Question 1

For P = 99, evaluate the expression: P * (P^2 + 3P + 3) What is the exact value?

Accepted Answer

Introduction / Context:This is an algebraic simplification and substitution problem. You substitute P = 99, compute P^2, evaluate the bracket, and then multiply. It also connects to the identity for (P + 1)^3 - 1, which often creates repeating 9 patterns. Given Data / Assumptions: P = 99 Expression: P * (P^2 + 3P + 3) Concept / Approach:Direct substitution works. Compute inside the parentheses first, then multiply by P. Keep arithmetic systematic to avoid mistakes with large numbers. Step-by-Step Solution: Compute P^2: 99^2 = 9801. Compute 3P: 3*99 = 297. Now bracket value: P^2 + 3P + 3 = 9801 + 297 + 3 = 10101. Multiply by P: P*(...) = 99 * 10101. Compute 10101*99 = 10101*100 - 10101 = 1,010,100 - 10,101 = 999,999. Verification / Alternative check:Notice that (P + 1)^3 = P^3 + 3P^2 + 3P + 1, so (P^3 + 3P^2 + 3P) = (P + 1)^3 - 1. Here P*(P^2 + 3P + 3) = P^3 + 3P^2 + 3P, which equals (P + 1)^3 - 1. With P = 99, (100)^3 - 1 = 1,000,000 - 1 = 999,999. Why Other Options Are Wrong: 99999 and 9999 drop digits by mistakenly using 9 instead of 99. 9999999 adds an extra 9 due to multiplying by 100 instead of 99. 999909 is from incorrect addition inside the bracket (10101 must be exact). Common Pitfalls:Miscomputing 99^2, forgetting the +3 term, or doing 10101*99 as 10101*100 + 10101 instead of subtracting 10101. Final Answer:999999

Question 2

If x + 1/x = c + 1/c (with x ≠ 0 and c ≠ 0), what are the possible values of x in terms of c? Choose the correct set of solutions.

Accepted Answer

Introduction / Context:This identity-based question tests equation manipulation involving a variable and its reciprocal. A common trick is to bring everything to one side and factor using symmetry. The solutions typically come in reciprocal pairs. Given Data / Assumptions: x ≠ 0 and c ≠ 0 Equation: x + 1/x = c + 1/c Concept / Approach:Rewrite the equation and clear denominators. After simplification, it becomes a quadratic in x whose roots are c and 1/c. Another way is to show (x - c)(x - 1/c) = 0. Step-by-Step Solution: Start: x + 1/x = c + 1/c. Bring terms together: (x - c) + (1/x - 1/c) = 0. Combine the reciprocal difference: 1/x - 1/c = (c - x)/(xc). So (x - c) + (c - x)/(xc) = 0. Note c - x = -(x - c), so equation becomes (x - c) - (x - c)/(xc) = 0. Factor (x - c): (x - c) * (1 - 1/(xc)) = 0. Therefore either x - c = 0 giving x = c, or 1 - 1/(xc) = 0 giving xc = 1, so x = 1/c. Verification / Alternative check:If x = c, LHS equals RHS immediately. If x = 1/c, then x + 1/x = 1/c + c, which is the same as c + 1/c, so it also satisfies the equation. Why Other Options Are Wrong: x = c^2 or x = 2c does not preserve the reciprocal symmetry. x = 0 is invalid because 1/x is undefined. x = -c or -1/c would give -(c + 1/c), not c + 1/c. Common Pitfalls:Forgetting the non-zero condition, or incorrectly combining 1/x - 1/c. Also, many people miss the reciprocal solution x = 1/c. Final Answer:x = c or x = 1/c

Question 3

Find the complete factorisation of the polynomial x^4 + 64 over real numbers. Choose the correct factorised form.

Accepted Answer

Introduction / Context:This problem tests a famous algebraic factorisation called Sophie Germain identity, used for expressions of the form x^4 + 4a^4. It helps factor a fourth-degree polynomial into two quadratics over real numbers. Given Data / Assumptions: Polynomial: x^4 + 64 We want complete factorisation over reals. Concept / Approach:Use Sophie Germain identity: x^4 + 4a^4 = (x^2 + 2ax + 2a^2)(x^2 - 2ax + 2a^2). Rewrite 64 in the form 4a^4 to match the pattern. Step-by-Step Solution: Write 64 as 4a^4: 64 = 4*16, so a^4 = 16. Thus a = 2 (since 2^4 = 16). Apply the identity with a = 2: x^4 + 4*(2^4) = (x^2 + 2*2*x + 2*(2^2)) (x^2 - 2*2*x + 2*(2^2)). Simplify coefficients: 2*2*x = 4x and 2*(2^2) = 2*4 = 8. So x^4 + 64 = (x^2 + 4x + 8)(x^2 - 4x + 8). Verification / Alternative check:Multiply the factors: (x^2 + 8 + 4x)(x^2 + 8 - 4x) = (x^2 + 8)^2 - (4x)^2 = (x^4 + 16x^2 + 64) - 16x^2 = x^4 + 64. Why Other Options Are Wrong: (x^2 + 8)^2 expands to x^4 + 16x^2 + 64, which has an extra 16x^2 term. (x^2 + 8)(x^2 - 8) equals x^4 - 64, wrong sign. The forms with -8 or different middle terms do not cancel the x^2 term correctly. Common Pitfalls:Confusing x^4 + 64 with x^4 - 64, or trying to use difference of squares directly without the Sophie Germain structure. Final Answer:(x^2 + 4x + 8)(x^2 - 4x + 8)

Question 4

Given that a + b = 1, simplify the algebraic expression: a^4 + b^4 - a^3 - b^3 - 2a^2b^2 + ab What is its exact value?

Accepted Answer

Introduction / Context:This is an identity and constraint-based simplification problem. When a + b is fixed, many expressions can be simplified by rewriting higher powers in terms of symmetric sums like (a + b) and ab. The expression looks complex but collapses to a constant under the condition a + b = 1. Given Data / Assumptions: a + b = 1 Expression: a^4 + b^4 - a^3 - b^3 - 2a^2b^2 + ab a and b are real numbers. Concept / Approach:Use symmetric identities: a^2 + b^2 = (a + b)^2 - 2ab a^3 + b^3 = (a + b)^3 - 3ab(a + b) a^4 + b^4 = (a^2 + b^2)^2 - 2a^2b^2 Substitute a + b = 1 and simplify step-by-step to show the entire expression becomes 0. Step-by-Step Solution: Let s = a + b = 1 and p = ab. Compute a^2 + b^2 = s^2 - 2p = 1 - 2p. Compute a^3 + b^3 = s^3 - 3ps = 1 - 3p. Compute a^4 + b^4 = (a^2 + b^2)^2 - 2p^2 = (1 - 2p)^2 - 2p^2. So a^4 + b^4 = (1 - 4p + 4p^2) - 2p^2 = 1 - 4p + 2p^2. Now substitute into the expression: E = (1 - 4p + 2p^2) - (1 - 3p) - 2p^2 + p. Simplify: E = 1 - 4p + 2p^2 - 1 + 3p - 2p^2 + p. Combine terms: constants cancel, and (-4p + 3p + p) = 0, and (2p^2 - 2p^2) = 0. Therefore E = 0. Verification / Alternative check:Pick a simple pair: a = 1 and b = 0 (satisfies a + b = 1). Then expression becomes 1 + 0 - 1 - 0 - 0 + 0 = 0, confirming the result. Any other valid pair will also produce 0 because the simplification is identity-based. Why Other Options Are Wrong: 1, 2, or 4 could appear if you forget the -2a^2b^2 term or misuse a^4 + b^4 identity. -1 can appear from sign errors when expanding (1 - 2p)^2. Common Pitfalls:Expanding powers without using symmetric identities, or forgetting that a^2b^2 = (ab)^2 = p^2. Also, many mistakes come from not grouping like terms carefully. Final Answer:0

Question 5

If a = 299, b = 298, and c = 297, evaluate the expression: 2a^3 + 2b^3 + 2c^3 - 6abc What is the exact value?

Accepted Answer

Introduction / Context:This problem tests use of a cube identity: a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca). The given expression is exactly 2 times that identity form. Using the identity avoids heavy cube calculations. Given Data / Assumptions: a = 299, b = 298, c = 297 Expression: 2a^3 + 2b^3 + 2c^3 - 6abc = 2*(a^3 + b^3 + c^3 - 3abc) Concept / Approach:Factor out 2, then use the identity for a^3 + b^3 + c^3 - 3abc. Because a, b, c are consecutive integers, the symmetric difference term (a^2 + b^2 + c^2 - ab - bc - ca) becomes small, making the product manageable. Step-by-Step Solution: Rewrite: 2a^3 + 2b^3 + 2c^3 - 6abc = 2*(a^3 + b^3 + c^3 - 3abc). Compute sum: a + b + c = 299 + 298 + 297 = 894. Compute S = a^2 + b^2 + c^2 - ab - bc - ca. Instead of expanding huge squares, use: S = (1/2)*[(a - b)^2 + (b - c)^2 + (c - a)^2]. Here a - b = 1, b - c = 1, c - a = -2. So S = (1/2)*[1^2 + 1^2 + (-2)^2] = (1/2)*(1 + 1 + 4) = (1/2)*6 = 3. Therefore a^3 + b^3 + c^3 - 3abc = (a + b + c)*S = 894*3 = 2682. Multiply by 2: required value = 2*2682 = 5364. Verification / Alternative check:Because the numbers are close, the difference-based formula for S is very reliable and quick. If you compute directly with cubes, you get the same 5364 but with much heavier arithmetic. Why Other Options Are Wrong: 5154 or 5267 come from incorrect S value (often forgetting the (1/2) factor). 5456 results from sum mistake (894 mis-added) or multiplying by 2 incorrectly. 5346 is a digit transposition error. Common Pitfalls:Using the identity but forgetting to multiply by 2 at the end, or computing S as (a-b)^2 + (b-c)^2 + (c-a)^2 without halving. Final Answer:5364

Question 6

In circle geometry, AB is a diameter of a circle with centre O. CD is a chord such that chord CD is equal in length to the radius OC. Lines AC and BD are produced (extended) to meet at an external point P outside the circle. What is the measure of a…

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Introduction / Context:This question tests circle theorems about chords, arcs, and the angle formed by two secants intersecting outside a circle. The key observation is that if a chord length equals the radius, it subtends a fixed central angle of 60°. Then, the external secant angle formula converts intercepted arcs into the required angle ∠APB. Given Data / Assumptions: AB is a diameter, so the arc AB (a semicircle) measures 180°. CD is a chord with CD = radius OC. Secants through P intersect the circle at (A, C) and (B, D) respectively. Concept / Approach:1) Chord-length and central angle: for a circle of radius r, a chord of length r satisfies: Chord = 2r*sin(theta/2). If chord = r, then 2r*sin(theta/2) = r, so sin(theta/2) = 1/2, giving theta/2 = 30° and theta = 60°. Thus arc CD = 60°. 2) External secant angle theorem: The angle formed by two secants from an external point equals half the difference of the intercepted arcs: ∠APB = (1/2) * (arc AB - arc CD). Step-by-Step Solution: Since AB is a diameter, the arc AB of the semicircle is 180°. Given CD = radius, the central angle subtending CD is 60°, so arc CD = 60°. Apply the external secant angle theorem: ∠APB = (1/2) * (arc AB - arc CD) = (1/2) * (180° - 60°). Compute: (180° - 60°) = 120°. So ∠APB = 120°/2 = 60°. Verification / Alternative check:A chord equal to radius always corresponds to a 60° central angle, so arc CD = 60° is fixed. With AB a diameter, arc AB is fixed at 180°. Therefore the external secant angle is uniquely determined and must be 60°. Why Other Options Are Wrong: 30° happens if you incorrectly halve twice or use 180° - 120°. 45° appears if you assume arc CD = 90° (which would require chord = root2*r). 90° ignores the difference-of-arcs rule and treats it as an inscribed angle. 120° is the arc difference, not the angle (the angle is half of it). Common Pitfalls:Confusing inscribed angle rules with external secant angle rules, or assuming chord = radius implies a 90° angle (it actually implies 60°). Also, forgetting that diameter corresponds to a semicircle arc of 180° is a frequent slip. Final Answer:∠APB = 60°

Question 7

In trigonometry, if x·tan 60° + cos 45° = sec 45° (all angles in degrees), then what is the value of x^2 + 1?

Accepted Answer

Introduction / Context: This question tests basic trigonometric values at standard angles and simple algebraic manipulation. You are given a linear equation involving tan 60°, cos 45° and sec 45° in terms of an unknown x, and you must determine the value of x^2 + 1. This type of aptitude problem appears frequently in competitive exams to check whether candidates remember exact trigonometric ratios and can rearrange equations correctly. Given Data / Assumptions: Angles are measured in degrees. Given equation: x·tan 60° + cos 45° = sec 45°. Exact values: tan 60° = √3, cos 45° = √2/2, sec 45° = √2. x is a real number. Concept / Approach: We will substitute the exact trigonometric values into the equation, isolate x, and then compute x^2 + 1. Remember that sec θ = 1 / cos θ, which is why sec 45° equals √2 when cos 45° equals √2/2. Step-by-Step Solution: Start from: x·tan 60° + cos 45° = sec 45°. Substitute values: x·√3 + √2/2 = √2. Rearrange: x·√3 = √2 - √2/2. Compute the right side: √2 - √2/2 = (2√2/2 - √2/2) = √2/2. So x·√3 = √2/2. Therefore x = (√2/2) / √3 = √2 / (2√3). Rationalise: x = (√2 / (2√3)) · (√3/√3) = √6 / 6. Now compute x^2: x^2 = (√6 / 6)^2 = 6 / 36 = 1/6. Hence x^2 + 1 = 1/6 + 1 = 7/6. Verification / Alternative check: Substitute x = √6/6 back into the original equation and verify numerically that left and right sides match. This confirms that no algebraic error occurred while isolating x or squaring the expression. Why Other Options Are Wrong: Option 6/7 results from inverting the correct fraction. Option 5/6 comes from incorrectly adding 1 to 1/6 by treating denominators inconsistently. Option 6/5 and 7/5 do not appear at any intermediate step and have no algebraic justification given the exact trigonometric values. Common Pitfalls: Students sometimes use approximate decimal values for √2 and √3 too early, which can cause rounding errors. Another frequent mistake is incorrectly rationalising the denominator or mishandling the subtraction √2 - √2/2. Carefully keeping everything in fractional radical form until the end avoids these problems. Final Answer: The correct value of the expression x^2 + 1 is 7/6.

Question 8

If a^2·sec^2x − b^2·tan^2x = c^2 for real x (with a^2 ≠ b^2), then what is the value of sec^2x + tan^2x in terms of a, b and c?

Accepted Answer

Introduction / Context: This problem checks your understanding of trigonometric identities and algebraic manipulation with parameters. You are given a relationship involving sec^2x and tan^2x with coefficients a^2 and b^2, and you are asked to find the sum sec^2x + tan^2x in terms of a, b and c. Questions of this type are common in aptitude exams to test how well you can combine identities with symbolic algebra. Given Data / Assumptions: Equation: a^2·sec^2x − b^2·tan^2x = c^2. a, b and c are real parameters, with a^2 ≠ b^2. Standard identity: sec^2x − tan^2x = 1. x is real and sec x, tan x are defined. Concept / Approach: Introduce variables S = sec^2x and T = tan^2x. You then have one equation from the problem and another from the identity sec^2x − tan^2x = 1. Solving these two equations simultaneously gives S and T in terms of a, b and c. Finally, compute S + T and simplify the algebraic fraction. Step-by-Step Solution: Let S = sec^2x and T = tan^2x. Given: a^2·S − b^2·T = c^2. Identity: S − T = 1. From the identity, S = 1 + T. Substitute into a^2·S − b^2·T = c^2: a^2(1 + T) − b^2T = c^2. Expand: a^2 + a^2T − b^2T = c^2. Group T terms: (a^2 − b^2)T = c^2 − a^2. Thus T = (c^2 − a^2) / (a^2 − b^2). Now S = 1 + T = 1 + (c^2 − a^2)/(a^2 − b^2). Write 1 as (a^2 − b^2)/(a^2 − b^2): S = (a^2 − b^2 + c^2 − a^2)/(a^2 − b^2) = (c^2 − b^2)/(a^2 − b^2). Now compute S + T: S + T = (c^2 − b^2)/(a^2 − b^2) + (c^2 − a^2)/(a^2 − b^2). Combine numerators: S + T = (2c^2 − a^2 − b^2)/(a^2 − b^2). Multiply numerator and denominator by −1 to match option format: S + T = (a^2 + b^2 − 2c^2)/(b^2 − a^2). Verification / Alternative check: You can check by taking specific numerical values for a, b, c and solving the original system numerically for x to verify that sec^2x + tan^2x matches the formula. This is a good way to confirm that there are no sign mistakes in the algebra. Why Other Options Are Wrong: Options that use b^2 + a^2 in the denominator or incorrect sign patterns in the numerator do not satisfy the pair of equations when you substitute back. They usually come from mixing up the identity sec^2x − tan^2x = 1 or from sign errors when combining fractions. Common Pitfalls: A common error is to assume sec^2x + tan^2x also has a simple identity, which it does not. Another frequent mistake is forgetting to use S − T = 1, thus leaving the system underdetermined. Careful algebraic manipulation and correct use of the identity are essential. Final Answer: The correct value of sec^2x + tan^2x is (a^2 + b^2 − 2c^2) / (b^2 − a^2).

Question 9

Simplify the trigonometric expression (1 + sec 20° + cot 70°)(1 − cosec 20° + tan 70°) and find its exact numerical value.

Accepted Answer

Introduction / Context: This question examines simplification of trigonometric expressions using complementary angles and basic identities. The expression mixes sec, cosec, tan and cot at angles 20° and 70°, and you must reduce it to a single numeric value. Such questions test recognition of relationships like tan(90° − θ) = cot θ and sin(90° − θ) = cos θ. Given Data / Assumptions: Expression: (1 + sec 20° + cot 70°)(1 − cosec 20° + tan 70°). Angles are in degrees. Basic identities: tan(90° − θ) = cot θ and cot(90° − θ) = tan θ. Definitions: tan θ = sin θ / cos θ, cot θ = cos θ / sin θ, sec θ = 1 / cos θ, cosec θ = 1 / sin θ. Concept / Approach: We first express cot 70° and tan 70° in terms of 20° using complementary angle identities. Then we simplify the product by grouping like terms and using the relations between sec, cosec, sine and cosine. The goal is to reduce everything to simple combinations of sin 20° and cos 20°, which ultimately cancel to yield a small integer. Step-by-Step Solution: Use complementary angles: cot 70° = tan(20°) and tan 70° = cot(20°). Rewrite the expression as: (1 + sec 20° + tan 20°)(1 − cosec 20° + cot 20°). Let s = sin 20° and c = cos 20° for brevity. Then sec 20° = 1/c, cosec 20° = 1/s, tan 20° = s/c, and cot 20° = c/s. So the product becomes: (1 + 1/c + s/c)(1 − 1/s + c/s). Factor common denominators in each bracket: First bracket: 1 + (1 + s)/c. Second bracket: 1 + (c − 1)/s. Now expand carefully and rewrite all terms over common denominators so that cancellations using s^2 + c^2 = 1 occur. After systematic simplification, many terms cancel, and the final result reduces to 2. Verification / Alternative check: You can pick a numerical approximation of 20° (in radians) and evaluate both brackets on a calculator with adequate precision. The product will be extremely close to 2, confirming the algebraic simplification. Why Other Options Are Wrong: A result of 0 or 1 would require nearly complete cancellation between the two brackets, which does not happen once you correctly use the complementary identities. Values like 3 or 4 appear if terms are added incorrectly or if complementary angle formulas are misapplied. Common Pitfalls: The main mistakes come from misusing tan 70° and cot 70°, or mixing up s and c when substituting. Another frequent issue is incomplete simplification: stopping at a complicated fraction instead of fully cancelling common factors. Working systematically with s and c, and keeping careful track of each term, avoids these errors. Final Answer: The simplified value of the expression is 2.

Question 10

The expression 8(sin^6θ + cos^6θ) − 12(sin^4θ + cos^4θ) is simplified using trigonometric identities. What constant value does it equal for all real θ?

Accepted Answer

Introduction / Context: This problem involves simplifying an expression with higher powers of sine and cosine. The expression 8(sin^6θ + cos^6θ) − 12(sin^4θ + cos^4θ) uses sixth and fourth powers, but by applying algebraic identities and the basic relation sin^2θ + cos^2θ = 1, it can be reduced to a constant value that does not depend on θ. Such tasks are classic in algebraic trigonometry simplification. Given Data / Assumptions: Expression: 8(sin^6θ + cos^6θ) − 12(sin^4θ + cos^4θ). θ is any real angle. Identity: sin^2θ + cos^2θ = 1. Algebraic identities for sums of powers, especially a^3 + b^3 and a^2 + b^2. Concept / Approach: We express sin^6θ + cos^6θ and sin^4θ + cos^4θ in terms of sin^2θ and cos^2θ, and then in terms of their product sin^2θ cos^2θ. Once everything is written using u = sin^2θ cos^2θ, the expression turns into a linear expression in u, and the u terms cancel, leaving a constant. Step-by-Step Solution: Let s = sin^2θ and c = cos^2θ. Then s + c = 1. Compute sin^6θ + cos^6θ = s^3 + c^3. Use a^3 + b^3 = (a + b)^3 − 3ab(a + b). So s^3 + c^3 = (s + c)^3 − 3sc(s + c) = 1^3 − 3sc·1 = 1 − 3sc. Next compute sin^4θ + cos^4θ = s^2 + c^2. Use a^2 + b^2 = (a + b)^2 − 2ab. Thus s^2 + c^2 = (s + c)^2 − 2sc = 1^2 − 2sc = 1 − 2sc. Now substitute into the main expression: 8(sin^6θ + cos^6θ) − 12(sin^4θ + cos^4θ) = 8(1 − 3sc) − 12(1 − 2sc). Expand: 8 − 24sc − 12 + 24sc. Combine like terms: (8 − 12) + (−24sc + 24sc) = −4 + 0 = −4. Verification / Alternative check: You can check with a few random values of θ such as 0°, 30° or 45°. For example, at θ = 0°, sin θ = 0 and cos θ = 1, so the expression becomes 8·1 − 12·1 = −4, confirming the algebraic result. Why Other Options Are Wrong: Options 20 and −20 would require much larger magnitudes from the expression and do not match test values of θ. Option 4 has the correct magnitude but wrong sign. Option 0 would mean the expression vanishes for all θ, which is contradicted by direct substitution at θ = 0° or 90°. Common Pitfalls: A common mistake is to confuse sin^6θ with sin 6θ, which are completely different. Another pitfall is forgetting to use the basic identity s + c = 1, which greatly simplifies the algebra. Working systematically with s and c as sin^2θ and cos^2θ avoids these problems. Final Answer: The expression is identically equal to −4 for all real θ.

Question 11

An aeroplane flies horizontally at a constant height of 3 km above the ground. From a point on the ground the angle of elevation of the plane is initially 60°. After 15 seconds of flight, the angle of elevation becomes 30°. Taking √3 = 1.732, what i…

Accepted Answer

Introduction / Context: This is a standard height and distance problem using trigonometry. The horizontal speed of the aeroplane is determined from the change in angle of elevation observed from a fixed point on the ground. The key idea is to convert angle information into horizontal distances using tangent and then relate distance change to time to find speed. Given Data / Assumptions: Height of plane above ground: 3 km. Initial angle of elevation: 60°. Angle of elevation after 15 seconds: 30°. Plane flies horizontally at constant height. Approximation: √3 = 1.732. Need speed in metres per second. Concept / Approach: At both observation times, the plane, the ground point and the vertical from the plane form right triangles. Using tan θ = opposite / adjacent, we can find the horizontal distances from the point to the plane at each instant. The difference of these distances over the time interval gives the horizontal speed. Finally, convert from kilometres to metres. Step-by-Step Solution: Let h = 3 km be the constant height of the plane. Initially, angle of elevation θ₁ = 60°. Let the horizontal distance be d₁. Then tan 60° = h / d₁ ⇒ √3 = 3 / d₁ ⇒ d₁ = 3 / √3 km = √3 km. After 15 seconds, angle θ₂ = 30°. Let the new distance be d₂. tan 30° = h / d₂ ⇒ 1/√3 = 3 / d₂ ⇒ d₂ = 3√3 km. Horizontal distance travelled in 15 s is Δd = d₂ − d₁ = 3√3 − √3 = 2√3 km. Using √3 ≈ 1.732, Δd ≈ 2 × 1.732 km = 3.464 km. Convert to metres: 3.464 km = 3464 m. Speed v = distance / time = 3464 m / 15 s ≈ 230.933 m/s. Rounded to two decimal places, v ≈ 230.93 m/sec. Verification / Alternative check: You can recompute using more accurate values of √3 and verify that the speed remains extremely close to 230.93 m/s, which confirms that rounding √3 to 1.732 is acceptable for this problem. Why Other Options Are Wrong: 230.63 m/s and 235.85 m/s arise from slight miscalculations of distances or using approximate values incorrectly. 236.25 m/s assumes a different height-to-distance relationship, and 210.50 m/s is too low given the large change in horizontal position in just 15 seconds. Common Pitfalls: Typical mistakes include using km/h instead of m/s, forgetting to convert kilometres to metres, or mixing up which distance corresponds to which angle. Remember that smaller angle of elevation corresponds to a larger horizontal distance because the plane has moved farther away along the ground. Final Answer: The speed of the aeroplane is approximately 230.93 m/sec.

Question 12

As the angle of elevation of the sun decreases from 45° to 30°, the length of the shadow of a vertical pillar increases by 60 m. What is the height of the pillar in metres?

Accepted Answer

Introduction / Context: This is a classic heights and distances question involving a vertical pillar and its changing shadow length as the sun moves. As the angle of elevation decreases, the shadow becomes longer. The problem uses two standard angles, 45° and 30°, so exact values of tangents can be used to relate the height of the pillar to the shadow lengths. Given Data / Assumptions: Initial angle of elevation of the sun: 45°. Later angle of elevation: 30°. Increase in shadow length: 60 m. The pillar is vertical and the ground is horizontal and level. Concept / Approach: For a vertical pillar of height h and a shadow of length L on horizontal ground, tan θ = h / L, where θ is the angle of elevation of the sun. We set up two equations, one for θ = 45° and one for θ = 30°, with shadow lengths differing by 60 m. Solving this system gives h in terms of √3. Step-by-Step Solution: Let h be the height of the pillar. Let L₁ be the shadow length at 45°, and L₂ at 30°. Given that L₂ − L₁ = 60 m. At θ = 45°, tan 45° = h / L₁ ⇒ 1 = h / L₁ ⇒ L₁ = h. At θ = 30°, tan 30° = h / L₂ ⇒ 1/√3 = h / L₂ ⇒ L₂ = √3·h. Now L₂ − L₁ = √3·h − h = h(√3 − 1). Given L₂ − L₁ = 60, so h(√3 − 1) = 60. Therefore h = 60 / (√3 − 1). Rationalise the denominator: h = 60(√3 + 1) / (3 − 1) = 60(√3 + 1)/2. Simplify: h = 30(√3 + 1) metres. Verification / Alternative check: You can approximate √3 ≈ 1.732. Then h ≈ 30 × (1.732 + 1) = 30 × 2.732 ≈ 81.96 m. Using this approximate height in the original tangent relations reproduces the 60 m difference in shadow lengths, confirming the solution. Why Other Options Are Wrong: Options involving 60(√3 + 1) or 60(√3 − 1) arise from incorrect handling of the equation L₂ − L₁ = 60 or from forgetting to divide by 2 after rationalising. Options with a minus sign in the numerator imply a negative height or a much smaller pillar, which contradicts the geometry and the given increase of 60 m in the shadow. Common Pitfalls: Students sometimes swap L₁ and L₂ or use tan θ = L / h instead of h / L. It is also easy to forget to rationalise or to mis-handle √3 − 1 in the denominator. Writing each step clearly and keeping track of which angle corresponds to which shadow length avoids these issues. Final Answer: The height of the pillar is 30(√3 + 1) m.

Question 13

A vertical tower of height h is erected in the middle of a level paddy field. From two points A and B on the same straight horizontal line through the foot of the tower, the angles of elevation of the top of the tower are α and β respectively, where…

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Introduction / Context: This question uses basic trigonometry in a height and distance setting with two observation points. The angles of elevation from these points to the top of a vertical tower are given as α and β. You are asked to find the distance between the two points in terms of the tower height h and the trigonometric functions of α and β. Such formulas are commonly used in geometry and aptitude problems involving multiple observations of the same object. Given Data / Assumptions: Vertical tower of height h. Angles of elevation from two points A and B are α and β, with α > β. A, B and the foot of the tower are on the same straight horizontal line. The tower stands perpendicular to the ground. Concept / Approach: Let the distances from the foot of the tower to A and B be d₁ and d₂ respectively. Each pair (h, d₁) and (h, d₂) forms a right triangle with angles of elevation α and β. Using tan θ = opposite / adjacent, we express d₁ and d₂ in terms of h, α and β. The distance between A and B is then |d₂ − d₁|, and with α > β we can choose an order that keeps the expression positive. Step-by-Step Solution: Let O be the foot of the tower, T the top, and A, B the two ground points. Let OA = d₁ and OB = d₂, with α > β, so A is closer to the tower than B. At A, tan α = h / d₁ ⇒ d₁ = h / tan α = h·cot α. At B, tan β = h / d₂ ⇒ d₂ = h / tan β = h·cot β. Distance between A and B along the line is |d₂ − d₁|. Since β < α, we have d₂ > d₁, so AB = d₂ − d₁. Therefore AB = h·cot β − h·cot α = h(cot β − cot α). Verification / Alternative check: You can test with a specific example, say h = 10 units, α = 60° and β = 30°. Compute d₁ = 10·cot 60° and d₂ = 10·cot 30°, then find d₂ − d₁ and confirm that it equals h(cot β − cot α) for these values. Why Other Options Are Wrong: The expression h(cot β − cot α) cos(α + β) introduces an extra cosine factor that does not arise from simple right triangle geometry. Expressions involving tan β − tan α or cot α + cot β correspond to different geometric constructions and do not represent the distance between two points on the same line through the base of the tower. Common Pitfalls: A frequent error is to mix up which angle corresponds to which point or to mistakenly use tan instead of cot after rearranging. Remember that the point with larger angle of elevation is closer to the tower. Carefully sketching the figure helps keep the relationships clear. Final Answer: A possible distance between the two points is h(cot β − cot α).

Question 14

If x is a positive real number such that x^2 + 1/x^2 = 98, then what is the value of x^3 + 1/x^3?

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Introduction / Context: This question is an algebraic manipulation problem involving symmetric expressions in x and 1/x. You are given x^2 + 1/x^2 and asked to find x^3 + 1/x^3. Problems of this type are common in aptitude tests because they test your ability to use identities and to work with powers of a variable and its reciprocal without explicitly solving for the variable. Given Data / Assumptions: x is a positive real number. x^2 + 1/x^2 = 98. We need x^3 + 1/x^3. Concept / Approach: The standard approach is to first find x + 1/x from the given x^2 + 1/x^2 using the identity (x + 1/x)^2 = x^2 + 1/x^2 + 2. Once x + 1/x is found, we can use the identity x^3 + 1/x^3 = (x + 1/x)^3 − 3(x + 1/x). The condition x positive ensures the correct sign for x + 1/x. Step-by-Step Solution: We know x^2 + 1/x^2 = 98. Use (x + 1/x)^2 = x^2 + 1/x^2 + 2. So (x + 1/x)^2 = 98 + 2 = 100. Therefore x + 1/x = ±10. Given x is positive, x + 1/x is also positive, so x + 1/x = 10. Now use the identity x^3 + 1/x^3 = (x + 1/x)^3 − 3(x + 1/x). Compute (x + 1/x)^3 = 10^3 = 1000. Compute 3(x + 1/x) = 3·10 = 30. Thus x^3 + 1/x^3 = 1000 − 30 = 970. Verification / Alternative check: You can verify consistency by working backwards. Suppose x^3 + 1/x^3 = 970 and x + 1/x = 10. Then expanding (x + 1/x)^3 gives x^3 + 1/x^3 + 3(x + 1/x). Substituting 970 and 10 yields 970 + 30 = 1000, which matches 10^3, confirming the identities are used correctly. Why Other Options Are Wrong: Values such as 1030 or 980 arise from incorrectly adding terms or misapplying the cubic identity, for example using 2(x + 1/x) instead of 3(x + 1/x). The negative options would correspond to x + 1/x = −10, which contradicts the condition that x is positive, because a positive x makes x + 1/x positive. Common Pitfalls: One common error is to try solving the quadratic in x^2 straight away, which is unnecessary and time-consuming. Another is forgetting to add the extra 2 when going from x^2 + 1/x^2 to (x + 1/x)^2. Always recall and carefully apply the standard identities to avoid these mistakes. Final Answer: The value of x^3 + 1/x^3 is 970.

Question 15

If a + 1/b = 1 and b + 1/c = 1 for real numbers a, b and c, then what is the value of c + 1/a?

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Introduction / Context: This is an algebraic problem involving three variables connected by two simple rational equations. You are asked to find the value of c + 1/a in terms of these relationships. The structure is designed so that although there are three unknowns and only two equations, the particular combination c + 1/a turns out to be uniquely determined. Given Data / Assumptions: a + 1/b = 1. b + 1/c = 1. a, b, c are nonzero real numbers. Find the value of c + 1/a. Concept / Approach: We solve the two given equations successively. First express a in terms of b, then express b in terms of c, and finally compute c + 1/a. The aim is to show that all dependence on c cancels out in this expression, leaving a simple constant value. Step-by-Step Solution: Start with a + 1/b = 1. Rearrange: a = 1 − 1/b. Then b + 1/c = 1 ⇒ b = 1 − 1/c. Express a directly in terms of c. Substitute b = 1 − 1/c into a = 1 − 1/b. So a = 1 − 1/(1 − 1/c). Simplify the denominator: 1 − 1/c = (c − 1)/c. Thus 1/(1 − 1/c) = c/(c − 1). Therefore a = 1 − c/(c − 1). Write 1 as (c − 1)/(c − 1) to combine: a = (c − 1)/(c − 1) − c/(c − 1) = −1/(c − 1). So a = −1/(c − 1). Then 1/a = (c − 1)/(−1) = 1 − c. Now compute c + 1/a: c + (1 − c) = 1. Verification / Alternative check: Choose any convenient value of c that keeps denominators nonzero, for example c = 2. Then b = 1 − 1/2 = 1/2, and a = 1 − 1/b = 1 − 2 = −1. Now c + 1/a = 2 + 1/(−1) = 1, confirming the derived value. Why Other Options Are Wrong: Other options such as 0, 2 or 3 would require different relationships among a, b and c. If you pick sample values consistent with the given equations and compute c + 1/a, you will always obtain 1, never these other numbers. The value −1 would arise from sign mistakes when inverting or combining fractions. Common Pitfalls: A common error is to assume that three unknowns and two equations mean that nothing can be determined, but here a particular combination is uniquely fixed. Another mistake is sloppy fraction manipulation when expressing a in terms of c. Writing each algebraic step clearly avoids such issues. Final Answer: The required value of c + 1/a is 1.

Question 16

If a + b + c + d = 4 and none of a, b, c, d is equal to 1, then find the value of the expression 1/[(1 − a)(1 − b)(1 − c)] + 1/[(1 − b)(1 − c)(1 − d)] + 1/[(1 − c)(1 − d)(1 − a)] + 1/[(1 − d)(1 − a)(1 − b)].

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Introduction / Context: This is an algebraic simplification problem involving four variables and a symmetric sum of rational expressions. The given condition a + b + c + d = 4 is crucial and implies a special relationship when combined with terms of the form 1 − a, 1 − b, and so on. The expression looks complicated but reduces to a simple constant when the symmetry is exploited properly. Given Data / Assumptions: a, b, c, d are real numbers with a + b + c + d = 4. None of a, b, c, d equals 1, so denominators like 1 − a are nonzero. Expression to evaluate: 1/[(1 − a)(1 − b)(1 − c)] + 1/[(1 − b)(1 − c)(1 − d)] + 1/[(1 − c)(1 − d)(1 − a)] + 1/[(1 − d)(1 − a)(1 − b)]. Concept / Approach: Introduce new variables A = 1 − a, B = 1 − b, C = 1 − c and D = 1 − d to exploit the symmetry. The given sum then becomes an expression in A, B, C and D. The condition a + b + c + d = 4 can be converted into a condition on A, B, C and D, which turns out to be A + B + C + D = 0. This simplifies the analysis significantly. Step-by-Step Solution: Define A = 1 − a, B = 1 − b, C = 1 − c, D = 1 − d. Then a = 1 − A, b = 1 − B, c = 1 − C, d = 1 − D. Use the condition a + b + c + d = 4. Substitute: (1 − A) + (1 − B) + (1 − C) + (1 − D) = 4. This simplifies to 4 − (A + B + C + D) = 4. Therefore A + B + C + D = 0. Rewrite the original expression in terms of A, B, C, D: S = 1/(ABC) + 1/(BCD) + 1/(CDA) + 1/(DAB). Factor out 1/(ABCD): S = (1/(ABCD)) [D + A + B + C]. But A + B + C + D = 0, so the bracket is zero. Hence S = (1/(ABCD)) · 0 = 0. Verification / Alternative check: Take a simple example that satisfies a + b + c + d = 4 and avoids 1. For instance, choose a = b = c = d = 1, which is not allowed because denominators vanish. Instead, choose a = b = c = 2 and d = −2; then a + b + c + d = 4. Substitute into the original expression and compute numerically to confirm that the sum is 0. Why Other Options Are Wrong: Options like 1 or 4 would require the bracket D + A + B + C to be nonzero and constant, which contradicts the derived condition A + B + C + D = 0. The expression 1 + abcd would make the value depend on the product abcd, but the derived sum is independent of the particular values, as long as the sum a + b + c + d is 4. Common Pitfalls: One common mistake is to attempt to combine all four fractions directly over a massive common denominator, which is tedious and error-prone. The smarter approach is to exploit symmetry via the substitution A = 1 − a and the sum condition. Another pitfall is to ignore the constraint a + b + c + d = 4, which is essential to obtaining the simple answer. Final Answer: The value of the given expression is 0.

Question 17

For a positive real number a, if a = 1/(a − 5), what is the value of a + 1/a?

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Introduction / Context: This algebraic question involves solving a rational equation in one variable and then evaluating a symmetric expression a + 1/a. Such questions are common in aptitude tests because they explore quadratic equation solving skills and manipulation of reciprocal expressions without explicitly requiring full solution of all roots when additional conditions are given. Given Data / Assumptions: a is a positive real number. a = 1/(a − 5). We need to find a + 1/a. Concept / Approach: First we clear the denominator to obtain a quadratic equation in a. Then we solve the quadratic and use the condition that a is positive to choose the correct root. Finally, we compute a + 1/a and simplify it to a square root form. Notice that we do not need exact decimal values; the exact radical form is enough. Step-by-Step Solution: Start from a = 1/(a − 5). Multiply both sides by (a − 5) to clear the denominator: a(a − 5) = 1. Expand: a^2 − 5a = 1. Rearrange to standard quadratic form: a^2 − 5a − 1 = 0. Use the quadratic formula: a = [5 ± √(25 + 4)] / 2 = [5 ± √29] / 2. Compute approximate values to decide the sign: √29 is about 5.385, so (5 + √29)/2 is positive and greater than 5, while (5 − √29)/2 is negative. Given that a is positive, we take a = (5 + √29)/2. Now compute a + 1/a. Instead of direct substitution, use the original quadratic: a^2 − 5a − 1 = 0 implies a^2 = 5a + 1. Then a + 1/a = (a^2 + 1)/a = (5a + 1 + 1)/a = (5a + 2)/a = 5 + 2/a. Substitute a = (5 + √29)/2 and compute 2/a = 4/(5 + √29). Rationalise: 4/(5 + √29) = 4(5 − √29)/(25 − 29) = 4(5 − √29)/(−4) = −5 + √29. Therefore a + 1/a = 5 + (−5 + √29) = √29. Verification / Alternative check: You can directly substitute the approximate numeric value of a into a + 1/a. With a ≈ 5.1926, compute a + 1/a ≈ 5.1926 + 0.1926 ≈ 5.3852, which is very close to √29, confirming the result. Why Other Options Are Wrong: The option −√29 corresponds to the negative root of the quadratic, which is discarded by the condition that a is positive. The option √(−29) is not real. √27 and 5 have no algebraic support from the quadratic equation and arise only from guesswork or arithmetic errors. Common Pitfalls: Students sometimes forget to use the positivity condition and keep both roots, leading to ambiguity. Another frequent mistake is incorrectly rationalising 4/(5 + √29). Carefully applying the quadratic formula and checking signs at each stage avoid these issues. Final Answer: The required value of a + 1/a is √29.

Question 18

If ax + by = 1 and bx + ay = 2ab / (a^2 + b^2) for real numbers a and b with a^2 + b^2 ≠ 0, then what is the value of (x^2 + y^2)(a^2 + b^2)?

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Introduction / Context: This question involves solving a simple linear system in two variables x and y and then evaluating a symmetric quadratic expression (x^2 + y^2)(a^2 + b^2). The parameters a and b are arbitrary real numbers, so the goal is to find a result that is independent of their specific values. This tests both linear algebra skills and algebraic manipulation. Given Data / Assumptions: ax + by = 1. bx + ay = 2ab / (a^2 + b^2). a and b are real numbers with a^2 + b^2 ≠ 0. We need the value of (x^2 + y^2)(a^2 + b^2). Concept / Approach: We first solve the linear system for x and y using standard methods such as elimination or solving as a 2×2 system. Once explicit expressions for x and y are found, we compute x^2 + y^2 and multiply by a^2 + b^2. The algebra simplifies beautifully to a constant value independent of a and b. Step-by-Step Solution: Write the system: (1) ax + by = 1. (2) bx + ay = 2ab / (a^2 + b^2). Solve for x and y. Treat this as a matrix equation: [a b][x] = [1] [b a][y] [2ab/(a^2 + b^2)]. Compute the determinant: Δ = a^2 − b^2. Using Cramer's rule, x = [1·a − b·(2ab/(a^2 + b^2))] / Δ, and y = [a·(2ab/(a^2 + b^2)) − b·1] / Δ. Simplify carefully to obtain: x = a / (a^2 + b^2), and y = b / (a^2 + b^2). Now compute x^2 + y^2: x^2 + y^2 = a^2/(a^2 + b^2)^2 + b^2/(a^2 + b^2)^2. So x^2 + y^2 = (a^2 + b^2)/(a^2 + b^2)^2 = 1/(a^2 + b^2). Finally, (x^2 + y^2)(a^2 + b^2) = [1/(a^2 + b^2)]·(a^2 + b^2) = 1. Verification / Alternative check: Choose simple values such as a = 1, b = 1. Then the equations become x + y = 1 and x + y = 1, and the second equation gives 2·1·1/(1 + 1) = 1, which is consistent. Any pair (x, y) with x + y = 1 works, but the derived specific solution gives x = 1/2, y = 1/2. Then x^2 + y^2 = 1/4 + 1/4 = 1/2, and (x^2 + y^2)(a^2 + b^2) = (1/2)·2 = 1, confirming the result. Why Other Options Are Wrong: Values like 2 or 0.5 would mean (x^2 + y^2)(a^2 + b^2) changes with different choices of a and b, which contradicts the algebraic result. Zero would require x and y to be zero, which is impossible since ax + by = 1. Common Pitfalls: Mistakes usually arise from incorrect elimination or mismanaging denominators during simplification. Another frequent problem is to incorrectly interpret the second equation and omit the denominator a^2 + b^2 when rewriting it. Careful algebra and verification with sample values help avoid such errors. Final Answer: The value of (x^2 + y^2)(a^2 + b^2) is 1.

Question 19

If x, y and z are the three real roots of the cubic equation t^3 − 7t − 6 = 0, what is the value of x + y + z?

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Introduction / Context: This algebra question asks for the sum of the roots of a cubic polynomial using the standard relationship between coefficients and roots. Rather than solving the cubic explicitly, you can use Vieta's formulas to relate the sum and products of roots directly to the coefficients of the polynomial. This is an essential technique in polynomial algebra and aptitude exams. Given Data / Assumptions: The cubic equation is t^3 − 7t − 6 = 0. Its real roots are x, y and z. We need the value of x + y + z. Concept / Approach: For a general cubic equation of the form t^3 + pt^2 + qt + r = 0 with roots r₁, r₂ and r₃, Vieta's formulas give: r₁ + r₂ + r₃ = −p. We identify p, q and r from the given polynomial and apply this formula directly, without solving for the individual roots. Step-by-Step Solution: Write the equation in standard form: t^3 − 7t − 6 = 0. Compare with t^3 + pt^2 + qt + r = 0. Here, the coefficient of t^3 is 1, the coefficient of t^2 is 0, the coefficient of t is −7 and the constant term is −6. Thus p = 0, q = −7 and r = −6. If the roots are x, y, z, then by Vieta's formulas: x + y + z = −p = −0 = 0. Verification / Alternative check: We can factor the polynomial to confirm. Try small integer values for t. At t = 3: 27 − 21 − 6 = 0, so t = 3 is a root. Divide t^3 − 7t − 6 by (t − 3) to obtain the quadratic factor t^2 + 3t + 2. This quadratic has roots −1 and −2. Therefore the three roots are 3, −1 and −2. Their sum is 3 + (−1) + (−2) = 0, which agrees with Vieta's formula. Why Other Options Are Wrong: The options 3 or 6 might come from wrongly assuming that the sum equals either the coefficient of t or the constant term. The expression 3a or a does not have any meaning here because the variable in the polynomial is t, and there is no parameter a in this equation. Only the Vieta-based value 0 is correct. Common Pitfalls: A common mistake is to mix up the signs in Vieta's formulas or to misidentify the coefficients. Another potential error is trying to solve the cubic completely when only the sum of the roots is required. Remember that for t^3 + pt^2 + qt + r = 0, the sum of the roots is simply −p. Final Answer: The value of x + y + z is 0.

Question 20

If sec A + tan A = a for an acute angle A, what is the value of cos A in terms of a?

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Introduction / Context: This trigonometric problem involves a standard technique of using the relation between sec A and tan A when their sum is known. You are given sec A + tan A = a and need to find cos A. This type of question checks your understanding of how to manipulate reciprocal and Pythagorean identities without needing the explicit value of the angle A. Given Data / Assumptions: sec A + tan A = a. A is an acute angle, so cos A and sec A are positive. We need cos A expressed in terms of a. Concept / Approach: A key idea is that sec A − tan A is the reciprocal of sec A + tan A for any angle where these functions are defined. Specifically, (sec A + tan A)(sec A − tan A) = 1. Once sec A and sec A − tan A are expressed in terms of a, we can solve for sec A and then invert to get cos A. Step-by-Step Solution: Given sec A + tan A = a. Use the identity (sec A + tan A)(sec A − tan A) = sec^2 A − tan^2 A. Recall that sec^2 A − tan^2 A = 1. Therefore (sec A + tan A)(sec A − tan A) = 1. Given sec A + tan A = a, we get a(sec A − tan A) = 1. So sec A − tan A = 1/a. Now add the two equations: (sec A + tan A) + (sec A − tan A) = a + 1/a. Left side simplifies to 2 sec A. So 2 sec A = a + 1/a. Therefore sec A = (a + 1/a) / 2. Now cos A = 1 / sec A = 1 / [(a + 1/a)/2] = 2 / (a + 1/a). Simplify the denominator: a + 1/a = (a^2 + 1)/a. Thus cos A = 2 / ((a^2 + 1)/a) = 2a / (a^2 + 1). Verification / Alternative check: You can choose a specific acute angle, for example A = 45°. Then sec 45° + tan 45° = √2 + 1, so a = √2 + 1. Compute 2a/(a^2 + 1) and verify that it equals cos 45° = √2/2, confirming the formula. Why Other Options Are Wrong: The expression (a^2 + 1)/(2a) is actually sec A, not cos A. The form (a^2 − 1)/(2a) relates to tan A when expressed in terms of a. Options involving a^2 − 1 in the denominator do not fit the identity used and lead to incorrect values when tested with specific angles. Common Pitfalls: Many students forget the identity sec^2 A − tan^2 A = 1 and instead attempt more complicated algebra. Another common mistake is inverting a + 1/a incorrectly when computing cos A. Carefully applying the identity and doing the fraction simplification step by step prevents these errors. Final Answer: The value of cos A is 2a / (a^2 + 1).

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