BCA Entrance Exam Notes
BcaMathematics•Updated: 5/17/2026
BCA ENTRANCE EXAM
Complete Study Notes
Mathematics •
Total: 100 Marks
📐 MATHEMATICS
Marks: 50 | 25 Topics
Algebra • Arithmetic • Logic • Geometry
1. Set and Relation
Definition: A set is a well-defined collection of distinct objects called elements.
Notation: Sets are written in curly braces: A = {1, 2, 3}
Types of Sets: Empty ({}), Finite, Infinite, Subset, Universal, Equal sets
Set Operations: Union (A∪B), Intersection (A∩B), Complement (A'), Difference (A-B)
De Morgan's Law: (A∪B)' = A'∩B' | (A∩B)' = A'∪B'
n(A∪B) = n(A) + n(B) - n(A∩B)
Relation: A relation R from set A to B is a subset of A×B (Cartesian product).
Types: Reflexive, Symmetric, Transitive, Equivalence relation
2. Percentage
Percentage = (Value / Total Value) × 100
% Increase = (Increase / Original) × 100
% Decrease = (Decrease / Original) × 100
Value after x% increase = Original × (1 + x/100)
Value after x% decrease = Original × (1 - x/100)
To convert fraction to %: multiply by 100 → 1/4 = 25%
To convert % to fraction: divide by 100 → 35% = 35/100 = 7/20
Example: If 60 is 30% of a number, the number = 60 × 100/30 = 200
3. Integers / HCF & LCM
HCF (GCD): Largest number that divides both numbers exactly
LCM: Smallest number exactly divisible by both numbers
HCF × LCM = Product of the two numbers
HCF by Division Method: Divide larger by smaller, then remainder divides previous divisor
LCM by Prime Factorization: Take highest power of each prime factor
Integers: ..., -3, -2, -1, 0, 1, 2, 3, ...
Positive integers × Negative = Negative | Negative × Negative = Positive
Example: HCF(12,18)=6 | LCM(12,18)=36 | Verify: 6×36=216=12×18 ✓
4. Time and Work
Work = Rate × Time
If A completes work in 'n' days → A's 1-day work = 1/n
If A and B work together → combined 1-day work = 1/a + 1/b
Days to finish together = ab / (a+b)
If A is x times as fast as B → A takes 1/x of B's time
Example: A does work in 10 days, B in 15 days. Together = (10×15)/(10+15) = 6 days
Pipes & Cisterns: Inlet pipe fills (+), Outlet pipe drains (-)
5. Algebra
Identities: (a+b)² = a²+2ab+b² | (a-b)² = a²-2ab+b²
(a+b)(a-b) = a²-b²
(a+b)³ = a³+3a²b+3ab²+b³ | (a-b)³ = a³-3a²b+3ab²-b³
Linear Equation: ax + b = 0 → x = -b/a
Quadratic: ax²+bx+c=0 → x = [-b ± √(b²-4ac)] / 2a
Discriminant D = b²-4ac: D>0 (two real roots), D=0 (equal roots), D<0 (no real roots)
Factorization: Find two numbers whose product=ac and sum=b
6. Probability
P(Event) = Number of Favorable Outcomes / Total Outcomes
0 ≤ P(E) ≤ 1
P(E) + P(E') = 1 → P(not E) = 1 - P(E)
P(A∪B) = P(A) + P(B) - P(A∩B)
For mutually exclusive events: P(A∪B) = P(A) + P(B)
Sample space of a coin: {H, T} → P(Head) = 1/2
Sample space of a die: {1,2,3,4,5,6} → P(even) = 3/6 = 1/2
Independent Events: P(A∩B) = P(A) × P(B)
7. Ratio and Proportion
Ratio a:b = a/b
Direct Proportion: a/b = c/d → ad = bc (cross multiplication)
Inverse Proportion: a × b = c × d
Compound Ratio: (a:b) × (c:d) = ac:bd
Dividing N in ratio a:b → shares = Na/(a+b) and Nb/(a+b)
Mean Proportional of a and b = √(ab)
If A:B = 2:3 and B:C = 4:5, then A:B:C = 8:12:15
8. Sequence and Series
AP (Arithmetic Progression): a, a+d, a+2d, ...
nth term: Tn = a + (n-1)d
Sum of n terms: Sn = n/2 × [2a + (n-1)d] or n/2 × (first + last)
GP (Geometric Progression): a, ar, ar², ...
nth term: Tn = ar^(n-1)
Sum: Sn = a(rⁿ-1)/(r-1) for r≠1 | Sum to infinity = a/(1-r) if |r|<1
Common difference (d) of AP = T2 - T1
Common ratio (r) of GP = T2 / T1
Example AP: 2, 5, 8, 11 → d=3, T10 = 2+9×3 = 29
9. Law of Indices
aᵐ × aⁿ = aᵐ⁺ⁿ aᵐ ÷ aⁿ = aᵐ⁻ⁿ
(aᵐ)ⁿ = aᵐⁿ (ab)ⁿ = aⁿbⁿ
a⁰ = 1 a⁻ⁿ = 1/aⁿ
a^(1/n) = ⁿ√a a^(m/n) = ⁿ√(aᵐ)
Example: 2³ × 2⁴ = 2⁷ = 128
Example: (4)^(3/2) = (√4)³ = 2³ = 8
Simplify: x² × x⁻³ = x²⁻³ = x⁻¹ = 1/x
10. Simple Interest
SI = (P × R × T) / 100
Total Amount (A) = P + SI
P = (SI × 100) / (R × T)
R = (SI × 100) / (P × T)
T = (SI × 100) / (P × R)
Compound Interest: A = P(1 + R/100)ⁿ | CI = A - P
Example: P=1000, R=5%, T=3 yrs → SI = (1000×5×3)/100 = Rs.150
Difference: CI > SI for same principal, rate and time (except for 1 year)
11. Matrices and Determinants
Matrix: Rectangular array of numbers in rows (m) and columns (n) → m×n matrix Types: Row, Column, Square, Identity (I), Null, Diagonal, Symmetric Addition/Subtraction: Only for same-order matrices (element-wise) Multiplication: A(m×n) × B(n×p) = C(m×p) — inner dimensions must match Transpose: Rows become columns (Aᵀ) |
For 2×2 Matrix: |a b| → Determinant = ad - bc |c d| Inverse: A⁻¹ = (1/|A|) × Adj(A) | AA⁻¹ = I |
Identity Matrix: diagonal elements = 1, rest = 0
If |A| = 0, matrix is singular (no inverse exists)
12. Polynomial Equations
Polynomial: P(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀
Degree: highest power of x
Remainder Theorem: P(x) ÷ (x-a) → remainder = P(a)
Factor Theorem: (x-a) is a factor of P(x) if P(a) = 0
Linear (degree 1): ax+b=0 | Quadratic (degree 2): ax²+bx+c=0
Sum of roots (α+β) = -b/a | Product of roots (αβ) = c/a
Cubic roots: α+β+γ = -b/a | αβ+βγ+γα = c/a | αβγ = -d/a
13. Averages
Mean (Average) = Sum of all values / Number of values
Median: Middle value of sorted data (odd n → middle, even n → avg of two middle)
Mode: Most frequently occurring value
Weighted Mean = Σ(wᵢxᵢ) / Σwᵢ
If avg of n numbers is A, and one number x is replaced by y: New avg = A + (y-x)/n
Mean, Median, Mode relationship (for symmetric distribution): Mean = Median = Mode
Example: Data: 3,5,7,5,9 → Mean=29/5=5.8 | Median=5 | Mode=5
14. Profit and Loss
Profit = SP - CP Loss = CP - SP
Profit% = (Profit/CP) × 100 Loss% = (Loss/CP) × 100
SP = CP × (100 + P%) / 100 SP = CP × (100 - L%) / 100
CP = SP × 100 / (100 + P%) CP = SP × 100 / (100 - L%)
Discount = MP - SP Discount% = (Discount/MP) × 100
If shopkeeper cheats using false weights → Profit% = (True weight - False weight)/False weight × 100
Example: CP=200, SP=250 → Profit=50 → Profit%=25%
15. Trigonometry and Logarithm
sin θ = Opposite/Hypotenuse cos θ = Adjacent/Hypotenuse tan θ = sin θ / cos θ cosec θ = 1/sin θ sec θ = 1/cos θ cot θ = 1/tan θ sin²θ + cos²θ = 1 1 + tan²θ = sec²θ 1 + cot²θ = cosec²θ |
Key angles: θ=0°: sin=0, cos=1, tan=0 θ=30°: sin=1/2, cos=√3/2, tan=1/√3 θ=45°: sin=1/√2, cos=1/√2, tan=1 θ=60°: sin=√3/2, cos=1/2, tan=√3 θ=90°: sin=1, cos=0, tan=undefined |
Logarithm: log_a(x) = y means aʸ = x log(mn) = log m + log n log(m/n) = log m - log n log(mⁿ) = n·log m log_a(a) = 1 log_a(1) = 0 Change of base: log_a(b) = log(b)/log(a) |
16. Statistics
Mean (x̄) = Σx / n
Variance (σ²) = Σ(x - x̄)² / n
Standard Deviation (σ) = √Variance
Range = Maximum - Minimum
Coefficient of Variation (CV) = (σ / x̄) × 100
Grouped data mean: x̄ = Σfx / Σf (f = frequency, x = midpoint)
Less variability → smaller SD → more consistent data
Median in grouped data: L + [(n/2 - cf)/f] × h
17. Coordinate Geometry
Distance between (x₁,y₁) and (x₂,y₂) = √[(x₂-x₁)² + (y₂-y₁)²]
Midpoint = [(x₁+x₂)/2, (y₁+y₂)/2]
Slope (m) = (y₂-y₁) / (x₂-x₁)
Line equation: y = mx + c (slope-intercept form)
y - y₁ = m(x - x₁) (point-slope form)
Parallel lines: m₁ = m₂ | Perpendicular lines: m₁ × m₂ = -1
Distance from point (x₁,y₁) to line ax+by+c=0 = |ax₁+by₁+c| / √(a²+b²)
Circle: (x-h)² + (y-k)² = r² where (h,k) is center and r is radius
18. Area and Volume
Triangle: Area = ½ × base × height | Heron's: √[s(s-a)(s-b)(s-c)], s=(a+b+c)/2 Rectangle: Area = l × b | Perimeter = 2(l+b) Circle: Area = πr² | Circumference = 2πr Trapezium: Area = ½(a+b) × h |
Cube: Volume = a³ Surface Area = 6a² Cuboid: Volume = l×b×h Surface Area = 2(lb+bh+lh) Cylinder: Volume = πr²h Curved SA = 2πrh Total SA = 2πr(r+h) Sphere: Volume = (4/3)πr³ Surface Area = 4πr² Cone: Volume = (1/3)πr²h Slant height l = √(r²+h²) |
19. Permutation and Combination
Permutation (arrangement): nPr = n! / (n-r)!
Combination (selection): nCr = n! / [r!(n-r)!]
n! = n × (n-1) × (n-2) × ... × 1 | 0! = 1
nCr = nC(n-r) (symmetry property)
nCr + nC(r-1) = (n+1)Cr (Pascal's rule)
Permutation → order matters (arrangement) | Combination → order doesn't matter (selection)
Circular permutation of n objects = (n-1)!
Example: 5P2 = 5!/(5-2)! = 20 | 5C2 = 5!/(2!×3!) = 10
20. Coding and Decoding
Letter coding: Each letter replaced by another letter using a rule (shift forward/backward)
Number coding: Letters assigned numbers (A=1, B=2...Z=26 or A=26...Z=1)
Substitution coding: One word/symbol represents another
Example: CAT → DOG (each letter shifted +1 forward in alphabet)
If APPLE = 5, MANGO = 5 → count of letters
Reverse coding: A=Z, B=Y, C=X (mirror alphabet)
Tip: First find the pattern — check if shift is +1, +2, alternating, or positional
21. Letter and Symbol Series
Identify the pattern of letters or symbols in a given sequence
Types: Alphabetical order, Skip pattern, Reverse order, Repeat pattern
Example: A, C, E, G, ? → skip 1 letter each time → answer: I
Example: AZ, BY, CX, ? → forward + reverse → DW
Symbol series: Find the repeating cycle or rule
Steps: (1) Find gap between terms (2) Check if pattern repeats (3) Apply to find answer
22. Clock Problems
Clock has 12 hours → 360° → each hour = 30°
Minute hand speed = 6° per minute
Hour hand speed = 0.5° per minute
Angle between hands at h hours m minutes = |30h - 5.5m|
Hands coincide: every 65 5/11 minutes
Hands at right angle (90°): 22 times in 12 hours
Hands opposite (180°): 11 times in 12 hours
Example: Angle at 3:20 = |30×3 - 5.5×20| = |90-110| = 20°
23. Number Series
Identify and continue the pattern in a given number sequence
Types of series:
Prime: 2, 3, 5, 7, 11, 13, ...
Squares: 1, 4, 9, 16, 25, 36, ...
Cubes: 1, 8, 27, 64, 125, ...
Fibonacci: 1, 1, 2, 3, 5, 8, 13, ... (each = sum of previous two)
Difference series: 3, 6, 10, 15, 21 → differences: 3,4,5,6...
Strategy: Find difference between consecutive terms → check if differences form a pattern
24. Ranking Test
Used to determine position of a person/object in a row or rank list
If a person is rth from left and sth from right in a row:
Total persons = r + s - 1
If total is N, rank from left = r, then rank from right = N - r + 1
From top + From bottom = Total + 1
Example: Riya is 7th from left and 5th from right → Total = 7+5-1 = 11 students
Tip: Draw a mental line and place person at position, then count from other end
25. Missing Number
Find the missing number in a matrix, series, or figure using a pattern
Common patterns:
Row/column sum or product is constant
Diagonal pattern
Arithmetic or geometric relationship
Difference between adjacent elements
In a 3×3 matrix: often the middle number = average, product, or result of surrounding numbers
Example: 2 4 8 | 3 6 ? → each row multiplies by 2 → answer: 12