Algebra Practice Questions — JAMB UTME — Page 9

Question 161Question →

If $\frac{3\sqrt{5} - \sqrt{3}}{\sqrt{5} + \sqrt{3}} = a + b\sqrt{15}$ , where $a$ and $b$ are rational numbers, what is the value of $a - b$ ?

11

8

5

13

Show answer & explanation

Answer: 11

Answer

11

By multiplying the numerator and denominator by the conjugate of the denominator,

(\sqrt{5} - \sqrt{3})

, the fraction simplifies to

\frac{18 - 4\sqrt{15}}{2} = 9 - 2\sqrt{15}

. Equating this to

a + b\sqrt{15}

gives

a = 9

and

b = -2

. Calculating

a - b

gives

9 - (-2) = 11

.

Step-by-Step Solution

1

Multiply the numerator and denominator by the conjugate of the denominator

\frac{3\sqrt{5} - \sqrt{3}}{\sqrt{5} + \sqrt{3}} \times \frac{\sqrt{5} - \sqrt{3}}{\sqrt{5} - \sqrt{3}} = \frac{(3\sqrt{5} - \sqrt{3})(\sqrt{5} - \sqrt{3})}{(\sqrt{5})^2 - (\sqrt{3})^2}

Rationalising the denominator requires using the difference of squares identity

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

.

2

Expand the numerator and denominator

Denominator = 5 - 3 = 2. Numerator = 3(5) - 3\sqrt{15} - \sqrt{15} + 3 = 15 + 3 - 4\sqrt{15} = 18 - 4\sqrt{15}.

Apply distributive property to expand

(3\sqrt{5} - \sqrt{3})(\sqrt{5} - \sqrt{3})

carefully combining like terms.

3

Simplify the fraction to match the form

a + b\sqrt{15}

\frac{18 - 4\sqrt{15}}{2} = 9 - 2\sqrt{15}

Divide each term in the numerator by 2.

4

Identify

a

and

b

and evaluate

a - b

a = 9, b = -2 \implies a - b = 9 - (-2) = 11

Subtracting negative 2 is equivalent to adding 2.

Key Concept

Rationalisation of Binomial Surd Denominators

Estimated Time:1m 30s

Question 162Question →

An arithmetic progression (A.P.) has a fifth term of $17$ and a common difference of $3$ . What is the first term of the progression?

5

2

-1

29

Show answer & explanation

Answer: $5$

Answer

The first term of the progression is

5

.

Using the

n^{\text{th}}

term formula for an arithmetic progression,

T_n = a + (n - 1)d

, substituting

T_5 = 17

,

n = 5

, and

d = 3

gives

17 = a + 4(3) \implies 17 = a + 12 \implies a = 5

.

Step-by-Step Solution

1

Identify the given parameters and formula for the

n^{\text{th}}

term of an A.P.

Formula:

T_n = a + (n - 1)d

, where

T_5 = 17

,

n = 5

, and

d = 3

.

The standard formula connects the

n^{\text{th}}

term, first term, term position, and common difference.

2

Substitute the given values into the formula.

17 = a + (5 - 1) \times 3 \implies 17 = a + 4 \times 3 \implies 17 = a + 12

.

Evaluating

(n - 1)d

gives the total difference added to the first term.

3

Solve for the first term

a

.

a = 17 - 12 = 5

.

Subtracting

12

from both sides isolates

a

.

Key Concept

Arithmetic Progression

n^{\text{th}}

term calculation

Estimated Time:45s

Question 163Question →

The $3^{\text{rd}}$ term of a geometric progression (G.P.) is $18$ and its common ratio is $3$ . What is the first term of the progression?

Show answer & explanation

Answer: 2

Answer

The first term of the geometric progression is 2.

In a geometric progression, the

n^{\text{th}}

term is given by

T_n = a r^{n-1}

. For the

3^{\text{rd}}

term (

n = 3

) with common ratio

r = 3

and term value

18

, the equation is

18 = a \cdot 3^{2} = 9a

. Dividing by

9

yields the first term

a = 2

.

Step-by-Step Solution

1

Identify the formula for the

n^{\text{th}}

term of a geometric progression.

T_n = a r^{n-1}

This formula connects the

n^{\text{th}}

term

T_n

to the first term

a

, common ratio

r

, and term index

n

.

2

Substitute

T_3 = 18

,

r = 3

, and

n = 3

into the formula.

18 = a \cdot 3^{3-1} \implies 18 = 9a

Evaluating

3^{3-1} = 3^2 = 9

simplifies the equation.

3

Solve for the first term

a

.

a = \frac{18}{9} = 2

Dividing both sides of the equation by 9 isolates the first term.

Key Concept

Geometric Progression nth term calculation

Question 164Question →

Given the matrices $A = \begin{pmatrix} 2 & -1 \\ 3 & 4 \end{pmatrix}$ and $B = \begin{pmatrix} 1 & x \\ -2 & 5 \end{pmatrix}$ , if the determinant of the product matrix $AB$ is equal to $121$ , find the value of $x$ .

Show answer & explanation

Answer: 3

Answer

The value of

x

is

3

.

By applying the property

\det(AB) = \det(A) \cdot \det(B)

, we compute

\det(A) = (2)(4) - (-1)(3) = 11

and

\det(B) = (1)(5) - (x)(-2) = 5 + 2x

. Substituting into the equation gives

11(5 + 2x) = 121

, which yields

5 + 2x = 11

and leads to

x = 3

.

Step-by-Step Solution

1

Find the determinant of matrix

A

\det(A) = (2)(4) - (-1)(3) = 11

The determinant of a

2 \times 2

matrix

\begin{pmatrix} a & b \\ c & d \end{pmatrix}

is calculated as

ad - bc

.

2

Express the determinant of matrix

B

in terms of

x

\det(B) = (1)(5) - (x)(-2) = 5 + 2x

Apply the

2 \times 2

determinant formula to matrix

B

.

3

Apply the determinant product rule

\det(AB) = \det(A) \cdot \det(B) = 11(5 + 2x) = 121

For any square matrices

A

and

B

,

\det(AB) = \det(A) \det(B)

.

4

Solve the resulting linear equation for

x

x = 3

Dividing

121

by

11

gives

5 + 2x = 11

, which simplifies to

2x = 6

and

x = 3

.

Key Concept

Determinant of a Matrix Product

Estimated Time:1m 30s

Question 165Question →

Let the universal set be $\mathcal{U} = \{x \in \mathbb{Z} : 1 \le x \le 25\}$ . If $A = \{x \in \mathcal{U} : x \text{ is a multiple of } 3\}$ and $B = \{x \in \mathcal{U} : x \text{ is a perfect square}\}$ , what is the cardinality of $(A \cup B)'$ ?

13

12

14

17

Show answer & explanation

Answer: 13

Answer

13

The total number of elements in the universal set is 25. Set A has 8 elements (multiples of 3) and set B has 5 elements (perfect squares). The element 9 belongs to both sets. Therefore, the union of A and B contains 8 + 5 - 1 = 12 elements. Subtracting this from the universal set size gives 25 - 12 = 13 elements in the complement.

Step-by-Step Solution

1

Determine the elements of the universal set and its cardinality.

\mathcal{U} = \{1, 2, 3, \dots, 25\}

, so

n(\mathcal{U}) = 25$.

The universal set contains all integers from 1 to 25 inclusive.

2

List the elements of set A and set B.

A = \{3, 6, 9, 12, 15, 18, 21, 24\}

(so

n(A) = 8

) and

B = \{1, 4, 9, 16, 25\}

(so

n(B) = 5

).

Set A contains multiples of 3 within the domain, and set B contains perfect squares within the domain.

3

Find the intersection

A \cap B

and calculate the cardinality of the union

A \cup B

.

A \cap B = \{9\}

, so

n(A \cap B) = 1

. Thus,

n(A \cup B) = n(A) + n(B) - n(A \cap B) = 8 + 5 - 1 = 12

.

The principle of inclusion-exclusion avoids double-counting the common element 9.

4

Calculate the cardinality of the complement

(A \cup B)'

.

n((A \cup B)') = n(\mathcal{U}) - n(A \cup B) = 25 - 12 = 13

.

The complement consists of all elements in the universal set that do not belong to the union of A and B.

Key Concept

Complement of Set Union and Inclusion-Exclusion Principle

Question 166Question →

Given the matrices $A = \begin{pmatrix} 2 & x \\ -1 & 3 \end{pmatrix}$ and $B = \begin{pmatrix} 1 & 0 \\ 4 & 2 \end{pmatrix}$ , if $\det(A^{-1} B^T) = \frac{1}{5}$ , what is the value of $x$ ?

4

-4

14

-1

Show answer & explanation

Answer: 4

Answer

The value of

x

is 4.

Using the properties

\det(B^T) = \det(B) = 2

and

\det(A^{-1}) = \frac{1}{\det(A)} = \frac{1}{6+x}

, the product rule gives

\det(A^{-1}B^T) = \frac{2}{6+x}

. Equating this to

\frac{1}{5}

gives

6+x = 10

, so

x = 4

.

Step-by-Step Solution

1

Calculate the determinants of matrices

A

and

B

.

\det(A) = (2)(3) - (-1)(x) = 6 + x

and

\det(B) = (1)(2) - (0)(4) = 2$.

The determinant of a 2x2 matrix is computed using

ad - bc

.

2

Apply determinant rules for inverse and transpose matrices.

\det(B^T) = \det(B) = 2

and

\det(A^{-1}) = \frac{1}{\det(A)} = \frac{1}{6+x}$.

The determinant of a transpose equals the original determinant, and the determinant of an inverse is its reciprocal.

3

Express

\det(A^{-1} B^T)

using the product rule of determinants.

\det(A^{-1} B^T) = \det(A^{-1}) \cdot \det(B^T) = \frac{1}{6+x} \cdot 2 = \frac{2}{6+x}$.

For square matrices,

\det(MN) = \det(M)\det(N)

.

4

Equate the expression to

\frac{1}{5}

and solve for

x

.

\frac{2}{6+x} = \frac{1}{5} \implies 10 = 6 + x \implies x = 4$.

Cross-multiplying and simplifying gives

x = 4

.

Key Concept

Properties of Determinants (Product, Inverse, and Transpose)

Question 167Question →

The operating cost $C$ (in Naira per hour) of a speed boat is partly constant and partly varies directly as the square of its speed $v$ (in km/h). At a speed of $20\text{ km/h}$ , the operating cost is $₦18,000$ per hour, and at a speed of $30\text{ km/h}$ , the operating cost is $₦33,000$ per hour. What is the operating cost per hour (in Naira) when the boat travels at a speed of $40\text{ km/h}$ ?

Show answer & explanation

Answer: 54000

Answer

The operating cost per hour when the speed boat travels at

40\text{ km/h}

is

54,000

Naira.

By writing the partial variation relationship as

C = k_1 + k_2 v^2

and substituting the given pairs

(20, 18000)

and

(30, 33000)

, we set up simultaneous equations

k_1 + 400k_2 = 18000

and

k_1 + 900k_2 = 33000

. Solving these yields

k_2 = 30

and

k_1 = 6000

. Evaluating the model at

v = 40\text{ km/h}

gives

C = 6000 + 30(40^2) = 54,000

Naira per hour.

Step-by-Step Solution

1

Set up the general formula for partial variation

C = k_1 + k_2 v^2

The total cost consists of a fixed constant component

k_1

and a variable component

k_2 v^2

that varies directly with speed squared.

2

Form simultaneous linear equations from the given conditions

18,000 = k_1 + 400 k_2

and

33,000 = k_1 + 900 k_2

Substituting

v = 20, C = 18,000

and

v = 30, C = 33,000

creates a solvable system of equations in

k_1

and

k_2

.

3

Solve for the variation constants

k_1

and

k_2

k_2 = 30

and

k_1 = 6,000

Subtracting the first equation from the second eliminates

k_1

, giving

500 k_2 = 15,000 \implies k_2 = 30

. Substituting

k_2 = 30

into the first equation yields

k_1 = 6,000

.

4

Calculate the operating cost at

v = 40\text{ km/h}

C = 54,000

Evaluating

C = 6,000 + 30(40)^2 = 6,000 + 30(1,600) = 6,000 + 48,000 = 54,000

.

Key Concept

Partial Variation with Simultaneous Equations

Question 168Question →

When the polynomial $P(x) = 3x^4 - 2x^3 + ax^2 + bx - 12$ is divided by $(x^2 - 4)$ , the remainder is $5x - 4$ . What is the value of $a + b$ ?

Show answer & explanation

Answer: 3

Answer

The value of

a + b

is

3

.

By using the Remainder Theorem for the quadratic divisor

(x^2 - 4) = (x - 2)(x + 2)

, evaluating

P(2) = 6

gives

2a + b = -7

, and evaluating

P(-2) = -14

gives

2a - b = -33

. Solving these linear equations simultaneously yields

a = -10

and

b = 13

, which sums to

a + b = 3

.

Step-by-Step Solution

1

Set up the polynomial division relation.

P(x) = (x - 2)(x + 2)Q(x) + (5x - 4)

By the Remainder Theorem and Division Algorithm, dividing by

(x^2 - 4)

yields a remainder of

R(x) = 5x - 4

.

2

Find the values of

P(2)

and

P(-2)

from the remainder.

P(2) = 6

and

P(-2) = -14

Substituting the roots of the divisor

x = 2

and

x = -2

eliminates the quotient term

(x^2 - 4)Q(x)

.

3

Substitute

x = 2

into the polynomial

P(x)

and set equal to

6

.

2a + b = -7

3(16) - 2(8) + 4a + 2b - 12 = 20 + 4a + 2b = 6 \implies 4a + 2b = -14

.

4

Substitute

x = -2

into the polynomial

P(x)

and set equal to

-14

.

2a - b = -33

3(16) - 2(-8) + 4a - 2b - 12 = 52 + 4a - 2b = -14 \implies 4a - 2b = -66

.

5

Solve the system of equations for

a

and

b

.

a = -10

,

b = 13

, and

a + b = 3

Adding the two linear equations gives

4a = -40 \implies a = -10

. Substituting

a = -10

into

2a + b = -7

gives

b = 13

.

Key Concept

Polynomial Remainder Theorem for Non-Linear Divisors

Estimated Time:2m 30s

Question 169Question →

An arithmetic progression (A.P.) has a first term of $7$ and a common difference of $5$ . What is the $12^{\text{th}}$ term of the progression?

Show answer & explanation

Answer: 62

Answer

The

12^{\text{th}}

term of the progression is

62

.

Using the formula for the

n^{\text{th}}

term of an arithmetic progression,

T_n = a + (n - 1)d

, with

a = 7

,

d = 5

, and

n = 12

, we calculate

T_{12} = 7 + (12 - 1) \times 5 = 7 + 55 = 62

.

Step-by-Step Solution

1

Identify known parameters from the question

a = 7

,

d = 5

,

n = 12

These parameters are given in the problem statement.

2

Use the general formula for the

n^{\text{th}}

term of an arithmetic progression

T_n = a + (n - 1)d

This formula connects the first term, common difference, and term number to the value of the term.

3

Substitute values and evaluate

T_{12} = 7 + (12 - 1) \times 5 = 7 + 55 = 62

Performing basic arithmetic gives the final result.

Key Concept

Finding the

n^{\text{th}}

term of an Arithmetic Progression

Estimated Time:45s

Question 170Question →

A binary operation $\star$ defined on the set of real numbers $\mathbb{R}$ is given by $a \star b = 2a - 3b + ab$ . If $(2 \star x) \star 3 = 16$ , what is the value of $x$ ?

\frac{11}{4}

-1

-\frac{1}{5}

-\frac{3}{5}

Show answer & explanation

Answer: $-1$

Answer

-1

Applying the binary operation rule

a \star b = 2a - 3b + ab

step-by-step gives

2 \star x = 4 - x

. Operating on this with

3

yields

(4 - x) \star 3 = 2(4 - x) - 3(3) + 3(4 - x) = 11 - 5x

. Setting

11 - 5x = 16

gives

x = -1

.

Step-by-Step Solution

1

Evaluate the inner expression

2 \star x

using the given definition

a \star b = 2a - 3b + ab

.

2 \star x = 2(2) - 3(x) + (2)(x) = 4 - 3x + 2x = 4 - x

The expression inside the parentheses must be simplified first.

2

Substitute

4 - x

into the outer operation

(4 - x) \star 3

.

(4 - x) \star 3 = 2(4 - x) - 3(3) + (4 - x)(3) = 8 - 2x - 9 + 12 - 3x = 11 - 5x

Apply the operation rule with first element

a = 4 - x

and second element

b = 3

.

3

Equate the resulting expression to 16 and solve for

x

.

11 - 5x = 16 \implies -5x = 5 \implies x = -1

Solve the linear equation to determine the value of

x

.

Key Concept

Non-commutative nested binary operation evaluation

Estimated Time:2m 0s

Question 171Question →

An arithmetic progression (A.P.) has a first term of $8$ and a common difference of $6$ . What is the $9^{\text{th}}$ term of the progression?

56

62

50

68

Show answer & explanation

Answer: $56$

Answer

The

9^{\text{th}}

term of the arithmetic progression is

56

.

The

n^{\text{th}}

term of an arithmetic progression is given by

T_n = a + (n - 1)d

. Substituting

a = 8

,

d = 6

, and

n = 9

yields

T_9 = 8 + (9 - 1) \times 6 = 8 + 48 = 56

.

Step-by-Step Solution

1

Identify the given terms from the problem

First term

a = 8

, common difference

d = 6

, and term position

n = 9

.

These are the parameters required for the

n^{\text{th}}

term formula of an A.P.

2

Apply the

n^{\text{th}}

term formula

T_n = a + (n - 1)d

T_9 = 8 + (9 - 1) \times 6

The

n^{\text{th}}

term requires multiplying the common difference by

(n - 1)

.

3

Simplify the expression

T_9 = 8 + 8 \times 6 = 8 + 48 = 56

Perform multiplication before addition according to standard order of operations.

Key Concept

General term of an Arithmetic Progression

Question 172Question →

If $P = \begin{pmatrix} 3 & 1 \\ 2 & 4 \end{pmatrix}$ and $Q = \begin{pmatrix} 1 & -2 \\ 0 & 5 \end{pmatrix}$ , what is the product matrix $PQ$ ?

\begin{pmatrix} 3 & -1 \\ 2 & 16 \end{pmatrix}

\begin{pmatrix} 3 & -2 \\ 0 & 20 \end{pmatrix}

\begin{pmatrix} -1 & -7 \\ 10 & 20 \end{pmatrix}

\begin{pmatrix} 3 & 11 \\ 2 & 16 \end{pmatrix}

Show answer & explanation

Answer: $\begin{pmatrix} 3 & -1 \\ 2 & 16 \end{pmatrix}$

Answer

The product matrix

PQ

is

\begin{pmatrix} 3 & -1 \\ 2 & 16 \end{pmatrix}

.

The matrix containing elements 3, -1 in the top row and 2, 16 in the bottom row correctly applies the row-by-column multiplication rule for matrices

P

and

Q

.

Step-by-Step Solution

1

Set up the row-by-column matrix multiplication

PQ

PQ = \begin{pmatrix} 3 & 1 \\ 2 & 4 \end{pmatrix} \begin{pmatrix} 1 & -2 \\ 0 & 5 \end{pmatrix}

Matrix multiplication requires computing dot products of rows of the first matrix with columns of the second matrix.

2

Calculate the first row elements of the product matrix

Row 1, Column 1:

3(1) + 1(0) = 3 + 0 = 3

; Row 1, Column 2:

3(-2) + 1(5) = -6 + 5 = -1

Multiply elements of Row 1 of

P

by corresponding elements of Columns 1 and 2 of

Q

and sum them.

3

Calculate the second row elements of the product matrix

Row 2, Column 1:

2(1) + 4(0) = 2 + 0 = 2

; Row 2, Column 2:

2(-2) + 4(5) = -4 + 20 = 16

Multiply elements of Row 2 of

P

by corresponding elements of Columns 1 and 2 of

Q

and sum them.

4

Assemble the resulting

2 \times 2

matrix

PQ = \begin{pmatrix} 3 & -1 \\ 2 & 16 \end{pmatrix}

Combine all computed entry values into matrix form.

Key Concept

Matrix Multiplication (Row-by-Column Rule)

Estimated Time:1m 30s

Question 173Question →

If $a$ and $b$ are real numbers satisfying the simultaneous equations $a + b = 10$ and $a^2 - b^2 = 40$ , what is the value of $a$ ?

Show answer & explanation

Answer: 7

Answer

The value of

a

is 7.

Using the difference of squares identity,

a^2 - b^2 = (a+b)(a-b)

. Substituting

a+b = 10

into

a^2 - b^2 = 40

gives

10(a-b) = 40

, which simplifies to

a-b = 4

. Adding the two linear equations

a+b = 10

and

a-b = 4

eliminates

b

, giving

2a = 14

, hence

a = 7

.

Step-by-Step Solution

1

Factorize the quadratic expression

a^2 - b^2

.

(a + b)(a - b) = 40

Apply the difference of two squares identity.

2

Substitute

a + b = 10

into the factorized equation.

10(a - b) = 40 \implies a - b = 4

Simplifying yields a second linear equation.

3

Solve the system of linear equations

a + b = 10

and

a - b = 4

for

a

.

(a + b) + (a - b) = 10 + 4 \implies 2a = 14 \implies a = 7

Adding the two equations eliminates

b

directly.

Key Concept

Simultaneous linear and quadratic equations involving difference of squares

Question 174Question →

Given the matrices $A = \begin{pmatrix} x & 1 \\ 4 & 2 \end{pmatrix}$ and $B = \begin{pmatrix} 2 & 3 \\ 1 & x \end{pmatrix}$ , what is the sum of all real values of $x$ for which the matrix $C = AB - B^T$ is singular?

\frac{5}{2}

\frac{13}{2}

1

-\frac{5}{2}

Show answer & explanation

Answer: $\frac{5}{2}$

Answer

The sum of all real values of

x

is

\frac{5}{2}

.

Evaluating

AB

yields

\begin{pmatrix} 2x + 1 & 4x \\ 10 & 2x + 12 \end{pmatrix}

and subtracting

B^T = \begin{pmatrix} 2 & 1 \\ 3 & x \end{pmatrix}

gives matrix

C = \begin{pmatrix} 2x - 1 & 4x - 1 \\ 7 & x + 12 \end{pmatrix}

. Setting

\det(C) = (2x - 1)(x + 12) - 7(4x - 1) = 0

results in the quadratic equation

2x^2 - 5x - 5 = 0

. Since the discriminant

\Delta = 65 > 0

, both roots are real, and their sum is given by

-\frac{b}{a} = \frac{5}{2}

.

Step-by-Step Solution

1

Compute the matrix product

AB

AB = \begin{pmatrix} x & 1 \\ 4 & 2 \end{pmatrix} \begin{pmatrix} 2 & 3 \\ 1 & x \end{pmatrix} = \begin{pmatrix} 2x + 1 & 4x \\ 10 & 2x + 12 \end{pmatrix}

Matrix multiplication requires taking the dot product of rows of the first matrix with columns of the second matrix.

2

Find the transpose of matrix

B

, denoted

B^T

B^T = \begin{pmatrix} 2 & 1 \\ 3 & x \end{pmatrix}

The transpose of a matrix is formed by swapping its rows and columns.

3

Compute matrix

C = AB - B^T

C = \begin{pmatrix} 2x + 1 & 4x \\ 10 & 2x + 12 \end{pmatrix} - \begin{pmatrix} 2 & 1 \\ 3 & x \end{pmatrix} = \begin{pmatrix} 2x - 1 & 4x - 1 \\ 7 & x + 12 \end{pmatrix}

Perform element-wise subtraction of matrix

B^T

from matrix

AB

.

4

Set the determinant of matrix

C

to zero for singularity

\det(C) = (2x - 1)(x + 12) - 7(4x - 1) = 2x^2 + 23x - 12 - (28x - 7) = 2x^2 - 5x - 5 = 0

A matrix is singular if and only if its determinant is zero.

5

Find the sum of the real roots of the quadratic equation

2x^2 - 5x - 5 = 0

Discriminant

\Delta = (-5)^2 - 4(2)(-5) = 65 > 0

. By Vieta's formula,

x_1 + x_2 = -\frac{b}{a} = -\frac{-5}{2} = \frac{5}{2}

.

Since the discriminant is positive, two distinct real roots exist, and their sum is given by

-\frac{b}{a}

.

Key Concept

Singular matrix condition, matrix multiplication, transpose operations, and Vieta's formulas.

Question 175Question →

When the polynomial $P(x) = x^3 - 3x^2 + kx + 12$ is divided by $(x - 2)$ , the remainder is $6$ . What is the value of the constant $k$ ?

Show answer & explanation

Answer: -1

Answer

The value of the constant

k

is

-1

.

By the Remainder Theorem, the remainder when

P(x)

is divided by

(x - 2)

is

P(2)

. Substituting

x = 2

into

P(x) = x^3 - 3x^2 + kx + 12

gives

P(2) = 8 - 12 + 2k + 12 = 8 + 2k

. Setting

8 + 2k = 6

yields

2k = -2

, so

k = -1

.

Step-by-Step Solution

1

Apply the Remainder Theorem

P(2) = 6

Dividing

P(x)

by

(x - 2)

leaves a remainder equal to evaluating

P(x)

at

x = 2

.

2

Substitute

x = 2

into

P(x)

and set equal to

6

(2)^3 - 3(2)^2 + 2k + 12 = 6

Set the evaluated polynomial equal to the given remainder.

3

Simplify the arithmetic terms

8 + 2k = 6

Calculate powers and products:

8 - 12 + 12 = 8

.

4

Solve the linear equation for

k

k = -1

Subtract 8 from both sides to get

2k = -2

, then divide by 2.

Key Concept

Polynomial Remainder Theorem

Question 176Question →

If $\frac{2 + \sqrt{3}}{2 - \sqrt{3}} = x + y\sqrt{3}$ , where $x$ and $y$ are integers, what is the value of $x + y$ ?

Show answer & explanation

Answer: 11

Answer

The value of

x + y

is 11.

Multiplying the given fraction by

\frac{2 + \sqrt{3}}{2 + \sqrt{3}}

rationalises the denominator to 1 and simplifies the numerator to

7 + 4\sqrt{3}

. Comparing coefficients yields

x = 7

and

y = 4

, making

x + y = 11

.

Step-by-Step Solution

1

Multiply numerator and denominator by the conjugate of the denominator

\frac{(2 + \sqrt{3})(2 + \sqrt{3})}{(2 - \sqrt{3})(2 + \sqrt{3})}

Rationalising the denominator eliminates the surd from the denominator using the identity (a-b)(a+b) = a^2 - b^2.

2

Expand both the numerator and the denominator

\frac{4 + 4\sqrt{3} + 3}{4 - 3} = \frac{7 + 4\sqrt{3}}{1} = 7 + 4\sqrt{3}

Simplifying algebraic surd multiplication gives integer and surd terms.

3

Compare terms with x + y\sqrt{3} and solve for x and y

x = 7, y = 4 \implies x + y = 11

Matching rational components and coefficients of \sqrt{3} yields x and y.

Key Concept

Rationalisation of Binomial Denominators containing Surds

Estimated Time:1m 30s

Question 177Question →

Given the $2 \times 2$ matrix $M = \begin{pmatrix} 7 & -2 \\ 4 & 3 \end{pmatrix}$ , what is the value of $\det(M)$ ?

Show answer & explanation

Answer: 29

Answer

The determinant of matrix

M

is 29.

The determinant of a

2 \times 2

matrix

\begin{pmatrix} a & b \\ c & d \end{pmatrix}

is calculated as

ad - bc

. For the matrix

M = \begin{pmatrix} 7 & -2 \\ 4 & 3 \end{pmatrix}

, we compute

(7)(3) - (-2)(4) = 21 - (-8) = 29

.

Step-by-Step Solution

1

Identify the entries

a, b, c, d

from the matrix

M = \begin{pmatrix} 7 & -2 \\ 4 & 3 \end{pmatrix}

.

a = 7

,

b = -2

,

c = 4

, and

d = 3

.

These entries correspond to the standard

2 \times 2

matrix representation

\begin{pmatrix} a & b \\ c & d \end{pmatrix}

.

2

Compute the determinant using the formula

\det(M) = ad - bc

.

\det(M) = (7)(3) - (-2)(4) = 21 + 8 = 29

.

Multiplying the main diagonal entries and subtracting the product of the off-diagonal entries yields the determinant.

Key Concept

Determinant of a 2x2 Matrix

Question 178Question →

Given the simultaneous equations $2x - y = 4$ and $x^2 + y^2 = 13$ , where both $x$ and $y$ are positive real numbers, calculate the value of $x + y$ .

Show answer & explanation

Answer: 5

Answer

The correct value of

x + y

is 5.

Substituting

y = 2x - 4

into

x^2 + y^2 = 13

gives

5x^2 - 16x + 3 = 0

, which yields

x = 3

or

x = 0.2

. The corresponding

y

-values are

y = 2

and

y = -3.6

. Since both

x

and

y

must be positive, the valid pair is

(3, 2)

, giving

x + y = 3 + 2 = 5

.

Step-by-Step Solution

1

Express

y

in terms of

x

using the linear equation

2x - y = 4

.

y = 2x - 4

Substitution is the standard method for solving simultaneous linear and quadratic equations.

2

Substitute

y = 2x - 4

into the quadratic equation

x^2 + y^2 = 13

.

x^2 + (2x - 4)^2 = 13

This reduces the system to a single quadratic equation in one variable.

3

Expand and collect like terms.

5x^2 - 16x + 3 = 0

Expanding

(2x - 4)^2 = 4x^2 - 16x + 16

and subtracting 13 puts the quadratic in standard form

ax^2 + bx + c = 0

.

4

Solve the quadratic equation

5x^2 - 16x + 3 = 0

for

x

.

x = 3

or

x = 0.2

Factoring

(5x - 1)(x - 3) = 0

yields two real solutions for

x

.

5

Find corresponding values of

y

and apply the positivity constraint

x > 0

and

y > 0

.

x = 3, y = 2

When

x = 0.2

,

y = -3.6

, which is not positive. Hence,

(3, 2)

is the only valid solution pair.

6

Calculate

x + y

.

5

3 + 2 = 5

.

Key Concept

Solving simultaneous linear and quadratic equations using substitution

Estimated Time:1m 30s

Question 179Question →

Given the matrices $A = \begin{pmatrix} x & 2 & 1 \\ 3 & 0 & 2 \\ 1 & 4 & 1 \end{pmatrix}$ and $B = \begin{pmatrix} 2 & -1 \\ 5 & 3 \end{pmatrix}$ , find the value of $x$ if $\det(A) + \det(B) = 5$ .

Show answer & explanation

Answer: 2

Answer

The value of x is 2.

Evaluating \det(B) gives 11, and cofactor expansion of \det(A) along the top row yields 10 - 8x. Summing these determinants gives (10 - 8x) + 11 = 5, which simplifies to 8x = 16, resulting in x = 2.

Step-by-Step Solution

1

Calculate the determinant of the 2x2 matrix B

\det(B) = 11

For a 2x2 matrix \begin{pmatrix} a & b \\ c & d \end{pmatrix}, the determinant is ad - bc. Thus, \det(B) = (2)(3) - (-1)(5) = 6 + 5 = 11.

2

Calculate the determinant of the 3x3 matrix A in terms of x

\det(A) = 10 - 8x

Expanding along the first row: \det(A) = x\begin{vmatrix} 0 & 2 \\ 4 & 1 \end{vmatrix} - 2\begin{vmatrix} 3 & 2 \\ 1 & 1 \end{vmatrix} + 1\begin{vmatrix} 3 & 0 \\ 1 & 4 \end{vmatrix} = x(0 - 8) - 2(3 - 2) + 1(12 - 0) = -8x - 2 + 12 = 10 - 8x.

3

Set up and solve the linear equation \det(A) + \det(B) = 5

x = 2

Substituting the computed determinants gives (10 - 8x) + 11 = 5, which simplifies to 21 - 8x = 5. Rearranging yields 8x = 16, so x = 2.

Key Concept

Determinants of 2x2 and 3x3 matrices and linear equation solving

Question 180Question →

A binary operation $\star$ is defined on the set of real numbers $\mathbb{R}$ by $a \star b = a + b + 7$ . What is the identity element of the operation?

-7

7

0

-1

Show answer & explanation

Answer: $-7$

Answer

The identity element of the operation is

-7

.

For an identity element

e

, the condition

a \star e = a

must hold for all real numbers

a

. Substituting the operation definition gives

a + e + 7 = a

. Subtracting

a

from both sides results in

e + 7 = 0

, which solves to

e = -7

.

Step-by-Step Solution

1

Set up the defining equation for the identity element

e

.

a \star e = a

By definition, an identity element

e

leaves any element

a

unchanged under the operation.

2

Apply the given rule for the binary operation.

a + e + 7 = a

The binary operation is defined as

a \star b = a + b + 7

.

3

Solve for

e

.

e = -7

Subtracting

a + 7

from both sides gives

e = -7

.

Key Concept

Identity element of a binary operation

Estimated Time:45s

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