◆ QA · Algebra

Quadratic Equations , formulas + CAT PYQs

Focused Algebra kit. The full chapter formula sheet (with explanations & basic examples) is tucked below; every CAT PYQ for Quadratic Equations is here.

29CAT PYQs
Algebrachapter
Formulas PYQs (29)↩ Full Algebra chapter

Algebra, formula sheet

Show the full Algebra formula sheet (explanations + basic examples)
1Polynomials & zeroes
  • Plain English: a polynomial is just a sum of x-powers; its "zeroes" are the x-values that make it equal 0.
  • A polynomial of degree n: aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₁x + a₀
  • k is a zero of p(x) if p(k) = 0. Zeroes are the x-coordinates where y = p(x) cuts the x-axis.
  • Max zeroes = degree: linear → 1, quadratic → 2, cubic → 3, degree n → n.
  • e.g. p(x) = x² − 9 has degree 2, so at most 2 zeroes: x = 3 and x = −3.
2Sum & product of roots
  • Plain English: you can read the sum and product of the roots straight off the coefficients, no need to solve.
  • Quadratic ax²+bx+c: α + β = −b/a, αβ = c/a
  • Cubic ax³+bx²+cx+d: α+β+γ = −b/a, αβ+βγ+γα = c/a, αβγ = −d/a
  • Build the equation: x² − (sum)x + (product) = 0.
  • e.g. x² − 5x + 6 = 0: sum = 5, product = 6 ⇒ roots 2 and 3 (2+3=5, 2×3=6).
3Discriminant & nature of roots
  • Plain English: the discriminant D is a single number that tells you how many real roots a quadratic has, before solving.
  • For ax²+bx+c (a≠0): D = b² − 4ac
  • D > 0 → two distinct real roots; D = 0 → equal real roots; D < 0 → no real roots (complex).
  • D a perfect square (a,b,c rational) → roots are rational.
  • Roots: x = (−b ± √D)/2a
  • e.g. x² + x + 1: D = 1 − 4 = −3 < 0 ⇒ no real roots.
4Sum of squares of roots (trick)
  • Plain English: you can get α²+β² from the sum and product alone, never solve for the roots first.
  • α² + β² = (α+β)² − 2αβ
  • To minimise a sum-of-squares-of-roots expression in a parameter, complete the square, minimum is at the vertex.
  • e.g. if α+β = 3 and αβ = 2, then α²+β² = 9 − 4 = 5.
5Algebraic identities
  • Plain English: memorised expand/factor templates that turn ugly expressions into products (or vice-versa) instantly.
  • (a±b)² = a² ± 2ab + b²
  • a² − b² = (a+b)(a−b)
  • (a+b+c)² = a²+b²+c² + 2(ab+bc+ca)
  • a³ ± b³ = (a±b)(a² ∓ ab + b²)
  • a³+b³+c³ − 3abc = (a+b+c)(a²+b²+c² − ab−bc−ca)
  • e.g. 97×103 = (100−3)(100+3) = 100² − 3² = 9991.
6Linear equations in two variables
  • Plain English: comparing the coefficient ratios tells you whether two lines cross once, never, or lie on top of each other.
  • a₁x+b₁y+c₁=0 and a₂x+b₂y+c₂=0.
  • Unique solution (intersecting): a₁/a₂ ≠ b₁/b₂
  • No solution (parallel): a₁/a₂ = b₁/b₂ ≠ c₁/c₂
  • Infinite solutions (coincident): a₁/a₂ = b₁/b₂ = c₁/c₂
  • e.g. x+y=2 and 2x+2y=5: ratios 1/2 = 1/2 ≠ 2/5 ⇒ parallel, no solution.
7Inequalities, basic rules
  • Plain English: inequalities behave like equations, except multiplying or dividing by a negative reverses the arrow.
  • Adding/subtracting keeps direction; multiplying by a negative flips the sign.
  • If X > Y > 0 then 1/X < 1/Y.
  • For x > 0: x + 1/x ≥ 2 (equality at x = 1).
  • e.g. −2x > 6 ⇒ divide by −2 and flip ⇒ x < −3.
8Quadratic inequalities
  • Plain English: factor it, then "< 0" means between the roots and "> 0" means outside the roots.
  • (x−m)(x−n) < 0, n > m ⇒ m < x < n (between the roots).
  • (x−m)(x−n) > 0 ⇒ x < m or x > n (outside the roots).
  • Sign-of-product / wavy-curve method handles higher degree.
  • e.g. x² − 5x + 6 < 0 ⇒ (x−2)(x−3) < 0 ⇒ 2 < x < 3.
9Modulus (absolute value)
  • Plain English: |x| is the distance of x from 0, so it strips the sign and is never negative.
  • |x| = max(x, −x); −|x| ≤ x ≤ |x|.
  • |a+b| ≤ |a|+|b| and |a|−|b| ≤ |a−b|; |ab| = |a||b|.
  • |x| ≤ k ⇒ −k ≤ x ≤ k. |x| ≥ k ⇒ x ≥ k or x ≤ −k.
  • |f| + |g| = |f+g| only when f, g have the same sign.
  • e.g. |x| ≤ 3 ⇒ −3 ≤ x ≤ 3; |x − 4| = 2 ⇒ x = 6 or x = 2.
10AM-GM-HM inequality
  • Plain English: for positive numbers the plain average is always ≥ the geometric average, the go-to tool for "find the minimum".
  • For positive reals: AM ≥ GM ≥ HM, equality when all equal.
  • Two numbers: AM = (a+b)/2, GM = √(ab), HM = 2ab/(a+b).
  • AM × HM = GM²
  • If a₁a₂…aₙ = 1 then a₁+a₂+…+aₙ ≥ n.
  • e.g. for a = 2, b = 8: AM = 5 ≥ GM = √16 = 4. ✓
11Maxima & minima of a quadratic
  • Plain English: a parabola's turning point is at x = −b/2a; that's where the min (opens up) or max (opens down) lives.
  • ax²+bx+c: vertex at x = −b/2a; extreme value = (4ac − b²)/4a = −D/4a.
  • a > 0 → opens up → minimum; a < 0 → opens down → maximum.
  • min/max of max-of-two / min-of-two lines occurs where the two graphs intersect.
  • e.g. x² − 6x + 5: vertex at x = 3, minimum value = 9 − 18 + 5 = −4.
12Functions, domain, range, even/odd
  • Plain English: domain is what you may feed in, range is what comes out; even/odd describe the graph's symmetry.
  • Domain = allowed inputs; range = resulting outputs.
  • Even: f(−x) = f(x) (graph symmetric about y-axis), e.g. x², |x|.
  • Odd: f(−x) = −f(x) (symmetric about origin), e.g. x³, 1/x.
  • Inverse exists only if f is one-to-one.
  • e.g. f(x) = x³ is odd: f(−2) = −8 = −f(2). ✓
13Functional equations
  • Plain English: the form of a functional rule reveals the function, "turns + into ×" means exponential, etc.
  • f(x+y) = f(x)·f(y) ⇒ exponential type, f(x) = aˣ.
  • f(xy) = f(x)·f(y) ⇒ power/multiplicative; f(1) = 1.
  • If f(a+x) = f(a−x), the graph is symmetric about x = a; roots pair around a (sum of 4 roots = 4a).
  • e.g. f(x+y) = f(x)f(y) with f(1) = 3 ⇒ f(2) = f(1)² = 9.
14Graph shifting
  • Plain English: changes outside f() move the graph vertically; changes inside f() move it horizontally (and oppositely).
  • f(x)+c → shift up c; f(x)−c → shift down c.
  • f(x+c) → shift left c; f(x−c) → shift right c.
  • −f(x) → reflect in x-axis; f(−x) → reflect in y-axis.
  • e.g. y = (x−2)² is y = x² shifted 2 units right.
15Logarithm, definition
  • Plain English: log_b x just asks "what power of b gives x?", it's the inverse of raising to a power.
  • y = log_b x ⇔ x = bʸ (b > 0, b ≠ 1, x > 0).
  • log_a a = 1; log_a 1 = 0; a^(log_a m) = m.
  • e.g. log₂8 = 3 because 2³ = 8.
16Logarithm laws
  • Plain English: logs turn multiplication into addition, division into subtraction, and powers into multipliers.
  • log_a(xy) = log_a x + log_a y
  • log_a(x/y) = log_a x − log_a y
  • log_a(xᵐ) = m·log_a x
  • log_(aⁿ)(xᵐ) = (m/n)·log_a x
  • Change of base: log_a x = (log x)/(log a); log_a x = 1/log_x a
  • e.g. log₂40 = log₂(8×5) = log₂8 + log₂5 = 3 + log₂5.
17Indices (laws of exponents)
  • Plain English: same base, add exponents when multiplying, subtract when dividing, multiply when raising a power to a power.
  • pᵐ·pⁿ = pᵐ⁺ⁿ; pᵐ/pⁿ = pᵐ⁻ⁿ; (pᵐ)ⁿ = pᵐⁿ
  • pⁿ·qⁿ = (pq)ⁿ; (p/q)ⁿ = pⁿ/qⁿ
  • p⁻ⁿ = 1/pⁿ; p⁰ = 1; p^(1/n) = ⁿ√p
  • e.g. 2³·2⁴ = 2⁷ = 128; 8^(2/3) = (∛8)² = 2² = 4.
18Surds & rationalisation
  • Plain English: a surd is an unresolved root like √2; "rationalising" clears it from a denominator using the conjugate.
  • √(ab) = √a·√b; √(a/b) = √a/√b.
  • Rationalise a/(b+√c) by multiplying top & bottom by the conjugate (b−√c).
  • If a+√b is a root of a rational quadratic, so is its conjugate a−√b.
  • e.g. 1/(√3 − 1) × (√3 + 1)/(√3 + 1) = (√3 + 1)/2.
19Arithmetic Progression (AP)
  • Plain English: an AP adds the same step d each time; its sum is just "how many terms × the average of first and last".
  • Constant difference d. nth term: Tₙ = a + (n−1)d
  • Sum: Sₙ = n/2 · [2a + (n−1)d] = n/2 · (first + last)
  • Arithmetic mean of a, b: A = (a+b)/2. Middle term = average of an odd count of AP terms.
  • e.g. 2, 5, 8, …: T₄ = 2 + 3×3 = 11; sum of first 4 = 4/2·(2+11) = 26.
20Geometric Progression (GP)
  • Plain English: a GP multiplies by the same ratio r each time; if |r| < 1 the infinite sum settles to a finite value.
  • Constant ratio r. nth term: Tₙ = a·rⁿ⁻¹
  • Sum: Sₙ = a(rⁿ − 1)/(r − 1), r ≠ 1.
  • Infinite sum (|r| < 1): S∞ = a/(1 − r)
  • Geometric mean: G = √(ab).
  • e.g. 1 + ½ + ¼ + … = 1/(1 − ½) = 2.
21Harmonic Progression (HP)
  • Plain English: an HP is just an AP flipped, take reciprocals and you're back to a normal AP.
  • a, b, c… in HP ⇔ 1/a, 1/b, 1/c… in AP.
  • Harmonic mean of a, b: H = 2ab/(a+b)
  • nth term of HP = 1/(nth term of the corresponding AP).
  • e.g. 1, ½, ⅓, ¼ is an HP (reciprocals 1, 2, 3, 4 form an AP).
22Standard summation formulas
  • Plain English: ready-made closed forms for adding up the first n numbers, their squares, and their cubes.
  • Σn = n(n+1)/2
  • Σn² = n(n+1)(2n+1)/6
  • Σn³ = [n(n+1)/2]²
  • Telescoping: 1/(k·(k+1)) = 1/k − 1/(k+1).
  • e.g. 1 + 2 + … + 10 = 10×11/2 = 55.
23Common terms of two APs
  • Plain English: numbers shared by two APs themselves form an AP whose step is the LCM of the two steps.
  • Common terms of two APs form a new AP with common difference = LCM of the two differences.
  • Find the first common term, then count multiples of the LCM up to the smaller upper limit.
  • e.g. 2,5,8,… and 3,7,11,…: first common term 11, new step = LCM(3,4) = 12 ⇒ 11, 23, 35, …
24Recurrence & tₙ from Sₙ
  • Plain English: if you know the running total Sₙ, each term is just this total minus the previous total.
  • If Sₙ given: aₙ = Sₙ − Sₙ₋₁ (and a₁ = S₁).
  • Alternating-sum sequences: subtract consecutive defining equations to isolate a term.
  • e.g. Sₙ = n² ⇒ a₅ = S₅ − S₄ = 25 − 16 = 9.
25Integer / Diophantine solutions
  • Plain English: once you spot one whole-number solution, all the rest come by stepping x and y in fixed jumps.
  • ax + by = c with one integer solution (x₀, y₀): all others are x₀ + (b/g)t, y₀ − (a/g)t, where g = gcd(a,b).
  • Bound the count using the given ranges on x and y.
  • e.g. 2x + 3y = 12: (x,y) = (3,2) works; next is (0,4), then (6,0), x jumps by 3, y by 2.
26When does Aᴮ = 1?
  • Plain English: a power equals 1 in exactly three situations, check all three or you'll miss cases.
  • Base = 1 (any exponent), or
  • Exponent = 0 (base ≠ 0), or
  • Base = −1 with an even exponent.
  • e.g. (−1)⁴ = 1 (base −1, even power); 7⁰ = 1 (zero power); 1⁹⁹ = 1 (base 1).
27Three terms in AP / GP
  • Plain English: centering three terms on a middle value makes their sum (AP) or product (GP) collapse to one symbol.
  • Three in AP: take a−d, a, a+d (their sum = 3a).
  • Three in GP: take a/r, a, ar (product = a³).
  • Three consecutive integers as roots: n−1, n, n+1.
  • e.g. three numbers in AP summing to 18 ⇒ middle = 6, so 6−d, 6, 6+d.
28|x − a| as distance (modulus sums)
  • Plain English: read |x−a| as "distance from a", and sums of such distances are smallest when x sits among the points.
  • |x−a| = distance of x from a on the number line.
  • |x−p|+|x−q| is minimised for any x between p and q; minimum value = |p−q|.
  • |x−p| = |x−q| at the midpoint x = (p+q)/2.
  • e.g. |x−2| + |x−7| ≥ 5, achieved for any x in [2, 7].
29Sum of squares identity trick
  • Plain English: squares can't be negative, so if a bunch of squares add to 0 every single one must be 0.
  • If a sum of squares equals 0, each square = 0: e.g. (x−2y)² + (y−z)² = 0 ⇒ x = 2y and y = z.
  • Group given expressions into perfect squares to pin exact values.
  • e.g. (a−3)² + (b+1)² = 0 forces a = 3 and b = −1.
30Cauchy / vector identity
  • Plain English: this identity links two "sum-of-squares" products to two cross-terms, handy when three of the four pieces are given.
  • (a²+b²)(x²+y²) = (ax+by)² + (ay−bx)².
  • Useful when given a²+b², x²+y² and ax+by to find ay−bx.
  • e.g. (1²+2²)(3²+4²) = 5·25 = 125 = 11² + 2² = (1·3+2·4)² + (1·4−2·3)².
29 CAT questions

Quadratic Equations, CAT PYQs

Quadratic Equations

ModerateCAT 1996

Given the quadratic equation x² − (A − 3)x − (A − 2), for what value of A will the sum of the squares of the roots be zero?

  • (1) − 2
  • (2) 3
  • (3) 6
  • (4) None of these
Show solution
(4) None of these. α+β = A−3, αβ = −(A−2). α²+β² = (A−3)² + 2(A−2) = A² − 4A + 5 = 0 has no real solution for A. So none of the choices works.
ModerateCAT 1997

If the roots x₁ and x₂ of the quadratic equation x² − 2x + c = 0 also satisfy the equation 7x₂ − 4x₁ = 47, then which of the following is true?

  • (1) c = − 15
  • (2) x₁ = − 5, x₂ = 3
  • (3) x₁ = 4.5, x₂ = − 2.5
  • (4) None of these
Show solution
(1) c = −15. Sum of roots x₁+x₂ = 2. Solving with 7x₂ − 4x₁ = 47 gives x₁ = −3, x₂ = 5. Product of roots = (−3)(5) = −15 = c.
ModerateCAT 2003

Let p and q be the roots of the quadratic equation x² − (α − 2)x − α − 1 = 0. What is the minimum possible value of p² + q²?

  • (1) 0
  • (2) 3
  • (3) 4
  • (4) 5
Show solution
(4) 5. p+q = α−2, pq = −α−1. p²+q² = (p+q)² − 2pq = (α−2)² + 2(α+1) = α² − 2α + 5 = (α−1)² + 5. Minimum is 5 (at α = 1).
ModerateCAT 2008

If the roots of the equation x³ − ax² + bx − c = 0 are three consecutive integers, then what is the smallest possible value of b?

  • (1) − 1/√3
  • (2) − 1
  • (3) 0
  • (4) 1
Show solution
(2) −1. Roots n−1, n, n+1. b = sum of products in pairs = (n−1)n + n(n+1) + (n−1)(n+1) = 3n² − 1. Minimum at n = 0 gives b = −1.
ModerateCAT 2017

The minimum possible value of the sum of the squares of the roots of the equation x² + (a + 3)x − (a + 5) = 0 is

  • (1) 1
  • (2) 2
  • (3) 3
  • (4) 4
Show solution
(3) 3. Sum = −(a+3), product = −(a+5). Sum of squares = (a+3)² + 2(a+5) = a² + 8a + 19 = (a+4)² + 3. Minimum is 3.
ModerateCAT 2018TITA

If a and b are integers such that 2x² − ax + 2 > 0 and x² − bx + 8 ≥ 0 for all real numbers x, then the largest possible value of 2a − 6b is

Show solution
36. 2x²−ax+2 > 0 ⇒ a² < 16 ⇒ a ≤ 3. x²−bx+8 ≥ 0 ⇒ b² ≤ 32 ⇒ −5 ≤ b ≤ 5. Max of 2a − 6b = 2(3) − 6(−5) = 36.
HardCAT 2019

The product of the distinct roots of |x² − x − 6| = x + 2 is

  • (1) − 16
  • (2) − 4
  • (3) − 24
  • (4) − 8
Show solution
(1) −16. Case x²−x−6 = x+2 ⇒ x²−2x−8 = 0 ⇒ x = 4, −2. Case x²−x−6 = −(x+2) ⇒ x² = 4 ⇒ x = ±2. Distinct roots: {4, −2, 2}. Product = 4 × (−2) × 2 = −16.
ModerateCAT 2019

The quadratic equation x² + bx + c = 0 has two roots 4a and 3a, where a is an integer. Which of the following is a possible value of b² + c?

  • (1) 3721
  • (2) 361
  • (3) 427
  • (4) 549
Show solution
(4) 549. Sum = 7a = −b ⇒ b² = 49a². Product = 12a² = c. b² + c = 61a². Among the options only 549 = 61 × 9 = 61 × 3². So b² + c = 549.
ModerateCAT 2020TITA

The number of distinct real roots of the equation (x + 1/x)² − 3(x + 1/x) + 2 = 0 equals:

Show solution
1. Let y = x + 1/x: y² − 3y + 2 = 0 ⇒ y = 1 or 2. But |x + 1/x| ≥ 2 for real x, so y = 1 is impossible; only y = 2 ⇒ x = 1. One distinct real root.
ModerateCAT 2020

Let f(x) = x² + ax + b and g(x) = f(x + 1) − f(x − 1). If f(x) ≥ 0 for all real x, and g(20) = 72, then the smallest possible value of b is:

  • (1) 1
  • (2) 4
  • (3) 0
  • (4) 16
Show solution
(2) 4. g(20) = f(21) − f(19) = 72 ⇒ 4a + 80 = 72... solving gives a = −4. f(x) ≥ 0 needs D ≤ 0: a² − 4b ≤ 0 ⇒ 16 ≤ 4b ⇒ b ≥ 4. Smallest b = 4.
HardCAT 2020

Let m and n be positive integers, If x² + mx + 2n = 0 and x² + 2nx + m = 0 have real roots, then the smallest possible value of m + n is:

  • (1) 7
  • (2) 6
  • (3) 8
  • (4) 5
Show solution
(2) 6. Real roots need m² ≥ 8n and (2n)² ≥ 4m ⇒ n² ≥ m. n = 1 ⇒ m ≤ 1 but m² ≥ 8 fails. n = 2 ⇒ m ≤ 4 and m² ≥ 16 ⇒ m = 4. So m + n = 6.
HardCAT 2021 · Slot 1

If r is a constant such that |x² − 4x − 13| = r has exactly three distinct real roots, then the value of r is

  • (1) 15
  • (2) 21
  • (3) 18
  • (4) 17
Show solution
(4) 17. Write x²−4x−13 = (x−2)² − 17. The graph of |(x−2)² − 17| touches zero (the vertex value) at r = 17, which makes exactly three intersection points with y = r. So r = 17.
HardCAT 2022 · Slot 3

Suppose k is any integer such that the equation 2x² + kx + 5 = 0 has no real roots and the equation x² + (k − 5)x + 1 = 0 has two distinct real roots for x. Then, the number of possible values of k is:

  • (1) 7
  • (2) 9
  • (3) 8
  • (4) 13
Show solution
(2) 9. No real roots: k² < 40 ⇒ −6.3 < k < 6.3. Two distinct real roots: (k−5)² > 4 ⇒ k < 3 or k > 7. Intersecting integer values: k = −6,…,2, i.e. 9 values.
HardCAT 2023 · Slot 1TITA

Let α and β be the two distinct roots of the equation 2x² − 6x + k = 0, such that (α + β) and αβ are the distinct roots of the equation x² + px + p = 0. Then the value of 8(k − p) is

Show solution
6. α+β = 3 and αβ = k/2. These are roots of x²+px+p = 0, so their sum 3 + k/2 = −p and product 3·(k/2) = p. Solving gives p and k, and 8(k − p) = 6.
HardCAT 2023 · Slot 3TITA

A quadratic equation x² + bx + c = 0 has two real roots. If the difference between the reciprocals of the roots is 1/3, and the sum of the reciprocals of the squares of the roots is 5/9, then the largest possible value of (b + c) is

Show solution
9. 1/α − 1/β = (β−α)/(αβ) and 1/α² + 1/β² are expressed via b = −(α+β) and c = αβ. Solving the two conditions and taking the larger valid case gives b + c = 9.

CAT 2024 & 2025, recent

ModerateCAT 2024 · Slot 1

Let x, y, and z be real numbers satisfying 4(x² + y² + z²) = a, and 4(x − y − z) = 3 + a. Then a equals

  • (A) 3
  • (B) 1⅓
  • (C) 1
  • (D) 4
Show solution
(A) 3. Combining gives (2x−1)² + (2y+1)² + (2z+1)² = 0, so x = ½, y = z = −½. Then a = 4(¾) = 3.
ModerateCAT 2024 · Slot 2

The roots α, β of the equation 3x² + λx − 1 = 0, satisfy 1/α² + 1/β² = 15. The value of (α³ + β³)², is

  • (A) 1
  • (B) 4
  • (C) 9
  • (D) 16
Show solution
(B) 4. αβ = −1/3, so (αβ)² = 1/9; 1/α² + 1/β² = (α²+β²)/(αβ)² = 15 ⇒ α²+β² = 5/3. Then (α+β)² = 1, and α³+β³ = (α+β)[(α+β)²−3αβ] = ±2 ⇒ square = 4.
ModerateCAT 2024 · Slot 2 TITA

If x and y are real numbers such that 4x² + 4y² − 4xy − 6y + 3 = 0, then the value of (4x + 5y) is

Show solution
7. Rewrite as (2x − y)² + 3(y − 1)² = 0 ⇒ y = 1 and 2x − y = 0 ⇒ x = ½. So 4x + 5y = 2 + 5 = 7.
ModerateCAT 2025 · Slot 1 TITA

The number of non-negative integer values of k for which the quadratic equation x² − 5x + k = 0 has only integer roots, is

Show solution
3. Integer roots p, q with p + q = 5, pq = k ≥ 0. Pairs: (0,5)→k=0, (1,4)→k=4, (2,3)→k=6. Three values.
HardCAT 2025 · Slot 1

A value of c for which the minimum value of f(x) = x² − 4cx + 8c is greater than the maximum value of g(x) = −x² + 3cx − 2c, is

  • (A) 2
  • (B) 1/2
  • (C) −1/2
  • (D) −2
Show solution
(B) 1/2. min f = 8c − 4c² (at x = 2c); max g = 9c²/4 − 2c (at x = 3c/2). Require 8c − 4c² > 9c²/4 − 2c → 5c² − 8c < 0 → 0 < c < 8/5. Only c = 1/2 fits the options.
HardCAT 2025 · Slot 1

Let 3 ≤ x ≤ 6 and [x²] = [x]², where [x] is the greatest integer not exceeding x. If set S represents all feasible values of x, then a possible subset of S is

  • (A) (3, √10) ∪ [5, √26) ∪ {6}
  • (B) [3, √10] ∪ [5, √26]
  • (C) [3, √10] ∪ [4, √17] ∪ {6}
  • (D) (4, √18) ∪ [5, √27) ∪ {6}
Show solution
(A). [x] = n needs n² ≤ x² < n²+1, i.e. n ≤ x < √(n²+1). So S = [3,√10) ∪ [4,√17) ∪ [5,√26) ∪ {6}. The subset in option (A), (3,√10) ∪ [5,√26) ∪ {6}, lies entirely within S.
HardCAT 2025 · Slot 2

If log₆₄ x² + log₈ √y + 3·log₅₁₂(√y·z) = 4, where x, y and z are positive real numbers, then the minimum possible value of (x + y + z) is

  • (A) 48
  • (B) 36
  • (C) 24
  • (D) 96
Show solution
(A) 48. Convert to base 2; the equation reduces to xyz = constant; by AM-GM the minimum of x+y+z is 48.
HardCAT 2025 · Slot 2

If 9^(x²+2x−3) − 4(3^(x²+2x−2)) + 27 = 0 then the product of all possible values of x is

  • (A) 30
  • (B) 20
  • (C) 5
  • (D) 15
Show solution
(B) 20. Let t = 3^(x²+2x−3). Equation becomes t² − 12t + 27 = 0 ⇒ t = 3 or 9, giving x²+2x = 4 (product of roots −4) and x²+2x = 5 (product −5). Product of all four roots = (−4)(−5) = 20.
HardCAT 2025 · Slot 2

The set of all real values of x for which (x² − |x + 9| + x) > 0, is

  • (A) (−∞, −3) ∪ (3, ∞)
  • (B) (−∞, −9) ∪ (3, ∞)
  • (C) (−9, −3) ∪ (3, ∞)
  • (D) (−∞, −9) ∪ (9, ∞)
Show solution
(A) (−∞, −3) ∪ (3, ∞). Case x ≥ −9: x² − 9 > 0 → x < −3 or x > 3. Case x < −9: x² + 2x + 9 > 0 always. Combined: (−∞, −3) ∪ (3, ∞).
ModerateCAT 2025 · Slot 2

The equations 3x² − 5x + p = 0 and 2x² − 2x + q = 0 have one common root. The sum of the other roots of these equations is

  • (A) 8/3 − p + 3q/2
  • (B) 2/3 − p + 3q/2
  • (C) 8/3 + p + q/3
  • (D) 2/3 − 2p + 2q/3
Show solution
(A) 8/3 − p + 3q/2. Sum of all roots = 5/3 + 1; subtract the common root (expressed via p, q) to get the sum of the other two as 8/3 − p + 3q/2.
HardCAT 2025 · Slot 3

If (x² + 1/x²) = 25 and x > 0, then the value of (x⁷ + 1/x⁷) is

  • (A) 44853√3
  • (B) 44856√3
  • (C) 44859√3
  • (D) 44850√3
Show solution
(A) 44853√3. x + 1/x = √27 = 3√3 (since (x+1/x)² = 27). Use recurrences up to the 7th power: x⁷ + 1/x⁷ = 44853√3.
HardCAT 2025 · Slot 3

The sum of all possible real values of x for which log₍ₓ₋₃₎(x² − 9) = log₍ₓ₋₃₎(x + 1) + 2, is

  • (A) −3
  • (B) √33
  • (C) 3
  • (D) (3 + √33)/2
Show solution
(D) (3 + √33)/2. RHS = log₍ₓ₋₃₎[(x+1)(x−3)²]. So x²−9 = (x+1)(x−3)² with domain x > 3, x ≠ 4. The valid root(s) sum to (3 + √33)/2.
HardCAT 2025 · Slot 3

If f(x) = (x² + 3x)(x² + 3x + 2) then the sum of all real roots of the equation √(f(x) + 1) = 9701, is

  • (A) −6
  • (B) 6
  • (C) 3
  • (D) −3
Show solution
(D) −3. Let t = x² + 3x. Then f + 1 = (t+1)². So |t+1| = 9701 → t = 9700 or t = −9702. x² + 3x = 9700 has roots summing to −3; the other case has no real roots. Sum = −3.
HardCAT 2025 · Slot 3

For real values of x, the range of the function f(x) = (2x − 3)/(2x² + 4x − 6) is

  • (A) (−∞, 1/8] ∪ [1, ∞)
  • (B) (−∞, 1/4] ∪ [1, ∞)
  • (C) (−∞, 1/8] ∪ [1/2, ∞)
  • (D) (−∞, 1/4] ∪ [1/2, ∞)
Show solution
(C) (−∞, 1/8] ∪ [1/2, ∞). Set y = f(x), cross-multiply to a quadratic in x, require discriminant ≥ 0: (8y − 1)(2y − 1) ≥ 0, with critical points y = 1/8 and y = 1/2. Range = (−∞, 1/8] ∪ [1/2, ∞).