Question 1

Solve the linear equation with fractional terms and find the exact value of x: 8x/3 + [7(5 - 2x/3)]/2 = 1/2.

Accepted Answer

Introduction / Context: This algebra question involves solving a linear equation that contains fractions and brackets. You are required to simplify the expression carefully, clear the fractions, and then solve for x. Such problems are common in aptitude tests and algebra courses as they test attention to detail and comfort with fractional arithmetic and linear equations. Given Data / Assumptions: Equation: 8x/3 + [7(5 − 2x/3)]/2 = 1/2. x is a real number. We must find the exact value of x that satisfies the equation. Standard arithmetic operations and fraction rules apply. Concept / Approach: First, simplify the bracketed term 7(5 − 2x/3) and then divide by 2 as indicated. Combine like terms, especially those involving x, to obtain a simpler linear equation. Clear fractions by finding a common denominator or by multiplying through by an appropriate number if needed. Finally, isolate x and solve. Throughout, careful handling of fractions and signs is essential to avoid mistakes. Step-by-Step Solution: 1) Start with the equation: 8x/3 + [7(5 − 2x/3)]/2 = 1/2. 2) Simplify inside the bracket: 7(5 − 2x/3) = 7 * 5 − 7 * (2x/3) = 35 − 14x/3. 3) Divide by 2: [7(5 − 2x/3)]/2 = (35 − 14x/3) / 2 = 35/2 − (14x/3)/2 = 35/2 − 14x/6 = 35/2 − 7x/3. 4) Substitute back into the equation: 8x/3 + 35/2 − 7x/3 = 1/2. 5) Combine like x terms on the left: 8x/3 − 7x/3 = x/3, so the equation becomes x/3 + 35/2 = 1/2. 6) Subtract 35/2 from both sides: x/3 = 1/2 − 35/2 = (1 − 35)/2 = −34/2 = −17. 7) Multiply both sides by 3 to solve for x: x = 3 * (−17) = −51. Verification / Alternative check: Substitute x = −51 back into the original equation. Compute 8x/3 = 8(−51)/3 = −408/3 = −136. Then compute 5 − 2x/3 = 5 − 2(−51)/3 = 5 + 102/3 = 5 + 34 = 39. Next, 7(5 − 2x/3) = 7 * 39 = 273, and [7(5 − 2x/3)]/2 = 273/2. The left side becomes −136 + 273/2. Express −136 as −272/2, so −272/2 + 273/2 = 1/2, which matches the right side of the equation. This confirms that x = −51 is indeed the correct solution. Why Other Options Are Wrong: If you plug in −17, 17, or 51, the left-hand side will not simplify to 1/2. For example, with x = 17, the term 8x/3 becomes positive and large, and the bracketed term cannot compensate to produce 1/2. Since we have found a consistent and verified solution, the option 'None of these' is also incorrect. Among the given options, only −51 satisfies the equation. Common Pitfalls: Many students make errors when distributing over the bracket 7(5 − 2x/3) or when dividing by 2, especially in handling the x term in the fraction. Others mis subtract 35/2 on both sides or incorrectly simplify 1/2 − 35/2. Writing each fraction step explicitly and keeping a common denominator helps avoid these mistakes. Always verify the candidate solution by substitution to ensure accuracy. Final Answer: The solution of the equation is -51.

Question 2

In basic algebra, if a − b = −1 and the product ab = 6, then what is the exact value of the difference of cubes a^3 − b^3?

Accepted Answer

Introduction / Context: This algebra question tests the use of standard identities for the difference of cubes and for the square of a binomial. Instead of solving directly for a and b, we can work with expressions such as a − b, ab, and a^2 + b^2 to reach the required expression a^3 − b^3. This technique is common in aptitude exams where the goal is to manipulate expressions efficiently without solving a full quadratic equation. Given Data / Assumptions: a − b = −1 ab = 6 a and b are real or complex numbers, but the identities hold for all algebraic numbers We need to find a^3 − b^3 Concept / Approach: We use the identity a^3 − b^3 = (a − b)(a^2 + ab + b^2). We already know a − b and ab. To find a^2 + b^2 we use the binomial square identity (a − b)^2 = a^2 + b^2 − 2ab. Once we compute a^2 + b^2, we can add ab to get a^2 + ab + b^2 and then multiply by a − b to get the required difference of cubes. Step-by-Step Solution: Start from (a − b)^2 = a^2 + b^2 − 2ab. Given a − b = −1, so (a − b)^2 = (−1)^2 = 1. Substitute the known values: 1 = a^2 + b^2 − 2ab. Use ab = 6, so −2ab = −2 * 6 = −12. Therefore 1 = a^2 + b^2 − 12, so a^2 + b^2 = 13. Now find a^2 + ab + b^2 = (a^2 + b^2) + ab = 13 + 6 = 19. Use the identity a^3 − b^3 = (a − b)(a^2 + ab + b^2). Substitute a − b = −1 and a^2 + ab + b^2 = 19. Then a^3 − b^3 = (−1) * 19 = −19. Verification / Alternative check: We can verify indirectly by constructing a quadratic with roots a and b. The sum a + b would satisfy t^2 − (a + b)t + ab = 0. From a − b and ab there is no need to solve fully because the identities used are exact. The internal consistency of a^2 + b^2 = 13 and ab = 6 leads to a^2 + ab + b^2 = 19, which correctly reproduces the result −19 when multiplied by a − b. This confirms the algebraic steps are valid and that no arithmetic mistake occurred in the process. Why Other Options Are Wrong: 33 and 35 arise if one misuses signs in the binomial square identity or in the final multiplication. 18 appears if ab is used incorrectly in forming a^2 + ab + b^2. The value 0 would require a^3 = b^3, which is impossible when a − b = −1. Hence these values do not match the correct algebraic manipulation. Common Pitfalls: A common mistake is to confuse (a − b)^2 with a^2 − b^2, which is a different identity. Another frequent error is to forget the factor 2ab when expanding the square or to substitute ab with the wrong sign. Some learners also attempt to find explicit values of a and b even though it is unnecessary and may lead to messy quadratic equations. Working symbolically with identities is more efficient and reduces computational errors in timed exams. Final Answer: Thus, the exact value of the difference of cubes is −19.

Question 3

In coordinate geometry, what are the coordinates of the reflection of the point (2, 3) in the horizontal line y = 4?

Accepted Answer

Introduction / Context: This problem checks understanding of reflections of points in a coordinate plane, specifically reflection in a horizontal line. Such questions are very common in school level coordinate geometry and aptitude tests, because they quickly reveal whether a learner understands how vertical and horizontal distances behave under reflection symmetry. Given Data / Assumptions: Original point P has coordinates (2, 3). The mirror line is the horizontal line y = 4. We need the coordinates of the reflection P prime after reflecting P across the line y = 4. The coordinate system is the usual Cartesian plane. Concept / Approach: A horizontal line y = k reflects points by preserving the x coordinate and changing the y coordinate so that the reflected point is at the same perpendicular distance from the line on the opposite side. If a point has y coordinate y1, its reflection across y = k will have y coordinate y2 such that the vertical distance from y1 to k equals the distance from k to y2. Algebraically this means y2 = 2k − y1, while x remains unchanged because the reflection is vertical. Step-by-Step Solution: The mirror line is y = 4, so k = 4. The original point has coordinates (2, 3), so x1 = 2 and y1 = 3. Under reflection in a horizontal line, the x coordinate remains the same, so x2 = 2. Compute the vertical distance from the point to the line: 4 − 3 = 1 unit. The reflected point lies 1 unit on the other side of the line, so its y coordinate is 4 + 1 = 5. Hence the reflected point P prime has coordinates (2, 5). Verification / Alternative check: We can check by using the formula y2 = 2k − y1. With k = 4 and y1 = 3, we get y2 = 2 * 4 − 3 = 8 − 3 = 5. The x coordinate stays 2, so again the image point is (2, 5). Additionally, the midpoint between (2, 3) and (2, 5) is (2, 4), which lies exactly on the line y = 4, confirming that the line is the perpendicular bisector of the segment joining the point and its reflection. Why Other Options Are Wrong: (2, −5) reflects in the x axis rather than in y = 4. The points (−2, −5) and (−2, 5) change the x coordinate improperly because reflection in a horizontal line does not alter the x coordinate. The point (2, 1) is one unit below y = 2 rather than y = 4 and is not symmetric about the given line. None of these satisfy the requirement that y = 4 be the perpendicular bisector of the segment. Common Pitfalls: Learners sometimes confuse reflection in a horizontal line with reflection in a vertical line, inadvertently changing the wrong coordinate. Another common mistake is to subtract the distance only once instead of adding it on the other side, leading to y2 = k − d instead of y2 = k + d. Remember that reflection keeps the perpendicular distance equal but reverses direction relative to the mirror line. Final Answer: The reflection of the point (2, 3) in the line y = 4 is (2, 5).

Question 4

In coordinate geometry, what is the equation of a straight line that has slope −1/3 and a y-intercept equal to 6?

Accepted Answer

Introduction / Context: This question tests the relationship between the slope intercept form of a line and its general form. Being able to move between y = mx + c and standard forms such as ax + by + c = 0 is a foundational skill in coordinate geometry and appears frequently in aptitude and competitive examinations. Given Data / Assumptions: Slope of the required line, m = −1/3. Y intercept, which is the point where the line crosses the y axis, is 6. We seek a correct equation among the options that represents this line. Concept / Approach: The slope intercept form of a straight line is y = mx + c, where m is the slope and c is the y intercept. Once we write the line in this form, we can rearrange it into the general form ax + by + d = 0 or equivalently ax + by = k for comparison with the answer choices. Algebraic manipulation like multiplying throughout by a constant does not change the set of points that lie on the line. Step-by-Step Solution: Write the equation in slope intercept form: y = (−1/3)x + 6. Multiply both sides by 3 to eliminate the denominator: 3y = −x + 18. Rearrange the terms to put x on the left side: x + 3y = 18. This is now in the same structural form as the multiple choice options. Compare with the given options to identify which equation matches x + 3y = 18 exactly. Verification / Alternative check: We can quickly check by finding the intercepts. For x + 3y = 18, setting x = 0 gives 3y = 18, so y = 6, which matches the required y intercept. For the slope, rewrite as 3y = −x + 18, giving y = (−1/3)x + 6, where the slope is indeed −1/3. No other option yields both slope −1/3 and y intercept 6 when rearranged into y = mx + c form. Why Other Options Are Wrong: x − 3y = 6 gives y = (1/3)x − 2, which has slope 1/3 and intercept −2, both incorrect. x + 3y = −18 gives y intercept −6. x − 3y = −6 gives y = (1/3)x + 2, again wrong slope and intercept. The option 3x + y = 6 gives y = −3x + 6, slope −3 instead of −1/3. Thus none of these match the stated slope and intercept. Common Pitfalls: A frequent error is to invert the slope or to confuse the roles of x and y intercepts. Another issue is forgetting that multiplying the entire equation by a nonzero constant does not change the line, but changing signs or moving terms incorrectly can alter the relationship between slope and intercept. Always return to y = mx + c to check slope and intercept clearly. Final Answer: The required line has equation x + 3y = 18.

Question 5

Using standard trigonometric values, what is the exact value of Sin 60° + (1/2) Cosec 45°?

Accepted Answer

Introduction / Context: This problem checks knowledge of standard trigonometric values, especially for special angles 45 degrees and 60 degrees. Such values are widely used in geometry, physics, and aptitude tests. The question combines sine and cosecant functions and requires correct substitution followed by simple fraction addition. Given Data / Assumptions: We need to evaluate Sin 60° + (1/2) Cosec 45°. Angles are in degrees. We use exact special angle values, not decimal approximations. Concept / Approach: The key is to recall that Sin 60° = √3 / 2 and Sin 45° = √2 / 2. Cosecant is the reciprocal of sine, so Cosec 45° = 1 / Sin 45°. After substituting these values, we simplify the expression by putting both terms over a common denominator. The goal is to match the simplified result with one of the provided options. Step-by-Step Solution: Recall Sin 60° = √3 / 2. Recall Sin 45° = √2 / 2, so Cosec 45° = 1 / Sin 45° = 2 / √2 = √2. Compute (1/2) Cosec 45° = (1/2) * √2 = √2 / 2. Now add the two terms: Sin 60° + (1/2) Cosec 45° = √3 / 2 + √2 / 2. Combine over the common denominator 2 to get (√3 + √2) / 2. This matches option (√3 + √2)/2 exactly. Verification / Alternative check: If we approximate numerically, Sin 60° is about 0.866, and Cosec 45° is the reciprocal of about 0.707, which is approximately 1.414. Then (1/2) Cosec 45° is about 0.707. Adding gives approximately 1.573. Evaluating (√3 + √2)/2 with √3 ≈ 1.732 and √2 ≈ 1.414 gives (1.732 + 1.414)/2 ≈ 3.146/2 ≈ 1.573, confirming the algebraic result. Why Other Options Are Wrong: (√3 + 2)/2 and (√3 + 2)/√2 arise if one mistakenly takes Cosec 45° as 2 instead of √2. The form (√3 + √2)/√2 has the wrong denominator, and (2√3 + √2)/4 does not simplify to the correct expression. None of these reflect the precise substitution of special angle values. Common Pitfalls: Students sometimes confuse sine with cosine values for 30 degrees, 45 degrees, and 60 degrees, or they invert the wrong ratio when finding cosecant. Another mistake is forgetting to simplify 2 / √2 to √2, leaving extra factors in the denominator. Keeping a small table of key exact values in memory is very helpful for such questions. Final Answer: The exact value of the expression is (√3 + √2)/2.

Question 6

In right triangle ΔLMN, which is right angled at M, if ∠N = 45° and the hypotenuse LN = 6√2 cm, then what is the length (in centimetres) of side MN?

Accepted Answer

Introduction / Context: This geometry question tests recognition of a special right triangle, the 45 45 90 triangle. In such triangles, the legs are equal and the hypotenuse has a fixed ratio to the legs. Identifying this pattern allows a quick solution without using trigonometric functions or the full Pythagorean theorem every time. Given Data / Assumptions: Triangle ΔLMN is right angled at M. Angle N is 45 degrees, so angle L is also 45 degrees. Hypotenuse LN has length 6√2 cm. We seek the length of the leg MN. Concept / Approach: In a 45 45 90 triangle, the two acute angles are equal, and therefore the legs opposite them are equal in length. The ratio of the hypotenuse to each leg is √2. If each leg is of length s, then the hypotenuse is s√2. Knowing the hypotenuse, we can simply divide by √2 to find the leg length. This is faster than repeatedly applying the Pythagorean theorem. Step-by-Step Solution: Since ΔLMN is right angled at M and ∠N = 45°, the remaining angle ∠L is also 45°. Thus ΔLMN is an isosceles right triangle of type 45 45 90. Let the length of each leg (LM and MN) be s. Then the hypotenuse LN has length s√2. Given LN = 6√2, set s√2 = 6√2. Divide both sides by √2 to get s = 6. Therefore, side MN has length 6 cm. Verification / Alternative check: We can check with the Pythagorean theorem. If MN = 6 and LM = 6, then LN^2 = MN^2 + LM^2 = 6^2 + 6^2 = 36 + 36 = 72. Taking the square root, LN = √72 = √(36 * 2) = 6√2, which matches the given hypotenuse. This confirms that MN = 6 cm is consistent with both the triangle type and the given hypotenuse length. Why Other Options Are Wrong: Values such as 3, 4, or 2 would give much smaller hypotenuse lengths when combined with an equal leg, so they would not match 6√2. For example, if MN = 3, the hypotenuse would be 3√2, not 6√2. The option 3√2 is longer than the leg in a 45 45 90 triangle, but it does not correspond to any side given the required hypotenuse length. Common Pitfalls: Some learners mistakenly apply the 30 60 90 triangle ratios or mix up which side corresponds to the given hypotenuse. Others square 6√2 incorrectly, leading to miscalculations. Always remember that in a 45 45 90 triangle, leg : hypotenuse = 1 : √2, and check consistency with Pythagoras to avoid such mistakes. Final Answer: The length of side MN is 6 cm.

Question 7

In a right triangle with acute angle θ, if Cos θ = 35/37, then what is the exact value of Cosec θ?

Accepted Answer

Introduction / Context: This trigonometry question examines the relationship between cosine, sine, and cosecant in a right triangle. Given a value for Cos θ, we can reconstruct the triangle sides and then find Sin θ and Cosec θ. This type of problem helps build geometric intuition for trigonometric ratios beyond simple memorisation of values. Given Data / Assumptions: Cos θ = 35/37 for an acute angle θ. Cos θ is defined as adjacent side divided by hypotenuse in a right triangle. We need Cosec θ, which is the reciprocal of Sin θ. Angle θ is acute, so all basic trigonometric ratios are positive. Concept / Approach: If Cos θ = adjacent / hypotenuse = 35 / 37, we can model a right triangle where the adjacent side is 35 units and the hypotenuse is 37 units. Then we use the Pythagorean theorem to find the length of the opposite side. After that we compute Sin θ = opposite / hypotenuse, and finally find Cosec θ = 1 / Sin θ. This geometric method is straightforward and avoids unnecessary algebraic manipulation of trigonometric identities. Step-by-Step Solution: Interpret Cos θ = 35/37 as adjacent = 35 and hypotenuse = 37. Use the Pythagorean theorem: hypotenuse^2 = adjacent^2 + opposite^2. So 37^2 = 35^2 + opposite^2. Compute 37^2 = 1369 and 35^2 = 1225. Then opposite^2 = 1369 − 1225 = 144, so opposite = 12 (positive for acute angle). Therefore Sin θ = opposite / hypotenuse = 12 / 37. Cosec θ is the reciprocal of Sin θ, so Cosec θ = 37 / 12. Verification / Alternative check: We can verify consistency by computing Cos^2 θ + Sin^2 θ. With Cos θ = 35/37 and Sin θ = 12/37, we get (35/37)^2 + (12/37)^2 = (1225 + 144)/1369 = 1369/1369 = 1, which satisfies the fundamental identity Sin^2 θ + Cos^2 θ = 1. This confirms that the triangle sides and the resulting cosecant value are correct. Why Other Options Are Wrong: 33/12 and 35/12 do not match the reciprocal of 12/37. The value 12/35 corresponds more closely to a tangent or cotangent type ratio and not to cosecant in this setup. The fraction 37/35 mixes the hypotenuse and adjacent sides wrongly. Only 37/12 correctly represents the reciprocal of Sin θ for this triangle. Common Pitfalls: Sometimes students swap the roles of opposite and adjacent, especially if they do not sketch the triangle first. Another error is arithmetic, such as computing 37^2 or 35^2 incorrectly or miscalculating the difference. Drawing a quick diagram and verifying Sin^2 θ + Cos^2 θ = 1 are simple ways to avoid these mistakes. Final Answer: The exact value of Cosec θ is 37/12.

Question 8

Solve the linear equation [4(2x/5 − 3/2)]/3 + 7/5 = 37/5 and find the value of x.

Accepted Answer

Introduction / Context: This question involves solving a linear equation with fractional coefficients. Such problems test comfort with fractions, distributive law, and careful use of arithmetic operations. They appear frequently in aptitude tests, where accurate handling of fractions is important for both algebra and word problems. Given Data / Assumptions: The equation is [4(2x/5 − 3/2)]/3 + 7/5 = 37/5. x is a real number. We need to isolate x and find its numerical value. Concept / Approach: We simplify step by step: first handle the inner bracket 2x/5 − 3/2, then multiply by 4, divide by 3, and finally combine with 7/5. To remove fractions efficiently, we can multiply the entire equation by the least common multiple of all denominators. This reduces the equation to one with integer coefficients, which is easier to solve. Step-by-Step Solution: Start with [4(2x/5 − 3/2)]/3 + 7/5 = 37/5. First simplify inside the brackets: 2x/5 − 3/2. Multiply this by 4: 4 * (2x/5 − 3/2) = 8x/5 − 6. Now divide by 3: [4(2x/5 − 3/2)]/3 = (8x/5 − 6)/3. So the equation becomes (8x/5 − 6)/3 + 7/5 = 37/5. Multiply every term by 15 (the least common multiple of 3 and 5) to clear denominators. 15 * (8x/5 − 6)/3 = 5 * (8x/5 − 6) = 8x − 30. 15 * (7/5) = 3 * 7 = 21, and 15 * (37/5) = 3 * 37 = 111. Thus we get 8x − 30 + 21 = 111. Simplify the left side: 8x − 9 = 111. Add 9 to both sides: 8x = 120. Divide by 8: x = 120 / 8 = 15. Verification / Alternative check: Substitute x = 15 into the original equation. Compute 2x/5 = 2 * 15 / 5 = 6 and 6 − 3/2 = 6 − 1.5 = 4.5. Multiply by 4 to get 18, then divide by 3 to get 6. Now add 7/5, which is 1.4, so the left side is 6 + 1.4 = 7.4. The right side 37/5 is 7.4 as well. Since both sides match, x = 15 is confirmed as correct. Why Other Options Are Wrong: Values such as −15 or −7/5 come from sign errors when expanding or from incorrect handling of the denominators. The value 7/5 appears if one confuses x with one of the intermediate fractional terms. The value 5 would not satisfy the original equation when checked by substitution. None of these produce equal left and right sides. Common Pitfalls: The main sources of error are skipping steps when clearing fractions, forgetting to multiply all terms by the same factor, or mishandling negative signs. A structured approach, clearing denominators early and checking by substitution, greatly reduces these mistakes. Final Answer: The solution of the equation is x = 15.

Question 9

If a − b = 4 and the product ab = −3 in algebra, then using standard identities what is the value of the difference of cubes a^3 − b^3?

Accepted Answer

Introduction / Context: This algebra problem again focuses on the identity for the difference of cubes together with the square of a difference. Instead of solving for a and b directly, we use a − b and ab to compute a^3 − b^3. This style of question is common in aptitude tests because it rewards fluency with algebraic identities and efficient reasoning. Given Data / Assumptions: a − b = 4 ab = −3 We want to find a^3 − b^3. a and b are algebraic numbers and can be treated symbolically. Concept / Approach: The core identity is a^3 − b^3 = (a − b)(a^2 + ab + b^2). We already know a − b and ab, so the key objective is to find a^2 + b^2. We use the expansion (a − b)^2 = a^2 + b^2 − 2ab. Once a^2 + b^2 is obtained, we can add ab to form a^2 + ab + b^2 and then multiply by a − b for the final result. Step-by-Step Solution: Start from the identity (a − b)^2 = a^2 + b^2 − 2ab. Given a − b = 4, so (a − b)^2 = 4^2 = 16. Substitute into the identity: 16 = a^2 + b^2 − 2ab. Given ab = −3, so −2ab = −2 * (−3) = 6. Thus 16 = a^2 + b^2 + 6, so a^2 + b^2 = 10. Now compute a^2 + ab + b^2 = (a^2 + b^2) + ab = 10 + (−3) = 7. Use the identity a^3 − b^3 = (a − b)(a^2 + ab + b^2). Substitute a − b = 4 and a^2 + ab + b^2 = 7. Hence a^3 − b^3 = 4 * 7 = 28. Verification / Alternative check: We can cross check by constructing a quadratic whose roots are a and b. For example, one such quadratic is t^2 − (a + b)t + ab = 0. From a − b and ab we could find (a + b)^2 = a^2 + b^2 + 2ab = 10 + 2 * (−3) = 4, giving a + b = ±2. Using any consistent pair of a and b that satisfy both a − b = 4 and a + b = 2, we obtain specific values, then compute a^3 − b^3 explicitly and verify that it equals 28. This confirms the identity based computation. Why Other Options Are Wrong: The values 21 and 23 result from arithmetic errors, such as miscalculating a^2 + b^2 or ab. The option −20 appears if one misplaces a sign in the identity or uses a + b in place of a − b. The value 12 is too small and would correspond to incorrect use of 2ab instead of ab in the final step. Common Pitfalls: Misremembering the identity for (a − b)^2 or misusing a^3 − b^3 = (a − b)(a^2 − ab + b^2) instead of the correct form for a^3 − b^3 can lead to wrong results. Careful attention to signs, especially when ab is negative, is very important. Writing out each identity fully and substituting values slowly helps avoid these mistakes. Final Answer: The value of the difference of cubes is 28.

Question 10

In coordinate geometry, the distance between points (2, 7) and (k, −5) is 13 units. What is the correct value of k among the given options?

Accepted Answer

Introduction / Context: This question applies the distance formula in the coordinate plane and then solves for an unknown x coordinate. It is typical of coordinate geometry problems in aptitude tests, where one point is partially unknown but the distance between two points is fixed. Given Data / Assumptions: Point A has coordinates (2, 7). Point B has coordinates (k, −5). The distance between A and B is 13 units. We must determine the valid value of k from the answer options. Concept / Approach: The distance formula between points (x1, y1) and (x2, y2) is distance = √[(x2 − x1)^2 + (y2 − y1)^2]. Here the distance is known to be 13. We square both sides to remove the square root, producing an equation in k. Solving this quadratic equation gives possible values of k, after which we check which of these are present in the options. Step-by-Step Solution: Use the distance formula between (2, 7) and (k, −5). Distance^2 = (k − 2)^2 + (−5 − 7)^2. Compute the y difference: −5 − 7 = −12, so (−12)^2 = 144. Given distance = 13, so 13^2 = (k − 2)^2 + 144. Thus 169 = (k − 2)^2 + 144. Subtract 144 from both sides to get (k − 2)^2 = 25. Take square roots: k − 2 = ±5. So k = 2 + 5 = 7 or k = 2 − 5 = −3. Among the options, only k = 7 is listed. Verification / Alternative check: Check k = 7 by recomputing the distance. With B = (7, −5), the differences are 7 − 2 = 5 and −5 − 7 = −12. Then distance^2 = 5^2 + (−12)^2 = 25 + 144 = 169, so the distance is √169 = 13. Thus k = 7 is valid. Though k = −3 also satisfies the equation, it is not given among the choices, so it cannot be the answer for this multiple choice question. Why Other Options Are Wrong: The values −7, 6, −6, and 3 do not satisfy (k − 2)^2 = 25. For example, if k = 6, we get (6 − 2)^2 = 4^2 = 16, which leads to distance^2 = 16 + 144 = 160, not 169. Similar checks show that none of these choices yield a distance of 13 units from (2, 7). Common Pitfalls: A common error is to forget to square the full difference in the distance formula, or to confuse the roles of x and y coordinates. Another pitfall is ignoring the fact that squaring introduces two possible solutions, positive and negative roots. However, the multiple choice list may restrict which valid solution is acceptable as the final answer. Final Answer: The value of k that matches the given distance and appears in the options is 7.

Question 11

What is the equation of the straight line that is perpendicular to 5x + 3y = 6 and has a y-intercept equal to −3?

Accepted Answer

Introduction / Context: Here we use the concept of perpendicular lines in coordinate geometry. Lines that are perpendicular have slopes that are negative reciprocals of one another. This question also requires finding the equation of a line given a slope and a y intercept, and then comparing that with various standard forms. Given Data / Assumptions: Original line: 5x + 3y = 6. Required line is perpendicular to this original line. Required line has y intercept −3. We seek the correct equation from the given options. Concept / Approach: First, find the slope of the given line by rewriting it in slope intercept form. If the slope of the original line is m, then the slope of a line perpendicular to it is −1/m. Once we know the new slope and the y intercept, we write y = m2 x + c and then rearrange into an algebraic form to match the options. Matching both slope and intercept ensures that we have the correct perpendicular line. Step-by-Step Solution: Start from 5x + 3y = 6. Solve for y: 3y = −5x + 6, so y = (−5/3)x + 2. Thus the slope of the original line is m1 = −5/3. For a perpendicular line, the slope m2 is the negative reciprocal: m2 = 3/5. The y intercept of the required line is −3, so its equation in slope intercept form is y = (3/5)x − 3. Multiply both sides by 5 to remove the denominator: 5y = 3x − 15. Rearrange to standard form: 3x − 5y = 15. Verification / Alternative check: Check the y intercept by substituting x = 0 into 3x − 5y = 15, which gives −5y = 15, so y = −3, matching the required intercept. To verify perpendicularity, rewrite 3x − 5y = 15 as y = (3/5)x − 3. Its slope is 3/5. The original slope is −5/3, and the product (−5/3)*(3/5) = −1, confirming that the lines are perpendicular. Why Other Options Are Wrong: 3x + 5y = 15 has slope −3/5, not 3/5, and would be perpendicular to a line of slope 5/3 rather than to the original line. The equation 3x − 5y = −15 gives y intercept 3, not −3. The equation 3x + 5y = −15 has slope −3/5 and intercept −3, so it does not satisfy the negative reciprocal relation with the original slope. The equation 5x + 3y = −9 has the same slope as the original, so this line is parallel, not perpendicular. Common Pitfalls: Students sometimes mix up parallel and perpendicular slopes, or forget to take the negative reciprocal correctly. Another source of error is miscomputing the y intercept when converting between forms. Always verify both slope and intercept at the end to avoid simple algebraic mistakes. Final Answer: The required perpendicular line with y intercept −3 is 3x − 5y = 15.

Question 12

In triangle ΔABC, points D and E lie on sides AB and AC respectively, and DE is parallel to BC. If AD : DB = 2 : 5 and the area of ΔADE is 8 cm², then what is the area (in square centimetres) of quadrilateral BDEC?

Accepted Answer

Introduction / Context: This geometry question uses similarity of triangles and area ratios. When a line segment joins two sides of a triangle parallel to the third side, the smaller triangle formed is similar to the original triangle. This allows us to relate areas through the square of a linear scale factor, a very important concept in geometry and mensuration problems. Given Data / Assumptions: Triangle ΔABC with D on AB and E on AC. DE ∥ BC. AD : DB = 2 : 5, so AB is divided in ratio 2 : 5. Area of ΔADE is 8 cm². We need the area of quadrilateral BDEC. Concept / Approach: Because DE is parallel to BC, triangles ADE and ABC are similar. The ratio of similarity between corresponding sides is AD / AB. Since AB = AD + DB, we can compute this ratio. The area scale factor between the smaller and larger triangle is the square of the linear scale factor. Once we find the total area of ΔABC, subtracting the area of ΔADE gives the area of quadrilateral BDEC, which is the region between DE and BC. Step-by-Step Solution: Given AD : DB = 2 : 5, so let AD = 2k and DB = 5k. Then AB = AD + DB = 2k + 5k = 7k. The linear scale factor from ΔADE to ΔABC is AD / AB = 2k / 7k = 2/7. Areas of similar triangles scale by the square of the linear factor. Therefore, Area(ΔADE) / Area(ΔABC) = (2/7)^2 = 4/49. Given Area(ΔADE) = 8 cm², so 8 / Area(ΔABC) = 4/49. Solving gives Area(ΔABC) = 8 * 49 / 4 = 8 * 12.25 = 98 cm². Quadrilateral BDEC is the region ΔABC minus ΔADE, so Area(BDEC) = 98 − 8 = 90 cm². Verification / Alternative check: We can view ABC as composed of the small triangle ADE plus trapezium BDEC. The ratio of corresponding heights from A to BC and from A to DE is also 7 : 2, consistent with the side ratio. Hence, the large area should be larger than the small one by a factor of 49/4 = 12.25. Multiplying 8 by 12.25 gives 98, which is consistent. Subtracting the known 8 cm² area of ΔADE from the total again gives 90 cm², confirming the calculation. Why Other Options Are Wrong: If one mistakenly uses 2/5 as the scale factor, incorrect areas such as 80 cm² or 86 cm² can appear. Using a linear ratio directly without squaring can also produce options like 94 cm² or 98 cm² for the quadrilateral instead of the whole triangle. Only 90 cm² correctly represents the remaining area after removing ΔADE from ΔABC. Common Pitfalls: A frequent error is to forget that area scales with the square of the similarity ratio, not linearly. Another mistake is to mix up the structure and subtract in the wrong order, or to interpret 2 : 5 as the ratio between DE and BC directly. Carefully identifying corresponding sides and remembering that area ratio equals the square of side ratio are essential for accuracy. Final Answer: The area of quadrilateral BDEC is 90 cm².

Question 13

Using exact trigonometric values, what is the simplified value of the expression Sin 30° − √2 Cos 30°?

Accepted Answer

Introduction / Context: This trigonometric simplification question combines exact values for sine and cosine at 30 degrees with a radical multiplier. It tests both recall of special angle values and algebraic handling of square roots in fractions, a common theme in trigonometry simplification exercises. Given Data / Assumptions: We need to simplify Sin 30° − √2 Cos 30°. Angles are measured in degrees. Standard special angle values are to be used, not approximations. Concept / Approach: Recall the exact values Sin 30° = 1/2 and Cos 30° = √3 / 2. Substitute these into the expression and then simplify carefully. The radical √2 multiplies Cos 30°, so we obtain a product of √2 and √3, which becomes √6. Finally, we combine the terms over a common denominator to match the given options. Step-by-Step Solution: Recall Sin 30° = 1/2. Recall Cos 30° = √3 / 2. Compute √2 Cos 30° = √2 * (√3 / 2) = √6 / 2. Now substitute into the expression: Sin 30° − √2 Cos 30° = 1/2 − √6 / 2. Both terms have denominator 2, so combine them: (1 − √6) / 2. Thus the simplified exact value is (1 − √6)/2. Verification / Alternative check: We can verify by approximate numerical evaluation. Sin 30° is 0.5. Cos 30° is approximately 0.866, and √2 is about 1.414. Then √2 Cos 30° ≈ 1.414 * 0.866 ≈ 1.225. So the expression is 0.5 − 1.225 ≈ −0.725. Now evaluate (1 − √6)/2. Since √6 is about 2.449, 1 − 2.449 ≈ −1.449, and dividing by 2 gives about −0.7245, which matches the earlier decimal value within rounding error. Why Other Options Are Wrong: Expressions with denominator √2 such as (1 − √6)/√2 or (3 − √6)/√2 have different magnitudes and do not match the simplified exact result. Options (3 − √6)/2 and (1 − √3)/√2 come from misusing the values of sine or cosine or from incorrectly combining terms. Only (1 − √6)/2 is consistent with the correct substitution and simplification. Common Pitfalls: One frequent error is using incorrect special angle values, such as confusing Sin 30° with Cos 30°. Another mistake is mishandling the product √2 * √3, sometimes incorrectly writing it as √5. Also, students may forget to take the common denominator properly and end up with mismatched numerators. Careful substitution and stepwise simplification prevent such issues. Final Answer: The simplified value of the expression is (1 − √6)/2.

Question 14

In right triangle ΔXYZ, which is right angled at Y, if ∠Z = 60° and the hypotenuse ZX = 9√3 cm, then what is the length (in centimetres) of side YZ?

Accepted Answer

Introduction / Context: This problem uses the properties of a 30 60 90 right triangle. In such a triangle, the lengths of the sides have fixed ratios, and recognising this pattern allows you to solve quickly without relying solely on the Pythagorean theorem or trigonometric functions every time. Given Data / Assumptions: Triangle ΔXYZ is right angled at Y. Angle Z is 60 degrees, so angle X is 30 degrees. Hypotenuse ZX has length 9√3 cm. We need the length of side YZ, which is opposite angle X (30 degrees). Concept / Approach: In a 30 60 90 right triangle, the side opposite 30 degrees is the shortest side, denoted s. The side opposite 60 degrees is s√3, and the hypotenuse is 2s. With the hypotenuse given, we first determine s, then identify which side in the given triangle corresponds to s to find YZ. This ratio based method is efficient and avoids long calculations. Step-by-Step Solution: Because the triangle is right angled at Y and ∠Z = 60°, the remaining angle ∠X is 30°. Thus ΔXYZ is a 30 60 90 triangle. In such a triangle, if the side opposite 30° is s, then the hypotenuse is 2s. The hypotenuse ZX is given as 9√3 cm. Therefore 2s = 9√3, so s = (9√3)/2. The side opposite 30° is YZ, so YZ = s = 9√3/2 cm. Verification / Alternative check: We can check using the Pythagorean theorem. The side opposite 60° (XY) is s√3 = (9√3/2) * √3 = (9 * 3)/2 = 27/2. Then YZ^2 + XY^2 must equal ZX^2. Compute YZ^2 = (9√3/2)^2 = 81 * 3 / 4 = 243/4. Compute XY^2 = (27/2)^2 = 729/4. Adding gives 243/4 + 729/4 = 972/4 = 243. Now ZX^2 = (9√3)^2 = 81 * 3 = 243. The equality holds, confirming the correctness of YZ = 9√3/2. Why Other Options Are Wrong: Values such as 3√3 or 3√3/2 are smaller than the correct side and would give a hypotenuse different from 9√3 when checked. The value √3 is far too small, and 9 cm would correspond to a different triangle ratio. Only 9√3/2 satisfies the 30 60 90 triangle ratios and passes the Pythagorean verification. Common Pitfalls: Learners sometimes mix up which side is opposite which angle, especially when drawing the triangle quickly. Others reverse the ratio, treating the hypotenuse as s and the short leg as 2s, which leads to incorrect lengths. Drawing a clear diagram and labelling sides by angle can significantly reduce confusion. Final Answer: The length of side YZ is 9√3/2 cm.

Question 15

If Sec θ = 13/12 for an acute angle θ in a right triangle, what is the exact value of Sin θ?

Accepted Answer

Introduction / Context: This question tests the ability to convert between different trigonometric ratios by using the right triangle definition and the Pythagorean identity. Knowing Sec θ allows us to find Cos θ, then Sin θ, which is crucial in many problems involving right triangles and trigonometric simplification. Given Data / Assumptions: Sec θ = 13/12. Angle θ is acute, so all basic trig ratios are positive. Secant is the reciprocal of cosine. We need to find Sin θ. Concept / Approach: Since Sec θ = 1 / Cos θ, we can find Cos θ as 12/13. In a right triangle, Cos θ = adjacent / hypotenuse. With these values, we can use the Pythagorean theorem to find the opposite side. Then Sin θ = opposite / hypotenuse. This geometric approach is direct and avoids unnecessary symbolic manipulation. Step-by-Step Solution: Given Sec θ = 13/12, so Cos θ = 1 / Sec θ = 12/13. Interpret Cos θ = adjacent / hypotenuse = 12/13. Let adjacent side be 12 units and hypotenuse be 13 units. Use the Pythagorean theorem: hypotenuse^2 = adjacent^2 + opposite^2. Compute 13^2 = 12^2 + opposite^2, so 169 = 144 + opposite^2. Thus opposite^2 = 169 − 144 = 25. So opposite side = 5 units (positive because θ is acute). Now Sin θ = opposite / hypotenuse = 5 / 13. Verification / Alternative check: Check the fundamental identity Sin^2 θ + Cos^2 θ = 1. With Sin θ = 5/13 and Cos θ = 12/13, we have (5/13)^2 + (12/13)^2 = 25/169 + 144/169 = 169/169 = 1. This confirms that the pair of ratios is consistent with a valid right triangle and that our computation is correct. Why Other Options Are Wrong: 12/13 is Cos θ, not Sin θ. The ratio 5/12 would correspond to opposite over adjacent, more like the tangent value in this triangle. The values 12/5 and 13/5 are greater than 1, which is impossible for the sine of an acute angle. Thus only 5/13 matches the correct definition for Sin θ in this context. Common Pitfalls: Some learners mistakenly treat Sec θ itself as opposite over hypotenuse or confuse it with Cos θ. Others directly guess numerical values without constructing the triangle or checking the identity. Drawing a quick sketch and labelling sides based on the ratio significantly improves accuracy in such problems. Final Answer: The exact value of Sin θ is 5/13.

Question 16

Evaluate the decimal sum 0.77777 + 0.7777 + 0.777 + 0.77 + 0.7 + 0.07 and give the result correct to five decimal places.

Accepted Answer

Introduction / Context: This arithmetic question involves adding several decimal numbers with recurring patterns in their digits. It assesses place value understanding, careful alignment of decimal points, and accurate addition. Such problems are common in aptitude exams to quickly test accuracy with decimal arithmetic. Given Data / Assumptions: We must compute 0.77777 + 0.7777 + 0.777 + 0.77 + 0.7 + 0.07. The final answer should be reported to five decimal places. All numbers are standard decimal fractions. Concept / Approach: The key idea is to align the decimals correctly and add them column by column or use careful stepwise addition. Because each number has a different count of digits after the decimal point, we can add trailing zeros so that each term has five decimal places, making the addition straightforward. This avoids errors caused by misaligned decimals. Step-by-Step Solution: Rewrite each term with five decimal places: 0.77777, 0.77770, 0.77700, 0.77000, 0.70000, 0.07000. Add the last decimal place: 7 + 0 + 0 + 0 + 0 + 0 = 7. Next decimal place: 7 + 7 + 0 + 0 + 0 + 0 = 14, write 4 and carry 1. Third decimal place: 7 + 7 + 7 + 0 + 0 + 0 = 21, plus carry 1 gives 22, write 2 and carry 2. Fourth decimal place: 7 + 7 + 7 + 7 + 0 + 7 = 35, plus carry 2 gives 37, write 7 and carry 3. Fifth decimal place: 7 + 7 + 7 + 7 + 7 + 0 = 35, plus carry 3 gives 38, write 8 and carry 3. Now add the whole number parts: 0 + 0 + 0 + 0 + 0 + 0 plus carry 3 gives 3. Thus the sum is 3.87247 when all carries and places are accounted for. Verification / Alternative check: We can approximate the total to see if the answer is reasonable. Two terms around 0.78 contribute about 1.56. The next term 0.777 adds about 0.78, taking the total near 2.34. Adding 0.77 gives about 3.11, then 0.7 takes it near 3.81, and finally 0.07 brings it around 3.88. The precise value 3.87247 is close to this approximation, confirming that the detailed addition is sensible. Why Other Options Are Wrong: The values 3.86274, 3.80247, 3.85274, and 3.77247 correspond to typical digit or carry mistakes during addition, such as forgetting a carry or misaligning decimal places. None of them match the carefully computed sum of the six decimals. Common Pitfalls: Misalignment of decimal points is the most common issue with such sums. Another mistake is dropping trailing digits when rewriting numbers to a common precision, or failing to carry correctly from one decimal column to the next. Writing all numbers in a neat column with equal decimal places before adding is the best defence against these errors. Final Answer: The sum of the decimals correct to five decimal places is 3.87247.

Question 17

A crate of eggs contains one rotten egg out of every 25 eggs, and of all the rotten eggs, 5 out of 8 are unusable. If there are exactly 10 unusable eggs in the crate, how many eggs are there in total in the crate?

Accepted Answer

Introduction / Context: This quantitative aptitude question combines ratios and fractions in a real life context. It models a crate of eggs with a certain proportion rotten and, among those, a proportion unusable. Understanding how to move from nested fractions to total quantities is very useful in percentage and ratio based word problems. Given Data / Assumptions: In the crate, 1 out of every 25 eggs is rotten. Of all rotten eggs, 5 out of 8 are unusable. There are 10 unusable eggs in total. We must find the total number of eggs in the crate. Concept / Approach: Let the total number of eggs be N. Then the number of rotten eggs is N/25. Of those rotten eggs, 5/8 are unusable, so the number of unusable eggs is (5/8) * (N/25). This quantity is given to be 10. We set up an equation and solve for N. The method is essentially proportion and fraction manipulation. Step-by-Step Solution: Let the total number of eggs be N. Rotten eggs are 1 out of every 25, so the number of rotten eggs is N / 25. Of these rotten eggs, 5 out of 8 are unusable, so unusable eggs = (5/8) * (N / 25). We are told there are exactly 10 unusable eggs. Therefore, set up the equation (5/8) * (N / 25) = 10. Simplify the left side: (5N) / (8 * 25) = (5N) / 200. So (5N) / 200 = 10. Multiply both sides by 200 to clear the denominator: 5N = 10 * 200 = 2000. Divide by 5: N = 2000 / 5 = 400. Verification / Alternative check: Check the result by reconstructing the situation. With 400 eggs, rotten eggs are 400 / 25 = 16. Among these, unusable eggs are (5/8) of 16, which is 16 * 5 / 8 = 10. This matches the given information exactly, so N = 400 is confirmed as correct. Why Other Options Are Wrong: If N = 380, rotten eggs would be 380 / 25 = 15.2, not an integer, which is inconsistent with the scenario. Similarly, N = 420 or 440 would not produce 10 unusable eggs when applying the same fractions. The value 360 gives 360 / 25 = 14.4 rotten eggs, again not a whole number. Only N = 400 results in exactly 16 rotten eggs and precisely 10 unusable eggs. Common Pitfalls: Learners sometimes misinterpret the phrase one rotten egg out of every 25 eggs as 1/25 of the total being unusable directly, ignoring the second filter of 5 out of 8. Others invert the fractions or forget to apply both ratios sequentially. Writing the process symbolically as a product of fractions applied to N helps avoid such confusions. Final Answer: The crate contains a total of 400 eggs.

Question 18

In coordinate geometry, what is the area (in square units) of the triangle formed by the x-axis, the y-axis, and the line x + 2y = 6?

Accepted Answer

Introduction / Context: This problem uses intercepts of a straight line to find the area of a triangle formed with the coordinate axes. Such questions are standard in coordinate geometry and test understanding of intercept form and basic area formulas for right triangles. Given Data / Assumptions: The line is x + 2y = 6. The triangle is formed by this line and the x axis and y axis. We need the area in square units. The coordinate axes intersect at the origin (0, 0). Concept / Approach: Where the line meets the x axis, y = 0. Where it meets the y axis, x = 0. These intercepts provide the base and height of a right triangle with the axes. The area of a right triangle is half the product of its perpendicular sides. So we find the intercepts and apply the formula (1/2) * base * height. Step-by-Step Solution: To find the x intercept, set y = 0 in x + 2y = 6. This gives x + 2 * 0 = 6, so x = 6. Thus the x intercept is (6, 0). To find the y intercept, set x = 0 in x + 2y = 6. This gives 0 + 2y = 6, so y = 3. Thus the y intercept is (0, 3). The triangle is formed by points (0, 0), (6, 0), and (0, 3). The base along the x axis has length 6 units. The height along the y axis has length 3 units. Area of the right triangle = (1/2) * base * height = (1/2) * 6 * 3 = 9 square units. Verification / Alternative check: We can use the formula for a triangle with intercepts a and b on the axes, where the line is x/a + y/b = 1. For x + 2y = 6, dividing through by 6 gives x/6 + y/3 = 1, so a = 6 and b = 3. The area of the triangle formed with the axes is (1/2) * a * b = (1/2) * 6 * 3 = 9. This matches the earlier calculation, confirming the answer. Why Other Options Are Wrong: Values such as 3 or 6 correspond to using only one of the intercepts or forgetting the factor 1/2. An area of 12 would be obtained if someone multiplies 6 and 3 directly without halving. The value 18 is twice the correct area, as if one accidentally doubled instead of halved. Only 9 square units is consistent with the correct formula for the area of a right triangle with legs 6 and 3. Common Pitfalls: A common mistake is to misidentify the intercepts if the equation is not first rewritten clearly, or to treat the intercepts as coordinates in a more complicated area formula. For right triangles aligned with the axes, it is much simpler to use (1/2) * base * height directly. Always check that the line indeed intersects both axes in the first quadrant for a positive area interpretation. Final Answer: The area of the triangle is 9 square units.

Question 19

Consider the surd inequalities: I. 4√3 > 3√4 and II. 8√2 > 2√8. Which of the following statements correctly describes which inequalities are true?

Accepted Answer

Introduction / Context: This question tests comparison of surds by converting them either to decimal approximations or to a common simplified form. Surd comparison is an important skill for quickly checking which of two root based expressions is larger in competitive examinations and numerical reasoning tests. Given Data / Assumptions: Inequality I: 4√3 > 3√4. Inequality II: 8√2 > 2√8. We must decide whether each inequality is true or false. Concept / Approach: One method is to approximate the square roots with decimal values and compare. Another is to simplify expressions by using known roots, such as √4 = 2 and √8 = 2√2, and then compare exactly. Squaring both sides can also help if we are careful to preserve direction when both sides are positive. Here, simplifying using known exact values is efficient and precise. Step-by-Step Solution: For inequality I: 4√3 > 3√4. Compute √4 exactly: √4 = 2, so 3√4 = 3 * 2 = 6. So inequality I becomes 4√3 > 6. Approximate √3 ≈ 1.732, so 4√3 ≈ 4 * 1.732 ≈ 6.928. Since 6.928 > 6, inequality I is true. For inequality II: 8√2 > 2√8. Simplify √8 as √(4 * 2) = 2√2, so 2√8 = 2 * 2√2 = 4√2. Thus inequality II becomes 8√2 > 4√2. Both sides have the same surd √2 as a factor, and 8 > 4, so the inequality holds. Verification / Alternative check: For a quick decimal check, approximate √2 ≈ 1.414. Then 8√2 ≈ 8 * 1.414 ≈ 11.312 and 4√2 ≈ 4 * 1.414 ≈ 5.656. Clearly 11.312 > 5.656, confirming inequality II. For I, 4√3 ≈ 6.928 and 3√4 = 6 exactly, which confirms I again. Both methods match and show that each inequality is true. Why Other Options Are Wrong: Options claiming that only one inequality holds or that neither holds contradict the numeric comparisons above. Since I is true and II is also true, any option that asserts otherwise is incorrect. The phrasing I is false and II is true is directly contradicted by evaluating 4√3 and 3√4. Common Pitfalls: A typical mistake is to treat √8 as 8/2 or to mix up properties of roots. Another issue is forgetting that multiplying or dividing by a common positive surd factor such as √2 is allowed when comparing. While decimal approximations are convenient, they should be used with enough precision to avoid rounding errors that could reverse an inequality if values are close. Final Answer: Both inequalities are true, so the correct statement is Both I and II.

Question 20

Solve the equation x/2 − [4(15/2 − x/3)]/3 = −x/18 and find the value of x.

Accepted Answer

Introduction / Context: This linear equation contains nested fractions and requires careful handling of arithmetic operations. Being able to simplify such expressions accurately is important in algebra and in many quantitative aptitude questions where fractional coefficients appear regularly. Given Data / Assumptions: The equation is x/2 − [4(15/2 − x/3)]/3 = −x/18. x is a real number. We need to isolate x and find its value. Concept / Approach: We handle the bracket 15/2 − x/3 first, multiply by 4, then divide by 3, and finally simplify the equation. To avoid errors with denominators, it is efficient to multiply through by the least common multiple of all denominators so that the resulting equation has integer coefficients. Afterwards, standard linear solving steps are applied. Step-by-Step Solution: Start with x/2 − [4(15/2 − x/3)]/3 = −x/18. First simplify the inner expression: 15/2 − x/3. Multiply by 4: 4(15/2 − x/3) = 4 * 15/2 − 4 * x/3 = 30 − 4x/3. Now divide by 3: [4(15/2 − x/3)]/3 = (30 − 4x/3)/3. Rewrite as 30/3 − (4x/3)/3 = 10 − 4x/9. So the equation becomes x/2 − (10 − 4x/9) = −x/18. Distribute the minus sign: x/2 − 10 + 4x/9 = −x/18. Take the least common multiple of denominators 2, 9, and 18, which is 18, and multiply through. 18 * (x/2) = 9x, 18 * (−10) = −180, 18 * (4x/9) = 8x, and 18 * (−x/18) = −x. Thus 9x − 180 + 8x = −x. Combine like terms: 17x − 180 = −x. Add x to both sides: 18x − 180 = 0. Add 180: 18x = 180, so x = 180 / 18 = 10. Verification / Alternative check: Substitute x = 10 back into the original equation. The left side becomes 10/2 − [4(15/2 − 10/3)]/3. Compute 10/2 = 5. Inside the bracket, 15/2 − 10/3 = 45/6 − 20/6 = 25/6. Multiplying by 4 gives 100/6 = 50/3. Dividing by 3 gives 50/9. So the left side is 5 − 50/9 = (45/9 − 50/9) = −5/9. The right side is −x/18 = −10/18 = −5/9. Both sides match, confirming that x = 10 is correct. Why Other Options Are Wrong: Values such as −10 or 9/8 result from sign errors or mismanagement of denominators during simplification. The value −9/8 comes from incorrect inversion of a fraction, and 15 arises from miscalculating when clearing denominators. None of these, when substituted back, satisfy the original equation. Common Pitfalls: Common mistakes include dropping a minus sign in front of the bracket, incorrectly distributing 4 over the fraction, or not multiplying every term by the least common multiple when clearing denominators. Taking the time to simplify step by step and checking the solution in the original equation helps avoid such errors. Final Answer: The solution of the equation is x = 10.

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