Pure Mathematics 2 Quizzes: Enhance Your Problem-Solving Skills

Question 1

Identify the invalid trigonometric identity:

Accepted Answer

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

Answer

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

Answer

tan(x) = sin(x) / cos(x)

Answer

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

Question 2

Which of the following is **not** a valid method for solving quadratic equations?

Accepted Answer

Guessing and checking

Answer

Factorization

Answer

Completing the square

Answer

Using the quadratic formula

Question 3

The equation of a parabola is y = ax^2 + bx + c. If the parabola has a minimum value at the point (-2, 1), what is the value of b?

Accepted Answer

-4

Answer

4

Answer

0

Answer

2

Question 4

Which characteristic does NOT apply to a parabola?

Accepted Answer

The slope of a parabola is constant.

Answer

A parabola has a U-shape and opens either upward or downward.

Answer

The vertex of a parabola is where it changes direction.

Answer

The axis of symmetry of a parabola is a vertical line that divides it into two symmetrical halves.

Question 5

Solve the system of equations:

```
x + 2y = 5
2x + 4y = 10
```
Find the value of x + y.

Accepted Answer

5

Answer

3

Answer

4

Answer

6

Question 6

Solve the differential equation: y'' - 2y' + y = e^(2x)

Accepted Answer

y = Ae^x + Bxe^x + 1/2e^(2x) - 1/2xe^(2x)

Answer

y = Ae^x + Bxe^x + 1/2e^(2x)

Answer

y = Ae^x + Bxe^x + Ce^(2x)

Answer

y = Ae^x + Bxe^x + 1/2e^(2x) + 1/2xe^(2x)

Question 7

Under which condition does the geometric series (sum_{n=1}^{infty} a r^n) converge?

Accepted Answer

|r| < 1

Answer

r = 1

Answer

a = 0

Answer

r = 0

Question 8

Which condition ensures the divergence of the geometric series  $sum_{n=0}^infty ar^n $?

Accepted Answer

$|r| > 1$

Answer

$|r| < 1$

Answer

$|r| = 1$

Answer

$a = 0$

Question 9

What property of a quadratic function is represented by the vertex on its graph?

Accepted Answer

Maximum or minimum value

Answer

X-axis intercept

Answer

Midpoint of the parabola

Question 10

Which of the following is a linear matrix equation?

Accepted Answer

Ax = b

Answer

A^2 = A

Answer

A^-1 = A

Answer

A + B = C

Question 11

Calculate the determinant of the matrix A = [2 1; 3 4].

Accepted Answer

5

Answer

7

Answer

-5

Answer

-7

Question 12

Solve the differential equation dy/dx = 2x + 1 to obtain its general solution.

Accepted Answer

y = x^2 + x + C

Answer

y = x^2 - x + C

Answer

y = 2x^2 + x + C

Answer

y = 2x^2 - x + C

Question 13

Identify the diagonalizable matrix among the given options:

Accepted Answer

[1 2; 3 3]

Answer

[1 2; 3 4]

Answer

[2 1; 3 2]

Answer

[2 1; 1 2]

Question 14

Obtain the complex conjugate of the complex number z = 3 + 4i.

Accepted Answer

3 - 4i

Answer

3 + 4i

Answer

-3 - 4i

Answer

-3 + 4i

Question 15

Which of the following differential equations is first order and linear?

Accepted Answer

y' = y + e^x

Answer

y' = x^2 + y

Answer

y'' = sin(x)y

Answer

y' = xy + y^2

Question 16

Which of the following is a characteristic equation of the differential equation y'' - 2y' + 5y = 0?

Accepted Answer

r^2 - 2r + 5 = 0

Answer

r^2 + 2r + 5 = 0

Answer

r^2 - 2r - 5 = 0

Answer

r^2 + 2r - 5 = 0

Question 17

Vector operations include all of the following EXCEPT:

Accepted Answer

Integration

Answer

Addition

Answer

Dot product

Answer

Multiplication by a scalar

Question 18

Find the scalar product of the vectors a = (1, 2, 3) and b = (2, 1, 4).

Accepted Answer

11

Answer

9

Answer

7

Answer

5

Question 19

Find the equation of a sphere with center (2, -1, 3) and radius 4.

Accepted Answer

(x - 2)^2 + (y + 1)^2 + (z - 3)^2 = 16

Answer

(x + 2)^2 + (y - 1)^2 + (z + 3)^2 = 16

Answer

(x - 2)^2 + (y + 1)^2 + (z - 3)^2 = 4

Answer

(x + 2)^2 + (y - 1)^2 + (z + 3)^2 = 4

Question 20

Find the area of the triangle formed by the points A(1, 2), B(3, 5), and C(5, 2).

Accepted Answer

6

Answer

4

Answer

8

Answer

10

Question 21

Find the derivative of the function f(x) = sin(2x).

Accepted Answer

2cos(2x)

Answer

cos(2x)

Answer

2sin(x)

Answer

sin(x)

Question 22

Given a vector v = (2, 3, -1), calculate its magnitude.

Accepted Answer

√14

Answer

14

Answer

7

Answer

√7

Question 23

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

Accepted Answer

y' + y = 0

Answer

(d^2)y/dx^2 + y = 0

Answer

(d^3)y/dx^3 + y = 0

Answer

y^4 + (d^2)y/dx^2 = 0

Question 24

Which of the following operations is commutative for matrices?

Accepted Answer

Addition

Answer

Multiplication

Answer

Subtraction

Answer

Division

Question 25

Which of the following is a characteristic of a diagonal matrix?

Accepted Answer

All off-diagonal elements are zero

Answer

All diagonal elements are zero

Answer

All elements are equal

Answer

It is always a square matrix

Question 26

Find the gradient of the curve: y = x^2 - 3x + 2

Accepted Answer

2x - 3

Answer

x^2 - 3

Answer

x - 3

Answer

2x + 3

Question 27

Which of the following is a defining characteristic of vectors?

Accepted Answer

Possessing both magnitude and direction

Answer

Being perpendicular to the plane they lie in

Answer

Being added or subtracted element-wise

Answer

Being multiplied by scalars to yield another vector

Question 28

Identify the solution to the differential equation y' + 2y = 0.

Accepted Answer

y = Ce^-2x

Answer

y = Cxe^-2x

Answer

y = Ccos(2x)

Answer

y = Csin(2x)

Question 29

Find the vector equation of the line passing through the points A(2, -1) and B(4, 3).

Accepted Answer

r = 2i - j + t(2i + 4j)

Answer

r = 4i + 2j - t(2i + 4j)

Answer

r = 2i - j - t(2i + 4j)

Answer

r = 4i - 2j + t(-2i + 4j)

Question 30

Determine the equation of the plane passing through the point P(1, -2, 3) and perpendicular to the vector 2i - j + k.

Accepted Answer

2x - y + z = 8

Answer

2x + y - z = 2

Answer

x + 2y - z = 0

Answer

x - 2y + z = 6

Question 31

Solve the second-order differential equation y'' - 4y' + 4y = 0.

Accepted Answer

y = c_1cos(2x) + c_2sin(2x)

Answer

y = c_1e^(2x) + c_2e^(-2x)

Answer

y = c_1x + c_2

Answer

y = c_1e^x + c_2e^(-x)

Question 32

Calculate the inverse of the matrix:

Accepted Answer

1/5 * [-2 1; 3 -2]

Answer

1/5 * [2 -1; -3 2]

Answer

1/5 * [-2 -1; 3 2]

Answer

1/5 * [2 1; -3 -2]

Question 33

Calculate the magnitude of vector “a” in 3-dimensional space, given its components: a_x, a_y, and a_z.

Accepted Answer

|a| = √ (a_x^2 + a_y^2 + a_z^2)

Answer

|a| = √ (a_x^2 + a_y^2)

Answer

|a| = √ (a_x^2 - a_y^2)

Answer

|a| = √ (a_y^2 - a_x^2)

Question 34

Identify which of the following functions is a linear transformation.

Accepted Answer

f(x) = 2x + 1

Answer

f(x) = x^3

Answer

f(x) = sin(x)

Answer

f(x) = x^2 - 1

Question 35

Solve the system of equations: 2x + 3y = 7 and x - y = 1.

Accepted Answer

x = 2, y = 1

Answer

x = 1, y = 2

Answer

x = -2, y = -1

Answer

x = -1, y = -2

Question 36

A matrix is considered singular if and only if its determinant is

Accepted Answer

Zero

Answer

Negative

Answer

Non-zero

Answer

Positive

Question 37

Two vectors are perpendicular if their dot product is

Accepted Answer

Zero

Answer

One

Answer

Negative one

Answer

Non-zero

Question 38

The sum of the eigenvalues of a matrix is equivalent to its

Accepted Answer

Trace

Answer

Rank

Answer

Multiplicative inverse

Answer

Determinant

Question 39

The domain of the logarithmic function f(x) = log(x - 1) is

Accepted Answer

(1, infinity)

Answer

(1, 0)

Answer

(0, 1)

Answer

(-infinity, 1)

Question 40

The inverse of a matrix, if it exists, can be calculated using the

Accepted Answer

Adjoint matrix

Answer

Co-factor matrix

Answer

Transpose matrix

Answer

Identity matrix

Question 41

What is the result of the dot product between two vectors?

Accepted Answer

Scalar

Answer

Tensor

Answer

Matrix

Answer

Vector

Question 42

Under which condition is a matrix said to be non-invertible?

Accepted Answer

A matrix with a determinant of 0

Answer

A square matrix

Answer

A diagonal matrix

Answer

An upper triangular matrix

Question 43

What is the general solution to the differential equation y' + 2y = 0?

Accepted Answer

y = Ce^(-2x)

Answer

y = Ce^(2x)

Answer

y = 2Ce^(-x)

Answer

y = Ce^2x

Question 44

Which attribute is characteristic of a normal distribution?

Accepted Answer

It is symmetric about the mean

Answer

It has a finite variance

Answer

It is skewed to the right

Answer

It is a discrete distribution

Question 45

What is the value of the integral over the entire real line of a probability density function?

Accepted Answer

1

Answer

0

Answer

Greater than 1

Answer

Less than 1

Question 46

For a system of linear equations to have infinitely many solutions, what condition must its coefficient matrix satisfy?

Accepted Answer

The coefficient matrix has a rank less than the number of variables.

Answer

The coefficient matrix has a rank greater than the number of variables.

Answer

The right-hand side vector is the zero vector

Question 47

What is the derivative of the function f(x) = x^3 - 2x?

Accepted Answer

3x^2 - 2

Answer

x^2 - 2

Answer

x^3 - 2

Answer

3x - 2

Question 48

What is the indefinite integral of the function f(x) = sin(x)?

Accepted Answer

-cos(x) + C

Answer

sin(x) + C

Answer

-sin(x) + C

Answer

cos(x) + C

Question 49

What is the necessary condition for a function to be continuous at a point?

Accepted Answer

The left-hand limit, right-hand limit, and function value are all equal at that point

Answer

The left-hand limit exists

Answer

The right-hand limit exists

Question 50

Determine the eigenvalues of the matrix B = [[1, 2], [3, 4]].

Accepted Answer

2, 5

Answer

1, 4

Answer

3, 6

Answer

-2, -5

Question 51

Calculate the magnitude of the vector w = [3, 4, 5].

Accepted Answer

sqrt(50)

Answer

12

Answer

sqrt(25)

Answer

7

Question 52

Calculate the area of the parallelogram spanned by vectors a = [2, 1] and b = [3, -4].

Accepted Answer

11

Answer

5

Answer

21

Answer

17

Question 53

Find the inverse of the matrix C = [[1, 2], [3, 5]].

Accepted Answer

[[5, -2], [-3, 1]]

Answer

[[1, -2], [-3, 5]]

Answer

[[5, 2], [3, -1]]

Answer

[[1, 2], [3, -5]]

Question 54

Identify the eigenvalues of the matrix D = [[2, -1], [1, 2]].

Accepted Answer

1, 3

Answer

-2, -4

Answer

2, 4

Answer

-1, -3

Question 55

Find the equation of the tangent line to the curve y = x^2 + 3x at the point (1, 4).

Accepted Answer

y = 5x - 1

Answer

y = 4x

Answer

y = x + 3

Answer

y = 2x + 2

Question 56

Evaluate the definite integral ∫(2x + 1) dx from 1 to 3.

Accepted Answer

12

Answer

8

Answer

14

Answer

10

Question 57

Find the area enclosed by the curve y = x^2, the x-axis, and the lines x = 1 and x = 2.

Accepted Answer

7/3

Answer

2/3

Answer

4/3

Question 58

The equation of a circle is given by (x - 2)^2 + (y + 1)^2 = 9. What is the radius of the circle?

Accepted Answer

3

Answer

1

Answer

2

Answer

9