If a positive integer N has prime factorization p1^a * p2^b * p3^c, what are the formulas for total factors and sum of factors?

Total number of factors = (a + 1)(b + 1)(c + 1). Sum of factors = [(p1^(a+1) - 1)/(p1 - 1)] * [(p2^(b+1) - 1)/(p2 - 1)] * [(p3^(c+1) - 1)/(p3 - 1)].

In CAT VARC Odd-Sentence-Out questions, what are the three primary traps that distinguish the outlier sentence?

1) Scope shift (discussing a tangential aspect or different timeframe), 2) Tone/perspective mismatch, or 3) Failing to fit into the cohesive logical chain formed by the remaining four sentences.

What is the Derangement formula D(n) for the number of ways to arrange n distinct items such that no item appears in its original position?

D(n) = n! * [1 - 1/1! + 1/2! - 1/3! + ... + (-1)^n / n!]. Key values: D(1)=0, D(2)=1, D(3)=2, D(4)=9, D(5)=44.

In CAT DILR Set Theory (Venn Diagrams), how do you maximize or minimize the intersection of 3 sets A, B, and C when total elements N and individual sizes are fixed?

Using N = |A| + |B| + |C| - (exact 2-set overlaps) - 2*(exact 3-set overlap) + None, to maximize the 3-set overlap (x), minimize the 2-set-only overlaps and maximize the union up to N.

In logarithms, what is the base-change formula and the value of log_b(a) * log_c(b) * log_a(c)?

log_b(a) = log_k(a) / log_k(b). Consequently, log_b(a) * log_c(b) * log_a(c) = 1.

In mixtures and alligations, what is the repeated dilution (replacement) formula?

Final Concentration = Initial Concentration * (1 - x/V)^n, where V is total volume, x is volume removed and replaced each time, and n is the number of operations.

In coordinate geometry, what is the shortest distance from a point (x1, y1) to the line Ax + By + C = 0?

Perpendicular distance d = |Ax1 + By1 + C| / sqrt(A^2 + B^2).

In CAT DILR, what is the tournament seeding rule and total matches formula in a single-elimination (knockout) tournament with N players?

Total matches to determine a single winner in a knockout tournament with N players = N - 1. If N is a power of 2 (2^k), there are k rounds and in each round the sum of seeds of paired players (without upsets) is 2^k + 1.

What is the effective percentage change formula for two successive percentage changes of a% and b%?

Net percentage change = (a + b + (a * b) / 100)%.

What is Bayes' Theorem in probability for conditional events?

P(A | B) = [P(B | A) * P(A)] / P(B), where P(B) = sum_i [P(B | A_i) * P(A_i)] for mutually exclusive and exhaustive events A_i.

What are the formulas for the inradius (r) and circumradius (R) of any triangle with area Delta, semi-perimeter s, and sides a, b, c?

Inradius r = Delta / s; Circumradius R = (a * b * c) / (4 * Delta). For a right-angled triangle, r = (a + b - hypotenuse)/2 and R = hypotenuse / 2.

What is the sum of the first n terms of an Arithmetic-Geometric Progression (AGP) and an infinite AGP with |r| < 1?

For infinite AGP: S_infinity = a / (1 - r) + d*r / (1 - r)^2, where a is the first term, d is the common difference, and r is the common ratio.

What is Ptolemy's Theorem for a cyclic quadrilateral with sides a, b, c, d and diagonals d1, d2?

In a cyclic quadrilateral, the product of the diagonals equals the sum of the products of opposite sides: d1 * d2 = (a * c) + (b * d).

What is the Alternate Segment Theorem in circle geometry?

The angle between a tangent to a circle and a chord drawn from the point of contact is equal to the angle subtended by the chord in the alternate segment.

In Time, Speed, and Distance, when two bodies start simultaneously from A and B towards each other and take t1 and t2 hours after meeting to reach B and A respectively, what is the ratio of their speeds?

Speed_A / Speed_B = sqrt(t2 / t1), and the time taken to meet is sqrt(t1 * t2).

In XAT Decision Making (DM), what is the golden hierarchy for selecting the right course of action in ethical vs business dilemmas?

Prioritize constitutional/legal compliance and universal human ethics first, institutional integrity and long-term stakeholder welfare second, and reject options that involve shortcuts, bribery, scapegoating, or knee-jerk extreme punishments.

What is Descarte's Rule of Signs for finding the maximum number of positive and negative real roots of a polynomial f(x) = 0?

The maximum number of positive real roots equals the number of sign changes in f(x), and the maximum number of negative real roots equals the number of sign changes in f(-x).

How do you calculate the number of trailing zeroes in n! (n factorial)?

Find the highest power of 5 in n! using Legendre's formula: floor(n/5) + floor(n/25) + floor(n/125) + ... until 5^k > n.

In Permutations & Combinations, what is the Stars and Bars (Beggar's Method) formula for the number of non-negative and positive integral solutions to x1 + x2 + ... + xr = n?

Non-negative integer solutions (xi >= 0): C(n + r - 1, r - 1). Positive integer solutions (xi >= 1): C(n - 1, r - 1).

What is the AM-GM-HM inequality for positive real numbers, and when does equality hold?

Arithmetic Mean (AM) >= Geometric Mean (GM) >= Harmonic Mean (HM), and for two numbers AM * HM = GM^2. Equality holds if and only if all numbers are equal.

In CAT Reading Comprehension (RC), how do you distinguish an 'Inference' question from an 'Assumption' question?

An Inference must be logically deducible as true *from* the stated premises in the passage (what follows from the text), whereas an Assumption is an unstated premise that *must be true* for the author's conclusion to hold (Pass the Negation Test).

What is Apollonius's Theorem for the length of a median in a triangle with sides a, b, c?

In triangle ABC, if AD is the median to side BC (of length a), then AB^2 + AC^2 = 2(AD^2 + BD^2), i.e., c^2 + b^2 = 2(m_a^2 + (a/2)^2).

In Time and Work, if A takes 'a' days more than (A+B) together and B takes 'b' days more than (A+B) together to finish a job, how many days do (A+B) take together?

Time taken by (A + B) together = sqrt(a * b) days.

What is Ceva's Theorem and Menelaus's Theorem in triangle geometry?

Ceva's Theorem (concurrency of cevians AD, BE, CF): (BD/DC)*(CE/EA)*(AF/FB) = 1. Menelaus's Theorem (collinearity of points D, E, F on sides/extended sides): (BD/DC)*(CE/EA)*(AF/FB) = 1.

In Boats and Streams, how are boat speed in still water (B) and stream speed (S) related to downstream speed (D) and upstream speed (U)?

B = (D + U) / 2 and S = (D - U) / 2.

What is Wilson's Theorem in CAT Number Systems?

For any prime number p, (p - 1)! ≡ (p - 1) ≡ -1 (mod p), and (p - 2)! ≡ 1 (mod p).

What is the Chinese Remainder Theorem shortcut for finding the smallest N that leaves remainders r1 and r2 when divided by coprime divisors d1 and d2?

N = d1*k + r1 = d2*m + r2; once the smallest positive solution N0 is found, the general solution is N = LCM(d1, d2)*n + N0.

What is Euler's Totient Theorem for finding the remainder of a^phi(n) divided by n?

If 'a' and 'n' are coprime (GCD(a, n) = 1), then a^phi(n) ≡ 1 (mod n), where phi(n) is Euler's totient function counting positive integers up to n that are relatively prime to n.

In CAT VARC Parajumbles (TITA - Type In The Answer), what is the Mandatory Pair & Pronoun-Antecedent strategy?

Identify an independent opening sentence introducing the main subject, link mandatory pairs using demonstrative pronouns (this/these/such), transition markers (however, moreover, consequently), and chronological or full-name-to-acronym progression.

What is the difference between Simple Interest (SI) and Compound Interest (CI) for 2 years and 3 years at rate R% on principal P?

For 2 years: CI - SI = P * (R/100)^2. For 3 years: CI - SI = P * (R/100)^2 * (3 + R/100).

For a quadratic equation ax^2 + bx + c = 0 with roots alpha and beta, what are the sum of squares and difference of roots?

alpha + beta = -b/a, alpha*beta = c/a; alpha^2 + beta^2 = (b^2 - 2ac)/a^2; |alpha - beta| = sqrt(b^2 - 4ac) / |a|.

CAT, XAT & CUET UG/PG Aptitude: Quantitative Shortcuts (Number System, Arithmetic, Algebra, Geometry, Modern Math), DILR Sets & VARC Parajumbles

Core quantitative shortcuts, number theory theorems, arithmetic ratios, algebraic inequalities, geometry theorems, DILR set-selection frameworks, and VARC parajumble strategies for IIM CAT, XAT, and CUET.

Level: basic
Languages: en → en
Cards: 32
Markdown: CAT, XAT & CUET UG/PG Aptitude: Quantitative Shortcuts (Number System, Arithmetic, Algebra, Geometry, Modern Math), DILR Sets & VARC Parajumbles (.md)

Flashcards (32)

#	Front (Term / Question)	Back (Definition / Answer)
1	If a positive integer N has prime factorization p1^a * p2^b * p3^c, what are the formulas for total factors and sum of factors?	Total number of factors = (a + 1)(b + 1)(c + 1). Sum of factors = [(p1^(a+1) - 1)/(p1 - 1)] * [(p2^(b+1) - 1)/(p2 - 1)] * [(p3^(c+1) - 1)/(p3 - 1)].
2	In CAT VARC Odd-Sentence-Out questions, what are the three primary traps that distinguish the outlier sentence?	1) Scope shift (discussing a tangential aspect or different timeframe), 2) Tone/perspective mismatch, or 3) Failing to fit into the cohesive logical chain formed by the remaining four sentences.
3	What is the Derangement formula D(n) for the number of ways to arrange n distinct items such that no item appears in its original position?	D(n) = n! * [1 - 1/1! + 1/2! - 1/3! + ... + (-1)^n / n!]. Key values: D(1)=0, D(2)=1, D(3)=2, D(4)=9, D(5)=44.
4	In CAT DILR Set Theory (Venn Diagrams), how do you maximize or minimize the intersection of 3 sets A, B, and C when total elements N and individual sizes are fixed?	Using N = \|A\| + \|B\| + \|C\| - (exact 2-set overlaps) - 2*(exact 3-set overlap) + None, to maximize the 3-set overlap (x), minimize the 2-set-only overlaps and maximize the union up to N.
5	In logarithms, what is the base-change formula and the value of log_b(a) * log_c(b) * log_a(c)?	log_b(a) = log_k(a) / log_k(b). Consequently, log_b(a) * log_c(b) * log_a(c) = 1.
6	In mixtures and alligations, what is the repeated dilution (replacement) formula?	Final Concentration = Initial Concentration * (1 - x/V)^n, where V is total volume, x is volume removed and replaced each time, and n is the number of operations.
7	In coordinate geometry, what is the shortest distance from a point (x1, y1) to the line Ax + By + C = 0?	Perpendicular distance d = \|Ax1 + By1 + C\| / sqrt(A^2 + B^2).
8	In CAT DILR, what is the tournament seeding rule and total matches formula in a single-elimination (knockout) tournament with N players?	Total matches to determine a single winner in a knockout tournament with N players = N - 1. If N is a power of 2 (2^k), there are k rounds and in each round the sum of seeds of paired players (without upsets) is 2^k + 1.
9	What is the effective percentage change formula for two successive percentage changes of a% and b%?	Net percentage change = (a + b + (a * b) / 100)%.
10	What is Bayes' Theorem in probability for conditional events?	P(A \| B) = [P(B \| A) * P(A)] / P(B), where P(B) = sum_i [P(B \| A_i) * P(A_i)] for mutually exclusive and exhaustive events A_i.
11	What are the formulas for the inradius (r) and circumradius (R) of any triangle with area Delta, semi-perimeter s, and sides a, b, c?	Inradius r = Delta / s; Circumradius R = (a * b * c) / (4 * Delta). For a right-angled triangle, r = (a + b - hypotenuse)/2 and R = hypotenuse / 2.
12	What is the sum of the first n terms of an Arithmetic-Geometric Progression (AGP) and an infinite AGP with \|r\| < 1?	For infinite AGP: S_infinity = a / (1 - r) + d*r / (1 - r)^2, where a is the first term, d is the common difference, and r is the common ratio.
13	What is Ptolemy's Theorem for a cyclic quadrilateral with sides a, b, c, d and diagonals d1, d2?	In a cyclic quadrilateral, the product of the diagonals equals the sum of the products of opposite sides: d1 * d2 = (a * c) + (b * d).
14	What is the Alternate Segment Theorem in circle geometry?	The angle between a tangent to a circle and a chord drawn from the point of contact is equal to the angle subtended by the chord in the alternate segment.
15	In Time, Speed, and Distance, when two bodies start simultaneously from A and B towards each other and take t1 and t2 hours after meeting to reach B and A respectively, what is the ratio of their speeds?	Speed_A / Speed_B = sqrt(t2 / t1), and the time taken to meet is sqrt(t1 * t2).
16	In XAT Decision Making (DM), what is the golden hierarchy for selecting the right course of action in ethical vs business dilemmas?	Prioritize constitutional/legal compliance and universal human ethics first, institutional integrity and long-term stakeholder welfare second, and reject options that involve shortcuts, bribery, scapegoating, or knee-jerk extreme punishments.
17	What is Descarte's Rule of Signs for finding the maximum number of positive and negative real roots of a polynomial f(x) = 0?	The maximum number of positive real roots equals the number of sign changes in f(x), and the maximum number of negative real roots equals the number of sign changes in f(-x).
18	How do you calculate the number of trailing zeroes in n! (n factorial)?	Find the highest power of 5 in n! using Legendre's formula: floor(n/5) + floor(n/25) + floor(n/125) + ... until 5^k > n.
19	In Permutations & Combinations, what is the Stars and Bars (Beggar's Method) formula for the number of non-negative and positive integral solutions to x1 + x2 + ... + xr = n?	Non-negative integer solutions (xi >= 0): C(n + r - 1, r - 1). Positive integer solutions (xi >= 1): C(n - 1, r - 1).
20	What is the AM-GM-HM inequality for positive real numbers, and when does equality hold?	Arithmetic Mean (AM) >= Geometric Mean (GM) >= Harmonic Mean (HM), and for two numbers AM * HM = GM^2. Equality holds if and only if all numbers are equal.
21	What is the harmonic mean shortcut for average speed over two equal distances traveled at speeds u and v?	Average Speed = 2uv / (u + v).
22	In CAT Reading Comprehension (RC), how do you distinguish an 'Inference' question from an 'Assumption' question?	An Inference must be logically deducible as true from the stated premises in the passage (what follows from the text), whereas an Assumption is an unstated premise that must be true for the author's conclusion to hold (Pass the Negation Test).
23	What is Apollonius's Theorem for the length of a median in a triangle with sides a, b, c?	In triangle ABC, if AD is the median to side BC (of length a), then AB^2 + AC^2 = 2(AD^2 + BD^2), i.e., c^2 + b^2 = 2(m_a^2 + (a/2)^2).
24	In Time and Work, if A takes 'a' days more than (A+B) together and B takes 'b' days more than (A+B) together to finish a job, how many days do (A+B) take together?	Time taken by (A + B) together = sqrt(a * b) days.
25	What is Ceva's Theorem and Menelaus's Theorem in triangle geometry?	Ceva's Theorem (concurrency of cevians AD, BE, CF): (BD/DC)(CE/EA)(AF/FB) = 1. Menelaus's Theorem (collinearity of points D, E, F on sides/extended sides): (BD/DC)(CE/EA)(AF/FB) = 1.
26	In Boats and Streams, how are boat speed in still water (B) and stream speed (S) related to downstream speed (D) and upstream speed (U)?	B = (D + U) / 2 and S = (D - U) / 2.
27	What is Wilson's Theorem in CAT Number Systems?	For any prime number p, (p - 1)! ≡ (p - 1) ≡ -1 (mod p), and (p - 2)! ≡ 1 (mod p).
28	What is the Chinese Remainder Theorem shortcut for finding the smallest N that leaves remainders r1 and r2 when divided by coprime divisors d1 and d2?	N = d1k + r1 = d2m + r2; once the smallest positive solution N0 is found, the general solution is N = LCM(d1, d2)*n + N0.
29	What is Euler's Totient Theorem for finding the remainder of a^phi(n) divided by n?	If 'a' and 'n' are coprime (GCD(a, n) = 1), then a^phi(n) ≡ 1 (mod n), where phi(n) is Euler's totient function counting positive integers up to n that are relatively prime to n.
30	In CAT VARC Parajumbles (TITA - Type In The Answer), what is the Mandatory Pair & Pronoun-Antecedent strategy?	Identify an independent opening sentence introducing the main subject, link mandatory pairs using demonstrative pronouns (this/these/such), transition markers (however, moreover, consequently), and chronological or full-name-to-acronym progression.
31	What is the difference between Simple Interest (SI) and Compound Interest (CI) for 2 years and 3 years at rate R% on principal P?	For 2 years: CI - SI = P * (R/100)^2. For 3 years: CI - SI = P * (R/100)^2 * (3 + R/100).
32	For a quadratic equation ax^2 + bx + c = 0 with roots alpha and beta, what are the sum of squares and difference of roots?	alpha + beta = -b/a, alpha*beta = c/a; alpha^2 + beta^2 = (b^2 - 2ac)/a^2; \|alpha - beta\| = sqrt(b^2 - 4ac) / \|a\|.