All practice questions

Question 7361Question →

Given that $\frac{\sqrt{5} + 2}{\sqrt{5} - 2} = a + b\sqrt{5}$ , where $a$ and $b$ are rational numbers, what is the value of $a + b$ ?

13

11

9

7

Show answer & explanation

Answer: 13

Answer

The value of

a + b

is 13.

To simplify

\frac{\sqrt{5} + 2}{\sqrt{5} - 2}

, multiply both numerator and denominator by the conjugate of the denominator, which is

\sqrt{5} + 2

. Expanding the numerator

(\sqrt{5} + 2)^2

gives

5 + 4\sqrt{5} + 4 = 9 + 4\sqrt{5}

. The denominator simplifies to

(\sqrt{5})^2 - 2^2 = 5 - 4 = 1

. Thus, the expression becomes

9 + 4\sqrt{5}

. Comparing this to

a + b\sqrt{5}

gives

a = 9

and

b = 4

. Therefore,

a + b = 13

.

Step-by-Step Solution

1

Identify the conjugate of the denominator

The denominator is

\sqrt{5} - 2

, so its conjugate is

\sqrt{5} + 2

.

Multiplying by the conjugate rationalizes the binomial denominator using the difference of two squares.

2

Multiply the numerator and denominator by the conjugate

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

This removes the radical from the denominator.

3

Expand both numerator and denominator

\frac{5 + 4\sqrt{5} + 4}{5 - 4} = \frac{9 + 4\sqrt{5}}{1} = 9 + 4\sqrt{5}

Using

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

for the numerator and

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

for the denominator.

4

Equate to

a + b\sqrt{5}

and solve for

a + b

a = 9

and

b = 4

, so

a + b = 9 + 4 = 13

.

Matching the rational part

a

and coefficient of the surd

b

gives the target sum.

Key Concept

Rationalization of Binomial Denominators

Estimated Time:1m 15s

Question 7362Question →

What is the indefinite integral $\int (6x^5 + 8\sin(4x)) \, dx$ ?

x^6 - 2\cos(4x) + C

x^6 - 2\cos(4x)

x^6 + 2\cos(4x) + C

x^6 - 8\cos(4x) + C

Show answer & explanation

Answer: $x^6 - 2\cos(4x) + C$

Answer

x^6 - 2\cos(4x) + C

Integrating

6x^5

using the power rule gives

\frac{6x^6}{6} = x^6

. Integrating

8\sin(4x)

gives

8 \left(-\frac{\cos(4x)}{4}\right) = -2\cos(4x)

. Summing these and appending the constant of integration

C

yields

x^6 - 2\cos(4x) + C

.

Step-by-Step Solution

1

Split the integral into two individual terms

\int (6x^5 + 8\sin(4x)) \, dx = \int 6x^5 \, dx + \int 8\sin(4x) \, dx

Linearity property of integration allows term-by-term integration.

2

Integrate the polynomial term

6x^5

\int 6x^5 \, dx = 6 \cdot \frac{x^{5+1}}{5+1} = \frac{6x^6}{6} = x^6

Apply the power rule of integration:

\int x^n \, dx = \frac{x^{n+1}}{n+1}

.

3

Integrate the trigonometric term

8\sin(4x)

\int 8\sin(4x) \, dx = 8 \cdot \left( -\frac{\cos(4x)}{4} \right) = -2\cos(4x)

Apply the standard integral rule:

\int \sin(kx) \, dx = -\frac{1}{k}\cos(kx)

.

4

Combine results and add the constant of integration

x^6 - 2\cos(4x) + C

Indefinite integration requires an arbitrary constant

C

.

Key Concept

Indefinite Integration of Polynomial and Trigonometric Functions

Estimated Time:1m 30s

Question 7363Question →

If $\log_5 x + 2\log_x 5 = 3$ , what is the sum of all possible real values of $x$ ?

15

30

100

125

Show answer & explanation

Answer: 30

Answer

The sum of all possible real values of

x

is 30.

Applying the change of base identity

\log_x 5 = \frac{1}{\log_5 x}

converts the equation to

\log_5 x + \frac{2}{\log_5 x} = 3

. Setting

y = \log_5 x

yields

y^2 - 3y + 2 = 0

, giving solutions

y = 1

and

y = 2

. Converting back to

x

gives

x = 5^1 = 5

and

x = 5^2 = 25

. The sum of these values is

5 + 25 = 30

.

Step-by-Step Solution

1

Apply the change of base identity

\log_x 5 = \frac{1}{\log_5 x}

.

The equation becomes

\log_5 x + \frac{2}{\log_5 x} = 3

.

Converting all logarithmic terms to base 5 allows substitution and simplification.

2

Substitute

y = \log_5 x

and form a quadratic equation.

y + \frac{2}{y} = 3 \implies y^2 - 3y + 2 = 0

.

Multiplying through by

y

transforms the logarithmic equation into a standard quadratic form.

3

Solve the quadratic equation for

y

.

(y - 1)(y - 2) = 0 \implies y = 1 \text{ or } y = 2

.

Factoring determines the values of

y = \log_5 x

.

4

Convert back to

x

using exponentiation and calculate the sum.

For

y = 1

,

x = 5^1 = 5

. For

y = 2

,

x = 5^2 = 25

. Sum

= 5 + 25 = 30

.

Definition of logarithm

\log_a b = c \iff a^c = b

gives the values of

x

.

Key Concept

Logarithm Change of Base Formula and Quadratic Substitution

Estimated Time:1m 30s

Question 7364Question →

The frequency distribution table below shows the number of books checked out daily at a public library over a period of $40$ days:

Number of Books	Number of Days ( $f$ )
$1 - 5$	$4$
$6 - 10$	$10$
$11 - 15$	$16$
$16 - 20$	$10$

What is the mean number of books checked out per day?

10.0

10.5

12.0

14.0

Show answer & explanation

Answer: $12.0$

Answer

The mean number of books checked out per day is

12.0

.

The correct answer of

12.0

is found by calculating the midpoint (

x

) for each class interval (

3, 8, 13, 18

), multiplying each midpoint by its respective frequency to find

f \cdot x

, summing these products to obtain

\sum fx = 480

, and dividing by the total frequency

\sum f = 40

.

Step-by-Step Solution

1

Determine the midpoint (

x

) for each class interval.

Midpoints:

x_1 = \frac{1+5}{2} = 3

,

x_2 = \frac{6+10}{2} = 8

,

x_3 = \frac{11+15}{2} = 13

,

x_4 = \frac{16+20}{2} = 18

.

Grouped data calculations require a single representative value (the midpoint) for each class interval.

2

Calculate the product of frequency and midpoint (

f \cdot x

) for each class interval.

f_1 x_1 = 4 \times 3 = 12

,

f_2 x_2 = 10 \times 8 = 80

,

f_3 x_3 = 16 \times 13 = 208

,

f_4 x_4 = 10 \times 18 = 180

.

This yields the total value contributed by each class interval.

3

Calculate total frequency (

\sum f

) and total sum of products (

\sum fx

).

\sum f = 4 + 10 + 16 + 10 = 40

, and

\sum fx = 12 + 80 + 208 + 180 = 480

.

These sums are needed to compute the weighted mean.

4

Apply the grouped mean formula

\bar{x} = \frac{\sum fx}{\sum f}

.

\bar{x} = \frac{480}{40} = 12.0

.

Dividing the sum of all values by the total frequency yields the grouped mean.

Key Concept

Grouped Mean Calculation

Question 7365Question →

In $\Delta ABC$ , the length of side $a$ is $10\text{ cm}$ , angle $\angle A = 30^\circ$ , and angle $\angle B = 45^\circ$ . What is the length of side $b$ ?

10\sqrt{2}\text{ cm}

5\sqrt{2}\text{ cm}

10\sqrt{3}\text{ cm}

20\text{ cm}

Show answer & explanation

Answer: $10\sqrt{2}\text{ cm}$

Answer

The length of side

b

is

10\sqrt{2}\text{ cm}

.

The option specifying

10\sqrt{2}\text{ cm}

is correct because applying the Sine Rule gives

\frac{10}{\sin 30^\circ} = \frac{b}{\sin 45^\circ}

, which simplifies directly to

b = 10\sqrt{2}\text{ cm}

.

Step-by-Step Solution

1

State the Sine Rule formula relating sides

a

,

b

and their opposite angles

A

,

B

.

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

The Sine Rule connects the ratio of side lengths to the sines of their opposite angles.

2

Substitute the known values (

a = 10

,

A = 30^\circ

,

B = 45^\circ

) into the formula.

\frac{10}{\sin 30^\circ} = \frac{b}{\sin 45^\circ}

Isolating the variable

b

requires substituting given numeric angle and side measures.

3

Evaluate exact values of trigonometric functions and solve for

b

.

b = \frac{10 \times \frac{\sqrt{2}}{2}}{\frac{1}{2}} = 10\sqrt{2}\text{ cm}

Simplifying the algebraic fraction yields the exact length of side

b

.

Key Concept

Application of the Sine Rule to find an unknown side length in a non-right-angled triangle.

Estimated Time:45s

Question 7366Question →

What is the product of all real values of

x

that satisfy the exponential equation

4^{x+1} - 17 \times 2^x + 4 = 0

?

-4

1

-1

4

Show answer & explanation

Answer: $-4$

Answer

The product of all real values of

x

satisfying the equation is

-4

.

Expressing

4^{x+1}

as

4(2^x)^2

allows setting

y = 2^x

, giving

4y^2 - 17y + 4 = 0

. Solving this quadratic equation yields

y = \frac{1}{4}

and

y = 4

. Solving

2^x = \frac{1}{4} = 2^{-2}

gives

x = -2

, and

2^x = 4 = 2^2

gives

x = 2

. The product of these two real solutions is

(-2) \times 2 = -4

.

Step-by-Step Solution

1

Rewrite

4^{x+1}

using index laws

4^{x+1} = 4^x \times 4^1 = (2^2)^x \times 4 = 4 \times (2^x)^2

Converting all exponential terms to base

2

allows substitution into a quadratic form.

2

Substitute

y = 2^x

into the equation

4y^2 - 17y + 4 = 0

This transforms the exponential equation into a standard quadratic equation in terms of

y

.

3

Solve the quadratic equation for

y

(4y - 1)(y - 4) = 0 \implies y = \frac{1}{4} \text{ or } y = 4

Factoring

4y^2 - 16y - y + 4 = 0

yields the two valid values for

y

.

4

Solve for

x

using

y = 2^x

For

y = \frac{1}{4}

:

2^x = 2^{-2} \implies x_1 = -2

. For

y = 4

:

2^x = 2^2 \implies x_2 = 2

.

Applying the law of indices

a^m = a^n \implies m = n

determines the real solutions for

x

.

5

Calculate the product of the solutions

x_1 \times x_2 = (-2) \times 2 = -4

The question asks for the product of the real values of

x

.

Key Concept

Quadratic form exponential equations and index transformation rules

Question 7367Question →

Determine the least non-negative integer congruent to $68 \pmod{9}$ .

Show answer & explanation

Answer: 5

Answer

The least non-negative integer congruent to

68 \pmod{9}

is 5.

Dividing 68 by 9 gives a quotient of 7 and a remainder of 5. Since

0 \le 5 < 9

, the value 5 is the standard non-negative remainder.

Step-by-Step Solution

1

Divide 68 by the modulus 9 using the division algorithm.

68 = 9 \times 7 + 5

Every integer

a

can be written uniquely as

a = nq + r

where

q

is the quotient and

0 \le r < n

.

2

Extract the non-negative remainder

r

.

r = 5

In modular arithmetic, the least non-negative integer congruent to

a \pmod{n}

is the remainder

r

when

a

is divided by

n

.

Key Concept

Modular Arithmetic Remainder

Question 7368Question →

Using differentiation from first principles, what is the derivative $\frac{dy}{dx}$ of the function $y = 3 - 2x^2$ ?

-4x

4x

-2x

-4x - 2

Show answer & explanation

Answer: $-4x$

Answer

-4x

Applying the definition of derivative

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

to

f(x) = 3 - 2x^2

gives

\frac{-4xh - 2h^2}{h} = -4x - 2h

. Taking the limit as

h \to 0

results in

-4x

.

Step-by-Step Solution

1

Express

f(x+h)

for the function

f(x) = 3 - 2x^2

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

Substitute

(x+h)

into the place of

x

and expand the squared binomial.

2

Set up the difference quotient

\frac{f(x+h) - f(x)}{h}

\frac{(3 - 2x^2 - 4xh - 2h^2) - (3 - 2x^2)}{h} = \frac{-4xh - 2h^2}{h}

Subtract

f(x)

from

f(x+h)

to isolate terms involving increment

h

.

3

Divide numerator terms by

h

\frac{h(-4x - 2h)}{h} = -4x - 2h

Factor out

h

to cancel the denominator.

4

Take the limit as

h \to 0

\lim_{h \to 0} (-4x - 2h) = -4x

Evaluating the limit gives the exact derivative function

\frac{dy}{dx}

.

Key Concept

Differentiation from First Principles

Question 7369Question →

The table below displays the distribution of scores obtained by $50$ candidates in a Mathematics assessment test:

Score Interval	Frequency ( $f$ )
$10 - 19$	$8$
$20 - 29$	$12$
$30 - 39$	$18$
$40 - 49$	$7$
$50 - 59$	$5$

Match each statistical feature of the charts representing this data on the left with its corresponding numerical value on the right.

Click a left item, then click its matching right item

Show answer & explanation

Answer

The correct pairings are: Sector angle for the modal class matches

129.6^\circ

; Frequency density of the

30 - 39

class interval matches

1.8

; Lower class boundary of the median class matches

29.5

; Cumulative frequency up to

29.5

matches

20

.

Each chart feature correctly aligns with its mathematically derived value: the modal class sector angle is

\frac{18}{50} \times 360^\circ = 129.6^\circ

, the frequency density is

\frac{18}{10} = 1.8

, the median class lower boundary is

29.5

, and the cumulative frequency up to

29.5

is

8 + 12 = 20

.

Step-by-Step Solution

1

Determine the total frequency and locate the modal and median classes.

Total frequency

N = 8 + 12 + 18 + 7 + 5 = 50

. The modal class is

30 - 39

(highest frequency =

18

). The median position is

\frac{50}{2} = 25^{\text{th}}

, which falls within the

30 - 39

class interval since cumulative frequency reaches

38

at the end of this class.

Identifying

N

, the modal class, and the median position is required for calculating chart parameters.

2

Calculate the pie chart sector angle for the modal class (

30 - 39

).

Sector angle

= \frac{\text{Frequency}}{N} \times 360^\circ = \frac{18}{50} \times 360^\circ = 129.6^\circ

.

Pie chart sectors represent relative frequencies scaled to

360^\circ

.

3

Compute the frequency density for the histogram bar of class

30 - 39

.

Class width

= 39.5 - 29.5 = 10

. Frequency density

= \frac{\text{Frequency}}{\text{Class width}} = \frac{18}{10} = 1.8

.

Histogram height represents frequency density, defined as frequency divided by class width.

4

Identify the lower class boundary of the median class (

30 - 39

) and cumulative frequency at upper boundary

29.5

.

Lower boundary of

30 - 39

is

30 - 0.5 = 29.5

. Cumulative frequency up to

29.5

is

8 + 12 = 20

.

Class boundaries eliminate gaps between intervals for continuous plots like ogives and histograms.

Key Concept

Calculating statistical chart parameters (pie chart sector angles, histogram frequency densities, class boundaries, and cumulative frequencies) from grouped frequency distributions.

Question 7370Question →

A solid right circular cylinder of radius $5\text{ cm}$ and height $12\text{ cm}$ has a conical cavity of the same radius and height hollowed out from its top face. What is the total surface area of the remaining solid in $\text{cm}^2$ , expressed as a multiple of $\pi$ (that is, find the value of $K$ where the total surface area is $K\pi\text{ cm}^2$ )?

Show answer & explanation

Answer: 210

Answer

The total surface area of the remaining solid is

210\pi\text{ cm}^2

, so the required numerical coefficient

K

is

210

.

The total surface area consists of three parts: the flat circular base at the bottom (

25\pi\text{ cm}^2

), the outer curved surface of the cylinder (

120\pi\text{ cm}^2

), and the newly created inner curved surface of the conical cavity (

65\pi\text{ cm}^2

). Adding these together yields

25\pi + 120\pi + 65\pi = 210\pi\text{ cm}^2

, giving the coefficient

K = 210

.

Step-by-Step Solution

1

Calculate the slant height (

l

) of the conical cavity.

l = \sqrt{5^2 + 12^2} = \sqrt{169} = 13\text{ cm}

.

The slant height forms the hypotenuse of the right triangle whose leg lengths are the base radius and the vertical height of the cone.

2

Determine the area of all exposed boundary surfaces of the remaining solid.

Flat bottom base area

= \pi(5)^2 = 25\pi\text{ cm}^2

; Outer curved cylindrical surface area

= 2\pi(5)(12) = 120\pi\text{ cm}^2

; Inner curved conical surface area

= \pi(5)(13) = 65\pi\text{ cm}^2

.

Hollowing out the cone creates an internal curved boundary while leaving the outer cylindrical boundary and the flat bottom base exposed.

3

Sum the areas of all exposed surfaces to calculate the total surface area.

Total Surface Area

= 25\pi + 120\pi + 65\pi = 210\pi\text{ cm}^2

.

The total surface area is the sum of all external and internal exposed surface areas.

Key Concept

Surface Area of Hollowed and Composite 3D Solids

Question 7371Question →

If $y = 5e^{2x} + \cos(4x)$ , find the value of $\frac{dy}{dx}$ at $x = 0$ .

Show answer & explanation

Answer: 10

Answer

10

Differentiating

y = 5e^{2x} + \cos(4x)

with respect to

x

yields

\frac{dy}{dx} = 10e^{2x} - 4\sin(4x)

. Substituting

x = 0

gives

10e^0 - 4\sin(0) = 10(1) - 0 = 10

.

Step-by-Step Solution

1

Differentiate

y = 5e^{2x} + \cos(4x)

with respect to

x

.

\frac{dy}{dx} = 10e^{2x} - 4\sin(4x)

The derivative of

e^{ax}

is

a e^{ax}

and the derivative of

\cos(ax)

is

-a \sin(ax)

.

2

Evaluate the derivative at

x = 0

.

10

Substitute

x = 0

into

10e^{2x} - 4\sin(4x)

to obtain

10(1) - 4(0) = 10

.

Key Concept

Differentiation of exponential and trigonometric functions

Question 7372Question →

In a library, the ratio of science books to literature books was initially $4 : 5$ . After $60$ new science books were added to the collection and $20\%$ of the literature books were donated to a local school, the ratio of science books to literature books became $3 : 2$ . What was the total number of books (science and literature combined) originally in the library?

120

150

270

300

Show answer & explanation

Answer: $270$

Answer

The original total number of books in the library was

270

.

Let the original number of science books be

4x

and literature books be

5x

. After adding

60

science books, there are

4x + 60

science books. Donating

20\%

of literature books leaves

80\%

of

5x

, which equals

4x

literature books. Setting up the ratio

\frac{4x + 60}{4x} = \frac{3}{2}

yields

8x + 120 = 12x

, so

4x = 120

and

x = 30

. The original total number of books was

4x + 5x = 9x = 9(30) = 270

.

Step-by-Step Solution

1

Define initial quantities using a ratio multiplier

x

.

Initial science books =

4x

, initial literature books =

5x

, and total original books =

9x

.

The initial ratio given is

4 : 5

.

2

Calculate the updated quantities after books are added and donated.

New science books =

4x + 60

; new literature books =

5x \times (1 - 0.20) = 4x

.

60

science books were added and

20\%

of literature books were donated.

3

Set up the proportion for the new ratio of

3 : 2

.

\frac{4x + 60}{4x} = \frac{3}{2}

The new ratio of science to literature books is stated as

3 : 2

.

4

Solve the algebraic proportion for

x

.

2(4x + 60) = 3(4x) \implies 8x + 120 = 12x \implies 4x = 120 \implies x = 30.

Cross-multiplication simplifies the proportional equation to find the value of

x

.

5

Compute the original total number of books.

9x = 9 \times 30 = 270.

The original total collection consists of

9x

books.

Key Concept

Solving multi-step ratio changes and proportions with percentage alterations

Estimated Time:1m 30s

Question 7373Question →

The sum of the interior angles of a convex polygon is three times the sum of its exterior angles. Calculate the number of sides of the polygon.

Show answer & explanation

Answer: 8

Answer

The polygon has 8 sides.

The sum of the exterior angles of any convex polygon is always

360^\circ

. Since the sum of the interior angles is three times this value, the interior angle sum is

3 \times 360^\circ = 1080^\circ

. Setting

(n - 2) \times 180^\circ = 1080^\circ

gives

n - 2 = 6

, which yields

n = 8

sides.

Step-by-Step Solution

1

Determine the exterior angle sum

The sum of exterior angles for any convex polygon is

360^\circ

.

The exterior angles of any convex polygon sum to a full turn (

360^\circ

).

2

Compute the sum of the interior angles

Sum of interior angles =

3 \times 360^\circ = 1080^\circ

.

The question states that the interior angle sum is three times the exterior angle sum.

3

Apply the interior angle sum formula

(n - 2) \times 180^\circ = 1080^\circ

.

The interior angle sum of an

n

-sided convex polygon is given by

(n - 2) \times 180^\circ

.

4

Solve for

n

n - 2 = \frac{1080^\circ}{180^\circ} = 6 \implies n = 8

.

Dividing

1080^\circ

by

180^\circ

gives

6

, and adding

2

gives

n = 8

.

Key Concept

Relationship between the sum of interior and exterior angles of a convex polygon

Estimated Time:1m 0s

Question 7374Question →

What is the simplified numerical value of $27^{\frac{2}{3}} \times 9^{-\frac{1}{2}}$ ?

Show answer & explanation

Answer: 3

Answer

The simplified numerical value of the expression is 3.

Evaluating

27^{\frac{2}{3}}

gives

(\sqrt[3]{27})^2 = 3^2 = 9

. Evaluating

9^{-\frac{1}{2}}

gives

\frac{1}{\sqrt{9}} = \frac{1}{3}

. Multiplying

9

by

\frac{1}{3}

yields

3

.

Step-by-Step Solution

1

Rewrite bases in terms of prime factors

27 = 3^3

and

9 = 3^2

Expressing bases in power-of-3 form allows direct application of index laws.

2

Simplify each indexed term using

(a^m)^n = a^{m \cdot n}

and

a^{-n} = \frac{1}{a^n}

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

and

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

Fractional indices denote roots and negative indices denote reciprocals.

3

Multiply the resulting numbers

9 \times \frac{1}{3} = 3

Simplifying the final product yields the single numeric answer.

Key Concept

Fractional and Negative Index Laws

Question 7375Question →

What is the smallest non-negative integer $x$ that satisfies the modular congruence $5x + 7 \equiv 2 \pmod{9}$ ?

8

4

1

7

Show answer & explanation

Answer: $8$

Answer

The smallest non-negative integer

x

is 8.

Subtracting

7

from both sides yields

5x \equiv -5 \pmod{9}

, which reduces to

5x \equiv 4 \pmod{9}

. Multiplying both sides by the modular multiplicative inverse of

5

modulo

9

(which is

2

, since

5 \times 2 = 10 \equiv 1 \pmod{9}

) gives

x \equiv 4 \times 2 = 8 \pmod{9}

. Thus, the smallest non-negative integer value is

8

.

Step-by-Step Solution

1

Isolate the term containing

x

by subtracting 7 from both sides of the congruence.

5x \equiv 2 - 7 \equiv -5 \pmod{9}

Standard algebraic balance operations apply under modular arithmetic.

2

Convert the negative remainder

-5

to its canonical non-negative representative in modulo 9.

-5 + 9 = 4 \implies 5x \equiv 4 \pmod{9}

Modular numbers are conventionally expressed in the non-negative range

[0, n-1]

.

3

Find the modular multiplicative inverse of 5 modulo 9 and multiply both sides by it.

Since

5 \times 2 = 10 \equiv 1 \pmod{9}

, the inverse is 2. Multiplying gives

x \equiv 4 \times 2 = 8 \pmod{9}

.

Multiplying by the modular inverse solves for

x

when

\gcd(5, 9) = 1

.

Key Concept

Linear Modular Congruences and Modular Inverses

Estimated Time:1m 30s

Question 7376Question →

What is the $x$ -coordinate of the stationary point of the curve $y = 3x^2 - 12x + 7$ ?

2

-2

4

7

Show answer & explanation

Answer: $2$

Answer

The

x

-coordinate of the stationary point is

2

.

To find the stationary point, take the derivative of

y = 3x^2 - 12x + 7

with respect to

x

, obtaining

\frac{dy}{dx} = 6x - 12

. Setting the derivative equal to zero gives

6x - 12 = 0

, which solves to

x = 2

.

Step-by-Step Solution

1

Differentiate the equation of the curve with respect to

x

\frac{dy}{dx} = 6x - 12

Stationary points occur where the first derivative (gradient) of the function is equal to zero.

2

Set the derivative to zero and solve for

x

6x - 12 = 0 \implies 6x = 12 \implies x = 2

Solving this linear equation gives the exact value of

x

at which the tangent to the curve is horizontal.

Key Concept

Stationary Points of a Curve

Estimated Time:45s

Question 7377Question →

In a stable mainland population of a wildflower species, the frequency of a recessive allele ( $a$ ) causing albino foliage is $0.3$ . A small group of 10 seed-bearing plants is accidentally carried by wind to an isolated oceanic island where no other members of the species exist. In the newly established island colony, the frequency of allele $a$ is found to be $0.7$ purely due to sampling chance during colonization. Which evolutionary force best accounts for this rapid change in allele frequency, and what is its primary consequence on the island's gene pool?

Genetic drift via the founder effect, which reduces overall genetic variation in the new population regardless of natural selection.

Directional natural selection, which favors the albino phenotype to enhance photosynthetic efficiency under island sunlight.

Environmentally induced mutation, which causes green plants to alter their germline DNA to cope with new atmospheric conditions.

Gene flow, which continuously transfers traits from the mainland to create continuous phenotypic variation across island habitats.

Show answer & explanation

Answer: Genetic drift via the founder effect, which reduces overall genetic variation in the new population regardless of natural selection.

Answer

Genetic drift via the founder effect, which reduces overall genetic variation in the new population regardless of natural selection.

The correct answer identifies the founder effect—a key mechanism of genetic drift in modern evolutionary theory. When a small group of individuals colonizes a new geographical area, the alleles carried by the founders determine the new gene pool. By chance alone, allele frequencies in the colonizing group differ markedly from the source population, which decreases overall genetic diversity independently of natural selection.

Step-by-Step Solution

1

Analyze the cause of the allele frequency shift in the population

A very small sample size (10 plants) colonized an isolated island, shifting the recessive allele frequency from

0.3

to

0.7

purely due to sampling chance.

When a small fragment of a population colonizes a new area, the allele frequencies of the founders rarely match the ancestral gene pool.

2

Identify the primary evolutionary mechanism

Random fluctuations in allele frequency resulting from small colony founding represent the founder effect, a form of genetic drift.

Genetic drift acts randomly on allele frequencies without regard to reproductive fitness or adaptive advantage.

3

Evaluate option claims against modern evolutionary genetics

The founder effect reduces overall genetic diversity because only a portion of the original population's alleles are represented in the new colony.

Natural selection, direct environmental mutation induction, and gene flow fail to explain an allele frequency shift caused by random sampling during reproductive isolation.

Key Concept

Genetic drift and the Founder effect in Modern Evolutionary Theory

Question 7378Question →

If $\theta$ is an acute angle such that $\frac{1 - \tan \theta}{1 + \tan \theta} = 2 - \sqrt{3}$ , what is the exact value of $2\sin \theta + \sqrt{3}\cos \theta$ ?

\frac{5}{2}

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

\frac{\sqrt{5}}{2}

\frac{3}{2}

Show answer & explanation

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

Answer

The exact value of

2\sin \theta + \sqrt{3}\cos \theta

is

\frac{5}{2}

.

Cross-multiplying the equation

\frac{1 - \tan \theta}{1 + \tan \theta} = 2 - \sqrt{3}

yields

\tan \theta = \frac{1}{\sqrt{3}}

, which identifies

\theta

as

30^\circ

. Evaluating

2\sin 30^\circ + \sqrt{3}\cos 30^\circ = 2\left(\frac{1}{2}\right) + \sqrt{3}\left(\frac{\sqrt{3}}{2}\right) = 1 + \frac{3}{2} = \frac{5}{2}

.

Step-by-Step Solution

1

Solve the given equation

\frac{1 - \tan \theta}{1 + \tan \theta} = 2 - \sqrt{3}

for

\tan \theta

.

1 - \tan \theta = (2 - \sqrt{3})(1 + \tan \theta) \implies 1 - \tan \theta = 2 - \sqrt{3} + (2 - \sqrt{3})\tan \theta

. Rearranging terms gives

(\sqrt{3} - 1) = (3 - \sqrt{3})\tan \theta \implies \tan \theta = \frac{\sqrt{3} - 1}{3 - \sqrt{3}} = \frac{\sqrt{3} - 1}{\sqrt{3}(\sqrt{3} - 1)} = \frac{1}{\sqrt{3}}

.

Isolating

\tan \theta

allows determination of the specific angle

\theta

.

2

Determine the acute angle

\theta

corresponding to

\tan \theta = \frac{1}{\sqrt{3}}

.

Since

\theta

is acute and

\tan 30^\circ = \frac{1}{\sqrt{3}}

,

\theta = 30^\circ

.

Special angle identities state that

\tan 30^\circ = \frac{1}{\sqrt{3}}

.

3

Evaluate the target trigonometric expression

2\sin \theta + \sqrt{3}\cos \theta

at

\theta = 30^\circ

.

2\sin 30^\circ + \sqrt{3}\cos 30^\circ = 2\left(\frac{1}{2}\right) + \sqrt{3}\left(\frac{\sqrt{3}}{2}\right) = 1 + \frac{3}{2} = \frac{5}{2}

.

Substituting the exact surd values

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

and

\cos 30^\circ = \frac{\sqrt{3}}{2}

simplifies directly to the final fraction.

Key Concept

Trigonometric ratio deduction using special angles and surd simplification.

Estimated Time:2m 0s

Question 7379Question →

The sum of the first $n$ terms of an arithmetic progression (A.P.) is given by $S_n = 3n^2 + 5n$ . A geometric progression (G.P.) has a first term of $1$ and a common ratio of $2$ . If the $5^{\text{th}}$ term of the A.P. is equal to the $k^{\text{th}}$ term of the G.P., what is the value of $k$ ?

5

6

4

7

Show answer & explanation

Answer: 6

Answer

The value of

k

is 6.

Evaluating the

5^{\text{th}}

term of the A.P. gives

32

. Equating this to the

k^{\text{th}}

term formula of the G.P.,

1 \cdot 2^{k-1} = 32

, leads to

2^{k-1} = 2^5

, which gives

k = 6

.

Step-by-Step Solution

1

Find the general formula for the

n^{\text{th}}

term

T_n

of the A.P. using

S_n

T_n = S_n - S_{n-1} = (3n^2 + 5n) - [3(n-1)^2 + 5(n-1)] = 6n + 2

The

n^{\text{th}}

term of a series is the difference between the sum of the first

n

terms and the sum of the first

n-1

terms.

2

Calculate the

5^{\text{th}}

term of the A.P.

T_5 = 6(5) + 2 = 32

Substitute

n = 5

into the derived expression for

T_n

.

3

Set up the equation for the

k^{\text{th}}

term of the G.P.

G_k = a \cdot r^{k-1} = 1 \cdot 2^{k-1} = 2^{k-1}

The standard formula for the

k^{\text{th}}

term of a G.P. is

G_k = a \cdot r^{k-1}

.

4

Equate

T_5

and

G_k

to solve for

k

2^{k-1} = 32 \implies 2^{k-1} = 2^5 \implies k - 1 = 5 \implies k = 6

Since the bases are equal (

2

), equate the exponents to find

k

.

Key Concept

Relating Arithmetic Progression sum formulas to term values and solving indexed Geometric Progression equations.

Estimated Time:2m 0s

Question 7380Question →

In a technology conference of $60$ software developers, $35$ write code in Python, $28$ write in Java, and $12$ write in both Python and Java. How many developers write in neither of these two languages?

Show answer & explanation

Answer: 9

Answer

The number of developers who write in neither Python nor Java is

9

.

The total number of developers writing in at least one language is given by

35 + 28 - 12 = 51

. Subtracting this from the universal set size of

60

gives

60 - 51 = 9

developers who write in neither language.

Step-by-Step Solution

1

Calculate the number of developers who write in at least one of the languages.

n(P \cup J) = 35 + 28 - 12 = 51

By the principle of inclusion-exclusion for two sets,

n(P \cup J) = n(P) + n(J) - n(P \cap J)

.

2

Subtract from the universal set total to get the complement.

n((P \cup J)') = 60 - 51 = 9

The number of elements outside the union is the universal set total minus the union cardinality.

Key Concept

Two-set inclusion-exclusion and complement cardinality

Estimated Time:1m 0s

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