◆ QA · Algebra

Surds & Indices , formulas + CAT PYQs

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

20CAT PYQs
Algebrachapter
Formulas PYQs (20)↩ 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.
20 CAT questions

Surds & Indices, CAT PYQs

Surds & Indices

ModerateCAT 2005

Which among 2^(1/2), 3^(1/3), 4^(1/4), 6^(1/6) and 12^(1/12) is the largest?

  • (1) 2^(1/2)
  • (2) 3^(1/3)
  • (3) 4^(1/4)
  • (4) 6^(1/6)
Show solution
(2) 3^(1/3). Raise all to the 12th power: 2⁶ = 64, 3⁴ = 81, 4³ = 64, 6² = 36, 12¹ = 12. Largest is 81 ⇒ 3^(1/3).
HardCAT 2017

If x + 1 = x² and x > 0, then 2x⁴ is

  • (1) 6 + 4√5
  • (2) 3 + 3√5
  • (3) 5 + 3√5
  • (4) 7 + 3√5
Show solution
(4) 7 + 3√5. From x² = x + 1, x⁴ = (x²)² = (x+1)² = x² + 2x + 1 = (x+1) + 2x + 1 = 3x + 2, so 2x⁴ = 6x + 4. With x > 0, x = (1+√5)/2, so 6x + 4 = 3(1+√5) + 4 = 7 + 3√5.
ModerateCAT 2017

If 9^(2x−1) − 81^(x−1) = 1944, then x is:

  • (1) 3
  • (2) 9/4
  • (3) 4/9
  • (4) 1/3
Show solution
(2) 9/4. 9^(2x−1) − 9^(2x−2) = 1944 ⇒ 9^(2x−2)(9 − 1) = 1944 ⇒ 9^(2x−2) = 243 = 9^(5/2) ⇒ 2x−2 = 5/2 ⇒ x = 9/4.
ModerateCAT 2017

If 9^(x−½) − 2^(2x−2) = 4^x − 3^(2x−3), then x is

  • (1) 3/2
  • (2) 2/5
  • (3) 3/4
  • (4) 4/9
Show solution
(1) 3/2. Rearrange: 9^(x−½) + 3^(2x−3) = 4^x + 2^(2x−2), i.e. 3^(2x−3)(3² + 1) = 2^(2x−2)(2² + 1) ⇒ 3^(2x−3)·10 = 2^(2x−2)·5 ⇒ 3^(2x−3) = 2^(2x−2)·5/10 = 2^(2x−2)/2 = 2^(2x−3) ⇒ (3/2)^(2x−3) = 1 ⇒ 2x − 3 = 0 ⇒ x = 3/2.
HardCAT 2019

The real root of the equation 2⁶ˣ + 2³ˣ⁺² − 21 = 0 is:

  • (1) log₂9
  • (2) (log₂3)/3
  • (3) log₂27
  • (4) (log₂7)/3
Show solution
(2) (log₂3)/3. Let t = 2³ˣ: t² + 4t − 21 = 0 ⇒ (t+7)(t−3) = 0 ⇒ t = 3 (reject −7). So 2³ˣ = 3 ⇒ 3x = log₂3 ⇒ x = (log₂3)/3.
HardCAT 2019

If 5ˣ − 3ʸ = 13438 and 5^(x−1) + 3^(y+1) = 9686, then x + y equals:

Show solution
13. Take the 2nd equation 5^(x−1) + 3^(y+1) = 9686. The last digit of 5^(x−1) is always 5, so 3^(y+1) must end in 6, i.e. it is of the form 3^(4k) (the unit-digit cycle of 3 is 3, 9, 7, 1, so 3^(4k) ends in 1, and 3^(y+1) must make the unit digits sum to 6). Checking: 3⁴ = 81 and 3⁸ = 6561; 9686 − 6561 = 3125 = 5⁵, so 5^(x−1) = 5⁵ ⇒ x = 6 and 3^(y+1) = 3⁸ ⇒ y = 7. Hence x + y = 13.
ModerateCAT 2020

How many distinct positive integer-valued solutions exist to the equation (x² − 7x + 11)^(x² − 13x + 42) = 1?

  • (1) 2
  • (2) 4
  • (3) 8
  • (4) 6
Show solution
(4) 6. Aᴮ = 1 when: (i) exponent = 0: x²−13x+42 = 0 ⇒ x = 6, 7; (ii) base = 1: x²−7x+11 = 1 ⇒ x = 2, 5; (iii) base = −1 with even exponent: x²−7x+11 = −1 ⇒ x = 3, 4 (and the exponent is even there). Six solutions.
ModerateCAT 2020

The number of integers that satisfy the equality (x² − 5x + 7)^(x+1) = 1 is:

  • (1) 2
  • (2) 3
  • (3) 4
  • (4) 5
Show solution
(2) 3. Exponent 0: x + 1 = 0 ⇒ x = −1. Base 1: x²−5x+7 = 1 ⇒ x²−5x+6 = 0 ⇒ x = 2, 3. (Base −1 case gives no new integers.) Total 3.
HardCAT 2021 · Slot 2

Suppose one of the roots of the equation ax² − bx + c = 0 is 2 + √3, where a, b and c are rational numbers and a ≠ 0. If b = c³, then |a| equals

  • (1) 3
  • (2) 1
  • (3) 4
  • (4) 2
Show solution
(4) 2. Rational coefficients ⇒ conjugate 2 − √3 is the other root. Sum = 4 = b/a, product = 1 = c/a ⇒ b = 4a, c = a. With b = c³: 4a = a³ ⇒ a² = 4 ⇒ |a| = 2.
ModerateCAT 2021 · Slot 2TITA

For all possible integers n satisfying 2.25 ≤ 2 + 2^(n+2) ≤ 202, the number of integer values of 3 + 3^(n+1) is

Show solution
7. 2.25 ≤ 2 + 2^(n+2) ≤ 202 ⇒ 0.25 ≤ 2^(n+2) ≤ 200 ⇒ 2⁻² ≤ 2^(n+2) ≤ 200. So n + 2 ≥ −2 and 2^(n+2) ≤ 200 ⇒ n + 2 ≤ 7 (2⁷ = 128 ≤ 200, 2⁸ = 256 > 200). Thus n + 2 ranges over −2…7. For 3 + 3^(n+1) to be an integer, n + 1 ≥ 0 ⇒ n ≥ −1, i.e. n + 2 ranges over 1…7, giving 7 integer values.
ModerateCAT 2021 · Slot 3TITA

If n is a positive integer such that (⁷√10)(⁷√10)²…(⁷√10)ⁿ > 999, then the smallest value of n is

Show solution
6. The product is 10^(1/7)·10^(2/7)·…·10^(n/7) = 10^((1+2+…+n)/7) = 10^(n(n+1)/14). The inequality 10^(n(n+1)/14) > 999 (just under 10³) needs n(n+1)/14 ≥ 3, i.e. n(n+1) ≥ 42. At n = 6, n(n+1) = 42 gives 10³ = 1000 > 999; n = 5 gives 10^(30/14) < 999. So the smallest n is 6.
HardCAT 2022 · Slot 2TITA

The number of integer solutions of the equation (x² − 10)^(x² − 3x − 10) = 1 is:

Show solution
4. Aᴮ = 1 in three cases. Exponent = 0: x² − 3x − 10 = 0 ⇒ x = 5, −2. Base = 1: x² − 10 = 1 ⇒ x² = 11, no integer. Base = −1 with even exponent: x² − 10 = −1 ⇒ x² = 9 ⇒ x = ±3, and the exponent x² − 3x − 10 is even at both x = 3 and x = −3. So there are 4 integer solutions.
HardCAT 2023 · Slot 3TITA

Let n be any natural number such that 5^(n−1) < 3^(n+1). Then, the least integer value of m that satisfies 3^(n+1) < 2^(n+m) for each such n, is

Show solution
5. The constraint 5^(n−1) < 3^(n+1) restricts n to small values (n = 1, 2, 3, 4); for each of these the smallest integer m making 3^(n+1) < 2^(n+m) hold for all of them is m = 5.
HardCAT 2023 · Slot 2

The sum of all possible values of x satisfying the equation 2^(4x²) − 2^(2x²+x+16) + 2^(2x+30) = 0, is

  • (1) 3
  • (2) 5/2
  • (3) 3/2
  • (4) 1/2
Show solution
(4) 1/2. For the three powers of 2 to balance, the largest two exponents must be equal, with the third equal too. Setting 4x² = 2x² + x + 16 and 2x² + x + 16 = 2x + 30 and taking the consistent roots, the valid x-values sum to 1/2.
HardCAT 2023 · Slot 2

Let a, b, m and n be natural numbers such that a > 1 and b > 1. If aᵐbⁿ = 144¹⁴⁵, then the largest possible value of n − m is

  • (1) 579
  • (2) 580
  • (3) 289
  • (4) 290
Show solution
(1) 579. 144¹⁴⁵ = (2⁴·3²)¹⁴⁵ = 2⁵⁸⁰·3²⁹⁰. To maximise n − m, make m as small as possible and n as large as possible: take a = 3²⁹⁰ (so m = 1) and b = 2 (so n = 580). Then n − m = 580 − 1 = 579.
HardCAT 2023 · Slot 3

Let n and m be two positive integers such that there are exactly 41 integers greater than 8ᵐ and less than 8ⁿ, which can be expressed as powers of 2. Then, the smallest possible value of n + m is

  • (1) 44
  • (2) 14
  • (3) 16
  • (4) 42
Show solution
(3) 16. 8ᵐ = 2³ᵐ and 8ⁿ = 2³ⁿ. The powers of 2 strictly between them are 2^(3m+1), 2^(3m+2), …, 2^(3n−1), a count of (3n − 1) − (3m + 1) + 1 = 3n − 3m − 1 = 41 ⇒ n − m = 14. With m, n positive integers and n − m = 14, the smallest n + m is at m = 1, n = 15, giving n + m = 16.
HardCAT 2023 · Slot 3

If x is a positive real number such that x⁸ + (1/x)⁸ = 47, then the value of x⁹ + (1/x)⁹ is

  • (1) 34√5
  • (2) 40√5
  • (3) 30√5
  • (4) 36√5
Show solution
(1) 34√5. From x⁸ + 1/x⁸ = 47, work down using a² = (a²−2)+2: x⁴ + 1/x⁴ = √(47 + 2) = 7, x² + 1/x² = √(7 + 2) = 3, x + 1/x = √(3 + 2) = √5. Then x³ + 1/x³ = (x + 1/x)³ − 3(x + 1/x) = 5√5 − 3√5 = 2√5. Now x⁹ + 1/x⁹ = (x³ + 1/x³)³ − 3(x³ + 1/x³) = (2√5)³ − 3(2√5) = 40√5 − 6√5 = 34√5.

CAT 2024 & 2025, recent

HardCAT 2024 · Slot 1

If (a + b√n) is the positive square root of (29 − 12√5), where a and b are integers, and n is a natural number, then the maximum possible value of (a + b + n) is

  • (A) 18
  • (B) 22
  • (C) 4
  • (D) 6
Show solution
(A) 18. Squaring: a² + b²n = 29 and 2ab√n = −12√5. Writing 29 − 12√5 = 29 − 6√20 = (√20)² + (−3)² + 2(−3)(√20) = (−3 + √20)², so a = −3, b = 1, n = 20 gives a + b + n = −3 + 1 + 20 = 18, the maximum.
ModerateCAT 2024 · Slot 2 TITA

If (x + 6√2)^(1/2) − (x − 6√2)^(1/2) = 2√2, then x equals

Show solution
11. Squaring: 2x − 2√(x² − 72) = 8 ⇒ x − 4 = √(x² − 72) ⇒ x² − 8x + 16 = x² − 72 ⇒ x = 11.
EasyCAT 2024 · Slot 3

If (a + b√3)² = 52 + 30√3, where a and b are natural numbers, then a + b equals

  • (A) 7
  • (B) 8
  • (C) 9
  • (D) 10
Show solution
(B) 8. Expanding: a² + 3b² + 2ab√3 = 52 + 30√3 ⇒ a² + 3b² = 52 and 2ab = 30 ⇒ ab = 15. Testing factor pairs, a = 5, b = 3 works (25 + 27 = 52), so a + b = 8.