Question 1

Solve a simple linear equation: find the value of x for which the two linear expressions 5x + 17 and 17x − 5 are equal to each other.

Accepted Answer

Introduction / Context: This is a straightforward algebra question involving a linear equation in one variable. You are given two linear expressions, 5x + 17 and 17x − 5, and asked to find the value of x that makes them equal. Such problems are among the most basic in algebra and are often used early in aptitude and entrance exams to test comfort with solving simple equations. Given Data / Assumptions: The two expressions are 5x + 17 and 17x − 5. We are told they are equal for a particular value of x. We must find that value of x. x is a real number. Concept / Approach: When two expressions are equal, we can set them equal and solve the resulting equation. Since both sides are linear in x, the equation will have a single solution. The key steps are to collect like terms and isolate x on one side of the equation using basic algebraic operations such as addition, subtraction, and division by a non zero constant. Step-by-Step Solution: Step 1: Set the expressions equal: 5x + 17 = 17x − 5.Step 2: Bring all x terms to one side by subtracting 5x from both sides: 17x − 5x − 5 = 17.Step 3: Simplify the x terms: 12x − 5 = 17.Step 4: Add 5 to both sides to isolate the term in x: 12x = 22.Step 5: Divide both sides by 12: x = 22/12.Step 6: Simplify the fraction by dividing numerator and denominator by 2: x = 11/6. Verification / Alternative check: Substitute x = 11/6 back into both expressions. For 5x + 17, we obtain 5 × 11/6 + 17 = 55/6 + 102/6 = 157/6. For 17x − 5, we have 17 × 11/6 − 5 = 187/6 − 30/6 = 157/6. Since both expressions give the same value when x = 11/6, this verifies that our solution is correct. Why Other Options Are Wrong: The values −11/6 and ±6/11 are obtained by mishandling signs or inverting the fraction improperly during solving. For example, taking 12x = 22 and solving as x = 12/22 gives 6/11, which is incorrect. The option 1 does not satisfy the original equation, since 5(1) + 17 = 22 and 17(1) − 5 = 12. Only x = 11/6 makes both sides equal. Common Pitfalls: Typical errors include forgetting to subtract 5x from both sides, misplacing signs when moving terms, or not simplifying the final fraction to lowest terms. Writing down intermediate steps clearly and checking each arithmetic operation helps avoid these mistakes. Always verify the solution by substituting it back into the original expressions. Final Answer: The value of x that makes 5x + 17 equal to 17x − 5 is 11/6.

Question 2

In an algebraic simplification problem, evaluate the expression obtained when 111 a^2 b^2 c^2 is divided by 37 a^2, and express the result in its simplest form.

Accepted Answer

Introduction / Context: This question checks your ability to simplify algebraic expressions that involve coefficients and powers of variables. The expression 111 a^2 b^2 c^2 divided by 37 a^2 requires you to handle both numerical division and cancellation of powers of a common variable. Such simplification problems are common in basic algebra and aptitude tests, where quick manipulation is important. Given Data / Assumptions: The original expression is 111 a^2 b^2 c^2. This expression is divided by 37 a^2. a, b, and c are assumed to be nonzero real numbers so that division by a^2 is valid. We must simplify the resulting expression completely. Concept / Approach: The key ideas are to divide coefficients (numerical parts) and to subtract exponents of the same base when dividing powers. For the coefficient, 111 divided by 37 should be simplified. For the variables, a^2 in the numerator divided by a^2 in the denominator results in a^0, which equals 1. The powers of b and c have no counterparts in the denominator and so remain unchanged. Combining these observations produces a very simple final expression. Step-by-Step Solution: Step 1: Write the division explicitly as a fraction: (111 a^2 b^2 c^2) / (37 a^2).Step 2: Separate the numerical part and the variable part: (111/37) × (a^2/a^2) × b^2 c^2.Step 3: Simplify the numerical coefficient: 111/37 = 3, because 111 = 3 × 37.Step 4: Simplify the powers of a using the rule a^m / a^n = a^(m − n): a^2/a^2 = a^0 = 1.Step 5: The remaining variables are b^2 and c^2, which are unchanged.Step 6: Therefore the expression simplifies to 3 × 1 × b^2 c^2 = 3b^2c^2. Verification / Alternative check: To double check, substitute simple nonzero values for the variables. For example, let a = 1, b = 2, and c = 3. The original numerator is 111 × 1^2 × 2^2 × 3^2 = 111 × 4 × 9 = 3996. The denominator is 37 × 1^2 = 37. Then 3996/37 = 108. Now compute 3b^2c^2 with the same values: 3 × 4 × 9 = 108. Since both methods give 108, the simplification is correct. Why Other Options Are Wrong: The expressions 2c^2 and 2b^2 ignore proper simplification of the coefficient and drop one of the variable factors. The option 3 omits b^2 and c^2 entirely and is incomplete. The option 3bc^2 keeps only one power of b and thus fails to match the original exponent b^2. Only 3b^2c^2 retains the correct powers of b and c while using the proper numerical coefficient after division by 37 a^2. Common Pitfalls: Students sometimes forget that powers subtract when dividing or mistakenly add exponents instead. Another frequent mistake is not simplifying the fraction 111/37 or miscalculating it. Carefully separating coefficient simplification from variable exponent rules prevents these errors. Always check that the final expression preserves the correct powers for each variable from the original term. Final Answer: The simplified result of dividing 111 a^2 b^2 c^2 by 37 a^2 is 3b^2c^2.

Question 3

Consider the compound inequality 5x − 4 ≤ 2 − x and 4x + 5 > 2x − 5. Determine which of the given values of x satisfies both inequalities simultaneously.

Accepted Answer

Introduction / Context: This question assesses your ability to solve and interpret compound linear inequalities. Instead of solving only one inequality, you have to find values of x that satisfy two conditions at the same time. Such problems are common in aptitude tests, as they combine algebraic manipulation with logical reasoning about intervals on the number line. Given Data / Assumptions: The first inequality is 5x − 4 ≤ 2 − x. The second inequality is 4x + 5 > 2x − 5. x is a real number. We must identify which option among the given choices satisfies both inequalities. Concept / Approach: We solve each inequality separately to obtain intervals for x. The solution to the compound inequality is the intersection (overlap) of these intervals. Once we know the combined range of x that satisfies both conditions, we test the given options to see which one lies within that range. Working systematically avoids errors and makes the reasoning clear. Step-by-Step Solution: Step 1: Solve the first inequality 5x − 4 ≤ 2 − x.Step 2: Add x to both sides: 6x − 4 ≤ 2.Step 3: Add 4 to both sides: 6x ≤ 6.Step 4: Divide by 6 to obtain x ≤ 1.Step 5: Now solve the second inequality 4x + 5 > 2x − 5.Step 6: Subtract 2x from both sides: 2x + 5 > −5.Step 7: Subtract 5 from both sides: 2x > −10.Step 8: Divide by 2: x > −5.Step 9: The combined solution is −5 < x ≤ 1. Verification / Alternative check: Now check each option. For x = 3, the first inequality fails because 5(3) − 4 = 11 is not less than or equal to 2 − 3 = −1. For x = 6 and x = −7, the inequalities also do not satisfy both conditions. For x = 0, both inequalities hold, but we must also test the options given in the problem. Among the listed options, x = −1 clearly lies between −5 and 1. Substituting x = −1 in the first inequality gives 5(−1) − 4 = −9 and 2 − (−1) = 3, so −9 ≤ 3 is true. In the second inequality, 4(−1) + 5 = 1 and 2(−1) − 5 = −7, so 1 > −7 is also true. Thus x = −1 satisfies both inequalities. Why Other Options Are Wrong: The value 3 is greater than 1 and therefore fails the first inequality. The value 6 is also outside the combined range. The value −7 is less than −5 and fails the second inequality because it does not satisfy x > −5. The value 0 does satisfy both inequalities, but if it is not among the provided options, it cannot be chosen. Among the listed choices, only −1 lies within the interval −5 < x ≤ 1 and satisfies both inequalities. Common Pitfalls: One common mistake is to solve only one inequality and forget to intersect the solution sets. Another error is reversing the inequality sign incorrectly when multiplying or dividing by negative numbers. Writing down each step and sketching the intervals on a number line can help you visualise the overlap and avoid mistakes when combining the solutions. Final Answer: The value of x that satisfies both inequalities is −1.

Question 4

The centroid of triangle ABC has coordinates (3, −2). If vertices A and B are at A(−2, 5) and B(6, −2) respectively, determine the exact coordinates of vertex C.

Accepted Answer

Introduction / Context: This coordinate geometry question uses the formula for the centroid of a triangle. The centroid is the point where the three medians intersect and is located at the average of the x coordinates and the average of the y coordinates of the three vertices. Given the centroid and two vertex coordinates, you can work backwards to find the third vertex. This idea is useful in analytic geometry and appears frequently in exam problems. Given Data / Assumptions: Triangle ABC has centroid G with coordinates (3, −2). Vertex A is at (−2, 5). Vertex B is at (6, −2). Vertex C has coordinates (x, y) which we need to determine. All coordinates are real numbers. Concept / Approach: The centroid G of triangle ABC with vertices A(x1, y1), B(x2, y2), and C(x3, y3) is given by G = ( (x1 + x2 + x3)/3 , (y1 + y2 + y3)/3 ). Here, G is known, and two vertices are known, so x3 and y3 can be solved by reversing the centroid formulas. We set up two equations, one for the x coordinates and one for the y coordinates, and solve them for the unknown coordinates of vertex C. Step-by-Step Solution: Step 1: Let C have coordinates (x, y).Step 2: Apply the centroid formula for the x coordinate: (xA + xB + xC)/3 = 3.Step 3: Substitute xA = −2 and xB = 6 to get (−2 + 6 + x)/3 = 3.Step 4: Simplify the numerator: (4 + x)/3 = 3, so 4 + x = 9 and x = 5.Step 5: Now use the centroid formula for the y coordinate: (yA + yB + yC)/3 = −2.Step 6: Substitute yA = 5 and yB = −2 to obtain (5 − 2 + y)/3 = −2.Step 7: Simplify: (3 + y)/3 = −2, so 3 + y = −6 and y = −9.Step 8: Thus C has coordinates (5, −9). Verification / Alternative check: Check by recomputing the centroid using A(−2, 5), B(6, −2), and C(5, −9). The average of the x coordinates is (−2 + 6 + 5)/3 = 9/3 = 3. The average of the y coordinates is (5 − 2 − 9)/3 = (−6)/3 = −2. These match the given centroid coordinates, confirming that C(5, −9) is correct. Why Other Options Are Wrong: Points such as (−5, −9), (5, 9), and (−5, 9) produce different averages for x or y when used in the centroid formula, so they cannot give G(3, −2). The point (3, −2) is the centroid itself rather than a vertex and therefore cannot be vertex C of triangle ABC. Only the coordinate (5, −9) reproduces the correct centroid when combined with A and B. Common Pitfalls: Some students confuse the centroid formula with the midpoint formula, averaging only two points instead of three. Others incorrectly multiply or divide by 3 when rearranging equations. Writing the centroid equations carefully for x and y separately and solving step by step helps avoid arithmetic mistakes and ensures the correct coordinates for the missing vertex. Final Answer: The coordinates of vertex C are (5, −9).

Question 5

A three dimensional solid is a triangular prism that has a total of 9 edges. Based on the properties of prisms, determine how many vertices this triangular prism has.

Accepted Answer

Introduction / Context: This geometry question is about basic properties of three dimensional solids, specifically a triangular prism. A triangular prism has two congruent triangular faces connected by three rectangular faces. Understanding the relationship between faces, edges, and vertices in common solids is important in spatial reasoning and is often tested in aptitude and school level exams. Given Data / Assumptions: The solid in question is a triangular prism. The prism has 9 edges. We are asked to find the total number of vertices. The prism is assumed to be a simple, right or oblique, triangular prism without any truncation. Concept / Approach: A triangular prism can be visualised as a triangle extruded along a straight line. It has two triangular bases and three rectangular lateral faces. Each triangular base has 3 vertices and 3 edges. The corresponding vertices of the two triangles are connected by 3 additional edges along the length of the prism. Counting these features carefully allows us to determine the total number of vertices, edges, and faces. Step-by-Step Solution: Step 1: Consider one triangular base. A triangle has 3 vertices and 3 edges.Step 2: Since the prism has two congruent triangular bases, the total number of vertices from both triangles is 3 + 3 = 6.Step 3: The edges on the two bases contribute 3 edges on the top triangle and 3 edges on the bottom triangle, giving 6 base edges.Step 4: The remaining edges of the prism connect corresponding vertices of the two triangles. There are 3 such connecting edges.Step 5: Therefore, the total number of edges is 3 + 3 + 3 = 9, which matches the given information.Step 6: We have already counted the number of vertices as 3 on each triangular base, so the total number of vertices is 6. Verification / Alternative check: We can also recall the general structure of a triangular prism: it always has 5 faces (2 triangular and 3 rectangular), 9 edges, and 6 vertices. This is a standard fact from solid geometry. Since the problem statement confirms that the solid has 9 edges, we can safely apply this known property to conclude that it has 6 vertices. Why Other Options Are Wrong: Eight vertices would be appropriate for a rectangular prism (a cuboid), not a triangular prism. Twelve vertices correspond to more complex polyhedra with additional vertices, not to a simple triangular prism. Ten and four vertices do not match any standard prism built on a triangle. Only six vertices are consistent with a prism whose base is a triangle and which has exactly 9 edges. Common Pitfalls: Some learners confuse triangular prisms with rectangular prisms or cubes and incorrectly assume 8 vertices. Others miscount by forgetting that each vertex on one triangular base has exactly one corresponding vertex on the other base. Drawing a quick sketch of a triangular prism and marking all vertices and edges is an effective way to prevent mistakes and confirm the counts. Final Answer: The triangular prism with 9 edges has 6 vertices.

Question 6

The diagonal of a square is equal in length to the side of an equilateral triangle. If the area of the square is 12 cm^2, find the exact area of the equilateral triangle in square centimetres.

Accepted Answer

Introduction / Context: This geometry question links the properties of a square and an equilateral triangle through a shared length. The diagonal of the square is given to be equal to the side of the equilateral triangle. Knowing the area of the square, you must determine the area of the equilateral triangle. This problem tests understanding of area formulas, the relationship between side and diagonal in a square, and the area formula for an equilateral triangle. Given Data / Assumptions: The area of the square is 12 cm^2. The diagonal of the square equals the side length of an equilateral triangle. We need the area of the equilateral triangle in exact form. All lengths are in centimetres and areas in square centimetres. Concept / Approach: First, use the area of the square to find its side length. Then use the relation between the side and diagonal of a square: diagonal = side × √2. This diagonal is given to be the side length of the equilateral triangle. Once we know that side, we use the area formula for an equilateral triangle, which is (√3/4) × side^2. Finally, we simplify the surds to obtain the exact area in terms of √3. Step-by-Step Solution: Step 1: Let the side of the square be s cm.Step 2: The area of the square is s^2 and is given as 12, so s^2 = 12.Step 3: Therefore s = √12 = 2√3 cm.Step 4: The diagonal d of the square is s√2, so d = 2√3 × √2 = 2√6 cm.Step 5: This diagonal equals the side of the equilateral triangle, so the side length a of the equilateral triangle is a = 2√6 cm.Step 6: The area of an equilateral triangle is (√3/4) × a^2.Step 7: Compute a^2: a^2 = (2√6)^2 = 4 × 6 = 24.Step 8: Substitute into the area formula: Area = (√3/4) × 24 = 6√3 cm^2. Verification / Alternative check: As a quick check, you can compute approximate decimal values. Since √3 is about 1.732, 6√3 is about 10.392. The square has area 12 cm^2, so it is reasonable that the equilateral triangle whose side equals the diagonal of the square has slightly smaller area than the square, because its shape is different even though one of its sides is longer than the side of the square. This rough comparison supports the reasonableness of the result. Why Other Options Are Wrong: The value 12√3 cm^2 is double the correct area and would arise if a^2 were incorrectly taken as 48 instead of 24. The value 12√2 cm^2 belongs to a different configuration and does not follow from the equilateral triangle area formula. The value 24 cm^2 ignores the square root factor and is too large. The value 3√3 cm^2 is exactly half of the correct area. Only 6√3 cm^2 matches the correct step by step computation. Common Pitfalls: Learners sometimes confuse the diagonal formula and mistakenly use s^2 instead of s√2. Another error is forgetting to square both 2 and √6 when computing a^2, which leads to wrong coefficients. Misremembering the area formula for an equilateral triangle is also common. Keeping these formulas clear and substituting carefully prevents such mistakes. Final Answer: The area of the equilateral triangle is 6√3 cm^2.

Question 7

Evaluate the exact value of x if sin(−4π/3) = x, working carefully with signs and the position of the angle on the unit circle.

Accepted Answer

Introduction / Context: This trigonometry question asks you to evaluate the sine of a negative angle measured in radians, specifically sin(−4π/3). To do this correctly, you need to understand the periodicity and symmetry of the sine function and be comfortable working with angles on the unit circle. Negative angles are measured clockwise, which can cause confusion if the reference angle and quadrant are not identified clearly. Given Data / Assumptions: The angle is −4π/3 radians. We are told that sin(−4π/3) = x and asked to find x. Trigonometric functions are understood in terms of the unit circle. We assume standard properties of sine for real angles. Concept / Approach: The sine function is odd, which means sin(−θ) = −sin θ for any real angle θ. Therefore, we can first find sin(4π/3) and then take the negative of that value to obtain sin(−4π/3). The angle 4π/3 lies in the third quadrant, where sine is negative. By identifying its reference angle and using known values for special angles, we can determine sin(4π/3) exactly. Step-by-Step Solution: Step 1: Use the identity sin(−θ) = −sin θ.Step 2: Therefore sin(−4π/3) = −sin(4π/3).Step 3: Consider the angle 4π/3 radians. It is equal to 240°, which lies in the third quadrant.Step 4: The reference angle for 4π/3 is 4π/3 − π = π/3 (that is, 60°).Step 5: In the third quadrant, sine is negative and has the same magnitude as sin(π/3) at this reference angle.Step 6: We know sin(π/3) = √3/2, so sin(4π/3) = −√3/2.Step 7: Now use the odd function property: sin(−4π/3) = −sin(4π/3) = −(−√3/2) = √3/2. Verification / Alternative check: To check the sign, note that −4π/3 can also be represented by adding 2π: −4π/3 + 2π = 2π/3. So sin(−4π/3) = sin(2π/3). The angle 2π/3 equals 120°, which is in the second quadrant where sine is positive. Its reference angle is again π/3, so sin(2π/3) = sin(π/3) = √3/2. This alternate viewpoint confirms that sin(−4π/3) = √3/2. Why Other Options Are Wrong: The value −√3/2 corresponds to sin(4π/3), not sin(−4π/3), and has the opposite sign. The options −2 and √2 are outside the range of sine, which is limited to values between −1 and 1 inclusive. The value −2/√3 is also outside the valid range for sine. Only √3/2 lies within the correct range and matches the unit circle analysis for sin(−4π/3). Common Pitfalls: A frequent mistake is to forget that sine is an odd function and to assume sin(−θ) = sin θ. Another error is to misidentify the quadrant for 4π/3 or 2π/3, which leads to incorrect signs. Confusing degrees and radians can also cause errors. Working systematically with reference angles and quadrant signs helps ensure the correct exact value. Final Answer: The exact value of x in sin(−4π/3) = x is √3/2.

Question 8

When a number is increased by 28, the result becomes 107% of the original number. Using percentage and simple algebra, find the original number.

Accepted Answer

Introduction / Context: This arithmetic and percentage question is typical in quantitative aptitude tests. You are told that when a certain number is increased by a fixed amount, the new value is 107 percent of the original number. From this information, you must work backwards to determine the original number. This type of question reinforces the connection between percentage increases and algebraic equations. Given Data / Assumptions: Let the original number be N. When the number is increased by 28, the result is N + 28. This result equals 107 percent of the original number, that is 1.07N. We must find the value of N. All quantities are real numbers, and N is positive. Concept / Approach: A percentage increase can be represented as a multiplier of the original. An increase to 107 percent of the original means the new value is 1.07 times the original. So we set up the equation N + 28 = 1.07N. This yields a simple linear equation in N, which we solve step by step. Using fractional form instead of decimals is another valid approach, but the decimal representation is convenient here. Step-by-Step Solution: Step 1: Let the original number be N.Step 2: After increasing N by 28, we obtain N + 28.Step 3: According to the statement, N + 28 = 107 percent of N = 1.07N.Step 4: Write the equation: N + 28 = 1.07N.Step 5: Subtract N from both sides: 28 = 1.07N − N = 0.07N.Step 6: Solve for N: N = 28 / 0.07.Step 7: Since 0.07 = 7/100, N = 28 ÷ (7/100) = 28 × (100/7) = 4 × 100 = 400. Verification / Alternative check: Check the result by substituting N = 400 back into the original relationship. Increasing 400 by 28 gives 400 + 28 = 428. Now compute 107 percent of 400: 1.07 × 400 = 428. Since both methods give 428 as the new value, N = 400 is confirmed as correct. Why Other Options Are Wrong: If N were 336, adding 28 would give 364, which is not equal to 1.07 × 336. Similarly, 420, 252, and 280 do not satisfy the equation N + 28 = 1.07N when checked directly. Each of these values produces a mismatch between the left and right sides of the equation. Only 400 yields the correct balance between the increase and the percentage representation. Common Pitfalls: A common mistake is to treat the 7 percent increase as 0.7 rather than 0.07, which causes errors in the equation. Another error is to set up the equation as 28 = 1.07N instead of N + 28 = 1.07N. To avoid these mistakes, always translate phrases like 107 percent of the original number into 1.07 times the original and carefully account for the fixed increase in the equation. Final Answer: The original number before the increase was 400.

Question 9

If sec A − tan A = x for an acute angle A, use trigonometric identities to find an equivalent expression for x in terms of sec A and tan A.

Accepted Answer

Introduction / Context: This trigonometry question checks your understanding of identities involving secant and tangent. You are given that sec A − tan A equals x for an acute angle A and asked to find another expression that represents the same quantity. Recognising and applying identities like sec^2 A − tan^2 A = 1 and related manipulations is essential for simplifying trigonometric expressions in competitive exams. Given Data / Assumptions: Angle A is acute, so sec A and tan A are well defined and positive. sec A − tan A = x. We must express x in an equivalent form using sec A and tan A. Standard trigonometric identities are valid: sec^2 A − tan^2 A = 1 and sec^2 A = 1 + tan^2 A. Concept / Approach: A key identity is (sec A − tan A)(sec A + tan A) = sec^2 A − tan^2 A. Since sec^2 A − tan^2 A equals 1, the product of sec A − tan A and sec A + tan A is 1. This means that each factor is the reciprocal of the other. Using this identity, we can express x as the reciprocal of sec A + tan A, which gives a compact alternative expression for x. Step-by-Step Solution: Step 1: Start with the identity sec^2 A − tan^2 A = 1.Step 2: Factor the left side as a difference of squares: sec^2 A − tan^2 A = (sec A − tan A)(sec A + tan A).Step 3: Therefore (sec A − tan A)(sec A + tan A) = 1.Step 4: From the problem statement, sec A − tan A = x.Step 5: Substitute x into the identity: x × (sec A + tan A) = 1.Step 6: Solve for x by dividing both sides by (sec A + tan A): x = 1 / (sec A + tan A). Verification / Alternative check: As a simple numerical example, take A = 60°. Then sec 60° = 2 and tan 60° = √3. Compute sec A − tan A = 2 − √3 and sec A + tan A = 2 + √3. The product (2 − √3)(2 + √3) equals 4 − 3 = 1, confirming the identity. Therefore sec A − tan A is indeed the reciprocal of sec A + tan A. This verifies that x = 1/(sec A + tan A) is correct. Why Other Options Are Wrong: The option 1/(sec^2 A − tan^2 A) simplifies to 1/1 = 1, which is not equal to sec A − tan A in general. The option 1/(sec^2 A + tan^2 A) does not simplify in a useful way and is not equivalent to x. The expression 1/√(sec^2 A − tan^2 A) becomes 1/1 = 1 again and fails for the same reason. The product sec A tan A has a completely different form and is not reciprocally related to sec A − tan A. Only 1/(sec A + tan A) follows directly from the factorised identity and equals x. Common Pitfalls: Some learners misremember the identity and think sec^2 A + tan^2 A equals 1, which leads them to choose an incorrect denominator. Others try to square or square root expressions without a clear plan, creating more complicated forms instead of simplifying. Focusing on the standard identity sec^2 A − tan^2 A = 1 and recognising it as a difference of squares helps quickly arrive at the correct reciprocal relationship. Final Answer: The expression equivalent to x when sec A − tan A = x is 1 / (sec A + tan A).

Question 10

If cos 45° − sec 60° = x, use exact trigonometric values for special angles to find the exact simplified value of x.

Accepted Answer

Introduction / Context: This problem involves evaluating an expression that combines cosine and secant values at special angles 45° and 60°. Exact trigonometric values for these angles are well known and should be memorised. The expression cos 45° − sec 60° requires converting secant to its reciprocal cosine form and then simplifying the resulting difference. This type of simplification is common in trigonometry sections of aptitude and entrance tests. Given Data / Assumptions: The expression is cos 45° − sec 60°. We define the result as x and must find its exact value. Standard exact values are used: cos 45° and cos 60°. Secant is defined as the reciprocal of cosine: sec θ = 1/cos θ. Concept / Approach: First recall exact values for cos 45° and cos 60°. Then compute sec 60° as the reciprocal of cos 60°. Substitute these values into the given expression to simplify. The resulting expression can be written in several algebraically equivalent forms. One of the provided options matches this exact simplified value, although it may not initially look identical until algebraic manipulation is performed. Step-by-Step Solution: Step 1: Recall that cos 45° = √2/2.Step 2: Recall that cos 60° = 1/2, so sec 60° = 1/(1/2) = 2.Step 3: Substitute into the given expression: x = cos 45° − sec 60° = √2/2 − 2.Step 4: Rewrite 2 with denominator 2: 2 = 4/2.Step 5: Then x = (√2/2) − (4/2) = (√2 − 4)/2.Step 6: Now compare with the option (1 − 2√2)/√2. Simplify this option: (1 − 2√2)/√2 = 1/√2 − 2.Step 7: Since 1/√2 = √2/2, we have 1/√2 − 2 = √2/2 − 2, which equals (√2 − 4)/2, identical to x. Verification / Alternative check: Compute an approximate decimal value to verify. Using √2 ≈ 1.414, we have cos 45° ≈ 0.707 and sec 60° = 2. So x ≈ 0.707 − 2 = −1.293. Now evaluate (1 − 2√2)/√2 numerically: the numerator is approximately 1 − 2 × 1.414 = 1 − 2.828 = −1.828. Dividing by √2 ≈ 1.414 gives about −1.293. This matches the earlier approximation, confirming the correctness of the expression. Why Other Options Are Wrong: The expression (√3 − 4)/(2√3) involves √3 and cannot be equal to a value expressed purely in terms of √2. The option 1 is clearly incorrect since cos 45° − 2 is negative. The expression (√3 + √2)/2 combines both radicals without the subtraction of 2 and yields a positive value, which contradicts the negative result. The option (−4 + √2)/2 is algebraically equal to (√2 − 4)/2, but among the given responses, the clearly equivalent simplified form is (1 − 2√2)/√2, which matches x exactly. Common Pitfalls: A common mistake is to misremember cos 45° or cos 60°, or to forget that sec 60° is the reciprocal of cos 60°. Another error is to combine terms without using a common denominator, which leads to incorrect numerators. It is also easy to confuse algebraically equivalent expressions and overlook that two forms represent the same number. Simplifying step by step and, if needed, checking with approximate decimals helps avoid these issues. Final Answer: The exact value of x = cos 45° − sec 60° is (1 − 2√2) / √2.

Question 11

For an acute angle A, suppose (1 + cos A) / 2 = x. Using half angle trigonometric identities, determine which expression in terms of A is equal to x.

Accepted Answer

Introduction / Context: This question tests your understanding of half angle identities in trigonometry. You are given that (1 + cos A) / 2 equals x for an acute angle A and asked which expression in terms of A also represents x. Knowing the standard half angle formulas that relate cosine of a double angle to squares of sine and cosine of half the angle is essential in simplifying trigonometric expressions and solving equations. Given Data / Assumptions: A is an acute angle, so all basic trigonometric functions are positive. The expression (1 + cos A) / 2 is defined to be x. We must identify an equivalent expression in terms of A/2. Half angle identities for sine and cosine are available and valid. Concept / Approach: The relevant identity is the cosine half angle formula. For any angle θ, cos^2(θ) can be written in terms of cos(2θ) as cos^2(θ) = (1 + cos(2θ)) / 2. Here, if we let θ = A/2, then cos(2θ) becomes cos A. This gives a direct match between cos^2(A/2) and (1 + cos A) / 2. Recognising this pattern allows us to identify x as cos^2(A/2) without any complicated manipulation. Step-by-Step Solution: Step 1: Recall the identity for double angles: cos(2θ) = 2cos^2 θ − 1.Step 2: Rearrange this identity to express cos^2 θ: 2cos^2 θ = 1 + cos(2θ), so cos^2 θ = (1 + cos(2θ)) / 2.Step 3: Set θ = A/2. Then 2θ = A, and the identity becomes cos^2(A/2) = (1 + cos A) / 2.Step 4: Compare this with the given expression x = (1 + cos A) / 2.Step 5: It follows directly that x = cos^2(A/2). Verification / Alternative check: Consider a simple acute angle, for example A = 60°. Then cos 60° = 1/2, so (1 + cos 60°)/2 = (1 + 1/2)/2 = 3/2 ÷ 2 = 3/4. On the other hand, A/2 = 30°, and cos 30° = √3/2. Therefore cos^2(30°) = (√3/2)^2 = 3/4. The two values match, confirming that x equals cos^2(A/2) for this choice of A, and by identity it holds in general. Why Other Options Are Wrong: The expression sin(A/2) is not equal to (1 + cos A) / 2; instead, sin^2(A/2) is related to (1 − cos A) / 2. The square root expressions √(sin(A/2)) and √(cos(A/2)) involve fourth roots of trigonometric functions and do not match the given squared form. The expression sin^2(A/2) corresponds to (1 − cos A) / 2, not (1 + cos A) / 2. Only cos^2(A/2) exactly matches the formula for (1 + cos A) / 2. Common Pitfalls: Learners often mix up the half angle formulas, confusing the identities for sine and cosine. Another common error is to forget that the given expression is a square of a trigonometric function rather than the function itself. Careful use of the double angle identity and systematic substitution of θ = A/2 help to avoid these mistakes and to identify the correct equivalent expression. Final Answer: When (1 + cos A) / 2 = x, the equivalent expression is cos^2(A/2).

Question 12

For an acute angle A, suppose sec A + tan A = x. Using standard identities, determine which expression in terms of sine and cosine of A is equal to x.

Accepted Answer

Introduction / Context: This trigonometry question examines your ability to manipulate expressions involving secant and tangent and rewrite them in terms of sine and cosine. You are told that sec A + tan A equals x for an acute angle A and asked to identify another expression that is equal to x. This requires both algebraic manipulation and knowledge of trigonometric identities, which are important skills in higher level mathematics and aptitude examinations. Given Data / Assumptions: A is an acute angle. sec A + tan A = x. sec A = 1/cos A and tan A = sin A/cos A. sine and cosine satisfy sin^2 A + cos^2 A = 1. We must find an expression in terms of sin A and cos A that is equal to x. Concept / Approach: First rewrite sec A and tan A using sine and cosine. This gives a single fraction with cos A as the denominator. Next, rationalise this expression by multiplying the numerator and denominator by a suitable conjugate in order to transform the denominator into a simple expression in sin A. After simplification, we look for a match with one of the given options. Using the identity sin^2 A + cos^2 A = 1 is crucial for reducing the expression to its most useful form. Step-by-Step Solution: Step 1: Express sec A and tan A in terms of sine and cosine: sec A = 1/cos A and tan A = sin A/cos A.Step 2: Add them: sec A + tan A = 1/cos A + sin A/cos A = (1 + sin A)/cos A.Step 3: So x = (1 + sin A)/cos A.Step 4: Multiply numerator and denominator by (1 − sin A) to transform the expression: x = [(1 + sin A)(1 − sin A)] / [cos A(1 − sin A)].Step 5: Use the identity (1 + sin A)(1 − sin A) = 1 − sin^2 A.Step 6: Since 1 − sin^2 A = cos^2 A, the numerator becomes cos^2 A.Step 7: Therefore x = cos^2 A / [cos A(1 − sin A)] = cos A / (1 − sin A). Verification / Alternative check: Pick a simple acute angle, such as A = 30°. Then sec 30° = 1/cos 30° = 2/√3 and tan 30° = 1/√3. Their sum is (2/√3 + 1/√3) = 3/√3 = √3. Now evaluate cos A/(1 − sin A): cos 30° = √3/2 and sin 30° = 1/2, so cos A/(1 − sin A) = (√3/2)/(1 − 1/2) = (√3/2)/(1/2) = √3. The two results match, confirming that sec A + tan A equals cos A/(1 − sin A). Why Other Options Are Wrong: The expression cos A/(1 + sin A) is the reciprocal of cos A/(1 − sin A) after rationalisation and corresponds more closely to sec A − tan A. The square root expressions √(cos A/(1 − sin A)) and √(cos A/(1 + sin A)) introduce a square root that is not present in the original x and therefore cannot be equal. The option (1 + sin A)/cos A is the unreduced form we obtained before rationalisation, but among the listed choices, the fully simplified identity that matches standard forms is cos A/(1 − sin A). Common Pitfalls: Students sometimes stop at the intermediate form (1 + sin A)/cos A and do not recognise that rationalising further can produce a more useful equivalent expression. Others may incorrectly multiply numerator and denominator or misapply the identity 1 − sin^2 A = cos^2 A. Carefully applying each algebraic step and checking with a specific example angle helps solidify the understanding of these manipulations. Final Answer: When sec A + tan A = x, an equivalent expression is cos A / (1 − sin A).

Question 13

A straight line passes through the origin and is perpendicular to the line 3x − 2y = 6. This perpendicular line intersects 3x − 2y = 6 at point M. Find the exact coordinates of point M.

Accepted Answer

Introduction / Context: This coordinate geometry question involves slopes of perpendicular lines and their point of intersection. You are given a line in standard form and asked to find the point where it is intersected by another line that passes through the origin and is perpendicular to it. Problems of this type are common in analytic geometry and test understanding of slopes, perpendicularity, and solving simultaneous linear equations. Given Data / Assumptions: Line L1 has equation 3x − 2y = 6. Line L2 passes through the origin (0, 0). Line L2 is perpendicular to L1. The lines intersect at point M, whose coordinates we must find. All coordinates are real numbers. Concept / Approach: First, rewrite the given line 3x − 2y = 6 in slope intercept form to find its slope. The slope of a line perpendicular to this line is the negative reciprocal of that slope. Using the fact that L2 passes through the origin, we can write its equation immediately. Then we solve the system of equations formed by L1 and L2 to find their intersection point M. This approach combines slope concepts with solving linear systems. Step-by-Step Solution: Step 1: Rewrite L1: 3x − 2y = 6. Solve for y: −2y = 6 − 3x, so y = (3/2)x − 3.Step 2: The slope of L1 is 3/2.Step 3: The slope of a line perpendicular to L1 is the negative reciprocal, which is −2/3.Step 4: Since L2 passes through the origin, its equation is y = (−2/3)x.Step 5: To find M, solve the system consisting of y = (3/2)x − 3 and y = (−2/3)x.Step 6: Set the right sides equal: (3/2)x − 3 = (−2/3)x.Step 7: Add (2/3)x to both sides: (3/2x + 2/3x) − 3 = 0.Step 8: Find a common denominator for the coefficients: 3/2 = 9/6 and 2/3 = 4/6, so 3/2x + 2/3x = 13/6x.Step 9: The equation becomes 13/6 x − 3 = 0, so 13/6 x = 3 and x = 3 × 6/13 = 18/13.Step 10: Substitute x = 18/13 into y = (−2/3)x: y = (−2/3) × 18/13 = −36/39 = −12/13.Step 11: Thus M has coordinates (18/13, −12/13). Verification / Alternative check: Check that M lies on both lines. For line L1, substitute into 3x − 2y = 6: 3 × 18/13 − 2 × (−12/13) = 54/13 + 24/13 = 78/13 = 6, so M lies on L1. For line L2, check y = (−2/3)x: (−12/13) = (−2/3) × 18/13 = −36/39 = −12/13, so M lies on L2 as well. This confirms that the coordinates of M are correct. Why Other Options Are Wrong: The point (18/13, 12/13) has the wrong sign for y and does not satisfy y = (−2/3)x. The points (−18/13, −12/13) and (−18/13, 12/13) use incorrect signs for x and do not satisfy the equation 3x − 2y = 6. The origin (0, 0) lies on the perpendicular line but not on the given line, so it cannot be the intersection point. Only (18/13, −12/13) satisfies both line equations simultaneously. Common Pitfalls: Some learners confuse the slope of the perpendicular line and take it as the reciprocal rather than the negative reciprocal. Others make arithmetic errors when combining fractional slopes or solving for x. Carefully converting to slope intercept form, correctly determining the perpendicular slope, and working step by step with fractions helps avoid these mistakes. Final Answer: The coordinates of point M are (18/13, −12/13).

Question 14

Find the length of the arc of a circle whose central angle is 45° and whose radius is 28 cm, giving your answer in centimetres using the standard arc length formula.

Accepted Answer

Introduction / Context: This geometry question uses the standard formula for the length of an arc of a circle. Given the central angle in degrees and the radius, you need to compute the arc length, which is a portion of the full circumference. Such problems are very common in school mathematics and quantitative exams and help reinforce the relationship between degrees, circumference, and proportional reasoning. Given Data / Assumptions: The radius of the circle is r = 28 cm. The central angle is θ = 45°. We are asked to find the length of the corresponding arc. The entire circle corresponds to 360° around the centre. Concept / Approach: The length of an arc corresponding to a central angle θ (in degrees) is given by the fraction θ/360 of the entire circumference. The full circumference of a circle is 2πr. Therefore, the arc length s can be written as s = (θ/360) × 2πr. Once we substitute θ and r into this formula, the calculation becomes straightforward, and we can either leave the answer in terms of π or approximate it to a nearby integer value if the options are given in that form. Step-by-Step Solution: Step 1: Recall the arc length formula for degrees: s = (θ/360) × 2πr.Step 2: Substitute θ = 45° and r = 28 cm into the formula.Step 3: Compute the fractional factor: θ/360 = 45/360 = 1/8.Step 4: Compute the full circumference: 2πr = 2π × 28 = 56π cm.Step 5: Now calculate s = (1/8) × 56π = 7π cm.Step 6: If we approximate π as 22/7, then 7π ≈ 7 × 22/7 = 22 cm. Verification / Alternative check: Since 45° is one eighth of a full revolution (360°/8 = 45°), the arc length corresponding to 45° must be one eighth of the full circumference. The full circumference for radius 28 cm is 56π cm, so one eighth of that is 7π cm, confirming our earlier computation. Using π ≈ 3.14 also gives 7 × 3.14 ≈ 21.98 cm, which is very close to 22 cm and aligns with the nearest centimetre value in the options. Why Other Options Are Wrong: The value 11 cm is about half of the correct approximate value and would correspond to a much smaller central angle. The values 33 cm and 44 cm are too large; 44 cm is approximately two full radii and would require a larger sector. The exact value 7π cm is mathematically correct, but if the options use integer approximations, the matching choice is 22 cm. Among the numerical options provided, only 22 cm matches the correct approximate arc length. Common Pitfalls: Students sometimes mistakenly use θ/180 instead of θ/360 when working in degrees, which doubles the arc length. Others forget to multiply by 2πr and instead use πr, which is half the circumference. Keeping the correct formula in mind and recognising that 45° is one eighth of a full circle helps prevent such errors and leads quickly to the correct result. Final Answer: The length of the arc for a 45° central angle in a circle of radius 28 cm is 22 cm (approximately, corresponding to the exact value 7π cm).

Question 15

In basic algebra, solve for x in the linear equation in one variable: (9 − 3x) − (17x − 10) = −1. What is the value of x that satisfies this equation exactly?

Accepted Answer

Introduction / Context: This question tests your ability to solve a basic linear equation in one variable that involves brackets and negative signs. Such equations are extremely common in school algebra, aptitude tests, and many real life applications where you need to isolate an unknown. The main skills required are expanding brackets correctly, combining like terms, and performing the same operation on both sides of the equation to preserve equality. Given Data / Assumptions: The equation is (9 − 3x) − (17x − 10) = −1. x is a real number. Standard rules of algebra apply, including distribution over brackets and the equality principle. We seek the exact value of x that satisfies this equation. Concept / Approach: To solve this equation, we first remove brackets using distribution. The key idea is that when you subtract a bracket, you change the sign of every term inside that bracket. After expansion, we collect all x terms together and all constant terms together. Finally, we isolate x by performing simple addition, subtraction, or division steps. The equation is linear, so it will have at most one solution for x. Step-by-Step Solution: 1) Start with (9 − 3x) − (17x − 10) = −1. 2) Expand the first part: 9 − 3x remains as it is; there is no leading coefficient to distribute. 3) Subtract the second bracket by changing every sign inside it: −(17x − 10) becomes −17x + 10. 4) Combine terms on the left side: (9 − 3x) − 17x + 10 = 9 + 10 − 3x − 17x = 19 − 20x. 5) The equation becomes 19 − 20x = −1. 6) Subtract 19 from both sides to move constants to the right: −20x = −1 − 19 = −20. 7) Divide both sides by −20: x = (−20) / (−20) = 1. Verification / Alternative check: Substitute x = 1 back into the original equation. Compute 9 − 3x = 9 − 3 * 1 = 6. Next, compute 17x − 10 = 17 * 1 − 10 = 7. The left side becomes (9 − 3x) − (17x − 10) = 6 − 7 = −1, which matches the right side exactly. This confirms that x = 1 is indeed the correct solution and that there is no arithmetic mistake in the steps above. Why Other Options Are Wrong: Option b (−1) would give (9 − 3(−1)) − (17(−1) − 10) = 12 − (−17 − 10) which does not simplify to −1. Option c (9/10), option d (−9/10), and option e (2) likewise produce left side values different from −1 when substituted into the equation. Only x = 1 satisfies the equality exactly. Common Pitfalls: A common mistake is to forget to change the sign of both terms in the second bracket when subtracting (17x − 10). Some learners also incorrectly combine constants or mishandle negative numbers when moving terms across the equality sign. Writing each step carefully, especially the sign changes when removing brackets, helps avoid these errors and leads quickly to the correct value of x. Final Answer: The linear equation is satisfied when x equals 1.

Question 16

In algebra, two real numbers a and b satisfy a − b = −7 and a^2 + b^2 = 85. Using these conditions, find the exact value of the product ab.

Accepted Answer

Introduction / Context: This algebra question focuses on using identities that relate sums and differences of numbers to their squares and products. Instead of asking you to solve directly for a and b, it requires you to manipulate expressions like a^2 + b^2 and a − b to deduce the product ab. Such questions appear frequently in aptitude exams and help develop fluency with algebraic identities and symmetric expressions. Given Data / Assumptions: a and b are real numbers. a − b = −7. a^2 + b^2 = 85. We are asked to find the value of ab. No additional restrictions on a and b are given. Concept / Approach: The key identity is based on squaring a difference: (a − b)^2 = a^2 + b^2 − 2ab. We already know both a − b and a^2 + b^2, so we can substitute these values into the identity and solve the resulting linear equation for ab. This avoids having to solve a quadratic equation for a and b individually and makes the computation quick and efficient. Step-by-Step Solution: 1) Start with the identity (a − b)^2 = a^2 + b^2 − 2ab. 2) Substitute the given value a − b = −7. Then (a − b)^2 = (−7)^2 = 49. 3) Substitute a^2 + b^2 = 85 into the identity: 49 = 85 − 2ab. 4) Rearrange to isolate ab: move 85 to the left or 49 to the right. Subtract 85 from both sides to get 49 − 85 = −2ab. 5) Compute 49 − 85 = −36, so −36 = −2ab. 6) Divide both sides by −2: ab = (−36) / (−2) = 18. Verification / Alternative check: We can verify the result by constructing a quadratic whose roots are a and b. Let t be a variable with roots a and b, so t^2 − (a + b)t + ab = 0. While a + b is not given directly, we can find it using (a − b)^2 + 4ab = (a + b)^2. We already know (a − b)^2 = 49 and ab = 18, so (a + b)^2 = 49 + 4 * 18 = 49 + 72 = 121, giving a + b = 11 or −11. Using these values, you can solve for a and b explicitly and check that a^2 + b^2 indeed equals 85, which confirms that ab = 18 is consistent. Why Other Options Are Wrong: Option b (−18) would require (a − b)^2 to equal 85 + 2 * (−18) = 49, but this sign choice conflicts with the identity when computed carefully. Options c (30), d (−30), and e (44) all give values of a^2 + b^2 that do not match 85 when substituted into the identity. Only ab = 18 makes the identity (a − b)^2 = a^2 + b^2 − 2ab hold with the given numerical values. Common Pitfalls: A frequent mistake is to mis remember the identity as a^2 + b^2 = (a − b)^2 + 2ab, which is actually correct but often rearranged incorrectly when solving for ab. Another common issue is incorrect arithmetic when computing 49 − 85. Some learners also forget to keep track of negative signs and may end up with ab = −18 by accident. Careful substitution and step by step rearrangement prevent these errors. Final Answer: The product of the two numbers that satisfies the given conditions is 18.

Question 17

Solve the compound system of linear inequalities in one variable: 2x − 2(3 + 4x) < −1 − 2x and −1 − 2x > (−5/3) − (x/3). Among the given options, which value of x is the greatest integer that satisfies both inequalities?

Accepted Answer

Introduction / Context: This problem involves solving a system of two linear inequalities in a single variable x. Instead of just solving one inequality, you must find the intersection of the solution sets of both inequalities and then decide which of the given options fits an additional condition. Here, you are asked specifically for the greatest integer value among the options that satisfies both inequalities. This tests algebraic manipulation, understanding of inequalities, and reasoning about integer solutions. Given Data / Assumptions: The first inequality is 2x − 2(3 + 4x) < −1 − 2x. The second inequality is −1 − 2x > (−5/3) − (x/3). x is a real variable, but the options represent integer candidates. We must find which option is the greatest integer that satisfies both inequalities simultaneously. Concept / Approach: We solve each inequality separately and express the solution sets as intervals on the real number line. Then we intersect these intervals to obtain the combined solution region. Finally, we look at the given answer choices and select the largest integer that lies within this intersection. This approach mirrors how inequality systems are handled in algebra and aptitude tests, where understanding overlapping conditions is important. Step-by-Step Solution: 1) Solve the first inequality: 2x − 2(3 + 4x) < −1 − 2x. 2) Expand the bracket: 2x − 2 * 3 − 2 * 4x = 2x − 6 − 8x = −6 − 6x. 3) The inequality becomes −6 − 6x < −1 − 2x. 4) Add 6x to both sides: −6 < −1 + 4x. 5) Add 1 to both sides: −5 < 4x. 6) Divide by 4 (positive, so inequality direction stays): x > −5/4. 7) Now solve the second inequality: −1 − 2x > (−5/3) − (x/3). 8) Multiply every term by 3 to clear denominators: −3 − 6x > −5 − x. 9) Add 6x to both sides: −3 > −5 + 5x. 10) Add 5 to both sides: 2 > 5x. 11) Divide by 5: x < 2/5. 12) Combined, we have −5/4 < x < 2/5. Verification / Alternative check: The interval −5/4 < x < 2/5 corresponds approximately to −1.25 < x < 0.4. The integer values that lie in this open interval are x = −1 and x = 0. The question specifically asks for the greatest integer from the options that satisfies both inequalities. Among the options −2, −1, 0, 1, and 2, only −1 and 0 lie in the interval, and the greater of these two integers is 0. Therefore, x = 0 is the correct choice based on both inequalities and the wording of the question. Why Other Options Are Wrong: Option a (−2) is less than −1.25 and does not satisfy the first inequality. Option b (−1) is within the interval and is a valid solution, but it is not the greatest integer that works. Option d (1) and option e (2) are greater than 0.4 and violate the second inequality. Only option c (0) both satisfies the system and is the greatest integer among the choices that does so. Common Pitfalls: Students sometimes forget to compute the intersection of the two solution sets and instead choose any value that satisfies only one inequality. Another frequent mistake is mishandling signs when clearing denominators or moving terms across the inequality, or forgetting that multiplying by a negative number reverses the inequality direction. In this problem, we always divide or multiply by positive numbers, so the inequality directions remain unchanged. Carefully solving each inequality and then intersecting the results avoids these errors. Final Answer: The greatest integer among the given options that satisfies both inequalities is 0.

Question 18

Using the standard product-to-sum trigonometric identities, simplify the expression 2 cos A * sin B. If 2 cos A * sin B = x, which equivalent expression correctly represents x?

Accepted Answer

Introduction / Context: This trigonometry question focuses on the product-to-sum identities, which convert products of sine and cosine functions into sums or differences of sines or cosines. These identities are very useful in simplifying expressions, solving trigonometric equations, and working with integrals in calculus. Here, you are asked to simplify 2 cos A * sin B and express it in terms of sine functions using the correct identity. Given Data / Assumptions: The expression is 2 cos A * sin B. A and B are real angles (no special restriction on quadrants is needed for the identity itself). We define x = 2 cos A * sin B. We must express x using a sum or difference of sine functions based on a standard identity. Concept / Approach: The standard product-to-sum identities include: 2 sin X * cos Y = sin(X + Y) + sin(X − Y) and symmetric variants. In this problem, the factors are arranged as cos A * sin B, but multiplication is commutative, so we can think of the product as 2 sin B * cos A. Then we can apply the identity with X = B and Y = A. After that, we may need to rewrite terms like sin(B − A) in a form involving A − B, but we must be careful with sign changes because sine is an odd function: sin(−θ) = −sin θ. Step-by-Step Solution: 1) Start with x = 2 cos A * sin B. 2) Rewrite the product using commutativity: x = 2 sin B * cos A. 3) Apply the identity 2 sin X * cos Y = sin(X + Y) + sin(X − Y) with X = B and Y = A. 4) This gives x = sin(B + A) + sin(B − A). 5) Since addition is commutative, B + A = A + B, so sin(B + A) = sin(A + B). 6) Thus x = sin(A + B) + sin(B − A). 7) Note that sin(B − A) = −sin(A − B), because sine is an odd function. 8) Substitute sin(B − A) = −sin(A − B) to obtain x = sin(A + B) − sin(A − B). Verification / Alternative check: You can test the identity with specific angle values, for example A = 30 degrees and B = 60 degrees. Compute the left side: 2 cos 30 * sin 60 = 2 * (√3 / 2) * (√3 / 2) = 2 * (3/4) = 3/2. Now compute the right side for option a: sin(A + B) − sin(A − B) = sin 90 − sin(−30). Using standard values, sin 90 = 1 and sin(−30) = −1/2, so the expression becomes 1 − (−1/2) = 1 + 1/2 = 3/2. The two sides match, confirming that the identity is correct for this choice of A and B, and supporting option a as the correct expression. Why Other Options Are Wrong: Option b, sin(A + B) + sin(A − B), corresponds to 2 sin A * cos B instead of 2 cos A * sin B. Option c and option d involve cosine functions and match the product-to-sum identities for cos X * cos Y, not sin and cos products. Option e rearranges terms in a way that introduces a sign pattern inconsistent with the underlying identity. Only option a correctly represents 2 cos A * sin B as a difference of sine terms. Common Pitfalls: A common error is to mis remember which product corresponds to which sum identity, for example confusing the formulas for 2 sin X * cos Y and 2 cos X * cos Y. Another pitfall is failing to notice that sin(B − A) and sin(A − B) differ by a sign, leading to an incorrect choice between a sum and a difference. Writing down the general identity first and then substituting X and Y carefully helps avoid these mistakes. Final Answer: Using the product-to-sum identity, the expression 2 cos A * sin B is equal to sin(A + B) - sin(A - B).

Question 19

In algebraic factorization, completely factorize the cubic polynomial 48x^3 − 8x^2 − 93x − 45 into a product of three linear factors. Which of the following options gives the correct factorization?

Accepted Answer

Introduction / Context: This algebra question tests your ability to factorize a cubic polynomial into linear factors. Factorization of higher degree polynomials is an important skill in algebra, as it helps in solving equations, simplifying expressions, and understanding polynomial graphs. Here, the polynomial 48x^3 − 8x^2 − 93x − 45 is already presented together with several candidate factorizations, and you must determine which product of three linear factors reproduces the given cubic exactly. Given Data / Assumptions: The polynomial is P(x) = 48x^3 − 8x^2 − 93x − 45. We are given multiple options, each expressing P(x) as a product of three linear binomials. The task is to identify which option is the exact factorization of P(x). All arithmetic is over real numbers (the factors shown are real and rational). Concept / Approach: There are two main ways to tackle this problem. One way is to use the factor theorem by trying to find rational roots of P(x) using possible factors of the constant term and the leading coefficient. Another way, which fits well with the multiple choice structure, is to expand each candidate factorization and check which one matches the original polynomial term by term. Because the coefficients here are relatively small integers, expanding is manageable and allows us to verify the correct option systematically. Step-by-Step Solution: 1) Consider option c: (4x + 3)(4x + 3)(3x − 5). 2) First multiply (4x + 3)(4x + 3) to obtain a quadratic factor: (4x + 3)^2 = 16x^2 + 24x + 9. 3) Now multiply this quadratic by (3x − 5): (16x^2 + 24x + 9)(3x − 5). 4) Distribute 3x across the quadratic: 3x * 16x^2 = 48x^3, 3x * 24x = 72x^2, and 3x * 9 = 27x. 5) Distribute −5 across the quadratic: −5 * 16x^2 = −80x^2, −5 * 24x = −120x, and −5 * 9 = −45. 6) Combine like terms: the x^3 term is 48x^3. The x^2 terms are 72x^2 − 80x^2 = −8x^2. The x terms are 27x − 120x = −93x. The constant term is −45. 7) The resulting polynomial is 48x^3 − 8x^2 − 93x − 45, which matches P(x) exactly. Verification / Alternative check: To be fully confident, we can briefly check one other option to see that it does not match. For instance, option a is (4x + 3)(4x − 3)(3x − 5). Multiplying (4x + 3)(4x − 3) first gives 16x^2 − 9. Multiplying by (3x − 5) then yields (16x^2 − 9)(3x − 5) = 48x^3 − 80x^2 − 27x + 45. This cubic clearly does not match the original because the x^2 and constant terms differ in both sign and value. Similar mismatches occur if you expand the other incorrect options, confirming that only option c matches P(x) exactly. Why Other Options Are Wrong: Option a produces 48x^3 − 80x^2 − 27x + 45, which differs in three coefficients. Option b has a repeated factor (4x − 3) and leads to a very different x^2 coefficient. Option d includes (3x + 5) as a factor, which gives a positive constant term when expanded, whereas P(x) has a negative constant term. Option e uses factors that generate degrees and coefficients inconsistent with 48x^3 − 8x^2 − 93x − 45. Only option c produces exactly the original polynomial upon expansion. Common Pitfalls: Learners sometimes try to factor by guessing only one factor, leading to time consuming algebra. Another common issue is making small arithmetic mistakes when expanding products, especially with negative signs and combined like terms. Working systematically, expanding one pair of factors at a time, and double checking coefficients for x^3, x^2, x, and the constant term help avoid such errors. In multiple choice settings, careful expansion of the most promising option is often enough to identify the correct factorization quickly. Final Answer: The correct complete factorization of 48x^3 − 8x^2 − 93x − 45 is (4x + 3)(4x + 3)(3x - 5).

Question 20

Let x and y be positive real numbers such that xy = 56 and x^2 + y^2 = 113. Using these conditions, find the exact value of the sum (x + y).

Accepted Answer

Introduction / Context: This algebra question tests your understanding of how sums and products of two numbers relate to the squares of those numbers. Instead of solving directly for x and y, you are given xy and x^2 + y^2 and asked to find x + y. This kind of problem commonly appears in algebra and aptitude tests, often leading to a neat answer via a simple identity rather than lengthy calculations. Given Data / Assumptions: x and y are positive real numbers. Their product is xy = 56. The sum of their squares is x^2 + y^2 = 113. We must find x + y. Concept / Approach: The main identity to use is: (x + y)^2 = x^2 + y^2 + 2xy. We are given both x^2 + y^2 and xy, so we can compute (x + y)^2 directly, then take the positive square root to find x + y. Because x and y are specified to be positive, x + y must also be positive, which resolves any ambiguity in taking square roots. This method is far faster than trying to form and solve a quadratic for x or y individually. Step-by-Step Solution: 1) Start with the identity (x + y)^2 = x^2 + y^2 + 2xy. 2) Substitute the given values: x^2 + y^2 = 113 and xy = 56. 3) Compute 2xy = 2 * 56 = 112. 4) So (x + y)^2 = 113 + 112 = 225. 5) To find x + y, take the square root of both sides: x + y = ±√225. 6) Since x and y are positive, their sum x + y must also be positive, so x + y = √225 = 15. Verification / Alternative check: We can verify this result by explicitly solving for x and y. If x + y = 15 and xy = 56, then x and y are roots of the quadratic equation t^2 − 15t + 56 = 0. Factorizing gives (t − 7)(t − 8) = 0, so t = 7 or t = 8. Hence, {x, y} = {7, 8}. Now check the given conditions. The product xy = 7 * 8 = 56, and x^2 + y^2 = 7^2 + 8^2 = 49 + 64 = 113. Both constraints are satisfied, confirming that x + y = 7 + 8 = 15 is correct. Why Other Options Are Wrong: Option b (−15) would require x + y to be negative, which contradicts the fact that both x and y are positive. Options c (21), d (29), and e (13) lead to incorrect values of x^2 + y^2 or xy when plugged into the identity or used to form a quadratic equation. Only x + y = 15 satisfies both given conditions.consistently. Common Pitfalls: Some learners forget the identity for (x + y)^2 and attempt to solve for x and y directly, which is slower and more error prone. Others may mistakenly use x^2 + y^2 = (x + y)^2 − 2xy but then make sign errors when rearranging. Another frequent issue is forgetting to choose the positive square root for x + y, even though x and y are positive. Keeping the identity in mind and using the positivity condition avoids these mistakes. Final Answer: The sum of the two positive real numbers is 15.

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