Pure Mathematics 3 Quizzes: Test Your Skills and Prepare for Exams

Question 1

Find the inverse of the function f(x) = 2x + 1 and prove that your answer is correct.

Accepted Answer

f^-1(x) = (x - 1) / 2

Answer

f^-1(x) = 2x - 1

Answer

f^-1(x) = 1 / (2x + 1)

Answer

f^-1(x) does not exist

Question 2

Sketch the graph of the function f(x) = |x-1| + |x+1|. Analyze its continuity and differentiability, explaining your reasoning.

Accepted Answer

The graph is continuous but not differentiable at x = -1 and x = 1.

Answer

The graph is continuous and differentiable everywhere.

Answer

The graph is differentiable everywhere but not continuous at x = 0.

Answer

The graph is neither continuous nor differentiable anywhere.

Question 3

The function f(x) = |x² - 1| is defined at x = 0. Is it continuous and/or differentiable at this point?

Accepted Answer

f(x) is continuous but not differentiable at x = 0

Answer

f(x) is continuous and differentiable at x = 0

Answer

f(x) is neither continuous nor differentiable at x = 0

Question 4

What is the derivative of the function f(x) = x * e^x using the product rule?

Accepted Answer

(x + 1) * e^x

Answer

x * e^x

Answer

x + e^x

Answer

e^x

Question 5

Solve the differential equation dy/dx + y = e^x using the integrating factor method.

Accepted Answer

y(x) = e^x(1 + Ce^-x)

Answer

y(x) = e^x + Ce^(-x)

Answer

y(x) = e^-x(1 + Ce^x)

Answer

y(x) = Ce^x

Question 6

Find the equation of the straight line passing through the points (2, 5) and (1, 3).

Accepted Answer

y = -2x + 7

Answer

y = 2x + 1

Answer

y = x + 3

Answer

y = -x + 5

Question 7

Evaluate the value of tan(π/6).

Accepted Answer

√3

Answer

1

Answer

-1

Answer

-√3

Question 8

Find the sum of the first 10 terms of the geometric series 1 + 1/2 + 1/4 + 1/8 + ... ?

Accepted Answer

10.24

Answer

10.23

Answer

10.25

Answer

10.26

Question 9

Find the value of the determinant of the matrix A = [[1, 2], [3, 4]].

Accepted Answer

2

Answer

-2

Answer

-4

Answer

4

Question 10

The eigenvectors of the matrix A = [[2, 1], [1, 2]] are:

Accepted Answer

[(1, 1), (1, -1)]

Answer

[(1, -1), (1, 1)]

Answer

[(1, 1), (-1, 1)]

Answer

[(1, -1), (-1, 1)]

Question 11

The derivative of the function f(x) = x^3 + 2x^2 - 1 is:

Accepted Answer

3x^2 + 4x

Answer

x^2 + 4x

Answer

3x^2 - 4x

Answer

x^3 + 4x^2

Question 12

The area of the region bounded by the curves y = x^2 and y = 4 is:

Accepted Answer

16/3

Answer

8/3

Answer

32/3

Answer

64/3

Question 13

The value of the definite integral ∫[0, π/2] sin(x) dx is:

Accepted Answer

1

Answer

0

Answer

-1

Answer

π/2

Question 14

Find the inverse function of f(x) = 2x + 3.

Accepted Answer

f^-1(x) = (x - 3)/2

Answer

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

Answer

f^-1(x) = 2x - 3

Answer

f^-1(x) = 2x + 3

Question 15

State a necessary condition for a function to be continuous at a given point.

Accepted Answer

The left-hand limit and right-hand limit exist and are equal.

Answer

The function is defined at that point.

Answer

The function is differentiable at that point.

Answer

The function is monotonic at that point.

Question 16

Which of the following trigonometric identities is always true?

Accepted Answer

sin^2(x) + cos^2(x) = 1

Answer

sin(x + y) = sin(x) + sin(y)

Answer

cos(x + y) = cos(x) + cos(y)

Answer

tan(x + y) = tan(x) + tan(y)

Question 17

Which of the following numbers is a counterexample to the conjecture "All prime numbers are odd"?

Accepted Answer

2

Answer

-3

Answer

5

Answer

7

Question 18

What is the value of the discriminant of the quadratic equation 2x^2 - 5x + 3 = 0?

Accepted Answer

11

Answer

-11

Answer

25

Answer

-25

Question 19

Simplify the expression (2x - 3)(x + 5).

Accepted Answer

2x² + 7x - 15

Answer

2x² - 7x + 15

Answer

2x² + 13x - 15

Answer

2x² - 13x + 15

Question 20

Calculate the limit: lim_(x->π/2) (sin(x) - cos(x)) / (x - π/2).

Accepted Answer

0

Answer

1

Answer

-1

Answer

The limit does not exist.

Question 21

Calculate the derivative of f(x) = e^(sin(x)) using the chain rule.

Accepted Answer

cos(x) * e^(sin(x))

Answer

e^(sin(x))

Answer

sin(x) * e^(cos(x))

Question 22

Determine the continuity and differentiability of the function f(x) = |x^2 - 1| at x = 0.

Accepted Answer

Continuous but not differentiable at x = 0

Answer

Differentiable but not continuous at x = 0

Answer

Both continuous and differentiable at x = 0

Answer

Neither continuous nor differentiable at x = 0

Question 23

Find the derivative of the function f(x) = ln(x² + 1).

Accepted Answer

2x / (x² + 1)

Answer

2x * ln(x² + 1)

Answer

1 / (x² + 1)

Answer

1 * ln(x² + 1)

Question 24

Prove that for any positive integers n and k, n^k is even if n is even and k is odd.

Accepted Answer

Let n be even. Then n = 2m for some integer m. Therefore, n^k = (2m)^k = 2^(k*m), which is even because k*m is an integer.

Answer

Let n be odd. Then n = 2m + 1 for some integer m. Therefore, n^k = (2m + 1)^k = 2^(k*m) + 1, which is odd.

Question 25

Find the derivative of the function f(x) = 2x + 3x^2 + 5x^3

Accepted Answer

2 + 6x + 15x^2

Answer

2 + 6x - 15x^2

Answer

x + 2x^2 + 3x^3

Answer

2 - 6x + 15x^2

Question 26

Which of the following sequences converges?

Accepted Answer

aₙ = 1/n

Answer

aₙ = 2^n

Answer

aₙ = (-1)^n

Answer

aₙ = n³

Question 27

What is the indefinite integral of ∫(x² + 2x - 1) dx?

Accepted Answer

x³/3 + x² - x + C

Answer

x³ + 2x² - x + C

Answer

x² + x - 1 + C

Question 28

Which of the following transformations does not preserve the area of a figure?

Accepted Answer

Scaling

Answer

Reflection

Answer

Rotation

Answer

Translation

Question 29

Which of the following is a valid method of mathematical proof?

Accepted Answer

Mathematical induction

Answer

Trial and error

Answer

Analogy

Answer

Guess and check

Question 30

What is the vertex of the parabola with equation y = -x^2 + 6x + 5?

Accepted Answer

(3, 14)

Answer

(0, 5)

Answer

(-3, -14)

Answer

(6, 5)

Question 31

Simplify the trigonometric expression: sin(x)cos(x) + cos(x)sin(x)

Accepted Answer

1

Answer

sin(2x)

Answer

0

Answer

cos(2x)

Question 32

Find the derivative of the function: f(x) = x^4 - 3x^2 + 2x + 5

Accepted Answer

4x^3 - 6x + 2

Answer

x^3 - 6x + 2

Answer

4x^3 + 6x + 2

Answer

x^4 - 6x^2 + 2x + 5

Question 33

Expand (a + b)^4 using the binomial theorem.

Accepted Answer

a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + b^4

Answer

a^4 + 6a^3b + 3a^2b^2 + 2ab^3 + b^4

Answer

a^4 + 2a^2b^2 + b^4

Answer

a^4 + 4a^3b + 2a^2b^2 + b^4

Question 34

Find the general term of the sequence 1, 3, 6, 10, 15, ...

Accepted Answer

n^2 + 1

Answer

n^2 - 1

Answer

n + 1

Answer

2n - 1

Question 35

Differentiate the function f(x) = x^3 - 2x + 5.

Accepted Answer

f'(x) = 3x^2 - 2

Answer

f'(x) = x^2 - 2

Answer

f'(x) = x^3 - 2

Answer

f'(x) = 3x + 2

Question 36

Solve the logarithmic equation log2(x + 3) - log2(x - 1) = 3.

Accepted Answer

x = 7

Answer

x = 3

Answer

x = 1

Answer

x = 5

Question 37

Differentiate the function f(x) = x^3 - 2x^2 + 5

Accepted Answer

3x^2 - 4x

Answer

3x - 2

Answer

x^2 - 2

Answer

x^2 + 2x - 5

Question 38

Find the area of the region bounded by the curves y = x^2 and y = 4

Accepted Answer

32/3

Answer

8

Answer

24

Answer

16

Question 39

Which of the following is NOT a valid method of proof in mathematics?

Accepted Answer

Guess and check

Answer

Mathematical induction

Answer

Direct proof

Answer

Proof by contradiction

Question 40

Expand the expression (x + y)^3.

Accepted Answer

x^3 + 3x^2y + 3xy^2 + y^3

Answer

x^3 - 3x^2y + 3xy^2 - y^3

Answer

x^3 + y^3

Answer

x^2y + 2xy^2 + y^3

Question 41

What is the formula for the sum of the first 'n' terms of an arithmetic sequence, where 'a' is the first term and 'd' is the common difference?

Accepted Answer

n/2(2a + (n - 1)d)

Answer

n/2(a + (n - 1)d)

Answer

(2a + (n - 1)d)/n

Answer

n(a + d)

Question 42

What is the value of sin(π/3)?

Accepted Answer

√3/2

Answer

1/2

Answer

√2/2

Question 43

What is the derivative of the function f(x) = e^x?

Accepted Answer

e^x

Answer

log e^x

Answer

1/e^x

Answer

x^e

Question 44

What is the indefinite integral of (x^2 + 1) with respect to x?

Accepted Answer

(x^3)/3 + x + C

Answer

(x^2)/2 + C

Answer

x^3 + C

Question 45

Which of the following is a solution to the equation x^3 - 8 = 0?

Accepted Answer

2

Answer

-2

Answer

4

Answer

-4

Question 46

Find the sum of the infinite geometric series 1 + 1/2 + 1/4 + 1/8 + ...

Accepted Answer

2

Answer

1

Answer

3/2

Answer

5/4

Question 47

Prove by mathematical induction that 1^2 + 2^2 + ... + n^2 = n(n+1)(2n+1)/6 for all positive integers n.

Accepted Answer

The proof follows the principle of mathematical induction.

Answer

The formula is only true for even values of n.

Answer

The formula is only true for odd values of n.

Question 48

Find the equation of the circle with center (3, -2) and passing through the point (1, 4).

Accepted Answer

(x - 3)^2 + (y + 2)^2 = 40

Answer

(x + 3)^2 + (y - 2)^2 = 36

Answer

(x - 3)^2 + (y + 2)^2 = 36

Answer

(x + 3)^2 + (y - 2)^2 = 40

Question 49

Solve the equation 2^(2x - 1) = 8.

Accepted Answer

x = 2

Answer

x = 0

Answer

x = 1

Answer

x = 3

Question 50

Find the derivative of the function f(x) = ln(x^2 + 1).

Accepted Answer

f'(x) = 2x / (x^2 + 1)

Answer

f'(x) = 1 / (x^2 + 1)

Answer

f'(x) = 1 / (2x + 1)

Answer

f'(x) = 2x

Question 51

What is the derivative of the function f(x) = ln(x^2 + 1)?

Accepted Answer

2x / (x^2 + 1)

Answer

ln(2x)

Answer

2x

Answer

1 / (x^2 + 1)

Question 52

The sum of the first n terms of an arithmetic series is given by Sn = 2n^2 + 3n. Find the common difference.

Accepted Answer

4

Answer

2

Answer

3

Answer

6

Question 53

Find the equation of the tangent to the curve y = x^3 - 2x + 1 at the point (1, 0).

Accepted Answer

y = x - 1

Answer

y = x + 1

Answer

y = 2x - 2

Question 54

Solve the following system of equations: 2x + y = 5, x - 3y = -2

Accepted Answer

x = 13/7, y = 9/7

Answer

x = 1, y = 3

Answer

x = 2, y = 1

Answer

x = 3, y = -1

Question 55

Evaluate the integral ∫(x^2 + 1)dx from 0 to 1.

Accepted Answer

4/3

Answer

2/3

Answer

1

Question 56

If sin(θ) = 1/2 and θ is in the first quadrant, what is the value of cos(θ)?

Accepted Answer

√3/2

Answer

1/2

Answer

-√3/2

Answer

-1/2

Question 57

Find the domain of the function f(x) = √(x - 2).

Accepted Answer

x ≥ 2

Answer

x < 2

Answer

x > 2

Answer

x ≤ 2

Question 58

Prove by mathematical induction that 1 + 2 + 3 + ... + n = n(n+1)/2 for all positive integers n.

Accepted Answer

The statement holds true for n = 1, and assuming it holds true for n = k, we can prove it also holds for n = k+1.

Answer

The statement only holds true for even numbers.

Question 59

Find the equation of the line passing through the points (2, 3) and (4, 7).

Accepted Answer

y = 2x - 1

Answer

y = 3x - 3

Answer

y = -2x + 7

Answer

y = x + 1

Question 60

Solve the equation 2^x = 8.

Accepted Answer

x = 3

Answer

x = 1

Answer

x = 4

Answer

x = 2