All practice questions

Question 581Question →

When the polynomial $P(x) = x^3 - 2x^2 + ax + 8$ is divided by $(x - 3)$ , the remainder is $14$ . What is the value of the constant $a$ ?

Show answer & explanation

Answer: -1

Answer

The value of the constant

a

is

-1

.

According to the Remainder Theorem, dividing

P(x)

by

(x - 3)

means the remainder is

P(3)

. Evaluating

P(3) = 3^3 - 2(3)^2 + 3a + 8 = 27 - 18 + 3a + 8 = 17 + 3a

. Setting this equal to the remainder

14

gives

17 + 3a = 14

, which simplifies to

3a = -3

and yields

a = -1

.

Step-by-Step Solution

1

Apply the Remainder Theorem

The remainder when

P(x)

is divided by

(x - 3)

is equal to

P(3)

.

The Remainder Theorem states that dividing a polynomial

P(x)

by

(x - c)

yields a remainder equal to

P(c)

.

2

Substitute

x = 3

into the polynomial and set equal to the given remainder

3^3 - 2(3)^2 + a(3) + 8 = 14

Setting the value of

P(3)

equal to

14

allows us to form a linear equation for the unknown constant

a

.

3

Simplify numerical terms in the equation

27 - 18 + 3a + 8 = 14 \implies 17 + 3a = 14

Evaluate exponents and multiplication to isolate the term containing

a

.

4

Solve for

a

3a = -3 \implies a = -1

Subtract

17

from both sides and divide by

3

.

Key Concept

Polynomial Remainder Theorem

Question 582Question →

In triangle $XYZ$ , side length $x = 3\text{ cm}$ , side length $y = 8\text{ cm}$ , and the included angle $\angle Z = 60^\circ$ . What is the length of side $z$ in cm?

Show answer & explanation

Answer: 7

Answer

The length of side

z

is

7\text{ cm}

.

Using the Cosine Rule

z^2 = x^2 + y^2 - 2xy \cos Z

, substituting

x = 3

,

y = 8

, and

\angle Z = 60^\circ

yields

z^2 = 3^2 + 8^2 - 2(3)(8)(0.5) = 9 + 64 - 24 = 49

. Taking the positive square root gives

z = 7\text{ cm}

.

Step-by-Step Solution

1

Identify known triangle components and select the appropriate rule

Two sides and the included angle (SAS) are given:

x = 3

,

y = 8

,

\angle Z = 60^\circ

, requiring the Cosine Rule.

When given two sides and the included angle (SAS), the Cosine Rule is used to find the third side.

2

Substitute values into the Cosine Rule formula

z^2 = x^2 + y^2 - 2xy \cos Z

z^2 = 3^2 + 8^2 - 2(3)(8)\cos 60^\circ

Direct algebraic substitution of side lengths and angle measure into the Cosine Rule.

3

Evaluate the trigonometric term and simplify the arithmetic expression

z^2 = 9 + 64 - 48(0.5) = 73 - 24 = 49

The exact value of

\cos 60^\circ

is

0.5

.

4

Solve for side length

z

by taking the principal square root

z = \sqrt{49} = 7\text{ cm}

Side length must be positive.

Key Concept

Applying the Cosine Rule to find the third side of a non-right-angled triangle given two sides and an included angle (SAS).

Estimated Time:1m 30s

Question 583Question →

A piecewise function

f(x)

is defined by

f(x) = \begin{cases} \frac{2x^2 - 5x - 3}{x - 3}, & \text{for } x \neq 3 \ a^2 - 2, & \text{for } x = 3 \end{cases}

If

f(x)

is continuous at

x = 3

and

a > 0

, what is the numerical value of

a

?

Show answer & explanation

Answer: 3

Answer

The numerical value of a is 3.

For

f(x)

to be continuous at

x = 3

, the defined value

f(3) = a^2 - 2

must equal

\lim_{x \to 3} f(x)

. Factoring the numerator gives

\frac{(2x + 1)(x - 3)}{x - 3} = 2x + 1

for

x \neq 3

. Taking the limit as

x \to 3

yields

2(3) + 1 = 7

. Setting

a^2 - 2 = 7

leads to

a^2 = 9

, which gives

a = 3

under the constraint

a > 0

.

Step-by-Step Solution

1

Evaluate the limit of

f(x)

as

x \to 3

Factor the numerator

2x^2 - 5x - 3 = (2x + 1)(x - 3)

. For

x \neq 3

,

f(x) = 2x + 1

. Thus,

\lim_{x \to 3} f(x) = 2(3) + 1 = 7

.

Direct substitution gives an indeterminate form

\frac{0}{0}

, so canceling the common factor

(x - 3)

allows direct evaluation of the limit.

2

Apply the definition of continuity at

x = 3

f(3) = a^2 - 2 = 7

.

For a function to be continuous at a point

c

, the function value

f(c)

must equal the limit

\lim_{x \to c} f(x)

.

3

Solve for the parameter

a

a^2 = 9 \implies a = 3

(since

a > 0

).

Solving

a^2 = 9

gives solutions

3

and

-3

. The condition

a > 0

specifies the positive root.

Key Concept

Continuity of a Piecewise Function at a Point

Question 584Question →

The time, $t$ hours, required to complete a road maintenance project varies inversely as the number of workers, $w$ , assigned to the project. If 8 workers can finish the project in 15 hours, calculate the time, in hours, required for 12 workers to finish the same project.

Show answer & explanation

Answer: 10

Answer

10 hours

Because the time

t

varies inversely as the number of workers

w

, the total worker-hours required for the project is constant:

k = 8 \times 15 = 120

worker-hours. Dividing this total work by 12 workers gives

\frac{120}{12} = 10

hours.

Step-by-Step Solution

1

Set up the inverse variation equation

t = \frac{k}{w}

, where

k

is the constant of variation.

Inverse variation implies that as the number of workers increases, the time required decreases proportionally.

2

Determine the value of the constant of variation

k

k = t \times w = 15 \times 8 = 120

.

Substitute the known pair of values (

w = 8, t = 15

) into the equation.

3

Compute the new value of

t

for 12 workers

t = \frac{120}{12} = 10

hours.

Substitute

w = 12

and

k = 120

into

t = \frac{k}{w}

.

Key Concept

Inverse Variation

Question 585Question →

In a science competition, $3$ distinct prizes (first, second, and third place) are to be awarded to $3$ different students chosen from a group of $5$ finalists. In how many different ways can these $3$ prizes be awarded?

Show answer & explanation

Answer: 60

Answer

60 ways

Because the prizes are distinct, the order in which the finalists are selected matters. Calculating the number of arrangements of

3

students from

5

candidates is given by

^{5}P_{3} = 5 \times 4 \times 3 = 60

.

Step-by-Step Solution

1

Determine if order matters

Since the prizes are distinct (1st, 2nd, and 3rd place), the order of assignment matters, making this a permutation problem.

Assigning distinct positions to individuals requires calculating permutations rather than combinations.

2

Apply the permutation formula

^{n}P_{r} = \frac{n!}{(n-r)!}

^{5}P_{3} = \frac{5!}{(5-3)!} = \frac{5!}{2!}

There are

5

total candidates (

n = 5

) and

3

positions to fill (

r = 3

).

3

Calculate the numeric value

^{5}P_{3} = 5 \times 4 \times 3 = 60

Canceling

2!

from the numerator and denominator leaves

5 \times 4 \times 3 = 60

.

Key Concept

Linear permutation of r items selected from n distinct items

Estimated Time:45s

Question 586Question →

If $x$ and $y$ are real numbers greater than $1$ satisfying the system of equations $\log_x y + \log_y x = \frac{5}{2}$ and $xy = 64$ with $x > y$ , find the value of $x - y$ .

Show answer & explanation

Answer: 12

Answer

The value of

x - y

is 12.

Using the change of base identity

\log_y x = \frac{1}{\log_x y}

, the equation

\log_x y + \log_y x = \frac{5}{2}

converts to

u + \frac{1}{u} = \frac{5}{2}

for

u = \log_x y

. Solving the quadratic equation

2u^2 - 5u + 2 = 0

yields

u = \frac{1}{2}

or

u = 2

. Because

x > y > 1

, we must have

\log_x y < 1

, selecting

u = \frac{1}{2} \implies x = y^2

. Substituting into

xy = 64

gives

y^3 = 64 \implies y = 4

and

x = 16

. Therefore,

x - y = 16 - 4 = 12

.

Step-by-Step Solution

1

Apply the reciprocal change of base identity

Rewrite

\log_y x

as

\frac{1}{\log_x y}

, yielding

\log_x y + \frac{1}{\log_x y} = \frac{5}{2}

.

According to the change of base formula,

\log_y x = \frac{\log_x x}{\log_x y} = \frac{1}{\log_x y}

.

2

Solve the quadratic equation in terms of

u = \log_x y

Substituting

u = \log_x y

gives

u + \frac{1}{u} = \frac{5}{2} \implies 2u^2 - 5u + 2 = 0

, which factors into

(2u - 1)(u - 2) = 0

, yielding

u = \frac{1}{2}

or

u = 2

.

Multiplying through by

2u

clears fractions and forms a standard quadratic equation.

3

Select the valid root using given inequality constraints

Since

x > y > 1

, taking the logarithm base

x

yields

\log_x x > \log_x y \implies 1 > \log_x y

. Thus

u = \frac{1}{2}

, which means

y = x^{1/2}

or

x = y^2

.

The condition

x > y

restricts the logarithm of

y

base

x

to be strictly less than

1

.

4

Substitute into the product equation to find

x

and

y

Substituting

x = y^2

into

xy = 64

gives

y^3 = 64 \implies y = 4

. Consequently,

x = 4^2 = 16

.

Combining the relation

x = y^2

with

xy = 64

enables single-variable cubic solution.

5

Calculate the required difference

x - y

16 - 4 = 12

.

Direct subtraction of the derived values

x = 16

and

y = 4

.

Key Concept

Logarithmic Change of Base Reciprocal Property

Question 587Question →

A sector of a circle of radius $6\text{ cm}$ has a total perimeter whose numerical value is equal to the numerical value of its area. What is the area of the sector, in $\text{cm}^2$ ?

Show answer & explanation

Answer: 18

Answer

The area of the sector is

18\text{ cm}^2

.

For a sector of radius

r = 6\text{ cm}

, its total perimeter is

P = 2(6) + s = 12 + s

, where

s

is the arc length. Its area is

A = \frac{1}{2}(6)s = 3s

. Setting

P = A

gives

12 + s = 3s \implies 2s = 12 \implies s = 6\text{ cm}

. Substituting

s = 6

into the area expression gives

A = 3(6) = 18\text{ cm}^2

.

Step-by-Step Solution

1

Express the perimeter and area of the sector in terms of the arc length

s

P = 2r + s = 12 + s

and

A = \frac{1}{2}rs = 3s

The total perimeter of a sector includes two straight radii plus the arc length, while its area is given by

\frac{1}{2}rs

.

2

Equate the numerical values of perimeter and area

12 + s = 3s

The question specifies that the numerical value of the total perimeter equals the numerical value of its area.

3

Solve for the arc length

s

2s = 12 \implies s = 6\text{ cm}

Subtracting

s

from both sides yields

2s = 12

, so

s = 6\text{ cm}

.

4

Calculate the sector area

A = 3(6) = 18\text{ cm}^2

Substituting

s = 6

into

A = 3s

yields an area of

18\text{ cm}^2

.

Key Concept

Perimeter and Area of a Circular Sector

Question 588Question →

If $y = x^3(2x - 1)^4$ , find the value of $\frac{dy}{dx}$ at $x = 1$ .

Show answer & explanation

Answer: 11

Answer

The numerical value of

\frac{dy}{dx}

at

x = 1

is

11

.

Applying the product rule

\frac{dy}{dx} = u\frac{dv}{dx} + v\frac{du}{dx}

along with the chain rule for the expression

(2x - 1)^4

gives

\frac{dy}{dx} = 8x^3(2x - 1)^3 + 3x^2(2x - 1)^4

. Evaluating this at

x = 1

gives

8(1)^3(1)^3 + 3(1)^2(1)^4 = 8 + 3 = 11

.

Step-by-Step Solution

1

Set up the product rule for

y = u \cdot v

, where

u = x^3

and

v = (2x - 1)^4

.

u = x^3

and

v = (2x - 1)^4

The function is expressed as the product of two algebraic terms.

2

Find the derivative of each function component.

\frac{du}{dx} = 3x^2

and

\frac{dv}{dx} = 8(2x - 1)^3

The power rule gives

\frac{du}{dx} = 3x^2

, and applying the chain rule to

(2x - 1)^4

yields

4(2x - 1)^3 \cdot 2 = 8(2x - 1)^3

.

3

Substitute components into the product rule formula

\frac{dy}{dx} = u\frac{dv}{dx} + v\frac{du}{dx}

.

\frac{dy}{dx} = 8x^3(2x - 1)^3 + 3x^2(2x - 1)^4

Combining

u\frac{dv}{dx}

and

v\frac{du}{dx}

provides the full expression for the derivative.

4

Evaluate the derivative at

x = 1

.

\frac{dy}{dx}\Big|_{x=1} = 8(1)^3(2(1) - 1)^3 + 3(1)^2(2(1) - 1)^4 = 8 + 3 = 11

Substituting

x = 1

simplifies the terms to

8(1) + 3(1) = 11

.

Key Concept

Rules of Differentiation (Product and Chain Rules)

Question 589Question →

The total energy loss $E$ in Joules per minute in a magnetic core circuit is partly constant and partly varies directly as the square of the frequency $f$ in Hz of the alternating current. Given that $E = 120\text{ J}$ when $f = 10\text{ Hz}$ , and $E = 360\text{ J}$ when $f = 20\text{ Hz}$ , what is the value of $E$ in Joules when $f = 15\text{ Hz}$ ?

Show answer & explanation

Answer: 220

Answer

220

The relation describes a partial variation model

E = c + k f^2

. Substituting the two given conditions (

f=10, E=120

and

f=20, E=360

) yields the system of equations

c + 100k = 120

and

c + 400k = 360

. Solving this system gives

k = 0.8

and

c = 40

. Evaluating

E = 40 + 0.8(15)^2

results in

220\text{ J}

.

Step-by-Step Solution

1

Write the general equation for partial variation involving a constant term and a term proportional to

f^2

E = c + k f^2

Partial variation consists of a sum of a constant component and a variable component.

2

Substitute the known conditions into the variation equation to create a system of linear equations

Equation 1:

c + 100k = 120

; Equation 2:

c + 400k = 360

Two pairs of values are provided to determine the two unknown constants

c

and

k

.

3

Solve the system of simultaneous equations for

k

and

c

k = 0.8

and

c = 40

Subtracting Equation 1 from Equation 2 eliminates

c

, allowing direct calculation of

k

, after which

c

is found by substitution.

4

Calculate the required value of

E

when

f = 15

E = 40 + 0.8(15^2) = 40 + 0.8(225) = 220

Applying the discovered constants to the target frequency yields the final energy loss value.

Key Concept

Partial Variation and Simultaneous Equations

Estimated Time:2m 0s

Question 590Question →

If $y = \frac{4x + 1}{x - 2}$ , calculate the numerical value of $\frac{dy}{dx}$ at $x = 3$ .

Show answer & explanation

Answer: -9

Answer

The numerical value of

\frac{dy}{dx}

at

x = 3

is

-9

.

Using the quotient rule

\frac{dy}{dx} = \frac{v\frac{du}{dx} - u\frac{dv}{dx}}{v^2}

for

y = \frac{4x + 1}{x - 2}

gives

\frac{dy}{dx} = \frac{(x - 2)(4) - (4x + 1)(1)}{(x - 2)^2} = \frac{-9}{(x - 2)^2}

. Substituting

x = 3

produces

\frac{-9}{(3 - 2)^2} = -9

.

Step-by-Step Solution

1

Identify

u(x)

and

v(x)

for the quotient rule formula.

u = 4x + 1

and

v = x - 2

.

The given equation

y = \frac{4x + 1}{x - 2}

is a quotient of two functions of

x

.

2

Calculate the individual derivatives

\frac{du}{dx}

and

\frac{dv}{dx}

.

\frac{du}{dx} = 4

and

\frac{dv}{dx} = 1

.

These derivatives are required components of the quotient rule.

3

Substitute the expressions into the quotient rule formula

\frac{dy}{dx} = \frac{v u' - u v'}{v^2}

and simplify.

\frac{dy}{dx} = \frac{(x - 2)(4) - (4x + 1)(1)}{(x - 2)^2} = \frac{4x - 8 - 4x - 1}{(x - 2)^2} = \frac{-9}{(x - 2)^2}

.

Simplifying the numerator yields the general derivative function.

4

Evaluate the derivative at

x = 3

.

\frac{dy}{dx}\Big|_{x=3} = \frac{-9}{(3 - 2)^2} = \frac{-9}{1} = -9

.

Substituting

x = 3

gives the requested numerical value.

Key Concept

Quotient Rule of Differentiation

Question 591Question →

Evaluate the limit $\lim_{x \to 3} \frac{x^2 - 5x + 6}{x - 3}$ . What is the numerical value of this limit?

Show answer & explanation

Answer: 1

Answer

The numerical value of the limit is 1.

Substituting x = 3 into the original expression produces the indeterminate form 0/0. Factoring the numerator gives (x - 3)(x - 2). Canceling the common factor (x - 3) yields x - 2. Substituting x = 3 into x - 2 gives 3 - 2 = 1.

Step-by-Step Solution

1

Check the expression using direct substitution at x = 3.

The substitution yields \frac{3^2 - 5(3) + 6}{3 - 3} = \frac{0}{0}, an indeterminate form.

Direct substitution results in 0/0, indicating that algebraic factorization is needed.

2

Factorize the quadratic polynomial in the numerator.

x^2 - 5x + 6 = (x - 3)(x - 2)

Finding factors of +6 that sum to -5 allows cancellation of the denominator term.

3

Cancel the factor (x - 3) and evaluate the remaining linear expression at x = 3.

\lim_{x \to 3} (x - 2) = 3 - 2 = 1

The factor causing zero in the denominator is removed, leaving a continuous function.

Key Concept

Evaluating indeterminate limits (0/0) by algebraic factorization

Question 592Question →

Find the smallest positive integer $k$ for which the inequality $(k - 2)x^2 + 8x + k + 4 > 0$ holds for all real values of $x$ .

Show answer & explanation

Answer: 5

Answer

5

For the quadratic expression

(k - 2)x^2 + 8x + k + 4

to be positive for all real values of

x

, two conditions must be satisfied simultaneously: the leading coefficient must be positive (

k - 2 > 0 \implies k > 2

) and the discriminant must be strictly negative (

\Delta < 0

). Calculating the discriminant gives

\Delta = 8^2 - 4(k - 2)(k + 4) = 96 - 8k - 4k^2

. Setting

96 - 8k - 4k^2 < 0

and dividing by

-4

(reversing the inequality) yields

k^2 + 2k - 24 > 0

, which factors as

(k + 6)(k - 4) > 0

. This gives

k < -6

or

k > 4

. Intersecting with

k > 2

results in

k > 4

. The smallest integer greater than 4 is 5.

Step-by-Step Solution

1

Determine the conditions for positivity for all real numbers

k - 2 > 0

and

\Delta < 0

A quadratic

Ax^2 + Bx + C

remains above the x-axis for all real

x

if and only if its parabola opens upwards (

A > 0

) and has no real roots (

\Delta < 0

).

2

Set up and solve the discriminant inequality

k^2 + 2k - 24 > 0 \implies (k + 6)(k - 4) > 0

Expanding

8^2 - 4(k - 2)(k + 4) < 0

gives

64 - 4(k^2 + 2k - 8) < 0

, which simplifies to

k^2 + 2k - 24 > 0

after dividing by

-4

and reversing the inequality sign.

3

Intersect solution sets and find the smallest integer

k = 5

The intersection of

k > 2

and

(k < -6 \text{ or } k > 4)

gives

k > 4

. The smallest integer strictly greater than 4 is 5.

Key Concept

Condition for Positive Definite Quadratic Inequalities

Question 593Question →

A rhombus has diagonals of lengths $12\text{ cm}$ and $16\text{ cm}$ . What is the perimeter of the rhombus in $\text{cm}$ ?

Show answer & explanation

Answer: 40

Answer

The perimeter of the rhombus is

40\text{ cm}

.

The diagonals of a rhombus bisect each other at right angles, dividing the rhombus into four congruent right-angled triangles. Each triangle has legs measuring

6\text{ cm}

and

8\text{ cm}

. Applying the Pythagorean theorem, the hypotenuse (which is the side length of the rhombus) is

\sqrt{6^2 + 8^2} = \sqrt{100} = 10\text{ cm}

. Since all four sides of a rhombus are equal, the perimeter is

4 \times 10 = 40\text{ cm}

.

Step-by-Step Solution

1

Calculate the lengths of the semi-diagonals

The semi-diagonals are

6\text{ cm}

and

8\text{ cm}

.

The diagonals of a rhombus bisect each other perpendicularly.

2

Determine the side length of the rhombus using the Pythagorean theorem

Side length

s = \sqrt{6^2 + 8^2} = \sqrt{100} = 10\text{ cm}

.

Each side of the rhombus forms the hypotenuse of a right-angled triangle formed by the semi-diagonals.

3

Calculate the total perimeter

Perimeter

P = 4 \times 10 = 40\text{ cm}

.

All four sides of a rhombus are equal in length.

Key Concept

Perimeter of a rhombus derived from diagonal lengths using right-triangle properties

Estimated Time:1m 0s

Question 594Question →

A piecewise function

f(x)

is defined by

f(x) = \begin{cases} \frac{\sin(3x) + \tan(5x)}{2x}, & x \neq 0 \\ a^2 - 5, & x = 0 \end{cases}

If

f(x)

is continuous at

x = 0

, where

a > 0

, determine the numerical value of

a

.

Show answer & explanation

Answer: 3

Answer

The numerical value of

a

is 3.

For the function to be continuous at

x = 0

, the limit as

x \to 0

must equal the function value

f(0)

. Splitting the trigonometric limit gives

\frac{3}{2} + \frac{5}{2} = 4

. Setting

a^2 - 5 = 4

yields

a^2 = 9

. Because

a > 0

, taking the positive square root gives

a = 3

.

Step-by-Step Solution

1

Evaluate the limit of the trigonometric expression as

x

approaches 0.

\lim_{x \to 0} \frac{\sin(3x) + \tan(5x)}{2x} = 4

Using standard trigonometric limit principles: \lim_{x \to 0} \frac{\sin(kx)}{x} = k and \lim_{x \to 0} \frac{\tan(kx)}{x} = k.

2

Equate the limit value to

f(0)

to ensure continuity at

x = 0

.

a^2 - 5 = 4

By definition, a function is continuous at x = c if and only if \lim_{x \to c} f(x) = f(c).

3

Solve the resulting quadratic equation for the positive constant

a

.

a = 3

Adding 5 to both sides gives a^2 = 9; taking the principal square root gives a = 3 since a > 0.

Key Concept

Limits and Continuity of Functions

Question 595Question →

A binary operation $\Delta$ defined on the set of integers modulo $11$ is given by $a \Delta b \equiv (2a^2 - 3ab + b^2) \pmod{11}$ . Find the smallest non-negative integer $x$ that satisfies the equation $4 \Delta x \equiv 5 \pmod{11}$ .

Show answer & explanation

Answer: 3

Answer

The smallest non-negative integer

x

that satisfies

4 \Delta x \equiv 5 \pmod{11}

is 3.

Evaluating

4 \Delta x

gives

32 - 12x + x^2 \equiv x^2 - x + 10 \pmod{11}

. Setting this congruent to

5 \pmod{11}

yields

x^2 - x + 5 \equiv 0 \pmod{11}

, which converts to

x^2 - x - 6 \equiv 0 \pmod{11}

. Factoring gives

(x - 3)(x + 2) \equiv 0 \pmod{11}

, which yields solutions

x \equiv 3 \pmod{11}

and

x \equiv 9 \pmod{11}

. The smallest non-negative integer among these solutions is

3

.

Step-by-Step Solution

1

Substitute the given value

a = 4

into the operation definition.

4 \Delta x = 2(4)^2 - 3(4)x + x^2 = 32 - 12x + x^2

This establishes the explicit algebraic polynomial in terms of

x

.

2

Reduce coefficients modulo

11

.

32 \equiv 10 \pmod{11}

and

-12x \equiv -x \pmod{11}

, giving

x^2 - x + 10 \pmod{11}

.

Simplifying coefficients reduces computational complexity during equation solving.

3

Form the modular quadratic equation and set it to zero.

x^2 - x + 10 \equiv 5 \pmod{11} \implies x^2 - x + 5 \equiv 0 \pmod{11} \implies x^2 - x - 6 \equiv 0 \pmod{11}

.

Expressing

5 \equiv -6 \pmod{11}

allows standard integer factorisation.

4

Factor the quadratic polynomial and solve for

x

.

(x - 3)(x + 2) \equiv 0 \pmod{11} \implies x \equiv 3 \text{ or } x \equiv -2 \equiv 9 \pmod{11}

.

Since

11

is prime, a product congruent to

0 \pmod{11}

implies at least one factor is congruent to

0 \pmod{11}

.

5

Select the smallest non-negative integer from the valid solution set

\{3, 9\}

.

x = 3

3

is non-negative and strictly smaller than

9

.

Key Concept

Modular Arithmetic Binary Operations and Quadratic Congruences

Estimated Time:3m 0s

Question 596Question →

A normal line is drawn to the curve $y = \frac{x^2 - 5}{x - 3}$ at the point where $x = 2$ . What is the $x$ -intercept of this normal line?

Show answer & explanation

Answer: -1

Answer

The

x

-intercept of the normal line is

-1

.

Substituting

x = 2

into the curve yields the point

(2, 1)

. Differentiation via the quotient rule gives

\frac{dy}{dx} = \frac{x^2 - 6x + 5}{(x - 3)^2}

. At

x = 2

, the tangent slope is

-3

, making the normal slope

\frac{1}{3}

. The line equation

y - 1 = \frac{1}{3}(x - 2)

simplifies to

x - 3y + 1 = 0

. Setting

y = 0

gives

x = -1

.

Step-by-Step Solution

1

Calculate the y-coordinate of the point of tangency

y = 1

Substitute

x = 2

into the curve equation

y = \frac{x^2 - 5}{x - 3}

to get the point

(2, 1)

.

2

Differentiate the function using the quotient rule

\frac{dy}{dx} = \frac{x^2 - 6x + 5}{(x - 3)^2}

Applying

\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{u'v - uv'}{v^2}

to

u = x^2 - 5

and

v = x - 3

.

3

Find the tangent gradient at

x = 2

m_t = -3

Evaluate

\frac{dy}{dx}

at

x = 2

to obtain

m_t = \frac{2^2 - 6(2) + 5}{(2-3)^2} = -3

.

4

Calculate the gradient of the normal line

m_n = \frac{1}{3}

The normal line is perpendicular to the tangent line, so

m_n = -\frac{1}{m_t} = \frac{1}{3}

.

5

Derive the line equation for the normal

x - 3y + 1 = 0

Use point-slope form

y - y_1 = m_n(x - x_1)

with

(2, 1)

and

m_n = \frac{1}{3}

.

6

Find the x-intercept

x = -1

Set

y = 0

in the normal equation

x - 3(0) + 1 = 0

, giving

x = -1

.

Key Concept

Equation of normal line to a rational curve and finding its axis intercepts

Question 597Question →

A pie chart is used to display the distribution of $720$ candidates registered for a competitive examination across five subjects. If the sector representing Further Mathematics has a central angle of $45^\circ$ , what is the total number of candidates who registered for Further Mathematics?

Show answer & explanation

Answer: 90

Answer

The total number of candidates who registered for Further Mathematics is 90.

To calculate the number of candidates represented by a pie chart sector, multiply the total count by the ratio of the sector's central angle to 360 degrees: (45 / 360) * 720 = 90 candidates.

Step-by-Step Solution

1

Determine the fraction of the total population represented by the Further Mathematics sector.

\frac{45^\circ}{360^\circ} = \frac{1}{8}

A complete pie chart circle corresponds to an angle of 360 degrees.

2

Calculate the actual number of candidates by multiplying the fraction by the total student population.

\frac{1}{8} \times 720 = 90

The number of items in a sector is directly proportional to its central sector angle relative to 360 degrees.

Key Concept

Calculating category frequencies from pie chart sector angles

Question 598Question →

If $(x + 3)$ is a factor of the polynomial $P(x) = x^3 + 4x^2 + mx - 6$ , what is the value of $m$ ?

Show answer & explanation

Answer: 1

Answer

The value of

m

is 1.

According to the Factor Theorem,

(x + 3)

is a factor of

P(x)

if

P(-3) = 0

. Substituting

x = -3

into

P(x) = x^3 + 4x^2 + mx - 6

yields

(-3)^3 + 4(-3)^2 + m(-3) - 6 = 0

, which simplifies to

-27 + 36 - 3m - 6 = 0

, giving

3 - 3m = 0

and thus

m = 1

.

Step-by-Step Solution

1

Apply the Factor Theorem

P(-3) = 0

By the Factor Theorem, for a linear divisor

(x - a)

to be a factor of

P(x)

,

P(a)

must equal zero. Here

x + 3 = 0 \implies x = -3

.

2

Substitute

x = -3

into

P(x) = x^3 + 4x^2 + mx - 6

(-3)^3 + 4(-3)^2 + m(-3) - 6 = 0

Evaluating

P(-3)

sets up an equation to find the unknown coefficient

m

.

3

Simplify and solve for

m

-27 + 36 - 3m - 6 = 0 \implies 3 - 3m = 0 \implies m = 1

Combine the constant terms

-27 + 36 - 6 = 3

and solve the linear equation in terms of

m

.

Key Concept

Factor Theorem

Question 599Question →

Using differentiation from first principles, evaluate the numerical value of the derivative of the polynomial function $f(x) = 2x^3 - 3x^2 + 4$ at $x = 2$ .

Show answer & explanation

Answer: 12

Answer

The numerical value of the derivative at

x = 2

is 12.

Applying the first principles limit formula

\lim_{h \to 0} \frac{f(x+h) - f(x)}{h}

to

f(x) = 2x^3 - 3x^2 + 4

yields

f'(x) = 6x^2 - 6x

. Evaluating this derivative function at

x = 2

yields

6(4) - 12 = 12

.

Step-by-Step Solution

1

Set up the difference quotient using the definition of differentiation from first principles

f'(x) = \lim_{h \to 0} \frac{[2(x+h)^3 - 3(x+h)^2 + 4] - [2x^3 - 3x^2 + 4]}{h}

Differentiation from first principles evaluates the limit of the rate of change as the increment

h

approaches zero.

2

Expand

(x+h)^3

and

(x+h)^2

and subtract

f(x)

f(x+h) - f(x) = 6x^2h + 6xh^2 + 2h^3 - 6xh - 3h^2

Expanding the terms allows cancellation of all terms not containing

h

.

3

Divide by

h

and evaluate the limit as

h \to 0

f'(x) = \lim_{h \to 0} (6x^2 + 6xh + 2h^2 - 6x - 3h) = 6x^2 - 6x

Dividing by

h

eliminates the indeterminate form

\frac{0}{0}

, allowing direct substitution of

h = 0

.

4

Substitute

x = 2

into the derivative function

f'(x)

f'(2) = 6(2)^2 - 6(2) = 24 - 12 = 12

Evaluating at the given point gives the slope of the tangent line at

x = 2

.

Key Concept

Differentiation from first principles using limit of difference quotient

Question 600Question →

What is the gradient of the tangent line to the curve $y = 4x^2 - 7x + 2$ at the point where $x = 3$ ?

Show answer & explanation

Answer: 17

Answer

The gradient of the tangent line to the curve at

x = 3

is 17.

Differentiating

y = 4x^2 - 7x + 2

gives

\frac{dy}{dx} = 8x - 7

. Substituting

x = 3

yields

8(3) - 7 = 17

.

Step-by-Step Solution

1

Find the derivative of the curve's equation

\frac{dy}{dx} = 8x - 7

The gradient of the curve at any point is given by its first derivative with respect to x.

2

Evaluate the derivative at x = 3

m = 8(3) - 7 = 17

Substituting the given x-coordinate into the derivative gives the specific slope of the tangent line at that point.

Key Concept

The gradient of the tangent to a curve

y = f(x)

at

x = a

is the value of the first derivative

f'(a)

.

Answer

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