Question 1

What expression must be added to 5(2x − y) so that the result becomes 4(2x − 3y) + 5(x + 4y)?

Accepted Answer

Introduction / Context: This algebra question asks you to find an expression that must be added to one linear combination of x and y to obtain another. It tests your understanding of algebraic expansion, combining like terms, and solving simple equations with expressions. Such manipulations are common in simplifying linear expressions and in forming equivalent equations. Given Data / Assumptions: - We start with the expression 5(2x − y). - We want to add an unknown expression E so that 5(2x − y) + E = 4(2x − 3y) + 5(x + 4y). - x and y are real variables. - We must solve for E and match it with one of the answer choices. Concept / Approach: The equation 5(2x − y) + E = 4(2x − 3y) + 5(x + 4y) can be rearranged as E = [4(2x − 3y) + 5(x + 4y)] − 5(2x − y). Hence, we simply expand all the brackets on the right, combine like terms, and the resulting simplified expression is E. This is a direct application of basic algebraic operations. Step-by-Step Solution: Step 1: Let E be the expression that needs to be added. Then 5(2x − y) + E = 4(2x − 3y) + 5(x + 4y). Step 2: Rearrange to isolate E: E = 4(2x − 3y) + 5(x + 4y) − 5(2x − y). Step 3: Expand 4(2x − 3y) to get 8x − 12y. Step 4: Expand 5(x + 4y) to get 5x + 20y. Step 5: Expand 5(2x − y) to get 10x − 5y. Step 6: Substitute these expansions into the expression for E to get E = (8x − 12y) + (5x + 20y) − (10x − 5y). Step 7: Combine like terms in stages. First add (8x − 12y) and (5x + 20y): this gives (8x + 5x) + (−12y + 20y) = 13x + 8y. Step 8: Now subtract (10x − 5y): E = (13x + 8y) − (10x − 5y) = 13x + 8y − 10x + 5y. Step 9: Combine the final like terms: (13x − 10x) + (8y + 5y) = 3x + 13y. Step 10: Therefore, the required expression E is 3x + 13y. Verification / Alternative check: To verify, plug E = 3x + 13y back into the original equation. Compute 5(2x − y) + E = (10x − 5y) + (3x + 13y) = 13x + 8y. Now evaluate the right hand side: 4(2x − 3y) + 5(x + 4y) = (8x − 12y) + (5x + 20y) = 13x + 8y. Since both sides give 13x + 8y, the equation holds, confirming that E is correct. Why Other Options Are Wrong: Option a, 3x − 13y, would lead to 10x − 5y + (3x − 13y) = 13x − 18y, which does not match 13x + 8y. Option c, 13x − 3y, and option d, 13x + 3y, produce different coefficients for x and y that do not align with the right hand side expression after expansion. Option e, x + 3y, is too small in both x and y coefficients to make the expressions equal. Common Pitfalls: Students sometimes mis distribute the negative sign when subtracting the expression 5(2x − y), writing −5(2x − y) incorrectly as −5(2x) − y instead of −10x + 5y. Another common error is mixing up coefficients when combining like terms. Writing each step clearly, particularly the expansion and the subtraction, helps avoid such mistakes. Always verify by substituting back into the original equation to confirm equality. Final Answer: The expression that must be added is 3x + 13y.

Question 2

In aptitude simplification with linear inequalities, solve the compound inequality and choose the value of x that satisfies both conditions: 3(2 - 3x) < (2 - 3x) and (2 - 3x) ≥ (4x - 6).

Accepted Answer

Introduction / Context: This question belongs to aptitude simplification with linear inequalities in one variable. You are given a pair of inequalities that must be satisfied at the same time, and you must choose a value of x from the options that satisfies both. Problems like this test your ability to manipulate inequalities, handle brackets, and understand how solution intervals overlap. Given Data / Assumptions: First inequality: 3(2 - 3x) < (2 - 3x). Second inequality: (2 - 3x) ≥ (4x - 6). x is a real number. We must choose one option that satisfies both inequalities simultaneously. Concept / Approach: For each inequality, we remove brackets, simplify, and isolate x. The solution to each inequality is an interval on the real line. The combined solution must satisfy both, so we take the intersection of these intervals. Finally, we test the given discrete options against this intersection and identify which value of x fits the combined condition. This systematic approach avoids guesswork and ensures an accurate result. Step-by-Step Solution: 1) Start with 3(2 - 3x) < (2 - 3x). 2) Expand the left side: 3 * (2 - 3x) = 6 - 9x. 3) The inequality becomes 6 - 9x < 2 - 3x. 4) Subtract 2 from both sides: 4 - 9x < -3x. 5) Add 9x to both sides: 4 < 6x, so x > 4/6 = 2/3. 6) Now consider (2 - 3x) ≥ (4x - 6). 7) Add 3x to both sides: 2 ≥ 4x + 3x - 6 = 7x - 6. 8) Add 6 to both sides: 8 ≥ 7x, so x ≤ 8/7. 9) Combine the results: x must satisfy x > 2/3 and x ≤ 8/7. So the solution interval is (2/3, 8/7]. 10) Among the options −2, −1, 1, 2, and 'None of these', only x = 1 lies in the interval (2/3, 8/7]. Verification / Alternative check: Substitute x = 1 into the original inequalities. For the first inequality, 3(2 - 3 * 1) = 3(2 - 3) = 3 * (−1) = −3 and (2 - 3 * 1) = −1, so −3 < −1, which is true. For the second inequality, (2 - 3 * 1) = −1 and (4 * 1 - 6) = 4 - 6 = −2, so −1 ≥ −2, which is also true. Thus x = 1 satisfies both inequalities exactly, confirming that it is a valid solution and consistent with the intersection found earlier. Why Other Options Are Wrong: For x = −2 or x = −1, the first inequality fails because 3(2 - 3x) becomes larger than (2 - 3x) when evaluated numerically. For x = 2, the second inequality fails because (2 - 3 * 2) is less than (4 * 2 - 6). Therefore none of these values satisfy both inequalities. The option 'None of these' is wrong because we have already found that x = 1 is a valid solution. Common Pitfalls: A common mistake is to divide or multiply both sides of an inequality by an expression involving x without checking its sign, which can reverse the inequality direction incorrectly. Another error is failing to compute the intersection of the solution sets and choosing a value that only satisfies one inequality. Carefully isolating x in each inequality and then intersecting the resulting intervals avoids these issues. Final Answer: The value of x that satisfies both inequalities is 1.

Question 3

In trigonometric simplification, if X = sec^2(A) + cosec^2(A), express X in terms of tan(A) and cot(A) and select the correct equivalent expression for X (for angles where all terms are defined).

Accepted Answer

Introduction / Context: This question tests your understanding of basic trigonometric identities and how to rewrite expressions involving secant and cosecant in terms of tangent and cotangent. Rewriting expressions in different trigonometric forms is very useful for simplification, solving equations, and proving identities in both school-level mathematics and aptitude exams. Given Data / Assumptions: X is defined as X = sec^2(A) + cosec^2(A). A is an angle for which all the trigonometric functions sec(A), cosec(A), tan(A), and cot(A) are defined. We must choose the expression written in terms of tan(A) and cot(A) that is exactly equal to X. Concept / Approach: We use the fundamental identities that relate secant and cosecant to tangent and cotangent. Specifically, sec^2(A) = 1 + tan^2(A) and cosec^2(A) = 1 + cot^2(A). By substituting these into the definition of X, we obtain X in a new form purely in terms of tan(A) and cot(A). Then we compare the resulting expression to the given options and pick the matching one. This approach relies only on standard identities and straightforward algebraic simplification. Step-by-Step Solution: 1) Start with the definition X = sec^2(A) + cosec^2(A). 2) Use the identity sec^2(A) = 1 + tan^2(A). 3) Use the identity cosec^2(A) = 1 + cot^2(A). 4) Substitute into X: X = [1 + tan^2(A)] + [1 + cot^2(A)]. 5) Combine like terms: X = 1 + tan^2(A) + 1 + cot^2(A) = 2 + tan^2(A) + cot^2(A). 6) Therefore the required expression for X in terms of tan(A) and cot(A) is X = 2 + tan^2(A) + cot^2(A). Verification / Alternative check: You can verify this identity numerically by choosing a specific angle A for which all functions are defined, for example A = 45 degrees. Then tan(45°) = 1 and cot(45°) = 1, so the right side becomes 2 + 1^2 + 1^2 = 4. On the left side, sec(45°) = √2, so sec^2(45°) = 2, and cosec(45°) = √2, so cosec^2(45°) = 2. The sum sec^2(45°) + cosec^2(45°) also equals 4. This confirms that the formula X = 2 + tan^2(A) + cot^2(A) is correct. Why Other Options Are Wrong: Option b gives 1 + tan^2(A) + cot^2(A), which would correspond to sec^2(A) + cosec^2(A) − 1 and therefore underestimates X by 1. Option c omits the constant 2 entirely. Option d, sec^2(A) * cosec^2(A), multiplies the squares of secant and cosecant and does not match the sum. Option e, tan^2(A) * cot^2(A), simplifies to 1, which is clearly different from X for most angles. Only option a matches the derived expression exactly. Common Pitfalls: Students sometimes confuse the identities sec^2(A) = 1 + tan^2(A) and cosec^2(A) = 1 + cot^2(A) with other formulas or forget the constant term 1. Another mistake is attempting to express sec(A) and cosec(A) directly in terms of sin(A) and cos(A) and then in terms of tan(A) and cot(A), which is longer and more error prone. Remembering the standard squared identities and applying them directly is the quickest and safest path to the correct answer. Final Answer: The expression X = sec^2(A) + cosec^2(A) can be written as 2 + tan^2(A) + cot^2(A).

Question 4

In basic algebraic simplification, solve the linear equation with brackets and find the exact value of x: (4x - 3) - (2x + 1) = 4.

Accepted Answer

Introduction / Context: This is a straightforward linear equation in one variable that involves subtracting a bracketed expression. Such questions appear frequently in aptitude and school exams and assess your ability to correctly expand brackets, combine like terms, and isolate the variable. Mastering these basics is essential before moving on to more advanced algebraic techniques. Given Data / Assumptions: The equation is (4x - 3) - (2x + 1) = 4. x is a real number. We must find the unique value of x that satisfies the equation. Standard algebraic operations apply (addition, subtraction, etc.). Concept / Approach: The key idea is to remove the parentheses carefully. When subtracting a bracket, you must change the sign of each term inside the bracket. After expansion, collect all x terms together and all constant terms together. Then solve the resulting simple linear equation by isolating x. This method is efficient and avoids errors that come from skipping steps or mis-handling signs. Step-by-Step Solution: 1) Start with the equation: (4x - 3) - (2x + 1) = 4. 2) Expand the first bracket: (4x - 3) stays as 4x - 3. 3) Subtract the second bracket: −(2x + 1) becomes −2x − 1. 4) Combine like terms on the left side: 4x − 3 − 2x − 1 = (4x − 2x) + (−3 − 1) = 2x − 4. 5) The equation is now 2x − 4 = 4. 6) Add 4 to both sides to isolate the x term: 2x = 8. 7) Divide both sides by 2: x = 8 / 2 = 4. Verification / Alternative check: Substitute x = 4 back into the original equation to ensure the solution is correct. Compute 4x − 3 = 4 * 4 − 3 = 16 − 3 = 13. Then compute 2x + 1 = 2 * 4 + 1 = 8 + 1 = 9. The left-hand side becomes 13 − 9 = 4, which matches the right-hand side exactly. This confirms that x = 4 is the correct and unique solution of the given linear equation. Why Other Options Are Wrong: If x = 2, then (4x − 3) − (2x + 1) becomes (8 − 3) − (4 + 1) = 5 − 5 = 0, not 4. If x = 1, we get (4 − 3) − (2 + 1) = 1 − 3 = −2. For x = 0, we get (0 − 3) − (0 + 1) = −3 − 1 = −4. For x = 3, we get (12 − 3) − (6 + 1) = 9 − 7 = 2. None of these equal 4, so those options are incorrect. Common Pitfalls: A common error is forgetting to distribute the negative sign when subtracting the second bracket, which leads to incorrect expressions like 4x − 3 − 2x + 1. Another mistake is performing incorrect arithmetic when combining the constant terms. Writing each step clearly and carefully handling signs will help you avoid these mistakes in similar problems. Final Answer: The exact value of x that satisfies the equation is 4.

Question 5

In quadratic equations, determine which of the following equations has real and distinct roots, that is, two different real solutions for x.

Accepted Answer

Introduction / Context: This question checks your understanding of the discriminant of a quadratic equation and how it determines the nature of the roots. Quadratic equations can have two distinct real roots, one repeated real root, or a pair of complex conjugate roots. Knowing how to quickly classify them using the discriminant is a standard skill in algebra and aptitude exams. Given Data / Assumptions: Each option is a quadratic equation of the form ax^2 + bx + c = 0. We must find which equation has two distinct real roots. The discriminant is defined as D = b^2 − 4ac. Real and distinct roots occur when D > 0. Concept / Approach: For a quadratic equation ax^2 + bx + c = 0: If D > 0, the equation has two distinct real roots. If D = 0, the equation has real and equal (repeated) roots. If D < 0, the equation has no real roots (complex roots). We will compute the discriminant for each equation and check which one yields a positive value. This approach allows us to decide without actually solving the equations for x. Step-by-Step Solution: 1) For option a: 3x^2 − 6x + 2 = 0, we have a = 3, b = −6, c = 2. 2) Compute the discriminant: D = (−6)^2 − 4 * 3 * 2 = 36 − 24 = 12. This is positive, so roots are real and distinct. 3) For option b: 3x^2 − 6x + 3 = 0, a = 3, b = −6, c = 3. Then D = (−6)^2 − 4 * 3 * 3 = 36 − 36 = 0, so roots are real and equal. 4) For option c: x^2 − 8x + 16 = 0, a = 1, b = −8, c = 16. D = (−8)^2 − 4 * 1 * 16 = 64 − 64 = 0, real and equal roots. 5) For option d: 4x^2 − 8x + 4 = 0, a = 4, b = −8, c = 4. D = (−8)^2 − 4 * 4 * 4 = 64 − 64 = 0, real and equal roots. 6) Only option a has D > 0, so only that equation has two distinct real roots. Verification / Alternative check: We can also factor or use the quadratic formula for option a. Using the quadratic formula for 3x^2 − 6x + 2 = 0 gives x = [6 ± √12] / 6 = [6 ± 2√3] / 6 = 1 ± (√3)/3, which clearly yields two different real values. For options b, c, and d, the repeated roots can be seen directly: for example, x^2 − 8x + 16 = (x − 4)^2 and 4x^2 − 8x + 4 = 4(x − 1)^2, each having a single repeated root. Why Other Options Are Wrong: Options b, c, and d all have discriminant equal to zero, meaning they have real but equal roots, not distinct. Option e, 'None of these', is incorrect because we have already identified that option a does indeed have two distinct real roots. Therefore, only option a satisfies the required condition of real and distinct roots. Common Pitfalls: Students often miscompute the discriminant by forgetting the factor 4ac or by mishandling the sign of b. Another common mistake is assuming that the presence of a square in the constant term automatically implies equal roots, which is not always correct. Always compute D carefully, and remember that only D > 0 guarantees two different real solutions. Final Answer: The quadratic equation with real and distinct roots is 3x^2 - 6x + 2 = 0.

Question 6

In coordinate geometry, triangle ABC has vertices A(-5, 4), B(-4, 0), and C(-2, 2). Find the equation of the median AD, where D is the midpoint of side BC and the median is drawn from vertex A to side BC.

Accepted Answer

Introduction / Context: This coordinate geometry question asks you to find the equation of a median of a triangle. A median is the line segment drawn from a vertex to the midpoint of the opposite side. Understanding how to find midpoints and then equations of lines joining two points is essential in analytic geometry and appears frequently in competitive exams and school syllabi. Given Data / Assumptions: Triangle ABC has vertices A(−5, 4), B(−4, 0), and C(−2, 2). D is the midpoint of side BC. AD is the median from vertex A to side BC. We need the equation of the line AD in standard form. Concept / Approach: First, we find the midpoint D of BC using the midpoint formula: D = ((x_B + x_C) / 2, (y_B + y_C) / 2). Then we find the equation of the line passing through A(−5, 4) and D using the two-point form or the slope-intercept method. Finally, we convert the result into a standard linear form and compare it with the given options to choose the correct one. Step-by-Step Solution: 1) Coordinates of B are (−4, 0) and of C are (−2, 2). 2) Compute the midpoint D of BC: x_D = (−4 + (−2)) / 2 = −6 / 2 = −3, y_D = (0 + 2) / 2 = 2 / 2 = 1. So D is (−3, 1). 3) Now we need the line through A(−5, 4) and D(−3, 1). 4) Find the slope m: m = (1 − 4) / (−3 − (−5)) = (−3) / (2) = −3/2. 5) Use point-slope form with point A: y − 4 = (−3/2)(x + 5). 6) Multiply both sides by 2 to clear the fraction: 2(y − 4) = −3(x + 5). 7) Expand: 2y − 8 = −3x − 15. 8) Rearrange to standard form: 3x + 2y + 7 = 0, or 3x + 2y = −7. Verification / Alternative check: Check whether both points A and D satisfy the equation 3x + 2y = −7. For A(−5, 4), 3(−5) + 2(4) = −15 + 8 = −7, so A lies on the line. For D(−3, 1), 3(−3) + 2(1) = −9 + 2 = −7, so D also lies on the line. Therefore, 3x + 2y = −7 correctly represents the median AD, and our calculations are consistent. Why Other Options Are Wrong: Option a, 3x − 2y = −11, and option d, 3x − 2y = 11, use a different sign pattern and slope. Option b, 3x + 2y = 7, gives positive 7 on the right-hand side and does not pass through the given points. Substituting A or D into these incorrect equations will not satisfy them. Option e, 'None of these', is not correct because we have found a matching option. Only option c, 3x + 2y = −7, fits the median AD exactly. Common Pitfalls: Typical mistakes include miscalculating the midpoint, incorrectly computing the slope due to sign errors, or rearranging to standard form incorrectly. Some students also mix up the coordinates when using the point-slope formula. Writing each step clearly, especially the midpoint and slope, helps avoid these issues and leads to the correct median equation quickly. Final Answer: The equation of the median AD from A to side BC is 3x + 2y = -7.

Question 7

In coordinate geometry, the distance between points (7, 7) and (k, -5) is 13 units. Using the distance formula, determine which option gives a possible value of k.

Accepted Answer

Introduction / Context: This question checks your understanding of the distance formula in the Cartesian plane. Given one point with fixed coordinates and another point with an unknown x-coordinate k, you are told the distance between them and asked to find which value of k from the options makes that distance equal to 13 units. This is a classic coordinate geometry and aptitude-style problem involving simple quadratic reasoning. Given Data / Assumptions: First point: (7, 7). Second point: (k, −5). The distance between the points is 13 units. We must find a value of k from the options that satisfies this condition. Concept / Approach: We apply the distance formula between two points (x1, y1) and (x2, y2): Distance^2 = (x2 − x1)^2 + (y2 − y1)^2. Here, the distance is 13, so its square is 169. We substitute x1 = 7, y1 = 7, x2 = k, and y2 = −5 into the formula, simplify to get an equation in k, and then solve for k. After finding the possible values, we see which one appears in the options given and choose that as the answer. Step-by-Step Solution: 1) Use the distance formula: (k − 7)^2 + (−5 − 7)^2 = 13^2. 2) Simplify the y-difference: −5 − 7 = −12, so the equation becomes (k − 7)^2 + (−12)^2 = 169. 3) Compute (−12)^2 = 144, giving (k − 7)^2 + 144 = 169. 4) Subtract 144 from both sides: (k − 7)^2 = 169 − 144 = 25. 5) Take square roots: k − 7 = ±5, so k = 7 + 5 = 12 or k = 7 − 5 = 2. 6) The possible k values from the equation are 2 and 12. Among the given options −4, −2, 2, 4, and 11, the only matching value is k = 2. Verification / Alternative check: Verify k = 2 by computing the distance between (7, 7) and (2, −5). The difference in x is 2 − 7 = −5, and the difference in y is −5 − 7 = −12. The distance squared is (−5)^2 + (−12)^2 = 25 + 144 = 169, so the distance is √169 = 13 units, which matches the requirement exactly. This confirms that k = 2 is indeed a valid solution consistent with the problem statement. Why Other Options Are Wrong: If k = −4, −2, 4, or 11, substituting into the distance formula will not produce a distance of 13 units. For example, if k = 4, then (4 − 7)^2 + (−12)^2 = (−3)^2 + 144 = 9 + 144 = 153, which gives a distance of √153, not 13. Similarly, the other values do not satisfy the equation (k − 7)^2 = 25. Therefore, they do not correspond to a distance of 13 units between the two points. Common Pitfalls: A common error is forgetting to square the differences or forgetting that the distance formula uses the square of the distance, not the distance itself, when setting up the equation. Another mistake is to take only the positive square root when solving (k − 7)^2 = 25, thereby missing one of the possible values. Careful application of the formula, along with proper handling of squares and square roots, ensures accurate results. Final Answer: The possible value of k among the options that makes the distance exactly 13 units is 2.

Question 8

In algebraic simplification, find the expression that must be added to 8(3x - 4y) so that the final result becomes exactly 18x - 18y.

Accepted Answer

Introduction / Context: This question tests your ability to manipulate algebraic expressions and match coefficients. You are given a scaled binomial 8(3x − 4y) and asked what must be added to it to obtain the target expression 18x − 18y. This is similar to balancing equations and is a useful technique in simplifying expressions and solving linear algebra problems in aptitude tests. Given Data / Assumptions: Starting expression: 8(3x − 4y). Final expression required: 18x − 18y. We must find an expression E such that 8(3x − 4y) + E = 18x − 18y. x and y are variables representing real numbers. Concept / Approach: First, expand 8(3x − 4y) to obtain a simpler form. Then set up an equation where the unknown expression E bridges the difference between this expanded form and the desired final expression. By subtracting the known expression from the target, we derive E directly. Finally, we compare E with the given options and select the matching one. Matching coefficients of x and y ensures correctness. Step-by-Step Solution: 1) Expand the given expression: 8(3x − 4y) = 24x − 32y. 2) We want 8(3x − 4y) + E = 18x − 18y, so substitute the expansion: 24x − 32y + E = 18x − 18y. 3) Solve for E by subtracting 24x − 32y from both sides: E = (18x − 18y) − (24x − 32y). 4) Distribute the negative sign: E = 18x − 18y − 24x + 32y. 5) Combine like terms: (18x − 24x) = −6x and (−18y + 32y) = 14y. 6) Therefore, E = −6x + 14y. Verification / Alternative check: Check by substitution: 8(3x − 4y) + (−6x + 14y) = 24x − 32y − 6x + 14y. Combine like terms to get (24x − 6x) + (−32y + 14y) = 18x − 18y, which matches the target expression exactly. This confirms that adding −6x + 14y to 8(3x − 4y) indeed produces 18x − 18y without any discrepancy. Why Other Options Are Wrong: Option a (6x − 14y) would give 24x − 32y + 6x − 14y = 30x − 46y, which does not match the required coefficients. Option b (14y + 6x) is the same as 6x + 14y and leads to the same incorrect result. Option c (14y − 6x) rearranges signs incorrectly and does not match E. Option d (6xy) introduces a new term xy, which is not present in the original or target expressions. Only option e, −6x + 14y, correctly balances the equation. Common Pitfalls: Students sometimes expand incorrectly, for example writing 8(3x − 4y) as 24x − 4y or mixing up the signs when subtracting expressions. Another mistake is to guess by inspection without actually matching coefficients, which can be misleading when several options look similar. Always expand completely and carefully subtract to find the exact expression required. Final Answer: The expression that must be added is -6x + 14y.

Question 9

In coordinate geometry, find the equation of the straight line that has y-intercept 3/4 and makes an angle of 45° with the positive x-axis. Choose the correct equation from the options.

Accepted Answer

Introduction / Context: This question combines two important ideas from coordinate geometry: the relationship between slope and angle with the x-axis, and the intercept form of a line. You are asked to find the equation of a line given its y-intercept and the angle it makes with the positive x-axis. Being able to translate geometric information into an equation of a line is a core analytic geometry skill. Given Data / Assumptions: The line makes an angle of 45° with the positive x-axis. The y-intercept of the line is 3/4. We must find the equation of this line in standard form. The coordinate system is the usual Cartesian plane. Concept / Approach: The slope m of a line that makes an angle θ with the positive x-axis is m = tan(θ). With θ = 45°, the slope is tan(45°) = 1. The general slope-intercept form of a line is y = mx + c, where c is the y-intercept. After writing the line in y = mx + c form, we can rearrange to obtain the standard form Ax + By = C, clear any fractions, and match the result with the given options. Step-by-Step Solution: 1) The slope of the line is m = tan(45°) = 1. 2) The y-intercept is 3/4, so the line passes through (0, 3/4). 3) Using the slope-intercept form y = mx + c, we get y = x + 3/4. 4) Rearrange this to standard form: y − x = 3/4, or x − y + 3/4 = 0. 5) Multiply the entire equation by 4 to eliminate the fraction: 4x − 4y + 3 = 0. 6) This can be written as 4x − 4y = −3, which matches option b exactly. Verification / Alternative check: Check the intercepts and slope from 4x − 4y = −3. Setting x = 0, we get −4y = −3, so y = 3/4, confirming the y-intercept. To check the slope, rearrange to y = x + 3/4. This shows the coefficient of x is 1, which matches the slope m = tan(45°) = 1. Therefore, both given conditions (angle 45° and y-intercept 3/4) are satisfied by this equation. Why Other Options Are Wrong: Option a, 4x − 4y = 3, rearranges to y = x − 3/4 and has y-intercept −3/4, not 3/4. Options c and d have different numeric coefficients and intercepts; for example, 3x − 3y = 4 leads to y = x − 4/3. Option e, y = x − 3/4, again has a y-intercept of −3/4. None of these match the required y-intercept and slope simultaneously. Only option b matches both conditions exactly. Common Pitfalls: A frequent mistake is to confuse the sign of the intercept when converting between standard and slope-intercept forms. Another error is miscalculating the slope from the angle, though tan(45°) = 1 is a well known special value. Carefully rearranging the equation and checking both slope and intercept ensures the correct choice. Final Answer: The equation of the line is 4x - 4y = -3.

Question 10

In divisibility tests for aptitude simplification, determine which of the following numbers is completely divisible by 99 (that is, divisible by both 9 and 11).

Accepted Answer

Introduction / Context: This question tests your understanding of divisibility rules, particularly for the composite number 99. Since 99 = 9 * 11, a number is divisible by 99 exactly when it is divisible by both 9 and 11. Using the divisibility tests for 9 and 11 allows us to check each option quickly without performing full long division. This is a common skill tested in quantitative aptitude exams. Given Data / Assumptions: We are given four candidate numbers: 57717, 57627, 55162, and 56982. We must find which number is divisible by 99. A number divisible by 99 must be divisible by both 9 and 11. Standard divisibility tests for 9 and 11 are used. Concept / Approach: The divisibility test for 9: a number is divisible by 9 if the sum of its digits is a multiple of 9. The test for 11: a number is divisible by 11 if the difference between the sum of the digits in odd positions and the sum of the digits in even positions (from left) is a multiple of 11 (including 0). We apply both tests to each option. A number that passes both tests is divisible by 99. Step-by-Step Solution: 1) Consider 57717. Sum of digits = 5 + 7 + 7 + 1 + 7 = 27, which is a multiple of 9, so it is divisible by 9. 2) For the 11-test, assign positions from the left: digits are 5(1), 7(2), 7(3), 1(4), 7(5). 3) Sum of digits in odd positions (1, 3, 5) = 5 + 7 + 7 = 19. Sum in even positions (2, 4) = 7 + 1 = 8. Difference = 19 − 8 = 11, which is a multiple of 11, so 57717 is divisible by 11. 4) Therefore 57717 is divisible by both 9 and 11, so it is divisible by 99. 5) Quickly check another option, 57627. Sum of digits = 5 + 7 + 6 + 2 + 7 = 27, so it is divisible by 9. 6) For 11-test on 57627: odd positions (1, 3, 5) → 5 + 6 + 7 = 18; even positions (2, 4) → 7 + 2 = 9; difference = 18 − 9 = 9, not a multiple of 11, so 57627 is not divisible by 11 and hence not by 99. Verification / Alternative check: We can confirm by dividing 57717 by 99. First divide 57717 by 9: 57717 / 9 = 6413. Then check 6413 / 11 = 583, which is an integer. Therefore 57717 = 99 * 583, confirming that it is exactly divisible by 99. Similar checks for other options would either fail at 9 or at 11, confirming that they are not divisible by 99. Why Other Options Are Wrong: For 57627, although the digit sum is 27 (divisible by 9), the 11-test fails, so it is not divisible by 99. The number 55162 has digit sum 5 + 5 + 1 + 6 + 2 = 19, which is not a multiple of 9, so it fails immediately. The number 56982 has digit sum 5 + 6 + 9 + 8 + 2 = 30, which is not a multiple of 9 either. Therefore, options b, c, and d are not divisible by 99. Option e, 'None of these', is wrong because we have already found a valid number, 57717. Common Pitfalls: Students sometimes misapply the 11-test by adding all digits instead of forming the alternating sum, or they miscount positions from the right instead of the left. It is also easy to miscalculate digit sums under exam pressure. Carefully assigning positions and computing sums helps avoid such mistakes. Always remember that divisibility by 99 requires divisibility by both 9 and 11, not just one of them. Final Answer: The number that is completely divisible by 99 is 57717.

Question 11

In aptitude inequalities, solve both inequalities together and choose a value of x that satisfies BOTH: 4(2x + 3) > 5 - x and 5x - 3(2x - 7) > 3x - 1.

Accepted Answer

Introduction / Context: This question involves solving a system of two linear inequalities in one variable. You must find values of x that satisfy both inequalities simultaneously and then select the correct value from the options. Such problems test your ability to simplify inequalities, handle brackets, and understand how solution intervals overlap on the real number line. Given Data / Assumptions: First inequality: 4(2x + 3) > 5 − x. Second inequality: 5x − 3(2x − 7) > 3x − 1. x is a real number. We must choose one value of x from the options that satisfies both inequalities. Concept / Approach: We solve each inequality separately by expanding brackets, collecting like terms, and isolating x. Each inequality yields an interval of solutions. The overall solution set is the intersection of these intervals, since both conditions must hold at the same time. Finally, we check the given discrete options and pick the value that lies within this intersection. This method is systematic and avoids guesswork. Step-by-Step Solution: 1) Solve the first inequality: 4(2x + 3) > 5 − x. 2) Expand: 4 * (2x + 3) = 8x + 12. 3) The inequality becomes 8x + 12 > 5 − x. 4) Add x to both sides: 9x + 12 > 5. 5) Subtract 12 from both sides: 9x > −7, so x > −7/9. 6) Now solve the second inequality: 5x − 3(2x − 7) > 3x − 1. 7) Expand the bracket: 5x − 3(2x − 7) = 5x − 6x + 21 = −x + 21. 8) The inequality becomes −x + 21 > 3x − 1. 9) Add x to both sides: 21 > 4x − 1. 10) Add 1 to both sides: 22 > 4x, so x < 22/4 = 11/2. 11) Combine both results: x must satisfy x > −7/9 and x < 11/2, so the solution interval is (−7/9, 11/2). Verification / Alternative check: Now test each option. The interval (−7/9, 11/2) is approximately (−0.78, 5.5). Among the options −6, −1, 5, 6, and 'None of these', only x = 5 lies within this interval. Check x = 5 in the original inequalities. For the first: 4(2 * 5 + 3) = 4(10 + 3) = 4 * 13 = 52, and 5 − 5 = 0, so 52 > 0 is true. For the second: 5 * 5 − 3(2 * 5 − 7) = 25 − 3(10 − 7) = 25 − 9 = 16, and 3 * 5 − 1 = 15 − 1 = 14, so 16 > 14 is also true. Thus x = 5 satisfies both inequalities. Why Other Options Are Wrong: x = −6 and x = −1 are less than −7/9, so they fail the first inequality. For x = 6, which is greater than 11/2, the second inequality fails because it lies outside the allowed interval. Since one valid option exists, 'None of these' is not correct. Therefore, only x = 5 satisfies both inequalities simultaneously. Common Pitfalls: Students sometimes solve each inequality correctly but forget to take the intersection of the solution sets, or they mis-handle the expansion step and get incorrect coefficients. Another common mistake is to reverse the inequality sign when it is not required; in this problem we divide only by positive numbers, so the inequality direction remains the same. Drawing a quick number line with both bounds can help visualize the final interval clearly. Final Answer: The value of x that satisfies both inequalities is 5.

Question 12

In trigonometric identities, express tan(A - B) in terms of tan(A) and tan(B). Select the correct formula for X where X = tan(A - B).

Accepted Answer

Introduction / Context: This question tests your knowledge of trigonometric angle subtraction formulas, specifically the identity for tan(A − B). Being able to express tan(A − B) in terms of tan(A) and tan(B) is essential in solving many trigonometric equations and simplifying expressions, and it also illustrates the structure of tangent addition and subtraction formulas used widely in mathematics. Given Data / Assumptions: A and B are angles for which tan(A) and tan(B) are defined. We must find the correct expression for tan(A − B) in terms of tan(A) and tan(B). Several candidate formulas are provided; exactly one matches the standard identity. Concept / Approach: The standard tangent addition and subtraction identities are: tan(A + B) = (tan A + tan B) / (1 − tan A * tan B) tan(A − B) = (tan A − tan B) / (1 + tan A * tan B) We recall these formulas or derive them from the sine and cosine addition formulas. Then, we match the expression for tan(A − B) to the given options and select the one that agrees with the standard identity. Step-by-Step Solution: 1) Start from the known identity: tan(A − B) = (tan A − tan B) / (1 + tan A * tan B). 2) Compare this general identity with the multiple-choice options. 3) Option a uses tan(A) + tan(B) in the numerator and 1 − tan(A)*tan(B) in the denominator, which corresponds to tan(A + B), not tan(A − B). 4) Option b still uses tan(A) + tan(B) and an incorrect denominator pattern for tan(A − B). 5) Option c has tan(A) − tan(B) in the numerator but 1 − tan(A)*tan(B) in the denominator, which does not match the subtraction identity. 6) Option d, (tan(A) − tan(B)) / (1 + tan(A)*tan(B)), matches exactly the known formula for tan(A − B). 7) Option e has a completely different structure and does not correspond to any standard addition or subtraction formula. Verification / Alternative check: To verify, take specific angles, such as A = 60° and B = 30°. Then tan(A − B) = tan(30°) = 1/√3. Evaluate option d: tan(60°) = √3 and tan(30°) = 1/√3, so the numerator is √3 − 1/√3 and the denominator is 1 + √3 * (1/√3) = 1 + 1 = 2. Numerically simplifying the numerator and denominator yields a result equal to 1/√3, confirming that option d matches tan(A − B). The other options give different values when evaluated with the same angles. Why Other Options Are Wrong: Option a is the identity for tan(A + B), not tan(A − B). Option b incorrectly mixes signs in both the numerator and denominator and does not correspond to any standard identity. Option c uses tan(A) − tan(B) but pairs it with the denominator from tan(A + B), which is mismatched. Option e does not align with any of the standard formulas and behaves very differently for typical angle values. Only option d reflects the correct formula for tan(A − B). Common Pitfalls: A common mistake is to confuse the signs for the addition and subtraction formulas, especially in the denominator. Some learners incorrectly assume that both numerator and denominator simply follow the sign between A and B, which is not the case for tangent. Carefully memorizing or deriving the formula from sine and cosine definitions helps avoid these sign errors and solidifies understanding. Final Answer: The correct expression for tan(A − B) is (tan(A) - tan(B)) / (1 + tan(A)*tan(B)).

Question 13

In trigonometric simplification, find the exact value of sec(330°) using standard trigonometric values and quadrant knowledge.

Accepted Answer

Introduction / Context: This trigonometry question asks you to evaluate sec(330°), which is the reciprocal of cos(330°). To solve it, you need to use your knowledge of standard angles on the unit circle and the signs of trigonometric functions in different quadrants. Such questions commonly appear in aptitude tests and school exams to check conceptual clarity about angles and trigonometric ratios. Given Data / Assumptions: The angle is 330°, measured from the positive x-axis in standard position. We must find sec(330°) exactly, not approximately. Cosine and secant are reciprocal functions: sec(θ) = 1 / cos(θ). We assume standard trigonometric values for special angles like 30°, 45°, and 60° are known. Concept / Approach: First, express 330° as a related reference angle. Since 330° = 360° − 30°, the reference angle is 30°. In the fourth quadrant (between 270° and 360°), cos(θ) is positive and sin(θ) is negative. Therefore, cos(330°) has the same magnitude as cos(30°) but is positive. After finding cos(330°), we take its reciprocal to obtain sec(330°) and then match the result with the given options. Step-by-Step Solution: 1) Recognize that 330° = 360° − 30°, so the reference angle is 30° in the fourth quadrant. 2) In the fourth quadrant, the cosine of an angle is positive. 3) We know that cos(30°) = √3 / 2. 4) Therefore, cos(330°) = cos(360° − 30°) = cos(30°) = √3 / 2. 5) By definition, sec(θ) = 1 / cos(θ). Thus sec(330°) = 1 / (√3 / 2) = 2 / √3. 6) This matches option c exactly. Verification / Alternative check: We can verify sign and magnitude by considering coordinates on the unit circle. The point corresponding to 330° has coordinates (cos 330°, sin 330°). Because it is 30° below the positive x-axis, the x-coordinate is cos 330° = √3 / 2 and the y-coordinate is sin 330° = −1/2. The secant is the reciprocal of the x-coordinate, giving sec 330° = 1 / (√3 / 2) = 2 / √3. This matches the earlier calculation and confirms the correctness of the result. Why Other Options Are Wrong: Option a (2) would correspond to sec(60°) or sec(300°), not sec(330°). Option b (−2) suggests a negative cosine value, which is not correct for an angle in the fourth quadrant where cosine is positive. Option d (−2/√3) has the wrong sign. Option e (√3/2) is actually a cosine value (cos 30°), not a secant value. Only 2/√3 correctly represents sec(330°). Common Pitfalls: A typical error is to confuse 330° with 300° or another nearby angle, leading to the wrong standard value. Another mistake is misremembering which trigonometric functions are positive in each quadrant; for the fourth quadrant, cosine and secant are positive, while sine and tangent are negative. Carefully identifying the reference angle and quadrant before applying standard values helps avoid such errors. Final Answer: The exact value of sec(330°) is 2/√3.

Question 14

Simplify the algebraic expression by factorization: (4a^2 + 12ab + 9b^2) / (2a + 3b). Find the simplified result in lowest algebraic terms.

Accepted Answer

Introduction / Context: This algebra problem asks you to simplify a rational expression by factorization. The numerator is a quadratic expression in a and b, and the denominator is a binomial. Recognizing common patterns such as perfect square trinomials helps you factor quickly and cancel common factors, which is a key skill in algebra and aptitude tests involving algebraic fractions. Given Data / Assumptions: Expression: (4a^2 + 12ab + 9b^2) / (2a + 3b). a and b are real numbers such that the denominator (2a + 3b) is not zero. We must express the result in simplified form after factorization. Concept / Approach: Notice that the numerator 4a^2 + 12ab + 9b^2 resembles the expansion of a squared binomial. In particular, (2a + 3b)^2 = 4a^2 + 12ab + 9b^2. Once we factor the numerator into (2a + 3b)^2, we can simplify the fraction by canceling a common factor of (2a + 3b) in the numerator and denominator, assuming it is not zero. This process is a standard application of factorization and cancellation in rational expressions. Step-by-Step Solution: 1) Consider the numerator: 4a^2 + 12ab + 9b^2. 2) Recognize the pattern of a perfect square: (2a)^2 = 4a^2, 2 * (2a) * (3b) = 12ab, and (3b)^2 = 9b^2. 3) Therefore 4a^2 + 12ab + 9b^2 = (2a + 3b)^2. 4) Substitute into the original expression: (4a^2 + 12ab + 9b^2) / (2a + 3b) = (2a + 3b)^2 / (2a + 3b). 5) Cancel the common factor (2a + 3b) in numerator and denominator, as long as (2a + 3b) ≠ 0. 6) The simplified expression is 2a + 3b. Verification / Alternative check: To verify, choose specific values for a and b, for example a = 1 and b = 1. Then the numerator becomes 4(1)^2 + 12(1)(1) + 9(1)^2 = 4 + 12 + 9 = 25. The denominator is 2(1) + 3(1) = 5. The fraction is 25 / 5 = 5. Evaluating 2a + 3b for a = 1 and b = 1 gives 2(1) + 3(1) = 5. The two results agree, confirming that the simplification is correct. Why Other Options Are Wrong: Option a (2a − 3b) would result if the middle term were negative, as in 4a^2 − 12ab + 9b^2, which is not the case here. Options c (2a) and d (3b) ignore one of the variables and do not align with the factorization. Option e (4a + 9b) comes from incorrectly combining coefficients rather than properly factoring. Only option b, 2a + 3b, matches the factorization of the numerator divided by the denominator. Common Pitfalls: A frequent mistake is misidentifying the perfect square structure or trying to factor the numerator into incorrect binomials. Some students also cancel terms instead of factors, which is algebraically invalid. Always factor the numerator completely first and cancel only common factors that multiply the entire numerator and denominator, not individual terms inside a sum. Final Answer: The simplified value of the expression is 2a + 3b.

Question 15

In coordinate geometry, find the equation of the line with slope -1/2 that passes through the intersection point of the two lines: x - y = -1 and 3x - 2y = 0.

Accepted Answer

Introduction / Context: This coordinate geometry problem combines solving a system of linear equations to find their intersection point with writing the equation of a new line through that point with a specified slope. Such composite questions strengthen your understanding of line equations, intersections, and the slope concept, all of which are fundamental topics in analytic geometry and aptitude tests. Given Data / Assumptions: Lines: x − y = −1 and 3x − 2y = 0. The required line must pass through their intersection. The slope of the required line is −1/2. We must find its equation in standard form. Concept / Approach: First, solve the system of equations x − y = −1 and 3x − 2y = 0 to find their intersection point. Then, use the point-slope form of the equation of a line with the given slope and passing through that point. Finally, convert the resulting equation into standard form and compare with the answer choices. This two-step approach ensures both the correct point and correct slope are accounted for. Step-by-Step Solution: 1) Solve x − y = −1 for y: y = x + 1. 2) Substitute y = x + 1 into 3x − 2y = 0: 3x − 2(x + 1) = 0. 3) Expand: 3x − 2x − 2 = 0 → x − 2 = 0 → x = 2. 4) Substitute x = 2 into y = x + 1: y = 2 + 1 = 3. 5) The intersection point is (2, 3). 6) The required line has slope m = −1/2 and passes through (2, 3). 7) Use point-slope form: y − 3 = (−1/2)(x − 2). 8) Multiply both sides by 2: 2(y − 3) = −(x − 2). 9) Expand: 2y − 6 = −x + 2. 10) Rearrange to standard form: x + 2y − 8 = 0 or x + 2y = 8. Verification / Alternative check: Check that the point (2, 3) satisfies x + 2y = 8. Substituting, 2 + 2 * 3 = 2 + 6 = 8, which is correct. The slope of x + 2y = 8 can be found by writing it as y = −(1/2)x + 4, which clearly has slope −1/2. Both the point and the slope match the problem requirements, confirming that x + 2y = 8 is the correct equation. Why Other Options Are Wrong: Option b, 3x + y = 7, rearranges to y = −3x + 7, which has slope −3, not −1/2. Option c, x + 2y = −8, and option d, 3x + y = −7, both have incorrect intercepts and do not pass through (2, 3). Option e, y = −x/2, passes through the origin and has the correct slope but does not pass through the intersection point (2, 3). Only option a, x + 2y = 8, satisfies both conditions. Common Pitfalls: Errors commonly occur in solving the original system of equations, such as mis substituting y or algebraic sign mistakes. Another frequent issue is mis rearranging the point-slope form to standard form or misreading the slope from the final equation. Carefully solving for the intersection and double-checking both slope and point inclusion will prevent these errors. Final Answer: The equation of the line is x + 2y = 8.

Question 16

Find the coefficient of x^2 after expanding and simplifying the polynomial: (x + 9)(6 - 4x)(4x - 7).

Accepted Answer

Introduction / Context: This algebra question asks you to find the coefficient of x^2 in the expansion of a product of three binomials. Instead of fully expanding to get every term, you can strategically multiply and track only the contributions to the x^2 term. This technique is useful in polynomial algebra, where full expansion can be time consuming, especially in aptitude tests with limited time. Given Data / Assumptions: Expression: (x + 9)(6 − 4x)(4x − 7). We must expand and simplify, then identify the coefficient of x^2. x is a real variable. Concept / Approach: First, multiply the two binomials (6 − 4x) and (4x − 7) to get a quadratic in x. Then multiply that quadratic by (x + 9). During the second multiplication, we can focus on terms that produce x^2 in the final product. This reduces the amount of work compared with finding the entire expanded expression. Collecting all x^2 contributions yields the desired coefficient directly. Step-by-Step Solution: 1) Multiply (6 − 4x)(4x − 7) first. 2) Compute 6 * (4x − 7) = 24x − 42. 3) Compute −4x * (4x − 7) = −16x^2 + 28x. 4) Add these to get (6 − 4x)(4x − 7) = −16x^2 + (24x + 28x) − 42 = −16x^2 + 52x − 42. 5) Now multiply (x + 9) by this quadratic: (x + 9)(−16x^2 + 52x − 42). 6) Multiply x by the quadratic: x * (−16x^2 + 52x − 42) = −16x^3 + 52x^2 − 42x. 7) Multiply 9 by the quadratic: 9 * (−16x^2 + 52x − 42) = −144x^2 + 468x − 378. 8) Combine like terms. For x^2, the total coefficient is 52x^2 − 144x^2 = −92x^2. 9) Therefore, the coefficient of x^2 in the full expansion is −92. Verification / Alternative check: As a check, you can use a symbolic expansion approach or re multiply while tracking terms more briefly. Since any x^2 term in the final product arises either from x times the x^2 term in the quadratic or from 9 times the x^2 term in that quadratic, our calculation already accounts for all such contributions. We do not get x^2 terms from combinations that involve three x factors (which produce x^3) or constant terms. Carefully re checking the intermediate quadratic and both partial products confirms the coefficient −92 is consistent. Why Other Options Are Wrong: Options a (216) and b (108) are large positive numbers and can arise from mis tracking x^3 or constant terms, but they do not match the correct combined x^2 coefficient. Option d (−4) and option e (92) result from partial or sign errors, such as using 52 − 48 or incorrectly adding absolute values. Only −92 is consistent with correct multiplication and term collection. Common Pitfalls: A common error is to expand all three factors at once or to multiply in a rushed manner, leading to sign mistakes or missing terms. Some students also forget that x times 52x contributes to x^2, while 9 times −16x^2 also contributes, and they may overlook one of these. Breaking the problem into two clear multiplication steps and systematically collecting only x^2 terms helps prevent these mistakes. Final Answer: The coefficient of x^2 in the expansion of (x + 9)(6 − 4x)(4x − 7) is -92.

Question 17

In aptitude inequalities, solve the chain inequality correctly and choose a value of x that satisfies BOTH conditions: 5x - 3(2x - 7) > 3x - 1 and 3x - 1 < 7 + 4x.

Accepted Answer

Introduction / Context: This question involves a pair of linear inequalities in one variable that must both be satisfied. Such chain inequalities are common in aptitude tests and check your ability to simplify inequalities, distribute correctly, and combine solution sets using interval intersection. You must then pick a specific value of x from the options that lies within the common solution region. Given Data / Assumptions: First inequality: 5x − 3(2x − 7) > 3x − 1. Second inequality: 3x − 1 < 7 + 4x. x is a real number. We must choose a value of x that satisfies both inequalities simultaneously. Concept / Approach: We simplify each inequality step by step to isolate x. The first inequality will give an upper bound or lower bound on x, and the second inequality will give another bound. The intersection of these bounds determines the full solution set. Finally, we test each option to see which values lie in this intersection and satisfy both inequalities together. Step-by-Step Solution: 1) Solve the first inequality: 5x − 3(2x − 7) > 3x − 1. 2) Expand the bracket: 5x − 3(2x − 7) = 5x − 6x + 21 = −x + 21. 3) The inequality becomes −x + 21 > 3x − 1. 4) Add x to both sides: 21 > 4x − 1. 5) Add 1 to both sides: 22 > 4x, so x < 22 / 4 = 11/2. 6) Now solve the second inequality: 3x − 1 < 7 + 4x. 7) Subtract 3x from both sides: −1 < 7 + x. 8) Subtract 7 from both sides: −8 < x, which means x > −8. 9) Combine both results: −8 < x and x < 11/2, so the solution interval is (−8, 11/2). Verification / Alternative check: Now evaluate each option against the interval (−8, 11/2), which is approximately (−8, 5.5). Among the options 6, 9, −6, −9, and 'None of these', only x = −6 lies within the interval. Test x = −6 in both inequalities. First: 5(−6) − 3(2(−6) − 7) = −30 − 3(−12 − 7) = −30 − 3(−19) = −30 + 57 = 27, and the right side is 3(−6) − 1 = −18 − 1 = −19, so 27 > −19, which is true. Second: 3(−6) − 1 = −18 − 1 = −19, and 7 + 4(−6) = 7 − 24 = −17, so −19 < −17, which is also true. Thus x = −6 satisfies both inequalities. Why Other Options Are Wrong: Options 6 and 9 are greater than 11/2 (5.5) and therefore violate the first inequality, which requires x < 11/2. Option −9 is less than −8 and violates the second inequality, which requires x > −8. Since we have found that x = −6 is a valid solution, the option 'None of these' is incorrect. Only x = −6 lies in the intersection and satisfies both inequalities. Common Pitfalls: Common mistakes include sign errors when expanding −3(2x − 7), forgetting to change inequality signs when appropriate (though in this problem we never divide by a negative number), and failing to correctly interpret the combined inequality as an intersection of intervals. Drawing a quick number line and marking both bounds helps visualise the final solution and can prevent misreading of the interval endpoints. Final Answer: The value of x that satisfies both inequalities is -6.

Question 18

Simplify the trigonometric expression: (sec(A) - 1) / (sec(A) + 1) and choose an equivalent expression in terms of sin(A) or cos(A).

Accepted Answer

Introduction / Context: This trigonometric simplification question asks you to rewrite the expression (sec(A) − 1) / (sec(A) + 1) in terms of sine or cosine. Simplifying such expressions using algebraic manipulation and fundamental identities is a common task in trigonometry and aptitude exams. The goal is to recognize how to convert secant into cosine and then simplify the resulting fraction to match one of the given equivalent forms. Given Data / Assumptions: Expression: (sec(A) − 1) / (sec(A) + 1). A is an angle for which sec(A) is defined and sec(A) + 1 ≠ 0. We must express this fraction in terms of sin(A) or cos(A) and match it to one of the options. Concept / Approach: We start by expressing sec(A) in terms of cos(A) using sec(A) = 1 / cos(A). Then we simplify the resulting expression. A useful technique is to multiply the numerator and denominator by (sec(A) − 1), creating a difference of squares in the denominator. After simplification, we convert the result into a form involving cosine and possibly sine by using the identity sin^2(A) + cos^2(A) = 1. Finally, we compare the simplified form with the options to identify the correct match. Step-by-Step Solution: 1) Let E = (sec(A) − 1) / (sec(A) + 1). 2) Multiply numerator and denominator by (sec(A) − 1) to simplify: E = (sec(A) − 1)^2 / (sec^2(A) − 1). 3) Use the identity sec^2(A) − 1 = tan^2(A). 4) So E = (sec(A) − 1)^2 / tan^2(A). 5) Now write sec(A) and tan(A) in terms of sin(A) and cos(A): sec(A) = 1 / cos(A) and tan(A) = sin(A) / cos(A). 6) Then (sec(A) − 1) = (1 / cos(A)) − 1 = (1 − cos(A)) / cos(A). 7) Thus (sec(A) − 1)^2 = (1 − cos(A))^2 / cos^2(A). 8) Also tan^2(A) = sin^2(A) / cos^2(A). 9) Substitute into E: E = [(1 − cos(A))^2 / cos^2(A)] / [sin^2(A) / cos^2(A)] = (1 − cos(A))^2 / sin^2(A). 10) Use sin^2(A) = 1 − cos^2(A) = (1 − cos(A))(1 + cos(A)) to get: E = (1 − cos(A))^2 / [(1 − cos(A))(1 + cos(A))] = (1 − cos(A)) / (1 + cos(A)). Verification / Alternative check: Pick a convenient angle, for example A = 60°. Then sec(60°) = 2, so the original expression (sec(A) − 1) / (sec(A) + 1) becomes (2 − 1) / (2 + 1) = 1 / 3. Now evaluate (1 − cos(A)) / (1 + cos(A)) at A = 60°. Here cos(60°) = 1/2, so (1 − 1/2) / (1 + 1/2) = (1/2) / (3/2) = 1/3, which matches the original value. This numerical check supports the algebraic simplification. Why Other Options Are Wrong: Option a, (1 − sin(A)) / (1 + sin(A)), gives a different value when tested with A = 60°, as does option c, (1 + sin(A)) / (1 − sin(A)). Option b, (1 + cos(A)) / (1 − cos(A)), is the reciprocal of the correct expression. Option e, sin(A) / cos(A), is simply tan(A) and does not match the simplified result. Only option d, (1 − cos(A)) / (1 + cos(A)), matches the derived form exactly. Common Pitfalls: Students sometimes try to simplify directly without multiplying by the conjugate, which can make the algebra messy. Another common mistake is to cancel terms incorrectly instead of factoring and canceling common factors. Misusing the identity sin^2(A) + cos^2(A) = 1 is also frequent. Following a structured approach—rewrite in terms of cos(A), use the difference of squares, and apply the Pythagorean identity—helps avoid these errors. Final Answer: The expression (sec(A) − 1) / (sec(A) + 1) simplifies to (1 - cos(A)) / (1 + cos(A)).

Question 19

In trigonometry, evaluate the exact value of tan(7π/6) using reference angles and quadrant signs.

Accepted Answer

Introduction / Context: This trigonometric evaluation problem asks you to find the exact value of tan(7π/6). To solve it, you must recognize 7π/6 as a standard angle on the unit circle and use reference angles and quadrant information to determine the correct trigonometric value and sign. Such questions are typical in trigonometry courses and aptitude exams to test understanding of radian measure and unit circle properties. Given Data / Assumptions: Angle: 7π/6 radians. We must find tan(7π/6) exactly. Standard special angles in radians, such as π/6, π/4, and π/3, and their tangent values are assumed known. Concept / Approach: First, express 7π/6 in terms of π to identify its position on the unit circle. Since 7π/6 = π + π/6, it lies in the third quadrant with reference angle π/6. In the third quadrant, both sine and cosine are negative, so tangent, as sin(θ)/cos(θ), is positive. Therefore, tan(7π/6) has the same magnitude as tan(π/6) but is positive. Using the known value of tan(π/6), we obtain the exact value of tan(7π/6). Step-by-Step Solution: 1) Write 7π/6 as π + π/6. This shows that 7π/6 is π/6 beyond π. 2) When an angle is written as π + α, it lies in the third quadrant and has reference angle α. 3) Therefore, the reference angle for 7π/6 is π/6. 4) In the third quadrant, both sine and cosine are negative, but their ratio, tangent, is positive. 5) We know the special angle value tan(π/6) = 1 / √3. 6) Since tan(π + α) = tan(α) for any angle α, we get tan(7π/6) = tan(π + π/6) = tan(π/6) = 1 / √3. Verification / Alternative check: You can cross-check using the unit circle. The coordinates for angle π/6 are (√3 / 2, 1 / 2). For angle 7π/6 = π + π/6, both coordinates become negative: (−√3 / 2, −1 / 2). Tangent is given by y / x, so tan(7π/6) = (−1/2) / (−√3/2) = 1 / √3, confirming the earlier result. This verifies both the magnitude and the sign of the tangent value. Why Other Options Are Wrong: Option b (−1/√3) would be correct for a fourth quadrant angle with reference π/6, such as 11π/6, where tangent is negative. Option c (√3) and option d (−√3) correspond to reference angles of π/3 in different quadrants, not π/6. Option e (0) would be the tangent of angles like 0, π, or 2π, where the sine is zero. None of these match the situation for 7π/6, which clearly has a positive tangent of magnitude 1/√3. Common Pitfalls: A common error is misidentifying the quadrant for 7π/6 or confusing it with another angle such as 5π/6 or 11π/6. Another mistake is forgetting that tangent repeats every π radians, leading some students to compute tan(7π/6) incorrectly. Remembering that tan(π + α) = tan(α) and confirming the quadrant sign are key to avoiding such errors. Final Answer: The exact value of tan(7π/6) is 1/√3.

Question 20

In coordinate geometry with reflections, a point R(a, b) is first reflected in the origin to get R₁, and then R₁ is reflected in the x-axis to become the point (-5, 1). Find the coordinates of the original point R.

Accepted Answer

Introduction / Context: This coordinate geometry question involves successive reflections of a point in the origin and then in the x-axis. You are given the final image after the two reflections and must determine the original point. Understanding how reflections change coordinates is important for geometry, transformations, and many aptitude-style reasoning problems involving symmetry. Given Data / Assumptions: Original point: R(a, b). Reflection of R in the origin gives R₁. Reflection of R₁ in the x-axis gives the point (−5, 1). We assume standard reflection rules in the Cartesian plane. Concept / Approach: Reflection in the origin transforms a point (x, y) to (−x, −y). Reflection in the x-axis transforms (x, y) to (x, −y). We apply these transformations step by step starting from R(a, b). First, we obtain R₁ by reflecting in the origin. Then we reflect R₁ in the x-axis to match the given final coordinates, and we solve for a and b. This reverse engineering process from the final point leads to the original coordinates. Step-by-Step Solution: 1) Let the original point be R(a, b). 2) Reflect R in the origin to get R₁. Reflection in the origin changes (x, y) to (−x, −y). Thus R₁ = (−a, −b). 3) Reflect R₁ in the x-axis. Reflection in the x-axis changes (x, y) to (x, −y). So the reflection of (−a, −b) becomes (−a, b). 4) We are told this final point equals (−5, 1). Therefore, (−a, b) = (−5, 1). 5) Equate coordinates: −a = −5 and b = 1. 6) From −a = −5, we get a = 5. 7) Therefore the original point R is (a, b) = (5, 1). Verification / Alternative check: Check the transformations explicitly using R(5, 1). Reflecting R in the origin gives R₁ = (−5, −1). Reflecting this in the x-axis changes (−5, −1) to (−5, 1). This matches the given final point, confirming that R(5, 1) is correct. No other option, when processed through the same two reflections, produces (−5, 1). Why Other Options Are Wrong: Option a, (5, −1), reflected in the origin becomes (−5, 1), and reflecting that in the x-axis gives (−5, −1), which is not the given final point. Option b, (−1, 5), and option c, (1, −5), similarly produce different final points after the two reflections. Option e, (−5, 1), is the final image itself, not the original R. Only option d, (5, 1), yields (−5, 1) after the prescribed sequence of reflections. Common Pitfalls: A common mistake is reversing the order of reflections or mixing up the rules for each reflection. For example, some students mistakenly think reflection in the origin is the same as reflection in the x-axis or y-axis alone, which is not true. Another error is to try to guess the answer by visual intuition instead of applying the coordinate rules. Systematically applying the transformation formulas avoids these issues. Final Answer: The original point R has coordinates (5, 1).

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