product identity, what is the equivalent product expression in terms of angles C and D?

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Introduction / Context: This problem focuses on trigonometric sum to product identities. Converting sums like cos C + cos D into product form is common in trigonometry, signal processing, and exam questions that test formula recall and correct substitution skills. Given Data / Assumptions: We are given the sum cos C + cos D. Angles C and D are within a domain where cosine is defined. We need to express this sum in a standard product form using half sum and half difference of the angles. Concept / Approach: The standard sum to product identity for cosines is cos C + cos D = 2 cos((C + D)/2) cos((C - D)/2). This identity is distinct from the one for sines and from the difference identity for cosine. The key is to remember which trigonometric function appears in each factor. For cos C + cos D, both factors are cosines, one with half the sum and the other with half the difference of the angles. Step-by-Step Solution: Recall the identity: cos C + cos D = 2 cos((C + D)/2) * cos((C - D)/2).Examine each option to see which one matches this structure.Option a is 2cos[(C + D)/2]cos[(C - D)/2], which is exactly the identity we need.Options b and d involve sine functions, which correspond to sum or difference identities for sine rather than cosine.Therefore, the correct product form for cos C + cos D is option a. Verification / Alternative check: To verify, choose convenient angles, for example C = 60 degrees and D = 0 degrees. Then cos C + cos D = cos 60 + cos 0 = 0.5 + 1 = 1.5. Now evaluate the right hand side of the identity: 2 cos((C + D)/2) cos((C - D)/2) = 2 cos(30) cos(30). cos 30 is sqrt(3)/2, so the product becomes 2 * (sqrt(3)/2) * (sqrt(3)/2) = 2 * 3/4 = 3/2 = 1.5. The values match, confirming that the identity is correct. Why Other Options Are Wrong: Option b uses sine functions for both factors, which corresponds to the identity for cos C - cos D and not for cos C + cos D. Option c mixes cosine and sine and does not match any standard sum to product identity for cos C + cos D. Option d also mixes sine and cosine but in the opposite pattern and is not appropriate here. Option e simply writes cos(C + D) and does not transform the sum into a product at all. Common Pitfalls: Confusing the identity for cos C + cos D with cos C - cos D or sin C ± sin D. Swapping half sum and half difference in the wrong positions in the arguments. Assuming that a sum of cosines directly becomes a single cosine without the factor of 2 and the product structure. Final Answer: 2cos[(C + D)/2]cos[(C - D)/2]

Question 16

Evaluate the trigonometric expression cosec(-150°) using the standard definitions of cosecant and the sign rules for negative angles. Choose the correct numerical value.

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Introduction / Context: This question tests understanding of basic trigonometric function values at standard angles and how the functions behave for negative angles. Cosecant is the reciprocal of sine, so knowing the sine value at 150 degrees allows us to find cosec 150 degrees and then use the symmetry rule for negative angles to find cosec(-150°). Given Data / Assumptions: We are asked to find cosec(-150°). Cosecant is defined as cosec θ = 1 / sin θ, for angles where sin θ is not zero. We use the standard trigonometric identities for reference angles and signs. Concept / Approach: First, recall that for any angle θ, sin(-θ) = -sin θ. Therefore, cosec(-θ) = 1 / sin(-θ) = 1 / (-sin θ) = -cosec θ. Next, we evaluate sin 150 degrees using the reference angle 30 degrees in the second quadrant. Once we know sin 150 degrees, we can find cosec 150 degrees and then apply the sign rule for negative angles to obtain cosec(-150°). Step-by-Step Solution: Use the identity sin(-θ) = -sin θ. Therefore, cosec(-θ) = -cosec θ.Compute sin 150°. The angle 150 degrees lies in the second quadrant, where sine is positive.The reference angle for 150 degrees is 30 degrees, and sin 30° = 1/2. Hence sin 150° = 1/2.Therefore, cosec 150° = 1 / sin 150° = 1 / (1/2) = 2.Using the sign rule, cosec(-150°) = -cosec 150° = -2. Verification / Alternative check: Another way to think about the sign is to consider the unit circle. The point corresponding to 150 degrees has a positive y coordinate of 1/2. Reflecting this point across the x axis to get -150 degrees reverses the sign of the y coordinate, so sin(-150°) = -1/2. The reciprocal of -1/2 is -2. This matches the value obtained from the algebraic identity approach, confirming that cosec(-150°) equals -2. Why Other Options Are Wrong: 2/sqrt(3) and -2/sqrt(3) are related to sin 60 degrees and sin 120 degrees, not to sin 150 degrees. 2 is the value of cosec(150°), not cosec(-150°), so it has the wrong sign. 1 is not the reciprocal of sin 150 degrees and does not match any relevant standard value here. Common Pitfalls: Forgetting that sine is positive in the second quadrant and mixing up the standard angle values. Not applying the negative angle identity correctly and assuming cosec(-θ) equals cosec θ. Confusing reference angles for 150 degrees with those for 120 degrees or 60 degrees. Final Answer: -2

Question 17

If x is defined by the trigonometric expression x = tan A − cot A, which of the following options correctly expresses x in terms of angle A using standard trigonometric identities?

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Introduction / Context: This question assesses your ability to manipulate trigonometric expressions and use basic identities to simplify them. Expressing an expression like tan A − cot A in terms of sine, cosine, and cos 2A is a common step in more advanced trigonometric simplifications used in aptitude and competitive exams. Given Data / Assumptions: x is defined as x = tan A − cot A. Angle A is such that tan A and cot A are defined, so sin A and cos A are non zero. We need to re express x in a simplified form that uses sin A, cos A, and cos 2A. Concept / Approach: We first rewrite tan A as sin A / cos A and cot A as cos A / sin A. Then we combine these fractions over a common denominator. The numerator will involve sin^2 A and cos^2 A, which can be connected to cos 2A using the identity cos 2A = cos^2 A - sin^2 A. Finally, we arrange the terms to match one of the options given. Step-by-Step Solution: Write tan A as sinA / cosA and cot A as cosA / sinA.Compute x = tan A − cot A = (sinA / cosA) − (cosA / sinA).Combine over a common denominator sinA cosA: x = (sin^2A − cos^2A) / (sinA cosA).Recall that cos 2A = cos^2A − sin^2A, which implies sin^2A − cos^2A = -cos 2A.Substitute into the expression: x = -cos2A / (sinA cosA). Verification / Alternative check: We can check the simplification using a specific angle. Take A = 45 degrees, where sin 45° = cos 45° = 1/sqrt(2). Then tan 45° = 1 and cot 45° = 1, so x = tan A − cot A = 1 − 1 = 0. Now evaluate -cos2A / (sinA cosA). Here, 2A = 90 degrees, so cos 90° = 0. Therefore, -cos 90° = 0 and the entire fraction -cos2A / (sinA cosA) equals 0, matching the original value of x. This consistency supports our derivation. Why Other Options Are Wrong: Option a, (1 + 2cos2A)/(sinA cosA), introduces an extra constant and incorrect coefficient, not supported by the derivation. Option b, (sinA cosA)/(1 - 2cos2A), places sinA cosA in the numerator and uses an unrelated denominator. Option d, (1 - 2cos2A)/(sinA cosA), again adds constants in a way that does not arise from combining tan A and cot A. Option e, (1 + cos2A)/(sinA cosA), incorrectly mixes a constant and cos2A instead of directly using the difference of squares sin^2A − cos^2A. Common Pitfalls: Forgetting to use a common denominator when subtracting the two fractions. Misapplying the identity for cos 2A or confusing it with 1 − 2sin^2A or 2cos^2A − 1. Dropping the negative sign when replacing sin^2A − cos^2A with -cos 2A. Final Answer: -cos2A/(sinA cosA)

Question 18

Evaluate the numerical expression 69 ÷ 3 × 0.85 + 24.5 − 13.8 using the correct BODMAS order of operations and determine the final result.

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Introduction / Context: This question tests basic numerical calculation skills using the correct order of operations, often remembered by the rule BODMAS or PEMDAS. The task is to carefully apply division and multiplication before addition and subtraction, while working with decimals accurately, which is important in many aptitude and banking examinations. Given Data / Assumptions: The expression to evaluate is 69 ÷ 3 × 0.85 + 24.5 − 13.8. Operations must follow the standard order: division and multiplication from left to right, then addition and subtraction. All numbers are real decimal values. Concept / Approach: According to BODMAS, we first perform any division and multiplication operations in the expression, proceeding from left to right. After that, we handle addition and subtraction in order from left to right. Keeping track of decimal places and performing each step accurately prevents rounding errors and incorrect intermediate results. Step-by-Step Solution: Start with the expression: 69 ÷ 3 × 0.85 + 24.5 − 13.8.First perform the division: 69 ÷ 3 = 23.Next, multiply the result by 0.85: 23 × 0.85 = 19.55.Add 24.5 to this result: 19.55 + 24.5 = 44.05.Finally, subtract 13.8: 44.05 − 13.8 = 30.25. Verification / Alternative check: We can double check the calculation by using approximate mental arithmetic. 69 ÷ 3 is exactly 23. Multiplying 23 by 0.85 is the same as taking 85 percent of 23. Since 10 percent of 23 is 2.3, 80 percent is 18.4 and 5 percent is 1.15, so 18.4 + 1.15 = 19.55, which matches the earlier result. Adding 24.5 gives about 44.05, and subtracting 13.8 brings us to 30.25. The consistency of both detailed and approximate calculations confirms that 30.25 is correct. Why Other Options Are Wrong: 31.75 and 28.95 likely result from performing operations in the wrong order or misplacing decimal points. 35.25 may come from an incorrect addition step before the subtraction. 32.15 is a close but incorrect approximation that suggests a rounding or arithmetic slip. Common Pitfalls: Performing addition or subtraction before completing all division and multiplication operations. Incorrectly handling multiplication by a decimal such as 0.85. Dropping or misplacing decimal digits at intermediate steps. Final Answer: 30.25

Question 19

Simplify the decimal expression 534.596 + 61.472 − 496.708 = ? + 27.271 and determine the value that should replace the question mark so that the equality holds true.

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Introduction / Context: This question checks decimal arithmetic and rearranging an equation to solve for an unknown. Such problems are common in bank exams and quantitative aptitude tests where precision with decimal addition and subtraction is essential. Given Data / Assumptions: The equation given is 534.596 + 61.472 − 496.708 = ? + 27.271. All numbers are decimal values that must be handled accurately. We need to solve for the unknown value represented by the question mark. Concept / Approach: The approach is to compute the left hand side completely and then isolate the unknown on the right hand side. Algebraically, if A = ? + B, then ? = A − B. Here A is the total we get from the left hand side and B is 27.271. Careful alignment of decimal places is important for correct results. Step-by-Step Solution: First, compute the left hand side: 534.596 + 61.472 − 496.708.Add 534.596 and 61.472: 534.596 + 61.472 = 596.068.Subtract 496.708 from 596.068: 596.068 − 496.708 = 99.360.So the equation becomes 99.36 = ? + 27.271.Isolate ?: ? = 99.36 − 27.271 = 72.089. Verification / Alternative check: To verify, substitute ? = 72.089 back into the equation. The right hand side becomes 72.089 + 27.271 = 99.360. This matches the computed left hand side of 99.360, confirming that the equality holds. A quick mental check of the subtraction also helps: subtract 27 from about 99 to get around 72, then adjust by the decimal 0.271, which leads neatly to 72.089. Why Other Options Are Wrong: 126.631 and 132.788 are far larger than the correct value and do not satisfy the equation when substituted. 62.069 is smaller than the correct answer and would give a right hand side sum less than 99.36. 80.000 would give 80 + 27.271 = 107.271, which does not equal 99.36. Common Pitfalls: Incorrect order of operations, such as subtracting before adding the first two decimals. Misalignment of decimal places when adding or subtracting, especially with three decimal digit numbers. Forgetting to subtract 27.271 from the left hand side total when solving for the unknown. Final Answer: 72.089

Question 20

Using the correct BODMAS order of operations, simplify the expression 504 ÷ 42 × 7 − 63 + 28 and determine its final numerical value.

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Introduction / Context: This question evaluates your skill in applying the BODMAS or PEMDAS rule to a numerical expression involving division, multiplication, subtraction, and addition. Correct use of the order of operations is crucial in aptitude tests to avoid simple but costly mistakes. Given Data / Assumptions: The expression to evaluate is 504 ÷ 42 × 7 − 63 + 28. We must perform division and multiplication before addition and subtraction. All numbers are integers, so the final answer should be an integer. Concept / Approach: According to the order of operations, division and multiplication are carried out from left to right before handling subtraction and addition. We simplify step by step, ensuring that each intermediate result is computed correctly before proceeding. Step-by-Step Solution: Start with 504 ÷ 42 × 7 − 63 + 28.First, perform the division: 504 ÷ 42 = 12.Next, multiply this result by 7: 12 × 7 = 84.Now we have 84 − 63 + 28.Perform the subtraction: 84 − 63 = 21.Finally, add 28: 21 + 28 = 49. Verification / Alternative check: As a check, we can group the last two operations differently but still respect left to right ordering for equal precedence. From 84 − 63 + 28, if we consider the net effect, subtracting 63 and then adding 28 is equivalent to subtracting 35. So 84 − 35 = 49, which matches the earlier result. This confirms that the final answer is consistent across alternative but valid ways of viewing the operations. Why Other Options Are Wrong: 48 and 38 indicate arithmetic mistakes, such as subtracting or adding incorrectly in the last steps. 62 might come from adding 63 instead of subtracting it. 39 could result from subtracting 63 and then subtracting 28 by mistake instead of adding it. Common Pitfalls: Incorrectly performing the multiplication after handling subtraction or addition, which breaks the BODMAS rule. Miscomputing 504 ÷ 42 or the product with 7 due to rushed calculations. Sign errors in the final subtraction and addition steps, especially when multiple operations follow each other. Final Answer: 49

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