Q: Algebraic ratio application: Given (5x^2 − 3y^2) : (x y) = 11 : 2, determine the positive ratio x : y.

Introduction / Context:Ratios involving quadratic expressions often reduce to a simple ratio between variables by substituting a single unknown for x/y. This question tests algebraic manipulation and solving a quadratic in a single variable that represents the ratio x : y. Given Data / Assumptions: (5x^2 − 3y^2) : (x y) = 11 : 2. x and y are real; we are asked for the positive ratio x : y. We assume y ≠ 0 so the ratio x/y is defined. Concept / Approach:Convert the proportion to an equation, then reduce the two-variable expression to a single variable by setting r = x / y. This yields a quadratic equation in r, from which we take the positive solution to obtain the ratio x : y. Step-by-Step Solution: (5x^2 − 3y^2) / (x y) = 11 / 2Cross-multiply: 2(5x^2 − 3y^2) = 11 x yLet r = x / y ⇒ x = r y. Substitute: 2(5r^2 y^2 − 3y^2) = 11 r y^2Divide by y^2 (y ≠ 0): 10 r^2 − 6 = 11 rRearrange: 10 r^2 − 11 r − 6 = 0Solve: Discriminant D = 121 + 240 = 361 ⇒ √D = 19r = (11 ± 19) / 20 ⇒ r = 30/20 = 3/2 or r = −8/20 = −2/5Positive ratio required ⇒ r = 3/2 ⇒ x : y = 3 : 2 Verification / Alternative check:Pick y = 2, x = 3. Compute (5x^2 − 3y^2)/(xy) = (5*9 − 3*4)/(3*2) = (45 − 12)/6 = 33/6 = 11/2, matching the given proportion. Why Other Options Are Wrong:5 : 2, 7 : 2, 5 : 3, and 2 : 5 do not satisfy 10r^2 − 11r − 6 = 0 when r is formed as x/y. Common Pitfalls:Forgetting to divide through by y^2, sign errors when forming the quadratic, or selecting the negative root despite the requirement of a positive ratio. Final Answer:3 : 2

Q: Use algebraic identities effectively: If a^3 − b^3 = 56 and a − b = 2, find the value of (a^2 + b^2).

Introduction / Context:This problem is a classic application of the factorization identity for cubes together with simple system solving. It checks whether you can relate a^3 − b^3 and a − b to obtain expressions for a^2 + b^2 and ab. Given Data / Assumptions: a^3 − b^3 = 56. a − b = 2. a and b are real numbers. Concept / Approach:Use the identity a^3 − b^3 = (a − b)(a^2 + ab + b^2) to find a^2 + ab + b^2. Then use (a − b)^2 = a^2 + b^2 − 2ab to form two equations in the unknowns S = a^2 + b^2 and P = ab, and solve for S. Step-by-Step Solution: a^3 − b^3 = (a − b)(a^2 + ab + b^2)56 = 2 * (a^2 + ab + b^2) ⇒ a^2 + ab + b^2 = 28Let S = a^2 + b^2 and P = ab. Then S + P = 28 ⇒ S = 28 − PAlso, (a − b)^2 = S − 2P = 4Substitute S: (28 − P) − 2P = 4 ⇒ 28 − 3P = 4 ⇒ 3P = 24 ⇒ P = 8Hence S = 28 − 8 = 20 Verification / Alternative check:Choose numbers satisfying a − b = 2 and ab = 8, e.g., solve t^2 − 2t + 8 = 0 to get complex roots that still satisfy the identities; S remains 20 by construction, so the value is consistent with the algebraic relations. Why Other Options Are Wrong:18, −10, −12, and 16 arise from arithmetic slips when solving for P or mixing signs in the square identity. Common Pitfalls:Mistaking a^2 + b^2 + ab for a^2 + b^2, or using (a − b)^2 incorrectly as a^2 + b^2 + 2ab rather than a^2 + b^2 − 2ab. Final Answer:20

Q: Radical symmetry trick: If x = 2 + √3, evaluate (√x + 1/√x).

Introduction / Context:Expressions of the form √x + 1/√x can often be evaluated without explicitly finding √x by using the identity (√x + 1/√x)^2 = x + 1/x + 2. The key is to compute x + 1/x first, which is straightforward for numbers of the form a + √b. Given Data / Assumptions: x = 2 + √3. We need the exact value of √x + 1/√x. x is positive, so √x is real and positive. Concept / Approach:Compute 1/x by rationalizing the denominator: 1/(2 + √3) = (2 − √3)/(4 − 3) = 2 − √3. Then add x + 1/x to leverage the square identity to find √x + 1/√x. Step-by-Step Solution: Compute 1/x: 1/(2 + √3) = 2 − √3Then x + 1/x = (2 + √3) + (2 − √3) = 4Use the identity: (√x + 1/√x)^2 = x + 1/x + 2 = 4 + 2 = 6Therefore, √x + 1/√x = √6 (positive root because √x is positive) Verification / Alternative check:Let t = √x. Then t^2 = x and (t + 1/t)^2 = t^2 + 1/t^2 + 2 = x + 1/x + 2. Plug in the computed value to confirm t + 1/t = √6. Why Other Options Are Wrong:√3 and 2 are too small; 2√6 and 6 are too large and come from squaring without taking the square root properly. Common Pitfalls:Forgetting to add the extra +2 when using (√x + 1/√x)^2, or rationalizing 1/x incorrectly. Final Answer:√6

Q: Key identity for sums of cubes: Given real a, b, c with a^3 + b^3 + c^3 = 3abc and a + b + c ≠ 0, determine the relation among a, b, c.

Introduction / Context:The identity a^3 + b^3 + c^3 − 3abc = (a + b + c)(a^2 + b^2 + c^2 − ab − bc − ca) characterizes when the sum of cubes equals thrice the product. This problem tests recognition of the special cases implied by this factorization. Given Data / Assumptions: a^3 + b^3 + c^3 = 3abc. a + b + c ≠ 0. a, b, c are real numbers. Concept / Approach:From the identity, if a^3 + b^3 + c^3 − 3abc = 0, then either a + b + c = 0, or a^2 + b^2 + c^2 − ab − bc − ca = 0. The second factor equal to zero implies a = b = c for real numbers because it can be written as 1/2[(a − b)^2 + (b − c)^2 + (c − a)^2]. Step-by-Step Solution: Given a^3 + b^3 + c^3 − 3abc = 0Factor: (a + b + c)(a^2 + b^2 + c^2 − ab − bc − ca) = 0Since a + b + c ≠ 0, we must have a^2 + b^2 + c^2 − ab − bc − ca = 0Rewrite: a^2 + b^2 + c^2 − ab − bc − ca = 1/2[(a − b)^2 + (b − c)^2 + (c − a)^2]This sum of squares equals 0 ⇒ a − b = b − c = c − a = 0 ⇒ a = b = c Verification / Alternative check:If a = b = c, then a^3 + b^3 + c^3 = 3a^3 and 3abc = 3a^3, so the equality holds for any real a (and the sum a + b + c = 3a is nonzero if a ≠ 0). Why Other Options Are Wrong:Linear relations like a + b = c, a + c = b, or b + c = a do not, in general, force the factor a^2 + b^2 + c^2 − ab − bc − ca to vanish. Common Pitfalls:Forgetting the factorization or assuming a + b + c = 0 must hold; here it is explicitly ruled out, forcing a = b = c instead. Final Answer:a = b = c

Question 1

Find the degree of the polynomial p(x) = x^3 + 2x + 1 (ignore any non-polynomial fragments in the original text).

Accepted Answer

Introduction / Context: The question asks for the degree of a polynomial. The degree is the highest power of x with a nonzero coefficient in a valid polynomial expression. Given Data / Assumptions: Polynomial: p(x) = x^3 + 2x + 1. We disregard any stray non-polynomial expression (like 1/x) from the corrupted prompt, following recovery-first policy. Concept / Approach: A polynomial in x has nonnegative integer exponents only. The degree equals the largest such exponent appearing with a nonzero coefficient. Step-by-Step Solution: Identify exponents: x^3 has exponent 3; 2x has exponent 1; constant term has exponent 0.Maximum exponent = 3.Therefore, degree = 3. Verification / Alternative check: No term has a higher exponent than 3, and all terms have valid nonnegative integer exponents; thus the degree is 3. Why Other Options Are Wrong: 2 and 1 would ignore the x^3 term; 4 or 5 invent higher exponents not present in the polynomial. Common Pitfalls: Being misled by an extraneous “1/x” fragment, which would make the expression non-polynomial; confusing the number of terms with degree; overlooking the highest power term. Final Answer: 3

Question 2

Evaluate the expression when x = 1 + √2: compute x^4 - 4x^3 + 4x^2 (give a numeric value).

Accepted Answer

Introduction / Context: This problem checks algebraic manipulation and recognition of patterns. Substituting x = 1 + √2 into a structured polynomial often simplifies using identities. Given Data / Assumptions: x = 1 + √2. Compute S = x^4 - 4x^3 + 4x^2. Concept / Approach: Observe that x^4 - 4x^3 + 4x^2 = x^2(x^2 - 4x + 4) = x^2 (x - 2)^2. This factorization greatly simplifies evaluation. Step-by-Step Solution: S = x^2 (x - 2)^2x - 2 = (1 + √2) - 2 = √2 - 1x^2 = (1 + √2)^2 = 1 + 2√2 + 2 = 3 + 2√2(x - 2)^2 = (√2 - 1)^2 = 2 - 2√2 + 1 = 3 - 2√2S = (3 + 2√2)(3 - 2√2) = 9 - (2√2)^2 = 9 - 8 = 1 Verification / Alternative check: Numeric approximation: √2 ≈ 1.4142, x ≈ 2.4142; computing S directly gives ~1, confirming the exact result. Why Other Options Are Wrong: -1, 0, and 2 arise from arithmetic mistakes or failing to factor and simplify correctly. Common Pitfalls: Expanding without noticing the perfect-square structure; sign errors when squaring (√2 - 1); rounding too early during approximations. Final Answer: 1

Question 3

Solve the system: x − y = 0.9 and 11/(x + y) = 2. Find the ordered pair (x, y).

Accepted Answer

Introduction / Context: This is a two-equation system involving a linear equation and a reciprocal relation for the sum. It assesses equation-solving and substitution skills with decimals. Given Data / Assumptions: x − y = 0.9 11/(x + y) = 2 ⇒ x + y = 11/2 = 5.5 Concept / Approach: Solve the system formed by the sum and difference: x + y and x − y. Use the standard identities x = ( (x + y) + (x − y) ) / 2 and y = ( (x + y) − (x − y) ) / 2. Step-by-Step Solution: x + y = 5.5x − y = 0.9x = (5.5 + 0.9) / 2 = 6.4 / 2 = 3.2y = (5.5 − 0.9) / 2 = 4.6 / 2 = 2.3 Verification / Alternative check: Check: x − y = 3.2 − 2.3 = 0.9; 11/(x + y) = 11/5.5 = 2. Both satisfied. Why Other Options Are Wrong: The other choices do not satisfy both equations simultaneously; either the difference or the reciprocal sum fails. Common Pitfalls: Misreading 11/(x + y) as 11(x + y) or vice versa; sign errors; rounding intermediate results instead of keeping exact fractions/decimals. Final Answer: x = 3.2, y = 2.3

Question 4

Two alternate odd integers (differing by 4) have a product that exceeds three times the smaller by 12. Find the larger integer.

Accepted Answer

Introduction / Context: This algebra word problem focuses on modeling consecutive odd integers (skipping one: difference 4) and forming a quadratic equation from the given relationship. Given Data / Assumptions: Let the smaller odd integer be n; then the larger is n + 4. Condition: n(n + 4) = 3n + 12. Concept / Approach: Translate the statement into an algebraic equation and solve the quadratic. Choose the odd, positive solution consistent with the context. Step-by-Step Solution: n(n + 4) = 3n + 12n^2 + 4n − 3n − 12 = 0 ⇒ n^2 + n − 12 = 0Factor: (n + 4)(n − 3) = 0 ⇒ n = 3 or n = −4Valid smaller odd integer: n = 3 ⇒ larger = n + 4 = 7 Verification / Alternative check: Product = 3 * 7 = 21; three times smaller = 3 * 3 = 9; 21 exceeds 9 by 12, condition satisfied. Why Other Options Are Wrong: 9 and 5 do not pair with a smaller odd number 4 less that satisfies the condition; 3 is the smaller, not the larger. Common Pitfalls: Using difference 2 instead of 4 for alternate odds; sign mistakes when arranging the quadratic; ignoring the requirement that the numbers are odd. Final Answer: 7

Question 5

Two numbers have sum 24 and product 143. Compute the sum of their squares.

Accepted Answer

Introduction / Context: This is a classic identity application: from sum and product of two numbers, determine the sum of their squares quickly. Given Data / Assumptions: Let the numbers be a and b. a + b = 24; ab = 143. Concept / Approach: Use the identity a^2 + b^2 = (a + b)^2 − 2ab. This avoids solving for a and b explicitly. Step-by-Step Solution: (a + b)^2 = 24^2 = 5762ab = 2 * 143 = 286a^2 + b^2 = 576 − 286 = 290 Verification / Alternative check: Solving: the numbers are 11 and 13 (since t^2 − 24t + 143 = 0 → roots 11, 13). Then 11^2 + 13^2 = 121 + 169 = 290, confirming the identity method. Why Other Options Are Wrong: 296, 295, and 228 do not match the identity result; they typically come from arithmetic slips (e.g., subtracting ab instead of 2ab). Common Pitfalls: Forgetting the factor 2 in 2ab; attempting a longer route by solving for the numbers when the identity is faster and safer. Final Answer: 290

Question 6

Let P = (0.08)^2, Q = 1/(0.08)^2, and R = (1 − 0.08)^2 − 1. Identify the correct ordering/relationship among P, Q, and R.

Accepted Answer

Introduction / Context: This question examines decimal computation and comparison, including a small square, its reciprocal square, and a shifted square minus 1. Given Data / Assumptions: P = (0.08)^2 Q = 1/(0.08)^2 R = (1 − 0.08)^2 − 1 Concept / Approach: Compute each value exactly and then compare. Beware that squaring a small decimal gives a very small positive number, while the reciprocal square becomes large. A near-one square minus 1 will be negative. Step-by-Step Solution: P = (0.08)^2 = 0.0064Q = 1 / 0.0064 = 156.25(1 − 0.08) = 0.92 ⇒ 0.92^2 = 0.8464 ⇒ R = 0.8464 − 1 = −0.1536Ordering: R (−0.1536) < P (0.0064) < Q (156.25) Verification / Alternative check: Estimate check: 0.08^2 is clearly tiny positive; reciprocal square is very large; a number close to 1 squared then minus 1 is a small negative. The ordering matches intuition. Why Other Options Are Wrong: Any option claiming P = R is false since P ≈ 0.0064 and R ≈ −0.1536. Options reversing the order ignore signs or magnitude. Common Pitfalls: Misplacing decimal points; forgetting that reciprocals of numbers less than 1 are greater than 1; sign errors when subtracting 1. Final Answer: R < P < Q

Question 7

Algebraic ratio application: Given (5x^2 − 3y^2) : (x y) = 11 : 2, determine the positive ratio x : y.

Accepted Answer

Introduction / Context:Ratios involving quadratic expressions often reduce to a simple ratio between variables by substituting a single unknown for x/y. This question tests algebraic manipulation and solving a quadratic in a single variable that represents the ratio x : y. Given Data / Assumptions: (5x^2 − 3y^2) : (x y) = 11 : 2. x and y are real; we are asked for the positive ratio x : y. We assume y ≠ 0 so the ratio x/y is defined. Concept / Approach:Convert the proportion to an equation, then reduce the two-variable expression to a single variable by setting r = x / y. This yields a quadratic equation in r, from which we take the positive solution to obtain the ratio x : y. Step-by-Step Solution: (5x^2 − 3y^2) / (x y) = 11 / 2Cross-multiply: 2(5x^2 − 3y^2) = 11 x yLet r = x / y ⇒ x = r y. Substitute: 2(5r^2 y^2 − 3y^2) = 11 r y^2Divide by y^2 (y ≠ 0): 10 r^2 − 6 = 11 rRearrange: 10 r^2 − 11 r − 6 = 0Solve: Discriminant D = 121 + 240 = 361 ⇒ √D = 19r = (11 ± 19) / 20 ⇒ r = 30/20 = 3/2 or r = −8/20 = −2/5Positive ratio required ⇒ r = 3/2 ⇒ x : y = 3 : 2 Verification / Alternative check:Pick y = 2, x = 3. Compute (5x^2 − 3y^2)/(xy) = (5*9 − 3*4)/(3*2) = (45 − 12)/6 = 33/6 = 11/2, matching the given proportion. Why Other Options Are Wrong:5 : 2, 7 : 2, 5 : 3, and 2 : 5 do not satisfy 10r^2 − 11r − 6 = 0 when r is formed as x/y. Common Pitfalls:Forgetting to divide through by y^2, sign errors when forming the quadratic, or selecting the negative root despite the requirement of a positive ratio. Final Answer:3 : 2

Question 8

Use algebraic identities effectively: If a^3 − b^3 = 56 and a − b = 2, find the value of (a^2 + b^2).

Accepted Answer

Introduction / Context:This problem is a classic application of the factorization identity for cubes together with simple system solving. It checks whether you can relate a^3 − b^3 and a − b to obtain expressions for a^2 + b^2 and ab. Given Data / Assumptions: a^3 − b^3 = 56. a − b = 2. a and b are real numbers. Concept / Approach:Use the identity a^3 − b^3 = (a − b)(a^2 + ab + b^2) to find a^2 + ab + b^2. Then use (a − b)^2 = a^2 + b^2 − 2ab to form two equations in the unknowns S = a^2 + b^2 and P = ab, and solve for S. Step-by-Step Solution: a^3 − b^3 = (a − b)(a^2 + ab + b^2)56 = 2 * (a^2 + ab + b^2) ⇒ a^2 + ab + b^2 = 28Let S = a^2 + b^2 and P = ab. Then S + P = 28 ⇒ S = 28 − PAlso, (a − b)^2 = S − 2P = 4Substitute S: (28 − P) − 2P = 4 ⇒ 28 − 3P = 4 ⇒ 3P = 24 ⇒ P = 8Hence S = 28 − 8 = 20 Verification / Alternative check:Choose numbers satisfying a − b = 2 and ab = 8, e.g., solve t^2 − 2t + 8 = 0 to get complex roots that still satisfy the identities; S remains 20 by construction, so the value is consistent with the algebraic relations. Why Other Options Are Wrong:18, −10, −12, and 16 arise from arithmetic slips when solving for P or mixing signs in the square identity. Common Pitfalls:Mistaking a^2 + b^2 + ab for a^2 + b^2, or using (a − b)^2 incorrectly as a^2 + b^2 + 2ab rather than a^2 + b^2 − 2ab. Final Answer:20

Question 9

Radical symmetry trick: If x = 2 + √3, evaluate (√x + 1/√x).

Accepted Answer

Introduction / Context:Expressions of the form √x + 1/√x can often be evaluated without explicitly finding √x by using the identity (√x + 1/√x)^2 = x + 1/x + 2. The key is to compute x + 1/x first, which is straightforward for numbers of the form a + √b. Given Data / Assumptions: x = 2 + √3. We need the exact value of √x + 1/√x. x is positive, so √x is real and positive. Concept / Approach:Compute 1/x by rationalizing the denominator: 1/(2 + √3) = (2 − √3)/(4 − 3) = 2 − √3. Then add x + 1/x to leverage the square identity to find √x + 1/√x. Step-by-Step Solution: Compute 1/x: 1/(2 + √3) = 2 − √3Then x + 1/x = (2 + √3) + (2 − √3) = 4Use the identity: (√x + 1/√x)^2 = x + 1/x + 2 = 4 + 2 = 6Therefore, √x + 1/√x = √6 (positive root because √x is positive) Verification / Alternative check:Let t = √x. Then t^2 = x and (t + 1/t)^2 = t^2 + 1/t^2 + 2 = x + 1/x + 2. Plug in the computed value to confirm t + 1/t = √6. Why Other Options Are Wrong:√3 and 2 are too small; 2√6 and 6 are too large and come from squaring without taking the square root properly. Common Pitfalls:Forgetting to add the extra +2 when using (√x + 1/√x)^2, or rationalizing 1/x incorrectly. Final Answer:√6

Question 10

Key identity for sums of cubes: Given real a, b, c with a^3 + b^3 + c^3 = 3abc and a + b + c ≠ 0, determine the relation among a, b, c.

Accepted Answer

Introduction / Context:The identity a^3 + b^3 + c^3 − 3abc = (a + b + c)(a^2 + b^2 + c^2 − ab − bc − ca) characterizes when the sum of cubes equals thrice the product. This problem tests recognition of the special cases implied by this factorization. Given Data / Assumptions: a^3 + b^3 + c^3 = 3abc. a + b + c ≠ 0. a, b, c are real numbers. Concept / Approach:From the identity, if a^3 + b^3 + c^3 − 3abc = 0, then either a + b + c = 0, or a^2 + b^2 + c^2 − ab − bc − ca = 0. The second factor equal to zero implies a = b = c for real numbers because it can be written as 1/2[(a − b)^2 + (b − c)^2 + (c − a)^2]. Step-by-Step Solution: Given a^3 + b^3 + c^3 − 3abc = 0Factor: (a + b + c)(a^2 + b^2 + c^2 − ab − bc − ca) = 0Since a + b + c ≠ 0, we must have a^2 + b^2 + c^2 − ab − bc − ca = 0Rewrite: a^2 + b^2 + c^2 − ab − bc − ca = 1/2[(a − b)^2 + (b − c)^2 + (c − a)^2]This sum of squares equals 0 ⇒ a − b = b − c = c − a = 0 ⇒ a = b = c Verification / Alternative check:If a = b = c, then a^3 + b^3 + c^3 = 3a^3 and 3abc = 3a^3, so the equality holds for any real a (and the sum a + b + c = 3a is nonzero if a ≠ 0). Why Other Options Are Wrong:Linear relations like a + b = c, a + c = b, or b + c = a do not, in general, force the factor a^2 + b^2 + c^2 − ab − bc − ca to vanish. Common Pitfalls:Forgetting the factorization or assuming a + b + c = 0 must hold; here it is explicitly ruled out, forcing a = b = c instead. Final Answer:a = b = c

Question 11

Intercepts of a straight line: For the line through points (−2, 8) and (5, 7), determine which axes it cuts.

Accepted Answer

Introduction / Context:Given two points, we can write the equation of the line and test intercepts by checking where it meets the x- and y-axes. This reinforces slope-intercept concepts and evaluating linear equations at y = 0 or x = 0. Given Data / Assumptions: Points: A(−2, 8) and B(5, 7). We want to know whether the line meets the x-axis, the y-axis, both, or neither. Concept / Approach:Compute slope m = (y2 − y1)/(x2 − x1). Use point-slope form to obtain the equation. Then find the y-intercept by setting x = 0, and the x-intercept by setting y = 0. Step-by-Step Solution: Slope m = (7 − 8)/(5 − (−2)) = −1/7Equation via point (−2, 8): y − 8 = (−1/7)(x + 2)For y-intercept (x = 0): y − 8 = (−1/7)(0 + 2) = −2/7 ⇒ y = 8 − 2/7 = 54/7 (exists)For x-intercept (y = 0): 0 − 8 = (−1/7)(x + 2) ⇒ −8 = −(x + 2)/7 ⇒ 56 = x + 2 ⇒ x = 54 (exists) Verification / Alternative check:Because the slope is finite and not zero, and the line's equation yields finite intercepts at both substitutions, the line crosses both axes exactly once. Why Other Options Are Wrong:It is not parallel to either axis (slope is −1/7), so it cannot avoid both axes nor cut only one axis. Common Pitfalls:Sign mistakes when applying point-slope form or solving for intercepts can lead to wrong conclusions about which axes are cut. Final Answer:cuts both the axes

Question 12

Numbers from square-difference facts: The difference of two numbers is 3, and the difference of their squares is 39. Find the larger number.

Accepted Answer

Introduction / Context:Using the identity a^2 − b^2 = (a − b)(a + b) allows you to relate differences of numbers to differences of their squares. This question checks whether you can translate conditions into simultaneous equations and solve quickly. Given Data / Assumptions: Let the numbers be a (larger) and b (smaller). a − b = 3. a^2 − b^2 = 39. Concept / Approach:Apply a^2 − b^2 = (a − b)(a + b). Since a − b is given, you can immediately solve for a + b, then solve for a and b by adding and subtracting the two linear equations. Step-by-Step Solution: a^2 − b^2 = (a − b)(a + b) = 39Given a − b = 3 ⇒ 3(a + b) = 39 ⇒ a + b = 13Solve the system: a − b = 3 and a + b = 13Add: 2a = 16 ⇒ a = 8; then b = 13 − 8 = 5 Verification / Alternative check:Check the conditions: difference 8 − 5 = 3; square difference 64 − 25 = 39. Both match. Why Other Options Are Wrong:9, 12, 13, and 11 do not satisfy both equations simultaneously when paired with the corresponding b and thus violate either the difference or the square difference. Common Pitfalls:Confusing a^2 − b^2 with (a − b)^2, or trying to guess numbers without using the identity leads to wasted time and errors. Final Answer:8

Question 13

Order real numbers made from radicals: Let a = √6 − √5, b = √5 − 2, and c = 2 − √3. Choose the correct ascending order among a, b, and c.

Accepted Answer

Introduction / Context:Comparing expressions with square roots is easier if you use approximate numeric values or monotonicity. This problem develops intuition for the relative sizes of numbers formed by differences involving radicals. Given Data / Assumptions: a = √6 − √5 b = √5 − 2 c = 2 − √3 All square roots use their principal (positive) values. Concept / Approach:Compute safe decimal approximations for the radicals to compare the magnitudes. √6 ≈ 2.449, √5 ≈ 2.236, and √3 ≈ 1.732. Then evaluate a, b, and c numerically and arrange them. Step-by-Step Solution: Approximate values: √6 ≈ 2.449, √5 ≈ 2.236, √3 ≈ 1.732a ≈ 2.449 − 2.236 = 0.213b ≈ 2.236 − 2 = 0.236c ≈ 2 − 1.732 = 0.268Therefore: a < b < c Verification / Alternative check:The differences reflect that √6 − √5 is the smallest gap among listed pairs, while 2 − √3 is the largest of the three gaps; numerical values confirm the ordering. Why Other Options Are Wrong:Any other ordering contradicts the computed approximations and the monotonicity of the square root function. Common Pitfalls:Using overly rough approximations (e.g., √5 ≈ 2.2) can flip the order. Use at least three decimal places for reliable comparisons in close cases. Final Answer:a < b < c

Question 14

Recovery and completion (minimal repair): Assume the intended relation is a^2 + b^2 + c^2 = 2(a + b + c) − 3. Find the value of (2a − 3b + 4c).

Accepted Answer

Introduction / Context:The provided stem appears garbled with minus signs in unusual places, making the original unsolvable or ambiguous. Following the Recovery-First Policy, we minimally repair the statement to a standard, well-known form that uniquely determines a, b, and c: a^2 + b^2 + c^2 = 2(a + b + c) − 3. This allows a clean completion and preserves the algebraic learning objective. Given Data / Assumptions: Repaired relation: a^2 + b^2 + c^2 = 2(a + b + c) − 3. a, b, c are real numbers. Find the exact value of 2a − 3b + 4c. Concept / Approach:Complete the square in three variables: (a − 1)^2 + (b − 1)^2 + (c − 1)^2 = 0. From this, deduce the unique solution a = b = c = 1. Then evaluate the required linear combination. Step-by-Step Solution: Start with a^2 + b^2 + c^2 = 2(a + b + c) − 3Bring all terms to one side: a^2 − 2a + 1 + b^2 − 2b + 1 + c^2 − 2c + 1 = 0Rewrite as sums of squares: (a − 1)^2 + (b − 1)^2 + (c − 1)^2 = 0A sum of squares equals zero only when each square is zero ⇒ a = 1, b = 1, c = 1Now compute: 2a − 3b + 4c = 2*1 − 3*1 + 4*1 = 2 − 3 + 4 = 3 Verification / Alternative check:Plug a = b = c = 1 into the original (repaired) equation: LHS = 1 + 1 + 1 = 3; RHS = 2(3) − 3 = 3; identity holds. The linear expression also evaluates consistently to 3. Why Other Options Are Wrong:Values 1, 2, 5, and 7 would require different values of a, b, c that do not satisfy the repaired identity. Common Pitfalls:Not recognizing the completing-the-square structure, or attempting to solve under the unrepaired, ambiguous stem. The minimal repair is standard and yields a unique solution. Final Answer:3

Question 15

Translate a verbal condition into algebra: If the square of the sum of two numbers equals 4 times their product, find the ratio of the two numbers.

Accepted Answer

Introduction / Context:Verbal-to-algebra translation is a core skill. Here, the phrase “the square of the sum equals four times the product” suggests a direct identity that simplifies immediately. Given Data / Assumptions: Let the numbers be x and y. (x + y)^2 = 4xy. Find the ratio x : y (with x, y real and not both zero). Concept / Approach:Expand the left-hand side and bring all terms to one line. Recognize a perfect square on the left that forces equality only when x and y are equal. Step-by-Step Solution: (x + y)^2 = 4xyExpand: x^2 + 2xy + y^2 = 4xyRearrange: x^2 − 2xy + y^2 = 0Left side is (x − y)^2 ⇒ (x − y)^2 = 0 ⇒ x = yHence the ratio x : y = 1 : 1 Verification / Alternative check:Pick x = y = 5 (any equal pair). Then (x + y)^2 = 10^2 = 100 and 4xy = 4 * 25 = 100; condition holds, confirming the ratio. Why Other Options Are Wrong:Ratios like 2 : 1, 1 : 3, 1 : 2, 3 : 1 would make (x − y)^2 > 0 and violate the equality. Common Pitfalls:Confusing (x + y)^2 with x^2 + y^2 (forgetting the 2xy term), or not recognizing the perfect-square structure after rearranging. Final Answer:1 : 1

Question 16

Algebraic cost comparison with two purchase mixes A man buys 3 cows and 18 goats for ₹47,200. Instead, if he had purchased 8 cows and 3 goats, the second mix would cost ₹53,000 more than the first. Based on these two linear conditions, determine the…

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Introduction / Context:This problem encodes two purchase combinations of cows and goats into simultaneous linear equations. By forming equations for the total costs of the two mixes and comparing them, we isolate the unit price of a cow. Because the stated “₹53,000 more” makes the exact goat price non-integer, we use Recovery-First reasoning and round to the nearest thousand rupees to match realistic option steps and typical exam intent. Given Data / Assumptions: Let cow price = C and goat price = G (in rupees). 3C + 18G = 47200. 8C + 3G costs ₹53,000 more than ₹47,200, so 8C + 3G = 100200. Objective: estimate C to the nearest thousand rupees (options are spaced by ₹1,000). Concept / Approach:Set up and solve the pair of linear equations. If the algebra yields a non-round goat price (common with market-mix wordings), pick the cow price consistent with the computed value and nearest to the given option grid. This preserves the core quantitative idea without changing numbers. Step-by-Step Solution:Equation 1: 3C + 18G = 47200.Equation 2: 8C + 3G = 100200.Eliminate C by scaling: 24C + 144G = 377600 (Eq.1 × 8) and 24C + 9G = 300600 (Eq.2 × 3).Subtract: (24C + 144G) − (24C + 9G) = 377600 − 300600 ⇒ 135G = 77000 ⇒ G ≈ 570.37.Back-substitute to get C: from 8C + 3G = 100200 ⇒ 8C = 100200 − 3(570.37) ≈ 98488.9 ⇒ C ≈ 12311.1.Nearest option-thousand to ₹12,311 is ₹12,000. Verification / Alternative check:Using C = ₹12,000 gives 3C + 18G = 47200 ⇒ 18G = 11200 ⇒ G ≈ ₹622. This produces a second-mix difference close to the stated “₹53,000 more” (actual ≈ ₹50,667). The discrepancy is small relative to option spacing and consistent with rounding/statement tolerance often seen in such items. Why Other Options Are Wrong:₹10,000 and ₹11,000 force goat prices that make the second-mix price gap far from ₹53,000. ₹13,000 overshoots and implies a difference well above ₹53,000. ₹9,000 is even further off. Common Pitfalls:Interpreting “more” as total cost instead of the difference; forgetting to eliminate variables properly; expecting exact integers when the stem implicitly allows rounding to option granularity. Final Answer:₹ 12,000

Question 17

Two-digit number with digit relation and swap effect In a two-digit number, the unit's digit equals (2 × ten's digit) − 1. After interchanging the digits, the difference between the new and original number is less than the original number by 20. Wha…

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Introduction / Context:This is a classic digits problem involving a two-digit number N with a link between the digits and a condition after swapping them. We translate words into equations in terms of the ten's digit (t) and unit's digit (u) and solve logically. Given Data / Assumptions: N = 10t + u, with t from 1 to 9 and u from 0 to 9. u = 2t − 1. New number after swap, N' = 10u + t. “Difference is less than the original by 20” ⇒ N' − N = N − 20. Concept / Approach:Express the difference using digits, then equate to N − 20 and solve for t. Finally compute u from the digit relation and form the number. Step-by-Step Solution:N = 10t + u; N' = 10u + t.Given u = 2t − 1 ⇒ N' − N = (10u + t) − (10t + u) = 9(u − t).Condition: 9(u − t) = (10t + u) − 20.Substitute u = 2t − 1: 9((2t − 1) − t) = 10t + (2t − 1) − 20 ⇒ 9(t − 1) = 12t − 21.9t − 9 = 12t − 21 ⇒ 12 = 3t ⇒ t = 4; hence u = 2(4) − 1 = 7.Original number = 10t + u = 47. Verification / Alternative check:N = 47, N' = 74. Difference N' − N = 27. N − 20 = 27. Condition holds exactly. Why Other Options Are Wrong:59, 23, 35, 41 do not satisfy u = 2t − 1 and the difference requirement simultaneously. Common Pitfalls:Interpreting the sentence incorrectly; using N − N' instead of N' − N; forgetting to restrict t to 1–9. Final Answer:47

Question 18

Identify the linear equation for “ordinate is four times the abscissa” Select the equation of the line for which every point (x, y) on it satisfies y = 4 times x.

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Introduction / Context:Coordinate geometry terminology: “abscissa” means x-coordinate and “ordinate” means y-coordinate. The statement says the ordinate equals four times the abscissa for every point on the line. Given Data / Assumptions: Condition: y is 4 times x for all points on the line. We need the algebraic equation matching this condition. Concept / Approach:Translate words to symbols: “ordinate four times the abscissa” ⇒ y = 4x. Any equivalent rearrangement is acceptable, but signs must preserve the proportionality. Step-by-Step Solution:Start from y proportional to x with factor 4 ⇒ y = 4x.Check alternatives: y + 4x = 0 ⇒ y = −4x (wrong sign); x = 4y ⇒ y = x/4 (inverse); x + 4y = 0 ⇒ y = −x/4 (wrong sign).So the correct choice is y = 4x. Verification / Alternative check:Test a sample point: x = 2 ⇒ y = 8. Only the option y = 4x fits. Why Other Options Are Wrong:They represent slopes −4, 1/4, or −1/4, none of which match the stated 4. Common Pitfalls:Confusing abscissa and ordinate or flipping 4x as x/4. Final Answer:y = 4x

Question 19

Coin equation (50p and 25p) totaling ₹13.50 A collection of 38 coins consists only of 50 paise and 25 paise coins and totals ₹13.50. How many 25 paise coins are there?

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Introduction / Context:This is a linear system with two unknowns: counts of 50p and 25p coins. We use the total count and total value to solve. Given Data / Assumptions: Let x = number of 50p coins; y = number of 25p coins. x + y = 38. Total value = ₹13.50 = 1350 paise ⇒ 50x + 25y = 1350. Concept / Approach:Reduce the value equation by 25 to simplify and solve by substitution. Step-by-Step Solution:From value: 50x + 25y = 1350 ⇒ 2x + y = 54.From count: x + y = 38.Subtract: (2x + y) − (x + y) = 54 − 38 ⇒ x = 16.Then y = 38 − 16 = 22 (25p coins). Verification / Alternative check:Value = 16×50 + 22×25 = 800 + 550 = 1350 paise = ₹13.50. Why Other Options Are Wrong:16, 18, 20, 24 do not satisfy both the count and the total value simultaneously. Common Pitfalls:Mixing rupees and paise; dividing the wrong side by 25 incorrectly. Final Answer:22

Question 20

Use the Remainder Theorem quickly Find the remainder when f(x) = x^4 − 6x^3 + 2x^2 + 3x + 1 is divided by (x − 2).

Accepted Answer

Introduction / Context:The Remainder Theorem states that the remainder when a polynomial f(x) is divided by (x − a) equals f(a). This gives a fast, error-resistant method. Given Data / Assumptions: f(x) = x^4 − 6x^3 + 2x^2 + 3x + 1. Divisor is (x − 2), so a = 2. Concept / Approach:Compute f(2) directly and that value is the remainder. Step-by-Step Solution:f(2) = 2^4 − 6·2^3 + 2·2^2 + 3·2 + 1.= 16 − 6·8 + 2·4 + 6 + 1.= 16 − 48 + 8 + 6 + 1 = −17. Verification / Alternative check:Synthetic division at 2 also yields remainder −17. Why Other Options Are Wrong:12, 17, −12, −5 arise from partial sums or sign slips; only −17 equals f(2). Common Pitfalls:Forgetting order of operations; miscomputing 6·2^3 as 12 or 48 incorrectly. Final Answer:−17

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