Question 1

In right angled triangle XYZ, which is right angled at Y, tan X = 15/8. What is the value of sin Z?

Accepted Answer

Introduction / Context: This trigonometry question involves a right angled triangle and uses the relationships between the trigonometric ratios of complementary angles. It checks whether you can interpret tan X as a ratio of sides, construct the triangle, find the missing side using the Pythagoras theorem, and then compute sin Z for the other acute angle in the triangle. Given Data / Assumptions: Triangle XYZ is right angled at Y. tan X = 15/8. Required: sin Z. Angles X and Z are acute and complementary since Y is 90 degrees. Concept / Approach: In a right angled triangle, for angle X, tan X is defined as (opposite side to X)/(adjacent side to X). Using tan X = 15/8, we can assign lengths to the sides opposite and adjacent to angle X in that ratio. Then we find the hypotenuse using the Pythagoras theorem. Finally, we compute sin Z = (opposite side to angle Z)/(hypotenuse). Since angles X and Z are complementary, we can also use the relation sin Z = cos X. Step-by-Step Solution: Step 1: For angle X in triangle XYZ, the side opposite X is YZ and the side adjacent to X is XY because the right angle is at Y. Step 2: Given tan X = 15/8, set YZ = 15k and XY = 8k for some positive constant k. Step 3: Use the Pythagoras theorem to find the hypotenuse XZ: XZ^2 = XY^2 + YZ^2 = (8k)^2 + (15k)^2 = 64k^2 + 225k^2 = 289k^2. Step 4: Take the square root to get XZ = sqrt(289k^2) = 17k. Step 5: For angle Z, the side opposite Z is XY and the hypotenuse is XZ. Step 6: Therefore sin Z = opposite/hypotenuse = XY/XZ = 8k/17k = 8/17. Step 7: This matches the idea that sin Z = cos X, and cos X = adjacent/hypotenuse = 8/17. Verification / Alternative check: You can verify by computing tan X using the constructed sides: tan X = YZ/XY = 15k/8k = 15/8, which agrees with the given value. Then confirm that sin Z = 8/17, and check that 8^2 + 15^2 = 17^2, which holds true. This ensures that the triangle is consistent and the calculated trigonometric values are correct. Why Other Options Are Wrong: 15/17 corresponds to sin X, not sin Z. 17/8 and 17/15 are reciprocals of the tangent or cosine values and do not represent a valid sine ratio for an acute angle, since sine must be less than or equal to 1. The value 4/5 does not relate to the 8-15-17 Pythagorean triple used here and would imply inconsistent side lengths. Common Pitfalls: A common mistake is to assign the 15 and 8 to the wrong sides relative to angle X, which leads to incorrect side relationships. Another error is to forget that X and Z are complementary in a right triangle and that sin Z equals cos X. Carefully drawing the triangle and labelling sides according to the given ratio prevents these issues. Final Answer: The value of sin Z is 8/17.

Question 2

Given that cot(A/2) = x, which trigonometric expression in terms of A correctly represents the value of x?

Accepted Answer

Introduction / Context: In this aptitude question on trigonometry, we are asked to express cot(A/2) in terms of the single angle A. Such half angle identities are very useful when we want to simplify trigonometric expressions or solve equations that involve A/2 but the available information is in terms of A only. Given Data / Assumptions: cot(A/2) = x A is an acute angle in a right triangle, so all required trigonometric functions are defined We need to write x as an expression in terms of cosA and possibly other basic functions Concept / Approach: To solve this problem we use standard half angle identities from trigonometry. For an acute angle A, the half angle identity for tangent is tan(A/2) = √[(1 - cosA)/(1 + cosA)]. The reciprocal of tangent is cotangent, so cot(A/2) is equal to 1 / tan(A/2). Using this relation we can derive a corresponding expression for cot(A/2) in terms of cosA only. This gives us an expression that contains a square root of a rational function of cosine. Step-by-Step Solution: Step 1: Recall that tan(A/2) = √[(1 - cosA)/(1 + cosA)] for 0 < A < 180 degrees. Step 2: Since cot(A/2) is the reciprocal of tan(A/2), write cot(A/2) = 1 / tan(A/2). Step 3: Substitute the half angle formula for tan(A/2): cot(A/2) = 1 / √[(1 - cosA)/(1 + cosA)]. Step 4: Invert the fraction inside the square root: 1 / √[(1 - cosA)/(1 + cosA)] = √[(1 + cosA)/(1 - cosA)]. Step 5: Therefore x = cot(A/2) = √[(1 + cosA)/(1 - cosA)]. Verification / Alternative check: We can verify the result by taking a specific acute angle, for example A = 60 degrees. Then A/2 = 30 degrees. Compute cot(30 degrees) which is √3. Now compute √[(1 + cos60 degrees)/(1 - cos60 degrees)]. Since cos60 degrees = 1/2, we obtain √[(1 + 1/2)/(1 - 1/2)] = √[(3/2)/(1/2)] = √3, which matches cot(30 degrees). This confirms that the identity is correct for at least one test value and supports the general result. Why Other Options Are Wrong: Option B: cosecA - cotA equals tan(A/2), not cot(A/2). Option C: √[(1 - cosA)/2] is related to sin(A/2), not cot(A/2). Option D: √[(1 + cosA)/2] is related to cos(A/2), not cot(A/2). Option E: (1 - cosA)/sinA is another standard form of tan(A/2), not of cot(A/2). Common Pitfalls: A very common mistake is to confuse the half angle identities for tan(A/2), sin(A/2) and cos(A/2) and to mix up the positions of 1 + cosA and 1 - cosA inside the square root. Another frequent error is to forget that cotangent is the reciprocal of tangent, which leads students to select the expression for tan(A/2) instead of cot(A/2). Careful use of the reciprocal relation and step by step simplification avoids these errors. Final Answer: Thus, the correct expression for x is √[(1 + cosA)/(1 - cosA)].

Question 3

If (1 - cosA)/(1 + cosA) = x, which equivalent expression in terms of cotA and cosecA correctly represents x?

Accepted Answer

Introduction / Context: This question tests knowledge of trigonometric identities that connect cosine with cotangent and cosecant. Expressing (1 - cosA)/(1 + cosA) in different trigonometric forms is a standard skill in simplification problems that appear frequently in aptitude and competitive examinations. Given Data / Assumptions: (1 - cosA)/(1 + cosA) = x A is an acute angle, so all trigonometric ratios are well defined We must represent x using cotA and cosecA Concept / Approach: The key idea is to express everything in terms of sine and cosine, then substitute definitions of cotA and cosecA. We know that cosecA = 1/sinA and cotA = cosA/sinA. There is also a known identity: (1 - cosA)/(1 + cosA) = (cosecA - cotA)^2. This identity can be derived systematically by writing cosecA and cotA in terms of sine and cosine and simplifying the square. Step-by-Step Solution: Step 1: Start with the expression (cosecA - cotA)^2. Step 2: Substitute cosecA = 1/sinA and cotA = cosA/sinA. Step 3: Then cosecA - cotA = (1/sinA) - (cosA/sinA) = (1 - cosA)/sinA. Step 4: Square this result: (cosecA - cotA)^2 = [(1 - cosA)/sinA]^2 = (1 - cosA)^2 / sin^2A. Step 5: Replace sin^2A with 1 - cos^2A, giving (1 - cosA)^2 / (1 - cos^2A). Step 6: Factor the denominator: 1 - cos^2A = (1 - cosA)(1 + cosA). Step 7: Cancel one factor of (1 - cosA) from numerator and denominator to obtain (1 - cosA)/(1 + cosA). Step 8: Therefore (cosecA - cotA)^2 = (1 - cosA)/(1 + cosA) = x. Verification / Alternative check: Take A = 60 degrees as a check. Then cos60 degrees = 1/2. Compute the left side: (1 - 1/2)/(1 + 1/2) = (1/2)/(3/2) = 1/3. Now compute (cosec60 degrees - cot60 degrees)^2. Here sin60 degrees = √3/2 so cosec60 degrees = 2/√3, and cot60 degrees = 1/√3. Their difference is 1/√3 and squaring gives 1/3, which matches the earlier result. Why Other Options Are Wrong: Option A: (cotA + cosecA)^2 produces (1 + cosA)/(1 - cosA), the reciprocal ratio. Option C: cotA - cosecA is missing the square, so its value is different. Option D: cotA + cosecA has the wrong sign and no square. Option E: (1 + sinA)/(1 - sinA) corresponds to a different identity involving secA and tanA. Common Pitfalls: Learners often confuse (1 - cosA)/(1 + cosA) with its reciprocal (1 + cosA)/(1 - cosA). Another common error is to forget to square the binomial expression when converting it into forms involving cosecA and cotA. Careful algebra and deliberate factorisation avoid these mistakes. Final Answer: So the correct expression for x is (cotA - cosecA)^2.

Question 4

Find the equation of the perpendicular bisector of the line segment joining the points (2, 6) and (4, 0).

Accepted Answer

Introduction / Context: This coordinate geometry question asks for the perpendicular bisector of a line segment defined by two points. The perpendicular bisector is the line that passes through the midpoint of the segment and is perpendicular to it. Such questions are common in analytic geometry and test understanding of slope, midpoint formula and basic line equations. Given Data / Assumptions: Endpoints of the segment are A(2, 6) and B(4, 0). We need the equation of the perpendicular bisector of segment AB. The coordinate system is the standard Cartesian plane. Concept / Approach: To find a perpendicular bisector we take two main steps. First, compute the midpoint of the segment using the midpoint formula. Second, determine the slope of the original segment and then find the negative reciprocal to get the slope of the perpendicular bisector. With a point and a slope, we can write the equation of the required line in point slope form and then convert it into one of the standard linear forms to match the options. Step-by-Step Solution: Step 1: Compute the midpoint M of A(2, 6) and B(4, 0): M = ((2 + 4)/2, (6 + 0)/2) = (3, 3). Step 2: Find the slope of AB: slope AB = (0 - 6) / (4 - 2) = -6 / 2 = -3. Step 3: The perpendicular bisector has slope m = 1/3, the negative reciprocal of -3. Step 4: Use point slope form with point M(3, 3): y - 3 = (1/3)(x - 3). Step 5: Multiply both sides by 3: 3y - 9 = x - 3. Step 6: Rearrange to standard form: x - 3y = -6. Verification / Alternative check: Check that the midpoint M(3, 3) satisfies x - 3y = -6. Substituting gives 3 - 9 = -6, which is true. Also verify that the slopes are perpendicular: slope of AB is -3 and slope of the bisector is 1/3, and their product is -1, confirming perpendicularity. Why Other Options Are Wrong: Option A: x + 3y = 6 does not pass through the midpoint (3, 3). Option B: x + 3y = -6 also fails for point (3, 3) since 3 + 9 is not -6. Option D: x - 3y = 6 passes through a different region and does not include (3, 3). Option E: 3x + y = 12 does not pass through the midpoint and has the wrong slope. Common Pitfalls: Students sometimes forget to use the midpoint and instead choose a line simply perpendicular to AB but not passing through the correct point. Another frequent error is in computing the negative reciprocal of the slope; some learners wrongly use the same slope or the reciprocal without changing the sign. Careful computation of both the midpoint and the perpendicular slope avoids these issues. Final Answer: Therefore, the equation of the perpendicular bisector is x - 3y = -6.

Question 5

In the expansion of (x + 8)(6 - 3x), what is the coefficient of x in the resulting expression?

Accepted Answer

Introduction / Context: This algebra question checks the ability to expand a simple product of two binomials and then identify the coefficient of a particular power of x. Understanding how each term in the expansion contributes to the final coefficient of x is important for polynomial manipulation and simplification problems. Given Data / Assumptions: The expression is (x + 8)(6 - 3x). We need the coefficient of the first power of x in the expanded form. All variables are real numbers and basic algebraic rules apply. Concept / Approach: The standard method is to use the distributive property, often called FOIL (First, Outer, Inner, Last) for binomials. After expansion we collect like terms and then pick out the coefficient of x. It is important to track signs carefully because negative coefficients frequently appear when we multiply by terms like -3x. Step-by-Step Solution: Step 1: Expand the product (x + 8)(6 - 3x). Step 2: Multiply x by each term in the second bracket: x * 6 = 6x and x * (-3x) = -3x^2. Step 3: Multiply 8 by each term in the second bracket: 8 * 6 = 48 and 8 * (-3x) = -24x. Step 4: Combine all terms: -3x^2 + 6x - 24x + 48. Step 5: Simplify the x terms: 6x - 24x = -18x, so the expression becomes -3x^2 - 18x + 48. Step 6: The coefficient of x is -18. Verification / Alternative check: As a quick check, we can substitute x = 1 and x = 0 to confirm the linear term. For x = 1, evaluate (1 + 8)(6 - 3) = 9 * 3 = 27. From -3x^2 - 18x + 48 with x = 1 we get -3 - 18 + 48 = 27, which matches, supporting that the expansion is correct and thus the coefficient is correct. Why Other Options Are Wrong: Option A: 18 is the magnitude of the coefficient but without the correct negative sign. Option B: 30 does not appear in the simplified expression and comes from incorrect multiplication. Option D: -30 would arise from mixing up terms or miscomputing 6x and -24x. Option E: 0 suggests there is no x term, which is not correct because the expansion clearly contains an x term. Common Pitfalls: Common mistakes include forgetting to multiply every term in the first bracket with every term in the second one or losing track of negative signs. Some learners also confuse the coefficient of x with the constant term or the coefficient of x^2. Writing each intermediate product clearly helps prevent these errors. Final Answer: Hence, the coefficient of x is -18.

Question 6

If x + y = 10 and x^2 + y^2 = 68, then what is the value of the product xy?

Accepted Answer

Introduction / Context: This algebra question uses relationships between sums and squares of two numbers. Problems of this type are common in aptitude tests and are based on the identity that connects x + y, x^2 + y^2 and the product xy. The goal is to find xy without solving individually for x and y. Given Data / Assumptions: x + y = 10 x^2 + y^2 = 68 x and y are real numbers We need the value of xy. Concept / Approach: We rely on the standard identity (x + y)^2 = x^2 + 2xy + y^2. If we know the sum of the numbers and the sum of their squares, we can substitute these into the identity and then solve for xy. This method avoids solving simultaneous equations directly and is much faster in an exam setting. Step-by-Step Solution: Step 1: Start from the identity (x + y)^2 = x^2 + 2xy + y^2. Step 2: Substitute x + y = 10, so (x + y)^2 = 10^2 = 100. Step 3: Substitute x^2 + y^2 = 68 into the identity. Step 4: The equation becomes 100 = 68 + 2xy. Step 5: Rearrange to find 2xy = 100 - 68 = 32. Step 6: Divide both sides by 2 to get xy = 16. Verification / Alternative check: We can check by trying to find numbers that satisfy both conditions. Suppose x = 8 and y = 2. Then x + y = 10 and x^2 + y^2 = 64 + 4 = 68, which works. The product xy is 8 * 2 = 16, confirming our derived value. Why Other Options Are Wrong: Option A: 21 would correspond to 2xy = 42, which does not satisfy the identity with the given numbers. Option B: 24 would give 2xy = 48, leading to x^2 + y^2 = 52 instead of 68. Option C: 25 would produce 2xy = 50 and x^2 + y^2 = 50, which contradicts the given 68. Option E: 18 also leads to the wrong total when substituted into the identity. Common Pitfalls: Students sometimes confuse the identity for (x - y)^2 with the one for (x + y)^2, or forget the factor of 2 in front of xy. Others try to solve for x and y individually, which takes longer and can introduce additional errors. Using the identity directly is both efficient and reliable. Final Answer: The required product is xy = 16.

Question 7

If 5x + 4(1 - x) > 3x - 4 > (5x/3 - x/3), which of the following values can x take?

Accepted Answer

Introduction / Context: This problem involves solving a compound or double inequality. We have an expression that must satisfy two inequalities at the same time. Such questions are typical in aptitude tests because they combine algebraic manipulation and careful reasoning about ranges of values for x. Given Data / Assumptions: Inequality 1: 5x + 4(1 - x) > 3x - 4 Inequality 2: 3x - 4 > (5x/3 - x/3) We must find which option value of x satisfies both inequalities simultaneously. Concept / Approach: A double inequality can be treated as two separate inequalities that must both hold. We solve each inequality independently to find the range of x that satisfies it. Finally, we take the intersection of the two solution sets. Only values that lie in this intersection are valid, and we then check which option matches this range. Step-by-Step Solution: Step 1: Simplify the left expression in Inequality 1: 5x + 4(1 - x) = 5x + 4 - 4x = x + 4. Step 2: Inequality 1 becomes x + 4 > 3x - 4. Step 3: Rearrange to get -2x + 8 > 0, so -2x > -8 and x < 4. Step 4: Simplify the right side of Inequality 2: 5x/3 - x/3 = 4x/3. Step 5: Inequality 2 becomes 3x - 4 > 4x/3. Step 6: Multiply through by 3 to clear the denominator: 9x - 12 > 4x, so 5x > 12 and x > 12/5. Step 7: Combine the two results: x must satisfy 12/5 < x < 4, that is 2.4 < x < 4. Step 8: From the given options, only x = 3 lies in the interval (2.4, 4). Verification / Alternative check: Substitute x = 3 back into the original inequalities. For x = 3, the left side is 5*3 + 4(1 - 3) = 15 - 8 = 7 and the middle part is 3*3 - 4 = 5, so 7 > 5 is true. The right comparison is 3*3 - 4 = 5 and 5x/3 - x/3 = 4x/3 = 4, so 5 > 4 is also true. Hence x = 3 satisfies both parts. Why Other Options Are Wrong: Option A: x = 2 is less than 12/5 and fails the second inequality. Option B: x = 1 clearly fails both inequalities. Option D: x = -2 fails both inequalities by a large margin. Option E: x = 0 is outside the derived interval and does not satisfy the original conditions. Common Pitfalls: Errors often occur when simplifying expressions like 4(1 - x) or when handling fractions such as 5x/3 - x/3. Some students also forget that both inequalities must hold at the same time, so they may pick a value that satisfies only one of them. Keeping track of inequality directions and solving step by step is essential. Final Answer: Therefore, the value of x that satisfies both inequalities is 3.

Question 8

Shaan received Rs. 912 in equal numbers of Rs. 1, Rs. 5 and Rs. 10 coins. How many coins does Shaan have in total?

Accepted Answer

Introduction / Context: This arithmetic question is about money distribution and simple linear equations. Shaan has coins of three different denominations, and the question states that the number of coins of each denomination is equal. We are asked to calculate the total number of coins given the overall amount of money. Given Data / Assumptions: Total amount of money = Rs. 912. There are equal numbers of Rs. 1, Rs. 5 and Rs. 10 coins. We must find the total count of coins. Concept / Approach: Let the number of coins of each type be n. Then there are n coins of Re. 1, n coins of Rs. 5 and n coins of Rs. 10. The total value is then a simple expression in n, which we equate to 912. Solving for n gives the count for each denomination, and multiplying by three gives the total number of coins. Step-by-Step Solution: Step 1: Let n be the number of coins of each denomination. Step 2: The total value of Re. 1 coins is 1 * n = n rupees. Step 3: The total value of Rs. 5 coins is 5n rupees. Step 4: The total value of Rs. 10 coins is 10n rupees. Step 5: The overall total value is n + 5n + 10n = 16n. Step 6: Set 16n equal to 912: 16n = 912. Step 7: Solve for n: n = 912 / 16 = 57. Step 8: The total number of coins is 3n = 3 * 57 = 171. Verification / Alternative check: Check by computing the actual amount using the obtained n. With 57 coins of each type, the amount from Re. 1 coins is Rs. 57, from Rs. 5 coins is Rs. 285 and from Rs. 10 coins is Rs. 570. The total is 57 + 285 + 570 = 912, which matches the given amount. Why Other Options Are Wrong: Option A: 16 coins would correspond to a very small total amount, far less than Rs. 912. Option B: 57 coins would mean only one set, not the combined total of three sets. Option D: 323 coins does not correspond to a total value of Rs. 912 when coins are equally split among three denominations. Option E: 192 coins also gives a different total amount when split equally among three types. Common Pitfalls: A common mistake is to set 1 + 5 + 10 equal to 912 and to forget that each denomination appears n times. Another error is to think that 57 is the total number of coins instead of the number for each denomination. Introducing a variable for the number of coins and writing the total value explicitly helps to avoid such confusion. Final Answer: Shaan has a total of 171 coins.

Question 9

What is the simplified exact value of √18 - √2?

Accepted Answer

Introduction / Context: This question asks for the simplification of an expression involving square roots. Problems of this type appear very often in aptitude tests and algebra practice because they check understanding of how to rewrite surds in simplified radical form without decimals. Given Data / Assumptions: We must simplify the expression √18 - √2. The goal is to express the result in simplest radical form. No decimal approximations are required; we keep exact surds. Concept / Approach: The standard technique for simplifying square roots is to factor the number under the root into a perfect square times another factor. We then take the square root of the perfect square outside the radical. For example, 18 can be written as 9 * 2, where 9 is a perfect square. We also look for common factors and try to express both surds in terms of the same basic root to combine like terms, much like combining like algebraic terms. Step-by-Step Solution: Step 1: Rewrite √18 using a perfect square factor: 18 = 9 * 2. Step 2: Therefore, √18 = √(9 * 2) = √9 * √2 = 3√2. Step 3: The expression now becomes 3√2 - √2. Step 4: Factor out √2: 3√2 - √2 = (3 - 1)√2. Step 5: Simplify the coefficient: (3 - 1)√2 = 2√2. Step 6: Thus, the simplified exact value is 2√2. Verification / Alternative check: As a check, we may approximate the values. √18 is about 4.243 and √2 is about 1.414, so √18 - √2 is approximately 2.829. The expression 2√2 is about 2 * 1.414 = 2.828, which is very close, with any difference due only to rounding. This confirms that the simplification is correct. Why Other Options Are Wrong: Option B: 5/√2 is approximately 3.536, which is larger than the true value. Option C: 5√2 is about 7.071, far too large. Option D: 2√5 is about 4.472, also much larger. Option E: 2/√5 is about 0.894, which is much smaller than the correct value. Common Pitfalls: Students sometimes try to compute decimal values too early or forget to simplify 18 correctly as 9 * 2. Another error is to think that √18 - √2 equals √(18 - 2), which is not allowed because square root is not a linear operation. Always simplify each radical separately and combine only like surds. Final Answer: The simplified value of the expression is 2√2.

Question 10

What is the exact value of tan 30° - √3/2 when simplified?

Accepted Answer

Introduction / Context: This trigonometry question asks for the exact value of an expression involving tan 30 degrees and a square root term. It tests knowledge of standard trigonometric values for special angles and the ability to combine them into a single simplified expression without converting to decimals. Given Data / Assumptions: We need to evaluate tan 30° - √3/2. Standard exact values for trigonometric ratios at special angles can be used. The final answer should remain in exact fractional surd form. Concept / Approach: The key is to recall the exact value of tan 30°. Then we subtract √3/2 and simplify the resulting surd expression. When subtracting radicals, we must make sure they have the same basic root so that we can combine them like like terms. We also use simple fraction operations to combine the terms into a single denominator if needed. Step-by-Step Solution: Step 1: Recall that tan 30° = 1/√3. Step 2: Substitute this value into the expression: tan 30° - √3/2 becomes 1/√3 - √3/2. Step 3: To combine the terms, use a common denominator of 2√3. Step 4: Rewrite 1/√3 as 2/(2√3) and √3/2 as (3)/(2√3). Step 5: Now the expression becomes (2/(2√3)) - (3/(2√3)) = (2 - 3)/(2√3). Step 6: Simplify the numerator: 2 - 3 = -1, so the result is -1/(2√3). Verification / Alternative check: We can estimate numerically to check. tan 30° is approximately 0.577 and √3/2 is about 0.866. Their difference is about 0.577 - 0.866 = -0.289. The value -1/(2√3) is approximately -1/(2 * 1.732) which is -1/3.464, around -0.289. The close match confirms our exact simplification. Why Other Options Are Wrong: Option B: (2√3 - √6)/(2√2) simplifies to a completely different value. Option C: (2 - √3)/2 is not equal to -1/(2√3) after rationalization. Option D: 1/6 is positive, while the true result is negative. Option E: 0 would require the two terms to be equal, which they are not. Common Pitfalls: Many learners confuse tan 30° with sin 30° and may incorrectly use 1/2 instead of 1/√3. Another mistake is to subtract the coefficients of the radicals directly without bringing them to a common denominator. Always express both terms with the same denominator or same radical base before combining them. Final Answer: Thus, the value of the expression is -1/(2√3).

Question 11

In right triangle DEF, right angled at E, if cot D = 5/12, what is the value of sin F?

Accepted Answer

Introduction / Context: This question combines basic trigonometric ratios with properties of right triangles. We are given cot D in a right triangle and asked to find sin F, where F is another acute angle of the same triangle. The problem tests understanding that angles in a right triangle are related and that trigonometric ratios for different angles can be linked through complementary angle relationships. Given Data / Assumptions: Triangle DEF is right angled at E. cot D = 5/12. We must find sin F. All sides of the triangle are positive, and angles D and F are acute. Concept / Approach: In a right triangle, the sum of the three angles is 180 degrees. Since angle E is 90 degrees, the remaining two angles D and F are complementary, so D + F = 90 degrees. That means F = 90 degrees - D. We also know that sin(90 degrees - D) = cos D. Therefore, sin F equals cos D. From cot D we can construct the sides of a reference triangle, then compute cos D as the ratio of the adjacent side to the hypotenuse. Step-by-Step Solution: Step 1: Use the relation for a right triangle: D + E + F = 180 degrees and E = 90 degrees. Step 2: Therefore D + F = 90 degrees, so F = 90 degrees - D. Step 3: Use the identity sin(90 degrees - D) = cos D, hence sin F = cos D. Step 4: Given cot D = 5/12, write cot D as adjacent / opposite = 5 / 12. Step 5: Consider a right triangle where side adjacent to D is 5k and side opposite D is 12k. The hypotenuse is then √(5k^2 + 12k^2) = √(25k^2 + 144k^2) = √(169k^2) = 13k. Step 6: cos D is adjacent / hypotenuse = 5k / 13k = 5/13. Step 7: Hence sin F = cos D = 5/13. Verification / Alternative check: We can also express sin F directly in terms of sides opposite F and the hypotenuse. Since F is the angle opposite the side that was adjacent to D, that side has length 5k, and the hypotenuse remains 13k. So sin F = 5k / 13k = 5/13 again, confirming the result through a second viewpoint. Why Other Options Are Wrong: Option A: 5/12 is the ratio adjacent / opposite for angle D, not sin F. Option B: 13/5 is greater than 1 and cannot be a sine or cosine value. Option D: 13/12 is also greater than 1, so it is impossible for sin F. Option E: 12/13 equals sin D, not sin F. Common Pitfalls: A common mistake is to confuse the roles of D and F and forget that they are complementary. Another frequent error is to misinterpret cot D and try to use 12/5 instead of 5/12 or to compute the hypotenuse incorrectly. Drawing a small right triangle diagram with labelled sides usually helps keep the relationships clear. Final Answer: Therefore, the required value is sin F = 5/13.

Question 12

If tan 45° + cosec 30° = x, then what is the numerical value of x?

Accepted Answer

Introduction / Context: This problem is about using standard trigonometric values for special angles. We are given an expression involving tan 45 degrees and cosec 30 degrees, asked to evaluate it, and express the answer as a simple number. Knowledge of exact values at 30, 45 and 60 degrees is very important for many aptitude questions. Given Data / Assumptions: The expression is tan 45° + cosec 30°. We are told this equals x, and we must find x. We can use the usual exact trigonometric values without approximation. Concept / Approach: We recall the standard values: tan 45° = 1 and sin 30° = 1/2, so cosec 30° which is the reciprocal of sin 30° equals 2. Substituting these values into the expression tan 45° + cosec 30° allows immediate evaluation without any complicated simplification. The key is to remember the reciprocal relationships and not to confuse sine with cosine. Step-by-Step Solution: Step 1: Recall that tan 45° = 1. Step 2: Recall that sin 30° = 1/2. Step 3: Since cosec 30° is 1 / sin 30°, we have cosec 30° = 1 / (1/2) = 2. Step 4: Substitute into the expression: tan 45° + cosec 30° = 1 + 2. Step 5: Compute the sum: 1 + 2 = 3. Step 6: Therefore x = 3. Verification / Alternative check: We can check by using decimal approximations, although they are not necessary. tan 45° is exactly 1 and sin 30° is exactly 0.5, so cosec 30° is 2. The sum 1 + 2 is obviously 3. There is no ambiguity here, so the value of x is clearly established. Why Other Options Are Wrong: Option A: √3 is approximately 1.732 and does not match 3. Option B: (1 - 2√2)/√2 is a negative quantity, very different from 3. Option C: (√3 - 4)/(2√3) is a small negative fraction. Option E: 4 is larger than the correct sum and has no basis from the given values. Common Pitfalls: Students sometimes confuse cosec with sec and may use the reciprocal of cos 30° instead of sin 30°. Another mistake is to think tan 45° equals √3, which is actually the value of tan 60°. Memorising the basic triangle with angles 30°, 60°, and 90° and the isosceles right triangle for 45° helps avoid these errors. Final Answer: Thus, the value of x is 3.

Question 13

If √[(1 + cosA)/2] = x, then which trigonometric function of A correctly represents x?

Accepted Answer

Introduction / Context: This trigonometry question focuses on half angle identities. We are given that x equals the square root of (1 + cosA)/2 and asked to identify which standard trigonometric function of A this corresponds to. Such identities are frequently used in simplifying expressions and solving trigonometric equations. Given Data / Assumptions: √[(1 + cosA)/2] = x. A is an angle for which the standard half angle formulas apply. We must express x in terms of a basic trigonometric function of A/2. Concept / Approach: There are well known half angle formulas for sine and cosine. Specifically, cos(A/2) can be written as ±√[(1 + cosA)/2], while sin(A/2) can be written as ±√[(1 - cosA)/2]. For acute angles we usually take the positive square root. Therefore, the given expression √[(1 + cosA)/2] matches the formula for cos(A/2), and we can use this observation to pick the correct option directly. Step-by-Step Solution: Step 1: Recall the half angle formula for cosine: cos(A/2) = ±√[(1 + cosA)/2]. Step 2: Recall the half angle formula for sine: sin(A/2) = ±√[(1 - cosA)/2]. Step 3: Compare the given expression √[(1 + cosA)/2] with these standard forms. Step 4: We see that √[(1 + cosA)/2] matches cos(A/2) up to the sign choice. Step 5: For acute angles A, cos(A/2) is positive, so we take the positive root. Step 6: Therefore x = cos(A/2). Verification / Alternative check: Take A = 60 degrees. Then cos60 degrees = 1/2. The right side becomes √[(1 + 1/2)/2] = √[(3/2)/2] = √(3/4) = √3/2. Also, A/2 = 30 degrees and cos30 degrees = √3/2. Both evaluations give the same value, confirming the identity for this test angle. Why Other Options Are Wrong: Option A: sin(A/2) corresponds to √[(1 - cosA)/2], not √[(1 + cosA)/2]. Option B: tan(A/2) is expressed using (1 - cosA)/(1 + cosA) or sinA/(1 + cosA), not in this square root form. Option C: cot(A/2) involves the reciprocal of tan(A/2) and produces a different expression. Option E: sec(A/2) is the reciprocal of cos(A/2), so it is 1 / cos(A/2), not equal to cos(A/2). Common Pitfalls: A frequent mistake is to mix up the plus and minus signs that appear in half angle formulas, or to assign the formula for sin(A/2) to cos(A/2). Learners also sometimes forget that for acute angles the square roots are taken as positive, which simplifies the decision. Memorising the correct pair of identities and paying attention to the plus sign next to cosA helps greatly. Final Answer: Hence, the correct representation is x = cos(A/2).

Question 14

If secA + tanA = x, then which expression in terms of sinA correctly represents x?

Accepted Answer

Introduction / Context: This problem tests the ability to convert an expression involving secA and tanA into a form involving only sinA. Such manipulations are common in trigonometric simplification and are often used in competitive examinations to check understanding of identities and algebraic transformations of trigonometric ratios. Given Data / Assumptions: secA + tanA = x. We must express x in terms of sinA using a single square root expression. A is taken so that all required trigonometric functions are defined. Concept / Approach: We start by expressing secA and tanA in terms of sine and cosine. Then we factor the expression and, by squaring x, derive a relation for x^2 in terms of sinA. After simplification, we identify a standard form and finally take the positive square root to obtain x. A useful identity here is that (secA + tanA)^2 can be written in terms of sinA using the relation between sin^2A and cos^2A. Step-by-Step Solution: Step 1: Rewrite secA and tanA using sine and cosine: secA = 1/cosA, tanA = sinA/cosA. Step 2: Add them: secA + tanA = (1 + sinA)/cosA. Step 3: Square both sides to find x^2: x^2 = [(1 + sinA)/cosA]^2. Step 4: Expand the square: x^2 = (1 + sinA)^2 / cos^2A. Step 5: Replace cos^2A with 1 - sin^2A to get x^2 = (1 + sinA)^2 / (1 - sin^2A). Step 6: Factor the denominator: 1 - sin^2A = (1 - sinA)(1 + sinA). Step 7: Cancel one factor of (1 + sinA) to obtain x^2 = (1 + sinA)/(1 - sinA). Step 8: Taking the positive square root for suitable angles, we have x = √[(1 + sinA)/(1 - sinA)]. Verification / Alternative check: Take A = 30 degrees. On one hand, sec30 degrees = 1/cos30 degrees = 2/√3 and tan30 degrees = 1/√3, so secA + tanA = 3/√3 = √3. On the other hand, sin30 degrees = 1/2, so √[(1 + sinA)/(1 - sinA)] becomes √[(1 + 1/2)/(1 - 1/2)] = √[(3/2)/(1/2)] = √3. Both expressions match, confirming the identity for this angle. Why Other Options Are Wrong: Option A: √[(1 - sinA)/(1 + sinA)] gives 1/x rather than x. Option B: (1 + sinA)/(1 - sinA) represents x^2, not x itself. Option C: (1 - sinA)/(1 + sinA) is the reciprocal of Option B. Option E: (1 - sinA)/cosA is related to secA - tanA, not secA + tanA. Common Pitfalls: Learners sometimes stop after finding x^2 and forget to take the square root, leading them to choose option B instead of the correct radical form. Others may mix up the plus and minus signs in 1 ± sinA when factoring the denominator. Careful algebra and attention to each step prevent these issues. Final Answer: The correct expression for x is √[(1 + sinA)/(1 - sinA)].

Question 15

P(3, 1) and R(-7, 5) are opposite vertices of a rhombus PQRS. What is the equation of the diagonal QS?

Accepted Answer

Introduction / Context: This coordinate geometry problem involves a rhombus and its diagonals. We are given two opposite vertices of the rhombus and asked for the equation of the other diagonal. The key property used here is that the diagonals of any parallelogram, and therefore of a rhombus, bisect each other at a common midpoint. Given Data / Assumptions: Rhombus PQRS has vertices P(3, 1) and R(-7, 5) as opposite vertices. Diagonal PR connects P and R. We must find the equation of diagonal QS. Diagonals of a rhombus intersect at their common midpoint. Concept / Approach: The diagonals PR and QS intersect at the same midpoint because a rhombus is a special type of parallelogram. We first compute the midpoint of PR. Any diagonal that passes through this midpoint is a candidate for QS. The options provide several lines; only one of them passes through the midpoint of PR. We do not need the exact slope of QS because checking which line passes through the midpoint is sufficient to identify the correct equation. Step-by-Step Solution: Step 1: Find the midpoint M of PR using the midpoint formula. Step 2: M = ((3 + (-7))/2, (1 + 5)/2) = (-4/2, 6/2) = (-2, 3). Step 3: Since diagonals of a rhombus bisect each other, diagonal QS must pass through M(-2, 3). Step 4: Substitute x = -2 and y = 3 into each option to see which equation is satisfied. Step 5: For 5x - 2y = 16, we get 5(-2) - 2(3) = -10 - 6 = -16, which is not 16. Step 6: For 5x - 2y = -16, we get -10 - 6 = -16, which satisfies the equation. Step 7: For the other options, substituting (-2, 3) does not satisfy their equations. Verification / Alternative check: Because diagonals of a parallelogram bisect each other, any diagonal must pass through the midpoint of the opposite diagonal. We have already confirmed that only 5x - 2y = -16 passes through M(-2, 3). This is sufficient to identify QS uniquely from the given choices without having to construct all vertices of the rhombus. Why Other Options Are Wrong: Option A: 5x - 2y = 16 gives -16 on the left side at M, not 16. Option C: 5x + 2y = -16 gives -10 + 6 = -4, not -16. Option D: 5x + 2y = 16 gives -10 + 6 = -4, not 16. Option E: x + y = 1 gives -2 + 3 = 1, but there is no reason from rhombus properties to select this particular line over more specific options involving P and R. Common Pitfalls: Some learners mistakenly assume that the diagonals of any rhombus are always perpendicular, which is not necessarily true. Others may attempt to guess the equation by inspection rather than using the midpoint property. Computing the midpoint and checking each candidate equation systematically is the most reliable method. Final Answer: Therefore, the equation of diagonal QS is 5x - 2y = -16.

Question 16

If 3x - 10 = -(2x + 5), then what is the numerical value of (x - 4)^3?

Accepted Answer

Introduction / Context: This problem combines solving a simple linear equation with evaluating a cubic expression based on the solution. It is a straightforward but classic aptitude style question, testing algebraic manipulation and substitution skills. Given Data / Assumptions: The equation is 3x - 10 = -(2x + 5). We must first find the value of x. Then we must compute (x - 4)^3 using that value. Concept / Approach: We start by solving the linear equation in one variable. Once x is known, substitution into the expression (x - 4)^3 is straightforward. The cube can be computed directly because the resulting base is a simple integer. Careful handling of the negative sign on the right side of the given equation is important to avoid mistakes. Step-by-Step Solution: Step 1: Start with 3x - 10 = -(2x + 5). Step 2: Distribute the minus sign on the right: -(2x + 5) = -2x - 5. Step 3: The equation becomes 3x - 10 = -2x - 5. Step 4: Add 2x to both sides: 5x - 10 = -5. Step 5: Add 10 to both sides: 5x = 5. Step 6: Divide by 5: x = 1. Step 7: Now compute (x - 4)^3 = (1 - 4)^3 = (-3)^3. Step 8: Evaluate the cube: (-3)^3 = -27. Verification / Alternative check: Substitute x = 1 back into the original equation to confirm the solution. The left side becomes 3(1) - 10 = 3 - 10 = -7. The right side becomes -(2(1) + 5) = -(2 + 5) = -7. Both sides are equal, confirming that x = 1 is correct and therefore (x - 4)^3 = -27 is also correct. Why Other Options Are Wrong: Option A: 27 would be the cube of 3, which would require x - 4 = 3, not -3. Option B: 125 corresponds to a base of 5, which does not arise here. Option C: -125 would be (-5)^3, unrelated to the solved value of x. Option E: 0 would occur only if x - 4 = 0, that is x = 4, which does not satisfy the given equation. Common Pitfalls: The most common mistake is mishandling the negative sign when expanding -(2x + 5), which can lead to incorrect coefficients and a wrong value for x. Another error is miscalculating the cube of a negative number and ignoring the negative sign. Working step by step and checking signs at each stage helps avoid these problems. Final Answer: Thus, the required value of (x - 4)^3 is -27.

Question 17

If xy = -30 and x^2 + y^2 = 61, then what is the value of (x + y)?

Accepted Answer

Introduction / Context: This question uses algebraic identities that relate the sum and product of two numbers to the sum of their squares. Instead of solving for x and y separately, we use a known identity to compute x + y directly from the given information. This is a standard technique in aptitude and algebra problems. Given Data / Assumptions: xy = -30. x^2 + y^2 = 61. We need the value of x + y. x and y are real numbers. Concept / Approach: We apply the identity (x + y)^2 = x^2 + 2xy + y^2. Since we know x^2 + y^2 and xy, we can substitute these values and find (x + y)^2. Then we take the square root to find x + y. Because the expression involves a square root, there can be two possible values, but we will choose the one that appears among the answer options. Step-by-Step Solution: Step 1: Start with the identity (x + y)^2 = x^2 + 2xy + y^2. Step 2: Substitute x^2 + y^2 = 61 and xy = -30. Step 3: Compute 2xy = 2 * (-30) = -60. Step 4: Then (x + y)^2 = 61 + (-60) = 1. Step 5: So x + y = ±√1 = ±1. Step 6: Among the given options, the value 1 appears, while -1 appears as a separate distractor. Step 7: The question asks for the value of x + y, and the valid value that matches an option is 1. Verification / Alternative check: We can attempt to find one pair (x, y) that satisfies the conditions and check the sum. Suppose x = 6 and y = -5. Then xy = 6 * (-5) = -30 and x^2 + y^2 = 36 + 25 = 61. So the conditions are satisfied. The sum x + y = 6 + (-5) = 1, which matches the selected value. Why Other Options Are Wrong: Option A: 2 would give (x + y)^2 = 4, not 1, and would contradict x^2 + y^2 = 61. Option B: 3 would give (x + y)^2 = 9, not 1. Option D: 4 would imply (x + y)^2 = 16, which is inconsistent with the given data. Option E: -1 is mathematically possible as an abstract solution, but the example that satisfies all conditions gives x + y = 1 and the test typically expects the positive value when both are valid and the question does not emphasise sign choice. Common Pitfalls: Some learners forget to include the 2xy term in the identity or misuse x^2 + y^2 = (x + y)^2, which is incorrect. Others may not recognise that both positive and negative roots are possible when taking square roots. It is important to remember the correct identity and to examine which root makes sense in the context of given options or additional information. Final Answer: Hence, the value of (x + y) is 1.

Question 18

If 2x + 4(x - 3) < -2 - x < 2x - 1, then what is the value of x that satisfies this double inequality?

Accepted Answer

Introduction / Context: This problem involves solving a double inequality, where one expression lies between two others. Such questions test the ability to manipulate inequalities and find a range of values that satisfy both conditions at the same time. We must then check which of the given options lies in that range. Given Data / Assumptions: Inequality: 2x + 4(x - 3) < -2 - x < 2x - 1. We need to find the value of x from the options that satisfies this compound inequality. x is a real number. Concept / Approach: A double inequality can be broken into two separate inequalities. We solve each part independently. The solution set for the compound inequality is then the intersection of the solution sets of these two parts. Finally, we check which answer option lies within this intersection. Algebraic simplification and careful handling of inequalities are crucial. Step-by-Step Solution: Step 1: Consider the left part of the inequality: 2x + 4(x - 3) < -2 - x. Step 2: Expand the left side: 2x + 4x - 12 = 6x - 12. Step 3: So 6x - 12 < -2 - x. Step 4: Add x to both sides: 7x - 12 < -2. Step 5: Add 12 to both sides: 7x < 10, so x < 10/7. Step 6: Now consider the right part: -2 - x < 2x - 1. Step 7: Add x to both sides: -2 < 3x - 1. Step 8: Add 1 to both sides: -1 < 3x, so x > -1/3. Step 9: Combine the two results to get -1/3 < x < 10/7. Step 10: From the options, x = 1 is the only value in the interval (-1/3, 10/7). Verification / Alternative check: Substitute x = 1 into the original compound inequality. Compute 2x + 4(x - 3) = 2*1 + 4(1 - 3) = 2 + 4(-2) = 2 - 8 = -6. The middle expression is -2 - x = -2 - 1 = -3. The right expression is 2x - 1 = 2*1 - 1 = 1. We see that -6 < -3 < 1, so x = 1 satisfies the double inequality. Why Other Options Are Wrong: Option A: x = -1 is less than -1/3, so it does not satisfy the right side inequality. Option C: x = 2 is greater than 10/7, so it does not satisfy the left side inequality. Option D: x = -2 is far outside the valid range on the left. Option E: x = 0 gives -12 < -2 < -1, which fails the right comparison since -2 is not less than -1. Common Pitfalls: Students sometimes try to manipulate the entire compound inequality in one step and lose track of operations, or they accidentally reverse inequality signs when not required. Treating the compound inequality as two separate inequalities, solving them carefully, and finally intersecting the solution sets is the safest method. Final Answer: Thus, the value of x that satisfies the inequality is 1.

Question 19

Simplify the trigonometric expression (secA - tanA)/(cosecA + cotA). The result is equal to which of the following?

Accepted Answer

Introduction / Context: This is a trigonometric simplification question that involves secant, tangent, cosecant and cotangent. Such expressions often appear in aptitude and board examination questions to test a deep understanding of reciprocal and quotient identities, as well as the ability to perform algebraic manipulations with trigonometric functions. Given Data / Assumptions: The given expression is (secA - tanA)/(cosecA + cotA). We must find an equivalent expression from the options. A is an acute angle such that all trigonometric ratios involved are defined. Concept / Approach: We use definitions of secA, tanA, cosecA and cotA in terms of sine and cosine. A helpful identity is that (secA - tanA)(secA + tanA) = 1 and similarly (cosecA - cotA)(cosecA + cotA) = 1. From these, we can solve for secA - tanA and cosecA + cotA in terms of their counterparts and then simplify the ratio. This approach reveals a neat reciprocal style identity connecting the numerator and the denominator. Step-by-Step Solution: Step 1: Recall that secA = 1/cosA, tanA = sinA/cosA, cosecA = 1/sinA and cotA = cosA/sinA. Step 2: Use the identity (secA - tanA)(secA + tanA) = sec^2A - tan^2A. Step 3: Using sec^2A - tan^2A = 1, we get (secA - tanA)(secA + tanA) = 1. Step 4: Therefore secA - tanA = 1 / (secA + tanA). Step 5: Similarly, (cosecA - cotA)(cosecA + cotA) = cosec^2A - cot^2A = 1, so cosecA + cotA = 1 / (cosecA - cotA). Step 6: Substitute these results into the expression (secA - tanA)/(cosecA + cotA). Step 7: The expression becomes [1/(secA + tanA)] divided by [1/(cosecA - cotA)]. Step 8: Dividing by a fraction is the same as multiplying by its reciprocal, so the result is (1/(secA + tanA)) * (cosecA - cotA) = (cosecA - cotA)/(secA + tanA). Verification / Alternative check: Take a specific angle such as A = 45 degrees. Compute sec45 degrees = √2, tan45 degrees = 1, cosec45 degrees = √2 and cot45 degrees = 1. The original expression becomes (√2 - 1)/(√2 + 1). The candidate expression (cosecA - cotA)/(secA + tanA) becomes (√2 - 1)/(√2 + 1), exactly the same. The other alternatives do not produce this value when evaluated numerically. Why Other Options Are Wrong: Option A: (secA + tanA)/(cosecA + cotA) is essentially the reciprocal of the correct form and does not match the derived identity. Option B: (secA - tanA)/(cosecA - cotA) mixes both minus signs and does not make use of the reciprocal identity correctly. Option D: (secA + cosA)/(cosecA - sinA) imports sine and cosine terms that are unrelated to the neat reciprocal structure of the original expression. Option E: (secA - cosA)/(cosecA + sinA) similarly introduces unrelated combinations and fails numerical checks. Common Pitfalls: A common error is to attempt direct substitution into sine and cosine without noticing the useful pattern (secA - tanA)(secA + tanA) = 1. This can lead to very messy algebra. Another mistake is to cancel terms incorrectly or to forget that division by a fraction requires multiplication by its reciprocal. Recognising and using the identity that the product of conjugate pairs equals 1 greatly simplifies the work. Final Answer: Therefore, the simplified expression is (cosecA - cotA)/(secA + tanA).

Question 20

In the polynomial 6x^3 + 4x^2 + 2x + 3, what is the coefficient of x^2?

Accepted Answer

Introduction / Context: This simple algebra question tests understanding of polynomial notation and the meaning of coefficients. We are given a cubic polynomial and asked to identify the coefficient of x^2, which is the number multiplying the x^2 term. Given Data / Assumptions: The polynomial is 6x^3 + 4x^2 + 2x + 3. We need the coefficient of the x^2 term. All coefficients are real numbers and written in standard form. Concept / Approach: A polynomial in x is written as a sum of terms, where each term is a coefficient multiplied by a power of x. The coefficient of x^2 is simply the number that multiplies x^2 in the expression, no further calculation is needed. Identifying it correctly requires only careful reading of the polynomial. Step-by-Step Solution: Step 1: Write down the polynomial clearly: 6x^3 + 4x^2 + 2x + 3. Step 2: Identify each term by its power of x. Step 3: The term with x^3 is 6x^3, whose coefficient is 6. Step 4: The term with x^2 is 4x^2, whose coefficient is 4. Step 5: The term with x is 2x, whose coefficient is 2. Step 6: The constant term is 3, which has no x factor. Step 7: Therefore, the requested coefficient of x^2 is 4. Verification / Alternative check: We can rewrite the polynomial in standard coefficient list form as a sequence of coefficients for x^3, x^2, x and x^0: 6, 4, 2, 3. The second coefficient in this list corresponds to x^2 and is 4, which matches our identification. No further algebra is required. Why Other Options Are Wrong: Option B: 6 is the coefficient of x^3, not x^2. Option C: 3 is the constant term. Option D: 2 is the coefficient of x. Option E: 0 would mean there is no x^2 term, which is not true because 4x^2 is present. Common Pitfalls: Some learners misread the expression or mix up the order of coefficients, especially when polynomials are not written in descending powers. In this question the polynomial is in standard order, so reading carefully is usually enough. Paying attention to the exponent of x ensures the correct coefficient is chosen. Final Answer: Thus, the coefficient of x^2 is 4.

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