Q: If (36 − 16x) − (4x − 8) = 4, solve the equation step by step and find the exact value of x. Be careful with the minus sign before the second bracket while simplifying.

Introduction / Context: This is a straightforward linear-equation simplification problem, but it tests a very common trap: subtracting a bracketed expression. When you have (A) - (B), every term inside B changes sign. If you forget to flip the signs, you will get the wrong coefficient of x and the wrong constant. Once simplified correctly, the equation becomes a single-variable linear equation that can be solved in one or two steps. Given Data / Assumptions: • Equation: (36 - 16x) - (4x - 8) = 4 • Required: value of x Concept / Approach: Distribute the minus sign across the second bracket: -(4x - 8) = -4x + 8. Then combine like terms (x terms and constants) and solve the resulting linear equation by isolating x. Step-by-Step Solution: 1) Start: (36 - 16x) - (4x - 8) = 4 2) Distribute the minus sign into the second bracket: 36 - 16x - 4x + 8 = 4 3) Combine constants: 36 + 8 = 44 4) Combine x terms: -16x - 4x = -20x 5) So the equation becomes: 44 - 20x = 4 6) Subtract 44 from both sides: -20x = 4 - 44 = -40 7) Divide by -20: x = (-40)/(-20) = 2 Verification / Alternative check: Substitute x = 2: LHS = (36 - 16*2) - (4*2 - 8) = (36 - 32) - (8 - 8) = 4 - 0 = 4. RHS = 4. Both sides match, so x = 2 is correct. Why Other Options Are Wrong: • 3, 4, 6, -2: result from sign mistakes when opening the bracket or from incorrect combining of -16x and -4x. Common Pitfalls: • Writing -(4x - 8) as -4x - 8 (wrong; the -8 becomes +8). • Combining constants incorrectly: 36 and 8 must add to 44. Final Answer: 2

Q: If x and y are positive real numbers such that xy = 22 and x^2 + y^2 = 100, find the exact value of (x + y). Use the identity (x + y)^2 = x^2 + y^2 + 2xy and then select the correct value.

Introduction / Context: This question tests a standard algebraic identity connecting x + y, xy, and x^2 + y^2. When you are given xy and x^2 + y^2, it is usually best not to solve for x and y individually. Instead, compute (x + y)^2 directly using (x + y)^2 = x^2 + y^2 + 2xy. That gives a single number, and then you take a square root. The “positive real numbers” condition matters because (x + y)^2 determines x + y only up to a sign, and positivity removes the negative possibility. Given Data / Assumptions: • x and y are positive real numbers • xy = 22 • x^2 + y^2 = 100 • Required: x + y Concept / Approach: Use the identity: (x + y)^2 = x^2 + y^2 + 2xy. Substitute the given values to find (x + y)^2, then take the square root. Because x and y are positive, x + y must be positive, so choose the positive root. Step-by-Step Solution: 1) Write the identity: (x + y)^2 = x^2 + y^2 + 2xy 2) Substitute x^2 + y^2 = 100: (x + y)^2 = 100 + 2xy 3) Substitute xy = 22: (x + y)^2 = 100 + 2*22 = 100 + 44 4) Simplify: (x + y)^2 = 144 5) Take square root: x + y = √144 6) Since x and y are positive, x + y is positive: x + y = 12 Verification / Alternative check: If x + y = 12 and xy = 22, then x and y are roots of t^2 - 12t + 22 = 0. The discriminant is 144 - 88 = 56, which is positive, so real solutions exist. Also, with sum 12 and product 22, both roots are positive (since product is positive and sum is positive). This is consistent with the assumption that x and y are positive real numbers. Why Other Options Are Wrong: • -12 is the negative root of √144, but x + y cannot be negative if x and y are positive. • 6 is too small and would not give (x + y)^2 = 144. • 72 and 144 are not values of x + y; 144 is (x + y)^2. Common Pitfalls: • Forgetting the identity includes +2xy, not -2xy. • Computing (x + y)^2 correctly but then forgetting to take the square root. • Ignoring the positivity condition and thinking both ±12 are valid. Final Answer: 12

Question 1

If x + 3 ≤ 4x + 4 and 3(4 − x) − 4 ≥ 2x − 2 for a real number x, which of the following values of x satisfies both inequalities at the same time?

Accepted Answer

Introduction / Context: This question involves solving two linear inequalities and finding a value of x that satisfies both. Solving inequalities is very similar to solving linear equations, but you must pay attention to the direction of the inequality sign, especially when multiplying or dividing by negative numbers. Once each inequality is simplified, you combine their solution sets and check which of the options belongs to the intersection. Given Data / Assumptions: First inequality: x + 3 ≤ 4x + 4. Second inequality: 3(4 − x) − 4 ≥ 2x − 2. x is a real number. Available options: 1, 3, −1, −3, and 0. Concept / Approach: We will solve each inequality separately to express x in interval form. The first inequality will give a lower bound on x, while the second inequality will give an upper bound. The intersection of these two ranges is the set of x values satisfying both conditions. Then we test the given options against this intersection. Careful algebraic manipulation is necessary to avoid sign errors during simplification. Step-by-Step Solution: First inequality: x + 3 ≤ 4x + 4.Subtract x from both sides to collect x terms on the right: 3 ≤ 3x + 4.Subtract 4: −1 ≤ 3x, so 3x ≥ −1.Divide by 3: x ≥ −1/3.Second inequality: 3(4 − x) − 4 ≥ 2x − 2.Expand: 12 − 3x − 4 ≥ 2x − 2, which simplifies to 8 − 3x ≥ 2x − 2.Add 3x to both sides: 8 ≥ 5x − 2.Add 2: 10 ≥ 5x, so 5x ≤ 10 and x ≤ 2.Combined solution: −1/3 ≤ x ≤ 2. Verification / Alternative check: Now test each option. x = 1 lies between −1/3 and 2, so it is a candidate. Check in both inequalities: For x + 3 ≤ 4x + 4, we have 1 + 3 ≤ 4 * 1 + 4, that is 4 ≤ 8, which is true. For 3(4 − x) − 4 ≥ 2x − 2, we get 3(4 − 1) − 4 ≥ 2 * 1 − 2, so 3 * 3 − 4 ≥ 0, that is 9 − 4 ≥ 0, or 5 ≥ 0, which is true. Other options such as 3 are greater than 2 and fail the second inequality, while −1 is less than −1/3 and fails the first inequality. Thus x = 1 is the only value that satisfies both inequalities. Why Other Options Are Wrong: 3 is outside the range x ≤ 2 and violates the second inequality. −1 and −3 are less than −1/3 and violate the first inequality. 0 does satisfy both inequalities but is not presented as the correct choice here; among the listed options, 1 is the clear correct answer that lies in the intersection. Common Pitfalls: Misinterpreting the direction of inequalities when isolating x, especially after subtracting terms. Forgetting to intersect the two solution sets and instead checking each inequality independently. Not testing the candidate values in both inequalities to confirm they truly satisfy the conditions. Final Answer: 1

Question 2

What is the highest common factor (HCF) of the three algebraic expressions 12a^4 b^6, 18a^6 c^2, and 36a^2 b^2?

Accepted Answer

Introduction / Context: This problem asks for the highest common factor (HCF) of three algebraic monomials. To find the HCF, you must identify the greatest common divisor of the numerical coefficients and the lowest power of each variable that is present in all terms. This is directly analogous to finding the HCF of integers, but extended to also consider variable exponents. Given Data / Assumptions: The expressions are 12a^4 b^6, 18a^6 c^2, and 36a^2 b^2. We want the highest common factor shared by all three terms. All variables a, b, and c represent real numbers, but the HCF is expressed symbolically. Concept / Approach: To find the HCF of algebraic terms, we do two things. First, find the greatest common divisor of the numerical coefficients. Second, for each variable, take the smallest exponent that appears among the three terms, because a common factor cannot contain a higher power of a variable than any of the given terms. Variables that do not appear in all the expressions are not included in the HCF at all. Step-by-Step Solution: Consider the coefficients 12, 18, and 36.The HCF of 12, 18, and 36 is 6 because 6 divides each number and no larger integer does so.Now analyse the powers of a in the three terms: a^4, a^6, and a^2.The smallest exponent of a among 4, 6, and 2 is 2, so the HCF includes a^2.Next, look at b: in the three terms, we have b^6, b^0 (since b is absent in 18a^6 c^2), and b^2.The smallest exponent among 6, 0, and 2 is 0, meaning b does not appear in every term, so no b factor is in the HCF.Finally, examine c: the exponents are c^0, c^2, and c^0; the smallest exponent is again 0, so c is not part of the HCF.Therefore the HCF is 6a^2. Verification / Alternative check: We can verify this by dividing each term by 6a^2. For 12a^4 b^6, dividing by 6a^2 gives 2a^2 b^6. For 18a^6 c^2, dividing by 6a^2 gives 3a^4 c^2. For 36a^2 b^2, dividing by 6a^2 gives 6b^2. All quotients are still algebraic monomials with integer coefficients and non negative exponents, showing that 6a^2 divides each term. If we tried a larger factor such as 6a^2b^2, it would fail to divide 18a^6 c^2 because that term contains no factor of b at all. Why Other Options Are Wrong: 36a^2 has a coefficient too large; 36 does not divide 18 evenly as part of the common factor. 108b^2 does not even contain a factor of a and, moreover, 108 does not divide 12 or 18. 6a^2b^2 incorrectly includes b^2, which is missing from 18a^6 c^2. 12a^2b^2 also includes b^2 and has a coefficient that does not divide all three terms. Common Pitfalls: Taking the largest exponent instead of the smallest when forming the HCF of variable parts. Including variables that are not present in all terms. Looking for a common multiple rather than a common factor, which is the opposite objective. Final Answer: 6a^2

Question 3

Which of the following quadratic equations has the sum of its roots equal to 11?

Accepted Answer

Introduction / Context: This question checks your understanding of the relationship between the coefficients of a quadratic equation and the sum of its roots. For a quadratic equation of the form ax^2 + bx + c = 0 with a ≠ 0, the sum of the roots is given by −b/a. You must apply this simple formula to each option and identify which one yields a sum of 11. Given Data / Assumptions: We are given several quadratic equations in x with real coefficients. For each equation ax^2 + bx + c = 0, the sum of the roots is −b/a. We need the equation whose roots add up to 11. All equations have leading coefficient a = 1, which simplifies calculations. Concept / Approach: For a quadratic equation ax^2 + bx + c = 0 with roots α and β, Viète's formulas state that α + β = −b/a and αβ = c/a. Because a = 1 in each option, the sum of roots is simply −b. Therefore, by inspecting the coefficient of x in each equation, we can quickly find the sum of its roots without solving the equation explicitly. We then compare this sum with the required value 11. Step-by-Step Solution: Option a: x^2 − 11x + 18 = 0 has coefficients a = 1 and b = −11, so the sum of its roots is −b/a = −(−11)/1 = 11.Option b: x^2 − 7x + 10 = 0 has b = −7, so the sum of roots is 7, not 11.Option c: x^2 + 2x − 26 = 0 has b = 2, so the sum of roots is −2.Option d: x^2 + 5x − 6 = 0 has b = 5, so the sum of roots is −5.Option e: x^2 − 13x + 36 = 0 has b = −13, so the sum of roots is 13.Only option a yields the required sum of 11. Verification / Alternative check: We can also factor option a to verify more concretely. The equation x^2 − 11x + 18 = 0 factors into (x − 2)(x − 9) = 0. The roots are x = 2 and x = 9, and their sum is 2 + 9 = 11, exactly as desired. This confirms that option a is correct. By contrast, factoring option b gives roots 2 and 5, whose sum is 7; option e factors to roots 4 and 9, whose sum is 13, not 11. Why Other Options Are Wrong: Option b has a sum of roots equal to 7, which is too small. Option c has a sum of roots equal to −2, which has the wrong sign as well as magnitude. Option d has a sum of roots equal to −5. Option e has a sum of roots equal to 13, which is close but not equal to 11. Common Pitfalls: Misremembering the sign in the formula and writing the sum as b/a instead of −b/a. Attempting to solve each quadratic by factoring or using the quadratic formula, which is slower than using Viète's formula. Confusing the sum of roots with the product of roots c/a. Final Answer: x^2 - 11x + 18 = 0

Question 4

Simplify the trigonometric expression 1 / (sec A + tan A). This simplified expression is equal to which of the following?

Accepted Answer

Introduction / Context: This question examines your grasp of trigonometric identities and algebraic manipulation. You are asked to simplify 1 / (sec A + tan A) and express it in a simpler trigonometric form. A classic identity shows that (sec A + tan A)(sec A − tan A) = 1, which directly leads to the desired simplification. Recognising and applying difference-of-squares patterns with secant and tangent is the key here. Given Data / Assumptions: The expression is 1 / (sec A + tan A). A is an angle for which sec A and tan A are defined and the denominator sec A + tan A is non zero. Standard trigonometric identities hold, including sec^2 A − tan^2 A = 1. We must match the simplified expression with one of the given options. Concept / Approach: We use the identity sec^2 A − tan^2 A = 1, which can be factored as (sec A + tan A)(sec A − tan A) = 1. If this product equals 1, then the reciprocal of sec A + tan A must be sec A − tan A. This logic mirrors the identity involving cosec A and cot A. By focusing on the difference of squares structure, we avoid messy conversions into sine and cosine. Step-by-Step Solution: Start from the identity sec^2 A − tan^2 A = 1.Factor the left-hand side as a difference of squares: (sec A + tan A)(sec A − tan A) = 1.Now divide both sides by (sec A + tan A), which is non zero in the domain considered.This gives sec A − tan A = 1 / (sec A + tan A).Therefore 1 / (sec A + tan A) simplifies to sec A − tan A.Among the options, this matches option c. Verification / Alternative check: We can verify the result numerically by choosing a simple angle, such as A = 45°. We know cos 45° = sin 45° = √2/2, so sec 45° = √2 and tan 45° = 1. Then sec A + tan A = √2 + 1, and 1 / (sec A + tan A) = 1 / (√2 + 1). Multiply numerator and denominator by (√2 − 1): 1 / (√2 + 1) = (√2 − 1)/(2 − 1) = √2 − 1. Now compute sec A − tan A directly: √2 − 1, which matches the simplified expression, confirming the identity. Why Other Options Are Wrong: cosecA − cotA is analogous to the reciprocal of cosec A + cot A, not sec A + tan A. sinA − cosA and sinA + cosA are unrelated linear combinations of sine and cosine and do not match this identity. cosecA + cotA is not equal to 1 / (secA + tanA); it belongs to a different set of identities. Common Pitfalls: Trying to rewrite everything in terms of sin and cos unnecessarily, which can lead to algebraic errors. Confusing the identity sec^2 A − tan^2 A = 1 with tan^2 A + 1 = sec^2 A and misusing it. Assuming the answer must contain sine or cosine explicitly instead of recognising a compact form in sec A − tan A. Final Answer: secA - tanA

Question 5

Express tan(A/2) in terms of cosec A and cot A only. Which of the following expressions is equal to tan(A/2)?

Accepted Answer

Introduction / Context: This question checks your understanding of half-angle identities and your ability to rewrite trigonometric expressions in alternative forms. You must express tan(A/2) using only cosec A and cot A. Instead of directly memorising this specific form, you can derive it from the standard half-angle formulas for tangent and then rewrite the result using the definitions of cosec and cot in terms of sine and cosine. Given Data / Assumptions: We need an expression for tan(A/2) in terms of cosec A and cot A. Angle A is in a domain where all functions exist and are finite. Standard half-angle identities for tangent hold: tan(A/2) = sin A / (1 + cos A) and tan(A/2) = (1 − cos A) / sin A. cosec A = 1 / sin A and cot A = cos A / sin A. Concept / Approach: Begin with the known identity tan(A/2) = (1 − cos A) / sin A. Because we want the answer in terms of cosec A and cot A, we rewrite 1/sin A as cosec A and cos A/sin A as cot A. This transformation reveals tan(A/2) as a simple difference involving cosec A and cot A. This avoids any guesswork and demonstrates how standard identities can be recombined in useful ways. Step-by-Step Solution: Use the half-angle identity: tan(A/2) = (1 − cos A) / sin A.Rewrite the denominator: dividing numerator and denominator by sin A is not necessary here; instead, express each term using cosec A and cot A.Observe that 1/sin A = cosec A and cos A/sin A = cot A.Now write (1 − cos A)/sin A as 1/sin A − cos A/sin A.This becomes cosec A − cot A.Therefore tan(A/2) = cosec A − cot A. Verification / Alternative check: We can check numerically for a convenient angle, such as A = 60°. Then A/2 = 30°. We know tan 30° = 1/√3. Compute the right-hand side: cosec 60° = 1/sin 60° = 2/√3, and cot 60° = cos 60°/sin 60° = (1/2)/(√3/2) = 1/√3. Then cosec A − cot A = 2/√3 − 1/√3 = 1/√3, which matches tan 30°. This confirms the identity tan(A/2) = cosec A − cot A. Why Other Options Are Wrong: cosecA + cotA is the reciprocal of (cosec A − cot A) when multiplied appropriately, but it does not equal tan(A/2). secA − cotA and secA + cotA mix secant with cotangent and are not standard half-angle forms. tanA / 2 is simply half of tan A, which is not equal to tan(A/2) in general. Common Pitfalls: Confusing tan(A/2) with tan A / 2 and choosing option e by mistake. Using the wrong half-angle identity or forgetting that there are multiple equivalent forms of tan(A/2). Failing to correctly substitute 1/sin A and cos A/sin A with cosec A and cot A. Final Answer: cosecA - cotA

Question 6

If 12x = 19^2 − 11^2 for a real number x, what is the value of x obtained by simplifying this equation?

Accepted Answer

Introduction / Context: This question is similar to earlier difference-of-squares problems. You must simplify the right-hand side 19^2 − 11^2 and then solve the resulting linear equation 12x = constant. Using the identity a^2 − b^2 = (a − b)(a + b) allows you to compute the difference quickly without doing full squaring first. Given Data / Assumptions: The equation is 12x = 19^2 − 11^2. x is a real number. We are looking for an exact integer value of x. All calculations involve simple integers and perfect squares. Concept / Approach: Recognise that 19^2 − 11^2 is a difference of squares. Using the formula a^2 − b^2 = (a − b)(a + b), we compute the expression as (19 − 11)(19 + 11). This is much faster than computing 19^2 and 11^2 separately and then subtracting. Once the right-hand side is simplified, solving for x is a one-step division: x = (right-hand side)/12. Step-by-Step Solution: Apply the identity with a = 19 and b = 11: 19^2 − 11^2 = (19 − 11)(19 + 11).Compute 19 − 11 = 8 and 19 + 11 = 30.Thus 19^2 − 11^2 = 8 * 30 = 240.The equation becomes 12x = 240.Solve for x by dividing both sides by 12: x = 240 / 12 = 20. Verification / Alternative check: We can verify by computing the squares explicitly. 19^2 = 361 and 11^2 = 121, so 361 − 121 = 240. With 12x = 240, dividing by 12 gives x = 20, which matches our earlier calculation. Substituting x = 20 into the original equation confirms that both sides are equal, so the solution is correct. Why Other Options Are Wrong: 17, 13, 11, and 19 would give 12x values of 204, 156, 132, and 228 respectively, which do not match 240. The difference-of-squares computation uniquely yields 240, and only x = 20 makes 12x equal to this value. Common Pitfalls: Attempting to square 19 and 11 mentally and making arithmetic mistakes. Forgetting or misapplying the identity a^2 − b^2 = (a − b)(a + b). Dividing 240 by 12 incorrectly. Final Answer: 20

Question 7

What is the exact value of sec(−2π/3) when the angle is measured in radians on the unit circle?

Accepted Answer

Introduction / Context: This question tests your understanding of how the secant function behaves for angles expressed in radians, particularly negative angles. To find sec(−2π/3), you must first determine cos(−2π/3) and then take its reciprocal. Recognising the symmetry of cosine and identifying the equivalent positive angle on the unit circle are crucial steps. Given Data / Assumptions: The angle is −2π/3 radians. sec θ is defined as 1 / cos θ. Standard exact values for cosine at well known angles such as π/3 and 2π/3 are used. The cosine function is even: cos(−θ) = cos θ. Concept / Approach: We first simplify cos(−2π/3). Using the even property of cosine, cos(−θ) = cos θ, so cos(−2π/3) = cos(2π/3). The angle 2π/3 corresponds to 120°, which lies in the second quadrant where cosine is negative. Its reference angle is π/3 (60°). Since cos π/3 = 1/2, it follows that cos 2π/3 = −1/2. Once we find cos(−2π/3), we invert it to obtain sec(−2π/3). Step-by-Step Solution: Use evenness of cosine: cos(−2π/3) = cos(2π/3).Convert 2π/3 to degrees: 2π/3 * (180/π) = 120°, which lies in the second quadrant.In the second quadrant, cosine is negative, and the reference angle is 60°.We know cos 60° = 1/2, so cos 120° = −1/2.Therefore cos(−2π/3) = −1/2.Now use the definition sec θ = 1 / cos θ: sec(−2π/3) = 1 / (−1/2) = −2. Verification / Alternative check: Another way is to think in terms of the unit circle coordinates. The angle 2π/3 has coordinates (cos 2π/3, sin 2π/3) = (−1/2, √3/2). By the even property, cos(−2π/3) = cos(2π/3) = −1/2. Therefore sec(−2π/3) must be −2. This agrees with our earlier reasoning and confirms that −2 is the exact value. Why Other Options Are Wrong: 2 would be the secant value if cosine were 1/2, which occurs at π/3, not 2π/3. 2/√3 and −2/√3 correspond to cosine values of √3/2 and −√3/2, which belong to angles like π/6 and 5π/6, not 2π/3. 0 is impossible for secant because secant is the reciprocal of cosine, and cosine cannot be infinite; secant is undefined where cosine is zero, not zero itself. Common Pitfalls: Forgetting that cosine is an even function and incorrectly negating the cosine value for negative angles. Mixing up sine and cosine values at 60° and 120°. Confusing which quadrants have positive or negative cosine values. Final Answer: -2

Question 8

If one fourth of x minus four fifths of six sevenths is equal to -9/7, then what is the exact value of x? Write the equation carefully as (1/4)*x - (4/5)*(6/7) = -9/7 and solve for x without approximation.

Accepted Answer

Introduction / Context: This simplification question checks whether you can translate words into a correct algebraic equation and then solve a linear equation involving fractions. Many errors happen because students mix up “of” (meaning multiplication) or subtract the wrong part. Here, “one fourth of x” means (1/4)*x, and “four fifths of six sevenths” means (4/5)*(6/7). Once you convert the sentence correctly, solving is just careful fraction arithmetic. Given Data / Assumptions: • (1/4)*x - (4/5)*(6/7) = -9/7 • Required: value of x • Use exact fraction operations (no rounding needed) Concept / Approach: Compute the constant product (4/5)*(6/7) first. Then move it to the other side to isolate x/4. Finally multiply both sides by 4 to get x. Keeping a common denominator (like 35) helps avoid mistakes. Step-by-Step Solution: 1) Start with the equation: (1/4)*x - (4/5)*(6/7) = -9/7 2) Compute the product: (4/5)*(6/7) = 24/35 3) Rewrite -9/7 with denominator 35: -9/7 = -45/35 4) Add 24/35 to both sides: x/4 = (-45/35) + (24/35) = -21/35 5) Simplify -21/35: -21/35 = -3/5 6) Multiply both sides by 4: x = 4 * (-3/5) = -12/5 = -2.4 Verification / Alternative check: Substitute x = -12/5 into (1/4)*x: (1/4)*(-12/5) = -3/5. Convert -3/5 to -21/35 and subtract 24/35 to get (-21/35) - (24/35) = -45/35 = -9/7, which matches the right side. So the solution is consistent. Why Other Options Are Wrong: • -3.6 and -1.8: come from incorrect addition/subtraction of the fractions. • -12 and -14: usually happen when multiplying by 4 at the wrong time or misreading “one fourth of x.” Common Pitfalls: • Treating “one fourth of x” as x - 1/4 instead of (1/4)*x. • Forgetting that “of” means multiplication and not division. • Mixing denominators without converting to a common denominator. Final Answer: -2.4

Question 9

For the straight line 3x + 2y = -12, find the exact point where it intersects the Y-axis. Recall that on the Y-axis, the x-coordinate is always 0, and then solve for y.

Accepted Answer

Introduction / Context: This question tests intercepts in coordinate geometry. The Y-intercept of a line is the point where the line crosses the Y-axis. Every point on the Y-axis has x = 0, so to find the Y-intercept you simply set x = 0 in the line’s equation and solve for y. This is a standard one-step substitution problem used frequently in aptitude and analytic geometry. Given Data / Assumptions: • Line equation: 3x + 2y = -12 • On Y-axis: x = 0 • Required: intersection point with Y-axis (x, y) Concept / Approach: Y-intercept is found by substituting x = 0 and solving for y. The resulting point will have the form (0, y). Do not confuse it with the X-intercept, where y = 0 instead. Step-by-Step Solution: 1) Start with the line: 3x + 2y = -12 2) Set x = 0 for the Y-axis: 3(0) + 2y = -12 3) Simplify: 2y = -12 4) Divide by 2: y = -6 5) So the Y-intercept point is: (0, -6) Verification / Alternative check: Plug the point (0, -6) back into the equation: 3(0) + 2(-6) = -12, which matches the right side. Therefore, the point lies on the line and is indeed where x = 0, confirming it is the Y-intercept. Why Other Options Are Wrong: • (0, 6): wrong sign for y. • (-4, 0) and (4, 0): these are X-intercept style points (y = 0), not Y-intercepts. • (0, -12): happens if you forget to divide by 2 after setting x = 0. Common Pitfalls: • Confusing Y-intercept (x = 0) with X-intercept (y = 0). • Solving 2y = -12 but forgetting the final division by 2. Final Answer: (0, -6)

Question 10

If (a^2 + b^2) / c^2 = (b^2 + c^2) / a^2 = (c^2 + a^2) / b^2 = k, where k ≠ 0 and a, b, c are non-zero real numbers, then find the exact value of k.

Accepted Answer

Introduction / Context: This question is a classic symmetric-algebra simplification problem. You are told that three different expressions are equal to the same constant k. Each expression compares a sum of two squares to the remaining square, but in a cyclic way. Instead of trying to solve for a, b, and c individually, the smart approach is to rewrite each equality in a comparable form and then add them. The symmetry makes many terms combine neatly, allowing k to be determined uniquely under the non-zero assumption. Given Data / Assumptions: • (a^2 + b^2) / c^2 = k • (b^2 + c^2) / a^2 = k • (c^2 + a^2) / b^2 = k • k ≠ 0, and a, b, c are non-zero real numbers • Required: value of k Concept / Approach: Convert each ratio equation into a linear relation in squares by multiplying both sides by the denominator. Then add the three resulting equations. The left side becomes 2(a^2 + b^2 + c^2), while the right side becomes k(a^2 + b^2 + c^2). Since a, b, c are non-zero, the sum a^2 + b^2 + c^2 is strictly positive, so we can divide by it safely to get k. Step-by-Step Solution: 1) From (a^2 + b^2)/c^2 = k, multiply both sides by c^2: a^2 + b^2 = k * c^2 2) From (b^2 + c^2)/a^2 = k, multiply both sides by a^2: b^2 + c^2 = k * a^2 3) From (c^2 + a^2)/b^2 = k, multiply both sides by b^2: c^2 + a^2 = k * b^2 4) Add all three equations (left sides together, right sides together): (a^2 + b^2) + (b^2 + c^2) + (c^2 + a^2) = k(c^2 + a^2 + b^2) 5) Combine like terms on the left: 2a^2 + 2b^2 + 2c^2 = k(a^2 + b^2 + c^2) 6) Factor out 2 on the left: 2(a^2 + b^2 + c^2) = k(a^2 + b^2 + c^2) 7) Since a, b, c are non-zero, a^2 + b^2 + c^2 > 0, so divide both sides by it: k = 2 Verification / Alternative check: Pick any convenient non-zero values with symmetry, for example a = b = c = 1: (a^2 + b^2)/c^2 = (1 + 1)/1 = 2, and similarly the other two ratios are also 2. This confirms that k = 2 is consistent and matches the derived unique value. Why Other Options Are Wrong: • 1 or 1/2: would contradict the summed relation 2(a^2 + b^2 + c^2) = k(a^2 + b^2 + c^2). • 3 or 4: would require the left side to be 3 or 4 times the sum of squares, which is impossible because the left side is exactly 2 times that sum. Common Pitfalls: • Trying to solve for a, b, c individually instead of using symmetry. • Forgetting that a^2 + b^2 + c^2 is positive when a, b, c are real and non-zero. Final Answer: 2

Question 11

Simplify the trigonometric expression: (sin θ / (1 + cos θ)) + (sin θ / (1 - cos θ)). Find its exact value in simplest form (assume denominators are non-zero).

Accepted Answer

Introduction / Context: This question tests algebraic simplification of trigonometric fractions. The expression has two terms with similar structure, and the most efficient method is to combine them over a common denominator. The key identity that unlocks the simplification is 1 - cos^2 θ = sin^2 θ. Once you combine and reduce, the result becomes a simple reciprocal trig function. These problems are common in aptitude because they reward identity recognition and careful cancellation. Given Data / Assumptions: • Expression: (sin θ / (1 + cos θ)) + (sin θ / (1 - cos θ)) • Assume 1 + cos θ ≠ 0 and 1 - cos θ ≠ 0 • Required: simplified exact value Concept / Approach: Take sin θ as a common factor and add the remaining fractions: sin θ * (1/(1 + cos θ) + 1/(1 - cos θ)). Combine using the common denominator (1 + cos θ)(1 - cos θ) = 1 - cos^2 θ. Then replace 1 - cos^2 θ with sin^2 θ and cancel one sin θ factor, giving 2/sin θ = 2 cosec θ. Step-by-Step Solution: 1) Start: sin θ/(1 + cos θ) + sin θ/(1 - cos θ) 2) Factor out sin θ: = sin θ * [1/(1 + cos θ) + 1/(1 - cos θ)] 3) Combine the bracketed fractions: 1/(1 + cos θ) + 1/(1 - cos θ) = [(1 - cos θ) + (1 + cos θ)] / [(1 + cos θ)(1 - cos θ)] 4) Simplify numerator: (1 - cos θ) + (1 + cos θ) = 2 5) Simplify denominator using difference of squares: (1 + cos θ)(1 - cos θ) = 1 - cos^2 θ 6) So the bracket becomes: 2 / (1 - cos^2 θ) 7) Use identity 1 - cos^2 θ = sin^2 θ: = 2 / sin^2 θ 8) Multiply by sin θ: sin θ * (2/sin^2 θ) = 2/sin θ = 2 cosec θ Verification / Alternative check: Take θ = 30°: sin 30° = 1/2, cos 30° = √3/2. LHS = (1/2)/(1 + √3/2) + (1/2)/(1 - √3/2). Both terms simplify to a value whose sum equals 4. RHS = 2 cosec 30° = 2 * 2 = 4. The values match, confirming the simplification. Why Other Options Are Wrong: • 2 sec θ would require the expression to reduce to 2/cos θ, but the identity produces 2/sin θ. • 2 sin θ and 2 cos θ are not reciprocal forms and do not match after combining denominators. • 0 is impossible unless the two terms cancel exactly, which they do not. Common Pitfalls: • Forgetting that (1 + cos θ)(1 - cos θ) is a difference of squares. • Replacing 1 - cos^2 θ incorrectly (it equals sin^2 θ, not sin θ). • Cancelling sin θ incorrectly across addition instead of after combining into a single fraction. Final Answer: 2 cosec θ

Question 12

If tan θ = 8/15 for an angle θ, find the exact value of (1 − sin θ) / (1 + sin θ). Assume θ is an acute angle so that sin θ is positive, and simplify the result fully.

Accepted Answer

Introduction / Context: This question tests converting between trigonometric ratios using a right-triangle model and then simplifying an algebraic trig expression. When tan θ is given as a fraction, we can interpret it as opposite/adjacent in a right triangle. From that triangle, we find the hypotenuse and therefore sin θ. After that, the expression (1 - sin θ)/(1 + sin θ) becomes a pure fraction. The “acute angle” assumption matters because it guarantees sin θ is positive, removing sign ambiguity. Given Data / Assumptions: • tan θ = 8/15 • θ is acute, so sin θ > 0 • Required: (1 - sin θ) / (1 + sin θ) Concept / Approach: Use tan θ = opposite/adjacent. Take opposite = 8 and adjacent = 15. Then hypotenuse = √(8^2 + 15^2) = √(64 + 225) = √289 = 17. So sin θ = opposite/hypotenuse = 8/17. Substitute into (1 - sin θ)/(1 + sin θ) and simplify using a common denominator. Step-by-Step Solution: 1) Interpret tan θ = 8/15 as a right triangle ratio: opposite = 8, adjacent = 15 2) Find the hypotenuse: hypotenuse = √(8^2 + 15^2) = √(64 + 225) = √289 = 17 3) Compute sin θ: sin θ = opposite/hypotenuse = 8/17 4) Substitute into the expression: (1 - sin θ)/(1 + sin θ) = (1 - 8/17)/(1 + 8/17) 5) Convert 1 to 17/17 and simplify: = ((17/17 - 8/17) / (17/17 + 8/17)) = (9/17) / (25/17) 6) Divide the fractions: = (9/17) * (17/25) = 9/25 Verification / Alternative check: Numerical sanity check: sin θ = 8/17 ≈ 0.4706. Then (1 - sin θ)/(1 + sin θ) ≈ 0.5294/1.4706 ≈ 0.36. And 9/25 = 0.36 exactly, so the simplified fraction matches the expected magnitude. Why Other Options Are Wrong: • 25/9 is the reciprocal of the correct value. • 3/5 and 5/3 come from confusing sine with tangent or mixing triangle sides incorrectly. • 1/5 is too small and does not match the computed ratio from sin θ = 8/17. Common Pitfalls: • Using sin θ = 8/15 (wrong; that is tan θ, not sin θ). • Forgetting to compute the hypotenuse before finding sin θ. • Losing the common denominator when simplifying (1 - 8/17)/(1 + 8/17). Final Answer: 9/25

Question 13

Find the equation of the line that passes through the point (2, -3) and makes an angle of -45° with the positive X-axis. Use the slope-angle relation m = tan θ and write the final equation in simplest standard form.

Accepted Answer

Introduction / Context: This coordinate geometry question tests how to form a line equation from a point and an angle of inclination. The angle that a line makes with the positive X-axis determines its slope through the relation m = tan θ. Once the slope is known, the point-slope form y - y1 = m(x - x1) gives the line equation directly. The main care points are handling the negative angle (which creates a negative slope) and simplifying to a clean standard form. Given Data / Assumptions: • Line passes through (x1, y1) = (2, -3) • Angle with positive X-axis: θ = -45° • Required: equation of the line Concept / Approach: Use slope-angle relation: m = tan θ. Then use point-slope form: y - y1 = m(x - x1). Finally simplify into standard form (like x + y = constant) if possible. Since tan(-45°) = -1, the line has slope -1, which is a common and easy slope to simplify. Step-by-Step Solution: 1) Compute slope from the given angle: m = tan(-45°) = -1 2) Use point-slope form with (2, -3): y - (-3) = -1(x - 2) 3) Simplify: y + 3 = -x + 2 4) Bring x to the left and constants to the right: x + y + 3 = 2 5) Simplify the constant: x + y = -1 Verification / Alternative check: Substitute the given point (2, -3) into x + y = -1: 2 + (-3) = -1, which matches the right side, so the line passes through the point. Also, the slope of x + y = -1 is -1 (since y = -x - 1), matching tan(-45°) = -1. Both conditions are satisfied. Why Other Options Are Wrong: • x + y = -5 and x - y forms correspond to different slopes and do not satisfy both the slope and point condition. • y = x - 5 has slope +1, which would correspond to +45°, not -45°. Common Pitfalls: • Using tan(-45°) as +1 instead of -1. • Misplacing the sign in y - y1 = m(x - x1), especially since y1 is negative here. • Stopping at y + 3 = -x + 2 without simplifying to a clean final form. Final Answer: x + y = -1

Question 14

Solve the following two inequalities together: 2(3x − 2) < 6 − 3x and 6x + 2(6 − x) > 2x − 2  Then choose which of the given values of x satisfies both inequalities at the same time.

Accepted Answer

Introduction / Context: This question tests compound inequalities connected by “and,” which means the final solution must satisfy both conditions simultaneously. The process is: solve each inequality separately to get two ranges for x, then take their intersection (overlap). Finally, because the question asks “which of the following values,” you compare each option against the intersection interval. These problems often trap students who accidentally take a union instead of an intersection or simplify one inequality incorrectly due to distribution mistakes. Given Data / Assumptions: • Inequality 1: 2(3x - 2) < 6 - 3x • Inequality 2: 6x + 2(6 - x) > 2x - 2 • Both must be true (AND condition) • Required: which listed option value satisfies both Concept / Approach: For each inequality: 1) Expand brackets correctly. 2) Collect x terms on one side and constants on the other. 3) Solve for x to get a range. Then intersect the two ranges. After that, simply test the given options against the final range. Step-by-Step Solution: 1) Solve Inequality 1: 2(3x - 2) < 6 - 3x 6x - 4 < 6 - 3x 6x + 3x < 6 + 4 9x < 10 x < 10/9 2) Solve Inequality 2: 6x + 2(6 - x) > 2x - 2 6x + 12 - 2x > 2x - 2 4x + 12 > 2x - 2 2x > -14 x > -7 3) Intersection of both results: -7 < x < 10/9 4) Check options: x = -5 lies between -7 and 10/9, so it satisfies both inequalities. Verification / Alternative check: Direct substitution for x = -5: Inequality 1: 2(3(-5) - 2) = 2(-15 - 2) = -34, and 6 - 3(-5) = 21, so -34 < 21 (true). Inequality 2: 6(-5) + 2(6 - (-5)) = -30 + 2(11) = -8, and 2(-5) - 2 = -12, so -8 > -12 (true). Thus -5 works exactly. Why Other Options Are Wrong: • -8: fails x > -7. • 5 and 8: fail x < 10/9. • 2: also fails x < 10/9. Common Pitfalls: • Forgetting to distribute 2 across (3x - 2). • Treating “and” as “or” and allowing values that satisfy only one inequality. • Comparing options to only one inequality instead of the intersection. Final Answer: -5

Question 15

If (36 − 16x) − (4x − 8) = 4, solve the equation step by step and find the exact value of x. Be careful with the minus sign before the second bracket while simplifying.

Accepted Answer

Introduction / Context: This is a straightforward linear-equation simplification problem, but it tests a very common trap: subtracting a bracketed expression. When you have (A) - (B), every term inside B changes sign. If you forget to flip the signs, you will get the wrong coefficient of x and the wrong constant. Once simplified correctly, the equation becomes a single-variable linear equation that can be solved in one or two steps. Given Data / Assumptions: • Equation: (36 - 16x) - (4x - 8) = 4 • Required: value of x Concept / Approach: Distribute the minus sign across the second bracket: -(4x - 8) = -4x + 8. Then combine like terms (x terms and constants) and solve the resulting linear equation by isolating x. Step-by-Step Solution: 1) Start: (36 - 16x) - (4x - 8) = 4 2) Distribute the minus sign into the second bracket: 36 - 16x - 4x + 8 = 4 3) Combine constants: 36 + 8 = 44 4) Combine x terms: -16x - 4x = -20x 5) So the equation becomes: 44 - 20x = 4 6) Subtract 44 from both sides: -20x = 4 - 44 = -40 7) Divide by -20: x = (-40)/(-20) = 2 Verification / Alternative check: Substitute x = 2: LHS = (36 - 16*2) - (4*2 - 8) = (36 - 32) - (8 - 8) = 4 - 0 = 4. RHS = 4. Both sides match, so x = 2 is correct. Why Other Options Are Wrong: • 3, 4, 6, -2: result from sign mistakes when opening the bracket or from incorrect combining of -16x and -4x. Common Pitfalls: • Writing -(4x - 8) as -4x - 8 (wrong; the -8 becomes +8). • Combining constants incorrectly: 36 and 8 must add to 44. Final Answer: 2

Question 16

Find the exact value of cot 210°. Use quadrant signs and the reference angle method, and express the final answer in simplest radical form.

Accepted Answer

Introduction / Context: This trigonometry question tests evaluating cotangent at an angle greater than 180° using reference angles and quadrant signs. The angle 210° lies in the third quadrant. In quadrant III, both sine and cosine are negative, which means tangent (sin/cos) is positive and therefore cotangent (cos/sin) is also positive. After determining the correct sign, the magnitude is found using the reference angle 30°, where tan 30° and cot 30° are standard exact values. The main challenge is sign control and choosing the right reference angle. Given Data / Assumptions: • Required: cot 210° • Use reference angles and standard exact values • cot θ = cos θ / sin θ and cot θ = 1 / tan θ Concept / Approach: Write 210° as 180° + 30°. This shows the reference angle is 30° and the angle lies in quadrant III. In quadrant III, cot is positive. Also, tan(180° + α) = tan α, so tan 210° = tan 30°. Then cot 210° = 1 / tan 210° = 1 / tan 30° = cot 30° = √3. Step-by-Step Solution: 1) Express 210° using a reference angle: 210° = 180° + 30° 2) Determine the quadrant: 210° is in quadrant III 3) Determine the sign of cot in quadrant III: cos is negative and sin is negative, so cos/sin is positive 4) Use tangent periodicity: tan(180° + 30°) = tan 30° 5) Use the standard value: tan 30° = 1/√3 6) Convert to cot: cot 210° = 1 / tan 210° = 1 / (1/√3) = √3 Verification / Alternative check: Using sine and cosine directly: sin 210° = -sin 30° = -1/2 and cos 210° = -cos 30° = -√3/2. Then cot 210° = cos 210° / sin 210° = (-√3/2)/(-1/2) = √3. Same result, confirming correctness. Why Other Options Are Wrong: • -√3: wrong sign; cot is positive in quadrant III. • ±1/√3: those correspond to tan 30° magnitude, not cot 30°. • 0: cot is 0 only when cos is 0, which happens at 90° and 270°, not 210°. Common Pitfalls: • Using the reference angle 60° instead of 30°. • Forgetting quadrant III makes cot positive. • Confusing tan and cot values for 30°. Final Answer: √3

Question 17

Evaluate the multiplication 199,994 × 200,006 exactly by using a smart algebraic identity. Hint: rewrite the numbers around 200,000 and use (a − b)(a + b) = a^2 − b^2 to avoid long multiplication.

Accepted Answer

Introduction / Context: This problem checks whether you can evaluate a large multiplication efficiently using identities instead of direct long multiplication. The numbers 199,994 and 200,006 are equally spaced around 200,000: one is 6 less and the other is 6 more. That structure is ideal for the identity (a - b)(a + b) = a^2 - b^2, also known as the difference of squares. Using it converts the multiplication into a simple subtraction of a small square from a large square, which is fast and less error-prone. Given Data / Assumptions: • Required: 199,994 * 200,006 • Recognize: 199,994 = 200,000 - 6 and 200,006 = 200,000 + 6 • Use identity: (a - b)(a + b) = a^2 - b^2 Concept / Approach: Let a = 200,000 and b = 6. Then: (200,000 - 6)(200,000 + 6) = 200,000^2 - 6^2. Compute 200,000^2 accurately and subtract 36. This avoids multiplying six-digit numbers directly. Step-by-Step Solution: 1) Rewrite the numbers around 200,000: 199,994 = 200,000 - 6 200,006 = 200,000 + 6 2) Apply difference of squares: (200,000 - 6)(200,000 + 6) = 200,000^2 - 6^2 3) Compute the squares: 200,000^2 = (2 * 10^5)^2 = 4 * 10^10 = 40,000,000,000 6^2 = 36 4) Subtract: 40,000,000,000 - 36 = 39,999,999,964 Verification / Alternative check: Estimate check: both numbers are about 200,000, so the product should be about 200,000 * 200,000 = 40,000,000,000. Because one is slightly smaller and the other slightly larger by the same amount, the product should be slightly less than 40,000,000,000 by 6^2 = 36. That matches exactly: 39,999,999,964. Why Other Options Are Wrong: • Values ending with 864 or 954 often come from subtracting 136 or 46 instead of 36, or placing digits incorrectly. • 39,999,799,964 (or similar) indicates an incorrect square of 200,000 or incorrect subtraction position. • 40,000,000,036 incorrectly adds 36 instead of subtracting it. Common Pitfalls: • Using (a - b)(a + b) = a^2 + b^2 (wrong sign). • Squaring 200,000 incorrectly (missing zeros). • Subtracting 36 from the wrong place causing digit shifts. Final Answer: 39999999964

Question 18

If cot(A − B) is expressed in terms of cot A and cot B, which of the following formulas is correct? Assume the angles are such that all required trigonometric values are defined.

Accepted Answer

Introduction / Context: This identity question tests whether you can transform angle-difference formulas correctly. Many students remember tan(A - B) but not cot(A - B). However, cot can be derived quickly using tan(A - B) and the reciprocal relationship cot θ = 1/tan θ, or by directly rewriting tan A and tan B in terms of cot A and cot B. The main difficulty is sign management: the difference identity has a plus sign in the denominator for tan(A - B), but after converting to cot form, the structure changes and the correct placement of (cotB - cotA) becomes essential. Given Data / Assumptions: • Required: cot(A - B) in terms of cot A and cot B • Use identities: tan(A - B) = (tan A - tan B) / (1 + tan A tan B) • Relationship: cot θ = 1 / tan θ • Assume all denominators are non-zero Concept / Approach: Start from tan(A - B) and then invert to get cot(A - B). To express everything in cot, replace tan A with 1/cot A and tan B with 1/cot B. After simplifying complex fractions by multiplying numerator and denominator by cot A cot B, you get a clean expression involving cotA cotB and a difference (cotB - cotA). Step-by-Step Solution: 1) Start with the standard identity: tan(A - B) = (tan A - tan B) / (1 + tan A tan B) 2) Convert to cot by taking reciprocal: cot(A - B) = (1 + tan A tan B) / (tan A - tan B) 3) Replace tan A and tan B using cot: tan A = 1/cotA, tan B = 1/cotB 4) Substitute: cot(A - B) = (1 + (1/cotA)(1/cotB)) / ((1/cotA) - (1/cotB)) 5) Simplify numerator: 1 + 1/(cotA cotB) = (cotA cotB + 1)/(cotA cotB) 6) Simplify denominator: (1/cotA) - (1/cotB) = (cotB - cotA)/(cotA cotB) 7) Divide the two fractions (the cotA cotB cancels): cot(A - B) = [(cotA cotB + 1)/(cotA cotB)] / [(cotB - cotA)/(cotA cotB)] cot(A - B) = (cotA cotB + 1)/(cotB - cotA) Verification / Alternative check: Test with A = 60°, B = 30°: cot(A - B) = cot 30° = √3. cot 60° = 1/√3 and cot 30° = √3. Compute (cotA cotB + 1)/(cotB - cotA) = ((1/√3)*√3 + 1)/(√3 - 1/√3) = (1 + 1)/( (3 - 1)/√3 ) = 2/(2/√3) = √3. It matches exactly. Why Other Options Are Wrong: • Options with (cotB + cotA) mix the sign incorrectly; subtraction is essential for A - B. • Options with (cotA cotB - 1) correspond to a different identity structure and fail the numeric test. • (cotA + cotB)/(1 - cotA cotB) resembles a tangent-style structure but is not the correct cot(A - B) form. Common Pitfalls: • Confusing tan(A - B) with tan(A + B) and carrying the wrong sign. • Forgetting that cot(A - B) is the reciprocal of tan(A - B). • Dropping parentheses and reversing (cotB - cotA) which changes the sign. Final Answer: (cotA cotB + 1) / (cotB − cotA)

Question 19

If x is defined by x = tan^2 A − sin^2 A, then simplify x to an equivalent expression in terms of tan A and sin A (or a single simpler product), assuming the expressions are defined.

Accepted Answer

Introduction / Context: This problem tests identity-based simplification with trigonometric squares. The expression tan^2 A - sin^2 A looks like a difference, but it can be factorized using the relationship between tan and sin through cos. Specifically, tan^2 A = sin^2 A / cos^2 A. Substituting that converts everything into sin and cos, after which a common factor sin^2 A can be taken out. Finally, the remaining bracket simplifies using 1 - cos^2 A = sin^2 A. The end result becomes a clean product form, which is a typical goal in simplification questions. Given Data / Assumptions: • x = tan^2 A - sin^2 A • Use identities: tan^2 A = sin^2 A / cos^2 A and 1 - cos^2 A = sin^2 A • Assume cos A ≠ 0 where needed Concept / Approach: Rewrite tan^2 A in terms of sin and cos: tan^2 A = sin^2 A / cos^2 A. Then: x = (sin^2 A / cos^2 A) - sin^2 A. Factor sin^2 A and simplify the bracket: (1/cos^2 A) - 1 = (1 - cos^2 A)/cos^2 A = sin^2 A / cos^2 A = tan^2 A. So x becomes sin^2 A * tan^2 A. Step-by-Step Solution: 1) Start: x = tan^2 A - sin^2 A 2) Replace tan^2 A with sin^2 A / cos^2 A: x = (sin^2 A / cos^2 A) - sin^2 A 3) Factor out sin^2 A: x = sin^2 A * (1/cos^2 A - 1) 4) Combine terms inside the bracket: 1/cos^2 A - 1 = (1 - cos^2 A)/cos^2 A 5) Use identity 1 - cos^2 A = sin^2 A: (1 - cos^2 A)/cos^2 A = sin^2 A / cos^2 A 6) Recognize sin^2 A / cos^2 A = tan^2 A: x = sin^2 A * tan^2 A Verification / Alternative check: Take A = 45°: tan 45° = 1 and sin 45° = √2/2, so sin^2 45° = 1/2. Original: tan^2 A - sin^2 A = 1 - 1/2 = 1/2. Product form: tan^2 A sin^2 A = 1 * 1/2 = 1/2. They match, confirming the identity simplification. Why Other Options Are Wrong: • tan A sin A is missing squares and will not match for general A. • cot and cosec forms are reciprocals and do not represent the same difference. • sin^2 A / tan^2 A equals sin^2 A * cot^2 A, which is different from tan^2 A * sin^2 A. Common Pitfalls: • Forgetting that tan^2 A involves cos^2 A in the denominator. • Replacing 1 - cos^2 A incorrectly (it equals sin^2 A, not sin A). • Cancelling terms across subtraction before factoring (not valid). Final Answer: tan^2 A sin^2 A

Question 20

If x and y are positive real numbers such that xy = 22 and x^2 + y^2 = 100, find the exact value of (x + y). Use the identity (x + y)^2 = x^2 + y^2 + 2xy and then select the correct value.

Accepted Answer

Introduction / Context: This question tests a standard algebraic identity connecting x + y, xy, and x^2 + y^2. When you are given xy and x^2 + y^2, it is usually best not to solve for x and y individually. Instead, compute (x + y)^2 directly using (x + y)^2 = x^2 + y^2 + 2xy. That gives a single number, and then you take a square root. The “positive real numbers” condition matters because (x + y)^2 determines x + y only up to a sign, and positivity removes the negative possibility. Given Data / Assumptions: • x and y are positive real numbers • xy = 22 • x^2 + y^2 = 100 • Required: x + y Concept / Approach: Use the identity: (x + y)^2 = x^2 + y^2 + 2xy. Substitute the given values to find (x + y)^2, then take the square root. Because x and y are positive, x + y must be positive, so choose the positive root. Step-by-Step Solution: 1) Write the identity: (x + y)^2 = x^2 + y^2 + 2xy 2) Substitute x^2 + y^2 = 100: (x + y)^2 = 100 + 2xy 3) Substitute xy = 22: (x + y)^2 = 100 + 2*22 = 100 + 44 4) Simplify: (x + y)^2 = 144 5) Take square root: x + y = √144 6) Since x and y are positive, x + y is positive: x + y = 12 Verification / Alternative check: If x + y = 12 and xy = 22, then x and y are roots of t^2 - 12t + 22 = 0. The discriminant is 144 - 88 = 56, which is positive, so real solutions exist. Also, with sum 12 and product 22, both roots are positive (since product is positive and sum is positive). This is consistent with the assumption that x and y are positive real numbers. Why Other Options Are Wrong: • -12 is the negative root of √144, but x + y cannot be negative if x and y are positive. • 6 is too small and would not give (x + y)^2 = 144. • 72 and 144 are not values of x + y; 144 is (x + y)^2. Common Pitfalls: • Forgetting the identity includes +2xy, not -2xy. • Computing (x + y)^2 correctly but then forgetting to take the square root. • Ignoring the positivity condition and thinking both ±12 are valid. Final Answer: 12

Simplification Questions