All practice questions

Question 7441Question →

A point $P$ moves in a plane such that its distance from a fixed point $O$ is always $7\text{ cm}$ . What is the diameter, in $\text{cm}$ , of the geometric locus traced out by point $P$ ?

Show answer & explanation

Answer: 14

Answer

The diameter of the locus traced out by point

P

is

14\text{ cm}

.

The locus of a point that maintains a constant distance from a fixed point is a circle. The fixed point

O

is the center of the circle, and the constant distance of

7\text{ cm}

is its radius (

r

). Since the diameter (

D

) of a circle is twice its radius (

D = 2r

), the diameter is

2 \times 7 = 14\text{ cm}

.

Step-by-Step Solution

1

Identify the shape of the geometric locus defined by the condition

A circle centered at point

O

with radius

r = 7\text{ cm}

By definition, the set of all points at a fixed distance from a single point forms a circle.

2

Calculate the diameter using the radius

D = 2 \times 7\text{ cm} = 14\text{ cm}

The diameter of a circle is equal to twice its radius.

Key Concept

Locus of a point at a constant distance from a fixed point

Estimated Time:45s

Question 7442Question →

The table below shows the distribution of scores obtained by a group of candidates in an examination:

Score ( $x$ )	2	4	6	8	10
Frequency ( $f$ )	1	2	6	2	1

Calculate the variance of the scores.

Show answer & explanation

Answer: 4

Answer

The variance of the scores is 4.

The mean score is

\bar{x} = 6

. Summing the weighted squared deviations gives

\sum f(x - 6)^2 = 48

. Dividing by the total frequency

N = 12

results in a variance of

\frac{48}{12} = 4

.

Step-by-Step Solution

1

Find the total frequency and calculate the mean score.

Total frequency

N = 12

and mean

\bar{x} = 6

.

The mean is given by

\bar{x} = \frac{\sum f x}{\sum f} = \frac{2 + 8 + 36 + 16 + 10}{12} = \frac{72}{12} = 6

.

2

Determine the sum of squared deviations multiplied by their frequencies.

\sum f(x - \bar{x})^2 = 48.

Evaluating each term:

1(2-6)^2 = 16

,

2(4-6)^2 = 8

,

6(6-6)^2 = 0

,

2(8-6)^2 = 8

, and

1(10-6)^2 = 16

. Summing these yields

16 + 8 + 0 + 8 + 16 = 48

.

3

Compute the variance by dividing the total squared deviations by the total frequency.

Variance = 4.

Variance is

\sigma^2 = \frac{\sum f(x - \bar{x})^2}{\sum f} = \frac{48}{12} = 4

.

Key Concept

Variance of a Frequency Distribution

Question 7443Question →

When constructing a standard cumulative frequency curve (ogive) for a grouped frequency distribution, against which statistical values on the horizontal axis are the cumulative frequencies plotted?

Upper class boundaries

Lower class boundaries

Class midpoints (class marks)

Raw class frequencies

Show answer & explanation

Answer: Upper class boundaries

Answer

Upper class boundaries

In statistics, an ogive is constructed by plotting cumulative frequency on the vertical axis against the upper class boundaries of each interval on the horizontal axis.

Step-by-Step Solution

1

Identify the definition of a cumulative frequency curve (ogive).

An ogive displays running totals of data values up to a given upper limit.

By definition, the cumulative frequency at any point represents all observations up to and including the upper boundary of that class.

2

Determine the correct variable for the horizontal axis.

The horizontal axis must represent the upper class boundaries (

x

-axis), while the vertical axis represents cumulative frequency (

y

-axis).

This ensures the curve accurately reflects cumulative proportions at the end of each class interval.

Key Concept

Plotting Cumulative Frequency Curves (Ogives)

Estimated Time:45s

Question 7444Question →

If $\sqrt{28 - 10\sqrt{3}} + \sqrt{7 + 4\sqrt{3}} = k$ , where $k$ is a rational number, find the value of $k$ .

Show answer & explanation

Answer: 7

Answer

The value of

k

is

7

.

Simplifying each nested surd into standard binomial surd form gives

\sqrt{28 - 10\sqrt{3}} = 5 - \sqrt{3}

and

\sqrt{7 + 4\sqrt{3}} = 2 + \sqrt{3}

. Combining these terms cancels out the irrational component

\sqrt{3}

, leaving the integer result

5 + 2 = 7

.

Step-by-Step Solution

1

Simplify the nested surd

\sqrt{28 - 10\sqrt{3}}

into binomial surd form

\sqrt{28 - 10\sqrt{3}} = 5 - \sqrt{3}

Assuming a form

x - y\sqrt{3}

and squaring both sides gives

x^2 + 3y^2 - 2xy\sqrt{3} = 28 - 10\sqrt{3}

. Matching components leads to

xy = 5

and

x^2 + 3y^2 = 28

, which yields integer values

x = 5

and

y = 1

.

2

Simplify the nested surd

\sqrt{7 + 4\sqrt{3}}

into binomial surd form

\sqrt{7 + 4\sqrt{3}} = 2 + \sqrt{3}

Assuming a form

u + v\sqrt{3}

and squaring both sides gives

u^2 + 3v^2 + 2uv\sqrt{3} = 7 + 4\sqrt{3}

. Matching components leads to

uv = 2

and

u^2 + 3v^2 = 7

, which yields integer values

u = 2

and

v = 1

.

3

Sum the simplified expressions to calculate

k

k = 7

Summing

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

results in the irrational parts

-\sqrt{3}

and

\sqrt{3}

cancelling out, leaving

5 + 2 = 7

.

Key Concept

Square Root of a Binomial Surd Expression

Question 7445Question →

What is the simplified form of the surd expression $\sqrt{45} - \sqrt{20} + \sqrt{80}$ ?

5\sqrt{5}

\sqrt{105}

9\sqrt{5}

3\sqrt{5}

Show answer & explanation

Answer: $5\sqrt{5}$

Answer

The simplified form of the expression is

5\sqrt{5}

.

Each radical is decomposed into a product involving a perfect square:

\sqrt{45} = 3\sqrt{5}

,

\sqrt{20} = 2\sqrt{5}

, and

\sqrt{80} = 4\sqrt{5}

. Combining the coefficients

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

yields

5\sqrt{5}

.

Step-by-Step Solution

1

Simplify each individual surd into basic radical form by finding perfect square factors.

\sqrt{45} = \sqrt{9 \times 5} = 3\sqrt{5}

,

\sqrt{20} = \sqrt{4 \times 5} = 2\sqrt{5}

, and

\sqrt{80} = \sqrt{16 \times 5} = 4\sqrt{5}

.

Expressing surds in terms of identical basic radicals allows like terms to be combined.

2

Substitute the simplified surds back into the original expression and combine like terms.

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

.

Perform addition and subtraction on the coefficients of similar surds.

Key Concept

Simplification of Surds

Question 7446Question →

The maximum safe load $L$ supported by a horizontal wooden beam varies directly as its width $w$ and the square of its depth $d$ , and inversely as its length $l$ . A beam of width $6\text{ cm}$ , depth $10\text{ cm}$ , and length $4\text{ m}$ can support a maximum safe load of $900\text{ kg}$ . What is the maximum safe load that can be supported by a beam of the same material having a width of $4\text{ cm}$ , depth $12\text{ cm}$ , and length $6\text{ m}$ ?

576\text{ kg}

480\text{ kg}

1,296\text{ kg}

864\text{ kg}

Show answer & explanation

Answer: $576\text{ kg}$

Answer

576\text{ kg}

The relationship is modeled by

L = \frac{k w d^2}{l}

. Using the initial parameters (

w=6

,

d=10

,

l=4

,

L=900

), we find

k = 6

. Substituting

w=4

,

d=12

, and

l=6

into the equation gives

L = \frac{6 \times 4 \times 144}{6} = 576\text{ kg}

.

Step-by-Step Solution

1

Formulate the variation equation

L = \frac{k \cdot w \cdot d^2}{l}

Direct variation means multiplying factors in the numerator, while inverse variation places the variable in the denominator.

2

Calculate the constant of variation

k

using initial conditions

900 = \frac{k \cdot 6 \cdot 10^2}{4} \implies 900 = \frac{600 k}{4} = 150 k \implies k = 6

Substitute

L = 900

,

w = 6

,

d = 10

, and

l = 4

to solve for

k

.

3

Calculate the new load

L

for the new dimensions

L = \frac{6 \cdot 4 \cdot 12^2}{6} = \frac{6 \cdot 4 \cdot 144}{6} = 576\text{ kg}

Substitute

k = 6

,

w = 4

,

d = 12

, and

l = 6

into the variation formula.

Key Concept

Joint and Inverse Variation

Estimated Time:1m 30s

Question 7447Question →

Three water pipes, $A$ , $B$ , and $C$ , are used to fill a storage tank. The filling rates of pipes $A$ and $B$ are in the ratio $3 : 2$ , while the filling rates of pipes $B$ and $C$ are in the ratio $4 : 3$ . Pipes $A$ and $B$ are opened together for $5\text{ hours}$ , after which pipe $A$ is closed. Pipes $B$ and $C$ are then opened together for $10\text{ hours}$ to completely fill the remaining capacity of the tank. How many hours would pipe $A$ alone take to fill the entire empty tank?

Show answer & explanation

Answer: 20

Answer

Pipe

A

alone will take

20\text{ hours}

to fill the empty tank.

Combining the given ratios

A : B = 3 : 2

and

B : C = 4 : 3

gives

A : B : C = 6 : 4 : 3

. Assigning relative rates of

6r

,

4r

, and

3r

per hour, the work completed in the first phase is

5 \times (6r + 4r) = 50r

, and in the second phase is

10 \times (4r + 3r) = 70r

. The total capacity of the tank is

50r + 70r = 120r

. Dividing the total capacity by pipe

A

's filling rate (

6r

) gives

120r / 6r = 20\text{ hours}

.

Step-by-Step Solution

1

Find the combined ratio of filling rates for pipes A, B, and C

A : B : C = 6 : 4 : 3

Given

A : B = 3 : 2 = 6 : 4

and

B : C = 4 : 3

, aligning the common term

B = 4

yields

A : B : C = 6 : 4 : 3

.

2

Express rates in terms of a constant

r

and calculate work done in Phase 1

Work in Phase 1 =

50r

Rates are

A = 6r

,

B = 4r

,

C = 3r

. In

5\text{ hours}

, pipes

A

and

B

fill

5 \times (6r + 4r) = 50r

.

3

Calculate work done in Phase 2

Work in Phase 2 =

70r

In

10\text{ hours}

, pipes

B

and

C

fill

10 \times (4r + 3r) = 70r

.

4

Determine total capacity of the tank

Total Capacity =

120r

Total work required to fill the tank is

50r + 70r = 120r

.

5

Calculate time taken by pipe A alone

20\text{ hours}

Time taken by pipe

A = \frac{\text{Total Capacity}}{\text{Rate of } A} = \frac{120r}{6r} = 20\text{ hours}

.

Key Concept

Compound Ratio and Rates of Work

Question 7448Question →

In $\Delta LMN$ , the length of side $l = 10\text{ cm}$ , side $m = 10\sqrt{3}\text{ cm}$ , and angle $\angle L = 30^\circ$ . Given that angle $\angle M$ is an obtuse angle, what is the measure of angle $\angle M$ ?

120^\circ

60^\circ

150^\circ

45^\circ

Show answer & explanation

Answer: $120^\circ$

Answer

The measure of angle

\angle M

is

120^\circ

.

Using the Sine Rule, we find

\sin M = \frac{10\sqrt{3} \cdot \sin 30^\circ}{10} = \frac{\sqrt{3}}{2}

. The inverse sine gives a reference angle of

60^\circ

. Since the problem explicitly states that angle

\angle M

is obtuse, we select

180^\circ - 60^\circ = 120^\circ

.

Step-by-Step Solution

1

Apply the Sine Rule relating sides

l, m

and their opposite angles

L, M

.

\frac{l}{\sin L} = \frac{m}{\sin M}

The Sine Rule allows us to find an unknown angle given two sides and one non-included opposite angle.

2

Substitute the given values into the formula and solve for

\sin M

.

\sin M = \frac{10\sqrt{3} \cdot \sin 30^\circ}{10} = \sqrt{3} \cdot 0.5 = \frac{\sqrt{3}}{2}

Since

\sin 30^\circ = \frac{1}{2}

, simplifying the fraction gives

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

.

3

Determine the obtuse angle solution for

\sin M = \frac{\sqrt{3}}{2}

.

\angle M = 180^\circ - 60^\circ = 120^\circ

The principal value is

60^\circ

, but because

\angle M

is specified to be obtuse (

90^\circ < \angle M < 180^\circ

), we take the supplementary angle in the second quadrant.

Key Concept

Sine Rule and the Ambiguous Case (SSA Condition)

Estimated Time:1m 30s

Question 7449Question →

If $(2x - 3)$ is a factor of the polynomial $P(x) = 2x^3 - 5x^2 + ax + 6$ , what is the value of the constant $a$ ?

-8

-5

-1

1

Show answer & explanation

Answer: -1

Answer

The value of the constant

a

is

-1

.

According to the Factor Theorem, if

(2x - 3)

is a factor of

P(x)

, then

P\left(\frac{3}{2}\right) = 0

. Substituting

x = \frac{3}{2}

gives

\frac{27}{4} - \frac{45}{4} + \frac{3}{2}a + 6 = 0

, which simplifies to

\frac{3}{2} + \frac{3}{2}a = 0

, resulting in

a = -1

.

Step-by-Step Solution

1

Apply the Factor Theorem by setting the linear factor equal to zero.

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

If

(2x - 3)

is a factor of

P(x)

, then

P\left(\frac{3}{2}\right) = 0

.

2

Substitute

x = \frac{3}{2}

into the polynomial expression

P(x)

.

P\left(\frac{3}{2}\right) = 2\left(\frac{3}{2}\right)^3 - 5\left(\frac{3}{2}\right)^2 + a\left(\frac{3}{2}\right) + 6 = 0

Setting the resulting expression equal to zero allows solving for

a

.

3

Simplify the powers and numerical terms.

2\left(\frac{27}{8}\right) - 5\left(\frac{9}{4}\right) + \frac{3}{2}a + 6 = 0 \implies \frac{27}{4} - \frac{45}{4} + \frac{3}{2}a + 6 = 0

Evaluate each fraction before combining terms.

4

Combine constant terms and solve for

a

.

-\frac{18}{4} + 6 + \frac{3}{2}a = 0 \implies -\frac{9}{2} + 6 + \frac{3}{2}a = 0 \implies \frac{3}{2} + \frac{3}{2}a = 0 \implies a = -1

Isolating

a

yields the correct constant value.

Key Concept

Factor Theorem for linear divisors of the form

(ax - b)

Question 7450Question →

If $\log_3(x + 5) - \log_{\frac{1}{3}}(x - 1) = 3$ , find the real value of $x$ .

Show answer & explanation

Answer: 4

Answer

The real value of

x

is 4.

Using the change of base formula,

\log_{\frac{1}{3}}(x - 1) = -\log_3(x - 1)

. Substituting this back into the equation transforms it into

\log_3(x + 5) + \log_3(x - 1) = 3

. Combining the logarithms using the product property gives

\log_3[(x + 5)(x - 1)] = 3

, which means

(x + 5)(x - 1) = 3^3 = 27

. Expanding leads to

x^2 + 4x - 32 = 0

, factoring into

(x + 8)(x - 4) = 0

. Since logarithms require positive arguments (

x > 1

),

x = -8

is invalid, giving the final answer

x = 4

.

Step-by-Step Solution

1

Apply change of base to express the equation in a single base

\log_{\frac{1}{3}}(x - 1) = \frac{\log_3(x - 1)}{\log_3(1/3)} = -\log_3(x - 1)

Change of base rule

\log_b a = \frac{\log_c a}{\log_c b}

with base

c = 3

, where

\log_3(1/3) = -1

.

2

Substitute back into the original equation and combine terms

\log_3(x + 5) - [-\log_3(x - 1)] = \log_3(x + 5) + \log_3(x - 1) = 3

Subtracting a negative logarithm equals adding the positive logarithm.

3

Apply the product rule of logarithms

\log_3[(x + 5)(x - 1)] = 3

Product rule:

\log_b M + \log_b N = \log_b(MN)

.

4

Convert logarithmic equation to quadratic form and solve

(x + 5)(x - 1) = 3^3 = 27 \implies x^2 + 4x - 32 = 0 \implies (x + 8)(x - 4) = 0

Logarithmic definition

\log_b Y = k \implies Y = b^k

.

5

Verify domain constraints

x = 4

Logarithmic domain requires

x - 1 > 0 \implies x > 1

. Therefore,

x = -8

is extraneous.

Key Concept

Logarithmic Change of Base and Algebraic Reduction

Question 7451Question →

An artisan deposited $\text{₦}20,000$ into a savings account that pays a compound interest rate of $10\%$ per annum compounded annually. What is the total compound interest earned at the end of $2$ years?

\text{₦}4,200

\text{₦}24,200

\text{₦}4,000

\text{₦}2,000

Show answer & explanation

Answer: $\text{₦}4,200$

Answer

The total compound interest earned after 2 years is

\text{₦}4,200

.

The correct answer is obtained by calculating the total amount after 2 years at

10\%

compound interest, which is

20,000 \times (1.10)^2 = \text{₦}24,200

, and then subtracting the principal of

\text{₦}20,000

to get

\text{₦}4,200

.

Step-by-Step Solution

1

Calculate the total accumulated amount (

A

) after 2 years using the compound interest formula

A = P\left(1 + \frac{R}{100}\right)^n

.

A = 20,000 \left(1 + \frac{10}{100}\right)^2 = 20,000 \times (1.1)^2 = 20,000 \times 1.21 = \text{₦}24,200

.

Determines the total value of the investment at the end of the 2-year period.

2

Subtract the principal (

P

) from the total accumulated amount (

A

) to find the interest earned (

I = A - P

).

I = 24,200 - 20,000 = \text{₦}4,200

.

Isolates the interest earned from the initial principal amount.

Key Concept

Compound Interest Calculation

Estimated Time:45s

Question 7452Question →

If $\frac{\sqrt{x} + \sqrt{3}}{\sqrt{x} - \sqrt{3}} = 2$ , find the value of $x$ .

Show answer & explanation

Answer: 27

Answer

The value of

x

is 27.

Cross-multiplying yields

\sqrt{x} + \sqrt{3} = 2\sqrt{x} - 2\sqrt{3}

. Rearranging terms gives

\sqrt{x} = 3\sqrt{3}

. Squaring both sides produces

x = 3^2 \times 3 = 27

.

Step-by-Step Solution

1

Multiply both sides by the denominator

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

\sqrt{x} + \sqrt{3} = 2(\sqrt{x} - \sqrt{3})

Clear the rational surd expression to form a linear relation in terms of radicals

2

Expand the terms and isolate

\sqrt{x}

\sqrt{x} = 3\sqrt{3}

Group terms involving

\sqrt{x}

on one side and constant surds on the other side

3

Square both sides of the simplified equation

x = (3\sqrt{3})^2 = 9 \times 3 = 27

Eliminate the radical over

x

to obtain the integer solution

Key Concept

Solving algebraic surd equations using rationalization concepts and properties of radicals

Estimated Time:1m 30s

Question 7453Question →

What is the numerical value of $4\sin 30^\circ + 2\cos 60^\circ - \tan 45^\circ$ ?

Show answer & explanation

Answer: 2

Answer

2

Substituting the exact trigonometric values for special angles into the expression gives

4\left(\frac{1}{2}\right) + 2\left(\frac{1}{2}\right) - 1 = 2 + 1 - 1 = 2

.

Step-by-Step Solution

1

Recall exact values for special trigonometric angles

\sin 30^\circ = \frac{1}{2}

,

\cos 60^\circ = \frac{1}{2}

, and

\tan 45^\circ = 1

These are standard special angle values derived from standard right-angled triangles.

2

Substitute the values into the original expression

4\left(\frac{1}{2}\right) + 2\left(\frac{1}{2}\right) - 1 = 2 + 1 - 1

Direct algebraic substitution.

3

Simplify the resulting numerical expression

2

Combining terms

2 + 1 - 1

gives

2

.

Key Concept

Basic Trigonometric Ratios and Special Angles

Question 7454Question →

What is the remainder when the polynomial $P(x) = x^3 - 4x^2 + 5x - 2$ is divided by $(x + 1)$ ?

-12

0

-4

-2

Show answer & explanation

Answer: $-12$

Answer

-12

According to the Remainder Theorem, dividing a polynomial

P(x)

by a linear divisor

(x - a)

yields a remainder equal to

P(a)

. Here, the divisor is

(x + 1)

, which corresponds to

a = -1

. Substituting

x = -1

into

P(x) = x^3 - 4x^2 + 5x - 2

gives

(-1)^3 - 4(-1)^2 + 5(-1) - 2 = -1 - 4 - 5 - 2 = -12

. Thus, the value

-12

is correct.

Step-by-Step Solution

1

Apply the Remainder Theorem

To find the remainder when

P(x)

is divided by

(x + 1)

, evaluate

P(-1)

by setting

x + 1 = 0 \implies x = -1

.

By the Remainder Theorem, dividing

P(x)

by

(x - a)

leaves a remainder equal to

P(a)

.

2

Substitute

x = -1

into the polynomial

P(x) = x^3 - 4x^2 + 5x - 2

P(-1) = (-1)^3 - 4(-1)^2 + 5(-1) - 2

Replace each instance of

x

with

-1

.

3

Simplify the powers and terms

P(-1) = -1 - 4(1) - 5 - 2 = -1 - 4 - 5 - 2 = -12

Compute arithmetic operations following standard order of operations.

Key Concept

Remainder Theorem

Estimated Time:45s

Question 7455Question →

If $y$ varies inversely as $x$ , and $y = 8$ when $x = 3$ , what is the value of $y$ when $x = 6$ ?

4

16

24

2

Show answer & explanation

Answer: 4

Answer

4

For inverse variation,

y = \frac{k}{x}

. Given

y = 8

when

x = 3

, the constant of variation

k = 8 \times 3 = 24

. Substituting

x = 6

gives

y = \frac{24}{6} = 4

.

Step-by-Step Solution

1

Set up the equation for inverse variation

y = \frac{k}{x}

Inverse variation means

y

is inversely proportional to

x

with constant

k

.

2

Solve for the variation constant

k

using

y = 8

and

x = 3

k = 8 \times 3 = 24

Multiply both sides of the equation by

x

.

3

Calculate

y

when

x = 6

y = \frac{24}{6} = 4

Substitute the constant

k = 24

and

x = 6

into the inverse variation equation.

Key Concept

Inverse Variation

Question 7456Question →

The line $3x - y + k = 0$ is a normal to the curve $y = \frac{x + 2}{x - 1}$ at a point $P$ located in the first quadrant. What is the value of the constant $k$ ?

-10

-2

10

6

Show answer & explanation

Answer: -10

Answer

The value of the constant

k

is

-10

.

Differentiating

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

yields

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

. The gradient of the normal line is

m_N = -\frac{1}{dy/dx} = \frac{(x - 1)^2}{3}

. Equating

m_N

to the gradient of

3x - y + k = 0

(which is

3

) gives

\frac{(x - 1)^2}{3} = 3 \implies (x - 1)^2 = 9

. Solving gives

x = 4

or

x = -2

. Since point

P

lies in the first quadrant,

x = 4

, which gives

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

. Substituting

(4, 2)

into

3x - y + k = 0

gives

3(4) - 2 + k = 0

, so

k = -10

.

Step-by-Step Solution

1

Differentiate the curve equation to find the tangent gradient function.

Using the quotient rule on

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

,

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

.

The derivative gives the gradient of the tangent line to the curve at any point

x

.

2

Determine the gradient function for the normal line.

The normal gradient is

m_N = -\frac{1}{dy/dx} = \frac{(x - 1)^2}{3}

.

The normal line is perpendicular to the tangent line, so its gradient is the negative reciprocal of

\frac{dy}{dx}

.

3

Equate the normal gradient to the gradient of the given line and solve for

x

.

The line

3x - y + k = 0

has gradient

3

. Setting

\frac{(x - 1)^2}{3} = 3 \implies (x - 1)^2 = 9 \implies x - 1 = \pm 3

, giving

x = 4

or

x = -2

.

The normal line at point

P

must be parallel to (and thus have the same gradient as) the given line

3x - y + k = 0

.

4

Select the coordinate corresponding to the first quadrant and calculate the

y

-coordinate.

Since

P

lies in the first quadrant (

x > 0, y > 0

),

x = 4

. Substituting

x = 4

into the curve equation gives

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

. Thus,

P = (4, 2)

.

The problem specifies that point

P

is located in the first quadrant.

5

Substitute point

P(4, 2)

into the line equation to solve for

k

.

Substituting

x = 4

and

y = 2

into

3x - y + k = 0

yields

3(4) - 2 + k = 0 \implies 10 + k = 0 \implies k = -10

.

Point

P

lies on the normal line, so its coordinates must satisfy the line equation.

Key Concept

The normal line to a curve at a point

P

is perpendicular to the tangent line at

P

, with gradient

m_{\text{normal}} = -\frac{1}{dy/dx}

. A point

P

on the curve must satisfy both the curve equation and the normal line equation.

Question 7457Question →

A solid right pyramid has a square base with a side length of $6\text{ cm}$ and a vertical height of $4\text{ cm}$ . What is the total surface area of the pyramid in square centimeters?

Show answer & explanation

Answer: 96

Answer

The total surface area of the pyramid is

96\text{ cm}^2

.

To determine the total surface area of a square pyramid, sum the base area and the total area of the four triangular faces. The base area is

6 \times 6 = 36\text{ cm}^2

. The slant height of each triangular face is found via the Pythagorean theorem using half the base side length (

3\text{ cm}

) and the vertical height (

4\text{ cm}

), yielding

\sqrt{3^2 + 4^2} = 5\text{ cm}

. The area of one triangular face is

\frac{1}{2} \times 6 \times 5 = 15\text{ cm}^2

, making four faces equal to

60\text{ cm}^2

. Adding the base area gives

36 + 60 = 96\text{ cm}^2

.

Step-by-Step Solution

1

Calculate the area of the square base

Base area =

36\text{ cm}^2

The base is a square of side

6\text{ cm}

, so

\text{Area} = 6^2 = 36\text{ cm}^2

.

2

Find the slant height of each triangular lateral face

Slant height

l = 5\text{ cm}

The slant height forms the hypotenuse of a right-angled triangle inside the pyramid with legs equal to the vertical height (

4\text{ cm}

) and half the base edge (

3\text{ cm}

):

l = \sqrt{4^2 + 3^2} = 5\text{ cm}

.

3

Calculate the combined area of the four triangular faces

Lateral area =

60\text{ cm}^2

Each triangle has base

6\text{ cm}

and height

5\text{ cm}

, giving an area of

\frac{1}{2} \times 6 \times 5 = 15\text{ cm}^2

. For 4 identical faces, the total is

4 \times 15 = 60\text{ cm}^2

.

4

Calculate total surface area

Total Surface Area =

96\text{ cm}^2

Sum the base area and the total lateral area:

36 + 60 = 96\text{ cm}^2

.

Key Concept

Total Surface Area of a Right Pyramid

Question 7458Question →

In $\Delta ABC$ , $\sin A = \frac{3}{5}$ , $\sin B = \frac{4}{5}$ , and the side opposite angle $A$ has length $a = 15\text{ cm}$ . What is the length of side $b$ , in centimeters?

Show answer & explanation

Answer: 20

Answer

The length of side

b

is 20 cm.

Using the Sine Rule

\frac{a}{\sin A} = \frac{b}{\sin B}

, substitute

a = 15

,

\sin A = \frac{3}{5}

, and

\sin B = \frac{4}{5}

. Evaluating

\frac{15}{3/5}

gives

25

. Multiplying

25

by

\frac{4}{5}

yields

20\text{ cm}

.

Step-by-Step Solution

1

Set up the Sine Rule relationship between sides

a

,

b

and their opposite angles

A

,

B

.

\frac{a}{\sin A} = \frac{b}{\sin B}

The Sine Rule relates the side lengths of a triangle to the sines of its angles.

2

Substitute

a = 15

,

\sin A = \frac{3}{5}

, and

\sin B = \frac{4}{5}

into the Sine Rule equation.

\frac{15}{3/5} = \frac{b}{4/5}

Direct substitution of known values allows us to solve for the unknown side

b

.

3

Simplify the left side of the equation.

15 \times \frac{5}{3} = 25

Dividing 15 by

\frac{3}{5}

is equivalent to multiplying 15 by

\frac{5}{3}

.

4

Multiply both sides by

\frac{4}{5}

to find

b

.

b = 25 \times \frac{4}{5} = 20\text{ cm}

Isolating

b

gives the final length of side

b

.

Key Concept

Sine Rule

Question 7459Question →

A school kitchen uses $150$ loaves of bread to feed $300$ students. If $120$ additional students join the school, how many loaves of bread in total will be needed to feed all the students at the same consumption rate?

60

loaves

210

loaves

270

loaves

375

loaves

Show answer & explanation

Answer: $210$ loaves

Answer

210

loaves of bread

The rate of bread consumption is

150 \div 300 = 0.5

loaves per student. With

120

additional students, the total number of students becomes

300 + 120 = 420

. Multiplying

420

students by

0.5

loaves per student gives

210

loaves.

Step-by-Step Solution

1

Calculate the rate of consumption per student.

Rate =

\frac{150\text{ loaves}}{300\text{ students}} = 0.5\text{ loaves per student}

.

Determining the unit rate simplifies finding the total quantity needed for any number of students.

2

Find the total number of students.

Total students =

300 + 120 = 420\text{ students}

.

The question asks for the total loaves required to feed all students after the addition.

3

Calculate the total loaves needed by multiplying the total number of students by the unit rate.

Total loaves =

420 \times 0.5 = 210\text{ loaves}

.

Applying direct proportion yields the total amount of bread required.

Key Concept

Direct Proportion and Rate

Question 7460Question →

In a survey of $120$ town residents regarding three newspapers—*The Herald* ( $H$ ), *The Express* ( $E$ ), and *The Nation* ( $N$ )—it was found that $52$ read *The Herald*, $45$ read *The Express*, and $60$ read *The Nation*. Furthermore, $15$ read both *The Herald* and *The Express*, $22$ read both *The Express* and *The Nation*, and $18$ read both *The Herald* and *The Nation*. If the number of residents who read none of these three newspapers is twice the number of those who read all three newspapers, how many residents read exactly two of the newspapers?

31

37

43

55

Show answer & explanation

Answer: $37$

Answer

The number of residents who read exactly two newspapers is

37

.

Using the principle of inclusion-exclusion,

n(H \cup E \cup N) = 52 + 45 + 60 - (15 + 22 + 18) + x = 102 + x

, where

x

is the number of residents reading all three newspapers. Since the number of residents reading none is

2x

, the total universe equation is

102 + x + 2x = 120

, giving

3x = 18

and

x = 6

. The number of residents reading exactly two newspapers is

(15 - 6) + (22 - 6) + (18 - 6) = 9 + 16 + 12 = 37

.

Step-by-Step Solution

1

Formulate the principle of inclusion-exclusion for three sets.

n(H \cup E \cup N) = n(H) + n(E) + n(N) - n(H \cap E) - n(E \cap N) - n(H \cap N) + n(H \cap E \cap N)

To express the total number of residents reading at least one newspaper in terms of the unknown number of residents who read all three.

2

Substitute the given values and set up the equation for the universal set.

n(H \cup E \cup N) = 52 + 45 + 60 - 15 - 22 - 18 + x = 102 + x

. Let

n(H \cup E \cup N)' = 2x

. Then

102 + x + 2x = 120 \implies 102 + 3x = 120 \implies 3x = 18 \implies x = 6

.

The sum of elements in the union and its complement must equal the universal set size of

120

.

3

Calculate the number of residents reading exactly two newspapers.

(n(H \cap E) - x) + (n(E \cap N) - x) + (n(H \cap N) - x) = (15 - 6) + (22 - 6) + (18 - 6) = 9 + 16 + 12 = 37

Subtracting the triple intersection

x

from each pairwise intersection isolates the regions corresponding to exactly two newspapers.

Key Concept

Three-set inclusion-exclusion principle and Venn diagram cardinal region decomposition

Estimated Time:1m 30s

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept