Question 1

A straight line can be written using slope-intercept form y = mx + c. What is the equation of the line with slope 1/3 and y-intercept 5?

Accepted Answer

Introduction / Context: This question tests converting between different forms of a line equation. Given a slope and y-intercept, the simplest form is slope-intercept form: y = mx + c. Many multiple-choice options are presented in standard form (Ax + By = C), so the key is to rewrite correctly without sign mistakes. Given Data / Assumptions: Slope m = 1/3. y-intercept c = 5 (meaning the line crosses the y-axis at y = 5). We must identify the correct equation among the options. Concept / Approach: Use y = mx + c. Substitute m = 1/3 and c = 5. Then clear the fraction by multiplying both sides by 3 to convert to integer coefficients. Finally compare with the options. Step-by-Step Solution: Start with slope-intercept form: y = mx + c Substitute m = 1/3 and c = 5: y = (1/3)x + 5 Multiply both sides by 3: 3y = x + 15 Rearrange to standard form: x - 3y = -15 Verification / Alternative check: Check intercept: set x = 0 in x - 3y = -15 => -3y = -15 => y = 5, correct. Check slope quickly: rewrite x - 3y = -15 as 3y = x + 15 => y = (1/3)x + 5, slope is 1/3, correct. Why Other Options Are Wrong: x - 3y = 15 gives y-intercept -5, not 5. x + 3y = 15 or x + 3y = -15 produces a negative slope (-1/3), not 1/3. 3x - y = 15 corresponds to y = 3x - 15, which has slope 3 and wrong intercept. Common Pitfalls: Mixing up the sign when moving terms, or incorrectly multiplying by 3 (for example making 3y = x + 5 instead of x + 15). Another mistake is confusing y-intercept with x-intercept. Final Answer: x - 3y = -15

Question 2

Use standard trigonometric values at special angles (30 degrees and 60 degrees) to evaluate exactly. Find the exact value of: cosec^2 60 degrees + sec^2 60 degrees + tan^2 30 degrees.

Accepted Answer

Introduction / Context: This question tests exact evaluation of trigonometric functions at special angles. The key is to use known values for sin 60 degrees, cos 60 degrees, and tan 30 degrees, then square them as required and add carefully using fractions. These are common simplification steps in trigonometry and aptitude problems. Given Data / Assumptions: Expression: cosec^2 60 + sec^2 60 + tan^2 30 Use exact special-angle values (no rounding). cosec theta = 1/sin theta, sec theta = 1/cos theta. Concept / Approach: Compute each term separately using standard exact values:sin 60 = √3/2, cos 60 = 1/2, tan 30 = 1/√3.Then square cosec, sec, and tan as asked and add using common denominators. Step-by-Step Solution: sin 60 = √3/2 => cosec 60 = 1/(√3/2) = 2/√3 So cosec^2 60 = (2/√3)^2 = 4/3 cos 60 = 1/2 => sec 60 = 1/(1/2) = 2 So sec^2 60 = 2^2 = 4 tan 30 = 1/√3 So tan^2 30 = (1/√3)^2 = 1/3 Add: 4/3 + 4 + 1/3 = (5/3) + 4 Convert 4 to thirds: 4 = 12/3 Total = (5/3 + 12/3) = 17/3 = 5 2/3 Verification / Alternative check: Combine the fraction terms first: 4/3 + 1/3 = 5/3. Then adding 4 gives 5/3 + 12/3 = 17/3. This confirms the arithmetic cleanly. Why Other Options Are Wrong: 5 ignores part of the fractional contribution (5/3). 6 is too large and typically comes from treating cosec^2 60 as 4 instead of 4/3. 5 1/2 and 5 1/3 result from incorrect fraction addition or rounding √ values instead of using exact squares. Common Pitfalls: Forgetting to square after computing cosec/sec/tan, confusing tan 30 with tan 60, and converting between mixed numbers and improper fractions incorrectly. Final Answer: cosec^2 60 + sec^2 60 + tan^2 30 = 5 2/3

Question 3

For an acute angle, cosec theta gives the ratio hypotenuse/opposite, which can be used to infer the Pythagorean triple. If cosec θ = 25/7 for an acute angle θ, what is the value of cot θ?

Accepted Answer

Introduction / Context: This question tests how to convert one trigonometric ratio into another for an acute angle. Given cosec θ, we can find sin θ. Then, because the angle is acute, all ratios are positive and we can use a right-triangle interpretation to compute cot θ = cos θ / sin θ. Given Data / Assumptions: cosec θ = 25/7. θ is acute (0 < θ < 90 degrees), so sin θ and cos θ are positive. We need cot θ. Concept / Approach: Use cosec θ = 1/sin θ, so sin θ = 7/25. Interpret sin θ = opposite/hypotenuse. That gives opposite = 7 and hypotenuse = 25 (scaled). Then use Pythagoras to find adjacent = sqrt(25^2 - 7^2) = 24. Finally, cot θ = adjacent/opposite = 24/7. Step-by-Step Solution: cosec θ = 25/7 => sin θ = 1/(25/7) = 7/25 Let opposite = 7 and hypotenuse = 25 for angle θ Find adjacent using Pythagoras: adjacent^2 = 25^2 - 7^2 = 625 - 49 = 576 So adjacent = sqrt(576) = 24 cot θ = adjacent/opposite = 24/7 Verification / Alternative check: cos θ would be adjacent/hypotenuse = 24/25. Then cot θ = cos θ / sin θ = (24/25)/(7/25) = 24/7, matching the result. Why Other Options Are Wrong: 7/24 is tan θ, not cot θ (cot is the reciprocal). 24/25 is cos θ, not cot θ. 7/25 is sin θ, not cot θ. 25/24 is sec θ, not cot θ. Common Pitfalls: Confusing cot with tan, or inverting the ratio incorrectly. Another mistake is forgetting to use Pythagoras to find the missing side and assuming adjacent equals hypotenuse minus opposite, which is invalid. Final Answer: cot θ = 24/7

Question 4

Use exponent rules to rewrite both sides with the same base. If (2^3)^2 = 4^x, what is the value of 3x?

Accepted Answer

Introduction / Context: This question tests basic exponent laws: power of a power, and rewriting numbers with the same base. The goal is to compare exponents once both sides are written as powers of 2. After finding x, we simply compute 3x. Given Data / Assumptions: (2^3)^2 = 4^x We need the value of 3x. Use rules: (a^m)^n = a^(m*n) and 4 = 2^2. Concept / Approach: Rewrite both sides with base 2. Then equate exponents because if 2^p = 2^q, then p = q. Finally multiply x by 3 to match the question requirement. Step-by-Step Solution: Left side: (2^3)^2 = 2^(3*2) = 2^6 Right side: 4^x = (2^2)^x = 2^(2x) So the equation becomes: 2^6 = 2^(2x) Equate exponents: 6 = 2x Solve: x = 3 Compute 3x: 3x = 3*3 = 9 Verification / Alternative check: Check numerically: (2^3)^2 = 8^2 = 64. If x = 3, then 4^x = 4^3 = 64. Both sides match, so x is correct and 3x = 9 is confirmed. Why Other Options Are Wrong: 6 is the exponent on 2 in 2^6, but the question asks for 3x, not 2x or 6. 12 comes from incorrectly doing 3x = 2x + something, or assuming x = 4. 27 comes from mistakenly cubing instead of multiplying by 3. 3 comes from incorrectly taking x itself as the final answer without multiplying by 3. Common Pitfalls: Forgetting that 4 = 2^2, or misusing (a^m)^n as a^(m+n) instead of a^(m*n). Also, some learners stop after finding x and forget to compute 3x. Final Answer: 3x = 9

Question 5

Use algebraic identities to avoid expanding fully when possible. If a - b = 2 and ab = 8, what is the value of a^3 - b^3?

Accepted Answer

Introduction / Context: This question tests the use of algebraic identities and relationships between a and b. Instead of solving for a and b explicitly, we can compute a^3 - b^3 using the factorization a^3 - b^3 = (a - b)(a^2 + ab + b^2). We are given a - b and ab, so we just need a^2 + b^2. Given Data / Assumptions: a - b = 2 ab = 8 We need a^3 - b^3 Concept / Approach: Use identities:1) a^3 - b^3 = (a - b)(a^2 + ab + b^2)2) (a - b)^2 = a^2 - 2ab + b^2, so a^2 + b^2 = (a - b)^2 + 2ab.Then substitute known values and multiply. Step-by-Step Solution: Use: a^3 - b^3 = (a - b)(a^2 + ab + b^2) We know (a - b) = 2 and ab = 8, so we need a^2 + b^2 Compute a^2 + b^2 from (a - b)^2: (a - b)^2 = a^2 - 2ab + b^2 So a^2 + b^2 = (a - b)^2 + 2ab = 2^2 + 2*8 = 4 + 16 = 20 Now a^2 + ab + b^2 = (a^2 + b^2) + ab = 20 + 8 = 28 Finally: a^3 - b^3 = (a - b)(a^2 + ab + b^2) = 2 * 28 = 56 Verification / Alternative check: You could solve a - b = 2 and ab = 8 by setting a = b + 2, giving b(b+2)=8 => b^2+2b-8=0, so b=2 or b=-4 and a=4 or a=-2. In both cases, a^3 - b^3 = 56, confirming the identity-based result. Why Other Options Are Wrong: 34 and 43 come from incorrect computation of a^2 + b^2 or forgetting the +ab term. 40 is a common result if you use (a - b)(a^2 + b^2) and omit ab. 65 is unrelated and typically comes from random expansion mistakes. Common Pitfalls: Forgetting that a^2 + ab + b^2 is needed (not just a^2 + b^2), and misusing (a - b)^2 as a^2 - b^2. Also, sign errors with 2ab are common. Final Answer: a^3 - b^3 = 56

Question 6

The slope between two points equals the change in y divided by the change in x. The slope of line segment AB is -2/3. If A = (x, -3) and B = (5, 2), what is the value of x?

Accepted Answer

Introduction / Context: This problem tests coordinate geometry basics: computing slope from two points and solving for an unknown coordinate. The slope formula is m = (y2 - y1)/(x2 - x1). Here, one x-coordinate is unknown, and we use the given slope to form an equation and solve it. Given Data / Assumptions: Slope m = -2/3 Point A = (x, -3) Point B = (5, 2) Use slope formula m = (y_B - y_A)/(x_B - x_A) Concept / Approach: Substitute coordinates into the slope formula. Then solve the resulting fraction equation by cross-multiplication. Carefully keep the order of subtraction consistent because changing point order changes the sign of both numerator and denominator, but the slope must remain the given value. Step-by-Step Solution: Slope between A(x, -3) and B(5, 2): m = (2 - (-3)) / (5 - x) Compute numerator: 2 - (-3) = 5 So m = 5 / (5 - x) Given m = -2/3, set 5 / (5 - x) = -2/3 Cross-multiply: 5*3 = -2*(5 - x) 15 = -10 + 2x Add 10 to both sides: 25 = 2x So x = 25/2 = 12.5 Verification / Alternative check: Plug x = 12.5 back: denominator 5 - 12.5 = -7.5, so slope = 5/(-7.5) = -2/3, correct. This confirms the solution. Why Other Options Are Wrong: 10 and 4 typically come from using (x - 5) instead of (5 - x) without adjusting signs properly. -4 and -14 come from sign errors when distributing -2 across (5 - x). Common Pitfalls: Forgetting that 5 - x can be negative, mishandling negative signs during cross-multiplication, or incorrectly computing 2 - (-3). Another common mistake is swapping the denominator to x - 5 but not changing the sign of the slope. Final Answer: x = 12.5

Question 7

Given a trigonometric ratio for an acute angle, you can build a right triangle and find other ratios using Pythagoras. If cosec θ = 17/8 for an acute angle θ, what is the value of cos θ?

Accepted Answer

Introduction / Context: This question tests converting from cosec θ to cos θ using a right-triangle model. Since θ is acute, all ratios are positive. From cosec θ, we obtain sin θ, interpret it as opposite/hypotenuse, use Pythagoras to find the adjacent side, and then compute cos θ = adjacent/hypotenuse. Given Data / Assumptions: cosec θ = 17/8 θ is acute, so sin θ > 0 and cos θ > 0 cosec θ = 1/sin θ Concept / Approach: Convert cosec to sin: sin θ = 8/17. Treat this as a right triangle where opposite = 8 and hypotenuse = 17. Then adjacent = sqrt(17^2 - 8^2) = 15, giving cos θ = 15/17. Step-by-Step Solution: cosec θ = 17/8 => sin θ = 1/(17/8) = 8/17 Let opposite = 8 and hypotenuse = 17 Adjacent^2 = hypotenuse^2 - opposite^2 = 17^2 - 8^2 = 289 - 64 = 225 Adjacent = sqrt(225) = 15 cos θ = adjacent/hypotenuse = 15/17 Verification / Alternative check: Since sin θ = 8/17 and cos θ = 15/17, check sin^2 + cos^2 = (64 + 225)/289 = 289/289 = 1, consistent with a right triangle. Why Other Options Are Wrong: 8/17 is sin θ, not cos θ. 17/15 is sec θ (reciprocal of cos). 8/15 is tan θ for the 8-15-17 triangle. 15/8 is cosec-related or an inverted ratio and cannot be cos because it is greater than 1. Common Pitfalls: Inverting incorrectly (mixing cosec and sec), or assuming cos θ = 1/cosec θ (which is wrong). Also, some forget to compute the adjacent side using Pythagoras and guess using the given numbers. Final Answer: cos θ = 15/17

Question 8

For an acute angle in a right triangle, tan θ = opposite/adjacent and sec θ = hypotenuse/adjacent. If tan θ = 7/24 for an acute angle θ, what is the value of sec θ?

Accepted Answer

Introduction / Context: This problem tests how to move between trigonometric ratios using a right-triangle interpretation. Given tan θ, we can treat the ratio as opposite/adjacent, then find the hypotenuse using Pythagoras, and finally compute sec θ = hypotenuse/adjacent. Because θ is acute, all values are positive. Given Data / Assumptions: tan θ = 7/24 θ is acute tan θ = opposite/adjacent sec θ = 1/cos θ = hypotenuse/adjacent Concept / Approach: Model a right triangle: set opposite = 7 and adjacent = 24. Compute hypotenuse = sqrt(7^2 + 24^2) = 25 (a known 7-24-25 triple). Then sec θ = hypotenuse/adjacent = 25/24. Step-by-Step Solution: tan θ = 7/24 => opposite = 7 and adjacent = 24 Hypotenuse^2 = opposite^2 + adjacent^2 = 7^2 + 24^2 = 49 + 576 = 625 Hypotenuse = sqrt(625) = 25 sec θ = hypotenuse/adjacent = 25/24 Verification / Alternative check: cos θ = adjacent/hypotenuse = 24/25. Then sec θ = 1/cos θ = 25/24, matching the computed result. Why Other Options Are Wrong: 24/25 is cos θ, not sec θ. 7/25 is sin θ for this triangle, not sec θ. 25/7 equals cosec θ or hypotenuse/opposite, not hypotenuse/adjacent. 24/7 is cot θ, not sec θ. Common Pitfalls: Confusing sec with cosec, or mistakenly using opposite in the denominator for sec. Another common mistake is taking the reciprocal of tan instead of computing sec via hypotenuse/adjacent. Final Answer: sec θ = 25/24

Question 9

Complete the square to convert a quadratic expression into a perfect-square form. If x^2 + y^2 + 6x + 5 = 4(x - y), what is the value of x - y?

Accepted Answer

Introduction / Context: This question tests algebraic transformation and completing the square in two variables. By moving all terms to one side and grouping x-terms and y-terms, we can rewrite the expression as a sum of squares. A sum of squares equals zero only when each square equals zero, letting us deduce exact values of x and y and then compute x - y. Given Data / Assumptions: Equation: x^2 + y^2 + 6x + 5 = 4(x - y) x and y are real numbers Goal: find x - y Concept / Approach: Bring RHS to the left, then complete squares separately for x and y:(x^2 + 2x + 1) = (x + 1)^2 and (y^2 + 4y + 4) = (y + 2)^2. If the equation becomes (x + 1)^2 + (y + 2)^2 = 0, both squares must be 0, giving x and y uniquely. Step-by-Step Solution: Start: x^2 + y^2 + 6x + 5 = 4(x - y) = 4x - 4y Move all terms to left: x^2 + y^2 + 6x + 5 - 4x + 4y = 0 Simplify: x^2 + y^2 + 2x + 4y + 5 = 0 Complete square in x: x^2 + 2x = (x + 1)^2 - 1 Complete square in y: y^2 + 4y = (y + 2)^2 - 4 Substitute: (x + 1)^2 - 1 + (y + 2)^2 - 4 + 5 = 0 Constants: -1 - 4 + 5 = 0, so (x + 1)^2 + (y + 2)^2 = 0 A sum of squares is 0 only if each square is 0: x + 1 = 0 => x = -1, and y + 2 = 0 => y = -2 Therefore x - y = (-1) - (-2) = 1 Verification / Alternative check: Substitute x = -1, y = -2 into original: LHS = 1 + 4 + 6*(-1) + 5 = 5 - 6 + 5 = 4. RHS = 4((-1) - (-2)) = 4*(1) = 4. Matches, so x - y = 1 is confirmed. Why Other Options Are Wrong: 0 and 2 come from mistakes in moving terms across the equal sign. -1 results from swapping y - x instead of x - y at the end. 4 is typically from forgetting to complete the square correctly in both variables. Common Pitfalls: Sign mistakes when expanding 4(x - y), and incorrect completion of squares (forgetting to add and subtract the same constant). Another error is assuming (x + 1)^2 + (y + 2)^2 = 0 has many solutions; it has exactly one real solution pair. Final Answer: x - y = 1

Question 10

Work with surds (square roots) carefully and use algebraic simplification to evaluate exactly. If x = 1 + √2 + √3, what is the exact value of the expression: 2x^4 - 8x^3 - 5x^2 + 26x - 28?

Accepted Answer

Introduction / Context: This question tests advanced simplification with surds (square roots). Rather than expanding x^4 directly (which becomes messy), the standard approach is to exploit structured cancellation: many contest-style polynomials are designed so that substituting x = 1 + √2 + √3 reduces to a simple surd after combining terms. The clean result indicates the expression was crafted for simplification. Given Data / Assumptions: x = 1 + √2 + √3 Evaluate: 2x^4 - 8x^3 - 5x^2 + 26x - 28 Exact value is required (no decimal approximation). Concept / Approach: A practical method is to compute x^2 and x^3 in simplified surd form, track like terms (constants, √2, √3, √6), and then observe cancellation when substituted into the polynomial. Because √2 * √3 = √6, the basis {1, √2, √3, √6} is enough to represent all intermediate results. Step-by-Step Solution: Let x = 1 + √2 + √3 Compute x^2 = (1 + √2 + √3)^2 = 1 + 2 + 3 + 2√2 + 2√3 + 2√6 So x^2 = 6 + 2√2 + 2√3 + 2√6 Use x^3 = x * x^2 and x^4 = (x^2)^2, but keep everything in 1, √2, √3, √6 form When substituting into 2x^4 - 8x^3 - 5x^2 + 26x - 28, the constant, √2, and √3 parts cancel by design The remaining non-cancelling term is the √6 component, which simplifies to 6√6 Verification / Alternative check: A quick numerical check supports the exact form: x ≈ 1 + 1.414 + 1.732 ≈ 4.146. Evaluating the polynomial numerically gives approximately 14.697, and 6√6 ≈ 6*2.449 ≈ 14.697, which matches closely, confirming the exact result. Why Other Options Are Wrong: 2√2, 4√2, and 3√3 imply the expression collapses to a single √2 or √3 term, but the structure of x includes both √2 and √3, producing √6 terms naturally. 5√5 introduces √5, which cannot appear from combining √2 and √3 (since √2*√3 = √6, not √5). Common Pitfalls: Expanding x^4 directly and losing track of terms, or incorrectly multiplying surds (for example treating √2*√3 as √5). Also, dropping cross terms like 2√6 in x^2 leads to wrong final simplification. Final Answer: 2x^4 - 8x^3 - 5x^2 + 26x - 28 = 6√6

Question 11

Use circle theorems relating tangents and chords (alternate segment theorem). PQ is a tangent to a circle at point T. Points R and S lie on the circle such that TR = TS, and ∠RST = 65 degrees. What is the value of angle ∠PTS?

Accepted Answer

Introduction / Context: This problem combines two classic geometry ideas: (1) isosceles triangles formed by equal chords or equal segments in a circle, and (2) the tangent-chord theorem (also called the alternate segment theorem). The goal is to find the angle between a tangent at T and the chord TS, which equals the angle in the opposite segment subtended by chord TS. Given Data / Assumptions: PQ is tangent to the circle at T. R and S are points on the circle. TR = TS, so triangle RST is isosceles with equal sides meeting at T. Angle RST = 65 degrees. We need angle PTS (angle between tangent PT and chord TS). Concept / Approach: First use isosceles triangle properties: if TR = TS, then base angles at R and S are equal. Next apply the tangent-chord theorem: the angle between the tangent at T and chord TS equals the angle in the alternate segment, which is the angle subtended by chord TS at the opposite point on the circle (here point R). So ∠PTS equals ∠TRS. Step-by-Step Solution: Given TR = TS, triangle RST is isosceles with base RS So base angles are equal: ∠TRS = ∠RST Given ∠RST = 65°, therefore ∠TRS = 65° Now use tangent-chord (alternate segment) theorem: Angle between tangent at T and chord TS equals the angle in the opposite segment subtended by TS That opposite-segment angle is ∠TRS So ∠PTS = ∠TRS = 65° Verification / Alternative check: We can also find ∠RTS = 180° - 65° - 65° = 50°, consistent with an isosceles triangle. The tangent-chord theorem does not depend on this 50°, but it supports internal consistency of the given triangle angles. Why Other Options Are Wrong: 130° is the exterior angle at S if incorrectly doubled, but ∠PTS is not defined that way. 115° and 55° come from mixing up which angle corresponds to the alternate segment. 45° is an unrelated guess and does not match any derived circle angle relationship here. Common Pitfalls: Confusing the tangent-chord theorem with the angle at the center theorem, or using the wrong point for the “alternate segment” angle. Another frequent mistake is forgetting that TR = TS implies angles at R and S are equal, not angles at T and S. Final Answer: ∠PTS = 65°

Question 12

Use parity (odd/even) reasoning and prime-number constraints to maximize one variable. x, y and z are prime numbers such that x + y + z = 38. What is the maximum possible value of x?

Accepted Answer

Introduction / Context: This question tests prime numbers along with a common number-theory trick: using odd/even parity to restrict possibilities quickly. Since most primes are odd (except 2), the sum of three primes has predictable parity. Once we identify that one of the primes must be 2, the problem becomes finding two primes that add to 36 while making x as large as possible. Given Data / Assumptions: x, y, z are prime numbers. x + y + z = 38. We want the maximum possible value of x. Prime numbers are integers greater than 1 with exactly two factors. Concept / Approach: Use parity:- 38 is even.- The sum of three odd numbers is odd.Therefore, all three primes cannot be odd. The only even prime is 2, so one of x, y, z must be 2. Then the remaining two primes must sum to 36. To maximize x, choose the smallest possible other prime so that the remaining number is still prime. Step-by-Step Solution: Since 38 is even, and odd + odd + odd = odd, at least one prime must be even The only even prime is 2, so one of the primes is 2 Then the other two primes must sum to 38 - 2 = 36 Let x be the largest prime among the two that sum to 36 Try smallest prime for the other: 3 gives 36 - 3 = 33 (not prime) Next 5 gives 36 - 5 = 31 (prime) So we can take primes 2, 5, and 31, which sum to 38 This makes x = 31 Any larger x would require the partner prime to be smaller than 5, but 3 fails, and 2 would force 34 (not prime) Verification / Alternative check: Other valid pairs for 36 include 7 + 29, 13 + 23, 17 + 19, but these give x values 29, 23, and 19 respectively, all smaller than 31. So 31 is indeed the maximum. Why Other Options Are Wrong: 29 occurs with (2, 7, 29). 23 occurs with (2, 13, 23). 19 occurs with (2, 17, 19). 17 is too small and cannot be the maximum given better combinations exist. Common Pitfalls: Forgetting that 2 is the only even prime, or assuming you can use three odd primes to make an even sum. Another mistake is not checking primality of the partner number (for example 33 or 34 are not prime). Final Answer: The maximum possible value of x is 31

Question 13

For a quadratic equation to have equal (repeated) roots, its discriminant must be zero. If the roots of a(b - c)x^2 + b(c - a)x + c(a - b) = 0 are equal, which relation between a, b and c must be true?

Accepted Answer

Introduction / Context: This question tests the condition for equal roots of a quadratic equation. For a general quadratic Ax^2 + Bx + C = 0, the roots are equal (a repeated root) when the discriminant is zero: B^2 - 4AC = 0. Here A, B, and C are expressions in a, b, and c, so we compute the discriminant symbolically and then simplify to obtain the required relationship among parameters. Given Data / Assumptions: Quadratic: a(b - c)x^2 + b(c - a)x + c(a - b) = 0 A = a(b - c), B = b(c - a), C = c(a - b) Equal roots condition: B^2 - 4AC = 0 Assume denominators used in relations are non-zero where required. Concept / Approach: Compute discriminant D = B^2 - 4AC using A, B, C above. Then factor the expression to identify the parameter constraint. After simplifying, the condition reduces neatly to b(a + c) = 2ac, which can be rearranged into the option form 2/b = 1/a + 1/c. Step-by-Step Solution: Identify coefficients: A = a(b - c), B = b(c - a), C = c(a - b) Equal roots require: D = B^2 - 4*A*C = 0 Compute: B^2 = b^2(c - a)^2 Compute: 4AC = 4 * a(b - c) * c(a - b) After algebraic simplification, D becomes a perfect square: D = (ab - 2ac + bc)^2 For D = 0, we need ab - 2ac + bc = 0 Factor b from the first and third terms: b(a + c) - 2ac = 0 So b(a + c) = 2ac Divide both sides by abc (assuming a, b, c non-zero for this step): (a + c)/(ac) = 2/b But (a + c)/(ac) = 1/a + 1/c Therefore: 2/b = 1/a + 1/c Verification / Alternative check: If 2/b = 1/a + 1/c holds, then b = 2ac/(a + c). Substituting into the discriminant form (ab - 2ac + bc)^2 makes the inside zero, so the discriminant is zero and the roots are equal. This confirms the necessity and sufficiency of the relation. Why Other Options Are Wrong: a + b + c = 0 does not generally force the discriminant to be zero for these coefficients. a b c = a b + b c + c a is a different symmetric condition and does not match the derived discriminant factor. 2 b = (1/a) + (1/c) has unit/structure mismatch and is not equivalent to b = 2ac/(a + c). b = (a + c)/(a c) equals 1/a + 1/c, but the required relation is 2/b equals that quantity, so it misses the factor 2. Common Pitfalls: Using the wrong equal-roots condition (some confuse it with B = 0), or expanding A, B, C incorrectly due to sign errors like mixing (c - a) with (a - c). Another common mistake is failing to factor the discriminant properly and missing that it is a perfect square. Final Answer: 2/b = (1/a) + (1/c)

Question 14

In coordinate geometry, a point Q(a, b) is first reflected in the y-axis to a point Q1, and then Q1 is reflected in the x-axis to reach the point (6, 2). What are the coordinates of the original point Q?

Accepted Answer

Introduction / Context: This question tests the concept of reflections of points in the Cartesian coordinate plane. Specifically, it involves successive reflections of a point in the y-axis and then in the x-axis, and asks you to work backwards to find the original coordinates of the point before any reflection took place. Given Data / Assumptions: Original point is Q(a, b). Q is reflected in the y-axis to obtain Q1. Q1 is then reflected in the x-axis to obtain the final point (6, 2). Standard rules of reflection in coordinate geometry apply. Concept / Approach: Reflection in the y-axis changes the sign of the x-coordinate but keeps the y-coordinate the same. Reflection in the x-axis changes the sign of the y-coordinate but keeps the x-coordinate the same. By applying these transformation rules step by step and then equating the final coordinates, we can solve for the original values of a and b. Step-by-Step Solution: Step 1: Original point Q has coordinates (a, b). Step 2: Reflect Q in the y-axis. This gives Q1 = (−a, b). Step 3: Reflect Q1 in the x-axis. Reflection in the x-axis changes y to −y, so the new point is (−a, −b). Step 4: We are told that this final point equals (6, 2). Therefore, (−a, −b) = (6, 2). Step 5: Equate coordinates: −a = 6 and −b = 2. Step 6: Solve for a and b: a = −6 and b = −2. Step 7: Hence, the original point Q has coordinates (−6, −2). Verification / Alternative check: Start with Q = (−6, −2). Reflect in the y-axis: x changes sign to get Q1 = (6, −2). Then reflect Q1 in the x-axis: y changes sign to get (6, 2), which matches the given final point. This confirms that the original coordinates are correct. Why Other Options Are Wrong: Option (−6, 2) would reflect to (6, 2) after only one reflection, not two successive reflections as described. Option (6, −2) is actually Q1, the intermediate point after the y-axis reflection, not the original point Q. Option (2, −6) does not produce (6, 2) under the given sequence of reflections. Option (6, 2) is the final point after both reflections, not the starting point. Common Pitfalls: A common mistake is to reverse the order of reflections or to change both coordinates at once without following the correct sequence. Another error is to confuse which axis affects which coordinate. Remember that the y-axis reflection affects only the x-coordinate, and the x-axis reflection affects only the y-coordinate. Final Answer: The original point Q has coordinates (−6, −2).

Question 15

If cosec(4π/3) = x in trigonometry, where the angle 4π/3 is measured in radians, what is the exact value of x expressed as a simplified surd?

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Introduction / Context: This question checks your understanding of trigonometric ratios for standard angles, as well as your ability to work with cosecant, which is the reciprocal of sine. The angle is given in radians as 4π/3, which corresponds to 240 degrees, a standard angle in the unit circle. Given Data / Assumptions: Angle θ = 4π/3 radians. cosec(θ) = x. cosec(θ) is defined as 1 / sin(θ). We are working with exact values on the unit circle. Concept / Approach: To find cosec(4π/3), we first need sin(4π/3). The angle 4π/3 is in the third quadrant, where sine is negative. Reference angles and the symmetry of the unit circle allow us to relate this to sin(π/3). Once sin(4π/3) is known, we simply take its reciprocal to obtain cosec(4π/3). Step-by-Step Solution: Step 1: Convert 4π/3 to degrees to understand the quadrant. 4π/3 radians corresponds to 240 degrees. Step 2: The reference angle for 240 degrees is 240 − 180 = 60 degrees, which is π/3 radians. Step 3: We know that sin(π/3) = √3 / 2. Step 4: In the third quadrant, sine is negative, so sin(4π/3) = −√3 / 2. Step 5: By definition, cosec(θ) = 1 / sin(θ). Therefore, cosec(4π/3) = 1 / (−√3 / 2). Step 6: Simplify the reciprocal: 1 / (−√3 / 2) = −2 / √3. Step 7: Thus, x = −2 / √3. Verification / Alternative check: You can confirm by remembering the pattern of signs in each quadrant: sine is positive in the first and second quadrants and negative in the third and fourth. Since 4π/3 lies in the third quadrant with reference angle π/3, the magnitude of sine is √3 / 2 and the sign is negative. Taking the reciprocal again gives −2 / √3, which matches our result. Why Other Options Are Wrong: 2/√3 is incorrect because it ignores the negative sign of sine in the third quadrant. 1/√3 and −1/2 do not correspond to the reciprocal of ±√3 / 2 and therefore are not correct values of cosec(4π/3). √3/2 is actually the sine of 60 degrees, not a cosecant value for 4π/3. Common Pitfalls: Many students forget to consider the quadrant and hence the sign of the trigonometric function. Another frequent error is to confuse sine and cosecant or to take the wrong reciprocal. Always identify the quadrant, find the reference angle, determine the correct sign for sine or cosine, and only then take the reciprocal if needed. Final Answer: The exact value of x is −2 / √3.

Question 16

The price of oranges is increased by 30%, and after this increase a person can buy 12 fewer oranges for Rs 208 than before. What was the original price in rupees of one orange before the 30% increase?

Accepted Answer

Introduction / Context: This question is a classic application of percentage increase and basic algebra in a real-life context involving the price of fruits. You are given how many fewer oranges can be bought after a price hike and asked to work backwards to find the original price per orange before the increase took place. Given Data / Assumptions: Total money spent each time = Rs 208. Original price per orange = Rs p (unknown). Price increased by 30%, so new price = 1.3 * p. After the increase, the buyer can purchase 12 fewer oranges with the same Rs 208. Concept / Approach: If a person spends the same amount of money but the price per unit increases, the number of units that can be bought decreases. Let the original number of oranges be N. After the price increase, the number becomes N − 12. Using the relationships money = price * quantity before and after the increase, we can form an equation in terms of p and solve it. Step-by-Step Solution: Step 1: Let the original price per orange be p rupees. Step 2: The original number of oranges that can be bought is 208 / p. Step 3: After a 30% increase, the new price is 1.3 * p. Step 4: With the new price, the number of oranges that can be bought is 208 / (1.3 * p). Step 5: According to the question, the new number of oranges is 12 less than the original number: 208 / (1.3 * p) = 208 / p − 12. Step 6: Express 1.3 as 13 / 10 and rewrite: 208 * 10 / (13p) = 208 / p − 12. Step 7: Multiply both sides by p to remove the denominator: 2080 / 13 = 208 − 12p. Step 8: Simplify 2080 / 13 = 160, so 160 = 208 − 12p. Step 9: Rearrange: 12p = 208 − 160 = 48, so p = 48 / 12 = 4. Step 10: Therefore, the original price per orange was Rs 4. Verification / Alternative check: At Rs 4 per orange, the buyer could originally buy 208 / 4 = 52 oranges. After a 30% increase, the new price is 4 * 1.3 = Rs 5.20. At Rs 5.20 per orange, the buyer can buy 208 / 5.20 = 40 oranges. The decrease is 52 − 40 = 12 oranges, which matches the condition in the question. Why Other Options Are Wrong: Rs 2, Rs 3, Rs 6 and Rs 8 do not satisfy the condition that the difference in quantity is exactly 12 when the price increases by 30%. Substituting these values into the equation gives fewer or more than 12 oranges difference. Common Pitfalls: Students sometimes forget to convert 30% into a multiplier of 1.3 or set up the relationship between the two quantities incorrectly. It is also easy to make algebraic errors while clearing denominators. Writing the equations clearly and simplifying step by step helps avoid mistakes. Final Answer: The original price of one orange was Rs 4.

Question 17

Find the two real roots of the quadratic equation x^2 + 3x − 154 = 0, giving the values of x that satisfy this equation.

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Introduction / Context: This problem is a standard quadratic equation question from algebra. You are asked to find the roots of a quadratic polynomial, which means finding the values of x that make the expression x^2 + 3x − 154 equal to zero. Such questions test basic skills in factorisation or use of the quadratic formula. Given Data / Assumptions: Quadratic equation: x^2 + 3x − 154 = 0. We are looking for real roots. Standard algebraic methods such as factorisation or the quadratic formula apply. Concept / Approach: A quadratic equation of the form ax^2 + bx + c = 0 can be solved either by factorisation, by completing the square, or by using the quadratic formula x = [−b ± √(b^2 − 4ac)] / (2a). In this case, the coefficients are simple integers and factorisation is efficient: we search for two numbers whose product is −154 and whose sum is 3. Step-by-Step Solution: Step 1: Identify coefficients: a = 1, b = 3, c = −154. Step 2: We need two integers m and n such that m * n = −154 and m + n = 3. Step 3: The factor pairs of 154 are: 1 and 154, 2 and 77, 7 and 22, 11 and 14. Step 4: To get a product of −154, one factor must be negative. Check 11 and −14: 11 * (−14) = −154 and 11 + (−14) = −3, which is not 3. Step 5: Reverse the signs: −11 and 14 give (−11) * 14 = −154 and (−11) + 14 = 3. This pair works. Step 6: Rewrite the quadratic as x^2 + 3x − 154 = (x − 11)(x + 14) = 0. Step 7: Set each factor equal to zero: x − 11 = 0 or x + 14 = 0. Step 8: Solve to obtain x = 11 and x = −14. Verification / Alternative check: Substitute x = 11 into the original equation: 11^2 + 3 * 11 − 154 = 121 + 33 − 154 = 154 − 154 = 0. This satisfies the equation. Substitute x = −14: (−14)^2 + 3 * (−14) − 154 = 196 − 42 − 154 = 154 − 154 = 0. Both values satisfy the quadratic, confirming that they are correct roots. Why Other Options Are Wrong: 11, 14 gives one correct root and one incorrect root because 14 does not satisfy the equation. 14, −11 is just a reordered incorrect pair, since the actual roots are 11 and −14. 14, −22 and 7, −22 do not satisfy the equation when substituted into x^2 + 3x − 154, so they cannot be the correct roots. Common Pitfalls: A common error is mixing up the signs of the factor pair. Because the constant term is negative, the factors must have opposite signs. Forgetting to check both possible sign assignments can lead to the wrong pair. Another pitfall is not verifying the roots by substitution, which is a quick and reliable check. Final Answer: The roots of the equation are x = 11 and x = −14.

Question 18

For triangle ABC with vertices A(2, 4), B(3, 0) and C(5, 2) in the Cartesian plane, find the equation of the median AD, where D is the midpoint of side BC.

Accepted Answer

Introduction / Context: This question is from coordinate geometry and focuses on the concept of a median of a triangle. A median is a line segment joining a vertex of a triangle to the midpoint of the opposite side. Here, you must find the equation of the median AD given the coordinates of the vertices of triangle ABC. Given Data / Assumptions: A(2, 4), B(3, 0), C(5, 2) are the vertices of triangle ABC. D is the midpoint of BC. We need the equation of line AD in standard form. All coordinates lie in the usual Cartesian plane. Concept / Approach: To find the equation of a median, we first compute the midpoint of the opposite side using the midpoint formula. Then we find the equation of the line passing through the given vertex and this midpoint. The two main tools are: Midpoint formula: midpoint of (x1, y1) and (x2, y2) is ((x1 + x2) / 2, (y1 + y2) / 2). Two point form or slope form of a straight line equation. Step-by-Step Solution: Step 1: Identify the coordinates of B and C: B(3, 0), C(5, 2). Step 2: Compute the midpoint D of BC: D = ((3 + 5) / 2, (0 + 2) / 2) = (8 / 2, 2 / 2) = (4, 1). Step 3: We now need the equation of the line joining A(2, 4) and D(4, 1). Step 4: Calculate the slope m of AD: m = (1 − 4) / (4 − 2) = (−3) / 2 = −3 / 2. Step 5: Use the point-slope form: y − y1 = m(x − x1) with point A(2, 4). Step 6: Substitute: y − 4 = (−3 / 2)(x − 2). Step 7: Multiply both sides by 2 to clear the denominator: 2y − 8 = −3x + 6. Step 8: Rearrange to standard form: 3x + 2y − 14 = 0 or equivalently 3x + 2y = 14. Verification / Alternative check: Substitute A(2, 4) into 3x + 2y = 14: 3 * 2 + 2 * 4 = 6 + 8 = 14, which satisfies the equation. Substitute D(4, 1): 3 * 4 + 2 * 1 = 12 + 2 = 14, also satisfied. Because both A and D lie on the line, and AD is the median by construction, 3x + 2y = 14 is the correct equation. Why Other Options Are Wrong: 3x − 2y = 14 and 3x − 2y = 2 do not pass through both A and D when tested by substitution. 3x + 2y = 2 is incorrect because substituting A or D does not satisfy the equation. 2x + 3y = 14 is a different line with a different slope and does not represent median AD. Common Pitfalls: Students sometimes make arithmetic mistakes in calculating the midpoint or the slope. Another common error is mishandling signs when rearranging the linear equation. Always verify your final equation by checking whether it passes through both the given vertex and the computed midpoint. Final Answer: The equation of the median AD is 3x + 2y = 14.

Question 19

In a rhombus ABCD, the diagonal AC is drawn and the measure of angle CAB is 35°. Using the properties of rhombus diagonals and interior angles, what is the measure of interior angle ABC of the rhombus?

Accepted Answer

Introduction / Context: This question comes from Euclidean geometry and tests your understanding of the properties of a rhombus, especially how its diagonals relate to its interior angles. You are given an angle formed by a diagonal and a side and asked to find one of the interior angles of the rhombus. Given Data / Assumptions: ABCD is a rhombus. Diagonal AC is drawn. Angle CAB (an angle at vertex A between CA and AB) is 35 degrees. All standard geometric properties of a rhombus hold: all sides are equal, and diagonals bisect angles. Concept / Approach: In a rhombus, each diagonal bisects the interior angle at the vertices it joins. Thus, diagonal AC cuts angle A into two equal parts. If one of those parts (angle CAB) is known, we can find the full angle at A. Then, using the fact that adjacent interior angles of a parallelogram (and hence a rhombus) are supplementary, we can find angle B. Step-by-Step Solution: Step 1: In rhombus ABCD, diagonal AC bisects angle A. Step 2: We are given angle CAB = 35 degrees. Step 3: Because AC bisects angle A, angle DAB and angle CAB are equal. Step 4: Therefore, angle A = angle DAB + angle CAB = 35 + 35 = 70 degrees. Step 5: A rhombus is a special parallelogram, so adjacent interior angles are supplementary. That is, angle A + angle B = 180 degrees. Step 6: Substitute angle A = 70 degrees: 70 + angle B = 180. Step 7: Solve for angle B: angle B = 180 − 70 = 110 degrees. Step 8: Thus, the measure of angle ABC is 110 degrees. Verification / Alternative check: Another way to think about it is that opposite angles of a rhombus are equal, and diagonals bisect these angles. If angle A is 70 degrees, then angle C is also 70 degrees. The total of all interior angles in any quadrilateral is 360 degrees. So angle B + angle D = 360 − (70 + 70) = 220 degrees. Since B and D are equal in a rhombus, each must be 110 degrees. This again confirms that angle ABC is 110 degrees. Why Other Options Are Wrong: 70 degrees is the measure of angle A, not angle B. 40 degrees and 50 degrees do not satisfy the supplementary relationship with 70 degrees and do not fit the properties of a rhombus. 90 degrees would indicate a right angle, which is not supported by the given data because it would give each half-angle as 45 degrees, not 35 degrees. Common Pitfalls: A common mistake is to assume that the angle given (35 degrees) is the interior angle itself rather than half of it. Another pitfall is forgetting that a rhombus is also a parallelogram and therefore adjacent angles are supplementary. Always use both diagonal and parallelogram properties when dealing with rhombus questions. Final Answer: The measure of angle ABC is 110°.

Question 20

The first and last terms of an arithmetic progression (A.P.) are 25 and 52 respectively. If the progression contains 12 terms in total, what is the sum of all the terms of this A.P.?

Accepted Answer

Introduction / Context: This problem is based on sequences and series, specifically arithmetic progressions. You are given the first term, the last term, and the number of terms of an A.P., and asked to compute the total sum of all the terms. This is a standard formula-based question often seen in aptitude and competitive exams. Given Data / Assumptions: First term a1 = 25. Last term a12 = 52. Number of terms n = 12. The sequence is arithmetic, meaning the difference between consecutive terms is constant. Concept / Approach: For an arithmetic progression with n terms, first term a1 and last term an, the sum S of the n terms is given by the formula: S = (n / 2) * (a1 + an). We do not even need the common difference if we know the first and last terms and the number of terms. Substituting the given values into this formula directly yields the required sum. Step-by-Step Solution: Step 1: Identify n, a1 and an. n = 12, a1 = 25, an = 52. Step 2: Use the sum formula for an arithmetic progression: S = (n / 2) * (a1 + an). Step 3: Substitute the values: S = (12 / 2) * (25 + 52). Step 4: Simplify: 12 / 2 = 6 and 25 + 52 = 77. Step 5: Multiply: S = 6 * 77 = 462. Step 6: Therefore, the sum of the 12 terms of the progression is 462. Verification / Alternative check: We can quickly check the reasonableness of the answer. The average of the first and last terms is (25 + 52) / 2 = 77 / 2 = 38.5. Since there are 12 terms, the sum should be 12 * 38.5 = 462. This matches the value obtained from the standard formula, confirming the correctness of the solution. Why Other Options Are Wrong: 372, 252, 110 and 520 do not satisfy the arithmetic progression sum formula when the first term is 25, the last term is 52, and n = 12. They arise from either miscounting the number of terms, misusing the formula, or arithmetic errors. Common Pitfalls: One common error is to use the wrong formula, such as multiplying the first term by the number of terms without considering the last term. Another frequent mistake is to miscalculate the average or to forget to divide n by 2. Always recall the correct sum formula and check that you have substituted the correct values. Final Answer: The sum of all the terms of the A.P. is 462.

Simplification Questions