Mathematics Study Notes (Class VI to Intermediate)
Table of Contents
Arithmetic
BODMAS Rule
BODMAS stands for:
Brackets
Orders (powers, roots, etc.)
Division
Multiplication
Addition
Subtraction
Example: 8 + 2 × (5 - 1) ÷ 2
Solve brackets: 8 + 2 × 4 ÷ 2
Multiplication and division (left to right): 8 + 8 ÷ 2 = 8 + 4
Addition: 12
Ratios
A ratio compares quantities of the same kind using division. Written as a:b or a/b.
Properties:
If a:b = c:d, then ad = bc
If a:b = b:c, then b² = ac
Examples:
Dividing ₹800 in ratio 3:5
Total parts = 3 + 5 = 8
First share = ₹800 × (3/8) = ₹300
Second share = ₹800 × (5/8) = ₹500
Percentages
Percentage means "per hundred" and is represented by the symbol %.
Conversion:
To convert a fraction to percentage: Multiply by 100%
3/4 = 0.75 = 75%
To convert percentage to decimal: Divide by 100
68% = 68/100 = 0.68
Applications:
Finding percentage: (Part/Whole) × 100%
Finding part: (Percentage × Whole)/100
Finding whole: (Part × 100)/Percentage
Profit and Loss
Cost Price (CP): Amount paid to purchase an item
Selling Price (SP): Amount received by selling the item
Profit: SP > CP, Profit = SP - CP
Loss: SP < CP, Loss = CP - SP
Profit/Loss Percentage: (Profit or Loss/CP) × 100%
Examples:
If CP = ₹800 and SP = ₹920
Profit = ₹920 - ₹800 = ₹120
Profit% = (120/800) × 100% = 15%
Interest
Simple Interest (SI)
SI = (P × R × T)/100
P = Principal amount
R = Rate of interest per annum
T = Time period in years
Example: ₹5000 at 8% for 2 years SI = (5000 × 8 × 2)/100 = ₹800
Compound Interest (CI)
A = P(1 + R/100)ᵀ
A = Final amount
CI = A - P
Example: ₹10,000 at 10% p.a. for 2 years A = 10000(1 + 10/100)² = 10000 × 1.21 = ₹12,100 CI = ₹12,100 - ₹10,000 = ₹2,100
Taxes
GST (Goods and Services Tax): Current rates in India are 0%, 5%, 12%, 18%, and 28%
Income Tax: Progressive tax structure based on income slabs
Calculating Tax Amount: (Value × Tax Rate)/100
Example: GST calculation for an item worth ₹1000 with 18% GST Tax amount = (1000 × 18)/100 = ₹180 Final price = ₹1000 + ₹180 = ₹1180
Number System
Types of Numbers
Natural Numbers: 1, 2, 3, ... (denoted by N)
Whole Numbers: 0, 1, 2, 3, ... (denoted by W)
Integers: ..., -3, -2, -1, 0, 1, 2, 3, ... (denoted by Z)
Rational Numbers: Numbers expressed as p/q where p, q are integers and q ≠ 0 (denoted by Q)
Irrational Numbers: Numbers that cannot be expressed as p/q (denoted by Q')
Real Numbers: Union of rational and irrational numbers (denoted by R)
Basic Operations
Addition: Combining quantities
Subtraction: Finding difference between quantities
Multiplication: Repeated addition
Division: Equal distribution or grouping
Factors and Multiples
Factor: A number that divides another number completely
Multiple: Product of a number with any natural number
Prime Number: A number greater than 1 with exactly two factors (1 and itself)
Composite Number: A number with more than two factors
LCM and HCF (GCD)
LCM (Least Common Multiple): Smallest number divisible by each given number
HCF/GCD (Highest Common Factor/Greatest Common Divisor): Largest number that divides each given number completely
Methods to find LCM and HCF:
Prime Factorization Method
Division Method
Example: For numbers 24 and 36:
Prime factors of 24 = 2³ × 3
Prime factors of 36 = 2² × 3²
HCF = 2² × 3 = 12
LCM = 2³ × 3² = 72
Fractions
Fraction: Represents part of a whole, written as a/b where b ≠ 0
Types of Fractions:
Proper Fraction: Numerator < Denominator
Improper Fraction: Numerator ≥ Denominator
Mixed Fraction: Combination of a whole number and a proper fraction
Operations on Fractions:
Addition/Subtraction: Find LCM of denominators, convert to like fractions, then add/subtract numerators
Multiplication: Multiply numerators and denominators separately
Division: Multiply by the reciprocal of the divisor
Decimals
Decimal: Another way to write fractions with denominators as powers of 10
Types: Terminating and Non-terminating recurring decimals
Operations:
Align decimal points for addition and subtraction
Multiply as regular numbers, then adjust decimal point based on total decimal places
In division, make divisor a whole number by multiplying both numbers by appropriate powers of 10
Exponents and Logarithms
Exponent: a^n means 'a' multiplied by itself 'n' times
Laws of Exponents:
a^m × a^n = a^(m+n)
a^m ÷ a^n = a^(m-n)
(a^m)^n = a^(m×n)
a^0 = 1 (a ≠ 0)
a^(-n) = 1/(a^n)
Logarithm: If a^x = y, then log_a(y) = x
Laws of Logarithms:
log_a(xy) = log_a(x) + log_a(y)
log_a(x/y) = log_a(x) - log_a(y)
log_a(x^n) = n log_a(x)
log_a(1) = 0
log_a(a) = 1
Sets
Set: Well-defined collection of distinct objects
Representation: Roster form {a, b, c, ...} or Set-builder form {x | x has property P}
Types:
Finite Set: Contains a countable number of elements
Infinite Set: Contains uncountably many elements
Empty Set (∅): Contains no elements
Universal Set (U): Contains all elements under consideration
Set Operations:
Union (A ∪ B): Elements in either A or B or both
Intersection (A ∩ B): Elements common to both A and B
Complement (A'): Elements in U but not in A
Difference (A - B): Elements in A but not in B
Cartesian Product (A × B): Set of all ordered pairs (a, b) where a ∈ A and b ∈ B
Algebra
Algebraic Expressions
Expression: Combination of variables, constants, and operations
Terms: Parts of expression separated by + or - signs
Like Terms: Terms with identical variable parts
Coefficient: Numerical factor in a term
Operations:
Addition/Subtraction: Combine like terms
Multiplication: Use distributive property and laws of exponents
Division: Factorize and cancel common factors
Equations
Equation: Statement of equality containing variables
Linear Equation: Highest power of variable is 1 (ax + b = 0)
Quadratic Equation: Highest power of variable is 2 (ax² + bx + c = 0)
Solving Linear Equations:
Simplify both sides
Collect variable terms on one side, constants on the other
Divide both sides by coefficient of variable
Solving Quadratic Equations:
Standard form: ax² + bx + c = 0
Solutions using quadratic formula: x = (-b ± √(b² - 4ac))/2a
Other methods: factorization, completing the square
Functions
Function: Relation between two sets where each input has exactly one output
Domain: Set of all possible input values
Range: Set of all possible output values
Types:
One-to-one: Different inputs have different outputs
Onto: Every element in range is an output for some input
Bijective: Both one-to-one and onto
Common Functions:
Linear: f(x) = mx + c
Quadratic: f(x) = ax² + bx + c
Polynomial: f(x) = a₀ + a₁x + a₂x² + ... + aₙxⁿ
Exponential: f(x) = aˣ
Logarithmic: f(x) = log_a(x)
Trigonometric: sin(x), cos(x), tan(x), etc.
Matrices
Matrix: Rectangular array of numbers arranged in rows and columns
Order: m × n (m rows, n columns)
Types:
Row Matrix: Only one row
Column Matrix: Only one column
Square Matrix: Equal number of rows and columns
Diagonal Matrix: Non-zero elements only on main diagonal
Identity Matrix (I): Diagonal matrix with all 1's on diagonal
Null Matrix: All elements are zero
Operations:
Addition/Subtraction: Add/subtract corresponding elements
Scalar Multiplication: Multiply each element by scalar
Matrix Multiplication: (A × B)ᵢⱼ = Σ aᵢₖ × bₖⱼ
Determinant: Scalar value associated with square matrices
Inverse: A⁻¹ such that A × A⁻¹ = A⁻¹ × A = I
Geometry
Points, Lines, and Angles
Point: No dimension, represented by a dot
Line: One-dimensional, extends infinitely in both directions
Line Segment: Part of a line with two endpoints
Ray: Part of a line with one endpoint, extends infinitely in one direction
Angle: Formed by two rays with a common endpoint
Types of Angles:
Acute: < 90°
Right: = 90°
Obtuse: > 90° but < 180°
Straight: = 180°
Reflex: > 180° but < 360°
Complete: = 360°
Angle Relationships:
Complementary Angles: Sum = 90°
Supplementary Angles: Sum = 180°
Adjacent Angles: Share a common side
Vertically Opposite Angles: Equal angles formed by intersecting lines
Triangles
Triangle: Three-sided polygon
Types by Sides:
Equilateral: All sides equal
Isosceles: Two sides equal
Scalene: All sides different
Types by Angles:
Acute: All angles < 90°
Right: One angle = 90°
Obtuse: One angle > 90°
Properties:
Sum of angles = 180°
Triangle inequality: Sum of any two sides > third side
Area = (1/2) × base × height
Pythagorean theorem (for right triangles): a² + b² = c²
Quadrilaterals
Quadrilateral: Four-sided polygon
Types:
Parallelogram: Opposite sides parallel and equal
Rectangle: Parallelogram with all angles = 90°
Square: Rectangle with all sides equal
Rhombus: Parallelogram with all sides equal
Trapezium: Exactly one pair of parallel sides
Kite: Two pairs of adjacent sides equal
Properties:
Sum of angles = 360°
Area of parallelogram = base × height
Area of trapezium = (1/2) × (sum of parallel sides) × height
Circles
Circle: Locus of points equidistant from a fixed point (center)
Radius: Distance from center to any point on circle
Diameter: Line segment passing through center connecting two points on circle
Chord: Line segment connecting two points on circle
Secant: Line intersecting circle at two points
Tangent: Line touching circle at exactly one point
Properties:
Diameter = 2 × radius
Circumference = 2πr
Area = πr²
Angle in semicircle = 90°
Angles in same segment are equal
Euclidean Geometry
Congruence: Same size and shape
Similarity: Same shape, possibly different size
Criteria for Triangle Congruence:
SSS (Side-Side-Side)
SAS (Side-Angle-Side)
ASA (Angle-Side-Angle)
RHS (Right angle-Hypotenuse-Side)
Criteria for Triangle Similarity:
AAA (Angle-Angle-Angle)
SSS (Side-Side-Side in same ratio)
SAS (Side-Angle-Side in same ratio)
3D Shapes
Common 3D Shapes:
Cube: 6 square faces
Cuboid: 6 rectangular faces
Sphere: All points equidistant from center
Cylinder: Two circular bases joined by curved surface
Cone: Circular base with tapering to a point (apex)
Prism: Polygonal bases connected by rectangular faces
Pyramid: Polygonal base connected to apex by triangular faces
Mensuration
Area of Plane Figures
Rectangle: Length × Width
Square: Side²
Triangle: (1/2) × Base × Height
Parallelogram: Base × Height
Rhombus: (1/2) × Product of diagonals
Trapezium: (1/2) × (Sum of parallel sides) × Height
Circle: πr²
Surface Area
Cube: 6 × (Side)²
Cuboid: 2(lb + bh + hl) where l = length, b = breadth, h = height
Sphere: 4πr²
Hemisphere: 3πr²
Cylinder: 2πr(r + h) where r = radius, h = height
Cone: πr(r + l) where r = radius, l = slant height
Volume
Cube: (Side)³
Cuboid: Length × Breadth × Height
Sphere: (4/3)πr³
Hemisphere: (2/3)πr³
Cylinder: πr²h
Cone: (1/3)πr²h
Prism: Base area × Height
Pyramid: (1/3) × Base area × Height
Coordinate Geometry
Cartesian Coordinate System
Coordinates: Ordered pair (x, y) representing position
Quadrants: Four regions divided by coordinate axes
Distance Formula: d = √[(x₂ - x₁)² + (y₂ - y₁)²]
Section Formula:
Midpoint: ((x₁ + x₂)/2, (y₁ + y₂)/2)
Point dividing in ratio m:n: ((mx₂ + nx₁)/(m+n), (my₂ + ny₁)/(m+n))
Area of Triangle: (1/2)|x₁(y₂ - y₃) + x₂(y₃ - y₁) + x₃(y₁ - y₂)|
Straight Line
General Form: ax + by + c = 0
Slope-Intercept Form: y = mx + c (m = slope, c = y-intercept)
Point-Slope Form: y - y₁ = m(x - x₁)
Two-Point Form: (y - y₁)/(y₂ - y₁) = (x - x₁)/(x₂ - x₁)
Slope: m = tan θ = (y₂ - y₁)/(x₂ - x₁)
Angle Between Lines:
If m₁, m₂ are slopes: tan θ = |(m₂ - m₁)/(1 + m₁m₂)|
Parallel lines: m₁ = m₂
Perpendicular lines: m₁m₂ = -1
Pair of Straight Lines
Combined Equation: ax² + 2hxy + by² + 2gx + 2fy + c = 0
Condition for representing pair of straight lines: h² = ab
Angle between lines: tan θ = 2√(h² - ab)/(a + b)
Circle
Standard Form: (x - h)² + (y - k)² = r² (center (h, k), radius r)
General Form: x² + y² + 2gx + 2fy + c = 0 (center (-g, -f), radius √(g² + f² - c))
Tangent to Circle: Perpendicular from center to tangent = radius
Conic Sections
Parabola
Standard Forms:
y² = 4ax (axis along x, vertex at origin)
x² = 4ay (axis along y, vertex at origin)
Focus: (a, 0) or (0, a)
Directrix: x = -a or y = -a
Ellipse
Standard Form: (x²/a²) + (y²/b²) = 1 (a > b)
Foci: (±c, 0) where c² = a² - b²
Semi-major axis: a
Semi-minor axis: b
Eccentricity: e = c/a
Hyperbola
Standard Form: (x²/a²) - (y²/b²) = 1
Foci: (±c, 0) where c² = a² + b²
Asymptotes: y = ±(b/a)x
Eccentricity: e = c/a
Permutations and Combinations
Fundamental Principles
Multiplication Principle: If an event can occur in m ways, and another independent event can occur in n ways, then the two events together can occur in m × n ways.
Addition Principle: If an event can occur in m ways, and another mutually exclusive event can occur in n ways, then either event can occur in m + n ways.
Factorial
Notation: n! = n × (n-1) × (n-2) × ... × 3 × 2 × 1
Special Cases:
0! = 1
1! = 1
Permutations
Definition: Arrangement of objects in a specific order
Formula: ₚPᵣ = n!/(n-r)!
Permutation with repetition: n^r
Permutation of n objects with repetition: n!/(n₁!n₂!...nₖ!) where n₁, n₂, ..., nₖ are frequencies
Combinations
Definition: Selection of objects without regard to order
Formula: ₙCᵣ = n!/[r!(n-r)!]
Properties:
ₙC₀ = ₙCₙ = 1
ₙCᵣ = ₙCₙ₋ᵣ
ₙCᵣ + ₙCᵣ₊₁ = ₙ₊₁Cᵣ₊₁
Three Dimensional Coordinates
3D Coordinate System
Coordinates: Ordered triplet (x, y, z)
Distance Formula: d = √[(x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)²]
Section Formula:
Midpoint: ((x₁ + x₂)/2, (y₁ + y₂)/2, (z₁ + z₂)/2)
Point dividing in ratio m:n: ((mx₂ + nx₁)/(m+n), (my₂ + ny₁)/(m+n), (mz₂ + nz₁)/(m+n))
Direction Cosines and Ratios
Direction Cosines: cos α, cos β, cos γ (angles with positive x, y, z axes)
Relation: cos²α + cos²β + cos²γ = 1
Direction Ratios: l, m, n proportional to direction cosines
Equation of Line
Vector Form: r = a + λb (a is position vector of point on line, b is direction vector)
Cartesian Form: (x - x₁)/l = (y - y₁)/m = (z - z₁)/n
Equation of Plane
Vector Form: r·n = d (n is normal vector, d is distance from origin)
Cartesian Form: ax + by + cz + d = 0
Intercept Form: x/a + y/b + z/c = 1
Vector Algebra
Vectors
Definition: Quantity with magnitude and direction
Representation: Bold letter (a) or arrow over letter (→)
Magnitude: |a| or |→|
Unit Vector: Vector with magnitude 1, denoted by â = a/|a|
Types of Vectors
Zero Vector: Magnitude = 0, direction undefined
Unit Vector: Magnitude = 1
Position Vector: Vector from origin to a point
Equal Vectors: Same magnitude and direction
Collinear Vectors: Parallel to same line
Coplanar Vectors: Lie in same plane
Vector Operations
Addition: Triangle or parallelogram law
Scalar Multiplication: Scales magnitude, may change direction
Dot Product: a·b = |a||b|cos θ (scalar result)
Cross Product: a×b = |a||b|sin θ n̂ (vector result perpendicular to both)
Vector Applications
Work Done: W = F·d
Torque: τ = r×F
Projection: proj_b a = (a·b)/|b| (scalar projection)
Vector Projection: (a·b/|b|²)b
Calculus
Limits
Definition: Value a function approaches as input approaches a specific value
Notation: lim[x→a] f(x) = L
Properties:
lim[x→a] [f(x) ± g(x)] = lim[x→a] f(x) ± lim[x→a] g(x)
lim[x→a] [f(x) × g(x)] = lim[x→a] f(x) × lim[x→a] g(x)
lim[x→a] [f(x)/g(x)] = lim[x→a] f(x)/lim[x→a] g(x) if lim[x→a] g(x) ≠ 0
Common Limits:
lim[x→0] (sin x)/x = 1
lim[x→0] (e^x - 1)/x = 1
lim[x→∞] (1 + 1/x)^x = e
Continuity
Definition: Function f is continuous at x = a if:
f(a) is defined
lim[x→a] f(x) exists
lim[x→a] f(x) = f(a)
Types of Discontinuities:
Removable: lim[x→a] f(x) exists but ≠ f(a) or f(a) undefined
Jump: Left and right limits exist but are unequal
Infinite: Function approaches infinity as x approaches a
Differentiation
Definition: Rate of change of function with respect to variable
Notation: dy/dx, f'(x), D_x[f(x)]
Basic Rules:
d/dx(c) = 0 (c is constant)
d/dx(x^n) = nx^(n-1)
d/dx(e^x) = e^x
d/dx(ln x) = 1/x
d/dx(sin x) = cos x
d/dx(cos x) = -sin x
d/dx(tan x) = sec^2 x
Combination Rules:
d/dx[f(x) ± g(x)] = f'(x) ± g'(x)
d/dx[f(x) × g(x)] = f'(x)g(x) + f(x)g'(x) (Product Rule)
d/dx[f(x)/g(x)] = [f'(x)g(x) - f(x)g'(x)]/[g(x)]² (Quotient Rule)
d/dx[f(g(x))] = f'(g(x)) × g'(x) (Chain Rule)
Applications of Derivatives
Rate of Change
Tangent and Normal: Slope of tangent = f'(x)
Increasing/Decreasing Function: f'(x) > 0 (increasing), f'(x) < 0 (decreasing)
Maxima and Minima: Critical points where f'(x) = 0 or f'(x) undefined
Second Derivative Test:
If f'(a) = 0 and f''(a) > 0, local minimum at x = a
If f'(a) = 0 and f''(a) < 0, local maximum at x = a
Point of Inflection: f''(x) = 0 and changes sign
Integration
Indefinite Integration: ∫f(x)dx = F(x) + C where F'(x) = f(x)
Definite Integration: ∫[a to b]f(x)dx = F(b) - F(a)
Basic Rules:
∫x^n dx = x^(n+1)/(n+1) + C (n ≠ -1)
∫1/x dx = ln|x| + C
∫e^x dx = e^x + C
∫sin x dx = -cos x + C
∫cos x dx = sin x + C
∫tan x dx = ln|sec x| + C
Methods of Integration:
Substitution: Let u = g(x), then ∫f(g(x))g'(x)dx = ∫f(u)du
By Parts: ∫u(x)v'(x)dx = u(x)v(x) - ∫v(x)u'(x)dx
Partial Fractions: Decompose rational function into simpler fractions
Applications of Integration
Area under Curve: A = ∫[a to b]f(x)dx
Area between Curves: A = ∫[a to b][f(x) - g(x)]dx
Volume of Revolution: V = π∫[a to b][f(x)]²dx (about x-axis)
Length of Curve: L = ∫[a to b]√[1 + (f'(x))²]dx
Differential Equations
Order: Highest derivative in equation
Degree: Power of highest derivative
General Solution: Solution with arbitrary constants
Particular Solution: Solution with specific values for constants
First Order Differential Equations:
Variable Separable: dy/dx = f(x)g(y)
Homogeneous: dy/dx = f(y/x)
Linear: dy/dx + P(x)y = Q(x)
Exact: M(x,y)dx + N(x,y)dy = 0 where ∂M/∂y = ∂N/∂x
Second Order Differential Equations:
With Constant Coefficients: a(d²y/dx²) + b(dy/dx) + cy = 0
Auxiliary Equation: am² + bm + c = 0
Particular Integral: Special solution for non-homogeneous equations
1. What is the value of \( 7 \times 8 \)?
Answer: 56
2. What is the area of a triangle with a base of 10 cm and a height of 5 cm?
Answer: 25 cm²
3. What is the value of \( \sqrt{49} \)?
Answer: 7
4. What is the sum of the angles in a triangle?
Answer: 180°
5. What is the value of \( 2^3 \)?
Answer: 8
6. If \( x + 5 = 12 \), what is the value of \( x \)?
Answer: 7
7. What is the perimeter of a square with side length 4 cm?
Answer: 16 cm
8. What is the value of \( 15 \div 3 \)?
Answer: 5
9. What is the HCF of 12 and 18?
Answer: 6
10. What is the value of \( \frac{3}{4} + \frac{1}{4} \)?
Answer: 1
11. What is the slope of a line passing through the points \( (1,2) \) and \( (3,6) \)?
Answer: 2
12. What is the value of \( \pi \) approximately?
Answer: 3.14
13. What is the volume of a cube with side length 3 cm?
Answer: 27 cm³
14. What is the value of \( 5! \) (5 factorial)?
Answer: 120
15. What is the value of \( \frac{1}{2} \times \frac{2}{3} \)?
Answer: \( \frac{1}{3} \)
100 Mathematics Multiple Choice Questions with Answers
Arithmetic
Question 1: According to the BODMAS rule, what is the value of 24 ÷ 4 × 2 + 3² - 5? A) 8 B) 14 C) 18 D) 20
Answer: C) 18 Explanation: Following BODMAS, 24 ÷ 4 × 2 + 3² - 5 = 6 × 2 + 9 - 5 = 12 + 9 - 5 = 16 + 4 = 18
Question 2: If a sum of money is divided in the ratio 2:3:5, and the smallest share is ₹1200, find the total sum. A) ₹5000 B) ₹6000 C) ₹7200 D) ₹8400
Answer: B) ₹6000 Explanation: If 2 parts = ₹1200, then 1 part = ₹600. Total parts = 2 + 3 + 5 = 10. Total sum = 10 × ₹600 = ₹6000
Question 3: If the cost price of an article is ₹800 and the selling price is ₹920, what is the profit percentage? A) 12% B) 15% C) 18% D) 20%
Answer: B) 15% Explanation: Profit = SP - CP = ₹920 - ₹800 = ₹120. Profit% = (Profit/CP) × 100% = (120/800) × 100% = 15%
Question 4: Calculate the simple interest on ₹4500 at 8% per annum for 3 years. A) ₹1080 B) ₹1100 C) ₹1200 D) ₹1350
Answer: A) ₹1080 Explanation: SI = (P × R × T)/100 = (4500 × 8 × 3)/100 = 108000/100 = ₹1080
Question 5: In what ratio must a shopkeeper mix two varieties of tea costing ₹200 per kg and ₹350 per kg to get a mixture worth ₹250 per kg? A) 2:1 B) 3:2 C) 5:2 D) 4:3
Answer: A) 2:1 Explanation: Using the rule of alligation, (350 - 250):(250 - 200) = 100:50 = 2:1
Question 6: A shirt marked at ₹1200 is sold at a discount of 15%. What is the selling price? A) ₹980 B) ₹1000 C) ₹1020 D) ₹1050
Answer: C) ₹1020 Explanation: Discount = 15% of ₹1200 = ₹180. Selling price = ₹1200 - ₹180 = ₹1020
Question 7: Find the compound interest on ₹8000 for 2 years at 10% per annum compounded annually. A) ₹1600 B) ₹1640 C) ₹1680 D) ₹1720
Answer: C) ₹1680 Explanation: A = P(1 + R/100)^T = 8000(1 + 10/100)^2 = 8000 × 1.21 = ₹9680. CI = ₹9680 - ₹8000 = ₹1680
Question 8: If GST on an item is 18%, what would be the final price of an item marked at ₹500? A) ₹550 B) ₹590 C) ₹600 D) ₹618
Answer: B) ₹590 Explanation: GST amount = 18% of ₹500 = ₹90. Final price = ₹500 + ₹90 = ₹590
Number System
Question 9: Which of the following is an irrational number? A) 0.25 B) 0.333... C) √9 D) √7
Answer: D) √7 Explanation: √7 cannot be expressed as a ratio of two integers, making it irrational.
Question 10: The HCF and LCM of two numbers are 12 and 336 respectively. If one number is 48, find the other number. A) 84 B) 96 C) 72 D) 108
Answer: A) 84 Explanation: Using the formula: Product of numbers = HCF × LCM. Second number = (HCF × LCM)/First number = (12 × 336)/48 = 84
Question 11: What is the value of (2^3)^2 × 2^-4? A) 2^2 B) 2^8 C) 2^6 D) 2^4
Answer: A) 2^2 Explanation: (2^3)^2 × 2^-4 = 2^6 × 2^-4 = 2^(6-4) = 2^2
Question 12: Express 0.8 as a fraction in its simplest form. A) 4/5 B) 8/10 C) 7/9 D) 4/9
Answer: A) 4/5 Explanation: 0.8 = 8/10, which simplifies to 4/5
Question 13: If log₁₀(x) = 3, then the value of x is: A) 10 B) 100 C) 1000 D) 30
Answer: C) 1000 Explanation: If log₁₀(x) = 3, then x = 10^3 = 1000
Question 14: If A = {1, 2, 3, 4} and B = {3, 4, 5, 6}, then A ∩ B equals: A) {1, 2, 3, 4, 5, 6} B) {3, 4} C) {1, 2, 5, 6} D) { }
Answer: B) {3, 4} Explanation: A ∩ B contains elements common to both sets, which are 3 and 4.
Question 15: Which of the following is NOT a prime number? A) 31 B) 51 C) 71 D) 97
Answer: B) 51 Explanation: 51 = 3 × 17, so it's not a prime number.
Question 16: Find the value of 2log₂5 + log₂4. A) log₂40 B) log₂100 C) log₂20 D) log₂50
Answer: D) log₂50 Explanation: 2log₂5 + log₂4 = log₂(5²) + log₂4 = log₂25 + log₂4 = log₂(25×4) = log₂100 = log₂50
Algebra
Question 17: Simplify the expression: 3(x - 2) - 2(x + 5) A) x - 16 B) x - 4 C) -x - 4 D) 5x - 16
Answer: A) x - 16 Explanation: 3(x - 2) - 2(x + 5) = 3x - 6 - 2x - 10 = x - 16
Question 18: Solve for x: 3x - 7 = 5x + 9 A) x = -8 B) x = 8 C) x = -7 D) x = 7
Answer: A) x = -8 Explanation: 3x - 7 = 5x + 9 → 3x - 5x = 9 + 7 → -2x = 16 → x = -8
Question 19: For the quadratic equation x² - 5x + 6 = 0, the roots are: A) 2 and 3 B) -2 and -3 C) -2 and 3 D) 2 and -3
Answer: A) 2 and 3 Explanation: x² - 5x + 6 = 0 can be factored as (x - 2)(x - 3) = 0, giving roots x = 2 and x = 3.
Question 20: If f(x) = 2x² - 3x + 1 and g(x) = x + 2, find f(g(2)). A) 15 B) 17 C) 19 D) 21
Answer: C) 19 Explanation: g(2) = 2 + 2 = 4. f(g(2)) = f(4) = 2(4)² - 3(4) + 1 = 2(16) - 12 + 1 = 32 - 12 + 1 = 21
Question 21: If A = [2 3; 1 4] and B = [1 0; 2 3], find A + B. A) [3 3; 3 7] B) [3 0; 3 7] C) [3 3; 0 7] D) [2 0; 3 7]
Answer: A) [3 3; 3 7] Explanation: A + B = [2+1 3+0; 1+2 4+3] = [3 3; 3 7]
Question 22: If A = [1 2; 3 4], find det(A). A) -2 B) 2 C) -5 D) 5
Answer: A) -2 Explanation: det(A) = 1×4 - 2×3 = 4 - 6 = -2
Question 23: Which of the following is a one-to-one function? A) f(x) = x² B) f(x) = x³ C) f(x) = |x| D) f(x) = sin x
Answer: B) f(x) = x³ Explanation: For a one-to-one function, each output corresponds to exactly one input. f(x) = x³ satisfies this property.
Question 24: The domain of the function f(x) = √(x-4) is: A) x ≥ 0 B) x ≥ 4 C) x > 4 D) All real numbers
Answer: B) x ≥ 4 Explanation: For the square root to be defined, the expression inside must be non-negative: x - 4 ≥ 0, thus x ≥ 4.
Geometry
Question 25: In a triangle ABC, if angle A = 65° and angle B = 45°, what is angle C? A) 60° B) 65° C) 70° D) 75°
Answer: C) 70° Explanation: In a triangle, the sum of angles is 180°. So, angle C = 180° - 65° - 45° = 70°
Question 26: What is the measure of each interior angle of a regular hexagon? A) 108° B) 120° C) 135° D) 144°
Answer: B) 120° Explanation: For a regular polygon with n sides, each interior angle = (n-2) × 180° / n. For hexagon (n=6), angle = (6-2) × 180° / 6 = 4 × 30° = 120°
Question 27: In a right-angled triangle, if one of the acute angles is 30°, the other acute angle is: A) 30° B) 45° C) 60° D) 90°
Answer: C) 60° Explanation: In a right-angled triangle, the sum of all angles is 180°. If one angle is 90° and another is 30°, then the third angle = 180° - 90° - 30° = 60°
Question 28: Two angles are supplementary if their sum equals: A) 90° B) 180° C) 270° D) 360°
Answer: B) 180° Explanation: Supplementary angles are those whose sum equals 180°.
Question 29: Which of the following is NOT a property of a rhombus? A) All sides are equal B) Diagonals bisect each other C) All angles are equal D) Diagonals bisect opposite angles
Answer: C) All angles are equal Explanation: In a rhombus, all sides are equal but all angles are equal only if it's also a square.
Question 30: The angle in a semicircle is: A) 30° B) 60° C) 90° D) 180°
Answer: C) 90° Explanation: The angle in a semicircle is always 90° (right angle).
Question 31: Two triangles are similar if: A) All corresponding sides are equal B) All corresponding angles are equal C) Area is the same D) Perimeter is the same
Answer: B) All corresponding angles are equal Explanation: Two triangles are similar if all their corresponding angles are equal (AAA criterion).
Question 32: What is the sum of interior angles of a polygon with 7 sides? A) 540° B) 720° C) 900° D) 1080°
Answer: C) 900° Explanation: Sum of interior angles of an n-sided polygon = (n-2) × 180°. For n=7, sum = (7-2) × 180° = 5 × 180° = 900°
Mensuration
Question 33: Find the area of a circle with radius 7 cm. A) 49π cm² B) 14π cm² C) 154 cm² D) 22 cm²
Answer: A) 49π cm² Explanation: Area of circle = πr² = π × 7² = 49π cm²
Question 34: The surface area of a cube with side length 5 cm is: A) 25 cm² B) 125 cm² C) 150 cm² D) 175 cm²
Answer: C) 150 cm² Explanation: Surface area of cube = 6a² = 6 × 5² = 6 × 25 = 150 cm²
Question 35: The volume of a cylinder with radius 3 cm and height 10 cm is: A) 30π cm³ B) 90π cm³ C) 120π cm³ D) 150π cm³
Answer: B) 90π cm³ Explanation: Volume of cylinder = πr²h = π × 3² × 10 = 9π × 10 = 90π cm³
Question 36: The total surface area of a hemisphere with radius 7 cm is: A) 147π cm² B) 294π cm² C) 98π cm² D) 196π cm²
Answer: A) 147π cm² Explanation: Total surface area of hemisphere = 3πr² = 3 × π × 7² = 3 × π × 49 = 147π cm²
Question 37: The area of a trapezium with parallel sides 8 cm and 12 cm, and height 6 cm is: A) 48 cm² B) 60 cm² C) 72 cm² D) 96 cm²
Answer: B) 60 cm² Explanation: Area of trapezium = (1/2) × (sum of parallel sides) × height = (1/2) × (8 + 12) × 6 = (1/2) × 20 × 6 = 60 cm²
Question 38: A cone has a height of 12 cm and base radius 5 cm. Its volume is: A) 100π cm³ B) 200π cm³ C) 300π cm³ D) 400π cm³
Answer: A) 100π cm³ Explanation: Volume of cone = (1/3) × πr²h = (1/3) × π × 5² × 12 = (1/3) × π × 25 × 12 = (1/3) × 300π = 100π cm³
Question 39: The area of an equilateral triangle with side 6 cm is: A) 9√3 cm² B) 6√3 cm² C) 3√3 cm² D) 12√3 cm²
Answer: A) 9√3 cm² Explanation: Area of equilateral triangle = (√3/4) × side² = (√3/4) × 6² = (√3/4) × 36 = 9√3 cm²
Question 40: If the perimeter of a square is 40 cm, its area is: A) 25 cm² B) 100 cm² C) 400 cm² D) 625 cm²
Answer: B) 100 cm² Explanation: Perimeter = 4s = 40 cm, so s = 10 cm. Area = s² = 10² = 100 cm²
Coordinate Geometry
Question 41: The distance between points (3, 4) and (7, 1) is: A) 5 units B) 6 units C) 7 units D) 8 units
Answer: A) 5 units Explanation: Distance = √[(x₂-x₁)² + (y₂-y₁)²] = √[(7-3)² + (1-4)²] = √[16 + 9] = √25 = 5 units
Question 42: The midpoint of the line segment joining (2, -3) and (6, 5) is: A) (4, 1) B) (4, 2) C) (3, 1) D) (4, -1)
Answer: A) (4, 1) Explanation: Midpoint = ((x₁+x₂)/2, (y₁+y₂)/2) = ((2+6)/2, (-3+5)/2) = (4, 1)
Question 43: The slope of the line passing through points (2, 3) and (5, 9) is: A) 2 B) 3 C) 1/2 D) 2/3
Answer: A) 2 Explanation: Slope = (y₂-y₁)/(x₂-x₁) = (9-3)/(5-2) = 6/3 = 2
Question 44: The equation of a line with slope 2 and y-intercept 3 is: A) 2x + y = 3 B) y = 2x + 3 C) y = 3x + 2 D) x + 2y = 3
Answer: B) y = 2x + 3 Explanation: For a line with slope m and y-intercept c, the equation is y = mx + c. Here, y = 2x + 3.
Question 45: The center and radius of the circle x² + y² - 6x + 8y + 9 = 0 are: A) (3, -4) and 4 B) (-3, 4) and 4 C) (3, -4) and 2 D) (-3, 4) and 2
Answer: A) (3, -4) and 4 Explanation: Rearranging to standard form: (x - 3)² + (y + 4)² = 25. So center = (3, -4) and radius = 5.
Question 46: The focus of the parabola y² = 8x is: A) (0, 2) B) (2, 0) C) (0, 8) D) (8, 0)
Answer: B) (2, 0) Explanation: For y² = 4ax, the focus is at (a, 0). Here, 4a = 8, so a = 2. Focus = (2, 0).
Question 47: If two lines have slopes 2 and -1/2 respectively, then they are: A) Parallel B) Perpendicular C) Neither parallel nor perpendicular D) Coincident
Answer: B) Perpendicular Explanation: For perpendicular lines, the product of slopes = -1. Here, 2 × (-1/2) = -1, so they are perpendicular.
Question 48: The eccentricity of an ellipse with semi-major axis 5 and semi-minor axis 3 is: A) 3/5 B) 4/5 C) 5/3 D) 5/4
Answer: B) 4/5 Explanation: For an ellipse, e = √(1 - b²/a²) = √(1 - 3²/5²) = √(1 - 9/25) = √(16/25) = 4/5
Permutations and Combinations
Question 49: How many 4-digit numbers can be formed using the digits 1, 2, 3, 4, 5 without repetition? A) 24 B) 120 C) 625 D) 720
Answer: B) 120 Explanation: This is a permutation problem: ₅P₄ = 5!/(5-4)! = 5!/1! = 5! = 120
Question 50: In how many ways can 5 different books be arranged on a shelf? A) 20 B) 60 C) 120 D) 720
Answer: C) 120 Explanation: This is a permutation of 5 distinct objects: 5! = 120
Question 51: How many ways can a committee of 3 be selected from 8 people? A) 24 B) 56 C) 336 D) 6720
Answer: B) 56 Explanation: This is a combination problem: ₈C₃ = 8!/(3!(8-3)!) = 8!/(3!5!) = 56
Question 52: In how many ways can 4 boys and 3 girls be seated in a row if the boys and girls must alternate? A) 144 B) 288 C) 576 D) 720
Answer: B) 288 Explanation: Boys and girls must alternate, so either BGBGBGB or GBGBGBG. For the first case: 4! × 3! = 24 × 6 = 144. Similarly for the second case. Total = 144 + 144 = 288.
Question 53: How many diagonals does a polygon with 8 sides have? A) 16 B) 20 C) 28 D) 32
Answer: B) 20 Explanation: For an n-sided polygon, number of diagonals = n(n-3)/2. For n=8, diagonals = 8(8-3)/2 = 8×5/2 = 20
Question 54: From a pack of 52 cards, in how many ways can 5 cards be selected? A) 2,598,960 B) 311,875 C) 2,598 D) 52
Answer: A) 2,598,960 Explanation: This is a combination problem: ₅₂C₅ = 52!/(5!(52-5)!) = 52!/(5!47!) = 2,598,960
Question 55: If ₁₀C₃ = 120, then find the value of ₁₀C₇. A) 90 B) 120 C) 210 D) 360
Answer: B) 120 Explanation: Using property ₙCᵣ = ₙCₙ₋ᵣ, we get ₁₀C₇ = ₁₀C₃ = 120
Question 56: How many 6-letter words can be formed from the letters of the word "SQUARE" if repetition is not allowed? A) 6 B) 36 C) 720 D) 46656
Answer: C) 720 Explanation: The word "SQUARE" has 6 distinct letters. Number of arrangements = 6! = 720
Three Dimensional Coordinates
Question 57: The distance between points (1, 2, 3) and (4, 5, 1) in 3D space is: A) 3√2 units B) 3√3 units C) 5 units D) 7 units
Answer: C) 5 units Explanation: Distance = √[(x₂-x₁)² + (y₂-y₁)² + (z₂-z₁)²] = √[(4-1)² + (5-2)² + (1-3)²] = √[9 + 9 + 4] = √25 = 5 units
Question 58: The midpoint of the line segment joining (2, -1, 3) and (6, 3, 5) is: A) (4, 1, 4) B) (4, 2, 4) C) (4, 2, 8) D) (4, 1, 8)
Answer: A) (4, 1, 4) Explanation: Midpoint = ((x₁+x₂)/2, (y₁+y₂)/2, (z₁+z₂)/2) = ((2+6)/2, (-1+3)/2, (3+5)/2) = (4, 1, 4)
Question 59: If A = (1, -2, 3), B = (2, 0, 4) and C = (3, 1, 2), which of these points are collinear? A) A, B, C B) Only A and B C) Only B and C D) None of them
Answer: D) None of them Explanation: For collinearity, the direction ratios should be proportional. Direction ratio AB = (1, 2, 1) and BC = (1, 1, -2), which are not proportional.
Question 60: The direction cosines of a line satisfy the equation: A) cos²α + cos²β + cos²γ = 0 B) cos²α + cos²β + cos²γ = 1 C) cos²α + cos²β + cos²γ = 2 D) cos²α + cos²β + cos²γ = 3
Answer: B) cos²α + cos²β + cos²γ = 1 Explanation: For direction cosines of a line, the sum of squares is always 1.
Question 61: The equation of a plane passing through the point (1, 2, 3) with normal vector (2, -1, 4) is: A) 2x - y + 4z = 15 B) 2x - y + 4z = 16 C) 2x - y + 4z = 17 D) 2x - y + 4z = 18
Answer: A) 2x - y + 4z = 15 Explanation: Equation of plane with normal (a, b, c) through point (x₀, y₀, z₀) is a(x-x₀) + b(y-y₀) + c(z-z₀) = 0. Substituting: 2(x-1) - (y-2) + 4(z-3) = 0 → 2x - y + 4z - 2 + 2 - 12 = 0 → 2x - y + 4z = 12
Question 62: The equation of a line passing through (1, 2, 3) with direction ratios (2, 1, -1) is: A) (x-1)/2 = (y-2)/1 = (z-3)/-1 B) (x-1)/1 = (y-2)/2 = (z-3)/-1 C) (x-1)/2 = (y-2)/1 = (z-3)/1 D) x/1 = y/2 = z/3
Answer: A) (x-1)/2 = (y-2)/1 = (z-3)/-1 Explanation: Equation of line through (x₀, y₀, z₀) with direction ratios (a, b, c) is (x-x₀)/a = (y-y₀)/b = (z-z₀)/c. Substituting gives (x-1)/2 = (y-2)/1 = (z-3)/-1.
Vector Algebra
Question 63: If vectors a = 2i + 3j - k and b = i - 2j + 2k, find |a + b|. A) 3 B) √14 C) √13 D) 5
Answer: B) √14 Explanation: a + b = (2+1)i + (3-2)j + (-1+2)k = 3i + j + k. |a + b| = √(3² + 1² + 1²) = √(9 + 1 + 1) = √11
Question 64: The dot product of vectors a = 3i - 2j + k and b = i + j - k is: A) 0 B) 1 C) 2 D) 3
Answer: A) 0 Explanation: a·b = 3×1 + (-2)×1 + 1×(-1) = 3 - 2 - 1 = 0
Question 65: If vectors a = 2i + j and b = i - 3j, then the angle between them is: A) 30° B) 45° C) 60° D) 90°
Answer: D) 90° Explanation: a·b = 2×1 + 1×(-3) = 2 - 3 = -1. Since dot product is zero, the angle is 90°.
Question 66: The vector product of i and j is: A) i B) j C) k D) 0
Answer: C) k Explanation: i × j = k as per the right-hand rule of vector cross product.
Question 67: If a = 3i + 4j and b = i + j, the scalar projection of a onto b is: A) 5 B) 5√2 C) 7/√2 D) 7
Answer: C) 7/√2 Explanation: Scalar projection = a·b/|b| = (3×1 + 4×1)/√(1² + 1²) = 7/√2
Question 68: If a = 2i - j + 3k and b = i + 2j - k, then a × b equals: A) -7i - 5j - 5k B) 7i + 5j + 5k C) -5i - 7j - 5k D) 5i + 7j + 5k
Answer: A) -7i - 5j - 5k Explanation: a × b = |2 -1 3; 1 2 -1| = ((-1)×(-1) - 3×2)i - (2×(-1) - 3×1)j + (2×2 - (-1)×1)k
Question 69: Which of the following is NOT a property of vector addition? A) Commutative B) Associative C) Distributive over scalar multiplication D) Additive identity is the unit vector
Answer: D) Additive identity is the unit vector Explanation: The additive identity in vector addition is the zero vector, not the unit vector.
Calculus
Question 70: Evaluate lim[x→0] (sin x)/x A) 0 B) 1 C) -1 D) Does not exist
Answer: B) 1 Explanation: This is a standard limit with value equal to 1.
Question 71: A function f(x) is continuous at x = a if: A) f(a) is defined B) lim[x→a] f(x) exists C) lim[x→a] f(x) = f(a) D) All of the above
Answer: D) All of the above Explanation: For continuity at x = a, all three conditions must be satisfied.
Question 72: Find d/dx (x²sin x). A) 2x sin x + x²cos x B) 2x sin x - x²cos x C) 2x sin x D) x²cos x
Answer: A) 2x sin x + x²cos x Explanation: Use product rule: d/dx(uv) = u(dv/dx) + v(du/dx). Here, u = x² and v = sin x.
Question 73: If f(x) = 3x² - 12x + 7, then f'(2) equals: A) 0 B) 12 C) -12 D) -24
Answer: A) 0 Explanation: f'(x) = 6x - 12, so f'(2) = 6(2) - 12 = 12 - 12 = 0
Question 74: The derivative of e^x with respect to x is: A) e^x B) x·e^x C) e^x·ln x D) e^x/x
Answer: A) e^x Explanation: The derivative of e^x is e^x.
Question 75: Find the derivative of ln(sin x). A) cot x B) tan x C) cos x D) 1/sin x
Answer: A) cot x Explanation: d/dx[ln(sin x)] = (1/sin x) · (cos x) = cot x
Question 76: The critical points of the function f(x) = x³ - 3x² - 9x + 7 are: A) x = -1 and x = 3 B) x = 1 and x = 3 C) x = -3 and x = 1 D) x = 0 and x = 2
Answer: A) x = -1 and x = 3 Explanation: f'(x) = 3x² - 6x - 9 = 3(x² - 2x - 3) = 3(x - 3)(x + 1). Setting f'(x) = 0 gives x = -1 or x = 3.
Question 77: If f'(x) > 0 for all x ∈ (a, b), then f(x) is: A) Decreasing on (a, b) B) Increasing on (a, b) C) Constant on (a, b) D) None of these
Answer: B) Increasing on (a, b) Explanation: When the first derivative is positive, the function is increasing.
Question 78: Evaluate ∫ x·cos x dx A) sin x - x·cos x + C B) sin x + x·cos x + C C) x·sin x - cos x + C D) x·sin x + cos x + C
Answer: D) x·sin x + cos x + C Explanation: Using integration by parts with u = x and dv = cos x dx.
Question 79: Find ∫₁³ (2x + 3) dx A) 10 B) 12 C) 15 D) 18
Answer: C) 15 Explanation: ∫₁³ (2x + 3) dx = [x² + 3x]₁³ = (9 + 9) - (1 + 3) = 18 - 4 = 14
Question 80: ∫ (1/x) dx equals: A) ln|x| + C B) log₁₀|x| + C C) e^x + C D) 1/(x+1) + C
Answer: A) ln|x| + C Explanation: The integral of 1/x is ln|x| + C.
Question 81: The area under the curve y = x² between x = 0 and x = 2 is: A) 8/3 B) 4 C) 8 D) 16/3
Answer: A) 8/3 Explanation: Area = ∫₀² x² dx = [x³/3]₀² = 8/3 - 0 = 8/3
Question 82: The solution of the differential equation dy/dx = 2x with the initial condition y(0) = 3 is: A) y = x² + 3 B) y = 2x² + 3 C) y = x² + 2 D) y = 2x + 3
Answer: A) y = x² + 3 Explanation: Integrating dy/dx = 2x gives y = x² + C. Using initial condition y(0) = 3: 3 = 0 + C, so C = 3. Thus, y = x² + 3.
Question 83: The integrating factor for the differential equation dy/dx + Py = Q is: A) e^∫P dx B) e^∫Q dx C) e^∫(P+Q) dx D) e^∫PQ dx
Answer: A) e^∫P dx Explanation: For the linear differential equation dy/dx + Py = Q, the integrating factor is e^∫P dx.
Question 84: If y = sin(x²), then dy/dx equals: A) cos(x²) B) 2x·cos(x²) C) sin(2x) D) 2·sin(x)·cos(x)
Answer: B) 2x·cos(x²) Explanation: Using the chain rule: dy/dx = (dy/du) · (du/dx) where u = x². dy/du = cos(u) and du/dx = 2x. Therefore, dy/dx = cos(x²) · 2x = 2x·cos(x²).
Mixed Topics
Question 85: If a = 4, b = 3, and c = 5, find the value of a² + b² + c² - ab - bc - ca. A) 0 B) 3 C) 9 D) 12
Answer: C) 9 Explanation: a² + b² + c² - ab - bc - ca = 16 + 9 + 25 - 12 - 15 - 20 = 50 - 47 = 3
Question 86: If f(x) = x² - 3x and g(x) = 2x + 1, find (f ∘ g)(x). A) 4x² + 4x - 2 B) 4x² + 4x - 2 C) 2x² - 6x + 1 D) 2x² - 6x - 3
Answer: B) 4x² + 4x - 2 Explanation: (f ∘ g)(x) = f(g(x)) = f(2x + 1) = (2x + 1)² - 3(2x + 1) = 4x² + 4x + 1 - 6x - 3 = 4x² - 2x - 2
Question 87: The solution of the equation log₃(x-2) = 2 is: A) 11 B) 8 C) 5 D) 4
Answer: A) 11 Explanation: log₃(x-2) = 2 means x - 2 = 3² = 9, thus x = 11.
Question 88: What is the slope of a line perpendicular to the line 3x - 4y + 5 = 0? A) 3/4 B) 4/3 C) -3/4 D) -4/3
Answer: B) 4/3 Explanation: The given line can be written as y = (3x + 5)/4, so its slope is 3/4. The slope of a perpendicular line is the negative reciprocal: -1/(3/4) = -4/3.
Question 89: A bag contains 5 white and 7 black balls. If two balls are drawn at random, the probability that both are of the same color is: A) 5/33 B) 7/33 C) 17/33 D) 19/33
Answer: C) 17/33 Explanation: P(both white) = (5C₂)/(12C₂) = 10/66 = 5/33. P(both black) = (7C₂)/(12C₂) = 21/66 = 7/22. P(same color) = 5/33 + 7/22 = 15/66 + 21/66 = 36/66 = 6/11.
Question 90: If A = {1, 2, 3} and B = {2, 3, 4, 5}, then A ∪ B equals: A) {1, 2, 3, 4, 5} B) {2, 3} C) {1, 4, 5} D) {1, 2, 3, 4}
Answer: A) {1, 2, 3, 4, 5} Explanation: A ∪ B contains all elements from both sets, without duplicates.
Question 91: If z = 2 - 3i, then |z| equals: A) √5 B) √13 C) 5 D) 13
Answer: B) √13 Explanation: |z| = √(a² + b²) = √(2² + (-3)²) = √(4 + 9) = √13
Question 92: The domain of the function f(x) = √(4 - x²) is: A) [-2, 2] B) (-2, 2) C) [0, 2] D) (-∞, ∞)
Answer: A) [-2, 2] Explanation: For the square root to be defined, 4 - x² ≥ 0, which gives -2 ≤ x ≤ 2.
Question 93: If A is a square matrix such that A² = A, then (I - A)² equals: A) I - A B) I + A C) I - A² D) I²
Answer: A) I - A Explanation: (I - A)² = I² - 2A + A² = I - 2A + A = I - A (since A² = A)
Question 94: The polar form of the complex number z = 1 + i is: A) √2(cos(π/4) + i sin(π/4)) B) √2(cos(π/2) + i sin(π/2)) C) 2(cos(π/4) + i sin(π/4)) D) 2(cos(π/2) + i sin(π/2))
Answer: A) √2(cos(π/4) + i sin(π/4)) Explanation: |z| = √(1² + 1²) = √2 and arg(z) = tan⁻¹(1/1) = π/4. So z = √2(cos(π/4) + i sin(π/4)).
Question 95: If cos θ = 3/5 and θ is in the first quadrant, then tan θ equals: A) 3/4 B) 4/3 C) 3/5 D) 5/3
Answer: B) 4/3 Explanation: If cos θ = 3/5, then sin θ = 4/5 (using sin² θ + cos² θ = 1). Thus, tan θ = sin θ/cos θ = (4/5)/(3/5) = 4/3.
Question 96: The point (3, 4) after rotation of 90° clockwise about the origin becomes: A) (4, -3) B) (-4, 3) C) (4, 3) D) (-3, -4)
Answer: A) (4, -3) Explanation: For 90° clockwise rotation, (x, y) becomes (y, -x). So (3, 4) becomes (4, -3).
Question 97: The circumcentre of a triangle with vertices (0, 0), (1, 0) and (0, 1) is: A) (0, 0) B) (1/2, 1/2) C) (1/3, 1/3) D) (1, 1)
Answer: B) (1/2, 1/2) Explanation: The circumcentre is equidistant from all vertices, which is at (1/2, 1/2) for this triangle.
Question 98: The equation of a circle passing through the origin and having center at (0, 3) is: A) x² + y² - 3y = 0 B) x² + y² - 6y = 0 C) x² + y² + 3y = 0 D) x² + y² - 3y + 9 = 0
Answer: B) x² + y² - 6y = 0 Explanation: Standard equation: (x - h)² + (y - k)² = r². With center (0, 3), we have (x - 0)² + (y - 3)² = r². The circle passes through origin, so r² = (0 - 0)² + (0 - 3)² = 9. Thus, x² + (y - 3)² = 9, which expands to x² + y² - 6y + 9 = 9, or x² + y² - 6y = 0.
Question 99: The period of the function y = sin 2x + cos 3x is: A) π B) 2π C) 6π D) 2π/3
Answer: B) 2π Explanation: Period of sin nx is 2π/n and period of cos nx is 2π/n. The period of sum is the LCM of individual periods: LCM(2π/2, 2π/3) = LCM(π, 2π/3) = 2π.
Question 100: If the position vectors of points A, B, and C are 2i + 3j - k, 4i + j + 2k, and i + 5j - 3k respectively, then the area of triangle ABC is: A) 5 square units B) 10 square units C) 15 square units D) 7.5 square units
Answer: D) 7.5 square units Explanation: Area of triangle = (1/2)|AB × AC|. AB = (4-2)i + (1-3)j + (2+1)k = 2i - 2j + 3k. AC = (1-2)i + (5-3)j + (-3+1)k = -i + 2j - 2k. AB × AC = |(2 -2 3; -1 2 -2)| = ((-2)×(-2) - 3×2)i - (2×(-2) - 3×(-1))j + (2×2 - (-2)×(-1))k = (4-6)i - (-4-3)j + (4-2)k = -2i + 7j + 2k. |AB × AC| = √((-2)² + 7² + 2²) = √(4+49+4) = √57 = 3√19/2. Area = (1/2) × |AB × AC| = (1/2) × 3√19/2 = 7.5