Algebra Alıştırma Soruları — JAMB UTME — Sayfa 3

Soru 41Soru →

Let $A = \begin{pmatrix} 2 & 1 \\ -1 & 3 \end{pmatrix}$ and $B = \begin{pmatrix} x & 2 \\ 1 & y \end{pmatrix}$ be $2 \times 2$ matrices with integer entries $x$ and $y$ . If the matrix product $AB$ is singular and $\det(A + B) = 0$ , what is the value of $x + y$ ?

-3

3

-5

7

Cevabı ve açıklamayı göster

Cevap: $-3$

Cevap

The value of

x + y

is

-3

.

Since matrix

AB

is singular,

\det(AB) = \det(A)\det(B) = 0

. Evaluating

\det(A) = 7 \neq 0

, we find

\det(B) = xy - 2 = 0

, giving

xy = 2

. Computing

A+B = \begin{pmatrix} 2+x & 3 \\ 0 & 3+y \end{pmatrix}

, its determinant is

(2+x)(3+y) = 6 + 3x + 2y + xy = 0

. Substituting

xy = 2

yields

3x + 2y = -8

. The integer pair satisfying both

xy = 2

and

3x + 2y = -8

is

x = -2

and

y = -1

. Therefore,

x + y = -3

.

Adım Adım Çözüm

1

Calculate the determinant of matrix

A

.

\det(A) = (2)(3) - (1)(-1) = 6 + 1 = 7

.

Knowing

\det(A)

helps simplify the condition

\det(AB) = 0

using determinant properties.

2

Use the singularity of

AB

to find a relationship between

x

and

y

.

\det(AB) = \det(A)\det(B) = 7(xy - 2) = 0 \implies xy = 2

.

The determinant of a product of matrices equals the product of their determinants.

3

Form the matrix

A + B

and compute its determinant.

A + B = \begin{pmatrix} 2+x & 3 \\ 0 & 3+y \end{pmatrix} \implies \det(A+B) = (2+x)(3+y) - (3)(0) = (2+x)(3+y) = 6 + 3x + 2y + xy

.

Setting this determinant to zero gives a second equation involving

x

and

y

.

4

Substitute

xy = 2

into

\det(A+B) = 0

and solve for integer values of

x

and

y

.

6 + 3x + 2y + 2 = 0 \implies 3x + 2y = -8

. Testing integer pairs

(x,y)

for

xy = 2

gives

x = -2

and

y = -1

.

3(-2) + 2(-1) = -8

is satisfied only by

x = -2, y = -1

among integer pairs.

5

Compute

x + y

.

x + y = -2 + (-1) = -3

.

This answers the question directly.

Anahtar Kavram

Determinants of matrix products and sums, matrix singularity, and integer solutions to matrix equations.

Soru 42Soru →

If the matrix $P = \begin{pmatrix} x & 3 \\ 8 & 6 \end{pmatrix}$ is a singular matrix, what is the value of $x$ ?

-4

4

5

16

Cevabı ve açıklamayı göster

Cevap: 4

Cevap

The value of

x

is

4

.

For a

2 \times 2

matrix to be singular, its determinant must be zero. For matrix

P

, the determinant is

(x \cdot 6) - (3 \cdot 8) = 6x - 24

. Setting

6x - 24 = 0

gives

6x = 24

, which simplifies to

x = 4

.

Adım Adım Çözüm

1

State the condition for a matrix to be singular.

A matrix is singular if its determinant is equal to zero:

\det(P) = 0

.

By definition, a singular matrix has a determinant equal to zero.

2

Calculate the determinant of matrix

P

.

\det(P) = (x \times 6) - (3 \times 8) = 6x - 24.

The determinant of a

2 \times 2

matrix

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

is calculated using the formula

ad - bc

.

3

Solve the linear equation

6x - 24 = 0

for

x

.

6x = 24 \implies x = 4.

Adding

24

to both sides and dividing by

6

isolates

x

.

Anahtar Kavram

Singular matrix condition and

2 \times 2

determinant calculation

Soru 43Soru →

Find the maximum integer value of $m$ for which the quadratic inequality $x^2 - mx + 9 > 0$ holds for all real values of $x$ .

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Cevap: 5

Cevap

The maximum integer value of

m

is

5

.

For the quadratic expression

x^2 - mx + 9

to remain strictly positive for all real values of

x

, the quadratic curve must lie completely above the x-axis. Because the coefficient of

x^2

is positive (

1 > 0

), this requires the discriminant to be strictly negative (

D < 0

). Evaluating

b^2 - 4ac < 0

gives

m^2 - 36 < 0

, which simplifies to

-6 < m < 6

. The largest integer strictly less than

6

is

5

.

Adım Adım Çözüm

1

Determine the condition for the quadratic expression to be positive for all real values of

x

.

Since the leading coefficient is

1 > 0

, the condition is that the discriminant

D = b^2 - 4ac < 0

.

A parabola opening upward lies entirely above the horizontal axis when it has no real roots.

2

Calculate the discriminant using the coefficients of the quadratic expression.

D = (-m)^2 - 4(1)(9) = m^2 - 36 < 0

.

Here

a = 1

,

b = -m

, and

c = 9

.

3

Solve the quadratic inequality for

m

.

m^2 - 36 < 0 \implies -6 < m < 6

.

The roots of

m^2 - 36 = 0

are

m = -6

and

m = 6

, and the expression is negative strictly between these boundary values.

4

Determine the maximum integer value within the open interval

(-6, 6)

.

The maximum integer value is

5

.

The boundary value

6

is excluded by the strict inequality

m < 6

.

Anahtar Kavram

Quadratic Inequalities and Discriminant Conditions for Positive Definiteness

Soru 44Soru →

The sum of the first three terms of an arithmetic progression (AP) with a positive common difference $d$ is $21$ . If $2$ is added to the first term, $3$ is added to the second term, and $9$ is added to the third term, the resulting three numbers form consecutive terms of a geometric progression (GP). What is the sum of the first $10$ terms of this arithmetic progression?

Cevabı ve açıklamayı göster

Cevap: 210

Cevap

The sum of the first 10 terms of the arithmetic progression is 210.

Representing the AP terms as

7-d, 7, 7+d

and adding the specified values produces GP terms

9-d, 10, 16+d

. Solving

10^2 = (9-d)(16+d)

yields

d=4

. Consequently, the first term of the AP is

3

. Using

S_{10} = \frac{10}{2}[2(3) + 9(4)]

gives the final answer 210.

Adım Adım Çözüm

1

Express AP terms symmetrically and solve for the middle term.

The middle term is

a = 7

, making the terms

7-d

,

7

, and

7+d

.

Choosing terms

a-d, a, a+d

allows the sum equation

3a = 21

to directly isolate the middle term.

2

Set up the geometric progression relation to determine common difference

d

.

The GP terms are

9-d

,

10

, and

16+d

. Solving

10^2 = (9-d)(16+d)

gives

d^2 + 7d - 44 = 0

, yielding

d = 4

.

In any geometric progression, the square of the middle term equals the product of the first and third terms.

3

Determine the first term

a_1

and calculate

S_{10}

.

The first term is

a_1 = 7 - 4 = 3

, and the sum

S_{10} = \frac{10}{2}[2(3) + (10-1)(4)] = 210

.

Applying the AP sum formula

S_n = \frac{n}{2}[2a_1 + (n-1)d]

with

n=10

,

a_1=3

, and

d=4

.

Anahtar Kavram

Integrating AP and GP structural relationships to solve for sequence parameters and evaluate finite sums

Soru 45Soru →

A binary operation $\ast$ on the set of real numbers is defined by $a \ast b = a + b + 5$ . What is the identity element of this operation?

Cevabı ve açıklamayı göster

Cevap: -5

Cevap

The identity element of the binary operation is

-5

.

By the definition of an identity element

e

, the relation

a \ast e = a

must hold for all real numbers

a

. Applying the rule

a \ast b = a + b + 5

gives

a + e + 5 = a

. Subtracting

a

from both sides leads to

e + 5 = 0

, which gives

e = -5

.

Adım Adım Çözüm

1

Set up the identity element equation

a \ast e = a \implies a + e + 5 = a

By definition of an identity element, operating any element

a

with

e

yields

a

.

2

Solve the equation for

e

e = -5

Subtracting

a

from both sides gives

e + 5 = 0

, which yields

e = -5

.

Anahtar Kavram

Identity Element of a Binary Operation

Soru 46Soru →

Given the matrices $A = \begin{pmatrix} k & 2 \\ 3 & 1 \end{pmatrix}$ and $B = \begin{pmatrix} 1 & -1 \\ 2 & 0 \end{pmatrix}$ , what is the smaller positive integer value of $k$ for which $\det(A^2 + B) = 4$ ?

5

6

4

8

Cevabı ve açıklamayı göster

Cevap: 5

Cevap

The smaller positive integer value of

k

is 5.

Evaluating

A^2

via matrix multiplication gives

\begin{pmatrix} k^2 + 6 & 2k + 2 \\ 3k + 3 & 7 \end{pmatrix}

. Adding

B

gives

A^2 + B = \begin{pmatrix} k^2 + 7 & 2k + 1 \\ 3k + 5 & 7 \end{pmatrix}

. Computing

\det(A^2 + B) = (k^2 + 7)(7) - (2k + 1)(3k + 5) = k^2 - 13k + 44

. Setting this to 4 results in

k^2 - 13k + 40 = 0

, which yields roots

k = 5

and

k = 8

. The smaller value is 5.

Adım Adım Çözüm

1

Compute matrix

A^2

using matrix multiplication (row by column).

A^2 = \begin{pmatrix} k & 2 \\ 3 & 1 \end{pmatrix} \begin{pmatrix} k & 2 \\ 3 & 1 \end{pmatrix} = \begin{pmatrix} k^2 + 6 & 2k + 2 \\ 3k + 3 & 7 \end{pmatrix}

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

2

Add matrix

B

to

A^2

.

A^2 + B = \begin{pmatrix} k^2 + 6 & 2k + 2 \\ 3k + 3 & 7 \end{pmatrix} + \begin{pmatrix} 1 & -1 \\ 2 & 0 \end{pmatrix} = \begin{pmatrix} k^2 + 7 & 2k + 1 \\ 3k + 5 & 7 \end{pmatrix}

Matrix addition is performed by adding corresponding entries.

3

Calculate the determinant of

A^2 + B

and set it equal to 4.

\det(A^2 + B) = (k^2 + 7)(7) - (2k + 1)(3k + 5) = 7k^2 + 49 - (6k^2 + 13k + 5) = k^2 - 13k + 44 = 4

The determinant of a

2 \times 2

matrix

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

is

ad - bc

.

4

Solve the quadratic equation for

k

.

k^2 - 13k + 40 = 0 \implies (k - 5)(k - 8) = 0 \implies k = 5 \text{ or } k = 8

Factoring the quadratic equation gives the two possible integer roots.

Anahtar Kavram

Matrix multiplication, matrix addition, determinant of a 2x2 matrix, and quadratic equation solving.

Soru 47Soru →

The $2^{\text{nd}}$ , $5^{\text{th}}$ , and $14^{\text{th}}$ terms of a non-constant arithmetic progression (A.P.) form three consecutive terms of a geometric progression (G.P.). If the first term of the A.P. is $3$ , what is the sum of the first $6$ terms of the A.P.?

108

126

99

90

Cevabı ve açıklamayı göster

Cevap: 108

Cevap

108

With first term

a=3

, the terms

T_2 = 3+d

,

T_5 = 3+4d

, and

T_{14} = 3+13d

form a geometric progression. Therefore,

(3+4d)^2 = (3+d)(3+13d)

. Expanding gives

9 + 24d + 16d^2 = 9 + 42d + 13d^2

, which simplifies to

3d^2 - 18d = 0

. Since the sequence is non-constant (

d \neq 0

),

d = 6

. Using the A.P. sum formula

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

,

S_6 = 3[2(3) + 5(6)] = 3[6 + 30] = 108

.

Adım Adım Çözüm

1

Express the

2^{\text{nd}}

,

5^{\text{th}}

, and

14^{\text{th}}

terms of the A.P. in terms of the first term

a=3

and common difference

d

.

T_2 = 3 + d

,

T_5 = 3 + 4d

, and

T_{14} = 3 + 13d

.

The

n^{\text{th}}

term of an A.P. is given by

T_n = a + (n-1)d

.

2

Set up the condition for consecutive terms of a G.P. and solve for

d

.

(3 + 4d)^2 = (3 + d)(3 + 13d) \implies 9 + 24d + 16d^2 = 9 + 42d + 13d^2 \implies 3d^2 - 18d = 0 \implies d = 6

.

For consecutive terms in a G.P., the middle term squared equals the product of the adjacent terms (

T_5^2 = T_2 \times T_{14}

).

3

Calculate the sum of the first

6

terms of the A.P. using

a=3

and

d=6

.

S_6 = \frac{6}{2}[2(3) + (6-1)(6)] = 3[6 + 30] = 3 \times 36 = 108

.

The sum formula for an A.P. is

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

.

Anahtar Kavram

Simultaneous conditions connecting A.P. and G.P. terms combined with sequence sum formulas.

Tahmini Süre:2m 0s

Soru 48Soru →

A binary operation $\ast$ on the set of real numbers is defined by $a \ast b = a + b - 2ab$ . If $(x \ast 3) \ast 2 = 38$ , find the value of $x$ .

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Cevap: 3

Cevap

The value of

x

is 3.

Applying the binary operation definition sequentially yields

x \ast 3 = 3 - 5x

for the inner expression, and

(3 - 5x) \ast 2 = 15x - 7

for the composite expression. Equating

15x - 7 = 38

leads to

15x = 45

, giving

x = 3

.

Adım Adım Çözüm

1

Evaluate the inner binary operation expression

x \ast 3

x \ast 3 = 3 - 5x

Apply the definition

a \ast b = a + b - 2ab

with

a = x

and

b = 3

.

2

Evaluate the outer binary operation

(3 - 5x) \ast 2

(3 - 5x) \ast 2 = 15x - 7

Substitute the result from step 1 into the outer operation definition with

a = 3 - 5x

and

b = 2

.

3

Set the resulting expression equal to 38 and solve the linear equation

x = 3

Solve

15x - 7 = 38

by adding 7 to both sides to get

15x = 45

, then dividing by 15.

Anahtar Kavram

Nested composition of defined binary operations

Soru 49Soru →

Find the positive value of $x$ that satisfies the simultaneous equations $y = x + 2$ and $y = x^2 - 4$ .

Cevabı ve açıklamayı göster

Cevap: 3

Cevap

The positive value of

x

is 3.

Equating the linear equation

y = x + 2

and the quadratic equation

y = x^2 - 4

yields

x^2 - x - 6 = 0

. Factorizing this quadratic equation gives

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

, which yields roots

x = 3

and

x = -2

. Selecting the positive value gives 3.

Adım Adım Çözüm

1

Equate the linear and quadratic equations

x + 2 = x^2 - 4

Since both expressions are equal to

y

, set them equal to each other to solve for

x

.

2

Rearrange into standard quadratic form

ax^2 + bx + c = 0

x^2 - x - 6 = 0

Subtract

x

and

2

from both sides of the equation.

3

Factorize the quadratic expression

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

Find two factors of

-6

that add up to

-1

, which are

-3

and

2

.

4

Determine the roots and select the positive value

x = 3

Setting each factor to zero gives

x = 3

or

x = -2

. Selecting the positive root yields

3

.

Anahtar Kavram

Solving simultaneous linear and quadratic equations by substitution

Soru 50Soru →

A binary operation $\circ$ defined on the set of real numbers $\mathbb{R} \setminus \{1\}$ is given by $a \circ b = a + b - ab$ . What is the inverse of $3$ under this operation?

\frac{3}{2}

-\frac{3}{2}

1

\frac{1}{3}

Cevabı ve açıklamayı göster

Cevap: $\frac{3}{2}$

Cevap

The inverse of

3

under the given operation is

\frac{3}{2}

.

To find the inverse of

3

under the operation

a \circ b = a + b - ab

, we must first determine the identity element

e

. Setting

a \circ e = a

yields

a + e - ae = a

, which simplifies to

e(1 - a) = 0

. For all

a \neq 1

, the identity element is

e = 0

. Next, using the definition of inverse

3 \circ x = 0

, we substitute into the operational formula to obtain

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

. Thus, the option specifying

\frac{3}{2}

is correct.

Adım Adım Çözüm

1

Find the identity element

e

of the operation

\circ

.

e = 0

By definition of identity,

a \circ e = a

. Substituting into the operational formula gives

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

. Since

a \neq 1

,

e = 0

.

2

Set up the inverse equation for

3

, letting

x

be the inverse of

3

.

3 \circ x = 0

By definition of an inverse element,

a \circ a^{-1} = e

.

3

Expand

3 \circ x

using the operation rule and solve for

x

.

x = \frac{3}{2}

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

.

Anahtar Kavram

Identity and Inverse Elements in Binary Operations

Soru 51Soru →

The sum of the first $n$ terms of an arithmetic progression is given by $S_n = 2n^2 + 3n$ . What is the value of the $7^{\text{th}}$ term of the progression?

Cevabı ve açıklamayı göster

Cevap: 29

Cevap

The

7^{\text{th}}

term of the arithmetic progression is

29

.

For any sequence, the

n

-th term is found using

T_n = S_n - S_{n-1}

. Substituting

n = 7

gives

S_7 = 2(7)^2 + 3(7) = 119

and

S_6 = 2(6)^2 + 3(6) = 90

. Thus,

T_7 = 119 - 90 = 29

.

Adım Adım Çözüm

1

State the relationship between the

n

-th term

T_n

and the sum of first

n

terms

S_n

T_n = S_n - S_{n-1}

The sum of the first

n

terms minus the sum of the first

n-1

terms equals the

n

-th term.

2

Calculate the sum of the first 7 terms (

S_7

)

S_7 = 2(7)^2 + 3(7) = 119

Substitute

n = 7

into the sum formula

S_n = 2n^2 + 3n

.

3

Calculate the sum of the first 6 terms (

S_6

)

S_6 = 2(6)^2 + 3(6) = 90

Substitute

n = 6

into the sum formula

S_n = 2n^2 + 3n

.

4

Compute the

7^{\text{th}}

term (

T_7

)

T_7 = 119 - 90 = 29

Subtract

S_6

from

S_7

.

Anahtar Kavram

Relationship between the nth term and the sum of first n terms of an AP

Soru 52Soru →

The sum of the first $n$ terms of an arithmetic progression (A.P.) is $210$ . If the first term is $3$ and the last term is $39$ , what is the value of $n$ ?

10

11

5

9

Cevabı ve açıklamayı göster

Cevap: 10

Cevap

The number of terms

n

is

10

.

Using the standard sum formula

S_n = \frac{n}{2}(a + l)

for an A.P. with known first term

a = 3

and last term

l = 39

, we set

210 = \frac{n}{2}(3 + 39) = 21n

. Solving for

n

gives

n = 10

.

Adım Adım Çözüm

1

Identify the given parameters of the arithmetic progression.

First term

a = 3

, last term

l = 39

, and sum

S_n = 210

.

These values are required to apply the sum formula for an A.P.

2

Apply the sum formula

S_n = \frac{n}{2}(a + l)

.

210 = \frac{n}{2}(3 + 39) = \frac{n}{2}(42) = 21n

.

The sum of

n

terms in an A.P. with a known first and last term is given by

\frac{n}{2}(a + l)

.

3

Solve for

n

.

n = \frac{210}{21} = 10

.

Dividing the total sum by

21

gives the exact number of terms.

Anahtar Kavram

Sum of an Arithmetic Progression using first and last terms

Tahmini Süre:1m 30s

Soru 53Soru →

Given the simultaneous equations $x + 2y = 7$ and $x^2 + 3xy + y^2 = 19$ , find the product of all possible values of $y$ that satisfy the system.

Cevabı ve açıklamayı göster

Cevap: -30

Cevap

The product of all possible values of y that satisfy the system is -30.

Rearranging the linear equation gives x = 7 - 2y. Substituting this into the quadratic equation x^2 + 3xy + y^2 = 19 produces (7 - 2y)^2 + 3(7 - 2y)y + y^2 = 19. Expanding and combining like terms yields y^2 + 7y - 30 = 0. Solving for y gives y = 3 and y = -10. Multiplying these values together gives a product of -30.

Adım Adım Çözüm

1

Isolate x in the linear equation

x = 7 - 2y

Expressing one variable in terms of the other enables substitution into the quadratic equation.

2

Substitute x into the quadratic equation and expand

(7 - 2y)^2 + 3(7 - 2y)y + y^2 = 19

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

3

Simplify the resulting expression into standard quadratic form

y^2 + 7y - 30 = 0

Expanding yields (49 - 28y + 4y^2) + (21y - 6y^2) + y^2 = 19, which reduces to y^2 + 7y - 30 = 0.

4

Calculate the product of the roots of y

y_1 * y_2 = -30

By Vieta's formulas, the product of roots for y^2 + ay + b = 0 is b/1 = -30 (or factoring gives y = 3 and y = -10, with product 3 * (-10) = -30).

Anahtar Kavram

Solving simultaneous linear and quadratic equations via substitution and applying quadratic root properties

Tahmini Süre:2m 0s

Soru 54Soru →

The sum of the first four terms of an arithmetic progression (A.P.) is $32$ , and the sum of the next four terms is $96$ . What is the common difference of the progression?

Cevabı ve açıklamayı göster

Cevap: 4

Cevap

The common difference of the arithmetic progression is

4

.

The sum of the first four terms yields

2a + 3d = 16

. The sum of the first eight terms is

32 + 96 = 128

, which gives

2a + 7d = 32

. Subtracting these two linear equations gives

4d = 16

, leading to

d = 4

.

Adım Adım Çözüm

1

Formulate an equation for the sum of the first 4 terms.

2a + 3d = 16

The sum of the first

n

terms of an A.P. is

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

. Substituting

n = 4

and

S_4 = 32

gives

2(2a + 3d) = 32

, which simplifies to

2a + 3d = 16

.

2

Formulate an equation for the sum of the first 8 terms.

2a + 7d = 32

The total sum of the first 8 terms is the sum of the first 4 terms plus the sum of the next 4 terms (

S_8 = 32 + 96 = 128

). Substituting

n = 8

gives

4(2a + 7d) = 128

, which simplifies to

2a + 7d = 32

.

3

Solve the system of simultaneous linear equations for

d

.

d = 4

Subtracting

(2a + 3d = 16)

from

(2a + 7d = 32)

eliminates

2a

, resulting in

4d = 16

, which yields

d = 4

.

Anahtar Kavram

Sum of an Arithmetic Progression

Soru 55Soru →

If $2^{x+y} = 32$ and $\log_3 x + \log_3 y = 1 + \log_3 2$ , what is the value of $x^2 + y^2$ ?

13

21

26

37

Cevabı ve açıklamayı göster

Cevap: $13$

Cevap

The value of

x^2 + y^2

is

13

.

Expressing

32

as

2^5

gives

x + y = 5

. Converting

1

to

\log_3 3

allows combining the right-hand side to

\log_3(3 \times 2) = \log_3 6

, so

\log_3(xy) = \log_3 6 \implies xy = 6

. Evaluating

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

yields

5^2 - 2(6) = 25 - 12 = 13

.

Adım Adım Çözüm

1

Simplify the exponential equation using index laws.

x + y = 5

Since

32 = 2^5

,

2^{x+y} = 2^5 \implies x + y = 5

.

2

Simplify the logarithmic equation using logarithm laws.

xy = 6

\log_3 x + \log_3 y = \log_3(xy)

and

1 + \log_3 2 = \log_3 3 + \log_3 2 = \log_3(3 \times 2) = \log_3 6

. Therefore,

xy = 6

.

3

Use the algebraic identity to find

x^2 + y^2

.

x^2 + y^2 = 13

Using

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

, we have

5^2 = x^2 + y^2 + 2(6) \implies 25 = x^2 + y^2 + 12 \implies x^2 + y^2 = 13

.

Anahtar Kavram

Solving simultaneous equations involving indices and logarithms using index laws and logarithmic identities.

Soru 56Soru →

The $3^{\text{rd}}$ and $6^{\text{th}}$ terms of an Arithmetic Progression (A.P.) are $13$ and $28$ respectively. What is the sum of the first $10$ terms of the progression?

255

280

205

48

Cevabı ve açıklamayı göster

Cevap: $255$

Cevap

The sum of the first

10

terms of the progression is

255

.

The

3^{\text{rd}}

term is

a + 2d = 13

and the

6^{\text{th}}

term is

a + 5d = 28

. Subtracting these equations gives

3d = 15

, so

d = 5

, which leads to

a = 3

. Using the sum formula

S_{10} = \frac{10}{2}[2(3) + (10 - 1)5]

, we obtain

5(6 + 45) = 255

.

Adım Adım Çözüm

1

Set up equations for the given terms using the nth term formula

T_n = a + (n - 1)d

.

a + 2d = 13

and

a + 5d = 28

.

The

3^{\text{rd}}

term corresponds to

n=3

and the

6^{\text{th}}

term corresponds to

n=6

.

2

Solve the simultaneous equations for

a

(first term) and

d

(common difference).

Subtracting the first equation from the second gives

3d = 15 \Rightarrow d = 5

. Substituting

d = 5

into the first equation yields

a + 2(5) = 13 \Rightarrow a = 3

.

To find any property of an A.P., the first term

a

and common difference

d

must be determined.

3

Calculate the sum of the first

10

terms using

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

.

S_{10} = \frac{10}{2}[2(3) + (10 - 1)5] = 5[6 + 45] = 5(51) = 255

.

Applying the formula for the sum of the first

n

terms with

n = 10

,

a = 3

, and

d = 5

.

Anahtar Kavram

Finding the sum of the first

n

terms of an Arithmetic Progression given two specific terms.

Soru 57Soru →

If $\log_{10}(x) + \log_{10}(5) = 2$ , what is the value of $x$ ?

20

95

10

40

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Cevap: 20

Cevap

20

Using the logarithmic product identity

\log_b(A) + \log_b(B) = \log_b(AB)

, the left side simplifies to

\log_{10}(5x) = 2

. Converting to exponential form gives

5x = 10^2 = 100

, which simplifies directly to

x = 20

.

Adım Adım Çözüm

1

Apply the product law of logarithms

\log_{10}(5x) = 2

The sum of logarithms with the same base equals the logarithm of their product:

\log_b(A) + \log_b(B) = \log_b(AB)

.

2

Convert the logarithmic equation to exponential form

5x = 10^2 = 100

By definition of logarithms, if

\log_b(y) = c

, then

y = b^c

.

3

Solve for

x

x = \frac{100}{5} = 20

Divide both sides of the linear equation by

5

.

Anahtar Kavram

Logarithmic Product Law and Exponential Conversion

Tahmini Süre:45s

Soru 58Soru →

If $\sqrt{75} + \sqrt{12} - \sqrt{27} = k\sqrt{3}$ , what is the value of $k$ ?

Cevabı ve açıklamayı göster

Cevap: 4

Cevap

The value of

k

is 4.

Simplifying each surd into its basic form yields

\sqrt{75} = 5\sqrt{3}

,

\sqrt{12} = 2\sqrt{3}

, and

\sqrt{27} = 3\sqrt{3}

. Combining the coefficients gives

(5 + 2 - 3)\sqrt{3} = 4\sqrt{3}

. Equating

4\sqrt{3}

to

k\sqrt{3}

shows that

k = 4

.

Adım Adım Çözüm

1

Simplify each individual surd by factoring out perfect squares

\sqrt{75} = 5\sqrt{3}

,

\sqrt{12} = 2\sqrt{3}

,

\sqrt{27} = 3\sqrt{3}

To combine surds through addition or subtraction, they must be converted to similar surds.

2

Combine the coefficients of the like surds

5\sqrt{3} + 2\sqrt{3} - 3\sqrt{3} = (5 + 2 - 3)\sqrt{3} = 4\sqrt{3}

Like terms with the same radical factor

\sqrt{3}

can be added and subtracted directly.

3

Compare the resulting coefficient with

k\sqrt{3}

k = 4

By direct comparison of coefficients of

\sqrt{3}

,

k

equals 4.

Anahtar Kavram

Simplification and combining of similar surds

Soru 59Soru →

Two $2 \times 2$ matrices are given as $A = \begin{pmatrix} 2 & -1 \\ 4 & 3 \end{pmatrix}$ and $B = \begin{pmatrix} 1 & 0 \\ 2 & k \end{pmatrix}$ . If the determinant of the product matrix $AB$ is equal to $30$ , what is the value of $k$ ?

Cevabı ve açıklamayı göster

Cevap: 3

Cevap

The value of k is 3.

The determinant of a product of square matrices equals the product of their individual determinants:

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

. Evaluating

\det(A)

yields

(2)(3) - (-1)(4) = 10

, and

\det(B)

yields

(1)(k) - (0)(2) = k

. Substituting these into

\det(AB) = 30

gives

10k = 30

, leading directly to

k = 3

.

Adım Adım Çözüm

1

Evaluate the determinant of matrix A

\det(A) = 10

Using the

2 \times 2

determinant formula

\det\begin{pmatrix} a & b \\ c & d \end{pmatrix} = ad - bc

, we calculate

\det(A) = (2)(3) - (-1)(4) = 6 + 4 = 10

.

2

Evaluate the determinant of matrix B

\det(B) = k

Calculating the determinant of matrix

B

yields

\det(B) = (1)(k) - (0)(2) = k

.

3

Use the product property of determinants to solve for k

k = 3

Since

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

, we have

10k = 30

. Dividing both sides by 10 yields

k = 3

.

Anahtar Kavram

Determinant of Matrix Product

Tahmini Süre:1m 30s

Soru 60Soru →

Given the matrices $A = \begin{pmatrix} 2 & 3 \\ 1 & m \end{pmatrix}$ and $B = \begin{pmatrix} 4 \\ -2 \end{pmatrix}$ , if the matrix product $AB = \begin{pmatrix} 2 \\ 6 \end{pmatrix}$ , what is the value of $m$ ?

-1

1

-3

-5

Cevabı ve açıklamayı göster

Cevap: $-1$

Cevap

The value of

m

is

-1

.

Multiplying the

2 \times 2

matrix by the

2 \times 1

column vector yields a

2 \times 1

result where the second row entry is

1(4) + m(-2) = 4 - 2m

. Setting this equal to the second row element

6

gives

4 - 2m = 6

, which simplifies to

m = -1

.

Adım Adım Çözüm

1

Compute the matrix product

AB

AB = \begin{pmatrix} 2(4) + 3(-2) \\ 1(4) + m(-2) \end{pmatrix} = \begin{pmatrix} 8 - 6 \\ 4 - 2m \end{pmatrix} = \begin{pmatrix} 2 \\ 4 - 2m \end{pmatrix}

Matrix multiplication requires taking the dot product of each row of the first matrix with the column vector.

2

Equate the resulting vector to the given vector

\begin{pmatrix} 2 \\ 6 \end{pmatrix}

4 - 2m = 6

Corresponding elements of equal matrices must be equal.

3

Solve the linear equation for

m

-2m = 6 - 4 \Rightarrow -2m = 2 \Rightarrow m = -1

Isolating the variable

m

by subtracting

4

from both sides and dividing by

-2

.

Anahtar Kavram

Matrix Multiplication and Equality of Matrices

Algebra

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