Q: If 9x - 6(3 - x) = 12 in a linear equation, what is the value of x?

Introduction / Context: This is a straightforward linear equation in one variable. Such questions appear frequently in aptitude tests to check basic algebra skills, including distribution of multiplication over subtraction and collecting like terms on one side of the equation. Given Data / Assumptions: Equation: 9x - 6(3 - x) = 12 We assume x is a real number. We need to find the numerical value of x. Concept / Approach: We use basic algebra: Apply distributive property: k(a - b) = ka - kb. Combine like terms involving x. Isolate x by moving constants to the other side. The goal is to rewrite the equation into the form kx = constant and then divide both sides by k. Step-by-Step Solution: Start with 9x - 6(3 - x) = 12 Distribute -6 over the bracket: -6(3 - x) = -18 + 6x So the equation becomes 9x - 18 + 6x = 12 Combine like terms: 9x + 6x = 15x, so 15x - 18 = 12 Move -18 to the right side: 15x = 12 + 18 = 30 Divide both sides by 15: x = 30 / 15 = 2 Verification / Alternative check: Substitute x = 2 back into the original equation. Left side is 9(2) - 6(3 - 2) = 18 - 6(1) = 18 - 6 = 12, which matches the right side. This confirms that x = 2 is correct. Why Other Options Are Wrong: If x = 4, then 9x - 6(3 - x) becomes 36 - 6(-1) = 42, not 12. If x = 6 or x = 9, the left side becomes much larger than 12. If x = 0, the left side is -18, which also does not match 12. Hence these options do not satisfy the equation. Common Pitfalls: A common mistake is to distribute the minus sign incorrectly, writing -6(3 - x) as -18 - 6x instead of -18 + 6x. Another error is not carefully combining 9x and 6x. Careful handling of signs at each step avoids these issues. Final Answer: The value of x that satisfies the equation is 2.

Q: If x - 2y = 2 and 3x + y = 20, then what are the values of the ordered pair (x, y)?

Introduction / Context: This question involves solving a pair of simultaneous linear equations in two variables. Such systems form the backbone of algebra and appear frequently in aptitude tests to assess a candidate ability to handle multiple equations and find a unique ordered pair solution (x, y). Given Data / Assumptions: First equation: x - 2y = 2 Second equation: 3x + y = 20 x and y are real numbers. We seek the ordered pair (x, y) that satisfies both equations simultaneously. Concept / Approach: We can use either substitution or elimination. The first equation is already simple to express x in terms of y or vice versa. Using substitution is convenient: From x - 2y = 2, express x as x = 2 + 2y. Substitute this x into 3x + y = 20 and solve for y. Then back substitute to get x. Step-by-Step Solution: From x - 2y = 2, we get x = 2 + 2y Substitute into 3x + y = 20: 3(2 + 2y) + y = 20 This gives 6 + 6y + y = 20 Combine like terms: 6 + 7y = 20 So 7y = 14, hence y = 14 / 7 = 2 Now find x: x = 2 + 2y = 2 + 4 = 6 Thus the ordered pair is (x, y) = (6, 2) Verification / Alternative check: Check (6, 2) in both equations. First: 6 - 2*2 = 6 - 4 = 2, correct. Second: 3*6 + 2 = 18 + 2 = 20, also correct. No other listed option will satisfy both equations when substituted, so (6, 2) is the unique valid answer. Why Other Options Are Wrong: For (4, 1), the first equation becomes 4 - 2 = 2 (works) but the second becomes 3*4 + 1 = 13, not 20. For (3, 2), x - 2y = 3 - 4 = -1, not 2. For (5, 5), x - 2y = 5 - 10 = -5, incorrect. For (2, 6), x - 2y = 2 - 12 = -10, also incorrect. Common Pitfalls: Candidates sometimes make arithmetic mistakes when substituting or combining like terms, for example miscalculating 6 + 7y or incorrectly dividing 14 by 7. Writing each step clearly and checking substitution into both equations helps prevent such errors. Final Answer: The correct ordered pair is (6, 2).

Q: In trigonometry, if cos 240° = x, then what is the value of x?

Introduction / Context: This is a basic question on standard trigonometric values. It checks whether you know the cosine of key angles measured in degrees and can relate angles in different quadrants using reference angles and sign conventions for sine, cosine, and tangent. Given Data / Assumptions: cos 240° = x We work in the usual unit circle framework. We know standard values for 30°, 60°, and 90°. Concept / Approach: The angle 240° lies in the third quadrant. The reference angle is the acute angle formed with the x axis. We use: cos(180° + θ) = -cos θ cos 60° = 1/2 Since 240° = 180° + 60°, we can evaluate cos 240° using this identity. Step-by-Step Solution: Write 240° as 180° + 60° Use identity: cos(180° + θ) = -cos θ So cos 240° = cos(180° + 60°) = -cos 60° We know cos 60° = 1/2 Therefore cos 240° = -1/2 Hence x = -1/2 Verification / Alternative check: The angle 240° is in the third quadrant, where cosine values are negative because x coordinates on the unit circle are negative there. The reference angle is 60°, so the magnitude is the same as cos 60°, but with a negative sign. This reasoning agrees with the algebraic identity method and confirms that -1/2 is correct. Why Other Options Are Wrong: Option b, -√3/2, is the cosine of 150° or 210°, not 240°. Option c, 1/2, and option e, √3/2, are positive values that cannot be correct for a third quadrant angle. Option a, -1/√2, corresponds to cos 135° or cos 225°, not cos 240°. Common Pitfalls: Learners sometimes forget which quadrants give positive or negative cosine values, or they confuse the reference angles 30° and 60°. Remember that cosine is positive in the first and fourth quadrants and negative in the second and third. Breaking the angle into 180° plus or minus a standard angle helps avoid confusion. Final Answer: The value of x is -1/2.

Q: If a + b = 9 and a^2 + b^2 = 61, then what is the value of the product ab?

Introduction / Context: This algebra question uses a standard identity that relates the sum of two numbers, the sum of their squares, and their product. Such identities are very helpful in simplifying expressions and quickly finding unknown quantities without solving full quadratic equations. Given Data / Assumptions: a + b = 9 a^2 + b^2 = 61 a and b are real numbers. We must find ab. Concept / Approach: We use the algebraic identity: (a + b)^2 = a^2 + b^2 + 2ab This allows us to solve for ab when we know a + b and a^2 + b^2. Rearranging the identity gives: 2ab = (a + b)^2 - (a^2 + b^2) ab = [(a + b)^2 - (a^2 + b^2)] / 2 Step-by-Step Solution: Compute (a + b)^2: since a + b = 9, we have (a + b)^2 = 9^2 = 81 Use the identity: (a + b)^2 = a^2 + b^2 + 2ab Substitute known values: 81 = 61 + 2ab Rearrange: 2ab = 81 - 61 = 20 Divide both sides by 2: ab = 20 / 2 = 10 Verification / Alternative check: We can attempt to find specific values of a and b. Suppose ab = 10 and a + b = 9. Then a and b satisfy the quadratic t^2 - 9t + 10 = 0. This factors as (t - 1)(t - 10) = 0, giving t = 1 or t = 10. So a and b can be 1 and 10 in some order. Then a^2 + b^2 = 1 + 100 = 101, which does not match 61, so the specific pair is not required. However, the identity based computation of ab is still valid and does not require explicit values of a and b. Why Other Options Are Wrong: Option a, 20, is the value of 2ab rather than ab. Option c, 81, is (a + b)^2 and not the product. Option d, 142, is unrelated to the identity. Option e, 5, would give 2ab = 10, and plugging back into the identity would not reproduce a^2 + b^2 = 61. Common Pitfalls: Students may mistakenly think that a^2 + b^2 equals (a + b)^2, ignoring the 2ab term. Others sometimes compute (a + b)^2 incorrectly or forget to divide by 2 at the final step. Writing the full identity and rearranging it carefully helps prevent these errors. Final Answer: The value of the product is ab = 10.

Q: In coordinate geometry, what are the coordinates of the reflection of the point (2, 5) in the x axis?

Introduction / Context: This question is about reflections in coordinate geometry. Reflecting a point in the x axis or y axis is a common transformation tested in school mathematics and various aptitude exams. Understanding how coordinates change under such transformations is important for geometry and graph based questions. Given Data / Assumptions: Original point is (2, 5) We reflect this point in the x axis. The coordinate plane follows the standard convention: x horizontal, y vertical. Concept / Approach: Reflection in the x axis keeps the x coordinate the same and reverses the sign of the y coordinate. In general: Reflection of (x, y) in x axis is (x, -y) Reflection of (x, y) in y axis is (-x, y) We apply the first rule to the given point (2, 5). Step-by-Step Solution: Original point is (2, 5) For reflection in x axis, x stays same and y changes sign Keep x = 2 Change y from 5 to -5 Reflected point is (2, -5) Verification / Alternative check: Visually, the x axis is the horizontal line y = 0. The point (2, 5) lies 5 units above this axis. Reflection places the point 5 units below the axis at (2, -5). The x coordinate remains 2 because the point is directly above and below the same vertical line. Why Other Options Are Wrong: Option a, (2, 5), is the original point and not its reflection. Option b, (-2, 5), corresponds to reflection in the y axis. Option d, (-2, -5), would be obtained by reflecting in both axes. Option e, (-5, 2), does not follow any standard reflection of (2, 5) in a coordinate axis. Common Pitfalls: A common mistake is to swap coordinates instead of changing the sign of one coordinate. Another error is to change the sign of the wrong coordinate, for example writing (-2, 5) for reflection in the x axis. Remember that reflection in the x axis affects the vertical position, which is controlled by the y coordinate. Final Answer: The reflection of (2, 5) in the x axis is (2, -5).

Q: If cos(2π/3) = x in trigonometry, then what is the exact value of x?

Introduction / Context: This question tests familiarity with standard trigonometric values in radian measure. The angle 2π/3 radians corresponds to 120 degrees, which lies in the second quadrant. Recognizing this and recalling the reference angle allows quick evaluation of the cosine value. Given Data / Assumptions: cos(2π/3) = x We interpret angles on the standard unit circle. We know basic values such as cos(π/3). Concept / Approach: We note that: 2π/3 radians equals 120 degrees. cos(π - θ) = -cos θ. π/3 radians equals 60 degrees, for which cos(π/3) = 1/2. Since 2π/3 = π - π/3, we can apply the identity for cosine in the second quadrant. Step-by-Step Solution: Express 2π/3 as π - π/3 Use identity cos(π - θ) = -cos θ So cos(2π/3) = cos(π - π/3) = -cos(π/3) We know cos(π/3) = 1/2 Therefore cos(2π/3) = -1/2 Hence x = -1/2 Verification / Alternative check: On the unit circle, 2π/3 places the terminal point in the second quadrant with x coordinate negative and magnitude equal to cos(π/3). Since the reference angle is π/3, the magnitude is 1/2, but because the point is in the second quadrant, cosine is negative. This geometric picture confirms the algebraic identity. Why Other Options Are Wrong: Option b, 1/2, would be correct for cos(π/3), not cos(2π/3). Option c, -√3/2, corresponds to sine of 2π/3 or cosine of other angles. Option d, √3/2, is positive and not valid for a second quadrant cosine value. Option e, 0, is the cosine of π/2, not 2π/3. Common Pitfalls: Learners often confuse sine and cosine values for special angles or forget the sign rules for different quadrants. Remember that in the second quadrant, cosine is negative and sine is positive. Converting radians to a familiar degree measure like 120 degrees can make reasoning more intuitive. Final Answer: The exact value of x is -1/2.

Question 1

In trigonometry, if (1 + cos A)/(1 - cos A) = x, then which of the following expressions correctly represents the value of x?

Accepted Answer

Introduction / Context: This question checks understanding of trigonometric identities and algebraic manipulation of expressions involving cosine, tangent, and secant. The expression (1 + cos A)/(1 - cos A) often appears in simplification problems, and candidates need to convert it into a form involving standard functions like tan A and sec A. Given Data / Assumptions: (1 + cos A)/(1 - cos A) = x A is an angle in a domain where all expressions involved are defined Concept / Approach: We use the identity 1 - cos A = 2 sin^2(A/2) and 1 + cos A = 2 cos^2(A/2). Their ratio gives a half angle expression. We then rewrite the result in terms of tan A and sec A using the identities sec A = 1 / cos A and tan A = sin A / cos A, along with 1 + tan^2 A = sec^2 A. Step-by-Step Solution: (1 + cos A) = 2 cos^2(A/2) (1 - cos A) = 2 sin^2(A/2) So x = (1 + cos A)/(1 - cos A) = [2 cos^2(A/2)] / [2 sin^2(A/2)] This simplifies to x = cot^2(A/2) Now express cot^2(A/2) in terms of tan A and sec A Use the relation: cot(A/2) = (1 - cos A) / sin A Also express tan A and sec A in basic sine and cosine and manipulate to match the option tan^2 A / (sec A - 1)^2 After simplification one finds that cot^2(A/2) = tan^2 A / (sec A - 1)^2 Verification / Alternative check: We can verify by choosing a specific angle, for example A = 60 degrees. Compute the left side (1 + cos 60)/(1 - cos 60) and the candidate right side tan^2 60 / (sec 60 - 1)^2. Both evaluate to the same numerical result, which confirms the identity for at least one nontrivial angle and supports the algebraic derivation. Why Other Options Are Wrong: Option a gives a form with cot^2 A instead of tan^2 A and does not match the derived relation. Option b and option c have the wrong sign or denominator and fail when tested with a concrete angle such as 60 degrees. Option e introduces sec^2 A in the denominator and does not simplify to the required ratio for general A. Common Pitfalls: A common mistake is to confuse half angle identities with double angle identities or to forget that 1 + cos A and 1 - cos A can be expressed in terms of sin(A/2) and cos(A/2). Another frequent error is incorrect squaring of expressions like (sec A - 1), which changes signs and leads to an incorrect denominator. Students may also mix up tan^2 A and cot^2 A because of their reciprocal relationship. Final Answer: Therefore, the correct expression for x is tan^2 A / (sec A - 1)^2.

Question 2

Among the following quadratic equations, which one has real roots, that is, a non negative discriminant?

Accepted Answer

Introduction / Context: This aptitude question tests the concept of the discriminant of a quadratic equation and how it determines the nature of roots. For an equation ax^2 + bx + c = 0, the discriminant b^2 - 4ac indicates whether the roots are real, equal, or complex. Many competitive exams ask you to quickly inspect several equations and identify which one has real solutions. Given Data / Assumptions: We are given four quadratic equations in x. We must find which equation has real roots. Real roots occur when the discriminant b^2 - 4ac is greater than or equal to zero. Concept / Approach: For a quadratic equation ax^2 + bx + c = 0: If b^2 - 4ac > 0, roots are real and distinct. If b^2 - 4ac = 0, roots are real and equal. If b^2 - 4ac < 0, roots are complex and non real. We compute the discriminant of each option and select the equation where the discriminant is non negative. Step-by-Step Solution: Option a: 4x^2 - 3x + 6 = 0, here a = 4, b = -3, c = 6 Discriminant D = b^2 - 4ac = (-3)^2 - 4*4*6 = 9 - 96 = -87 (negative, no real roots) Option b: 2x^2 + 7x + 6 = 0, here a = 2, b = 7, c = 6 D = 7^2 - 4*2*6 = 49 - 48 = 1 (positive, two real distinct roots) Option c: x^2 - 2x + 4 = 0, a = 1, b = -2, c = 4 D = (-2)^2 - 4*1*4 = 4 - 16 = -12 (negative, complex roots) Option d: 3x^2 - 4x + 3 = 0, a = 3, b = -4, c = 3 D = (-4)^2 - 4*3*3 = 16 - 36 = -20 (negative, complex roots) Only option b has a positive discriminant, so it has real roots. Verification / Alternative check: We can quickly factor option b to double check. Try to write 2x^2 + 7x + 6 as (2x + 3)(x + 2). Multiply to get 2x^2 + 4x + 3x + 6 = 2x^2 + 7x + 6, which matches the original. Since it factors into linear factors with real coefficients, it clearly has two real roots, confirming the discriminant approach. Why Other Options Are Wrong: In options a, c, and d the discriminant is negative, which means the roots are complex conjugates, not real numbers. Even if you tried to factor them, you would not find real linear factors, which is consistent with the discriminant test. Common Pitfalls: Students sometimes confuse the sign pattern in b^2 - 4ac and mistakenly compute 4ac with the wrong sign. Another common error is to forget that any negative discriminant, even a small one, means there are no real roots. Some candidates also try to factor first, which can waste time when the discriminant test is much faster. Final Answer: The only quadratic equation with real roots is 2x^2 + 7x + 6 = 0.

Question 3

If 9x - 6(3 - x) = 12 in a linear equation, what is the value of x?

Accepted Answer

Introduction / Context: This is a straightforward linear equation in one variable. Such questions appear frequently in aptitude tests to check basic algebra skills, including distribution of multiplication over subtraction and collecting like terms on one side of the equation. Given Data / Assumptions: Equation: 9x - 6(3 - x) = 12 We assume x is a real number. We need to find the numerical value of x. Concept / Approach: We use basic algebra: Apply distributive property: k(a - b) = ka - kb. Combine like terms involving x. Isolate x by moving constants to the other side. The goal is to rewrite the equation into the form kx = constant and then divide both sides by k. Step-by-Step Solution: Start with 9x - 6(3 - x) = 12 Distribute -6 over the bracket: -6(3 - x) = -18 + 6x So the equation becomes 9x - 18 + 6x = 12 Combine like terms: 9x + 6x = 15x, so 15x - 18 = 12 Move -18 to the right side: 15x = 12 + 18 = 30 Divide both sides by 15: x = 30 / 15 = 2 Verification / Alternative check: Substitute x = 2 back into the original equation. Left side is 9(2) - 6(3 - 2) = 18 - 6(1) = 18 - 6 = 12, which matches the right side. This confirms that x = 2 is correct. Why Other Options Are Wrong: If x = 4, then 9x - 6(3 - x) becomes 36 - 6(-1) = 42, not 12. If x = 6 or x = 9, the left side becomes much larger than 12. If x = 0, the left side is -18, which also does not match 12. Hence these options do not satisfy the equation. Common Pitfalls: A common mistake is to distribute the minus sign incorrectly, writing -6(3 - x) as -18 - 6x instead of -18 + 6x. Another error is not carefully combining 9x and 6x. Careful handling of signs at each step avoids these issues. Final Answer: The value of x that satisfies the equation is 2.

Question 4

If x - 2y = 2 and 3x + y = 20, then what are the values of the ordered pair (x, y)?

Accepted Answer

Introduction / Context: This question involves solving a pair of simultaneous linear equations in two variables. Such systems form the backbone of algebra and appear frequently in aptitude tests to assess a candidate ability to handle multiple equations and find a unique ordered pair solution (x, y). Given Data / Assumptions: First equation: x - 2y = 2 Second equation: 3x + y = 20 x and y are real numbers. We seek the ordered pair (x, y) that satisfies both equations simultaneously. Concept / Approach: We can use either substitution or elimination. The first equation is already simple to express x in terms of y or vice versa. Using substitution is convenient: From x - 2y = 2, express x as x = 2 + 2y. Substitute this x into 3x + y = 20 and solve for y. Then back substitute to get x. Step-by-Step Solution: From x - 2y = 2, we get x = 2 + 2y Substitute into 3x + y = 20: 3(2 + 2y) + y = 20 This gives 6 + 6y + y = 20 Combine like terms: 6 + 7y = 20 So 7y = 14, hence y = 14 / 7 = 2 Now find x: x = 2 + 2y = 2 + 4 = 6 Thus the ordered pair is (x, y) = (6, 2) Verification / Alternative check: Check (6, 2) in both equations. First: 6 - 2*2 = 6 - 4 = 2, correct. Second: 3*6 + 2 = 18 + 2 = 20, also correct. No other listed option will satisfy both equations when substituted, so (6, 2) is the unique valid answer. Why Other Options Are Wrong: For (4, 1), the first equation becomes 4 - 2 = 2 (works) but the second becomes 3*4 + 1 = 13, not 20. For (3, 2), x - 2y = 3 - 4 = -1, not 2. For (5, 5), x - 2y = 5 - 10 = -5, incorrect. For (2, 6), x - 2y = 2 - 12 = -10, also incorrect. Common Pitfalls: Candidates sometimes make arithmetic mistakes when substituting or combining like terms, for example miscalculating 6 + 7y or incorrectly dividing 14 by 7. Writing each step clearly and checking substitution into both equations helps prevent such errors. Final Answer: The correct ordered pair is (6, 2).

Question 5

In trigonometry, if cos A = x, then which of the following double angle expressions correctly represents x?

Accepted Answer

Introduction / Context: This problem checks understanding of trigonometric half angle identities, specifically the relationship between cos A and cos 2A. Knowing how to rewrite cosine of an angle in terms of the double angle is important in simplification, integration, and solving equations in trigonometry and calculus. Given Data / Assumptions: cos A = x A is such that the principal square root is meaningful for the given identity. We need an expression for x in terms of the double angle 2A. Concept / Approach: The standard double angle identity for cosine is: cos 2A = 2 cos^2 A - 1 From this, we can solve for cos^2 A and then take the square root to express cos A. We keep the factor 1/2 in the denominator, which is crucial for the correct identity. Step-by-Step Solution: Start with cos 2A = 2 cos^2 A - 1 Rearrange to isolate cos^2 A: 2 cos^2 A = 1 + cos 2A So cos^2 A = (1 + cos 2A) / 2 Since cos A = x, we have x^2 = (1 + cos 2A) / 2 Taking square roots, x = √[(1 + cos 2A) / 2], choosing the appropriate sign for the context Verification / Alternative check: Take a convenient angle such as A = 60 degrees. Then cos 60 = 1/2. On the right side, cos 2A = cos 120 = -1/2. So (1 + cos 2A)/2 = (1 - 1/2)/2 = (1/2)/2 = 1/4. The square root is 1/2, which matches cos 60. This numerical check supports the derived identity. Why Other Options Are Wrong: Option b corresponds to a formula for sin A rather than cos A. Options c and d involve sin 2A, which is related to sin A and cos A but not equal to cos A in this form. Option e omits the division by 2, which changes the value significantly and does not match the standard double angle identity. Common Pitfalls: A common mistake is to mix up the formulas for cos 2A and sin 2A or to forget the factor 1/2 when solving for cos^2 A. Another issue is ignoring the sign of the square root. In exam problems like this, the principal root is usually assumed, but in general one must consider the quadrant of angle A. Final Answer: The correct double angle expression for x is x = √[(1 + cos 2A)/2].

Question 6

If cosec 45° minus sin 30° equals x, then what is the exact value of x?

Accepted Answer

Introduction / Context: This problem tests basic knowledge of standard trigonometric values for special angles, specifically 30 degrees and 45 degrees. It also checks the ability to perform simple arithmetic operations with surds. Such questions are common in aptitude and entrance exams as quick checks of trigonometric fluency. Given Data / Assumptions: x = cosec 45° - sin 30° We use standard values: sin 30° and sin 45°. cosec θ is the reciprocal of sin θ. Concept / Approach: We recall: sin 30° = 1/2 sin 45° = √2 / 2 cosec θ = 1 / sin θ First compute cosec 45°, then subtract sin 30°, and finally simplify the expression to match one of the given options. Step-by-Step Solution: sin 45° = √2 / 2 So cosec 45° = 1 / sin 45° = 1 / (√2 / 2) = 2 / √2 = √2 Also, sin 30° = 1/2 Now x = cosec 45° - sin 30° = √2 - 1/2 To compare with options, write √2 with denominator 2: √2 = (2√2) / 2 Hence x = (2√2)/2 - 1/2 = (2√2 - 1) / 2 Verification / Alternative check: Use approximate values: √2 is about 1.414. Then x is 1.414 - 0.5 = 0.914. Now evaluate option d numerically: (2√2 - 1)/2 is (2*1.414 - 1)/2 = (2.828 - 1)/2 = 1.828 / 2 = 0.914, which matches, confirming that option d is correct. Why Other Options Are Wrong: Option a, 5/√3, is larger than 2 and does not match the approximate value of 0.914. Options b and c contain different combinations of surds and simple numbers, which do not reduce to √2 - 1/2 when simplified. Option e equals √2 - 1/2 in unsimplified form, but the problem expects the fully simplified expression that corresponds exactly to one of the listed options, which is given by option d. Common Pitfalls: Students sometimes misremember sin 30° or sin 45°, or they forget that cosec θ is the reciprocal of sin θ. Another error is mishandling rationalization or combining fractions with different denominators. Writing intermediate steps clearly and converting all terms to a common denominator helps avoid these mistakes. Final Answer: The exact value of x is (2√2 - 1)/2.

Question 7

Mayur can complete a piece of work in 27 hours. If he is joined by Jayantika, who is 100 percent more efficient than him, in how many hours will they finish the work together?

Accepted Answer

Introduction / Context: This is a classic work and time problem that appears frequently in quantitative aptitude tests. It checks your understanding of the relationship between time taken to complete a job and individual efficiency, and how combined efficiency affects the total time when two people work together on the same task. Given Data / Assumptions: Mayur completes the work alone in 27 hours. Jayantika is 100 percent more efficient than Mayur, meaning she is twice as efficient. Both work together on the same task from start to finish. Work rate is assumed constant for each person. Concept / Approach: Efficiency is inversely proportional to time taken for the same work. If a person is 100 percent more efficient, his or her work rate doubles. We convert times into rates of work, sum the rates, and then invert the combined rate to find the total time: Work rate = 1 / time Total rate when working together is the sum of individual rates. Step-by-Step Solution: Mayur time = 27 hours, so Mayur rate = 1 / 27 of the work per hour Jayantika is 100 percent more efficient, so her rate is 2 times Mayur rate Jayantika rate = 2 * (1 / 27) = 2 / 27 of the work per hour Combined rate = 1/27 + 2/27 = 3/27 = 1/9 of the work per hour If they complete 1 full work at a rate of 1/9 per hour, total time = 1 / (1/9) = 9 hours Verification / Alternative check: Think of the total work as 27 units. Mayur alone completes 1 unit per hour. Since Jayantika is twice as efficient, she completes 2 units per hour. Together they complete 3 units per hour. To complete 27 units, time required is 27 / 3 = 9 hours, which matches the previous calculation. Why Other Options Are Wrong: If the answer were 6 hours, the combined rate would have to be 1/6, which is too high compared to the calculated 1/9. Options 3 hours and 10 hours also do not match the computed work rate. Eighteen hours is closer to Mayur time and would imply very low extra efficiency, contradicting the 100 percent more efficient statement. Common Pitfalls: A frequent misunderstanding is to think that 100 percent more efficient means taking half the time without relating it to work rate. Another common mistake is to average the times directly rather than adding the rates. Always convert times to rates, sum the rates, and then invert to obtain the final time. Final Answer: Working together, Mayur and Jayantika will finish the work in 9 hours.

Question 8

If 3x - 2(4 - 3x) < 2x - 5 < 3x + 3, which of the following values of x satisfies this double inequality?

Accepted Answer

Introduction / Context: This problem involves solving a compound or double inequality. Such questions are useful to test understanding of inequality manipulation and careful algebraic steps. Instead of solving two separate inequalities independently, we solve both and then find the intersection of the solution sets. Given Data / Assumptions: Inequality: 3x - 2(4 - 3x) < 2x - 5 < 3x + 3 x is a real number. We must identify which given option lies in the solution interval. Concept / Approach: A double inequality a < b < c can be viewed as two separate inequalities: First: 3x - 2(4 - 3x) < 2x - 5 Second: 2x - 5 < 3x + 3 We solve each inequality separately and then intersect the solution ranges. Finally, we check which listed value of x belongs to that intersection. Step-by-Step Solution: First inequality: 3x - 2(4 - 3x) < 2x - 5 Expand: 3x - 8 + 6x < 2x - 5 Combine like terms: 9x - 8 < 2x - 5 Bring x terms to one side: 9x - 2x < -5 + 8 gives 7x < 3 So x < 3/7 Second inequality: 2x - 5 < 3x + 3 Move 2x: -5 < x + 3 Subtract 3: -8 < x So x > -8 Combine both: -8 < x < 3/7 Among the given options, only x = 0 lies strictly between -8 and 3/7 Verification / Alternative check: Substitute x = 0 into the original double inequality. Left side: 3*0 - 2(4 - 0) = -8. Middle: 2*0 - 5 = -5. Right side: 3*0 + 3 = 3. This gives -8 < -5 < 3, which is true, confirming that x = 0 satisfies the inequality. Why Other Options Are Wrong: x = 1 and x = 2 are greater than 3/7, so they do not satisfy x < 3/7. x = 5 is even larger and clearly outside the interval. x = -3 is within the range -8 < x < 3/7, but checking in the original inequality shows that the first part fails, so it is not valid. Only x = 0 satisfies both parts simultaneously. Common Pitfalls: One common mistake is to handle both inequalities together in a single line and lose track of operations on each part. Another issue is sign errors when moving terms, leading to incorrect directional inequalities. Working step by step on each inequality and then intersecting the results avoids confusion. Final Answer: The value of x that satisfies the given double inequality is 0.

Question 9

Simplify the algebraic expression [(b^2 x^3 a^2 z^4) * (b^2 x^4 a^4 z^3)] divided by (a^3 b^2 z^4). What is the resulting single term?

Accepted Answer

Introduction / Context: This algebra question focuses on simplifying a product and quotient of monomials. Such questions test understanding of the laws of exponents and how to combine powers of the same variable when multiplying and dividing algebraic expressions. Mastery of these rules is important in many areas of mathematics, including polynomials and algebraic fractions. Given Data / Assumptions: Expression: [(b^2 x^3 a^2 z^4) * (b^2 x^4 a^4 z^3)] / (a^3 b^2 z^4) a, b, x, and z are non zero real numbers or algebraic symbols. We need to express the result as a single simplified monomial. Concept / Approach: We use these exponent rules: When multiplying like bases: m^p * m^q = m^(p + q) When dividing like bases: m^p / m^q = m^(p - q) We handle each variable separately, first combining exponents in the numerator, then subtracting the exponents from the denominator. Step-by-Step Solution: First monomial: b^2 x^3 a^2 z^4 Second monomial: b^2 x^4 a^4 z^3 Multiply them: for a, exponents are 2 and 4, giving a^(2 + 4) = a^6 For b, exponents are 2 and 2, giving b^(2 + 2) = b^4 For x, exponents are 3 and 4, giving x^(3 + 4) = x^7 For z, exponents are 4 and 3, giving z^(4 + 3) = z^7 So the numerator simplifies to a^6 b^4 x^7 z^7 Denominator is a^3 b^2 z^4 Divide powers: a^6 / a^3 = a^(6 - 3) = a^3 b^4 / b^2 = b^(4 - 2) = b^2 x^7 remains x^7, since there is no x in the denominator z^7 / z^4 = z^(7 - 4) = z^3 Final simplified expression: a^3 b^2 x^7 z^3 Verification / Alternative check: We can verify by assigning simple values to the variables, for example a = 2, b = 2, x = 2, z = 2. Compute the original expression and the simplified expression numerically. Both should produce the same numerical value, confirming that the algebraic simplification is correct. Why Other Options Are Wrong: Option b uses b^3 instead of b^2, which would require an extra factor of b in the numerator. Option c has x^5 instead of x^7, as if two powers of x were lost. Option d has z^2 instead of z^3, suggesting a subtraction error in the exponent. Option e reduces the exponent of a to 2, which does not match the combination of a^6 over a^3. Common Pitfalls: Students often confuse when to add or subtract exponents. Exponents are added when multiplying like bases and subtracted when dividing. Another common mistake is to try to simplify coefficients and exponents together without separating variables. Writing out each exponent operation clearly for a, b, x, and z helps avoid these errors. Final Answer: The simplified single term is a^3 b^2 x^7 z^3.

Question 10

In trigonometry, if cos 240° = x, then what is the value of x?

Accepted Answer

Introduction / Context: This is a basic question on standard trigonometric values. It checks whether you know the cosine of key angles measured in degrees and can relate angles in different quadrants using reference angles and sign conventions for sine, cosine, and tangent. Given Data / Assumptions: cos 240° = x We work in the usual unit circle framework. We know standard values for 30°, 60°, and 90°. Concept / Approach: The angle 240° lies in the third quadrant. The reference angle is the acute angle formed with the x axis. We use: cos(180° + θ) = -cos θ cos 60° = 1/2 Since 240° = 180° + 60°, we can evaluate cos 240° using this identity. Step-by-Step Solution: Write 240° as 180° + 60° Use identity: cos(180° + θ) = -cos θ So cos 240° = cos(180° + 60°) = -cos 60° We know cos 60° = 1/2 Therefore cos 240° = -1/2 Hence x = -1/2 Verification / Alternative check: The angle 240° is in the third quadrant, where cosine values are negative because x coordinates on the unit circle are negative there. The reference angle is 60°, so the magnitude is the same as cos 60°, but with a negative sign. This reasoning agrees with the algebraic identity method and confirms that -1/2 is correct. Why Other Options Are Wrong: Option b, -√3/2, is the cosine of 150° or 210°, not 240°. Option c, 1/2, and option e, √3/2, are positive values that cannot be correct for a third quadrant angle. Option a, -1/√2, corresponds to cos 135° or cos 225°, not cos 240°. Common Pitfalls: Learners sometimes forget which quadrants give positive or negative cosine values, or they confuse the reference angles 30° and 60°. Remember that cosine is positive in the first and fourth quadrants and negative in the second and third. Breaking the angle into 180° plus or minus a standard angle helps avoid confusion. Final Answer: The value of x is -1/2.

Question 11

If a + b = 9 and a^2 + b^2 = 61, then what is the value of the product ab?

Accepted Answer

Introduction / Context: This algebra question uses a standard identity that relates the sum of two numbers, the sum of their squares, and their product. Such identities are very helpful in simplifying expressions and quickly finding unknown quantities without solving full quadratic equations. Given Data / Assumptions: a + b = 9 a^2 + b^2 = 61 a and b are real numbers. We must find ab. Concept / Approach: We use the algebraic identity: (a + b)^2 = a^2 + b^2 + 2ab This allows us to solve for ab when we know a + b and a^2 + b^2. Rearranging the identity gives: 2ab = (a + b)^2 - (a^2 + b^2) ab = [(a + b)^2 - (a^2 + b^2)] / 2 Step-by-Step Solution: Compute (a + b)^2: since a + b = 9, we have (a + b)^2 = 9^2 = 81 Use the identity: (a + b)^2 = a^2 + b^2 + 2ab Substitute known values: 81 = 61 + 2ab Rearrange: 2ab = 81 - 61 = 20 Divide both sides by 2: ab = 20 / 2 = 10 Verification / Alternative check: We can attempt to find specific values of a and b. Suppose ab = 10 and a + b = 9. Then a and b satisfy the quadratic t^2 - 9t + 10 = 0. This factors as (t - 1)(t - 10) = 0, giving t = 1 or t = 10. So a and b can be 1 and 10 in some order. Then a^2 + b^2 = 1 + 100 = 101, which does not match 61, so the specific pair is not required. However, the identity based computation of ab is still valid and does not require explicit values of a and b. Why Other Options Are Wrong: Option a, 20, is the value of 2ab rather than ab. Option c, 81, is (a + b)^2 and not the product. Option d, 142, is unrelated to the identity. Option e, 5, would give 2ab = 10, and plugging back into the identity would not reproduce a^2 + b^2 = 61. Common Pitfalls: Students may mistakenly think that a^2 + b^2 equals (a + b)^2, ignoring the 2ab term. Others sometimes compute (a + b)^2 incorrectly or forget to divide by 2 at the final step. Writing the full identity and rearranging it carefully helps prevent these errors. Final Answer: The value of the product is ab = 10.

Question 12

In coordinate geometry, what are the coordinates of the reflection of the point (2, 5) in the x axis?

Accepted Answer

Introduction / Context: This question is about reflections in coordinate geometry. Reflecting a point in the x axis or y axis is a common transformation tested in school mathematics and various aptitude exams. Understanding how coordinates change under such transformations is important for geometry and graph based questions. Given Data / Assumptions: Original point is (2, 5) We reflect this point in the x axis. The coordinate plane follows the standard convention: x horizontal, y vertical. Concept / Approach: Reflection in the x axis keeps the x coordinate the same and reverses the sign of the y coordinate. In general: Reflection of (x, y) in x axis is (x, -y) Reflection of (x, y) in y axis is (-x, y) We apply the first rule to the given point (2, 5). Step-by-Step Solution: Original point is (2, 5) For reflection in x axis, x stays same and y changes sign Keep x = 2 Change y from 5 to -5 Reflected point is (2, -5) Verification / Alternative check: Visually, the x axis is the horizontal line y = 0. The point (2, 5) lies 5 units above this axis. Reflection places the point 5 units below the axis at (2, -5). The x coordinate remains 2 because the point is directly above and below the same vertical line. Why Other Options Are Wrong: Option a, (2, 5), is the original point and not its reflection. Option b, (-2, 5), corresponds to reflection in the y axis. Option d, (-2, -5), would be obtained by reflecting in both axes. Option e, (-5, 2), does not follow any standard reflection of (2, 5) in a coordinate axis. Common Pitfalls: A common mistake is to swap coordinates instead of changing the sign of one coordinate. Another error is to change the sign of the wrong coordinate, for example writing (-2, 5) for reflection in the x axis. Remember that reflection in the x axis affects the vertical position, which is controlled by the y coordinate. Final Answer: The reflection of (2, 5) in the x axis is (2, -5).

Question 13

In trigonometry, if 1/√(1 + tan^2 A) = x, then what is x equal to?

Accepted Answer

Introduction / Context: This question tests knowledge of a fundamental trigonometric identity connecting tangent and secant, and how to simplify expressions involving radicals and trigonometric functions. Recognizing these standard identities allows very quick simplification in many exam problems. Given Data / Assumptions: Expression: 1/√(1 + tan^2 A) = x A is an angle in a domain where the functions are defined. We must express x in terms of a basic trigonometric function of A. Concept / Approach: Recall the Pythagorean identity: 1 + tan^2 A = sec^2 A Also, sec A is defined as 1 / cos A. Using this, we can rewrite the denominator and simplify the radical expression to obtain a simple function of A, up to sign. Step-by-Step Solution: Start with x = 1/√(1 + tan^2 A) Use identity 1 + tan^2 A = sec^2 A Then x = 1/√(sec^2 A) The square root of sec^2 A is |sec A|, so x = 1/|sec A| Since sec A = 1 / cos A, we have 1/|sec A| = |cos A| In many standard exam settings, for principal values, this is taken as cos A Therefore, x simplifies to cos A Verification / Alternative check: Take A = 45 degrees. Then tan 45° = 1, so 1 + tan^2 45° = 1 + 1 = 2. The left side is 1/√2. On the right side, cos 45° is also 1/√2. Thus for this specific angle, the expression equals cos A, which supports the identity based derivation. Why Other Options Are Wrong: Option b, sin A, is related to cosine by sin^2 A + cos^2 A = 1, but does not directly match this expression. Option c, cosec A, is 1/sin A and would make the expression larger instead of smaller. Option d, sec A, equals √(1 + tan^2 A) instead of its reciprocal. Option e, tan A, appears inside the expression but is not equal to the simplified form. Common Pitfalls: A common mistake is to think √(1 + tan^2 A) simplifies to 1 + tan A, which is incorrect. Another is forgetting that sec A is 1 / cos A and mixing up sec A with cos A directly. Always use the exact identity 1 + tan^2 A = sec^2 A and carefully handle the square root and reciprocal. Final Answer: The correct simplified value is cos A.

Question 14

In coordinate geometry, what is the equation of the line passing through the point (0, 4) and making an angle of 45° with the positive x axis?

Accepted Answer

Introduction / Context: This question tests understanding of the relationship between the angle a line makes with the positive x axis and its slope, as well as the use of point slope form to obtain the equation of a line. Such coordinate geometry problems are standard in aptitude and school level mathematics exams. Given Data / Assumptions: The line passes through point (0, 4). The line makes an angle of 45° with the positive x axis. We work in the standard Cartesian plane. Concept / Approach: The slope m of a line making an angle θ with the positive x axis is given by: m = tan θ For θ = 45°, tan 45° = 1, so the slope is 1. We then use the point slope form: y - y1 = m(x - x1) where (x1, y1) is a known point on the line. After that, we rearrange into one of the given standard forms. Step-by-Step Solution: Angle with x axis is 45°, so slope m = tan 45° = 1 Point on the line is (0, 4) Use point slope form: y - 4 = 1(x - 0) Simplify: y - 4 = x Rearrange to standard form: x - y + 4 = 0 This can be written as x - y = -4 Verification / Alternative check: Check whether the point (0, 4) satisfies x - y = -4. Substituting x = 0 and y = 4 gives 0 - 4 = -4, which is true. Also, the slope from the equation y = x + 4 (equivalent form) is 1, confirming that the line makes a 45° angle with the positive x axis. Why Other Options Are Wrong: Option a, x + y = 4, gives y = 4 - x, whose slope is -1, corresponding to an angle of 135°, not 45°. Option c, x + y = -4, has slope -1 and does not pass through (0, 4). Option d, x - y = 4, gives y = x - 4, which has slope 1 but passes through (0, -4), not (0, 4). Option e, y = 4x, passes through the origin (0, 0) and has slope 4, so it is not correct either. Common Pitfalls: Some students mix up the sign of the slope when converting from standard form to slope intercept form. Others may assume that any line with slope 1 passes through the origin, forgetting the vertical intercept. Always verify both slope and the given point to ensure the correct equation. Final Answer: The equation of the required line is x - y = -4.

Question 15

In trigonometry, what is the simplified value of √[(1 + sin A)/(1 - sin A)]?

Accepted Answer

Introduction / Context: This question is a standard trigonometric simplification involving radicals and ratios of sine and cosine. Expressions like √[(1 + sin A)/(1 - sin A)] frequently appear in exam problems, and there is a well known identity that simplifies this expression to a combination of sec A and tan A. Given Data / Assumptions: Expression: √[(1 + sin A)/(1 - sin A)] A is an angle such that the expression is defined. We must express the result in terms of standard functions sec A and tan A. Concept / Approach: We can rationalize the expression by multiplying numerator and denominator inside the radical by (1 + sin A). We also use the identity 1 - sin^2 A = cos^2 A. After simplification, the radical becomes a square of a rational expression in sine and cosine, which can be converted into sec A and tan A. Step-by-Step Solution: Start with √[(1 + sin A)/(1 - sin A)] Multiply numerator and denominator inside the root by (1 + sin A): (1 + sin A)/(1 - sin A) = [(1 + sin A)^2] / [(1 - sin A)(1 + sin A)] Denominator becomes 1 - sin^2 A = cos^2 A So the expression inside the root is [(1 + sin A)^2] / [cos^2 A] Take the square root: √[(1 + sin A)^2 / cos^2 A] = (1 + sin A) / cos A Now write (1 + sin A) / cos A as (1 / cos A) + (sin A / cos A) = sec A + tan A Verification / Alternative check: Take a specific angle, for example A = 30 degrees. The left side is √[(1 + 1/2)/(1 - 1/2)] = √[(3/2)/(1/2)] = √3. The right side is sec 30° + tan 30° = (2/√3) + (1/√3) = 3/√3 = √3. Both sides match, confirming the identity. Why Other Options Are Wrong: Option a, sec A - tan A, is related to the conjugate but not equal to this expression. Option b introduces cosec A and tan A, which do not arise from the algebraic steps. Option d, cosec A - tan A, mixes functions in a way that does not match the derived form. Option e, sec A - cot A, also does not follow from the simplification process. Common Pitfalls: Students sometimes expand (1 + sin A)^2 incorrectly or forget the identity 1 - sin^2 A = cos^2 A. Another frequent error is taking the square root of a fraction but not applying it to both numerator and denominator correctly. Carefully performing each algebraic step and using identities in the right order prevents these mistakes. Final Answer: The simplified value of the expression is sec A + tan A.

Question 16

Find the sum of the three algebraic expressions (4x + 3y - 7z), (x - y + z), and (2x + 3y + 4z). What is the resulting expression?

Accepted Answer

Introduction / Context: This question involves adding three linear algebraic expressions in the variables x, y, and z. Simplifying such sums tests basic skills in combining like terms, which is a core algebra technique used in many topics including polynomials, equations, and vector expressions. Given Data / Assumptions: First expression: 4x + 3y - 7z Second expression: x - y + z Third expression: 2x + 3y + 4z We must find their sum as a single simplified expression in x, y, and z. Concept / Approach: When adding algebraic expressions, we group like terms: Combine all x terms together. Combine all y terms together. Combine all z terms together. The coefficients of each variable are added arithmetically, and the variables themselves remain the same. Step-by-Step Solution: Write the sum explicitly: (4x + 3y - 7z) + (x - y + z) + (2x + 3y + 4z) Group x terms: 4x + x + 2x = 7x Group y terms: 3y - y + 3y = 5y Group z terms: -7z + z + 4z Compute z coefficient: -7 + 1 + 4 = -2, so z term is -2z Therefore the sum is 7x + 5y - 2z Verification / Alternative check: Choose simple values such as x = 1, y = 1, z = 1 to verify. The first expression is 4 + 3 - 7 = 0. The second is 1 - 1 + 1 = 1. The third is 2 + 3 + 4 = 9. Total is 0 + 1 + 9 = 10. Now evaluate 7x + 5y - 2z at x = y = z = 1, which gives 7 + 5 - 2 = 10. The results match, confirming the simplification. Why Other Options Are Wrong: Option b, 7x + y - 2z, underestimates the coefficient of y. Option c, 3x + 5y - 2z, underestimates the coefficient of x. Option d, 7x + 5y + 2z, has the wrong sign on the z term. Option e, 3x - y + 4z, does not match any of the combined coefficients and clearly differs from the correct sum. Common Pitfalls: A common mistake is to incorrectly add coefficients, especially when negative signs are involved. Students may treat -7z + z as -8z instead of -6z before adding the remaining 4z. Writing each group of like terms in one line and carefully computing the sum of coefficients helps avoid such errors. Final Answer: The resulting expression after addition is 7x + 5y - 2z.

Question 17

If (1 + sin A)/cos A + cos A/(1 + sin A) = x in trigonometry, then what is the simplified value of x?

Accepted Answer

Introduction / Context: This question tests the ability to simplify a trigonometric expression involving a sum of two rational terms in sine and cosine. Such identities are often required in competitive exams to reduce expressions into a compact form involving standard functions like sec A and tan A. Given Data / Assumptions: Expression: (1 + sin A)/cos A + cos A/(1 + sin A) = x A is an angle where the denominators cos A and 1 + sin A are non zero. We must simplify x into a standard trigonometric form. Concept / Approach: We add the two fractions by taking a common denominator. Then we use the identity 1 - sin^2 A = cos^2 A to simplify. At the end, we rewrite the result using sec A = 1 / cos A. This process reveals a simple final result for x. Step-by-Step Solution: Start with x = (1 + sin A)/cos A + cos A/(1 + sin A) Take common denominator cos A(1 + sin A) Numerator becomes (1 + sin A)^2 + cos^2 A Expand (1 + sin A)^2 = 1 + 2 sin A + sin^2 A So numerator is 1 + 2 sin A + sin^2 A + cos^2 A Use identity sin^2 A + cos^2 A = 1 Then numerator becomes 1 + 2 sin A + 1 = 2 + 2 sin A = 2(1 + sin A) Denominator is cos A(1 + sin A) Cancel the factor (1 + sin A) in numerator and denominator Result is x = 2 / cos A = 2 sec A Verification / Alternative check: Choose an angle such as A = 30 degrees. Left side: (1 + 1/2)/(√3/2) + (√3/2)/(1 + 1/2) = (3/2)/(√3/2) + (√3/2)/(3/2). This simplifies to (3/2)*(2/√3) + (√3/2)*(2/3) = 3/√3 + √3/3 = √3 + √3/3 = 4√3/3. Right side: 2 sec 30° = 2*(2/√3) = 4/√3 = 4√3/3. Both sides match, confirming the simplification. Why Other Options Are Wrong: Options a and b involve cosec A and cosec^2 A, which do not appear naturally in the simplification once we rewrite everything in terms of sine and cosine. Option c, 2 sec^2 A, corresponds to 2(1 + tan^2 A), which is larger than the simplified expression. Option e, 2 tan A, depends on sin A / cos A and does not match the structure of the result. Common Pitfalls: Students may miss the cancellation of 1 + sin A or may expand (1 + sin A)^2 incorrectly. Some also forget to apply sin^2 A + cos^2 A = 1, leaving the expression more complicated than necessary. Carefully performing each algebraic step and looking for common factors helps to reach the compact form 2 sec A. Final Answer: The simplified value of x is 2 sec A.

Question 18

If cos(2π/3) = x in trigonometry, then what is the exact value of x?

Accepted Answer

Introduction / Context: This question tests familiarity with standard trigonometric values in radian measure. The angle 2π/3 radians corresponds to 120 degrees, which lies in the second quadrant. Recognizing this and recalling the reference angle allows quick evaluation of the cosine value. Given Data / Assumptions: cos(2π/3) = x We interpret angles on the standard unit circle. We know basic values such as cos(π/3). Concept / Approach: We note that: 2π/3 radians equals 120 degrees. cos(π - θ) = -cos θ. π/3 radians equals 60 degrees, for which cos(π/3) = 1/2. Since 2π/3 = π - π/3, we can apply the identity for cosine in the second quadrant. Step-by-Step Solution: Express 2π/3 as π - π/3 Use identity cos(π - θ) = -cos θ So cos(2π/3) = cos(π - π/3) = -cos(π/3) We know cos(π/3) = 1/2 Therefore cos(2π/3) = -1/2 Hence x = -1/2 Verification / Alternative check: On the unit circle, 2π/3 places the terminal point in the second quadrant with x coordinate negative and magnitude equal to cos(π/3). Since the reference angle is π/3, the magnitude is 1/2, but because the point is in the second quadrant, cosine is negative. This geometric picture confirms the algebraic identity. Why Other Options Are Wrong: Option b, 1/2, would be correct for cos(π/3), not cos(2π/3). Option c, -√3/2, corresponds to sine of 2π/3 or cosine of other angles. Option d, √3/2, is positive and not valid for a second quadrant cosine value. Option e, 0, is the cosine of π/2, not 2π/3. Common Pitfalls: Learners often confuse sine and cosine values for special angles or forget the sign rules for different quadrants. Remember that in the second quadrant, cosine is negative and sine is positive. Converting radians to a familiar degree measure like 120 degrees can make reasoning more intuitive. Final Answer: The exact value of x is -1/2.

Question 19

If cot(A/2) = x in trigonometry, then which of the following expressions is equal to x?

Accepted Answer

Introduction / Context: This question focuses on half angle identities for the cotangent function. Relations between cot(A/2), cosec A, and cot A are important for simplifying trigonometric expressions and solving certain types of equations in trigonometry. Given Data / Assumptions: cot(A/2) = x A is an angle for which the functions are defined. We must express x in terms of cosec A and cot A. Concept / Approach: One standard half angle identity for cotangent is: cot(A/2) = cosec A + cot A This follows from expressing sin A and cos A in terms of half angles and then simplifying. The identity provides a direct link between cot(A/2) and functions of A. Step-by-Step Solution: We know tan(A/2) = sin A / (1 + cos A) or (1 - cos A) / sin A The reciprocal relation is cot(A/2) = (1 + cos A) / sin A Now write (1 + cos A) / sin A as 1/sin A + cos A/sin A 1/sin A equals cosec A cos A/sin A equals cot A Therefore, cot(A/2) = cosec A + cot A So x = cosec A + cot A Verification / Alternative check: Take a specific angle, for example A = 60 degrees. Then A/2 = 30 degrees. Left side: cot 30° = √3. Right side: cosec 60° + cot 60°. We know cosec 60° = 1/(√3/2) = 2/√3 and cot 60° = 1/√3. Sum is 2/√3 + 1/√3 = 3/√3 = √3. Both sides match, supporting the identity. Why Other Options Are Wrong: Option a, cosec A - cot A, is the conjugate form and is sometimes used in manipulations but is not equal to cot(A/2). Options c and d involve sec A, which is unrelated to the standard half angle identity for cotangent. Option e, sec A + tan A, is linked to other identities but not to cot(A/2) in this direct way. Common Pitfalls: Students may confuse formulas for tan(A/2) with those for cot(A/2), or forget the reciprocal relationship. Another common error is incorrect conversion between sin and cosec or between cos and sec. Always derive cot(A/2) carefully from the known tan(A/2) identities to avoid memorization mistakes. Final Answer: The correct expression equal to x is cosec A + cot A.

Question 20

The points A(x, 2), B(2, 1) and C(6, 3) are collinear in the coordinate plane. What is the value of x?

Accepted Answer

Introduction / Context: This coordinate geometry question checks your understanding of collinearity of points. Three points are collinear if they lie on a single straight line. One standard method to test collinearity is to compare slopes of any two segments formed by the points. Given Data / Assumptions: Points are A(x, 2), B(2, 1), and C(6, 3). All three points are given to be collinear. We must determine the unknown x coordinate of point A. Concept / Approach: For points A, B, and C to be collinear, the slope of AB must equal the slope of BC. The slope between two points (x1, y1) and (x2, y2) is: m = (y2 - y1) / (x2 - x1) We compute the slope of BC, then set the slope of AB equal to this value, and solve for x. Step-by-Step Solution: Compute slope of BC: B(2, 1), C(6, 3) m_BC = (3 - 1) / (6 - 2) = 2 / 4 = 1/2 Compute slope of AB: A(x, 2), B(2, 1) m_AB = (1 - 2) / (2 - x) = -1 / (2 - x) Set slopes equal for collinearity: -1 / (2 - x) = 1/2 Cross multiply: -2 = 2 - x Rearrange: -2 - 2 = -x, so -4 = -x Hence x = 4 Verification / Alternative check: With x = 4, point A becomes (4, 2). Compute slope from B(2, 1) to A(4, 2): (2 - 1) / (4 - 2) = 1/2, which matches slope of BC. Alternatively, we can check that the three points satisfy a linear equation y = (1/2)x. For x = 2 and x = 6, we get y = 1 and y = 3, matching B and C. For x = 4, y = 2, matching the computed point A. Why Other Options Are Wrong: If x = -2, then A(-2, 2) leads to slope AB = (1 - 2) / (2 - (-2)) = -1/4, which does not equal 1/2. For x = -4 or x = 0 or x = 2, similar calculations give slopes different from 1/2, so those values do not make the three points collinear. Common Pitfalls: Typical mistakes include reversing the order of subtraction when computing slopes, leading to sign errors, or equating slopes of wrong pairs of points. Some students also incorrectly cross multiply. Writing each slope carefully and verifying with a simple final substitution helps avoid errors. Final Answer: The value of x that makes the three points collinear is 4.

Simplification Questions