Here’s a comprehensive Maths Formula Sheet for competitive exams like CAT, GMAT, GRE, SSC CGL, IBPS, and more. It includes essential formulas for key topics, which are typically tested in these exams.

1. Arithmetic
1.1 Percentages
Percentage = \( \frac{{\text{{Part}}}}{{\text{{Whole}}}} \times 100 \)
Percentage Increase = \( \frac{{\text{{Increase}}}}{{\text{{Original Value}}}} \times 100 \)
Percentage Decrease = \( \frac{{\text{{Decrease}}}}{{\text{{Original Value}}}} \times 100 \)

1.2 Profit, Loss, and Discount
Profit = Selling Price (SP) – Cost Price (CP)
Loss = Cost Price (CP) – Selling Price (SP)
Profit Percentage = \( \frac{{\text{{Profit}}}}{{\text{{CP}}}} \times 100 \)
Loss Percentage = \( \frac{{\text{{Loss}}}}{{\text{{CP}}}} \times 100 \)
Discount = Marked Price (MP) – Selling Price (SP)
Discount Percentage = \( \frac{{\text{{Discount}}}}{{\text{{MP}}}} \times 100 \)

1.3 Ratio and Proportion
Ratio: \( a:b = \frac{a}{b} \)
Proportion: \( \frac{a}{b} = \frac{c}{d} \)
If \( a:b = c:d \), then \( a \times d = b \times c \)

1.4 Average
Average = \( \frac{{\text{{Sum of Observations}}}}-{{\text{{Number of Observations}}}} \)

1.5 Simple and Compound Interest
Simple Interest (SI): \( SI = \frac{{P \times R \times T}}{100} \)
Compound Interest (CI): \( CI = P \left(1 + \frac{R}{100}\right)^T P \)
Where:
\( P \) = Principal
\( R \) = Rate of interest
\( T \) = Time in years

2. Algebra
2.1 Quadratic Equations
Standard form: \( ax^2 + bx + c = 0 \)
Solution: \( x = \frac{{b \pm \sqrt{{b^2 4ac}}}}{2a} \)
Discriminant: \( D = b^2 4ac \)
If \( D > 0 \), two distinct real roots
If \( D = 0 \), two equal real roots
If \( D < 0 \), no real roots

2.2 Linear Equations
General form of a linear equation in two variables: \( ax + by + c = 0 \)
Solution methods: Substitution, elimination, or crossmultiplication

2.3 Logarithms
\( \log(ab) = \log a + \log b \)
\( \log\left(\frac{a}{b}\right) = \log a – \log b \)
\( \log(a^b) = b \log a \)
\( \log 1 = 0 \), \( \log a^0 = 1 \)

3. Geometry and Mensuration
3.1 Circle
Circumference of a circle = \( 2\pi r \)
Area of a circle = \( \pi r^2 \)

3.2 Triangle
Area of a triangle = \( \frac{1}{2} \times \text{{Base}} \times \text{{Height}} \)
Pythagoras theorem: \( a^2 + b^2 = c^2 \) (where \( c \) is the hypotenuse)

3.3 Rectangle
Area of a rectangle = Length × Breadth
Perimeter of a rectangle = 2 × (Length + Breadth)

3.4 Square
Area of a square = Side²
Perimeter of a square = 4 × Side

3.5 Cylinder
Surface area of a cylinder = \( 2\pi r(r + h) \)
Volume of a cylinder = \( \pi r^2 h \)

3.6 Cone
Surface area of a cone = \( \pi r(l + r) \) (where \( l \) is the slant height)
Volume of a cone = \( \frac{1}{3} \pi r^2 h \)

3.7 Sphere
Surface area of a sphere = \( 4 \pi r^2 \)
Volume of a sphere = \( \frac{4}{3} \pi r^3 \)

4. Number System
4.1 Divisibility Rules
Divisible by 2: Last digit is even
Divisible by 3: Sum of digits is divisible by 3
Divisible by 5: Last digit is 0 or 5
Divisible by 11: Difference between the sum of digits in odd places and even places is divisible by 11

4.2 HCF and LCM
HCF (Highest Common Factor): Largest number that divides two or more numbers
LCM (Lowest Common Multiple): Smallest number that is a multiple of two or more numbers
\( \text{{HCF}} \times \text{{LCM}} = \text{{Product of Numbers}} \)

5. Trigonometry
5.1 Basic Trigonometric Ratios
\( \sin \theta = \frac{{\text{{Opposite}}}}{{\text{{Hypotenuse}}}} \)
\( \cos \theta = \frac{{\text{{Adjacent}}}}{{\text{{Hypotenuse}}}} \)
\( \tan \theta = \frac{{\sin \theta}}{{\cos \theta}} = \frac{{\text{{Opposite}}}}{{\text{{Adjacent}}}} \)

5.2 Common Angle Values
\( \sin 0^\circ = 0 \), \( \sin 30^\circ = \frac{1}{2} \), \( \sin 45^\circ = \frac{1}{\sqrt{2}} \), \( \sin 60^\circ = \frac{\sqrt{3}}{2} \), \( \sin 90^\circ = 1 \)
\( \cos 0^\circ = 1 \), \( \cos 30^\circ = \frac{\sqrt{3}}{2} \), \( \cos 45^\circ = \frac{1}{\sqrt{2}} \), \( \cos 60^\circ = \frac{1}{2} \), \( \cos 90^\circ = 0 \)

5.3 Trigonometric Identities
\( \sin^2 \theta + \cos^2 \theta = 1 \)
\( 1 + \tan^2 \theta = \sec^2 \theta \)
\( 1 + \cot^2 \theta = \csc^2 \theta \)

6. Time, Speed, and Distance
Speed = \( \frac{{\text{{Distance}}}}{{\text{{Time}}}} \)
Time = \( \frac{{\text{{Distance}}}}{{\text{{Speed}}}} \)
Distance = Speed × Time
Relative Speed (same direction) = Difference of speeds
Relative Speed (opposite direction) = Sum of speeds

6.1 Average Speed
Average Speed = \( \frac{{\text{{Total Distance}}}}{{\text{{Total Time}}}} \)

7. Work and Time
Work done = Rate of work × Time
If \( A \) can do a job in \( a \) days, \( B \) in \( b \) days, then together they can do the job in \( \frac{{ab}}{{a+b}} \) days
Work is inversely proportional to time when the amount of work is constant

This formula sheet provides a strong foundation for key math concepts and formulas frequently tested in competitive exams like CAT, GRE, and SSC CGL. For better preparation, practice applying these formulas to a variety of problems.