Geometry and Trigonometry Practice Questions — JAMB UTME — Page 6

Question 101Question →

A rhombus has an area of $120\text{ cm}^2$ . If the length of one of its diagonals exceeds the length of the other diagonal by $14\text{ cm}$ , what is the perimeter of the rhombus in centimeters?

Show answer & explanation

Answer: 52

Answer

The perimeter of the rhombus is 52 cm.

The area of a rhombus is given by A = \frac{1}{2} d_1 d_2. Setting \frac{1}{2} d_1(d_1 + 14) = 120$ yields the quadratic equation d_1^2 + 14d_1 - 240 = 0, which factors to (d_1 - 10)(d_1 + 24) = 0. Taking the positive solution d_1 = 10\text{ cm} gives d_2 = 24\text{ cm}. The diagonals intersect at right angles, dividing the rhombus into four congruent right triangles with legs of 5 cm and 12 cm. The hypotenuse (side length s) is \sqrt{5^2 + 12^2} = 13\text{ cm}. Therefore, the perimeter is 4 \times 13\text{ cm} = 52\text{ cm}.

Step-by-Step Solution

1

Set up the area formula for a rhombus in terms of its diagonals

d_1 \cdot d_2 = 240

The area of a rhombus is given by A = \frac{1}{2} d_1 d_2, so \frac{1}{2} d_1 d_2 = 120.

2

Substitute d_2 = d_1 + 14 into the area equation

d_1^2 + 14d_1 - 240 = 0

The difference between the diagonal lengths is 14 cm.

3

Solve the quadratic equation for d_1

d_1 = 10\text{ cm} \text{ and } d_2 = 24\text{ cm}

Factoring gives (d_1 - 10)(d_1 + 24) = 0. Discarding the negative root yields d_1 = 10 cm.

4

Calculate the side length s using the right-angled triangle formed by the perpendicular bisecting diagonals

s = 13\text{ cm}

s = \sqrt{(\frac{d_1}{2})^2 + (\frac{d_2}{2})^2} = \sqrt{5^2 + 12^2} = \sqrt{169} = 13\text{ cm}.

5

Multiply side length by 4 to get the total perimeter

P = 52\text{ cm}

All four sides of a rhombus are equal, so Perimeter = 4 \times s = 4 \times 13 = 52 cm.

Key Concept

Perimeter and Area of a Rhombus using Diagonals and Pythagorean Theorem

Question 102Question →

A solid metal trophy is formed by mounting a right circular cone of slant height $5\text{ cm}$ on top of a right circular cylinder of height $8\text{ cm}$ . Both the cone and the cylinder share a common base radius of $3\text{ cm}$ . If the trophy is completely melted down and recast to form a solid pyramid with a square base of side length $6\text{ cm}$ , what is the vertical height of the pyramid?

9\pi\text{ cm}

7\pi\text{ cm}

\frac{7\pi}{3}\text{ cm}

3\pi\text{ cm}

Show answer & explanation

Answer: $7\pi\text{ cm}$

Answer

7\pi\text{ cm}

First, find the height of the cone using the Pythagorean theorem:

h_{\text{cone}} = \sqrt{5^2 - 3^2} = 4\text{ cm}

. The volume of the conical section is

\frac{1}{3}\pi (3^2)(4) = 12\pi\text{ cm}^3

, and the volume of the cylindrical section is

\pi (3^2)(8) = 72\pi\text{ cm}^3

. The total volume melted is

12\pi + 72\pi = 84\pi\text{ cm}^3

. For the recast pyramid with square base area

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

, its volume is

\frac{1}{3}(36)h = 12h

. Equating the volumes (

12h = 84\pi

) yields a vertical height of

7\pi\text{ cm}

.

Step-by-Step Solution

1

Calculate the vertical height of the conical top

h_{\text{cone}} = \sqrt{5^2 - 3^2} = 4\text{ cm}

The slant height

l

, radius

r

, and vertical height

h_{\text{cone}}

form a right-angled triangle where

h_{\text{cone}} = \sqrt{l^2 - r^2}

.

2

Compute the total volume of the composite metal solid

V_{\text{total}} = 12\pi + 72\pi = 84\pi\text{ cm}^3

The total volume is the sum of the cone volume

V_{\text{cone}} = \frac{1}{3}\pi (3^2)(4) = 12\pi\text{ cm}^3

and the cylinder volume

V_{\text{cylinder}} = \pi (3^2)(8) = 72\pi\text{ cm}^3

.

3

Equate the total volume to the volume of the square pyramid and solve for its height

h = 7\pi\text{ cm}

The base area of the square pyramid is

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

. Its volume is

V_{\text{pyramid}} = \frac{1}{3}(36)h = 12h

. Setting

12h = 84\pi

gives

h = \frac{84\pi}{12} = 7\pi\text{ cm}

.

Key Concept

Conservation of volume during recasting of 3D solids and combined solid geometry

Estimated Time:2m 0s

Question 103Question →

In a circle of radius $13\text{ cm}$ , two parallel chords $AB$ and $CD$ are drawn on the same side of the center $O$ . If $AB = 24\text{ cm}$ and $CD = 10\text{ cm}$ , calculate the perpendicular distance between the two chords in centimeters.

Show answer & explanation

Answer: 7

Answer

The perpendicular distance between the two chords is

7\text{ cm}

.

The perpendicular line from the center

O

to a chord bisects the chord. Applying the Pythagorean theorem to the right triangles formed by the radius (

13\text{ cm}

) and half-chords (

12\text{ cm}

and

5\text{ cm}

) yields distances of

5\text{ cm}

and

12\text{ cm}

from the center, respectively. Since both parallel chords are on the same side of the center, the distance between them is

12\text{ cm} - 5\text{ cm} = 7\text{ cm}

.

Step-by-Step Solution

1

Find the perpendicular distance from center

O

to chord

AB

d_1 = 5\text{ cm}

A line drawn from the center of a circle perpendicular to a chord bisects the chord. Using the right triangle formed by the radius, half-chord (

12\text{ cm}

), and perpendicular distance:

d_1 = \sqrt{13^2 - 12^2} = 5\text{ cm}

.

2

Find the perpendicular distance from center

O

to chord

CD

d_2 = 12\text{ cm}

Using the perpendicular bisector property for chord

CD

(half-chord is

5\text{ cm}

):

d_2 = \sqrt{13^2 - 5^2} = 12\text{ cm}

.

3

Calculate the distance between the parallel chords

7\text{ cm}

Because both chords are on the same side of the center

O

, the distance between them is the difference of their individual distances from the center:

12\text{ cm} - 5\text{ cm} = 7\text{ cm}

.

Key Concept

Perpendicular from the center of a circle to a chord bisects the chord

Question 104Question →

Two boats depart simultaneously from a port $P$ . Boat $A$ travels along a straight path for $8\text{ km}$ , while Boat $B$ travels along another straight path for $15\text{ km}$ . If the angle between their paths at the port is $60^\circ$ , what is the distance between the two boats in kilometers?

Show answer & explanation

Answer: 13

Answer

The distance between the two boats is 13 km.

The distance between the two boats forms the third side of a triangle where two side lengths (

8\text{ km}

and

15\text{ km}

) and the included angle (

60^\circ

) are known. By the Cosine Rule,

d^2 = 8^2 + 15^2 - 2(8)(15)\cos(60^\circ) = 64 + 225 - 120 = 169

, so

d = \sqrt{169} = 13\text{ km}

.

Step-by-Step Solution

1

Formulate the geometric model

A triangle with two sides of length

8\text{ km}

and

15\text{ km}

, and an included angle of

60^\circ

.

The paths of the two boats and the distance between them form a triangle with two given side lengths and the included angle.

2

Set up the Cosine Rule formula

d^2 = 8^2 + 15^2 - 2(8)(15)\cos(60^\circ)

The Cosine Rule (

c^2 = a^2 + b^2 - 2ab\cos C

) is used when two sides and the included angle are known (SAS configuration).

3

Evaluate the trigonometric term and simplify

d^2 = 64 + 225 - 240(0.5) = 289 - 120 = 169

Since

\cos(60^\circ) = 0.5

, the subtraction term reduces to

120

.

4

Calculate the principal square root

d = 13

Taking the positive square root gives the distance in kilometers.

Key Concept

Applying the Cosine Rule to calculate the unknown side of a triangle given two sides and the included angle (SAS).

Question 105Question →

A trapezium has parallel sides of length $12\text{ cm}$ and $18\text{ cm}$ , and a perpendicular height of $7\text{ cm}$ . What is the area of the trapezium?

105\text{ cm}^2

210\text{ cm}^2

216\text{ cm}^2

37\text{ cm}^2

Show answer & explanation

Answer: $105\text{ cm}^2$

Answer

The area of the trapezium is

105\text{ cm}^2

.

The area of a trapezium is calculated using the formula

\text{Area} = \frac{1}{2}(a + b)h

, where

a

and

b

are the lengths of the parallel sides, and

h

is the perpendicular height. Substituting

a = 12\text{ cm}

,

b = 18\text{ cm}

, and

h = 7\text{ cm}

yields

\text{Area} = \frac{1}{2}(12 + 18) \times 7 = \frac{1}{2}(30) \times 7 = 105\text{ cm}^2

.

Step-by-Step Solution

1

Identify the given dimensions of the trapezium

Parallel sides

a = 12\text{ cm}

,

b = 18\text{ cm}

, and height

h = 7\text{ cm}

.

These are the necessary parameters for the area formula of a trapezium.

2

Sum the lengths of the parallel sides

a + b = 12 + 18 = 30\text{ cm}

.

The area formula requires the sum of the bases.

3

Multiply by the perpendicular height and divide by 2

\text{Area} = \frac{1}{2} \times 30 \times 7 = 15 \times 7 = 105\text{ cm}^2.

Applying the standard formula

\text{Area} = \frac{1}{2}(a + b)h

gives the plane region's total area.

Key Concept

Area of a Trapezium

Question 106Question →

In a right-angled triangle $PQR$ , where $\angle Q = 90^\circ$ and $\tan P = \frac{5}{12}$ , what is the exact value of $\sin P + \cos P$ ?

\frac{17}{13}

\frac{7}{13}

\frac{17}{12}

\frac{12}{5}

Show answer & explanation

Answer: $\frac{17}{13}$

Answer

\frac{17}{13}

Using

\tan P = \frac{5}{12}

, the triangle has opposite side

5

, adjacent side

12

, and hypotenuse

\sqrt{5^2+12^2} = 13

. Therefore,

\sin P = \frac{5}{13}

and

\cos P = \frac{12}{13}

, giving a sum of

\frac{17}{13}

.

Step-by-Step Solution

1

Identify the side lengths of the right triangle

PQR

using the given tangent ratio.

Opposite side to

P = 5

, adjacent side to

P = 12

.

By definition,

\tan P = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{5}{12}

.

2

Calculate the length of the hypotenuse

PR

using the Pythagorean theorem.

Hypotenuse

PR = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13

.

In a right triangle,

\text{hypotenuse}^2 = \text{opposite}^2 + \text{adjacent}^2

.

3

Determine

\sin P

and

\cos P

and compute their sum.

\sin P = \frac{5}{13}

,

\cos P = \frac{12}{13}

, so

\sin P + \cos P = \frac{5}{13} + \frac{12}{13} = \frac{17}{13}

.

\sin P = \frac{\text{Opposite}}{\text{Hypotenuse}}

and

\cos P = \frac{\text{Adjacent}}{\text{Hypotenuse}}

.

Key Concept

Basic Trigonometric Ratios and Pythagorean Triples

Estimated Time:1m 30s

Question 107Question →

A forest ranger station $Y$ is located on a bearing of $115^\circ$ from an observation tower $X$ . What is the bearing of the observation tower $X$ from the forest ranger station $Y$ ?

065^\circ

245^\circ

295^\circ

115^\circ

Show answer & explanation

Answer: $295^\circ$

Answer

295^\circ

The bearing of

Y

from

X

is

115^\circ

. To find the bearing of

X

from

Y

(the back bearing), add

180^\circ

to the forward bearing because

115^\circ < 180^\circ

. Calculating

115^\circ + 180^\circ

yields

295^\circ

.

Step-by-Step Solution

1

Identify the given forward bearing

The bearing of

Y

from

X

is

\theta = 115^\circ

.

This is the initial directional angle measured clockwise from True North at point

X

.

2

Calculate the back bearing of

X

from

Y

Back bearing

= 115^\circ + 180^\circ = 295^\circ

.

Because the forward bearing is less than

180^\circ

, the back bearing is obtained by adding

180^\circ

to find the opposite direction.

Key Concept

Back Bearing Calculation

Question 108Question →

A point $P(x, y)$ moves such that its perpendicular distance from the straight line $L_1: 4x - 3y + 5 = 0$ is equal to its perpendicular distance from the straight line $L_2: 3x + 4y - 10 = 0$ . Which of the following equations represents one of the straight lines constituting the locus of $P$ ?

x - 7y + 15 = 0

7x - y - 5 = 0

x + 7y + 15 = 0

7x + y + 5 = 0

Show answer & explanation

Answer: $x - 7y + 15 = 0$

Answer

The equation

x - 7y + 15 = 0

represents one of the lines constituting the locus.

The locus of a point equidistant from two intersecting straight lines is the pair of angle bisectors between those lines. Setting the perpendicular distance formulas equal yields

4x - 3y + 5 = \pm(3x + 4y - 10)

. Solving the positive branch yields

x - 7y + 15 = 0

, which correctly represents one of the component lines of the locus.

Step-by-Step Solution

1

Write the perpendicular distance formulas from point

P(x, y)

to both given lines.

d_1 = \frac{|4x - 3y + 5|}{\sqrt{4^2 + (-3)^2}} = \frac{|4x - 3y + 5|}{5}

and

d_2 = \frac{|3x + 4y - 10|}{\sqrt{3^2 + 4^2}} = \frac{|3x + 4y - 10|}{5}

.

The locus of points equidistant from two intersecting lines consists of the angle bisectors of the angles between the lines.

2

Set the two perpendicular distances equal to each other.

\frac{|4x - 3y + 5|}{5} = \frac{|3x + 4y - 10|}{5} \implies |4x - 3y + 5| = |3x + 4y - 10|.

Since the point is equidistant from both lines,

d_1 = d_2

.

3

Remove absolute values by considering both positive and negative cases.

4x - 3y + 5 = \pm(3x + 4y - 10)

.

Absolute value equality

|A| = |B|

implies

A = B

or

A = -B

.

4

Evaluate Case 1 (positive sign) to find the first line equation.

4x - 3y + 5 = 3x + 4y - 10 \implies (4x - 3x) + (-3y - 4y) + (5 + 10) = 0 \implies x - 7y + 15 = 0

.

Grouping like terms yields the linear equation for the first angle bisector.

5

Evaluate Case 2 (negative sign) to find the second line equation.

4x - 3y + 5 = -(3x + 4y - 10) \implies 4x - 3y + 5 = -3x - 4y + 10 \implies 7x + y - 5 = 0

.

Grouping like terms yields the linear equation for the second angle bisector.

Key Concept

Locus equidistant from two intersecting lines (Angle Bisectors)

Estimated Time:2m 0s

Question 109Question →

What is the exact numerical value of the trigonometric expression $\frac{6\sin^2 60^\circ - 2\cos^2 45^\circ}{\tan^2 30^\circ + \sec^2 45^\circ}$ ?

Show answer & explanation

Answer: 1.5

Answer

1.5

Substituting the exact special angle values gives a numerator of

6\left(\frac{3}{4}\right) - 2\left(\frac{1}{2}\right) = \frac{7}{2}

and a denominator of

\frac{1}{3} + 2 = \frac{7}{3}

. Dividing

\frac{7}{2}

by

\frac{7}{3}

yields

\frac{3}{2} = 1.5

.

Step-by-Step Solution

1

Substitute the exact values for the trigonometric ratios of the special angles.

\sin 60^\circ = \frac{\sqrt{3}}{2}

,

\cos 45^\circ = \frac{1}{\sqrt{2}}

,

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

, and

\sec 45^\circ = \sqrt{2}

.

Exact surd forms for special angles

30^\circ

,

45^\circ

, and

60^\circ

must be used.

2

Evaluate and simplify the numerator expression

6\sin^2 60^\circ - 2\cos^2 45^\circ

.

6\left(\frac{3}{4}\right) - 2\left(\frac{1}{2}\right) = \frac{9}{2} - 1 = \frac{7}{2}

.

Square each trigonometric ratio first, multiply by the coefficients, and then subtract.

3

Evaluate and simplify the denominator expression

\tan^2 30^\circ + \sec^2 45^\circ

.

\left(\frac{1}{\sqrt{3}}\right)^2 + (\sqrt{2})^2 = \frac{1}{3} + 2 = \frac{7}{3}

.

Square each trigonometric ratio and simplify the sum into a single improper fraction.

4

Divide the numerator result by the denominator result.

\frac{7/2}{7/3} = \frac{7}{2} \times \frac{3}{7} = \frac{3}{2} = 1.5

.

Dividing by a fraction is equivalent to multiplying by its reciprocal.

Key Concept

Evaluation of Trigonometric Expressions using Special Angles

Question 110Question →

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 111Question →

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 112Question →

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 113Question →

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 114Question →

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 115Question →

In $\Delta ABC$ , the lengths of the sides are $a = x\text{ cm}$ , $b = (x + 3)\text{ cm}$ , and $c = (x + 2)\text{ cm}$ . If $\angle C = 60^\circ$ , what is the value of $x$ ?

3

5

7

1

Show answer & explanation

Answer: 5

Answer

5

Applying the Cosine Rule

c^2 = a^2 + b^2 - 2ab\cos C

with

a = x

,

b = x + 3

,

c = x + 2

, and

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

gives

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

. Expanding both sides leads to

x^2 + 4x + 4 = x^2 + 3x + 9

. Subtracting

x^2

and isolating

x

yields

x = 5

.

Step-by-Step Solution

1

Apply the Cosine Rule for side

c

and angle

C

.

c^2 = a^2 + b^2 - 2ab\cos C

The Cosine Rule relates all three sides of a triangle to the cosine of an included angle.

2

Substitute

a = x

,

b = x + 3

,

c = x + 2

, and

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

into the formula.

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

Inserting the given algebraic side lengths and angle allows solving for

x

.

3

Expand both sides of the equation.

x^2 + 4x + 4 = x^2 + (x^2 + 6x + 9) - (x^2 + 3x)

Cancel the factor of

2

with

\frac{1}{2}

and expand

(x+2)^2

and

(x+3)^2

.

4

Simplify the right-hand side.

x^2 + 4x + 4 = x^2 + 3x + 9

Combine like terms:

(x^2 + x^2 - x^2) + (6x - 3x) + 9 = x^2 + 3x + 9

.

5

Subtract

x^2

from both sides and isolate

x

.

4x - 3x = 9 - 4 \implies x = 5

Subtracting

x^2 + 3x + 4

from both sides gives the linear solution

x = 5

.

Key Concept

Solving for unknown algebraic side lengths using the Cosine Rule.

Question 116Question →

A vertical flagpole of height $15\text{ m}$ standing on level ground casts a shadow of length $15\sqrt{3}\text{ m}$ . Calculate the angle of elevation of the sun in degrees.

Show answer & explanation

Answer: 30

Answer

The angle of elevation of the sun is 30 degrees.

The tangent of the angle of elevation is given by the height divided by the shadow length,

\tan\theta = \frac{15}{15\sqrt{3}} = \frac{1}{\sqrt{3}}

, which corresponds to an angle of

30^\circ

.

Step-by-Step Solution

1

Formulate the trigonometric relationship using the right triangle formed by the flagpole, the shadow, and the sunlight ray.

\tan\theta = \frac{\text{height of flagpole}}{\text{length of shadow}} = \frac{15}{15\sqrt{3}}

The tangent of an angle in a right-angled triangle is defined as the ratio of the opposite side to the adjacent side.

2

Simplify the ratio and evaluate the angle.

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

The standard acute angle whose tangent value is

\frac{1}{\sqrt{3}}

is

30^\circ

.

Key Concept

Angle of Elevation and Trigonometric Ratios

Question 117Question →

If $\sin \theta = \frac{4}{5}$ , where $\theta$ is an acute angle, what is the exact value of the expression $\frac{\sec^2 \theta - 1}{\cot \theta + \csc \theta}$ ?

\frac{8}{9}

\frac{16}{27}

\frac{8}{25}

\frac{4}{3}

Show answer & explanation

Answer: $\frac{8}{9}$

Answer

The exact value of the expression is

\frac{8}{9}

.

Given an acute angle

\theta

with

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

, the adjacent side is

3

. This gives

\tan \theta = \frac{4}{3}

,

\cot \theta = \frac{3}{4}

, and

\csc \theta = \frac{5}{4}

. Using the identity

\sec^2 \theta - 1 = \tan^2 \theta

, the numerator is

\left(\frac{4}{3}\right)^2 = \frac{16}{9}

. The denominator

\cot \theta + \csc \theta = \frac{3}{4} + \frac{5}{4} = 2

. Dividing

\frac{16}{9}

by

2

yields

\frac{8}{9}

.

Step-by-Step Solution

1

Determine all required trigonometric ratios for the acute angle

\theta

.

Given

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

, the opposite side is

4

and the hypotenuse is

5

. By the Pythagorean theorem, the adjacent side is

\sqrt{5^2 - 4^2} = 3

. Therefore,

\cos \theta = \frac{3}{5}

,

\tan \theta = \frac{4}{3}

,

\cot \theta = \frac{3}{4}

, and

\csc \theta = \frac{5}{4}

.

Defining the right-triangle side lengths allows direct evaluation of all six trigonometric ratios.

2

Simplify the numerator using standard trigonometric identities.

\sec^2 \theta - 1 = \tan^2 \theta = \left(\frac{4}{3}\right)^2 = \frac{16}{9}

.

Applying the fundamental identity

1 + \tan^2 \theta = \sec^2 \theta

simplifies the numerator.

3

Evaluate the denominator.

\cot \theta + \csc \theta = \frac{3}{4} + \frac{5}{4} = \frac{8}{4} = 2

.

Adding fractional values with a common denominator.

4

Divide the numerator by the denominator.

\frac{16/9}{2} = \frac{8}{9}

.

Dividing the simplified numerator by the simplified denominator yields the final result.

Key Concept

Evaluation of trigonometric expressions using fundamental identities and right-triangle ratio definitions.

Question 118Question →

A composite plane figure is formed by constructing a semicircle externally on one side of a square of side length $14\text{ cm}$ . What is the perimeter of the resulting figure? (Take $\pi = \frac{22}{7}$ )

64\text{ cm}

78\text{ cm}

86\text{ cm}

50\text{ cm}

Show answer & explanation

Answer: $64\text{ cm}$

Answer

64\text{ cm}

The outer perimeter of the composite shape consists of three straight edges of the square and one curved semicircular arc. Three sides of length

14\text{ cm}

yield

42\text{ cm}

. The arc length of a semicircle with radius

7\text{ cm}

is

\pi r = \frac{22}{7} \times 7 = 22\text{ cm}

. Adding these together gives

42\text{ cm} + 22\text{ cm} = 64\text{ cm}

.

Step-by-Step Solution

1

Identify the exposed straight sides of the square contributing to the perimeter.

The square has 4 sides, but 1 side is attached to the semicircle inside the figure. Thus, 3 sides are on the outer boundary:

3 \times 14\text{ cm} = 42\text{ cm}

.

Perimeter only includes the outer boundary of a composite shape.

2

Calculate the radius and arc length of the attached semicircle.

The diameter of the semicircle is equal to the side length of the square (

d = 14\text{ cm}

), so radius

r = 7\text{ cm}

. Semicircular arc length

= \pi r = \frac{22}{7} \times 7 = 22\text{ cm}

.

The curved boundary is half of the total circle circumference.

3

Sum the outer straight boundaries and the curved boundary.

Total perimeter

= 42\text{ cm} + 22\text{ cm} = 64\text{ cm}

.

Combining all outer segment lengths gives the total perimeter.

Key Concept

Perimeter of composite plane figures involving straight line segments and circular arcs.

Estimated Time:1m 30s

Question 119Question →

Given that $\cos x - \sin x = \frac{1}{\sqrt{5}}$ for an acute angle $x$ , what is the exact value of $\cos^3 x - \sin^3 x$ ?

\frac{7\sqrt{5}}{25}

\frac{3\sqrt{5}}{25}

\frac{9\sqrt{5}}{25}

\frac{7\sqrt{5}}{5}

Show answer & explanation

Answer: $\frac{7\sqrt{5}}{25}$

Answer

\frac{7\sqrt{5}}{25}

Squaring

\cos x - \sin x = \frac{1}{\sqrt{5}}

yields

1 - 2\sin x \cos x = \frac{1}{5}

, which gives

\sin x \cos x = \frac{2}{5}

. Using the difference of cubes factorization,

\cos^3 x - \sin^3 x = (\cos x - \sin x)(1 + \sin x \cos x) = \frac{1}{\sqrt{5}} \left(1 + \frac{2}{5}\right) = \frac{7}{5\sqrt{5}}

. Rationalizing the denominator produces

\frac{7\sqrt{5}}{25}

.

Step-by-Step Solution

1

Square both sides of the given equation to find the product

\sin x \cos x

.

(\cos x - \sin x)^2 = \left(\frac{1}{\sqrt{5}}\right)^2 \implies \cos^2 x - 2\sin x \cos x + \sin^2 x = \frac{1}{5}

.

Squaring allows us to use the Pythagorean identity

\sin^2 x + \cos^2 x = 1

to isolate

\sin x \cos x

.

2

Simplify using

\cos^2 x + \sin^2 x = 1

to solve for

\sin x \cos x

.

1 - 2\sin x \cos x = \frac{1}{5} \implies 2\sin x \cos x = 1 - \frac{1}{5} = \frac{4}{5} \implies \sin x \cos x = \frac{2}{5}

.

Finding the product of

\sin x

and

\cos x

is necessary for the algebraic expansion of the difference of cubes.

3

Apply the difference of cubes algebraic identity to

\cos^3 x - \sin^3 x

.

\cos^3 x - \sin^3 x = (\cos x - \sin x)(\cos^2 x + \sin x \cos x + \sin^2 x) = (\cos x - \sin x)(1 + \sin x \cos x)

.

The identity

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

breaks the target expression down into known terms.

4

Substitute the known values into the expression and rationalize the denominator.

\left(\frac{1}{\sqrt{5}}\right) \left(1 + \frac{2}{5}\right) = \frac{1}{\sqrt{5}} \times \frac{7}{5} = \frac{7}{5\sqrt{5}} = \frac{7\sqrt{5}}{25}

.

Evaluating the product and rationalizing

\frac{7}{5\sqrt{5}}

yields the final exact surd form.

Key Concept

Basic Trigonometric Ratios, Special Angles, and Identities

Estimated Time:2m 0s

Question 120Question →

Given that $\theta$ is an acute angle such that $\tan \theta = \frac{4}{3}$ , calculate the numerical value of the expression $\frac{3\sin \theta + 2\cos \theta}{3\sin \theta - \cos \theta}$ .

Show answer & explanation

Answer: 2

Answer

The exact numerical value of the given expression is 2.

Dividing both the numerator and denominator of

\frac{3\sin \theta + 2\cos \theta}{3\sin \theta - \cos \theta}

by

\cos \theta

gives

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

. Substituting

\tan \theta = \frac{4}{3}

yields

\frac{3(4/3) + 2}{3(4/3) - 1} = \frac{4 + 2}{4 - 1} = \frac{6}{3} = 2

.

Step-by-Step Solution

1

Express sine and cosine terms in terms of tangent or find individual ratio values

Divide every term in the numerator and denominator by

\cos \theta

to obtain

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

. Alternatively, using a right triangle with opposite side = 4 and adjacent side = 3 gives hypotenuse = 5, so

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

and

\cos \theta = \frac{3}{5}

.

Converting to

\tan \theta

simplifies the calculation directly without evaluating square roots or hypotenuse.

2

Substitute the value of

\tan \theta = \frac{4}{3}

into the expression

Numerator:

3\left(\frac{4}{3}\right) + 2 = 4 + 2 = 6

. Denominator:

3\left(\frac{4}{3}\right) - 1 = 4 - 1 = 3

.

Simplifies numerical fractions in both parts of the fraction.

3

Divide numerator by denominator

\frac{6}{3} = 2.

Yields the final integer solution.

Key Concept

Basic Trigonometric Ratios and Quotients

Geometry and Trigonometry

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