Pure Mathematics 3 Quizzes: Test Your Understanding

Question 1

Solve the differential equation with the given initial conditions: $$rac{d^2 y}{dx^2} - 7 rac{dy}{dx} + 10y = 0, y(0) = 1, y'(0) = 2$$

Accepted Answer

$y = 2e^{2x} - e^{5x}$

Answer

$y = e^{2x}$

Answer

$y = e^{5x}$

Answer

$y = e^{-2x} + 2e^{-5x}$

Question 2

Sketch the polar curve $r = 2 + sin(2	heta)$ and calculate the area enclosed by one loop of the curve.

Accepted Answer

The curve is a cardioid with an inner loop. The area enclosed by one loop is $rac{3pi}{2}$ square units.

Answer

The curve is a limaçon with an outer loop. The area enclosed by one loop is $2pi$ square units.

Answer

The curve is a circle with radius $3$. The area enclosed by one loop is $9pi$ square units.

Answer

The curve is a rose with $4$ petals. The area enclosed by one loop is $pi$ square units.

Question 3

A radioactive isotope has a half-life of 10 days. If you start with 100 grams of the isotope, how many grams will remain after 30 days?

Accepted Answer

6.25 grams

Answer

12.5 grams

Answer

25 grams

Answer

50 grams

Question 4

What is the derivative of cosh(x)?

Accepted Answer

sinh(x)

Answer

cosh(x)

Answer

tanh(x)

Answer

-sinh(x)

Question 5

Which of the following is a solution to the inequality |x - 2| < 1?

Accepted Answer

1 < x < 3

Answer

x < 1

Answer

x > 3

Answer

x < -1

Question 6

Find the inverse function of f(x) = 3x - 7.

Accepted Answer

f^(-1)(x) = (x + 7)/3

Answer

f^(-1)(x) = (x - 7)/3

Answer

f^(-1)(x) = 3x + 7

Answer

f^(-1)(x) = 3x - 7

Question 7

Which of the following is a property of a geometric sequence?

Accepted Answer

The ratio of any two consecutive terms is constant.

Answer

The sum of the first n terms is a finite value.

Answer

The sequence is always increasing.

Answer

The sequence is always decreasing.

Question 8

Find the acute angle between the curves y = sin(x) and y = cos(x) at the point (pi/4, sqrt(2)/2).

Accepted Answer

pi/8

Answer

pi/4

Answer

3pi/8

Answer

pi/12

Question 9

What is the derivative of the hyperbolic function y = sinh(x)?

Accepted Answer

cosh(x)

Answer

sinh(x)

Answer

tanh(x)

Answer

coth(x)

Question 10

Which of the following differential equations is a first-order linear homogeneous equation?

Accepted Answer

y' + 3y = 0

Answer

y' + y^2 = 0

Answer

y'' + y' - 2y = 0

Answer

y''' - 3y'' + 2y' - y = 0

Question 11

What is the domain of the function f(x) = cot(x)?

Accepted Answer

{x | x ≠ (2n + 1)π/2, n ∈ Z}

Answer

{x | x ≠ nπ, n ∈ Z}

Answer

{x | x ≠ π/2}

Answer

{x | x ≠ 0}

Question 12

Solve the differential equation dy/dx = 2x + 1, with the initial condition y(0) = 0.

Accepted Answer

y = x^2 + x + 1

Answer

y = x^2 + 1

Answer

y = 2x^2 + x

Answer

y = 2x^2 + 1

Question 13

Obtain the polar equation that represents a circle with radius 3 centered at the origin.

Accepted Answer

r = 3

Answer

r = 3cos(θ)

Answer

r = 3sin(θ)

Answer

r = 3(cos(θ) + sin(θ))

Question 14

Find the derivative of the hyperbolic sine function, sinh(x).

Accepted Answer

cosh(x)

Answer

sinh(x)

Answer

tanh(x)

Answer

sech(x)

Question 15

Find the derivative of the function y = sin(2x + 3).

Accepted Answer

2cos(2x + 3)

Answer

cos(2x + 3)

Answer

-2cos(2x + 3)

Answer

-cos(2x + 3)

Question 16

Find the Maclaurin series for the function f(x) = e^x.

Accepted Answer

1 + x + x^2/2! + x^3/3! + ...

Answer

x + x^2/2! + x^3/3! + x^4/4! + ...

Answer

1 + x + x^2 + x^3 + ...

Answer

x + x^2 + x^3 + x^4 + ...

Question 17

Use mathematical induction to prove the following identity for all positive integers (n):

$$1^3 + 2^3 + 3^3 + ... + n^3 = rac{n^2(n+1)^2}{4}$$

Accepted Answer

Base case: Verify that the identity holds for (n = 1).

Accepted Answer

Inductive step: Assume that the identity holds for some positive integer (k).

Accepted Answer

Substitute (n = k+1) into the identity and simplify.

Answer

The identity cannot be proven using mathematical induction.

Question 18

Which numerical method is suitable for approximating the solution to the non-linear equation f(x) = 0, where f(x) is a continuous function?

Accepted Answer

Newton-Raphson method

Answer

Bisection method

Answer

Euler's method

Answer

Runge-Kutta method

Question 19

Which of the following is a fundamental trigonometric identity?

Accepted Answer

sin(A + B) = sin(A)cos(B) + cos(A)sin(B)

Answer

cos(A + B) = cos(A)cos(B) - sin(A)sin(B)

Answer

tan(A + B) = tan(A) + tan(B)

Answer

cot(A + B) = cot(A) - cot(B)

Question 20

Solve the equation 2sin(x)cos(x) = 1 in the interval [0, π/2].

Accepted Answer

x = π/4

Answer

x = π/3

Answer

x = π/6

Answer

x = π/12

Question 21

Find the derivative of the function f(x) = tanh(3x).

Accepted Answer

3sech^2(3x)

Answer

sech^2(3x)

Answer

sech(3x)

Answer

3sech(3x)

Question 22

Find the general solution to the differential equation dy/dx = 2x + 1:

Accepted Answer

y = x^2 + x + C

Answer

y = x^2 + C

Answer

y = 2x^2 + x + C

Answer

y = 2x^2 + C

Question 23

Which numerical method uses the formula x_(n+1) = x_n - f(x_n)/f'(x_n) to find the root of an equation?

Accepted Answer

Newton-Raphson method

Answer

Bisection method

Answer

Regula Falsi method

Answer

Secant method

Question 24

Which condition is necessary for a function to have an inverse?

Accepted Answer

The function must be one-to-one

Answer

The function must be continuous

Answer

The function must be differentiable

Answer

The function must be monotonic

Question 25

Identify the differential equation that is both first-order and linear.

Accepted Answer

y' = y + e^x

Answer

y' + y^2 = x

Answer

y'' - 2y' + y = 0

Answer

y' = xy + sin(x)

Question 26

Find the general solution of the first-order linear differential equation dy/dx = x^2 + e^x.

Accepted Answer

y = (x^3)/3 + e^x + C

Answer

y = -(x^3)/3 - e^x + C

Answer

y = x^3 + e^x + C

Answer

y = -x^3 - e^x + C

Question 27

Which of the following is the graph of the function y = tanh(x)?

Accepted Answer

Image of a hyperbolic tangent graph

Answer

Image of a sine graph

Answer

Image of a cosine graph

Answer

Image of a parabola

Question 28

Which of the following is the derivative of y = tanh(x)?

Accepted Answer

sech²(x)

Answer

cosh²(x)

Answer

1

Answer

tanh(x)

Question 29

Which of the following is the integral of coth(x)?

Accepted Answer

ln|sinh(x)| + C

Answer

cosh(x) + C

Answer

tanh(x) + C

Answer

sech(x) + C

Question 30

Convert the point (√2/2, √2/2) from rectangular coordinates to polar coordinates.

Accepted Answer

(1, π/4)

Answer

(1, -π/4)

Answer

(-1, 3π/4)

Answer

(-1, π/4)

Question 31

Find the derivative of the hyperbolic cosine function, cosh(x).

Accepted Answer

sinh(x)

Answer

-sinh(x)

Answer

cosh(x)

Question 32

Determine the period of the function y = tan(2x).

Accepted Answer

π/2

Answer

π/4

Answer

π

Answer

2π

Question 33

Calculate the derivative of y = cosh(x).

Accepted Answer

sinh(x)

Answer

cosh(x)

Answer

-sinh(x)

Answer

tanh(x)

Question 34

Select the numerical method employed to approximate the roots of equations.

Accepted Answer

Bisection Method

Answer

Gauss-Jordan Elimination

Answer

Euler's Method

Answer

Trapezoidal Rule

Question 35

Which of the following identities is true for hyperbolic functions?

Accepted Answer

cosh^2(x) - sinh^2(x) = 1

Answer

cosh(x) + sinh(x) = 2

Answer

cosh(x) * sinh(x) = 1/2

Answer

cosh(x) - sinh(x) = 0

Question 36

The Newton-Raphson method is primarily used for:

Accepted Answer

Approximating the roots of equations iteratively.

Answer

Calculating derivatives numerically

Answer

Finding the area under a curve

Answer

Solving systems of linear equations

Question 37

Find the period of the trigonometric function y = sin(3x).

Accepted Answer

2π/3

Answer

π

Answer

π/3

Answer

3π/2

Question 38

Determine the value of sinh(0) according to the definition of hyperbolic functions.

Accepted Answer

0

Answer

-1

Answer

1

Question 39

Use the Trapezoidal Rule to approximate the integral of f(x) = x^2 from x = 0 to x = 2 with n = 4 subintervals.

Accepted Answer

16/3

Answer

16

Answer

8/3

Answer

32/3

Question 40

Find the derivative of the hyperbolic function y = tanh(x).

Accepted Answer

sech^2(x)

Answer

tanh^2(x)

Answer

None of the above

Answer

coth^2(x)

Question 41

Determine the range of the arcsine function, y = arcsin(x):

Accepted Answer

[-π/2, π/2]

Answer

[-1, 1]

Answer

[0, π]

Answer

[-π, π]

Question 42

Calculate the amplitude of the function y = 2sin(3x - π/2) + 1:

Accepted Answer

2

Answer

1

Answer

3

Answer

-2

Question 43

Convert the complex number 2 - 3i to polar form:

Accepted Answer

√(13) * (cos(arctan(-3/2)) + i*sin(arctan(-3/2)))

Answer

3 * (cos(arctan(2/3)) + i*sin(arctan(2/3)))

Answer

2 * (cos(arctan(-3/2)) + i*sin(arctan(-3/2)))

Answer

√(13) * (cos(arctan(2/3)) + i*sin(arctan(2/3)))

Question 44

Find the general solution to the system of differential equations x' = 2y, y' = -3x:

Accepted Answer

x = c1*cos(√13*t) + c2*sin(√13*t), y = c1*√13*sin(√13*t) - c2*√13*cos(√13*t)

Answer

x = c1*cos(t) + c2*sin(t), y = c1*sin(t) - c2*cos(t)

Answer

x = c1*e^(2t) + c2*e^(-2t), y = c1*3*e^(2t) - c2*3*e^(-2t)

Question 45

Calculate the exact value of cosh(ln3).

Accepted Answer

5/4

Answer

1/2

Answer

1

Answer

3/4

Question 46

A point P has polar coordinates (3, π/4). Determine its Cartesian coordinates:

Accepted Answer

((3√2)/2, (3√2)/2)

Answer

(3, π/4)

Answer

(3√2, 3√2)

Answer

(0, 3)

Question 47

Find the area of the region bounded by the curve $ C $ and the $ x $-axis, where $ C $ is defined by the parametric equations $ x = t^2 $, $ y = t^3 $, between the points where $ t = 0 $ and $ t = 2 $.

Accepted Answer

16/5

Answer

8/3

Answer

32/5

Answer

4/3

Question 48

Find the Maclaurin series expansion of $ f(x) = e^x $ up to the term in $ x^3 $.

Accepted Answer

1 + x + x^2/2 + x^3/6

Answer

1 + x/2 + x^2/4 + x^3/8

Answer

1 + x + x^2/3 + x^3/9

Answer

1 + x + x^2 + x^3

Question 49

Given $ \sinh(x) = (e^x - e^-x)/2 $, find the derivative of $ \sinh(x) $.

Accepted Answer

cosh(x)

Answer

-cosh(x)

Answer

-sinh(x)

Answer

sinh(x)

Question 50

Find the general solution of the differential equation $ \frac{dy}{dx} = \frac{y}{x} $.

Accepted Answer

y = Cx

Answer

y = C/x

Answer

y = x + C

Answer

y = x/C

Question 51

Find the length of the arc of the curve $ y = \ln(x) $ from $ x = 1 $ to $ x = e $.

Accepted Answer

sqrt(2)

Answer

e + 1

Answer

sqrt(e^2 + 1)

Answer

e - 1

Question 52

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

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 53

Which of the following is the graph of $ y = \cosh(x) $?

Accepted Answer

A curve that looks like a parabola, opening upwards.

Answer

A straight line with a positive slope.

Answer

A curve that looks like a parabola, opening downwards.

Question 54

What is the polar form of the complex number $ 2 + 3i $?

Accepted Answer

5(cos(56.3°) + i sin(56.3°))

Answer

2 + 3

Answer

-5(cos(56.3°) - i sin(56.3°))

Answer

5(cos(123.7°) + i sin(123.7°))

Question 55

Which of the following numerical methods is used to solve systems of equations?

Accepted Answer

Gauss-Jordan elimination

Answer

Newton's method

Answer

Bisection method

Answer

Euler's method

Question 56

Which convergence test can be used to determine that the series ∑(n=1 to ∞) (n/(n^2 + 1)) converges?

Accepted Answer

Comparison Test

Answer

Alternating Series Test

Answer

Ratio Test

Answer

Divergence Test

Question 57

What is the general solution of the differential equation dy/dx = ky?

Accepted Answer

y = Ce^(kt)

Answer

y = k/t

Answer

y = t^k

Answer

y = ke^t

Question 58

Which of the following matrices is invertible?

Accepted Answer

[[1, 2], [3, 4]]

Answer

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

Answer

[[1, 2, 3], [4, 5, 6], [7, 8, 9]]

Answer

[[2, 4], [4, 8], [8, 16]]

Question 59

What is the general solution to the differential equation dy/dx = y/x?

Accepted Answer

y = Cx

Answer

y = x + C

Answer

y = e^x + C

Answer

y = C/x

Question 60

What is the Maclaurin series expansion for cos(x)?

Accepted Answer

1 - x^2/2! + x^4/4! - ...

Answer

x + x^3/3! + x^5/5! + ...

Answer

1 + x/1! + x^2/2! + ...

Answer

x - x^3/3! + x^5/5! - ...

Question 61

The equation r = 2sin(θ) describes a curve in polar coordinates. What type of curve does this equation represent?

Accepted Answer

Circle

Answer

Ellipse

Answer

Parabola

Answer

Hyperbola

Question 62

What is the derivative of sinh(x)?

Accepted Answer

cosh(x)

Answer

-cosh(x)

Answer

sinh(2x)

Answer

cosh(2x)

Question 63

What is the exact value of sin(15°)?

Accepted Answer

(√6 - √2)/4

Answer

(√1 - √3)/4

Answer

(√3 - √1)/4

Answer

(√2 - √6)/4

Question 64

What is the general solution to the differential equation dy/dx = 2y?

Accepted Answer

y = Ce^(2x)

Answer

y = Ce^(x)

Answer

y = Ce^(-2x)

Answer

y = Ce^(-x)