Quiz: Relations and Functions (15 Questions)

1. The relation R = {(a, b) : a - b is even} on Z is:

Reflexive only
Symmetric only
Equivalence relation

2. A function f: A → B is said to be onto if:

Every element of B has a pre-image in A
Each element of A maps to a unique element of B
f is one-one

3. The function f(x) = x² is:

One-one
Many-one
Onto

4. If f: A → B and g: B → C, then (g∘f)(x) equals:

g(f(x))
f(g(x))
f(x) + g(x)

5. Domain of the function f(x) = √(x-2):

x > 2
x ≥ 2
All real numbers

6. Number of relations on a set with 3 elements is:

3
9
512

7. A function is one-one if:

Every element of codomain has a pre-image
Different elements of domain map to different elements in codomain
f(x) is increasing

8. If A has m elements and B has n elements, number of functions from A to B is:

m^n
n^m
m+n

9. Identity function f: R → R is defined as:

f(x) = x
f(x) = 1
f(x) = 0

10. The inverse of f(x) = 2x + 3 is:

f⁻¹(x) = (x - 3)/2
f⁻¹(x) = 2x - 3
f⁻¹(x) = (x + 3)/2

11. If a relation is symmetric and transitive, it may not be:

Reflexive
Symmetric
Transitive

12. A relation on A = {1, 2, 3} given by R = {(1,1), (2,2), (3,3)} is:

Only reflexive
Equivalence relation
Reflexive and symmetric

13. Range of the constant function f(x) = 7 is:

All real numbers
7
{7}

14. Which of the following is not a function?

f = {(1,2), (2,3)}
f = {(1,2), (1,3)}
f = {(2,4), (3,5)}

15. The number of one-one functions from set A with 3 elements to set B with 5 elements is:

60
125
15

Quiz: Inverse Trigonometric Functions (15 Questions)

1. The principal value branch of sin⁻¹x is:

[0, π]
[-π/2, π/2]
[0, 2π]

2. The domain of cos⁻¹x is:

(−∞, ∞)
[−1, 1]
[0, π]

3. Value of tan⁻¹(1) is:

0
π/4
π/2

4. sin⁻¹(sin(π/3)) =

π/3
2π/3
π

5. The value of cos⁻¹(−1/2) is:

2π/3
π/3
π/6

6. tan⁻¹(−√3) equals:

−π/3
π/3
−π/4

7. sin⁻¹(−1/2) =

−π/6
π/6
−π/3

8. cos⁻¹(cos(5π/6)) =

5π/6
π/6
π − 5π/6

9. Domain of tan⁻¹x is:

(−∞, ∞)
[−1, 1]
[0, π]

10. If sin⁻¹x + cos⁻¹x = α, then α equals:

0
π/2
π

11. The range of tan⁻¹x is:

[−π/2, π/2]
(−π/2, π/2)
[0, π]

12. sin⁻¹(1) + cos⁻¹(0) =

π
π/2
π/4

13. The function y = sin⁻¹x is:

Increasing
Decreasing
Constant

14. sin⁻¹(sin(5π/6)) =

5π/6
π/6
π/3

15. cos⁻¹(−1) =

0
π
π/2

Quiz: Matrices (15 Questions)

1. A matrix which has only one row is called:

Column matrix
Row matrix
Null matrix

2. The order of a matrix having 3 rows and 2 columns is:

2 × 3
3 × 2
3 + 2

3. The matrix A = [0] is called:

Scalar matrix
Zero matrix
Identity matrix

4. If A is a square matrix, then |A| is called:

Trace
Determinant
Transpose

5. A matrix A is symmetric if:

Aᵀ = −A
Aᵀ = A
AᵀA = I

6. If A is a 2×2 matrix, then A + Aᵀ is:

Always symmetric
Always skew-symmetric
Always zero matrix

7. The product of a matrix of order m×n with another of order n×p is of order:

m×p
m×n
n×p

8. If A = I (identity matrix), then A⁻¹ =

A
0
−A

9. (Aᵀ)ᵀ =

A
Aᵀ

10. Matrix multiplication is:

Commutative
Not always commutative
Always commutative

11. The inverse of a matrix exists only if:

It is a square matrix
It is symmetric
Its determinant is non-zero

12. The identity matrix of order 2 is:

[[0, 1], [1, 0]]
[[1, 0], [0, 1]]
[[1, 1], [1, 1]]

13. If A is a 2×2 matrix and k is a scalar, then (kA)ᵀ =

kAᵀ
Aᵀ + k
k + A

14. The matrix A = [[0, −1], [1, 0]] is:

Symmetric
Skew-symmetric
Identity matrix

15. The transpose of a symmetric matrix is:

Skew-symmetric
Symmetric
Diagonal

Quiz: Determinants (15 Questions)

1. The determinant of a 2×2 matrix [[a, b], [c, d]] is:

ad + bc
ad − bc
ab − cd

2. If any two rows (or columns) of a determinant are identical, then the determinant is:

Zero
One
Two

3. The value of determinant |1 0 0; 0 1 0; 0 0 1| is:

3
1
0

4. Determinant is defined only for:

Square matrices
Rectangular matrices
Row matrices

5. If determinant |A| = 0, then A is:

Invertible
Singular
Symmetric

6. The value of |2 3; 4 6| is:

0
6
12

7. If A is a matrix and k is a scalar, then |kA| =

k|A|
kⁿ|A| (where n = order)
|A|/k

8. Interchanging two rows of a determinant:

Doubles the value
Does not change the value
Changes the sign

9. The value of determinant |a b; 0 d| is:

ab
ad
ad − 0

10. Which property is used in Laplace expansion?

Minor and cofactor
Transpose
Adjoint

11. If A is a 3×3 matrix and |A| ≠ 0, then A⁻¹ exists and is given by:

adj(A)
Aᵀ/|A|
adj(A)/|A|

12. Which of the following is true for |AB|?

|AB| = |A| + |B|
|AB| = |A||B|
|AB| = |A| − |B|

13. The adjoint of a matrix is the:

Transpose of cofactor matrix
Inverse of matrix
Determinant

14. If |A| = 5, then |A⁻¹| =

1
0.2
5

15. Value of determinant |3 2; 1 4| is:

10
8
14

Quiz: Continuity and Differentiability (15 Questions)

1. A function f(x) is continuous at x = a if:

f(a) is undefined
Limit does not exist
lim(x→a) f(x) = f(a)

2. The function f(x) = |x| is:

Differentiable at x = 0
Not continuous at x = 0
Not differentiable at x = 0

3. If f(x) = x², then f'(x) is:

2x
x

4. The derivative of sin(x) is:

−cos(x)
cos(x)
sin(x)

5. A function can be continuous but not:

Defined
Differentiable
Periodic

6. If a function is differentiable at a point, then it is:

Always continuous at that point
Not continuous
Not defined

7. The derivative of tan(x) is:

sec²(x)
cot(x)
−cosec²(x)

8. The function f(x) = x³ is:

Not continuous
Continuous and differentiable everywhere
Discontinuous at x = 0

9. The function f(x) = |x| is not differentiable at:

x = 1
x = 0
x = −1

10. Chain rule is used for:

Integration
Product of functions
Composition of functions

11. The derivative of log(x) is:

1
1/x
x log(x)

12. Which of the following is NOT differentiable everywhere?

sin(x)
cos(x)
|x|

13. If f(x) = eˣ, then f'(x) =


x·eˣ
1

14. Differentiability implies:

Discontinuity
Continuity
Constant function

15. Derivative of a constant is:

Constant
Zero
One

Quiz: Application of Derivatives (15 Questions)

1. The derivative of a function gives:

The area under the curve
The slope of the tangent
The y-intercept

2. If f'(x) > 0 for all x in (a, b), then f(x) is:

Increasing
Decreasing
Constant

3. A function has a local maximum where:

f'(x) = 0 and f''(x) > 0
f'(x) = 0 and f''(x) < 0
f'(x) > 0

4. The minimum value of a function occurs at:

f''(x) < 0
f'(x) ≠ 0
f''(x) > 0

5. The point where the derivative is zero is called:

Point of inflection
Critical point
Asymptote

6. A function f(x) is concave upward if:

f''(x) > 0
f''(x) < 0
f''(x) = 0

7. The second derivative test helps in identifying:

Continuity
Differentiability
Maxima and minima

8. The slope of the tangent to the curve y = x² at x = 2 is:

2
4
8

9. For the function y = x³, the point of inflection is:

x = 1
x = 0
x = −1

10. The maximum or minimum value of a function in a given interval is called:

Absolute maximum/minimum
Local maximum/minimum
Average

11. The rate of change of displacement with respect to time is:

Acceleration
Velocity
Speed

12. If a function changes concavity, the point is called:

Maximum
Inflection point
Minimum

13. To find maxima/minima, we usually:

Set f(x) = 0
Set f'(x) = 0
Set f''(x) = 0

14. If f'(x) changes sign around x = a, then:

x = a is constant
x = a is an inflection point
x = a may be maximum or minimum

15. Derivative of x³ is:

2x
3x²

Quiz: Integrals (15 Questions)

1. The integral of a function is also called its:

Derivative
Antiderivative
Product

2. ∫x dx = ?


x²/2 + C
2x

3. The symbol ∫ represents:

Summation
Product
Integration

4. ∫e^x dx = ?

e^x + C
xe^x
ln x + C

5. ∫1/x dx = ?

x
ln|x| + C
1/(x²)

6. The process of finding an integral is called:

Differentiation
Integration
Equation

7. What is ∫cos x dx?

sin x + C
−sin x + C
cos x + C

8. Which of the following is the integral of sin x?

cos x + C
−cos x + C
tan x + C

9. The definite integral from a to b gives:

The slope
The area under the curve
The volume

10. ∫0^π sin x dx = ?

0
1
2

11. Integration by parts is used when:

The integrand is a product of two functions
The integrand is a single function
For definite integrals only

12. What is ∫sec²x dx?

tan x + C
sec x + C
cot x + C

13. If F'(x) = f(x), then F(x) is:

Derivative
Integral
Antiderivative

14. ∫tan x dx = ?

−ln|cos x| + C
ln|sec x| + C
ln|cos x| + C

15. The constant of integration is written as:

A
C
X

Quiz: Application of Integrals (15 Questions)

1. The area bounded by a curve y = f(x), the x-axis and vertical lines x = a and x = b is given by:

∫f(x) dx from b to a
∫a to b f(x) dx
∫f(y) dy from a to b

2. The area enclosed between two curves y = f(x) and y = g(x) from x = a to x = b is:

∫a to b [f(x) + g(x)] dx
∫a to b |f(x) − g(x)| dx
∫a to b [f(x) − g(x)] dx

3. Area under a curve can be negative if:

f(x) > 0
f(x) = 0
f(x) < 0

4. If the curve lies entirely above x-axis from a to b, then area =

∫a to b f(x) dx
−∫a to b f(x) dx
∫b to a f(x) dx

5. Area between curves y = sin x and y = cos x from x = 0 to x = π/2 is:

0
1
∫0 to π/2 |sin x − cos x| dx

6. Area enclosed between y = x and y = x² from x = 0 to x = 1 is:

∫0 to 1 (x - x²) dx
∫0 to 1 (x² - x) dx
∫1 to 0 (x - x²) dx

7. The graphical method to visualize area under curves is:

Derivative curve
Area bounded diagram
Tangent plot

8. The integral ∫0 to 2 (4 − x²) dx gives the area of:

Circle
Parabola
Region bounded by a downward parabola

9. If ∫a to b f(x) dx = A and ∫a to b g(x) dx = B, then area between f(x) and g(x) is:

A + B
A − B
|A − B|

10. The definite integral gives:

Slope of tangent
Area between curve and axis
Equation of line

11. If curve lies below x-axis, then ∫a to b f(x) dx is:

Positive
Negative
Zero

12. Which integral calculates the area between x = 0 and x = 1 for f(x) = x²?

∫0 to 1 x dx
∫0 to 1 x² dx
∫0 to 1 1/x dx

13. The area enclosed between curves y² = x and x = y is calculated by integrating:

w.r.t x
w.r.t y
w.r.t z

14. If the area under y = x from 0 to 2 is calculated, it equals:

4
2
∫0 to 2 x dx = 2

15. Which technique is essential to calculate area between curves?

Differentiation
Substitution
Definite Integration

Quiz: Differential Equations (15 Questions)

1. A differential equation is an equation involving:

Only variables
Derivatives of functions
Only constants

2. The order of a differential equation is the order of:

Highest power of derivative
Highest order derivative
Highest power of variable

3. The degree of a differential equation is defined only when:

Derivative is irrational
Derivative is not present
Differential equation is polynomial in derivatives

4. The solution of dy/dx = 3x² is:

x³ + C
x² + C
3x + C

5. dy/dx = ky represents which kind of growth?

Linear growth
Exponential growth
Quadratic growth

6. The general solution of a differential equation contains:

One constant only
Arbitrary constants
No constants

7. The process of finding the differential equation from a given solution is called:

Differentiation
Elimination
Formation

8. The method of separation of variables can be applied when:

Variables can be separated
Equation is linear
Equation is non-linear

9. Which of the following is a linear differential equation?

dy/dx + y = x
dy/dx = y²
y dy/dx = x

10. What is the order of d²y/dx² + 3 dy/dx + y = 0?

2
3
1

11. The degree of (d²y/dx²)³ + dy/dx = 0 is:

2
3
1

12. A solution without arbitrary constants is called:

Particular solution
General solution
Integral solution

13. dy/dx = (x + y) can be solved using:

Separation of variables
Homogeneous method
Linear equation method

14. A differential equation which cannot be expressed as polynomial in derivative is:

dy/dx = x
dy/dx = √(dy/dx)
dy/dx = x²

15. The equation dy/dx = y/t is separable in:

x and y
t and y
x and t

Quiz: Vector Algebra (15 Questions)

1. A vector quantity has:

Only magnitude
Only direction
Both magnitude and direction

2. The magnitude of vector **a = 3i + 4j** is:

7
5
1

3. Two vectors are equal if they have:

Same magnitude only
Same direction only
Same magnitude and direction

4. The unit vector in the direction of **a = 6i - 8j** is:

(3/5)i - (4/5)j
(4/5)i - (3/5)j
i - j

5. The dot product of **a = i + j** and **b = i - j** is:

0
1
2

6. If two vectors are perpendicular, their dot product is:

1
0
-1

7. The angle between two perpendicular vectors is:

90°

180°

8. If |a| = 2 and |b| = 3 and angle between them is 60°, then **a · b** is:

3
6
5

9. Cross product of two vectors is:

Scalar
Vector
Constant

10. The vector product of two parallel vectors is:

Maximum
Zero
Undefined

11. Which is true for i × j?

-k
k
i

12. The angle between vector and itself is:


90°
180°

13. The projection of vector a on b is given by:

|a × b|
a · b / |b|
a · b / |a|

14. The cross product of i and i is:

1
0
i

15. The value of (a × b) · c is called:

Scalar triple product
Vector triple product
Vector dot product

Quiz: Three Dimensional Geometry (15 Questions)

1. Direction cosines of a line are:

Always positive
Cosines of angles with x, y, z axes
Sines of angles with axes

2. The sum of the squares of direction cosines is:

0
1
2

3. Direction ratios of a vector are:

Unique
Not unique
Always zero

4. The angle between two lines with direction cosines (l₁, m₁, n₁) and (l₂, m₂, n₂) is given by:

cosθ = l₁l₂ + m₁m₂ + n₁n₂
sinθ = l₁l₂ + m₁m₂ + n₁n₂
tanθ = l₁l₂ + m₁m₂ + n₁n₂

5. A line passing through (x₁, y₁, z₁) and having direction ratios a, b, c is represented as:

(x−x₁)/a = (y−y₁)/b = (z−z₁)/c
(x+x₁)/a = (y+y₁)/b = (z+z₁)/c
(x·x₁)/a = (y·y₁)/b = (z·z₁)/c

6. The line x = 1 + 2t, y = 3 − t, z = 4t passes through:

(0, 0, 0)
(1, 3, 0)
(2, 4, 4)

7. The equation of x-axis in vector form is:

r = ai
r = aj + bk
r = a(i + j)

8. The angle between two lines is 90° when their dot product is:

1
0
-1

9. Two lines are parallel if their direction ratios are:

Equal or proportional
Perpendicular
Random

10. Distance between two skew lines is calculated using:

Dot product
Cross product
Scalar triple product

11. The shortest distance between two parallel lines is:

0
Perpendicular distance
Any arbitrary value

12. A plane can be represented as:

ax + by + cz = d
ax − by − cz = d
ax · by · cz = d

13. A plane perpendicular to z-axis has equation:

z = k
x = k
y = k

14. A line lies in a plane if:

Line is perpendicular to normal vector of plane
Line is parallel to x-axis
Line intersects x-axis

15. The equation of a plane containing point A(x₁, y₁, z₁) and perpendicular to vector n⃗ = ai + bj + ck is:

a(x−x₁) + b(y−y₁) + c(z−z₁) = 0
a(x+x₁) + b(y+y₁) + c(z+z₁) = 0
a·x + b·y + c·z = x₁y₁z₁

Quiz: Linear Programming (15 Questions)

1. Linear programming is used to:

Solve differential equations
Maximize or minimize a linear function
Compute limits

2. The function to be maximized or minimized in an LPP is called:

Objective function
Inequality
Constraint

3. The inequalities in an LPP are called:

Equations
Objective functions
Constraints

4. The solution set of constraints is known as:

Polygonal area
Feasible region
Objective plane

5. The feasible region of a linear programming problem is always:

A straight line
A triangle
A convex set

6. The optimal value of the objective function occurs at:

Any point
Corner point of feasible region
Midpoint

7. Which method is used for solving LPP graphically?

Newton’s method
Corner point method
Integration method

8. If feasible region is unbounded, then:

Minimum does not exist
Maximum does not exist
Check values at corner points only

9. LPP is applicable when:

Objective and constraints are linear
Only objective is linear
Constraints are quadratic

10. In graphical method of LPP, maximum number of decision variables allowed is:

1
2
3

11. A solution that satisfies all constraints is called:

Optimal solution
Feasible solution
Boundary point

12. What is not a possible outcome of a linear programming problem?

Unique solution
Multiple solutions
Complex solution

13. If all constraints are satisfied but not the objective function, then solution is:

Infeasible
Feasible
Optimal

14. LPP problems are used in:

Business and economics
Astronomy
Art

15. Which of the following is not a constraint?

x ≥ 0
Max Z = 3x + 4y
x + y ≤ 20

Quiz: Probability (15 Questions)

1. If A and B are two independent events, then P(A ∩ B) =

P(A) + P(B)
P(A) × P(B)
P(A) - P(B)

2. The probability of an event lies between:

0 and ∞
-1 and 1
0 and 1

3. If P(A) = 0.3, P(B) = 0.4 and A, B are mutually exclusive, then P(A ∪ B) =

0.7
0.1
0.12

4. If A and B are independent, then Ac and B are:

Not related
Also independent
Mutually exclusive

5. Bayes’ Theorem is used for:

Finding conditional probability
Calculating mean
Probability of mutually exclusive events

6. P(A ∪ B) =

P(A) + P(B)
P(A) + P(B) - P(A ∩ B)
P(A) - P(B)

7. Events which cannot occur together are called:

Exhaustive events
Mutually exclusive events
Independent events

8. If A and B are independent, then P(A|B) =

P(A)/P(B)
P(B)
P(A)

9. If two events are mutually exclusive, then P(A ∩ B) =

P(A) + P(B)
0
1

10. The total probability of sample space is:

0
1
2

11. Conditional probability is written as:

P(A ∩ B)
P(A|B)
P(B ∩ A)

12. If A and B are independent, then P(A ∩ Bc) =

P(A) - P(B)
P(A) × P(Bc)
P(B)

13. What is the probability of a sure event?

1
0.5
0

14. A fair die is rolled. What is the probability of getting a number less than 5?

1/2
2/3
5/6

15. The complement of an event A is:

A itself
Sample space
Ac