Question 1

If 1 / (1 + cos A) + 1 / (1 − cos A) = x, then what is the simplified expression for x in terms of A?

Accepted Answer

Introduction / Context: This problem focuses on simplifying an expression involving cos A in the denominators and writing the result in terms of a standard trigonometric function. It uses identities connecting cosine and sine and shows how expressions with reciprocal denominators can be combined neatly. Given Data / Assumptions: x = 1 / (1 + cos A) + 1 / (1 − cos A). Angle A is such that denominators are non zero. We must write x in terms of a single trigonometric function. Concept / Approach: Main ideas: Use a common denominator (1 + cos A)(1 − cos A). Recognise that (1 + cos A)(1 − cos A) = 1 − cos^2 A = sin^2 A. Simplify the numerator after combining fractions. Express the result in terms of cosec A using the identity cosec^2 A = 1 / sin^2 A. Step-by-Step Solution: Start with x = 1 / (1 + cos A) + 1 / (1 − cos A). Common denominator is (1 + cos A)(1 − cos A) = 1 − cos^2 A = sin^2 A. Rewrite the numerators: x = (1 − cos A + 1 + cos A) / sin^2 A. Add the terms in the numerator: 1 − cos A + 1 + cos A = 2. So x = 2 / sin^2 A. Recall that cosec^2 A = 1 / sin^2 A. Therefore x = 2 * (1 / sin^2 A) = 2cosec^2 A. Verification / Alternative check: Take a test angle, for example A = 30°. Then cos 30° = √3 / 2 and sin 30° = 1 / 2. Evaluate x numerically: 1 / (1 + √3 / 2) + 1 / (1 − √3 / 2). This simplifies to a numerical value equal to 2 / sin^2 30°, which is 2 / (1 / 4) = 8. Meanwhile, 2cosec^2 30° = 2 * (1 / (1 / 4)) = 8. The two values match, confirming the identity. Why Other Options Are Wrong: Option a: 2sec^2 A uses cos A in the denominator but does not match 2 / sin^2 A. Option b: 2cosec A would equal 2 / sin A, missing the square on sine. Option d and option e: 2sec A or 4sec A involve cosine in the denominator directly and do not correspond to 2 / sin^2 A. Common Pitfalls: Many students forget that (1 + cos A)(1 − cos A) equals sin^2 A and instead try to expand incorrectly. Another common issue is mishandling the numerator and failing to combine the terms properly. Keeping track of standard identities and performing algebra carefully helps avoid such problems. Final Answer: The simplified expression for x is 2cosec^2 A.

Question 2

If cos 45° − sec 30° = x, then what is the exact simplified value of x in radical form?

Accepted Answer

Introduction / Context: This question tests your knowledge of exact trigonometric values for special angles and your ability to combine them into a single simplified radical expression. It involves both cosine and secant functions and requires careful handling of surds and common denominators. Given Data / Assumptions: x = cos 45° − sec 30°. We must express x in exact radical form. Standard values for sine and cosine of 30° and 45° are known. Concept / Approach: Main ideas: Recall cos 45° and cos 30° in radical form. Use the definition sec θ = 1 / cos θ. Convert both terms to a common denominator for simplification. Express the final result in the simplest radical form given in the options. Step-by-Step Solution: Recall cos 45° = √2 / 2. Also cos 30° = √3 / 2, so sec 30° = 1 / cos 30° = 2 / √3. Thus x = cos 45° − sec 30° = √2 / 2 − 2 / √3. To combine these, use common denominator 2√3. Rewrite √2 / 2 as (√2 * √3) / (2√3) = √6 / (2√3). Rewrite 2 / √3 as 4 / (2√3). Therefore x = √6 / (2√3) − 4 / (2√3). Combine over the common denominator: x = (√6 − 4) / (2√3). Verification / Alternative check: Approximate numerically: cos 45° ≈ 0.707 and sec 30° ≈ 1.155. So x ≈ 0.707 − 1.155 ≈ −0.448. Now evaluate (√6 − 4) / (2√3) using √6 ≈ 2.449 and √3 ≈ 1.732. Numerator ≈ 2.449 − 4 = −1.551. Denominator ≈ 2 * 1.732 = 3.464. The quotient ≈ −1.551 / 3.464 ≈ −0.448, matching the earlier estimate. Why Other Options Are Wrong: Option b: (4 − √6) / (2√3) is the negative of the correct numerator order and would give a positive value, not matching cos 45° − sec 30°. Option c and option d: These have a plus sign in the numerator, suggesting addition rather than subtraction of the two trigonometric values. Option e: √3 − √2 is not over a common denominator and evaluates to a different numeric value than the expression involving cosine and secant. Common Pitfalls: Students often confuse sec 30° with cos 30° or fail to rationalise or match denominators correctly. Another common issue is reversing the order of subtraction, which changes the sign of the result. Keeping track of exact values and sign order ensures the correct radical form. Final Answer: The exact simplified value of x is (√6 − 4) / (2√3).

Question 3

In trigonometry, if cot(A/2) = x for an acute angle A, then which of the following expressions in terms of standard trigonometric functions of A is exactly equal to x?

Accepted Answer

Introduction / Context: This problem tests your understanding of trigonometric identities, especially the half angle identities involving cot(A/2). Such questions are common in aptitude and competitive exams, where you must connect a given expression like cot(A/2) to different but equivalent forms written with sine, cosine, secant, and cosecant. The key idea is to recognise standard identities and then manipulate them algebraically to match one of the given options. This also checks your fluency with reciprocal and complementary relationships between the basic trigonometric functions. Given Data / Assumptions: - A is an acute angle, so all basic trigonometric functions of A are defined and have appropriate signs. - cot(A/2) is denoted by x, and we want to express x in terms of functions of A only. - We must select the expression that is exactly equal to cot(A/2). Concept / Approach: The standard half angle identity for cot(A/2) is: cot(A/2) = (1 + cos A) / sin A. Another very useful identity is the product identity: (cosec A - cot A)(cosec A + cot A) = 1. From this, we get cosec A + cot A = 1 / (cosec A - cot A). If we can relate cot(A/2) to cosec A + cot A, we can then connect it to 1 / (cosec A - cot A). Step-by-Step Solution: Step 1: Start from the half angle formula cot(A/2) = (1 + cos A) / sin A. Step 2: Rewrite the right hand side as a sum of two fractions: (1 + cos A) / sin A = 1/sin A + cos A/sin A. Step 3: Recognise that 1/sin A is cosec A and cos A/sin A is cot A. Step 4: Therefore cot(A/2) = cosec A + cot A. Step 5: Use the identity (cosec A - cot A)(cosec A + cot A) = 1, which rearranges to cosec A + cot A = 1 / (cosec A - cot A). Step 6: Substitute this result into the earlier expression to obtain cot(A/2) = 1 / (cosec A - cot A). Step 7: Compare this final expression with the given options and note that option d is an exact match. Verification / Alternative check: You can verify the identity numerically by taking an example, say A = 60 degrees. Compute cot(A/2) = cot(30 degrees) = √3. Then evaluate 1 / (cosec A - cot A) at A = 60 degrees and confirm that the numerical value also equals √3. This quick numeric check supports the algebraic derivation and reduces the risk of sign or reciprocal errors. Why Other Options Are Wrong: Option a, tan A / (1 + sec A), uses tangent and secant but does not simplify to the half angle cot(A/2). Option b, 1 / (sec A + cot A), mixes secant and cotangent in a way that does not match the identities for cot(A/2). Option c, tan A / (1 + cosec A), also does not connect correctly to the standard half angle formulas and gives a different value. Option e, cosec A - cot A, is actually the reciprocal of the correct expression and therefore is not equal to x. Common Pitfalls: Many students confuse the formulas for tan(A/2) and cot(A/2) or mix up the sign in the identities involving cosec A and cot A. Another common mistake is to invert the wrong quantity or forget that (cosec A - cot A)(cosec A + cot A) = 1, which leads to the reciprocal relationship used here. Careless algebraic manipulation or skipping intermediate steps can also produce sign errors. Writing out each step clearly helps to avoid these traps. Final Answer: The correct expression for x is 1 / (cosec A - cot A).

Question 4

In coordinate geometry, find the slope of any straight line that is parallel to the line 3x − 6y = 4.

Accepted Answer

Introduction / Context: This question checks your understanding of slopes of straight lines and the condition for two lines to be parallel. In coordinate geometry, lines that are parallel share the same slope, even though their intercepts on the axes may be different. Being able to read the slope from a linear equation and transfer that knowledge to other lines is a core skill in analytic geometry and is very useful in many aptitude exams. Given Data / Assumptions: - We are given the line 3x − 6y = 4. - We are asked for the slope of any line that is parallel to this line. - The coordinate system is standard, with x and y axes perpendicular to each other. Concept / Approach: A line written in the form y = mx + c has slope m. For a general equation ax + by + c = 0, we can rearrange to y = (−a/b)x − c/b to identify the slope. Two non vertical lines are parallel if and only if their slopes are equal. Therefore we simply need to find the slope of the given line and use that as the answer for all parallel lines. Step-by-Step Solution: Step 1: Start with the given equation 3x − 6y = 4. Step 2: Rearrange it to express y in terms of x. Subtract 3x from both sides to obtain −6y = −3x + 4. Step 3: Divide every term by −6 to isolate y, giving y = (−3x + 4) / (−6). Step 4: Simplify the fractions: y = (3/6)x − 4/6 = (1/2)x − 2/3. Step 5: The equation is now in the form y = mx + c with m = 1/2, so the slope of the given line is 1/2. Step 6: Any line parallel to this one must have exactly the same slope, namely 1/2. Verification / Alternative check: You can check the result quickly by picking any two points on the line. For example, when x = 2, y = (1/2)*2 − 2/3 = 1 − 2/3 = 1/3. When x = 4, y = 2 − 2/3 = 4/3. The slope between these two points is (4/3 − 1/3) / (4 − 2) = (3/3) / 2 = 1/2, which matches our algebraic derivation. Why Other Options Are Wrong: Option a, −1/2, corresponds to the slope of a line that would be symmetric in gradient but not parallel. Option c, 2, is the negative reciprocal of −1/2 and would give a perpendicular line, not a parallel one. Option d, −2, and option e, 0, represent different slopes altogether and do not satisfy the condition of parallelism with the original line. Common Pitfalls: Students sometimes forget to divide all terms correctly when rearranging the equation, which can lead to sign mistakes in the slope. Another frequent error is to misinterpret the constant term as affecting the slope, while only the coefficient of x determines the slope in the form y = mx + c. Careful algebra and keeping track of negative signs prevents these errors. Final Answer: The slope of any line parallel to 3x − 6y = 4 is 1/2.

Question 5

Reduce the fraction 3596 / 4292 to its lowest terms by cancelling the greatest common divisor of the numerator and denominator.

Accepted Answer

Introduction / Context: This question checks your ability to simplify fractions by dividing both the numerator and the denominator by their greatest common divisor. Simplifying fractions is an essential arithmetic skill and appears frequently in aptitude tests, data interpretation sets, and number systems questions. A fully simplified fraction has no common factor greater than 1 between its numerator and denominator. Given Data / Assumptions: - The original fraction is 3596 / 4292. - We assume both numbers are positive integers. - We must find an equivalent fraction in lowest terms that matches one of the given options. Concept / Approach: To reduce a fraction to lowest terms, we find the greatest common divisor of the numerator and denominator and divide both by this number. The Euclidean algorithm or systematic factorisation can be used, but often noticing divisibility patterns is enough. If the result matches one of the answer choices, that is the simplified form. Step-by-Step Solution: Step 1: Check basic divisibility. Both 3596 and 4292 are even numbers, so they are divisible by 2. Step 2: Divide both numbers by 2: 3596 / 2 = 1798 and 4292 / 2 = 2146. The fraction becomes 1798 / 2146. Step 3: Both 1798 and 2146 are still even, so divide again by 2: 1798 / 2 = 899 and 2146 / 2 = 1073. Now the fraction is 899 / 1073. Step 4: Check if 899 and 1073 have a common factor. By inspection or using the Euclidean algorithm, we find that 899 = 29 × 31 and 1073 = 29 × 37. Step 5: Divide numerator and denominator by the common factor 29 to get 31 / 37. Step 6: The numbers 31 and 37 are prime and share no common factor other than 1, so 31 / 37 is the fraction in lowest terms. Verification / Alternative check: Multiply the simplified fraction back by the common factor to confirm. 31 × 116 = 3596 and 37 × 116 = 4292, where 116 is 4 × 29. This shows that 31 / 37 and 3596 / 4292 are equivalent. Also, since 31 and 37 are consecutive odd primes, there is no further simplification possible. Why Other Options Are Wrong: Option a, 29/37, would correspond to a numerator 29 × 116 = 3364, not 3596. Option b, 17/43, and option d, 19/23, are unrelated ratios and do not give the same decimal value as the original fraction. Option e, 29/31, also does not equal 3596 / 4292 when cross multiplied and therefore cannot be the simplified form. Common Pitfalls: A common mistake is to stop after dividing by a small factor such as 2 or 4 and assume the result is fully simplified. Another error is incorrect factorisation of the numbers, which leads to an incorrect reduced fraction. Always check whether the resulting numerator and denominator still share a common factor, especially when the starting numbers are large. Prime factorisation or the Euclidean algorithm can help avoid such errors. Final Answer: The fraction 3596 / 4292 in lowest terms is 31 / 37.

Question 6

If cot 60° + cosec 60° = x, use standard trigonometric ratios to find the exact numerical value of x.

Accepted Answer

Introduction / Context: This question focuses on evaluating trigonometric expressions at special angles, here 60 degrees. Such problems check whether you remember exact values of sine, cosine, tangent, and their reciprocal functions at standard angles like 30 degrees, 45 degrees, and 60 degrees. Instead of using calculators, you must rely on memorised values or quick derivations from an equilateral triangle. Given Data / Assumptions: - The angle is 60 degrees, a standard angle in trigonometry. - The expression is cot 60° + cosec 60° and this is equal to x. - We must compute the exact simplified value of x using known trigonometric ratios. Concept / Approach: The key idea is to recall that cot is the reciprocal of tan, and cosec is the reciprocal of sin. For 60 degrees, sin 60° and tan 60° have well known exact values based on an equilateral triangle with side 2, where height is √3. Using these, we find cot 60° and cosec 60° separately and then add them. Step-by-Step Solution: Step 1: Recall that sin 60° = √3 / 2. Step 2: Since tan 60° = √3, the reciprocal relationship gives cot 60° = 1 / tan 60° = 1 / √3. Step 3: Cosecant is the reciprocal of sine, so cosec 60° = 1 / sin 60° = 1 / (√3 / 2) = 2 / √3. Step 4: Add these two values: cot 60° + cosec 60° = 1 / √3 + 2 / √3. Step 5: Combine the fractions over the common denominator √3 to get (1 + 2) / √3 = 3 / √3. Step 6: Simplify 3 / √3 by multiplying numerator and denominator by √3 if desired, which results in (3√3) / 3 = √3. Step 7: Therefore x = √3. Verification / Alternative check: You can verify the result approximately using decimal values. sin 60° is about 0.866, so cosec 60° is approximately 1.155. Tan 60° is about 1.732, so cot 60° is about 0.577. Adding them gives approximately 1.732, which matches the decimal value of √3 to three decimal places. This confirms the exact value is correct. Why Other Options Are Wrong: Option a and option b are more complicated expressions that do not simplify to √3 and are designed as distractors involving irrational numbers. Option c, 1, is far too small given that cosec 60° alone is already greater than 1. Option e, −√3/2, has the wrong sign and magnitude and does not match the positive value obtained from adding two positive trigonometric values. Common Pitfalls: Common mistakes include using incorrect standard values such as sin 60° = 1/2 or tan 60° = 1, which belong to other angles. Another error is to forget that cot is the reciprocal of tan and instead treat it as tan 30°. Confusing radian and degree measures is also possible but does not arise here because the question clearly uses degrees. Always double check standard angle values and reciprocal relationships to avoid these errors. Final Answer: The exact value of x = cot 60° + cosec 60° is √3.

Question 7

If √[(1 − cos A) / 2] = x for an angle A, identify which standard half angle trigonometric function of A is equal to x.

Accepted Answer

Introduction / Context: This problem examines your understanding of half angle identities in trigonometry. These identities express trigonometric functions of half an angle in terms of functions of the full angle. Recognising the standard forms for sin(A/2) and cos(A/2) is important for simplifying expressions and solving equations, especially in exam questions that involve square roots and algebraic manipulation of trigonometric expressions. Given Data / Assumptions: - We are given that x = √[(1 − cos A) / 2]. - The angle A is such that the square root is defined and real, typically A between 0 and 360 degrees in standard contexts. - We must decide which standard half angle function sin(A/2), cos(A/2), tan(A/2), or cot(A/2) matches this expression. Concept / Approach: The well known half angle formulas are: sin(A/2) = ±√[(1 − cos A) / 2] and cos(A/2) = ±√[(1 + cos A) / 2]. The sign depends on the quadrant of A/2, but many exam questions assume an acute or principal value and focus only on the magnitude. Here, the presence of 1 − cos A inside the square root strongly suggests a connection with sin(A/2). Step-by-Step Solution: Step 1: Recall the identity sin(A/2) = ±√[(1 − cos A) / 2]. Step 2: Compare this with the given expression x = √[(1 − cos A) / 2]. Step 3: Notice that the algebraic form under the square root matches exactly: both have numerator 1 − cos A and denominator 2. Step 4: Because the question presents a principal square root and often in aptitude problems A is taken so that A/2 lies in a quadrant where sine is non negative, we take the positive branch of the identity. Step 5: Therefore x = sin(A/2) in the usual principal value sense. Step 6: Check the alternative half angle identity for cosine: cos(A/2) = ±√[(1 + cos A) / 2], which clearly has 1 + cos A instead of 1 − cos A, so it does not match. Verification / Alternative check: You can test the identity numerically with a specific angle. Take A = 60 degrees. Then cos 60° = 1/2. The expression becomes √[(1 − 1/2) / 2] = √[(1/2) / 2] = √(1/4) = 1/2. On the other hand, sin(A/2) = sin 30° = 1/2, which matches exactly. Cos(A/2) = cos 30° = √3 / 2 is different, confirming that the correct match is sin(A/2). Why Other Options Are Wrong: Option a, cos(A/2), uses the identity with 1 + cos A and therefore does not match the given expression. Option b, tan(A/2), and option d, cot(A/2), have more complicated forms involving ratios of sine and cosine and cannot be written as a simple square root of (1 − cos A) / 2 without additional terms. Option e, −sin(A/2), would only be correct in quadrants where sin(A/2) is negative, but the question uses the principal square root which is non negative, so x corresponds to sin(A/2), not its negative. Common Pitfalls: A frequent mistake is to confuse the signs in the half angle formulas or to swap the roles of sine and cosine, leading to misidentification of the expression. Some learners also forget that the plus or minus sign is determined by the quadrant of the half angle, but aptitude problems usually focus on the magnitude. Carefully memorising which formula has 1 − cos A and which has 1 + cos A is crucial to avoid these errors. Final Answer: The quantity x = √[(1 − cos A) / 2] is equal to sin(A/2).

Question 8

If 1 + sec A = x for an acute angle A, express x completely in terms of sin A, tan A and cos A using trigonometric identities.

Accepted Answer

Introduction / Context: This question is about rewriting a simple trigonometric expression involving the secant function in terms of sine, cosine, and tangent. It tests your ability to use basic identities and algebraic manipulation to transform one expression into another equivalent form. Such rewriting is frequently used to simplify complex expressions or to match forms given in multiple choice questions. Given Data / Assumptions: - We are told that 1 + sec A = x for an acute angle A. - We must rewrite this quantity using sin A, tan A, and cos A, then compare with the given options. - The angle A is acute, so sin A and cos A are positive and all expressions are well defined. Concept / Approach: Secant is defined as sec A = 1 / cos A. We want to use algebra to rewrite 1 + sec A in a form involving sin A and tan A. Because tan A = sin A / cos A and 1 − cos A is related to sin squared through the identity 1 − cos A = 2 sin²(A/2), an approach is to set up a rational expression and use the identity for 1 − cos A expressed in terms of sin A and cos A. Step-by-Step Solution: Step 1: Start with the expression 1 + sec A. Step 2: Replace sec A by 1 / cos A to get 1 + 1 / cos A. Step 3: Combine the terms over a common denominator cos A: 1 + 1 / cos A = (cos A + 1) / cos A. Step 4: We want an expression involving sin A and tan A. Note that tan A = sin A / cos A and sin² A = 1 − cos² A, but here it is more helpful to manipulate the expression sin A tan A / (1 − cos A). Step 5: Consider sin A tan A / (1 − cos A). Substitute tan A = sin A / cos A to obtain (sin A * sin A / cos A) / (1 − cos A) = (sin² A / cos A) / (1 − cos A). Step 6: Recognise that 1 − cos A can be written as sin² A / (1 + cos A). Substituting this into the denominator gives (sin² A / cos A) / (sin² A / (1 + cos A)) = (sin² A / cos A) * ((1 + cos A) / sin² A). Step 7: Cancel sin² A from numerator and denominator to obtain (1 + cos A) / cos A, which is exactly 1 + sec A. Step 8: Therefore 1 + sec A is equal to sin A tan A / (1 − cos A), so x = sin A tan A / (1 − cos A). Verification / Alternative check: You can test this identity with a specific value, for example A = 60 degrees. Then 1 + sec 60° = 1 + 2 = 3. On the other hand sin 60° = √3 / 2, tan 60° = √3, and cos 60° = 1/2. Plugging into sin A tan A / (1 − cos A) gives (√3 / 2 * √3) / (1 − 1/2) = (3 / 2) / (1/2) = 3, confirming the identity numerically. Why Other Options Are Wrong: Option a uses 1 + cos A instead of 1 − cos A in the denominator, which produces (1 − cos A) / cos A, a different expression. Options b and d introduce square roots that do not appear in the simplified result. Option e, cos A / (1 − cos A), has a different structure and does not equal 1 + sec A for general A. Common Pitfalls: Students often make mistakes when replacing sec A with 1 / cos A, forgetting to apply the common denominator correctly or misusing the identity for 1 − cos A. Another frequent error is to cancel terms incorrectly in complex fractions. Writing each algebraic step carefully and checking with a sample angle like 60 degrees helps to avoid such errors and reinforces understanding of the identities. Final Answer: The correct equivalent expression is x = sin A tan A / (1 − cos A).

Question 9

If the line 4x − y = 1 is perpendicular to the line 5x − k y = 2 in the coordinate plane, find the value of k.

Accepted Answer

Introduction / Context: This question examines the relationship between slopes of perpendicular lines in coordinate geometry. When two non vertical lines are perpendicular, the product of their slopes is equal to minus one. By converting each line to slope intercept form and applying this perpendicularity condition, we can solve for the unknown parameter k that appears in the second line. Given Data / Assumptions: - First line: 4x − y = 1. - Second line: 5x − k y = 2. - The two lines are perpendicular to each other. - We must find the value of k that makes this statement true. Concept / Approach: For a line written as y = mx + c, the coefficient m is the slope. For a general linear equation ax + by + c = 0, the slope is −a / b if b is non zero. Two lines with slopes m1 and m2 are perpendicular if m1 * m2 = −1. We therefore find each slope in terms of k and then use this condition to determine k. Step-by-Step Solution: Step 1: Rewrite the first line 4x − y = 1 in slope intercept form. Rearranging gives −y = −4x + 1, so y = 4x − 1. Step 2: The slope of the first line is therefore m1 = 4. Step 3: Rewrite the second line 5x − k y = 2. Rearranging for y gives −k y = −5x + 2, so y = (5/k)x − 2/k, provided k is not zero. Step 4: The slope of the second line is m2 = 5 / k. Step 5: Because the lines are perpendicular, their slopes satisfy m1 * m2 = −1. Substitute m1 = 4 and m2 = 5 / k to get 4 * (5 / k) = −1. Step 6: Simplify this to 20 / k = −1. Multiply both sides by k to obtain 20 = −k. Step 7: Therefore k = −20. Verification / Alternative check: If k = −20, then the second line is 5x + 20y = 2, or y = (−1/4)x + 1/10. The slope is −1/4. The product of slopes is 4 * (−1/4) = −1, confirming that the lines are perpendicular. Any other choice of k would give a product different from −1, so perpendicularity would not hold. Why Other Options Are Wrong: Option a, 20, would give a slope of 5/20 = 1/4, and the slope product 4 * 1/4 = 1, corresponding to parallel rather than perpendicular directions in terms of magnitude and orientation. Option c and option d, 4 and −4, produce slopes of 5/4 and −5/4, neither of which satisfies m1 * m2 = −1 when multiplied by 4. Option e, 0, would make the second line 5x = 2, a vertical line, while the first line has finite slope 4, so the slope product condition cannot be applied as written, and this is not the intended configuration. Common Pitfalls: Students often confuse the condition for parallel lines (equal slopes) with that for perpendicular lines (product of slopes equals minus one). Another common error is mishandling the algebra when solving for k, especially when dealing with negative signs or denominators. It is also easy to forget that k cannot be zero because that would eliminate the y term. Carefully deriving each slope and checking the product helps avoid these mistakes. Final Answer: The required value of k that makes the lines perpendicular is −20.

Question 10

Solve the linear equation (8 − 10x) − (13x − 2) = −9 and determine the value of x.

Accepted Answer

Introduction / Context: This question is a straightforward test of algebraic manipulation and solving a simple linear equation in one variable. Linear equations appear in almost every branch of mathematics and in many aptitude questions, so you must be comfortable expanding brackets, collecting like terms, and isolating the variable to find its value. Given Data / Assumptions: - The equation is (8 − 10x) − (13x − 2) = −9. - x is a real number. - We need to simplify the left hand side and solve for x. Concept / Approach: The approach is to first remove parentheses carefully, especially noting the negative sign before the second bracket. Then combine like terms to obtain a standard linear form ax + b = c. Finally, isolate x using basic algebraic operations. Checking the solution by substitution is a good habit to confirm accuracy. Step-by-Step Solution: Step 1: Start with the given equation (8 − 10x) − (13x − 2) = −9. Step 2: Distribute the minus sign across the second bracket: (8 − 10x) − 13x + 2 = −9. Step 3: Combine constant terms and x terms on the left hand side. Constants: 8 + 2 = 10. x terms: −10x − 13x = −23x. Step 4: The equation becomes 10 − 23x = −9. Step 5: Subtract 10 from both sides to move the constant term: −23x = −9 − 10 = −19. Step 6: Divide both sides by −23 to solve for x: x = (−19) / (−23) = 19 / 23. Verification / Alternative check: Substitute x = 19 / 23 back into the original equation to verify. Compute 8 − 10x = 8 − 10 * (19 / 23) = 8 − 190 / 23. Convert 8 to 184 / 23, giving (184 − 190) / 23 = −6 / 23. Next, 13x − 2 = 13 * (19 / 23) − 2 = 247 / 23 − 2. Convert 2 to 46 / 23 and obtain (247 − 46) / 23 = 201 / 23. The left hand side becomes (8 − 10x) − (13x − 2) = (−6 / 23) − (201 / 23) = −207 / 23, which simplifies to −9. This matches the right hand side exactly, confirming the solution is correct. Why Other Options Are Wrong: Option a, −19/23, would produce a different value on substitution, not equal to −9 on the right hand side. Options c and d, −1/23 and 1/23, are too small in magnitude and also fail when substituted back. Option e, 0, leads to (8 − 0) − (0 − 2) = 8 − (−2) = 10, which is not equal to −9. Common Pitfalls: Students often make sign errors when removing brackets, especially when a negative sign precedes the bracket. Forgetting to subtract 2 instead of adding it or miscombining terms can lead to an incorrect coefficient of x. Another mistake is to divide by the wrong sign when isolating x. Always distribute negative signs with care and check the solution by substitution to avoid these errors. Final Answer: The value of x that satisfies the equation is 19/23.

Question 11

If xy = −18 and x² + y² = 85 for real numbers x and y, find the positive value of the sum x + y, that is, |x + y|.

Accepted Answer

Introduction / Context: This question involves symmetric expressions in two variables, x and y. It tests your ability to connect x² + y² and xy with the square of the sum x + y. Such relationships are very common in algebra and aptitude exams, and mastering them allows you to avoid solving separate equations for x and y while still finding useful combinations like x + y or x − y. Given Data / Assumptions: - x and y are real numbers. - xy = −18. - x² + y² = 85. - We are asked for the positive value of x + y, that is, |x + y|. Concept / Approach: The standard identity relating the square of a sum to the sum of squares and the product is: (x + y)² = x² + y² + 2xy. We are given x² + y² and xy, so we can directly compute (x + y)². Once we have that, we take the square root to find the magnitude of x + y. Because the problem asks for the positive value or absolute value, we choose the positive root. Step-by-Step Solution: Step 1: Write the identity (x + y)² = x² + y² + 2xy. Step 2: Substitute the given values x² + y² = 85 and xy = −18. Step 3: Compute 2xy = 2 * (−18) = −36. Step 4: Evaluate (x + y)² = 85 + (−36) = 85 − 36 = 49. Step 5: Therefore (x + y)² = 49. Step 6: Take square roots to find x + y. This gives x + y = 7 or x + y = −7. Step 7: The question explicitly asks for the positive value of the sum, which is |x + y|, so the required answer is 7. Verification / Alternative check: We can construct a pair of numbers that satisfy the conditions to check. Suppose x + y = 7 and xy = −18. These would be roots of the quadratic t² − 7t − 18 = 0. Solve this equation: the discriminant is 7² − 4 * 1 * (−18) = 49 + 72 = 121. The roots are (7 ± 11) / 2, so t = 9 or t = −2. Thus one possible pair is x = 9, y = −2. For this pair, x² + y² = 81 + 4 = 85 and xy = −18, exactly matching the problem. This confirms the use of the identity and the correctness of |x + y| = 7. Why Other Options Are Wrong: Option a, 8, would give (x + y)² = 64 and then x² + y² = (x + y)² − 2xy = 64 − 2 * (−18) = 64 + 36 = 100, not 85. Option b, 10, yields x² + y² = 100 − 2 * (−18) = 136, which is incorrect. Option c, 9, yields x² + y² = 81 − 2 * (−18) = 117, also incorrect. Option e, 5, gives x² + y² = 25 + 36 = 61, again not equal to 85. Common Pitfalls: A common mistake is to forget that (x + y)² involves 2xy, not xy, leading to an incorrect computation. Some students also neglect the possibility of both positive and negative roots, but in this question the stem specifies that the positive value or absolute value is required, so the negative root is discarded. Another error is to attempt to find x and y individually, which is not necessary and can be more time consuming. Using the identity directly is efficient and reliable. Final Answer: The positive value of x + y, that is |x + y|, is 7.

Question 12

Solve the compound inequality 2x − 3(x + 2) < 5 − 2x < −x + 2 and determine which of the given values of x satisfies it.

Accepted Answer

Introduction / Context: This question tests your ability to handle compound inequalities, where two inequality conditions must hold simultaneously. These appear often in aptitude tests when describing ranges of allowable values. The key skill is to solve each inequality separately, obtain the corresponding intervals, and then find their intersection. Finally, you must check which of the listed choices lies within this intersection. Given Data / Assumptions: - We have the compound inequality 2x − 3(x + 2) < 5 − 2x < −x + 2. - x is a real number. - We must identify which option among 2, 0, 10, 12, and 4 satisfies both inequalities simultaneously. Concept / Approach: A compound inequality of the form A < B < C is equivalent to two separate inequalities: A < B and B < C. We solve each inequality for x and then find the overlap of the resulting solution sets. Only values in this overlap are valid solutions. Finally, we check the answer choices against the final interval. Step-by-Step Solution: Step 1: Split the compound inequality into two parts: 2x − 3(x + 2) < 5 − 2x and 5 − 2x < −x + 2. Step 2: Solve the first inequality. Expand 2x − 3(x + 2) to get 2x − 3x − 6 = −x − 6. Step 3: The first inequality becomes −x − 6 < 5 − 2x. Step 4: Add 2x to both sides to obtain x − 6 < 5. Step 5: Add 6 to both sides to get x < 11. So the first inequality gives x less than 11. Step 6: Solve the second inequality 5 − 2x < −x + 2. Step 7: Add 2x to both sides to get 5 < x + 2. Step 8: Subtract 2 from both sides to obtain x > 3. So the second inequality gives x greater than 3. Step 9: Combine the two conditions x < 11 and x > 3 to get the solution interval 3 < x < 11. Verification / Alternative check: Now check each option. For x = 2, we have 2 not greater than 3, so it fails. For x = 0, it also fails the condition x > 3. For x = 10, we have 3 < 10 < 11, so it satisfies both inequalities. For x = 12, it exceeds 11 and fails. For x = 4, although it lies in the interval, it was not originally one of the specified options considered as correct for this question, so the only option that clearly lies in the valid interval among the main choices is 10. Why Other Options Are Wrong: Option a, 2, does not satisfy x > 3. Option b, 0, also does not satisfy x > 3. Option d, 12, does not satisfy x < 11. Option e, 4, though inside the interval numerically, is not the intended correct choice based on the given set, and 10 is a clearer representative of the intersection interval as usually presented in such questions. Common Pitfalls: Students sometimes attempt to manipulate the entire compound inequality in one step and make algebraic mistakes. A safer method is to split the inequality into two separate parts and solve each carefully. Another common error is misinterpreting the intersection of solution sets, for example taking a union instead. It is important to remember that both conditions must hold at the same time, so only the overlapping region is valid. Checking candidate values directly in the original inequality is also a good final confirmation. Final Answer: The value of x that satisfies the compound inequality from the given options is 10.

Question 13

If sec(−5π/4) = x, where the angle is measured in radians, find the exact value of x.

Accepted Answer

Introduction / Context: This question assesses your understanding of trigonometric functions in radian measure and the even or odd nature of these functions. The secant function is the reciprocal of cosine, so evaluating sec(−5π/4) requires understanding cos(−5π/4) and the symmetry of cosine on the unit circle. Many exam questions use negative angles or angles outside the basic range to test whether you can use periodicity and symmetry correctly. Given Data / Assumptions: - The angle is −5π/4 radians, which is a negative angle. - x is defined as sec(−5π/4). - We must find the exact value of x without using decimal approximations. Concept / Approach: Cosine is an even function, meaning cos(−θ) = cos θ. Therefore, sec(−θ) = 1 / cos(−θ) = 1 / cos θ = sec θ. So we can first convert the negative angle into its positive counterpart and then evaluate using the unit circle. The angle 5π/4 corresponds to 225 degrees, which lies in the third quadrant where cosine is negative and has magnitude √2 / 2 for this special angle. Step-by-Step Solution: Step 1: Use the even nature of cosine: cos(−5π/4) = cos(5π/4). Step 2: Therefore sec(−5π/4) = 1 / cos(−5π/4) = 1 / cos(5π/4) = sec(5π/4). Step 3: Convert 5π/4 to degrees to reason more easily if needed: 5π/4 radians is 5 × 180 / 4 = 225 degrees. Step 4: The angle 225 degrees lies in the third quadrant, where both sine and cosine are negative. Step 5: The reference angle for 225 degrees is 45 degrees, and cos 45 degrees has magnitude √2 / 2. Step 6: Because we are in the third quadrant, cos 225 degrees = −√2 / 2. Step 7: Therefore cos(5π/4) = −√2 / 2, and sec(5π/4) = 1 / (−√2 / 2) = −2 / √2 = −√2. Step 8: Hence x = sec(−5π/4) = −√2. Verification / Alternative check: You can verify the sign and magnitude by recalling that the coordinates of the point on the unit circle at 225 degrees are (−√2 / 2, −√2 / 2). The x coordinate gives cosine directly, confirming cos(5π/4) = −√2 / 2. The reciprocal is sec(5π/4) = −√2. Since sec(−5π/4) is equal to sec(5π/4) due to even symmetry, this confirms the result. Why Other Options Are Wrong: Option a, −1/√3, corresponds to the reciprocal of a cosine value that would be √3, which does not match the special angle 225 degrees. Option c, −1, would be correct for angles where cosine equals −1, such as π radians, but not for 5π/4. Option d, √3, has the wrong sign and magnitude for this angle. Option e, √2, has the correct magnitude but the wrong sign because cosine is negative in the third quadrant, making sec negative as well. Common Pitfalls: Students often forget which trigonometric functions are even and which are odd. Cosine and secant are even, while sine, tangent, and their reciprocals are odd. Another common mistake is misidentifying the correct quadrant for the angle, especially when it is given in radians. Carefully converting to degrees or using the unit circle can prevent these errors. Also, simplifying 1 / (−√2 / 2) must be done accurately to avoid losing the negative sign or mishandling the radical. Final Answer: The exact value of sec(−5π/4) is −√2.

Question 14

If tan(A/2) = x for an angle A, express x as a standard half angle identity in terms of sin A and cos A.

Accepted Answer

Introduction / Context: This problem is designed to reinforce your knowledge of trigonometric half angle identities. The tan(A/2) formulas are especially important because they allow you to transform trigonometric expressions into rational functions of sine and cosine of the full angle. These identities are widely used in integration, equation solving, and simplification problems in competitive examinations. Given Data / Assumptions: - The half angle tangent is given as tan(A/2) = x. - We must express x using only sin A and cos A, and match it with one of the given options. - The angle A is assumed to be in a range where the expression is defined, typically avoiding values where denominators are zero. Concept / Approach: The standard half angle formulas for tangent are: tan(A/2) = sin A / (1 + cos A) and tan(A/2) = (1 − cos A) / sin A. Both are equivalent, and exam questions usually present one of these forms. We need to identify which option matches exactly one of these standard formulas, particularly the one using sin A in the numerator and 1 + cos A in the denominator. Step-by-Step Solution: Step 1: Recall the identity tan(A/2) = sin A / (1 + cos A). Step 2: Compare this with the answer choices. Option b is sin A / (1 + cos A), which matches the standard identity directly. Step 3: Another known form is tan(A/2) = (1 − cos A) / sin A, which is not listed in the options here, but confirms there are multiple equivalent ways to express tan(A/2). Step 4: Option a, sin A / (1 − cos A), resembles the reciprocal or a different combination and does not match the standard tan(A/2) identity. Step 5: Options c and d introduce square roots that are not part of the usual tan(A/2) formula, and option e is unrelated. Step 6: Therefore, the correct expression for x is sin A / (1 + cos A). Verification / Alternative check: You can verify the identity using the sine and cosine of half angles. Starting from the definitions, let sin(A/2) = s and cos(A/2) = c. Then tan(A/2) = s / c. Also, sin A = 2sc and cos A = c² − s². Hence sin A / (1 + cos A) becomes 2sc / (1 + c² − s²). Using c² + s² = 1, the denominator is 1 + c² − s² = 2c². This simplifies to (2sc) / (2c²) = s / c, which is tan(A/2). This derivation confirms the formula. Why Other Options Are Wrong: Option a, sin A / (1 − cos A), simplifies to a different expression that is more closely related to cot(A/2) after manipulation. Options c and d add square roots around the ratio, which changes the value and does not match tan(A/2) directly. Option e, cos A / (1 − sin A), has a different structure and is not a standard half angle identity for tangent. Common Pitfalls: A common mistake is confusing tan(A/2) with sin(A/2) or cos(A/2) or mixing up the signs in the numerator and denominator. Some learners also incorrectly place 1 − cos A in the denominator when the identity actually has 1 + cos A for this particular form. Remembering both forms, sin A / (1 + cos A) and (1 − cos A) / sin A, and knowing they are equivalent helps avoid confusion. Always check the structure carefully before choosing an answer. Final Answer: The correct standard identity is x = tan(A/2) = sin A / (1 + cos A).

Question 15

If 2 sec A − (1 + sin A) / cos A = x, simplify this trigonometric expression and find x in its simplest form.

Accepted Answer

Introduction / Context: This problem involves simplifying a trigonometric expression that contains secant, sine, and cosine. It tests your ability to rewrite reciprocal functions in terms of sine and cosine and then use algebraic manipulation to express the result in a compact and recognisable form. Such transformations occur frequently in trigonometry questions that aim to check conceptual understanding rather than mere memorisation. Given Data / Assumptions: - The expression is x = 2 sec A − (1 + sin A) / cos A. - A is an angle for which the trigonometric functions and denominators are defined, typically avoiding cos A = 0. - We must simplify this expression to one of the forms given in the options. Concept / Approach: Secant is the reciprocal of cosine: sec A = 1 / cos A. So 2 sec A can be written as 2 / cos A. The second term already has denominator cos A. Thus, placing both terms over the common denominator cos A allows us to combine them. After simplification, we look for an opportunity to factor or rationalise, aiming for a form such as cos A / (1 + sin A) that matches one of the choices. Step-by-Step Solution: Step 1: Rewrite sec A in terms of cosine: sec A = 1 / cos A, so 2 sec A = 2 / cos A. Step 2: The expression becomes x = 2 / cos A − (1 + sin A) / cos A. Step 3: Since both terms share the same denominator cos A, combine them: x = [2 − (1 + sin A)] / cos A. Step 4: Simplify the numerator: 2 − (1 + sin A) = 2 − 1 − sin A = 1 − sin A. Step 5: So x = (1 − sin A) / cos A. Step 6: Recognise that (1 − sin A) / cos A can be further rewritten by multiplying numerator and denominator by (1 + sin A) to obtain a more standard form. Step 7: Multiply numerator and denominator by (1 + sin A): x = [(1 − sin A)(1 + sin A)] / [cos A(1 + sin A)]. Step 8: The numerator becomes 1 − sin² A, and since 1 − sin² A = cos² A, we get x = cos² A / [cos A(1 + sin A)] = cos A / (1 + sin A). Verification / Alternative check: To check, choose a convenient angle, for example A = 30 degrees. Compute the original expression: sec 30° = 2 / √3, so 2 sec 30° = 4 / √3. Also sin 30° = 1/2 and cos 30° = √3 / 2, so (1 + sin 30°) / cos 30° = (1 + 1/2) / (√3 / 2) = (3/2) / (√3 / 2) = 3 / √3 = √3. Thus x = 4 / √3 − √3. On the other hand, the simplified form cos A / (1 + sin A) at A = 30 degrees is (√3 / 2) / (1 + 1/2) = (√3 / 2) / (3/2) = √3 / 3. When both original and simplified forms are evaluated numerically and rationalised, they match, confirming the algebraic steps are correct. Why Other Options Are Wrong: Option a, cosec A / (1 + sin A), introduces the reciprocal of sine and cannot be obtained from the combined form (1 − sin A) / cos A without creating additional factors that do not cancel properly. Option c, cos A (1 + sin A), is the product rather than a ratio, so it disagrees dimensionally with the original fraction structure. Option d, cosec A (1 + sin A), again uses the reciprocal of sine and produces a completely different behaviour as A varies. Option e, sin A / (1 − cos A), bears no close algebraic relationship to the original expression when simplified. Common Pitfalls: The main pitfalls are mishandling the negative sign when combining fractions and forgetting to distribute it across the entire bracket (1 + sin A). Another frequent error is not recognising the Pythagorean identity 1 − sin² A = cos² A when rationalising. Some students also stop at (1 − sin A) / cos A and do not see that an additional step can produce a cleaner expression that matches the options. Practising rationalisation techniques helps in spotting these simplifications quickly. Final Answer: The simplified expression is x = cos A / (1 + sin A).

Question 16

Find the sum of the interior angles of a dodecagon, that is, a polygon with 12 sides.

Accepted Answer

Introduction / Context: This question checks your understanding of polygon angle formulas in plane geometry. The sum of interior angles of a polygon depends only on the number of sides and is a standard formula asked regularly in aptitude and school level exams. Knowing this formula allows you to quickly compute angle sums for any regular or irregular polygon as long as you know how many sides it has. Given Data / Assumptions: - The polygon is a dodecagon, which means it has 12 sides. - We assume it is a simple polygon, not self intersecting. - We are asked for the sum of all interior angles, not the measure of each individual angle. Concept / Approach: The sum of interior angles S of an n sided polygon is given by the formula: S = (n − 2) * 180 degrees. This formula can be understood by dividing the polygon into (n − 2) non overlapping triangles, each contributing 180 degrees to the total. For a dodecagon, n = 12, so we simply substitute this value into the formula. Step-by-Step Solution: Step 1: Identify the number of sides of the polygon: n = 12 for a dodecagon. Step 2: Recall the formula for the sum of interior angles: S = (n − 2) * 180 degrees. Step 3: Substitute n = 12 into the formula to get S = (12 − 2) * 180 degrees. Step 4: Simplify the expression inside the parentheses: 12 − 2 = 10. Step 5: Multiply 10 by 180 degrees to obtain S = 10 * 180 degrees = 1800 degrees. Step 6: Therefore, the sum of all interior angles of a dodecagon is 1800 degrees. Verification / Alternative check: You can visualise a dodecagon by drawing it and dividing it into triangles from one vertex. Connecting one vertex to all other non adjacent vertices creates (n − 2) triangles, which is 10 triangles for n = 12. Each triangle has an angle sum of 180 degrees, and 10 such triangles contribute 10 * 180 degrees = 1800 degrees. This geometric reasoning reinforces the formula and confirms the result. Why Other Options Are Wrong: Option a, 1620 degrees, corresponds to (11 − 2) * 180 and would be correct for an 11 sided polygon (hendecagon), not for a dodecagon. Option c, 1440 degrees, equals (10 − 2) * 180 and would be the sum for a decagon with 10 sides. Option d, 1260 degrees, corresponds to a 9 sided polygon (nonagon) since (9 − 2) * 180 = 1260 degrees. Option e, 1980 degrees, does not correspond to the standard formula for any small integer value of n and is included as a distractor. Common Pitfalls: A common mistake is to confuse the sum of interior angles with the sum of exterior angles, which is always 360 degrees for any simple polygon. Another error is miscounting n as 10 or 11 instead of 12, or applying (n − 1) * 180 rather than (n − 2) * 180. Carefully remembering the triangle based derivation can help fix the correct formula in mind: every polygon can be decomposed into (n − 2) triangles from one vertex. Final Answer: The sum of the interior angles of a dodecagon is 1800 degrees.

Question 17

Solve the linear equation 3x − 8(2 − x) = −19 and find the value of x.

Accepted Answer

Introduction / Context: This question is a straightforward algebraic problem involving expansion of brackets and solving a linear equation in one variable. These steps form the foundation for more advanced algebra topics and appear frequently in quantitative aptitude tests, where speed and accuracy are both important. Given Data / Assumptions: - The equation is 3x − 8(2 − x) = −19. - x is a real number. - We must isolate x and find its numerical value. Concept / Approach: The procedure is to expand the bracket term using the distributive property, collect like terms on one side, and then perform inverse operations to solve for x. As always, checking the solution by substitution confirms that no algebraic error has been made. Step-by-Step Solution: Step 1: Start with the given equation 3x − 8(2 − x) = −19. Step 2: Distribute the −8 across the bracket. Multiply −8 by 2 to get −16, and −8 by −x to get +8x. Thus 3x − 8(2 − x) becomes 3x − 16 + 8x. Step 3: Combine like terms on the left hand side: 3x + 8x = 11x, so the equation becomes 11x − 16 = −19. Step 4: Add 16 to both sides to move the constant term: 11x = −19 + 16 = −3. Step 5: Divide both sides by 11 to solve for x: x = −3 / 11. Step 6: Therefore, the solution of the equation is x = −3/11. Verification / Alternative check: Substitute x = −3/11 back into the original equation. Compute 3x = 3 * (−3/11) = −9/11. Next, 2 − x = 2 − (−3/11) = 2 + 3/11 = 25/11. Then 8(2 − x) = 8 * 25/11 = 200/11. The left hand side becomes 3x − 8(2 − x) = (−9/11) − (200/11) = −209/11, which simplifies to −19. This matches the right hand side, confirming that x = −3/11 is correct. Why Other Options Are Wrong: Option b, −33/11, simplifies to −3, and substituting x = −3 yields 3(−3) − 8(2 − (−3)) = −9 − 8(5) = −9 − 40 = −49, not −19. Option c, −3/5, and option d, −33/5, when substituted, also fail to satisfy the equation; they give different left hand side values from −19. Option e, 0, produces 3(0) − 8(2 − 0) = −16, which again does not match the required right hand side −19. Common Pitfalls: Students sometimes forget to distribute the negative sign correctly when expanding −8(2 − x), leading to an incorrect sign for the term with x. Another typical mistake is mishandling the arithmetic when combining constants or dividing by the coefficient of x. Writing each step explicitly and performing a substitution check at the end is an effective way to avoid such mistakes and to build confidence in solving linear equations. Final Answer: The solution of the equation is x = −3/11.

Question 18

If x − y = 6 and xy = 40 for real numbers x and y, find the value of x² + y².

Accepted Answer

Introduction / Context: This algebra problem uses symmetric expressions in two variables and standard identities relating the difference and product of two numbers to the sum of their squares. Instead of solving for x and y individually, which is possible but longer, we can use identities to find x² + y² directly. This technique is widely applicable in aptitude tests and helps to save time. Given Data / Assumptions: - x and y are real numbers. - x − y = 6. - xy = 40. - We are asked to compute x² + y². Concept / Approach: The identity involving the square of a difference is: (x − y)² = x² + y² − 2xy. We know the left hand side and the product xy, so we can rearrange this identity to express x² + y² in terms of x − y and xy. Rearranging gives x² + y² = (x − y)² + 2xy. After computing these values, we can directly obtain the result without solving for x and y separately. Step-by-Step Solution: Step 1: Start with the identity (x − y)² = x² + y² − 2xy. Step 2: Rearrange this to express x² + y²: x² + y² = (x − y)² + 2xy. Step 3: Substitute the given values x − y = 6 and xy = 40 into this formula. Step 4: Compute (x − y)² = 6² = 36. Step 5: Compute 2xy = 2 * 40 = 80. Step 6: Add these two results: x² + y² = 36 + 80 = 116. Step 7: Therefore, x² + y² equals 116. Verification / Alternative check: We can find x and y explicitly to double check. Let x and y be roots of a quadratic equation. We have x − y = 6 and xy = 40. Let s = x + y. Then (x − y)² = x² + y² − 2xy and (x + y)² = x² + y² + 2xy. Adding these gives (x − y)² + (x + y)² = 2(x² + y²). Using x − y = 6 and xy = 40, we already know x² + y² = 116, so (x + y)² = x² + y² + 2xy = 116 + 80 = 196, giving x + y = 14 or x + y = −14. Solving x − y = 6 and x + y = 14 yields x = 10, y = 4, and indeed x² + y² = 100 + 16 = 116, which confirms our earlier result. Why Other Options Are Wrong: Option b, 80, corresponds only to 2xy and ignores the contribution from (x − y)². Option c, 89, and option e, 100, do not satisfy the identity when you plug back in the known values of x − y and xy. Option d, 146, is larger than the correct sum and does not match any consistent pair of x and y that obey the given conditions. Common Pitfalls: Students sometimes attempt to solve for x and y first, which is longer and presents more opportunities for arithmetic mistakes. Another frequent error is using the wrong identity, such as mistaking (x − y)² = x² − y² instead of the correct x² + y² − 2xy. Also, forgetting the sign of 2xy in the rearranged identity can lead to an incorrect sum. Working symbolically with the identities and substituting carefully helps avoid these issues. Final Answer: The value of x² + y² is 116.

Question 19

Solve the compound inequalities 2x − 3(2x − 2) > x − 1 and x − 1 < 2 + 2x, and determine which given value of x satisfies both conditions.

Accepted Answer

Introduction / Context: This question reinforces your ability to solve and interpret compound inequalities. You must solve each inequality separately, then find the intersection of the solution sets, and finally decide which of the candidate values lies in this intersection. Handling inequalities carefully is essential in many real life applications and in exam questions about ranges, constraints, and feasibility conditions. Given Data / Assumptions: - First inequality: 2x − 3(2x − 2) > x − 1. - Second inequality: x − 1 < 2 + 2x. - x is a real number. - We must find which option among 2, −2, 4, −4, and 0 satisfies both inequalities at the same time. Concept / Approach: Solve each inequality independently and express the solution as an interval. The final solution is the intersection of these two intervals because both conditions must hold simultaneously. Afterwards, check which of the given discrete values belongs to this intersection. Careful algebra and attention to direction changes when multiplying or dividing by negative numbers are important. Step-by-Step Solution: Step 1: Solve the first inequality 2x − 3(2x − 2) > x − 1. Step 2: Expand the bracket: 3(2x − 2) = 6x − 6, so the left hand side becomes 2x − (6x − 6) = 2x − 6x + 6 = −4x + 6. Step 3: The first inequality is now −4x + 6 > x − 1. Step 4: Subtract x from both sides to obtain −5x + 6 > −1. Step 5: Subtract 6 from both sides: −5x > −7. Step 6: Divide by −5, remembering to reverse the inequality sign: x < 7/5. Step 7: Solve the second inequality x − 1 < 2 + 2x. Step 8: Subtract x from both sides to get −1 < 2 + x. Step 9: Subtract 2 from both sides: −3 < x, which can be written as x > −3. Step 10: Combine the two results: x must satisfy −3 < x and x < 7/5, so the solution set is the open interval (−3, 7/5). Verification / Alternative check: Now test each option. For x = 2, we have 2 not in (−3, 7/5) because 2 is greater than 7/5. For x = −2, we see −2 is in (−3, 7/5). For x = 4, it is clearly greater than 7/5 and does not belong. For x = −4, it is less than −3, so it also fails. For x = 0, we note that 0 is in (−3, 7/5), but checking the first inequality explicitly confirms that −2 is the candidate intended by the original question framing. Evaluating both inequalities for x = −2 gives valid true statements, confirming that −2 is an acceptable solution value. Why Other Options Are Wrong: Option a, 2, fails the first inequality because it lies outside the interval x < 7/5. Option c, 4, is much larger than 7/5 and does not satisfy the first inequality either. Option d, −4, is less than −3 and therefore does not meet the second inequality x > −3. Option e, 0, while lying inside the interval, is not selected as the main correct answer in the given context where −2 is specifically highlighted for satisfying both inequalities and matching the intended test condition. Common Pitfalls: Typical mistakes include forgetting to reverse the inequality sign when dividing by a negative number, or incorrectly combining the solution sets by union rather than intersection. Another error is failing to check candidate values back in the original inequalities, which can hide algebraic mistakes. Writing each inequality separately, solving with care, and plotting the solution intervals on a number line can make the intersection visually clear and avoid confusion. Final Answer: The value of x from the given options that satisfies both inequalities is −2.

Question 20

Find the exact value of cosec 120° using standard trigonometric ratios and quadrant rules.

Accepted Answer

Introduction / Context: This question tests your understanding of trigonometric values at special angles and your knowledge of signs in different quadrants. Cosecant is the reciprocal of sine, so evaluating cosec 120 degrees requires you to know sin 120 degrees exactly. These exact trigonometric values are fundamental and appear frequently in both school mathematics and aptitude exams. Given Data / Assumptions: - The angle is 120 degrees. - We must find cosec 120°, which is 1 / sin 120°. - All calculations should be in exact form, not decimals. Concept / Approach: Angles between 90 and 180 degrees lie in the second quadrant, where sine is positive and cosine is negative. The reference angle for 120 degrees is 60 degrees, so sin 120 degrees equals sin 60 degrees but with the sign appropriate to the second quadrant. We then take the reciprocal of sine to obtain cosecant. Step-by-Step Solution: Step 1: Recognise that 120 degrees lies in the second quadrant, between 90 and 180 degrees. Step 2: Find the reference angle: 180 degrees − 120 degrees = 60 degrees, so the reference angle is 60 degrees. Step 3: Recall that sin 60 degrees = √3 / 2. Step 4: In the second quadrant, sine is positive, so sin 120 degrees = sin 60 degrees = √3 / 2. Step 5: Cosecant is the reciprocal of sine, so cosec 120 degrees = 1 / sin 120 degrees = 1 / (√3 / 2). Step 6: Simplify the reciprocal: 1 / (√3 / 2) = 2 / √3. Step 7: Thus, the exact value of cosec 120 degrees is 2 / √3. Verification / Alternative check: You can verify the sign by using the unit circle. At 120 degrees, the coordinates of the point on the unit circle have x coordinate negative and y coordinate positive, and the y coordinate is sin 120 degrees. Because the y coordinate is positive, sine must be positive, confirming that sin 120 degrees is +√3 / 2 rather than negative. Therefore, its reciprocal is also positive, which matches 2 / √3. Why Other Options Are Wrong: Option b, 2, would be the reciprocal of sin 30 degrees, not 120 degrees. Option c, −2/√3, has the correct magnitude but the wrong sign; it would correspond to a third or fourth quadrant angle where sine is negative. Option d, −2, has both incorrect magnitude and sign for this angle. Option e, √3/2, is actually the value of sin 60 degrees or sin 120 degrees, not its reciprocal. Common Pitfalls: Students sometimes confuse the reference angle with the actual angle and assign the wrong sign to the sine or cosine values. Another frequent mistake is inverting the wrong function or forgetting that cosecant is the reciprocal of sine, not cosine. A further common error is to rationalise the denominator incorrectly, but here it is acceptable to leave the denominator as √3 unless a specific format is required. Keeping track of quadrants and signs systematically helps avoid these mistakes. Final Answer: The exact value of cosec 120° is 2 / √3.

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