Algebra Practice Questions — JAMB UTME — Page 12

Question 221Question →

In a sports academy of 120 athletes, 70 play football, 60 play basketball, and 50 play tennis. If 10 athletes play none of these three sports and 15 athletes play all three sports, how many athletes play exactly two of these sports?

Show answer & explanation

Answer: 40

Answer

40 athletes play exactly two of the sports.

The correct answer is 40. Subtracting the 10 athletes who play no sports from the total of 120 leaves 110 athletes playing at least one sport. Using inclusion-exclusion, the sum of pairwise intersections is

S_2 = 70 + 60 + 50 + 15 - 110 = 85

. Since

S_2

contains the region of all three sports counted three times, subtracting

3 \times 15 = 45

gives 40 athletes who play exactly two sports.

Step-by-Step Solution

1

Determine the cardinality of the union of all three sets

n(F \cup B \cup T) = 120 - 10 = 110

Athletes who play none of the three sports are excluded from the total universal set.

2

Apply the Principle of Inclusion-Exclusion for three sets to find the sum of 2-set intersections

S_2 = n(F) + n(B) + n(T) + n(F \cap B \cap T) - n(F \cup B \cup T) = 70 + 60 + 50 + 15 - 110 = 85

The formula relates the total union, individual set cardinalities, pairwise intersections, and triple intersection.

3

Subtract three times the triple intersection from

S_2

to isolate regions corresponding to exactly two sports

Exactly two sports =

S_2 - 3 \times n(F \cap B \cap T) = 85 - 3(15) = 40

Each pairwise intersection sum

S_2

includes the triple intersection region three times.

Key Concept

Three-set inclusion-exclusion principle and region cardinality decomposition

Estimated Time:1m 30s

Question 222Question →

The sum of the first $n$ terms of an arithmetic progression (A.P.) is given by $S_n = 2n^2 + 3n$ . What is the $5^{\text{th}}$ term of the progression?

21

65

44

25

Show answer & explanation

Answer: $21$

Answer

The

5^{\text{th}}

term of the arithmetic progression is

21

.

The

n^{\text{th}}

term of a sequence can be determined from its sum formula using

T_n = S_n - S_{n-1}

. Evaluating

S_5 = 2(5)^2 + 3(5) = 65

and

S_4 = 2(4)^2 + 3(4) = 44

, the difference

T_5 = 65 - 44 = 21

gives the correct value of the

5^{\text{th}}

term.

Step-by-Step Solution

1

Calculate the sum of the first 5 terms (

S_5

)

S_5 = 2(5)^2 + 3(5) = 2(25) + 15 = 65

To find the sum up to the

5^{\text{th}}

term using the given formula

S_n = 2n^2 + 3n

.

2

Calculate the sum of the first 4 terms (

S_4

)

S_4 = 2(4)^2 + 3(4) = 2(16) + 12 = 44

To find the cumulative total of all terms prior to the

5^{\text{th}}

term.

3

Subtract

S_4

from

S_5

to isolate the

5^{\text{th}}

term (

T_5

)

T_5 = S_5 - S_4 = 65 - 44 = 21

The

n^{\text{th}}

term of any sequence is given by the relation

T_n = S_n - S_{n-1}

.

Key Concept

Relationship between the sum of

n

terms (

S_n

) and the

n^{\text{th}}

term (

T_n

) in an Arithmetic Progression

Estimated Time:1m 30s

Question 223Question →

What is the positive value of $x$ that satisfies the equation $2^{2x+1} - 9 \cdot 2^x + 4 = 0$ ?

Show answer & explanation

Answer: 2

Answer

The positive value of

x

that satisfies the equation is 2.

Applying the law of indices

a^{m+n} = a^m \cdot a^n

gives

2^{2x+1} = 2 \cdot (2^x)^2

. Setting

y = 2^x

yields the quadratic equation

2y^2 - 9y + 4 = 0

. Factoring this expression gives

(2y - 1)(y - 4) = 0

, which yields roots

y = \frac{1}{2}

and

y = 4

. Solving

2^x = \frac{1}{2}

gives

x = -1

, and solving

2^x = 4

gives

x = 2

. The positive value is

2

.

Step-by-Step Solution

1

Use index laws to express the equation in terms of

2^x

2 \cdot (2^x)^2 - 9 \cdot (2^x) + 4 = 0

By the product law of indices,

2^{2x+1} = 2^{2x} \cdot 2^1 = 2 \cdot (2^x)^2

.

2

Substitute

y = 2^x

to form a quadratic equation

2y^2 - 9y + 4 = 0

Replacing

2^x

with a single variable simplifies the exponential equation into quadratic form.

3

Solve the quadratic equation for

y

y = \frac{1}{2}

or

y = 4

Factoring

2y^2 - 9y + 4 = 0

gives

(2y - 1)(y - 4) = 0

.

4

Substitute back

y = 2^x

to solve for

x

x = -1

or

x = 2

Since

2^x = \frac{1}{2} = 2^{-1}

,

x = -1

. Since

2^x = 4 = 2^2

,

x = 2

.

5

Select the positive value requested by the question

x = 2

x = 2

is positive, whereas

x = -1

is negative.

Key Concept

Reducing exponential equations to quadratic form using index laws

Estimated Time:1m 30s

Question 224Question →

Determine the smallest integer value of $x$ that satisfies the compound inequality $3 < 2x - 5 \le 11$ .

Show answer & explanation

Answer: 5

Answer

The smallest integer value of

x

that satisfies the inequality is 5.

Adding 5 across the compound inequality

3 < 2x - 5 \le 11

gives

8 < 2x \le 16

. Dividing by 2 yields

4 < x \le 8

. The integer values satisfying this range are 5, 6, 7, and 8. Therefore, the smallest integer solution is 5.

Step-by-Step Solution

1

Add 5 to all parts of the compound inequality

8 < 2x <= 16

Isolates the variable term in the middle segment.

2

Divide all parts of the compound inequality by 2

4 < x <= 8

Solves for x without changing inequality signs since 2 is positive.

3

Identify integer solutions within the range (4, 8]

x in {5, 6, 7, 8}

Since the inequality at 4 is strict (<), 4 is excluded, but 8 is included (<=).

4

Find the minimum integer value in the solution set

5

5 is the smallest integer strictly greater than 4.

Key Concept

Solving Compound Linear Inequalities

Question 225Question →

Which of the following represents the solution set of real values of $x$ satisfying the inequality $4(1 - x) \le 3(x + 6)$ ?

x \ge -2

x \le -2

x \ge 2

x \le 2

Show answer & explanation

Answer: $x \ge -2$

Answer

The set of real values satisfying the inequality is

x \ge -2

.

Expanding the given inequality yields

4 - 4x \le 3x + 18

. Subtracting

3x

and

4

from both sides gives

-7x \le 14

. When dividing both sides by

-7

, the inequality sign must reverse direction, yielding

x \ge -2

.

Step-by-Step Solution

1

Expand both sides of the inequality

4 - 4x \le 3x + 18

Remove brackets to group like terms.

2

Rearrange terms by moving variable terms to the left side and constant terms to the right side

-4x - 3x \le 18 - 4 \implies -7x \le 14

Isolate the term containing the variable

x

.

3

Divide both sides by

-7

and reverse the inequality sign

x \ge -2

Dividing or multiplying an inequality by a negative number flips the direction of the inequality symbol.

Key Concept

Solving linear inequalities involving bracket expansion and division by negative numbers

Question 226Question →

How many integer values of $x$ satisfy the quadratic inequality $2x^2 - 7x - 4 \le 0$ ?

Show answer & explanation

Answer: 5

Answer

There are 5 integer values of x that satisfy the inequality.

Solving the quadratic inequality yields

-\frac{1}{2} \le x \le 4

. The integer solutions within this interval are

0, 1, 2, 3,

and

4

. Counting them gives a total of

5

valid integer values.

Step-by-Step Solution

1

Factor the quadratic expression.

(2x + 1)(x - 4) \le 0

Factoring allows determination of the critical boundary points.

2

Find the critical values (roots of the equation).

x = -\frac{1}{2}

and

x = 4

The roots divide the number line into test intervals.

3

Determine the solution set of the inequality.

-\frac{1}{2} \le x \le 4

Since the quadratic coefficient is positive, the quadratic expression is non-positive between its roots.

4

List and count the integers within the range.

The integers are

0, 1, 2, 3, 4

, making a total of

5

integers.

Counting only whole numbers in the closed interval

[ -0.5, 4 ]

.

Key Concept

Quadratic Inequalities and Integer Solution Counting

Question 227Question →

Given that $(x + 2)$ is a factor of the polynomial $P(x) = 2x^3 - x^2 + ax + b$ , and that dividing $P(x)$ by $(2x - 1)$ leaves a remainder of $15$ , what is the value of $a + b$ ?

6

11

14

-1

Show answer & explanation

Answer: $14$

Answer

The value of

a + b

is

14

.

Applying the Factor Theorem with

x = -2

gives the equation

-2a + b = 20

. Applying the Remainder Theorem with

x = \frac{1}{2}

gives the equation

a + 2b = 30

. Solving these simultaneous equations yields

a = -2

and

b = 16

. Adding

a

and

b

gives

14

.

Step-by-Step Solution

1

Apply the Factor Theorem for

(x + 2)

-2a + b = 20

Since

(x + 2)

is a factor of

P(x)

,

P(-2) = 0

. Substituting

x = -2

yields

2(-2)^3 - (-2)^2 + a(-2) + b = 0

, which simplifies to

-16 - 4 - 2a + b = 0

or

-2a + b = 20

.

2

Apply the Remainder Theorem for

(2x - 1)

a + 2b = 30

Dividing

P(x)

by

(2x - 1)

leaves a remainder of

15

, so

P\left(\frac{1}{2}\right) = 15

. Substituting

x = \frac{1}{2}

yields

2\left(\frac{1}{8}\right) - \frac{1}{4} + \frac{a}{2} + b = 15

, which simplifies to

\frac{a}{2} + b = 15

or

a + 2b = 30

.

3

Solve the system of linear equations for

a

and

b

a = -2

and

b = 16

From step 1,

b = 2a + 20

. Substituting this into step 2 gives

a + 2(2a + 20) = 30 \implies 5a + 40 = 30 \implies 5a = -10 \implies a = -2

. Substituting

a = -2

into

b = 2a + 20

gives

b = 16

.

4

Calculate

a + b

14

Summing the calculated values gives

a + b = -2 + 16 = 14

.

Key Concept

Factor Theorem and Remainder Theorem for Polynomials

Question 228Question →

A binary operation $\star$ defined on the set of real numbers $\mathbb{R}$ is given by $x \star y = \frac{x + y}{x - y + 1}$ for $x - y \neq -1$ . If $5 \star p = 3$ , what is the value of $p$ ?

Show answer & explanation

Answer: 3.25

Answer

The value of

p

is

3.25

(or

\frac{13}{4}

).

Applying the operation rule

x \star y = \frac{x + y}{x - y + 1}

with

x = 5

and

y = p

gives

\frac{5 + p}{6 - p} = 3

. Cross-multiplying gives

5 + p = 18 - 3p

, which simplifies to

4p = 13

or

p = 3.25

.

Step-by-Step Solution

1

Substitute

x = 5

and

y = p

into the given binary operation definition

\frac{5 + p}{5 - p + 1} = 3

This sets up the equation for the given condition

5 \star p = 3

.

2

Simplify the denominator in the algebraic fraction

\frac{5 + p}{6 - p} = 3

Combining the constants

5 + 1 = 6

simplifies the denominator.

3

Multiply both sides by

(6 - p)

and expand

5 + p = 18 - 3p

Eliminating the denominator allows linear terms in

p

to be collected.

4

Rearrange terms to solve for

p

4p = 13 \implies p = 3.25

Adding

3p

to both sides and subtracting

5

gives

4p = 13

.

Key Concept

Solving linear equations derived from non-commutative binary operations

Question 229Question →

Given the matrices $A = \begin{pmatrix} 2 & 1 \\ 0 & 3 \end{pmatrix}$ and $B = \begin{pmatrix} 1 & 4 \\ 2 & -1 \end{pmatrix}$ , what is the product matrix $AB$ ?

\begin{pmatrix} 4 & 7 \\ 6 & -3 \end{pmatrix}

\begin{pmatrix} 2 & 4 \\ 0 & -3 \end{pmatrix}

\begin{pmatrix} 2 & 13 \\ 4 & -1 \end{pmatrix}

\begin{pmatrix} 4 & 9 \\ 6 & -3 \end{pmatrix}

Show answer & explanation

Answer: $\begin{pmatrix} 4 & 7 \\ 6 & -3 \end{pmatrix}$

Answer

The matrix product

AB

is

\begin{pmatrix} 4 & 7 \\ 6 & -3 \end{pmatrix}

.

To find

AB

, each element

c_{ij}

is formed by taking the dot product of row

i

of matrix

A

and column

j

of matrix

B

. Performing these steps gives

c_{11} = 2(1) + 1(2) = 4

,

c_{12} = 2(4) + 1(-1) = 7

,

c_{21} = 0(1) + 3(2) = 6

, and

c_{22} = 0(4) + 3(-1) = -3

, yielding

\begin{pmatrix} 4 & 7 \\ 6 & -3 \end{pmatrix}

.

Step-by-Step Solution

1

Multiply the first row of

A

by the first column of

B

to find element

(1,1)

.

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

Matrix multiplication rule requires taking the dot product of rows from the first matrix and columns from the second matrix.

2

Multiply the first row of

A

by the second column of

B

to find element

(1,2)

.

2(4) + 1(-1) = 8 - 1 = 7

Evaluates the element in row 1, column 2 of the resulting matrix.

3

Multiply the second row of

A

by the first column of

B

to find element

(2,1)

.

0(1) + 3(2) = 0 + 6 = 6

Evaluates the element in row 2, column 1 of the resulting matrix.

4

Multiply the second row of

A

by the second column of

B

to find element

(2,2)

.

0(4) + 3(-1) = 0 - 3 = -3

Evaluates the element in row 2, column 2 of the resulting matrix.

Key Concept

Matrix Multiplication

Estimated Time:1m 0s

Question 230Question →

A binary operation $\Delta$ defined on the set of real numbers $\mathbb{R}$ is given by $a \Delta b = a + b - \frac{ab}{4}$ . What is the inverse of $2$ under this operation?

-4

4

-2

-\frac{4}{3}

Show answer & explanation

Answer: $-4$

Answer

The inverse of

2

under the binary operation is

-4

.

To find the inverse of

2

, we first determine the identity element

e

using

a \Delta e = a

, which yields

e = 0

. Setting

2 \Delta x = 0

gives

2 + x - \frac{2x}{4} = 0

, which simplifies to

2 + \frac{x}{2} = 0

, resulting in

x = -4

.

Step-by-Step Solution

1

Determine the identity element

e

of the operation.

e = 0

By definition,

a \Delta e = a

. Thus,

a + e - \frac{ae}{4} = a \implies e\left(1 - \frac{a}{4}\right) = 0

, which gives

e = 0

for all real

a

.

2

Set up the inverse equation for the element

2

.

2 + x - \frac{2x}{4} = 0

Let

x

be the inverse of

2

. By definition,

2 \Delta x = e

, where

e = 0

.

3

Simplify and solve for

x

.

x = -4

Simplify

2 + x - \frac{x}{2} = 0 \implies 2 + \frac{x}{2} = 0 \implies \frac{x}{2} = -2 \implies x = -4

.

Key Concept

Inverse Element in Binary Operations

Question 231Question →

An arithmetic progression (A.P.) and a geometric progression (G.P.) both have a first term of $2$ . The common difference of the A.P. is $4$ . If the $5^{\text{th}}$ term of the A.P. is equal to the $3^{\text{rd}}$ term of the G.P., and the common ratio of the G.P. is positive, what is the $4^{\text{th}}$ term of the G.P.?

54

162

18

14

Show answer & explanation

Answer: 54

Answer

54

First, evaluate the 5th term of the A.P. using

T_5 = 2 + (5-1)4 = 18

. Next, set the 3rd term of the G.P. equal to 18:

2 r^2 = 18

, giving

r^2 = 9

and

r = 3

. Finally, find the 4th term of the G.P. using

T_4 = 2 \times 3^3 = 54

.

Step-by-Step Solution

1

Calculate the 5th term of the A.P.

T_5^{(A.P.)} = a + (5-1)d = 2 + 4(4) = 18

The formula for the

n^{\text{th}}

term of an A.P. is

T_n = a + (n-1)d

.

2

Find the common ratio

r

of the G.P. by equating the 3rd term of the G.P. to 18

T_3^{(G.P.)} = a r^{3-1} = 2 r^2 = 18 \implies r^2 = 9 \implies r = 3

(since

r > 0

)

The formula for the

n^{\text{th}}

term of a G.P. is

T_n = a r^{n-1}

.

3

Compute the 4th term of the G.P.

T_4^{(G.P.)} = a r^{4-1} = 2 \times 3^3 = 2 \times 27 = 54

Substitute

a = 2

,

r = 3

, and

n = 4

into

T_n = a r^{n-1}

.

Key Concept

Connecting terms of Arithmetic and Geometric Progressions using their

n^{\text{th}}

term formulas

Question 232Question →

If the determinant of the matrix $A = \begin{pmatrix} k & 1 & 2 \\ 3 & -1 & 0 \\ 2 & 4 & 1 \end{pmatrix}$ is equal to $17$ , what is the value of $k$ ?

Show answer & explanation

Answer: 8

Answer

The value of

k

is

8

.

Expanding the matrix determinant along the first row gives

\det(A) = k(-1) - 1(3) + 2(14) = -k + 25

. Setting

-k + 25 = 17

leads to

-k = -8

, so

k = 8

.

Step-by-Step Solution

1

Perform cofactor expansion along the first row of matrix

A

.

\det(A) = k((-1)(1) - (0)(4)) - 1((3)(1) - (0)(2)) + 2((3)(4) - (-1)(2))

Expanding along the first row uses the formula \det(A) = a_{11}C_{11} + a_{12}C_{12} + a_{13}C_{13}.

2

Evaluate the products and simplify the algebraic expression for the determinant.

\det(A) = -k - 3 + 28 = -k + 25

Simplifying each sub-determinant term yields a linear expression in

k

.

3

Equate the expression to

17

and solve for

k

.

-k + 25 = 17 \implies k = 8

Subtracting 25 from both sides gives

-k = -8

, which simplifies to

k = 8

.

Key Concept

Determinant of a 3x3 Matrix

Question 233Question →

In an Arithmetic Progression (A.P.), the sum of the first $10$ terms is $120$ and the sum of the next $10$ terms is $320$ . What is the common difference of the progression?

1

2

4

20

Show answer & explanation

Answer: $2$

Answer

The common difference of the progression is

2

.

The sum of an A.P. is

S_n = \frac{n}{2}[2a + (n-1)d]

. For the first 10 terms,

S_{10} = 5(2a + 9d) = 120

, simplifying to

2a + 9d = 24

. The sum of the first 20 terms is

120 + 320 = 440

, so

S_{20} = 10(2a + 19d) = 440

, simplifying to

2a + 19d = 44

. Subtracting the two equations gives

10d = 20

, which yields the common difference

d = 2

.

Step-by-Step Solution

1

Express the sum of the first 10 terms using the formula

S_n = \frac{n}{2}[2a + (n-1)d]

.

S_{10} = \frac{10}{2}[2a + 9d] = 5(2a + 9d) = 120 \implies 2a + 9d = 24

.

Relating the given sum of the first 10 terms to the first term

a

and common difference

d

generates the first linear equation.

2

Determine the sum of the first 20 terms (

S_{20}

) and set up the second linear equation.

S_{20} = S_{10} + \text{sum of next 10 terms} = 120 + 320 = 440

. Thus,

S_{20} = \frac{20}{2}[2a + 19d] = 10(2a + 19d) = 440 \implies 2a + 19d = 44

.

The sum of the next 10 terms added to the sum of the first 10 terms gives the total sum of the first 20 terms.

3

Solve the system of simultaneous linear equations for

d

.

(2a + 19d) - (2a + 9d) = 44 - 24 \implies 10d = 20 \implies d = 2

.

Subtracting equation (1) from equation (2) eliminates

a

to solve directly for the common difference

d

.

Key Concept

Sum of an Arithmetic Progression and Simultaneous Equations

Question 234Question →

If $(x, y)$ is a pair of positive integers satisfying the simultaneous equations $x + 2y = 7$ and $x^2 + y^2 = 13$ , what is the value of $2x - y$ ?

Show answer & explanation

Answer: 4

Answer

The value of

2x - y

is 4.

Substituting

x = 7 - 2y

into

x^2 + y^2 = 13

yields the quadratic

5y^2 - 28y + 36 = 0

, which factors as

(5y - 18)(y - 2) = 0

. Since

x

and

y

must both be positive integers, the valid solution pair is

(x, y) = (3, 2)

. Evaluating

2x - y

gives

2(3) - 2 = 4

.

Step-by-Step Solution

1

Express

x

in terms of

y

using the linear equation

x + 2y = 7

.

x = 7 - 2y

Isolating

x

allows substitution into the quadratic equation.

2

Substitute

x = 7 - 2y

into

x^2 + y^2 = 13

and expand.

(7 - 2y)^2 + y^2 = 13 \implies 49 - 28y + 4y^2 + y^2 = 13

This reduces the system to a single quadratic equation in terms of

y

.

3

Rearrange and solve the resulting quadratic equation

5y^2 - 28y + 36 = 0

.

(5y - 18)(y - 2) = 0 \implies y = 2

or

y = 3.6

Factoring provides the possible values for

y

.

4

Identify the integer solution pair and compute the target expression

2x - y

.

Since

y

must be a positive integer,

y = 2

and

x = 3

. Therefore,

2x - y = 2(3) - 2 = 4

.

The non-integer solution is rejected based on the positive integer constraint.

Key Concept

Simultaneous Linear and Quadratic Equations

Estimated Time:1m 30s

Question 235Question →

A binary operation $\ast$ is defined on the set of real numbers $\mathbb{R}$ by $a \ast b = a + b - 4$ . If $x^{-1}$ represents the inverse of an element $x$ under the operation $\ast$ , what is the value of $x$ such that $x \ast 3 = 10^{-1}$ ?

-1

-5

-9

\frac{11}{10}

Show answer & explanation

Answer: $-1$

Answer

The value of

x

is

-1

.

First, the identity element

e

is determined using

a \ast e = a

, which gives

a + e - 4 = a \implies e = 4

. Next, the inverse

10^{-1}

is calculated from

10 \ast 10^{-1} = 4

, yielding

10 + 10^{-1} - 4 = 4 \implies 10^{-1} = -2

. Finally, setting

x \ast 3 = -2

gives

x + 3 - 4 = -2 \implies x - 1 = -2 \implies x = -1

.

Step-by-Step Solution

1

Find the identity element

e

under the operation

\ast

.

e = 4

By definition of an identity element,

a \ast e = a \implies a + e - 4 = a \implies e = 4

.

2

Determine the inverse of

10

, denoted

10^{-1}

, under

\ast

.

10^{-1} = -2

By definition of an inverse element,

10 \ast 10^{-1} = e \implies 10 + 10^{-1} - 4 = 4 \implies 6 + 10^{-1} = 4 \implies 10^{-1} = -2

.

3

Substitute

10^{-1} = -2

into the given equation

x \ast 3 = 10^{-1}

and solve for

x

.

x = -1

Using the definition of the operation,

x \ast 3 = x + 3 - 4 = x - 1

. Setting

x - 1 = -2

yields

x = -1

.

Key Concept

Identity and Inverse Elements in Binary Operations

Question 236Question →

A $2 \times 2$ matrix is given by $A = \begin{pmatrix} x + 2 & 3 \\ 4 & x - 2 \end{pmatrix}$ . If $A$ is a singular matrix, what is the positive value of $x$ ?

Show answer & explanation

Answer: 4

Answer

The positive value of

x

is

4

.

A matrix is singular when its determinant is zero. Expanding

\det(A) = (x+2)(x-2) - 12 = 0

yields

x^2 - 16 = 0

. Solving

x^2 = 16

gives

x = \pm 4

. The positive value is

4

.

Step-by-Step Solution

1

Apply the condition for a matrix to be singular

\det(A) = 0

By definition, a matrix is singular if and only if its determinant is zero.

2

Formulate the determinant equation for matrix

A

(x + 2)(x - 2) - (3)(4) = 0

The determinant of a

2 \times 2

matrix

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

is calculated as

ad - bc

.

3

Expand and simplify the algebraic equation

x^2 - 4 - 12 = 0 \implies x^2 - 16 = 0

Expanding

(x+2)(x-2)

gives

x^2 - 4

, and subtracting

12

yields

x^2 - 16

.

4

Solve for

x

and select the positive root

x^2 = 16 \implies x = 4 \text{ or } x = -4 \implies x = 4

Taking the square root gives two roots, and the problem explicitly specifies the positive value.

Key Concept

Singular Matrices and Determinants

Estimated Time:1m 15s

Question 237Question →

A binary operation $\circ$ defined on the set of real numbers $\mathbb{R}$ is given by $a \circ b = 3a + 2b - 1$ . What is the value of $x$ for which $(2 \circ x) \circ 3 = 26$ ?

1

\frac{11}{6}

0

\frac{5}{6}

Show answer & explanation

Answer: $1$

Answer

The value of

x

for which

(2 \circ x) \circ 3 = 26

is

1

.

Applying the binary operation rule

a \circ b = 3a + 2b - 1

to the inner expression gives

2 \circ x = 2x + 5

. Then applying the rule to

(2x + 5) \circ 3

gives

3(2x + 5) + 2(3) - 1 = 6x + 20

. Equating

6x + 20 = 26

yields

6x = 6

, giving

x = 1

.

Step-by-Step Solution

1

Evaluate the inner binary operation expression

2 \circ x

using the rule

a \circ b = 3a + 2b - 1

.

2 \circ x = 3(2) + 2(x) - 1 = 6 + 2x - 1 = 2x + 5

The definition of the binary operation replaces

a

with

2

and

b

with

x

.

2

Substitute the result

(2x + 5)

into the outer binary operation expression

(2 \circ x) \circ 3

.

(2x + 5) \circ 3 = 3(2x + 5) + 2(3) - 1 = 6x + 15 + 6 - 1 = 6x + 20

Apply the rule

a \circ b = 3a + 2b - 1

where

a = 2x + 5

and

b = 3

.

3

Equate the simplified expression to

26

and solve for

x

.

6x + 20 = 26 \implies 6x = 6 \implies x = 1

Setting the calculated expression equal to the given value allows isolation of

x

.

Key Concept

Evaluation of non-commutative composite binary operations

Question 238Question →

What is the sum of the $y$ -values of the solution pairs that satisfy the simultaneous equations $y - x = 2$ and $y = x^2 - 3x + 5$ ?

4

6

8

15

Show answer & explanation

Answer: $8$

Answer

The sum of the

y

-values is 8.

Rearranging the linear equation gives

y = x + 2

. Equating this to the quadratic expression gives

x + 2 = x^2 - 3x + 5

, which simplifies to

x^2 - 4x + 3 = 0

. Solving for

x

yields

x = 1

and

x = 3

. Substituting these into

y = x + 2

gives

y = 3

and

y = 5

. The sum of these

y

-values is

3 + 5 = 8

.

Step-by-Step Solution

1

Express

y

in terms of

x

from the linear equation.

y = x + 2

Isolating

y

facilitates substitution into the quadratic equation.

2

Substitute

y = x + 2

into the quadratic equation and rearrange into standard form.

x + 2 = x^2 - 3x + 5 \implies x^2 - 4x + 3 = 0

Setting the two expressions for

y

equal to each other forms a single quadratic equation in

x

.

3

Solve the quadratic equation for

x

.

(x - 1)(x - 3) = 0 \implies x_1 = 1, x_2 = 3

Factoring gives the

x

-coordinates of the intersection points.

4

Calculate the corresponding

y

-values using

y = x + 2

.

y_1 = 1 + 2 = 3

and

y_2 = 3 + 2 = 5

Each

x

-value must be substituted into the linear equation to find its paired

y

-value.

5

Find the sum of the

y

-values.

y_1 + y_2 = 3 + 5 = 8

The question asks for the sum of the

y

-coordinates.

Key Concept

Simultaneous Linear and Quadratic Equations

Alternative Method

Using Vieta's formulas: once the quadratic equation for

x

is found as

x^2 - 4x + 3 = 0

, we know

x_1 + x_2 = 4

. Since

y_1 = x_1 + 2

and

y_2 = x_2 + 2

, the sum of the

y

-values is

y_1 + y_2 = (x_1 + 2) + (x_2 + 2) = (x_1 + x_2) + 4 = 4 + 4 = 8

.

Estimated Time:1m 30s

Question 239Question →

A binary operation $\star$ defined on the set of real numbers $\mathbb{R} \setminus \left\{\frac{1}{2}\right\}$ is given by $a \star b = a + b - 2ab$ . If $x^{-1}$ represents the inverse of an element $x$ under $\star$ , what is the value of $x$ for which $x^{-1} = 3$ ?

\frac{3}{5}

\frac{2}{5}

\frac{1}{3}

-\frac{3}{5}

Show answer & explanation

Answer: $\frac{3}{5}$

Answer

The value of

x

is

\frac{3}{5}

.

First find the identity element

e

using

a \star e = a

, which gives

a + e - 2ae = a \implies e(1 - 2a) = 0 \implies e = 0

. Next, use the inverse definition

x \star x^{-1} = e \implies x + x^{-1} - 2x x^{-1} = 0

. Substituting

x^{-1} = 3

yields

x + 3 - 2x(3) = 0 \implies -5x + 3 = 0 \implies x = \frac{3}{5}

.

Step-by-Step Solution

1

Find the identity element

e

under the operation

\star

.

e = 0

By definition,

a \star e = a \implies a + e - 2ae = a \implies e(1 - 2a) = 0 \implies e = 0

for all

a \neq \frac{1}{2}

.

2

Set up the inverse equation for element

x

.

x \star x^{-1} = 0 \implies x + x^{-1} - 2x x^{-1} = 0

An element operating on its inverse yields the identity element

e = 0

.

3

Substitute

x^{-1} = 3

into the inverse equation and solve for

x

.

x + 3 - 6x = 0 \implies -5x + 3 = 0 \implies 5x = 3 \implies x = \frac{3}{5}

Substitute the given inverse value and perform simple linear algebraic rearrangement.

Key Concept

Identity and Inverse Elements in Binary Operations

Algebra

Answer

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Key Concept

Common Mistakes

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Common Mistakes

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Answer

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Key Concept

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