Q: In an arithmetic progression, the 3rd term is equal to −13 and the 8th term is equal to 2. Using the standard formulas for terms of an arithmetic progression, what is the value of the 14th term of this sequence?

Introduction / Context: This problem tests your understanding of arithmetic progressions, specifically how to use known terms to find the common difference and then any other term. Such questions are standard in quantitative aptitude sections of competitive exams. Given Data / Assumptions: The sequence is an arithmetic progression (A.P.). T3 = −13 (the 3rd term). T8 = 2 (the 8th term). We need to find the 14th term T14. Concept / Approach: For an arithmetic progression with first term a and common difference d, the nth term is given by: Tn = a + (n − 1) * d. We can form two equations using T3 and T8, solve for a and d, and then substitute into the formula for T14. Step-by-Step Solution: Let a be the first term and d be the common difference. T3 = a + 2d = −13. T8 = a + 7d = 2. Subtract the first equation from the second: (a + 7d) − (a + 2d) = 2 − (−13). This gives 5d = 15, so d = 3. Substitute d = 3 into a + 2d = −13: a + 2 * 3 = −13. Thus a + 6 = −13, so a = −19. Now find T14: T14 = a + 13d = −19 + 13 * 3. Compute 13 * 3 = 39, so T14 = −19 + 39 = 20. Verification / Alternative check: You can quickly check by listing some terms. T3 = −13, T4 = −10, and adding 3 repeatedly should eventually reach T8 = 2 and T14 = 20. This matches the algebraic solution, confirming the result. Why Other Options Are Wrong: Option a (23), option b (17), option d (26), and option e (11) do not match the arithmetic progression generated from a = −19 and d = 3. When you compute T14, only 20 is consistent with the derived formula. Common Pitfalls: Learners often confuse the position index n in Tn with the term value. Another error is to mix up formulas for A.P. with those for geometric progressions. Always remember that in an A.P. the difference between consecutive terms is constant and use Tn = a + (n − 1)d. Final Answer: Therefore, the 14th term of the arithmetic progression is 20.

Q: In coordinate geometry, consider the straight line given by the equation 3x + 2y = 12. At which coordinate point does this line intersect and cut the Y axis?

Introduction / Context: Finding intercepts of a line is a basic skill in coordinate geometry. The Y intercept is the point where the line crosses the Y axis, and it is found by setting x equal to zero. This question checks whether you are comfortable with this standard technique. Given Data / Assumptions: Equation of the line: 3x + 2y = 12. Y axis is defined by x = 0. We need the point where the line meets the Y axis. Concept / Approach: The Y intercept is the point on the line where x equals zero. Substituting x = 0 into the line equation and solving for y gives the Y intercept. This is a very fast method and is often used for sketching graphs of linear equations. Step-by-Step Solution: Start with 3x + 2y = 12. At the Y axis, x = 0, so substitute x = 0 into the equation. Then 3 * 0 + 2y = 12 gives 2y = 12. Divide both sides by 2 to get y = 6. So the point of intersection with the Y axis is (0, 6). Verification / Alternative check: You can also rewrite the equation in slope intercept form: 2y = 12 − 3x, so y = 6 − (3 / 2) x. When x = 0, y = 6, confirming the Y intercept is (0, 6). Why Other Options Are Wrong: Option b (0, −6) has the wrong sign of y. Option c (6, 0) and option d (−6, 0) are X intercept candidates, not Y intercepts. Option e (2, 3) lies on the line but does not correspond to x = 0, so it is not the Y intercept. Common Pitfalls: Students sometimes mix up X and Y intercepts, or they try to solve the equation for both variables unnecessarily. Always remember that for the Y intercept, x is zero, and for the X intercept, y is zero. This small reminder saves time and avoids confusion. Final Answer: The line 3x + 2y = 12 cuts the Y axis at the point (0, 6).

Question 1

Solve the linear equation 7x - [3(2x - 3)]/2 = 1/2 by clearing the fraction and simplifying step by step, then choose the correct value of x from the options.

Accepted Answer

Introduction / Context: This simplification problem presents a linear equation involving a bracket and a fraction. Questions like this are used in aptitude exams to test basic algebraic manipulation skills, such as expanding brackets, working with fractions, and collecting like terms properly. Once the equation is simplified, you can solve for x and match it with one of the given answer choices. Given Data / Assumptions: The equation is 7x - [3(2x - 3)]/2 = 1/2. x is a real number. Fractions should be handled exactly without rounding. Multiplication and distribution inside brackets follow normal algebraic rules. Concept / Approach: The main idea is to remove the bracket by distributing the factor 3, then handle the division by 2. After that, multiply the whole equation by 2 to clear the denominator and obtain a simpler linear equation in x. Using standard algebraic steps such as combining like terms and isolating x will lead you to the correct solution efficiently. Step-by-Step Solution: Start with 7x - [3(2x - 3)]/2 = 1/2. First expand the bracket: 3(2x - 3) = 6x - 9. Substitute: 7x - (6x - 9)/2 = 1/2. Multiply the entire equation by 2 to clear the denominator: 2 * 7x - (6x - 9) = 2 * (1/2). This gives 14x - (6x - 9) = 1. Distribute the minus sign: 14x - 6x + 9 = 1. Combine like terms: 8x + 9 = 1. Solve for x: 8x = 1 - 9 = -8, so x = -8 / 8 = -1. Verification / Alternative check: Substitute x = -1 back into the original equation to verify. Compute 2x - 3 = 2(-1) - 3 = -2 - 3 = -5, so 3(2x - 3) = 3 * (-5) = -15. Then [3(2x - 3)]/2 = -15/2. The left hand side becomes 7(-1) - (-15/2) = -7 + 15/2 = (-14/2 + 15/2) = 1/2, which matches the right hand side. This confirms that x = -1 is the correct solution. Why Other Options Are Wrong: Option b (1), option c (3), and option d (-3) all give left hand side values that are not equal to 1/2 when substituted into the equation. Option e (0) corresponds to ignoring some terms or incorrectly simplifying the expression. Only x = -1 satisfies the equation exactly after full substitution. Common Pitfalls: Common errors include failing to distribute the minus sign correctly when removing the bracket, such as writing 14x - 6x - 9 instead of 14x - 6x + 9. Another pitfall is to forget to multiply every term, including the right hand side, when clearing denominators. Keeping track of all terms while multiplying and simplifying step by step is essential to avoid these mistakes. Final Answer: The correct solution of the equation 7x - [3(2x - 3)]/2 = 1/2 is x = -1.

Question 2

If a + b = 4 and ab = 3, use the sum of cubes identity to compute the exact value of a^3 + b^3 without solving separately for a and b.

Accepted Answer

Introduction / Context: This algebra problem focuses on the identity for the sum of cubes. Rather than solving explicitly for a and b, you are expected to recognise that a^3 + b^3 can be expressed in terms of a + b and ab. This technique allows rapid simplification and is very useful in aptitude exams where direct computation of roots would be time consuming. Given Data / Assumptions: a + b = 4. ab = 3. a and b are real numbers. The goal is to find a^3 + b^3. We want to avoid solving for a and b individually. Concept / Approach: The identity for the sum of cubes is a^3 + b^3 = (a + b)^3 - 3ab(a + b). This identity expresses a^3 + b^3 in terms of the sum a + b and the product ab, both of which are given. By substituting these values and simplifying, we can directly obtain the value of a^3 + b^3 in a few steps. This method prevents unnecessary quadratic equation solving and reduces algebraic errors. Step-by-Step Solution: Use the identity: a^3 + b^3 = (a + b)^3 - 3ab(a + b). Substitute a + b = 4 and ab = 3 into the identity. Compute (a + b)^3 = 4^3 = 64. Compute 3ab(a + b) = 3 * 3 * 4 = 36. Therefore a^3 + b^3 = 64 - 36 = 28. Verification / Alternative check: As a check, you can solve for a and b explicitly. From a + b = 4 and ab = 3, the pair (a, b) satisfies t^2 - 4t + 3 = 0, whose roots are t = 1 and t = 3. So a and b are 1 and 3 in some order. Then a^3 + b^3 = 1^3 + 3^3 = 1 + 27 = 28, which matches the identity based result. This confirms that the formula has been applied correctly. Why Other Options Are Wrong: Option b (21), option c (17), option d (31), and option e (26) result from common errors such as using 3ab instead of 3ab(a + b), miscomputing 4^3 as 16, or getting 3*3*4 as 34. None of these values are consistent with both the identity and the explicit evaluation using actual values of a and b. Common Pitfalls: Learners sometimes confuse the identities for a^3 - b^3 and a^3 + b^3 or mistakenly use (a + b)^3 = a^3 + b^3 + 3ab(a + b) without rearranging. Another pitfall is to forget to multiply by a + b in the term 3ab(a + b), leading to partial expressions. Carefully writing down the entire identity and substituting the given values step by step helps avoid these mistakes. Final Answer: The exact value of a^3 + b^3, given a + b = 4 and ab = 3, is 28.

Question 3

Find the point where the line 2x + 5y = -6 cuts the x-axis by setting y = 0, and then choose the correct x-intercept coordinate from the options.

Accepted Answer

Introduction / Context: This coordinate geometry problem asks you to find the x-intercept of a line given in standard form. The x-intercept is the point where the line crosses the x-axis. This concept appears frequently in graph based questions and helps you draw quick sketches of lines or interpret their behaviour by looking at how they intersect the axes. Given Data / Assumptions: The equation of the line is 2x + 5y = -6. The x-axis is defined by y = 0. The x-intercept has coordinates (x, 0). We need to solve for x when y is set to 0. Concept / Approach: To find the x-intercept of a line, we set y = 0 in the equation, because any point on the x-axis has a y coordinate of zero. Substituting y = 0 into the given equation reduces it to a simple linear equation in x. Solving this gives the x coordinate of the intercept. The resulting point (x, 0) is the required x-intercept. Step-by-Step Solution: Start from the line equation: 2x + 5y = -6. For the x-axis, set y = 0. Substitute y = 0: 2x + 5(0) = -6, which simplifies to 2x = -6. Solve for x: x = -6 / 2 = -3. Thus, the line intersects the x-axis at the point (-3, 0). Verification / Alternative check: You can also rewrite the equation in intercept form if desired. Solve for y to get 5y = -6 - 2x, so y = (-6 - 2x)/5. When y = 0, the numerator must be zero, which gives -6 - 2x = 0 and leads again to x = -3. This confirms that the x-intercept is (-3, 0) and not any of the other candidate points. Why Other Options Are Wrong: Option b (3, 0) has the correct y coordinate but the opposite sign in x. Substituting x = 3, y = 0 into the line equation gives 2*3 + 5*0 = 6, which is not equal to -6. Options c and d, (0, 3) and (0, -3), are potential y-intercepts, not x-intercepts, because their x coordinate is zero. Option e (0, 6) also lies on the y-axis rather than the x-axis and does not satisfy the equation when substituted. Common Pitfalls: A common mistake is to confuse y-intercepts and x-intercepts and therefore set x = 0 instead of y = 0 when asked for the x-intercept. Another pitfall is mishandling the algebraic signs when solving for x, leading to 3 instead of -3. Carefully identifying which coordinate must be zero for each type of intercept helps avoid such errors. Final Answer: The line 2x + 5y = -6 cuts the x-axis at the point (-3, 0).

Question 4

Evaluate the exact value of tan 45° + (4/√3) × sec 60° by using standard trigonometric values for special angles and simplifying the final surd expression.

Accepted Answer

Introduction / Context: This trigonometric simplification question combines standard values of tan and sec for special angles and requires you to manage surds carefully. The expression involves tan 45 degrees and sec 60 degrees multiplied by a fraction in terms of √3. By recalling the exact values for these special angles and simplifying step by step, you can express the final result as a single surd fraction that matches one of the given options. Given Data / Assumptions: tan 45 degrees is a standard value. sec 60 degrees is the reciprocal of cos 60 degrees. cos 60 degrees = 1/2, sin 60 degrees = √3/2. The expression to evaluate is tan 45 degrees + (4/√3) × sec 60 degrees. All calculations are based on exact surd values, not decimal approximations. Concept / Approach: Begin by substituting the known exact values for tan 45 degrees and sec 60 degrees. Then multiply the fraction (4/√3) by sec 60 degrees. After this multiplication, you will have a sum of 1 and a surd fraction. Finally, express everything over a common denominator √3 and combine the terms to obtain a single surd fraction that can be compared with the options. Step-by-Step Solution: Recall tan 45 degrees = 1. Recall cos 60 degrees = 1/2, so sec 60 degrees = 1 / cos 60 degrees = 2. Substitute into the expression: tan 45 degrees + (4/√3) × sec 60 degrees = 1 + (4/√3) × 2. Compute the product: (4/√3) × 2 = 8/√3. So the expression becomes 1 + 8/√3. Write 1 as √3/√3 to get a common denominator: 1 = √3/√3. Add the fractions: √3/√3 + 8/√3 = (√3 + 8)/√3. Verification / Alternative check: Perform a quick numerical check using approximations: tan 45 degrees ≈ 1 and sec 60 degrees ≈ 2. Then (4/√3) is approximately 4/1.732 ≈ 2.309, and (4/√3) × 2 ≈ 4.618. Adding tan 45 degrees gives roughly 5.618. Evaluating (√3 + 8)/√3 numerically gives (1.732 + 8)/1.732 ≈ 9.732/1.732 ≈ 5.618, which matches the approximate value of the original expression. Why Other Options Are Wrong: Option b ( (√3 + 8)/3 ) places 3 in the denominator instead of √3, changing the value significantly. Options c and d use (√3 - 8) instead of (√3 + 8), which would result in a negative or much smaller value. Option e ( (8 - √3)/2 ) does not correspond to the correct combination of tan and sec values and gives a different numerical result. Common Pitfalls: A common mistake is to misremember sec 60 degrees as 1/2 instead of 2, by confusing it with cos 60 degrees. Another pitfall is to attempt premature rationalisation of denominators, which can complicate the algebra unnecessarily before combining terms. Keeping the denominators as √3 until you combine like terms makes the arithmetic straightforward and reduces mistakes. Final Answer: The exact simplified value of tan 45 degrees + (4/√3) × sec 60 degrees is (√3 + 8)/√3.

Question 5

For an acute angle θ, if sin θ = 20/29, model θ using a right triangle, apply the Pythagoras theorem to find the remaining side, and then determine the exact value of cos θ.

Accepted Answer

Introduction / Context: This trigonometry question asks you to find cos θ when sin θ is given for an acute angle. Instead of using inverse trigonometric functions, you can treat the sine value as a ratio of sides in a right triangle and then use the Pythagoras theorem to find the adjacent side. This approach is straightforward and commonly used in competitive exams for speed and accuracy. Given Data / Assumptions: sin θ = 20/29. θ is an acute angle, so 0 degrees < θ < 90 degrees. sin θ = opposite side / hypotenuse. cos θ = adjacent side / hypotenuse. Pythagoras theorem: hypotenuse^2 = opposite^2 + adjacent^2. Concept / Approach: Interpret sin θ = 20/29 by constructing a right triangle where the side opposite θ is 20 units and the hypotenuse is 29 units. Then use the Pythagoras theorem to find the adjacent side. Once all sides are known, cos θ is simply the ratio of adjacent side to hypotenuse. Because θ is acute, all lengths are positive and we select the positive root when applying square roots. Step-by-Step Solution: Let opposite side to θ be 20 and hypotenuse be 29. Apply Pythagoras theorem: hypotenuse^2 = opposite^2 + adjacent^2. Substitute: 29^2 = 20^2 + adjacent^2. Compute: 29^2 = 841 and 20^2 = 400. Thus adjacent^2 = 841 - 400 = 441. Take the square root: adjacent = 21, since 21^2 = 441. Now cos θ = adjacent / hypotenuse = 21/29. Verification / Alternative check: You can verify that sin^2 θ + cos^2 θ = 1, as required by the basic identity. Compute sin^2 θ = (20/29)^2 = 400/841 and cos^2 θ = (21/29)^2 = 441/841. Their sum is (400 + 441)/841 = 841/841 = 1, confirming that cos θ = 21/29 is consistent with the given sine value and the Pythagoras identity. Why Other Options Are Wrong: Option b (29/21) is simply the reciprocal of the correct answer and would correspond to sec θ, not cos θ. Option c (21/20) is a ratio of adjacent to opposite, not adjacent to hypotenuse. Option d (20/29) is sin θ itself, not cos θ. Option e (29/20) is the reciprocal of sin θ, that is cosec θ, and therefore not the required cosine value. Common Pitfalls: A frequent mistake is to assign the hypotenuse and opposite sides incorrectly, for example treating 20 as hypotenuse and 29 as opposite, which violates the basic property that the hypotenuse is the longest side. Another pitfall is arithmetic error when squaring 29 or subtracting 400 to find the adjacent squared value. Carefully computing 29^2 = 841 and subtracting 400 to get 441 ensures you obtain the correct adjacent length and thus the correct cosine. Final Answer: The exact value of cos θ when sin θ = 20/29 for an acute angle is 21/29.

Question 6

Solve the linear equation 5x - (1/2)(2x - 7) = 5.5 (that is, 11/2) by simplifying both sides and isolating x, then choose the correct value of x from the options.

Accepted Answer

Introduction / Context: This is a straightforward linear equation involving a fraction and a bracketed expression. Such problems are very common in aptitude tests and are designed to see if you can correctly expand and simplify expressions with fractions. Once the equation is simplified, you can solve for x in a few clear steps and then match the result with the given options. Given Data / Assumptions: The equation is 5x - (1/2)(2x - 7) = 5.5. 5.5 can be written as 11/2 to work with exact fractions. x is a real number. Standard algebraic rules apply when expanding brackets and combining like terms. Concept / Approach: First, rewrite 5.5 as 11/2 to keep everything in fractional form. Then expand the bracket (1/2)(2x - 7) to simplify the left hand side. After combining like terms, you will have a simple linear equation in x. Solving it involves isolating x on one side of the equation by basic operations such as addition and division. Step-by-Step Solution: Rewrite the equation with fractions: 5x - (1/2)(2x - 7) = 11/2. Expand the bracket: (1/2)(2x - 7) = x - 7/2. Substitute into the equation: 5x - (x - 7/2) = 11/2. Distribute the minus sign: 5x - x + 7/2 = 11/2. Combine x terms: 4x + 7/2 = 11/2. Isolate 4x by subtracting 7/2 from both sides: 4x = 11/2 - 7/2 = 4/2 = 2. Divide both sides by 4: x = 2 / 4 = 1/2. Verification / Alternative check: Substitute x = 1/2 back into the original equation. Compute 2x - 7 = 2*(1/2) - 7 = 1 - 7 = -6. Then (1/2)(2x - 7) = (1/2)*(-6) = -3. The left hand side becomes 5x - (1/2)(2x - 7) = 5*(1/2) - (-3) = 5/2 + 3 = 5/2 + 6/2 = 11/2, which equals 5.5. This confirms that x = 1/2 satisfies the equation exactly. Why Other Options Are Wrong: Option b (3/2), option c (-1/2), option d (-3/2), and option e (-5/2) do not satisfy the equation when substituted back, and they arise from typical algebraic mistakes such as forgetting to distribute the minus sign, miscomputing 11/2 - 7/2, or mishandling the division by 4. Only x = 1/2 makes both sides of the equation equal. Common Pitfalls: A common error is to expand (1/2)(2x - 7) incorrectly as (2x/2 - 7) or to forget the negative sign in front of the entire bracket. Another pitfall is to convert 5.5 incorrectly into a fraction or to mix decimals and fractions, leading to unnecessary complexity. Working consistently in fractions and checking each step for sign accuracy helps avoid these issues. Final Answer: The correct solution of the equation 5x - (1/2)(2x - 7) = 5.5 is x = 1/2.

Question 7

If a + b = 4 and ab = -5, use the sum of cubes identity to find the exact value of a^3 + b^3 without explicitly solving for a and b.

Accepted Answer

Introduction / Context: This algebra question is another application of the identity for the sum of cubes, but with the twist that the product ab is negative. You are given the sum of two numbers and their product and asked to compute a^3 + b^3. Using the identity that expresses a^3 + b^3 in terms of a + b and ab allows you to avoid solving a quadratic equation, which is both faster and less error prone in exam conditions. Given Data / Assumptions: a + b = 4. ab = -5. a and b are real numbers. The task is to find a^3 + b^3. We should use identities rather than explicit solutions for a and b. Concept / Approach: The key identity is a^3 + b^3 = (a + b)^3 - 3ab(a + b). This formula is directly in terms of a + b and ab, which are both known. By substituting these given values into the identity and simplifying carefully, we can determine the value of a^3 + b^3 in just a few steps. This method handles the negative product smoothly as part of the arithmetic. Step-by-Step Solution: Use the identity: a^3 + b^3 = (a + b)^3 - 3ab(a + b). Substitute a + b = 4 and ab = -5. Compute (a + b)^3 = 4^3 = 64. Compute 3ab(a + b) = 3 * (-5) * 4 = -60. Now a^3 + b^3 = 64 - (-60) = 64 + 60 = 124. Verification / Alternative check: To verify, you can solve for a and b explicitly by using the quadratic equation t^2 - (a + b)t + ab = 0, which becomes t^2 - 4t - 5 = 0. The roots are t = 5 and t = -1, so a and b are 5 and -1 in some order. Then a^3 + b^3 = 5^3 + (-1)^3 = 125 - 1 = 124, which matches the identity based calculation exactly. Why Other Options Are Wrong: Option b (126) might come from adding 64 and 62 by mistake or miscomputing 3 * (-5) * 4 as -58. Option c (34) and option d (36) correspond to much smaller values, possibly due to using (a + b)^2 or mixing the formula for a^2 + b^2 instead of a^3 + b^3. Option e (0) would require a^3 and b^3 to cancel each other completely, which does not occur for the numbers 5 and -1 or any other pair with sum 4 and product -5. Common Pitfalls: A frequent error is to forget the negative sign in ab when computing 3ab(a + b), leading to 3*5*4 = 60 instead of -60, and then subtracting incorrectly. Another mistake is to confuse the formula a^3 + b^3 = (a + b)^3 - 3ab(a + b) with a different identity such as (a + b)^3 = a^3 + b^3 + 3ab(a + b) and not rearranging it properly. Taking care with signs and writing the full identity before substitution helps avoid these issues. Final Answer: The exact value of a^3 + b^3, given a + b = 4 and ab = -5, is 124.

Question 8

In elementary algebra, the sum of a real number x and four times its reciprocal 1/x is equal to 5. If x is non zero, what is the value of the original number x that satisfies this equation?

Accepted Answer

Introduction / Context: This question tests basic algebraic manipulation involving reciprocals and quadratic equations. Many aptitude exams include problems where a number and its reciprocal appear together, so learning a systematic way to handle such equations is very useful. Given Data / Assumptions: x is a real number and x is not equal to zero. x + 4 * (1 / x) = 5. We must find the numerical value of x that satisfies this equation. Concept / Approach: The main idea is to clear the denominator by multiplying through by x, which converts the equation with a reciprocal into a quadratic equation. Once we obtain a quadratic, we can factorise it or use standard techniques to find the roots and then choose the correct one from the options. Step-by-Step Solution: Start with x + 4 * (1 / x) = 5. Multiply both sides by x (which is non zero): x * x + 4 = 5x. This gives x^2 + 4 = 5x. Rearrange to standard quadratic form: x^2 - 5x + 4 = 0. Factorise: x^2 - 5x + 4 = (x - 1)(x - 4) = 0. So the possible values are x = 1 or x = 4. Check both values in the original equation x + 4 / x = 5. For x = 1: 1 + 4 / 1 = 5, which is valid. For x = 4: 4 + 4 / 4 = 4 + 1 = 5, which is also valid. Both 1 and 4 satisfy the equation, but only 4 appears in the given options. Verification / Alternative check: We verified by direct substitution that x = 1 and x = 4 both satisfy the equation. As aptitude questions usually ask you to choose from listed options, we simply select the root that is available in the options, which is x = 4. Why Other Options Are Wrong: Option b (5): 5 + 4 / 5 = 5 + 0.8 = 5.8, which is not equal to 5. Option c (6): 6 + 4 / 6 = 6 + 2 / 3 = 6.666..., not equal to 5. Option d (7): 7 + 4 / 7 is greater than 7 and clearly not equal to 5. Option e (8): 8 + 4 / 8 = 8.5, not equal to 5. Common Pitfalls: A common mistake is to forget to multiply every term by x when clearing the denominator or to make sign errors when rearranging to quadratic form. Another error is to stop after finding both roots but not checking which root is actually present in the options. Always substitute the roots back into the original equation and then compare with the options provided. Final Answer: Hence, the number that satisfies the given condition and appears in the options is 4.

Question 9

In an arithmetic progression, the 3rd term is equal to −13 and the 8th term is equal to 2. Using the standard formulas for terms of an arithmetic progression, what is the value of the 14th term of this sequence?

Accepted Answer

Introduction / Context: This problem tests your understanding of arithmetic progressions, specifically how to use known terms to find the common difference and then any other term. Such questions are standard in quantitative aptitude sections of competitive exams. Given Data / Assumptions: The sequence is an arithmetic progression (A.P.). T3 = −13 (the 3rd term). T8 = 2 (the 8th term). We need to find the 14th term T14. Concept / Approach: For an arithmetic progression with first term a and common difference d, the nth term is given by: Tn = a + (n − 1) * d. We can form two equations using T3 and T8, solve for a and d, and then substitute into the formula for T14. Step-by-Step Solution: Let a be the first term and d be the common difference. T3 = a + 2d = −13. T8 = a + 7d = 2. Subtract the first equation from the second: (a + 7d) − (a + 2d) = 2 − (−13). This gives 5d = 15, so d = 3. Substitute d = 3 into a + 2d = −13: a + 2 * 3 = −13. Thus a + 6 = −13, so a = −19. Now find T14: T14 = a + 13d = −19 + 13 * 3. Compute 13 * 3 = 39, so T14 = −19 + 39 = 20. Verification / Alternative check: You can quickly check by listing some terms. T3 = −13, T4 = −10, and adding 3 repeatedly should eventually reach T8 = 2 and T14 = 20. This matches the algebraic solution, confirming the result. Why Other Options Are Wrong: Option a (23), option b (17), option d (26), and option e (11) do not match the arithmetic progression generated from a = −19 and d = 3. When you compute T14, only 20 is consistent with the derived formula. Common Pitfalls: Learners often confuse the position index n in Tn with the term value. Another error is to mix up formulas for A.P. with those for geometric progressions. Always remember that in an A.P. the difference between consecutive terms is constant and use Tn = a + (n − 1)d. Final Answer: Therefore, the 14th term of the arithmetic progression is 20.

Question 10

In coordinate geometry, a point P has coordinates (4, −3). This point is reflected in the horizontal line y = 1. What are the coordinates of the reflected image of P after reflection in the line y = 1?

Accepted Answer

Introduction / Context: This question checks your understanding of reflections in coordinate geometry, specifically reflection of a point in a horizontal line. Such transformations appear frequently in school level analytic geometry and in aptitude exams that test spatial reasoning. Given Data / Assumptions: Original point P has coordinates (4, −3). Reflection line is the horizontal line y = 1. We need the coordinates of the reflected point P prime. Concept / Approach: Reflection in a horizontal line y = k keeps the x coordinate unchanged while the y coordinate is mirrored across the line. The vertical distance from the point to the line is preserved on the other side. If the original y coordinate is y1, then the reflected point has y coordinate y2 such that the midpoint of y1 and y2 is k. Step-by-Step Solution: The original point is P(4, −3). The reflection line is y = 1, so k = 1. For reflection across y = 1, x stays the same: x2 = 4. Let the new y coordinate be y2. The line y = 1 is exactly midway between −3 and y2. So the midpoint relation is (−3 + y2) / 2 = 1. Multiply both sides by 2: −3 + y2 = 2. Thus y2 = 2 + 3 = 5. Hence the reflected point is (4, 5). Verification / Alternative check: The vertical distance from −3 to 1 is 4 units. From 1 to 5 is also 4 units. Since distances are equal on both sides of the reflecting line, this confirms the reflection is correct. Why Other Options Are Wrong: Option a (4, −5) goes further away from the line y = 1 on the same side instead of the opposite side. Options c and d change the x coordinate sign without justification. Option e (5, 4) changes both coordinates and does not preserve the horizontal alignment with the original point. Common Pitfalls: A common mistake is to reflect across the wrong axis, for example treating the reflection as if it were across the x axis (y = 0) instead of y = 1. Another frequent error is to change both coordinates when reflecting in a horizontal line, even though only the y coordinate should change. Final Answer: The reflected image of the point (4, −3) in the line y = 1 is (4, 5).

Question 11

In coordinate geometry, consider the straight line given by the equation 3x + 2y = 12. At which coordinate point does this line intersect and cut the Y axis?

Accepted Answer

Introduction / Context: Finding intercepts of a line is a basic skill in coordinate geometry. The Y intercept is the point where the line crosses the Y axis, and it is found by setting x equal to zero. This question checks whether you are comfortable with this standard technique. Given Data / Assumptions: Equation of the line: 3x + 2y = 12. Y axis is defined by x = 0. We need the point where the line meets the Y axis. Concept / Approach: The Y intercept is the point on the line where x equals zero. Substituting x = 0 into the line equation and solving for y gives the Y intercept. This is a very fast method and is often used for sketching graphs of linear equations. Step-by-Step Solution: Start with 3x + 2y = 12. At the Y axis, x = 0, so substitute x = 0 into the equation. Then 3 * 0 + 2y = 12 gives 2y = 12. Divide both sides by 2 to get y = 6. So the point of intersection with the Y axis is (0, 6). Verification / Alternative check: You can also rewrite the equation in slope intercept form: 2y = 12 − 3x, so y = 6 − (3 / 2) x. When x = 0, y = 6, confirming the Y intercept is (0, 6). Why Other Options Are Wrong: Option b (0, −6) has the wrong sign of y. Option c (6, 0) and option d (−6, 0) are X intercept candidates, not Y intercepts. Option e (2, 3) lies on the line but does not correspond to x = 0, so it is not the Y intercept. Common Pitfalls: Students sometimes mix up X and Y intercepts, or they try to solve the equation for both variables unnecessarily. Always remember that for the Y intercept, x is zero, and for the X intercept, y is zero. This small reminder saves time and avoids confusion. Final Answer: The line 3x + 2y = 12 cuts the Y axis at the point (0, 6).

Question 12

Two triangles ΔABC and ΔPQR are similar. The area of triangle ΔABC is 121 square centimetres and the area of triangle ΔPQR is 64 square centimetres. If the side PQ of the smaller triangle is 12 cm, what is the length, in centimetres, of side AB of t…

Accepted Answer

Introduction / Context: This question uses properties of similar triangles, which are common in geometry based aptitude problems. When two triangles are similar, the ratio of their areas is equal to the square of the ratio of corresponding sides. Here we must use the area ratio to find an unknown side length of the larger triangle. Given Data / Assumptions: ΔABC and ΔPQR are similar. Area of ΔABC = 121 cm^2. Area of ΔPQR = 64 cm^2. PQ corresponds to AB and PQ = 12 cm. Find AB in centimetres. Concept / Approach: For similar triangles, if k is the ratio of a side in ΔABC to the corresponding side in ΔPQR, then the area ratio is k^2. So area(ABC) / area(PQR) = (AB / PQ)^2. We can use this relation to find AB since the area ratio and PQ are known. Step-by-Step Solution: Compute the area ratio: area(ABC) / area(PQR) = 121 / 64. Let AB / PQ = r. Then r^2 = 121 / 64. Take square roots: r = √(121 / 64) = 11 / 8. So AB / PQ = 11 / 8, and PQ = 12 cm. Therefore AB = (11 / 8) * 12 = 11 * 12 / 8. Simplify: 12 / 8 = 3 / 2, so AB = 11 * (3 / 2) = 33 / 2 = 16.5 cm. Verification / Alternative check: We can check the logic by squaring the side ratio again. AB / PQ = 16.5 / 12 = 33 / 24 = 11 / 8, so the area ratio becomes (11 / 8)^2 = 121 / 64, which matches the given data. This confirms that AB = 16.5 cm is consistent. Why Other Options Are Wrong: Options a, b, and d are significantly smaller and would produce an area ratio different from 121 / 64 when squared. Option e (18.5) would yield a side ratio greater than 11 / 8, which again does not match the squared area ratio. Only 16.5 gives the correct similarity ratio. Common Pitfalls: One frequent mistake is to assume the area ratio is directly equal to the side ratio, instead of its square. Another pitfall is to confuse which triangle is larger. Always compare the numerical areas first and confirm which triangle has the bigger area and therefore the larger sides. Final Answer: The length of side AB of the larger triangle is 16.5 cm.

Question 13

Using standard trigonometric values in degrees, evaluate the expression Tan 60° plus Cosec 60°. What is the exact simplified value of Tan 60° + Cosec 60°?

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Introduction / Context: This question checks your memorisation and use of standard trigonometric ratios for special angles in degrees, here 60 degrees. Such values are widely used in school mathematics and aptitude tests, so being comfortable with them can save valuable time in exams. Given Data / Assumptions: The angle is 60°, measured in degrees. We need to compute Tan 60° + Cosec 60°. Standard trigonometric values for 30°, 45°, and 60° are assumed known. Concept / Approach: For the special angle 60°, we recall that Tan 60° = √3 and Sin 60° = √3 / 2. Since Cosec θ is defined as 1 / Sin θ, we can find Cosec 60° from Sin 60°. Then we add the two values and simplify the expression into a single fraction with a rational denominator if needed. Step-by-Step Solution: Recall that Tan 60° = √3. Also recall that Sin 60° = √3 / 2. Therefore, Cosec 60° = 1 / Sin 60° = 1 / (√3 / 2) = 2 / √3. Now compute Tan 60° + Cosec 60° = √3 + 2 / √3. To add these, take a common denominator √3: √3 = 3 / √3. So √3 + 2 / √3 = 3 / √3 + 2 / √3 = 5 / √3. Verification / Alternative check: You can quickly check numerically using approximate values. √3 is about 1.732, so √3 + 2 / √3 is approximately 1.732 + 2 / 1.732 which is about 1.732 + 1.155 = 2.887. The fraction 5 / √3 is also about 5 / 1.732 which is again close to 2.887. This confirms that 5 / √3 is correct. Why Other Options Are Wrong: Option a (5 / 3) is smaller than the approximate decimal 2.887. Option b (2 / √3) ignores the Tan 60° term. Option d (2 / 3) is far too small. Option e (√3) only accounts for Tan 60° and omits Cosec 60° completely. Common Pitfalls: Students sometimes confuse Tan 60° with 1 / √3, which is actually Tan 30°. Another mistake is mixing up Sin and Cosec or forgetting that Cosec θ is the reciprocal of Sin θ. Always check that you are using the correct special angle values and the right reciprocal relationships. Final Answer: The exact simplified value of Tan 60° + Cosec 60° is 5 / √3.

Question 14

In right triangle ΔXYZ, which is right angled at Y, suppose that ∠Z = 60°. Let X be the remaining acute angle. Evaluate the expression (1 / √2) × Sec X and choose its exact value from the options given.

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Introduction / Context: This question combines basic geometry of right triangles with trigonometric ratios. You must use the fact that the sum of angles in a triangle is 180 degrees and then apply standard trigonometric values for well known angles. Given Data / Assumptions: ΔXYZ is a right triangle right angled at Y. ∠Z = 60°. Therefore the remaining acute angle at X satisfies ∠X + 60° + 90° = 180°. We need to compute (1 / √2) * Sec X. Concept / Approach: In any triangle, the sum of the three interior angles is 180°. For a right triangle, one angle is 90°, so the two acute angles must add to 90°. Once we find ∠X, we use standard trigonometric values: Sec θ is the reciprocal of Cos θ. Finally we multiply Sec X by 1 / √2 and simplify the resulting surd expression. Step-by-Step Solution: Start with the angle sum: ∠X + ∠Y + ∠Z = 180°. Given ∠Y = 90° and ∠Z = 60°, so ∠X + 90° + 60° = 180°. Thus ∠X + 150° = 180°, so ∠X = 30°. Now use the standard value Cos 30° = √3 / 2. Sec 30° is 1 / Cos 30° = 1 / (√3 / 2) = 2 / √3. We must compute (1 / √2) * Sec X = (1 / √2) * (2 / √3). Multiply the numerators and denominators: (1 * 2) / (√2 * √3) = 2 / √6. Verification / Alternative check: You can approximate the values numerically. Cos 30° is about 0.866, so Sec 30° is approximately 1.155. Then (1 / √2) is about 0.707, and the product is roughly 0.707 * 1.155 which is about 0.817. The fraction 2 / √6 is also approximately 0.816, confirming the result. Why Other Options Are Wrong: Option a (2 / √3) is Sec 30° alone without the factor 1 / √2. Option b (1 / √6) and option c (1 / √3) are too small compared with the approximate numeric result. Option e (√2) is larger than 1 and does not match the computed value. Common Pitfalls: A frequent mistake is to assume X is 60° rather than 30°, ignoring the right angle at Y. Another pitfall is confusing Sec with Cos or using the wrong special angle values. Always check the triangle angle sum carefully before applying trigonometric ratios. Final Answer: The value of (1 / √2) × Sec X is 2 / √6.

Question 15

Solve the linear equation x/3 − [5(7x/5 − 4/3)] / 2 = −x/6 and determine the exact value of x that satisfies this simplification problem.

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Introduction / Context: This algebra problem involves solving a linear equation that contains nested fractions. Questions of this type test your ability to simplify complex fractional expressions and handle operations in the correct order without losing track of signs or denominators. Given Data / Assumptions: The equation is x/3 − [5(7x/5 − 4/3)] / 2 = −x/6. x is a real number. We want the unique value of x that satisfies this equation. Concept / Approach: The key idea is to simplify the nested expression step by step. First simplify the inner bracket, then multiply by 5, and then divide by 2. Once the left hand side is simplified, we will obtain a simple linear equation in x that we can solve using standard algebraic operations such as collecting like terms and isolating x. Step-by-Step Solution: Start with x/3 − [5(7x/5 − 4/3)] / 2 = −x/6. Simplify inside the bracket: 7x/5 − 4/3. Multiply by 5: 5(7x/5 − 4/3) = 7x − 20/3. Divide by 2: [5(7x/5 − 4/3)] / 2 = (7x − 20/3) / 2 = 7x/2 − 10/3. Now the equation becomes x/3 − (7x/2 − 10/3) = −x/6. Distribute the minus sign: x/3 − 7x/2 + 10/3 = −x/6. Combine the terms on the left using denominator 6: x/3 = 2x/6, 7x/2 = 21x/6, and 10/3 = 20/6. So left side is (2x/6 − 21x/6 + 20/6) = (−19x + 20) / 6. Thus (−19x + 20) / 6 = −x/6. Multiply both sides by 6: −19x + 20 = −x. Add 19x to both sides: 20 = 18x, so x = 20 / 18 = 10 / 9. Verification / Alternative check: Substitute x = 10 / 9 back into the original equation and simplify both sides. With careful fraction work, you will see that both sides become equal, confirming that x = 10 / 9 is correct. Why Other Options Are Wrong: Options b, c, and d produce different left hand side and right hand side values when substituted into the equation. Option e (0) clearly fails because the right hand side would be zero while the left hand side would not simplify to zero. Common Pitfalls: Common errors include incorrect distribution of the minus sign, mixing up denominators when combining fractions, and losing track of negative signs. Working carefully with a common denominator and checking each simplification step reduces these mistakes. Final Answer: The correct solution of the equation is x = 10 / 9.

Question 16

If a^3 + b^3 = 19 and a + b = 1 for real numbers a and b, what is the value of the product ab obtained using the standard cubic identity for the sum of two cubes?

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Introduction / Context: This algebra question focuses on using the identity for the sum of cubes to connect the sum a + b, the sum of cubes a^3 + b^3, and the product ab. Such identities are very useful in aptitude problems where direct factorisation may not be obvious but standard formulas apply neatly. Given Data / Assumptions: a^3 + b^3 = 19. a + b = 1. a and b are real numbers. We must find ab. Concept / Approach: The standard identity for the sum of cubes is a^3 + b^3 = (a + b)^3 − 3ab(a + b). We are already given a^3 + b^3 and a + b, so we can substitute these values into the identity and solve for ab. This transforms the problem into solving a simple linear equation in ab. Step-by-Step Solution: Use the identity a^3 + b^3 = (a + b)^3 − 3ab(a + b). Given a^3 + b^3 = 19 and a + b = 1. Substitute into the identity: 19 = (1)^3 − 3ab(1). So 19 = 1 − 3ab. Rearrange to isolate ab: 1 − 3ab = 19. Thus −3ab = 19 − 1 = 18. So ab = −18 / 3 = −6. Verification / Alternative check: As a quick consistency check, we could attempt to find specific values of a and b that satisfy a + b = 1 and ab = −6 by solving the quadratic t^2 − (a + b)t + ab = 0, that is t^2 − t − 6 = 0. This quadratic has roots t = 3 and t = −2. For a = 3 and b = −2, indeed a + b = 1 and a^3 + b^3 = 27 − 8 = 19, confirming that ab = −6 works. Why Other Options Are Wrong: Options a, c, d, and e do not satisfy the identity when substituted. For example, if ab were 5 or 7, the expression (a + b)^3 − 3ab(a + b) would not equal 19 with a + b = 1. Only ab = −6 maintains the given relationship between the sum of cubes and the sum of the numbers. Common Pitfalls: Some learners mistakenly use the difference of cubes identity or try to factor a^3 + b^3 directly as (a + b)(a^2 − ab + b^2) without connecting it to the known value of a + b. While that path can also work, it usually leads to more algebra. Using the cubic identity with the known sum is the most efficient approach here. Final Answer: The required product of the two numbers is ab = −6.

Question 17

A proper fraction is greater than its reciprocal by 72/77. If the fraction is denoted by x, satisfying x − 1/x = 72/77, then what is the value of this fraction x?

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Introduction / Context: This question uses algebra to describe a relationship between a fraction and its reciprocal. The problem is typical in quantitative aptitude where you must translate a verbal condition into an equation, solve a quadratic, and then select the fraction that meets the condition. Given Data / Assumptions: Let the fraction be x, with x not equal to zero. x − 1 / x = 72 / 77. We must find x from the options given. Concept / Approach: The equation x − 1 / x = 72 / 77 is nonlinear due to the reciprocal term. Multiplying through by x gives a quadratic equation. We then solve the quadratic using factorisation or the quadratic formula and check which solutions match the options. Because the condition mentions a fraction, we expect rational roots that can be simplified to one of the listed fractions. Step-by-Step Solution: Start with x − 1 / x = 72 / 77. Multiply both sides by x (x is non zero): x^2 − 1 = (72 / 77) x. Multiply through by 77 to clear denominators: 77x^2 − 77 = 72x. Rearrange to standard quadratic form: 77x^2 − 72x − 77 = 0. Solve 77x^2 − 72x − 77 = 0, which factorises to (7x − 11)(11x + 7) = 0. Hence, x = 11 / 7 or x = −7 / 11. From the list of options, 11 / 7 is present and is greater than 1, which is consistent with x being greater than its reciprocal. Verification / Alternative check: Check x = 11 / 7 directly: 1 / x = 7 / 11. Then x − 1 / x = 11 / 7 − 7 / 11. Using denominator 77, this is (121 − 49) / 77 = 72 / 77, which matches the given condition exactly. Why Other Options Are Wrong: Option a (7 / 11) leads to 7 / 11 − 11 / 7 which equals (49 − 121) / 77 = −72 / 77, the negative of the required value. Options c and d do not satisfy the equation when substituted and do not arise as roots of the quadratic. Option e (11 / 4) also fails the equation x − 1 / x = 72 / 77. Common Pitfalls: A common mistake is to perform the subtraction x − 1 / x incorrectly or to forget to multiply every term by x, leading to an incorrect quadratic. Also, some learners pick the first root they see without verifying it matches the wording that the fraction is greater than its reciprocal by a positive amount. Final Answer: The fraction that satisfies the given condition is 11 / 7.

Question 18

Using standard trigonometric values in degrees, evaluate the expression (1 / √2) · Cot 45° + (1 / √3) · Cosec 60°. What is the exact simplified value of this trigonometric sum?

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Introduction / Context: This question tests your ability to work with standard trigonometric values and surds, and to simplify expressions involving square roots. Such problems are common in quantitative aptitude tests, where exact simplification is more important than decimal approximations. Given Data / Assumptions: Angles are measured in degrees. We must compute (1 / √2) * Cot 45° + (1 / √3) * Cosec 60°. Standard values at 45° and 60° are assumed known. Concept / Approach: We recall that Cot 45° = 1 and Sin 60° = √3 / 2. Since Cosec 60° is the reciprocal of Sin 60°, we can write Cosec 60° = 2 / √3. Then we substitute these values into the expression, simplify each term carefully, and combine them into a single simplified surd expression. Step-by-Step Solution: Start with (1 / √2) * Cot 45° + (1 / √3) * Cosec 60°. Use Cot 45° = 1, so the first term becomes 1 / √2. Use Sin 60° = √3 / 2, hence Cosec 60° = 1 / (√3 / 2) = 2 / √3. The second term is (1 / √3) * (2 / √3) = 2 / 3. So the expression becomes 1 / √2 + 2 / 3. Take a common denominator 3√2: 1 / √2 = (3) / (3√2) and 2 / 3 = (2√2) / (3√2). Add them: (3 + 2√2) / (3√2). Verification / Alternative check: Approximate numerically: 1 / √2 is about 0.707 and 2 / 3 is about 0.667. Their sum is roughly 1.374. The expression (3 + 2√2) / (3√2) gives a similar decimal value when evaluated, confirming the algebraic simplification is correct. Why Other Options Are Wrong: Option a and option c are alternative surd forms that do not simplify to the same numeric value. Option b (√3) is about 1.732, which is significantly larger than the correct decimal value near 1.374. Option e also does not equal the computed value when simplified. Common Pitfalls: A frequent error is to mix up Cosec 60° with Cosec 30°, or to confuse Cot and Tan at 45°. Another pitfall is mishandling the surd rationalisation step, which can lead to incorrect numerators or denominators. Working systematically with common denominators helps keep the simplification correct. Final Answer: The exact value of the expression is (3 + 2√2) / (3√2).

Question 19

In right triangle ΔABC, which is right angled at B, suppose that ∠A = 60°. Let C be the remaining acute angle. Evaluate the expression Sec C · Sin A and choose its exact value from the options.

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Introduction / Context: This is a trigonometry question that uses the relationship between complementary angles in a right triangle. The problem combines standard angle values with simple identities to obtain a neat simplified result. Given Data / Assumptions: Triangle ΔABC is right angled at B, so ∠B = 90°. ∠A = 60°. The third angle ∠C is therefore 30°. We must compute Sec C * Sin A. Concept / Approach: In a right triangle, the two acute angles are complementary and add up to 90°. So if one is 60°, the other must be 30°. We use standard trigonometric values: Sin 60° and Cos 30°. Sec is the reciprocal of Cos, so Sec C = 1 / Cos C. Then we multiply Sec C by Sin A and simplify. Step-by-Step Solution: Given ∠A = 60° and ∠B = 90°, the remaining angle is ∠C = 180° − 90° − 60° = 30°. We need Sec C * Sin A = Sec 30° * Sin 60°. Recall that Cos 30° = √3 / 2, so Sec 30° = 1 / Cos 30° = 2 / √3. Also recall that Sin 60° = √3 / 2. Now compute Sec 30° * Sin 60° = (2 / √3) * (√3 / 2). The numerator becomes 2 * √3 and the denominator becomes √3 * 2, so the factors cancel. Thus Sec 30° * Sin 60° = 1. Verification / Alternative check: Using approximate decimals, Sin 60° is about 0.866 and Cos 30° is also about 0.866. Hence Sec 30° is about 1 / 0.866 which is roughly 1.155. The product 1.155 * 0.866 is close to 1, confirming our exact symbolic result. Why Other Options Are Wrong: Options a, b, c, and e correspond to individual values of common trigonometric ratios but not to the product Sec 30° * Sin 60°. None of them gives exactly 1 when evaluated, so they do not match the simplified expression. Common Pitfalls: Some learners mistakenly treat A and C as both 60° or both 30°, forgetting that the sum of angles in a triangle must be 180°. Another error is to confuse Cos 30° with Sin 30° or to misremember the special angle values. Always confirm the angle measures and recall the standard ratio table before substituting values. Final Answer: The value of Sec C × Sin A is 1.

Question 20

Solve the linear equation [2(4x/5 − 3/4)] / 3 − 5/3 = −1/6 and determine the exact value of x that satisfies this fractional simplification problem.

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Introduction / Context: This question reinforces skills in solving linear equations involving multiple fractions and brackets. Such problems appear frequently in aptitude tests because they check attention to detail, especially with signs and denominators. Given Data / Assumptions: The equation is [2(4x/5 − 3/4)] / 3 − 5/3 = −1/6. x is a real number. We must find the value of x that satisfies this equation. Concept / Approach: The strategy is to simplify the expression step by step. First simplify inside the brackets, then multiply by 2, then divide by 3. After that, we combine all fractions on one side and obtain a simple linear equation in x. Solving that equation gives the required value of x. Step-by-Step Solution: Start with [2(4x/5 − 3/4)] / 3 − 5/3 = −1/6. Simplify inside: 4x/5 − 3/4. Multiply by 2: 2(4x/5 − 3/4) = 8x/5 − 3/2. Now divide by 3: [2(4x/5 − 3/4)] / 3 = (8x/5 − 3/2) / 3 = 8x/15 − 1/2. So the equation becomes 8x/15 − 1/2 − 5/3 = −1/6. Combine constants −1/2 and −5/3 using common denominator 6: −1/2 = −3/6, −5/3 = −10/6, sum is −13/6. Thus 8x/15 − 13/6 = −1/6. Add 13/6 to both sides: 8x/15 = −1/6 + 13/6 = 12/6 = 2. So 8x/15 = 2, giving x = 2 * 15 / 8 = 30 / 8 = 15 / 4. Verification / Alternative check: Substitute x = 15 / 4 back into the original equation and simplify numerically. You find that both sides reduce to −1/6, confirming that 15 / 4 is indeed the correct solution. Why Other Options Are Wrong: Options a, b, and c produce mismatched left and right hand sides when substituted. Option e (0) clearly fails because the left side becomes negative while the right side is −1/6. Only x = 15 / 4 balances the equation exactly. Common Pitfalls: Learners frequently make mistakes when dealing with several fractions, especially in combining terms and handling negative signs. Sometimes they drop the division by 3 or incorrectly expand the numerator. Proceeding slowly and using a clear common denominator step helps avoid such errors. Final Answer: The solution of the linear equation is x = 15 / 4.

Simplification Questions