Geometry and Trigonometry Practice Questions — JAMB UTME — Page 5

Question 81Question →

In a convex polygon of $n$ sides, three of its interior angles are right angles, and each of the remaining interior angles is equal to $150^\circ$ . What is the total number of diagonals in this polygon?

9

15

6

18

Show answer & explanation

Answer: 9

Answer

9 diagonals

The total sum of interior angles of an

n

-sided polygon is

(n - 2) \times 180^\circ

. Setting this equal to

3(90^\circ) + (n - 3)(150^\circ)

simplifies to

180n - 360 = 150n - 180

, which solves to

n = 6

. Substituting

n = 6

into the diagonal formula

D = \frac{n(n - 3)}{2}

yields

\frac{6 \times 3}{2} = 9

diagonals.

Step-by-Step Solution

1

Formulate the interior angle sum equation for the

n

-sided convex polygon.

The total interior angle sum is

(n - 2) \times 180^\circ

. Summing the individual angles gives

3 \times 90^\circ + (n - 3) \times 150^\circ

. Thus:

(n - 2) \times 180^\circ = 270^\circ + (n - 3) \times 150^\circ

.

The sum of all interior angles of any

n

-sided polygon is given by

(n - 2) \times 180^\circ

.

2

Solve the equation for the number of sides

n

.

180n - 360 = 270 + 150n - 450 \implies 180n - 360 = 150n - 180 \implies 30n = 180 \implies n = 6

.

Expanding terms and isolating

n

reveals that the polygon has 6 sides (a hexagon).

3

Calculate the total number of diagonals using the formula

D = \frac{n(n - 3)}{2}

.

D = \frac{6(6 - 3)}{2} = \frac{6 \times 3}{2} = 9

.

Each of the

n

vertices can connect to

n - 3

non-adjacent vertices, and dividing by 2 avoids double-counting each diagonal line segment.

Key Concept

Polygon interior angle sum theorem and diagonal calculation

Question 82Question →

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

Show answer & explanation

Answer: 1.75

Answer

The exact numerical value of the trigonometric expression is 1.75.

Dividing every term in the expression by

\cos \theta

converts

\sin \theta

terms into

\tan \theta

. The quotient becomes

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

. Substituting

\tan \theta = 2

gives

\frac{3(2) + 1}{2 + 2} = \frac{7}{4} = 1.75

.

Step-by-Step Solution

1

Divide numerator and denominator by

\cos \theta

The expression becomes

\frac{3\left(\frac{\sin \theta}{\cos \theta}\right) + 1}{\frac{\sin \theta}{\cos \theta} + 2} = \frac{3\tan \theta + 1}{\tan \theta + 2}

.

Using the trigonometric identity

\tan \theta = \frac{\sin \theta}{\cos \theta}

simplifies the calculation without needing to evaluate the hypotenuse.

2

Substitute

\tan \theta = 2

\frac{3(2) + 1}{2 + 2} = \frac{7}{4}

Replace

\tan \theta

with its given numerical value of 2.

3

Convert fraction to decimal

1.75

Decimal representation of the exact fraction

\frac{7}{4}

.

Key Concept

Basic Trigonometric Ratios and Quotient Identity

Question 83Question →

A point $P(x, y)$ moves in a Cartesian plane such that the sum of the squares of its distances from two fixed points $A(0, 0)$ and $B(8, 0)$ is equal to $82$ , defining a locus $L_1$ . A second locus $L_2$ is the set of all points equidistant from the parallel lines $y = -1$ and $y = 7$ . Given that $L_1$ and $L_2$ intersect at two distinct points $M$ and $N$ , what is the length of the line segment $MN$ ?

Show answer & explanation

Answer: 8

Answer

The length of the line segment

MN

is 8 units.

The locus

L_1

simplifies to the circle

(x - 4)^2 + y^2 = 25

with center

(4,0)

and radius

5

. The locus

L_2

is the horizontal line

y = 3

. Substituting

y = 3

into the circle equation yields

(x - 4)^2 = 16

, giving intersection points at

(0,3)

and

(8,3)

. The distance between these points is

8

units.

Step-by-Step Solution

1

Formulate the algebraic expression for locus

L_1

The distance squared

PA^2 = x^2 + y^2

and

PB^2 = (x - 8)^2 + y^2

. Setting

PA^2 + PB^2 = 82

gives

x^2 + y^2 + (x - 8)^2 + y^2 = 82

.

By the distance formula, the sum of squared distances from

P(x, y)

to

A(0,0)

and

B(8,0)

equals 82.

2

Simplify and complete the square for locus

L_1

Expanding yields

2x^2 - 16x + 64 + 2y^2 = 82 \implies 2x^2 - 16x + 2y^2 = 18 \implies x^2 - 8x + y^2 = 9

. Completing the square gives

(x - 4)^2 + y^2 = 25

.

This shows

L_1

is a circle with center

(4, 0)

and radius

R = 5

.

3

Find the equation for locus

L_2

The locus of points equidistant from two parallel lines

y = -1

and

y = 7

is the line

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

.

The locus equidistant from two parallel lines is a parallel line running midway between them.

4

Find the intersection points

M

and

N

Substitute

y = 3

into

(x - 4)^2 + y^2 = 25

:

(x - 4)^2 + 3^2 = 25 \implies (x - 4)^2 = 16 \implies x - 4 = \pm 4

, so

x = 8

or

x = 0

. Thus,

M = (0, 3)

and

N = (8, 3)

.

Solving the circle and line equations simultaneously gives the

x

-coordinates of the intersection points.

5

Compute the distance between

M

and

N

Since both points share the ordinate

y = 3

, distance

MN = 8 - 0 = 8

.

The length of a horizontal segment is the difference between its endpoints'

x

-coordinates.

Key Concept

Intersection of loci involving circles and parallel lines

Question 84Question →

A point $P(x, y)$ moves such that it is equidistant from two parallel lines $3x - 4y + 11 = 0$ and $3x - 4y - 1 = 0$ , defining locus $L_1$ . A second locus $L_2$ consists of all points that are at a constant distance of $5$ units from the fixed point $(1, 7)$ . Calculate the distance between the two points of intersection of locus $L_1$ and locus $L_2$ .

Show answer & explanation

Answer: 6

Answer

The distance between the two points of intersection of locus L1 and locus L2 is 6 units.

Locus L1 is the line midway between the two given parallel lines, having the equation 3x - 4y + 5 = 0. Locus L2 is a circle centered at (1, 7) with radius 5. The perpendicular distance from the center (1, 7) to line L1 is 4 units. Applying Pythagoras' theorem to the right triangle formed by the radius, perpendicular distance, and half-chord gives a half-chord length of sqrt(5^2 - 4^2) = 3. Therefore, the distance between the two intersection points (the full chord length) is 2 * 3 = 6 units.

Step-by-Step Solution

1

Determine the equation of locus L1

3x - 4y + 5 = 0

The locus of points equidistant from two parallel lines ax + by + c1 = 0 and ax + by + c2 = 0 is a line parallel to both and midway between them, given by ax + by + (c1 + c2)/2 = 0.

2

Identify the geometric representation of locus L2

A circle centered at (1, 7) with radius r = 5

The locus of points at a constant distance r from a fixed point (x0, y0) is a circle.

3

Calculate the perpendicular distance d from the circle's center (1, 7) to line L1

d = 4 units

Using the perpendicular distance formula d = |ax0 + by0 + c| / sqrt(a^2 + b^2), we get d = |3(1) - 4(7) + 5| / sqrt(3^2 + (-4)^2) = |-20| / 5 = 4.

4

Compute the total distance between the two intersection points

6 units

The line intersects the circle along a chord. Using Pythagoras' theorem, half the chord length is sqrt(r^2 - d^2) = sqrt(5^2 - 4^2) = 3. Multiplying by 2 gives the full distance of 6 units.

Key Concept

Intersection of Loci (Parallel Line Bisector and Circle)

Estimated Time:2m 30s

Question 85Question →

The measures of the exterior angles of an convex hexagon are given as $x^\circ$ , $(x + 10)^\circ$ , $(2x - 5)^\circ$ , $(x + 25)^\circ$ , $(2x + 15)^\circ$ , and $(x - 5)^\circ$ . What is the measure of the largest interior angle of the hexagon?

95^\circ

85^\circ

145^\circ

135^\circ

Show answer & explanation

Answer: $145^\circ$

Answer

145^\circ

The sum of all exterior angles of a convex polygon is

360^\circ

. Summing the given expressions yields

8x + 40 = 360^\circ

, which gives

x = 40^\circ

. Evaluating each exterior angle shows that the smallest exterior angle is

(40 - 5)^\circ = 35^\circ

. Since interior and exterior angles on a straight line are supplementary, the largest interior angle is

180^\circ - 35^\circ = 145^\circ

.

Step-by-Step Solution

1

Set up equation using the sum of exterior angles

x + (x + 10) + (2x - 5) + (x + 25) + (2x + 15) + (x - 5) = 360^\circ

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

360^\circ

.

2

Simplify and solve for

x

8x + 40 = 360 \implies 8x = 320 \implies x = 40^\circ

Combining like terms gives

8x + 40 = 360

.

3

Find the smallest exterior angle

x - 5 = 40 - 5 = 35^\circ

The largest interior angle corresponds to the smallest exterior angle because an interior angle and its adjacent exterior angle add up to

180^\circ

.

4

Calculate the largest interior angle

180^\circ - 35^\circ = 145^\circ

Subtracting the smallest exterior angle from

180^\circ

yields the largest interior angle.

Key Concept

Exterior and Interior Angle Relationship in Polygons

Estimated Time:1m 30s

Question 86Question →

Given that $\sin \theta + \cos \theta = \frac{\sqrt{6}}{2}$ where $\theta$ is an acute angle, what is the exact value of $\tan \theta + \cot \theta$ ?

4

2

\frac{2}{3}

\frac{1}{2}

Show answer & explanation

Answer: 4

Answer

The exact value of

\tan \theta + \cot \theta

is 4.

By squaring both sides of

\sin \theta + \cos \theta = \frac{\sqrt{6}}{2}

, we obtain

\sin^2 \theta + \cos^2 \theta + 2\sin \theta \cos \theta = \frac{6}{4} = \frac{3}{2}

. Applying the Pythagorean identity

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

simplifies this to

1 + 2\sin \theta \cos \theta = \frac{3}{2}

, which yields

\sin \theta \cos \theta = \frac{1}{4}

. Since

\tan \theta + \cot \theta = \frac{\sin \theta}{\cos \theta} + \frac{\cos \theta}{\sin \theta} = \frac{\sin^2 \theta + \cos^2 \theta}{\sin \theta \cos \theta} = \frac{1}{\sin \theta \cos \theta}

, the exact value is

\frac{1}{1/4} = 4

.

Step-by-Step Solution

1

Square both sides of the given equation

\sin \theta + \cos \theta = \frac{\sqrt{6}}{2}

.

(\sin \theta + \cos \theta)^2 = \left(\frac{\sqrt{6}}{2}\right)^2 \implies \sin^2 \theta + 2\sin \theta \cos \theta + \cos^2 \theta = \frac{6}{4} = \frac{3}{2}

.

Squaring allows the application of the fundamental Pythagorean identity

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

to isolate the product

\sin \theta \cos \theta

.

2

Substitute

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

into the equation and solve for

\sin \theta \cos \theta

.

1 + 2\sin \theta \cos \theta = \frac{3}{2} \implies 2\sin \theta \cos \theta = \frac{1}{2} \implies \sin \theta \cos \theta = \frac{1}{4}

.

Simplifying the algebraic equation isolates the product term.

3

Express

\tan \theta + \cot \theta

in terms of

\sin \theta

and

\cos \theta

.

\tan \theta + \cot \theta = \frac{\sin \theta}{\cos \theta} + \frac{\cos \theta}{\sin \theta} = \frac{\sin^2 \theta + \cos^2 \theta}{\sin \theta \cos \theta} = \frac{1}{\sin \theta \cos \theta}

.

Using basic trigonometric identities simplifies the target sum into a reciprocal product.

4

Substitute

\sin \theta \cos \theta = \frac{1}{4}

into the simplified expression.

\tan \theta + \cot \theta = \frac{1}{1/4} = 4

.

Evaluating the reciprocal fraction gives the final numerical answer.

Key Concept

Pythagorean identity

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

and reciprocal relation

\tan \theta + \cot \theta = \frac{1}{\sin \theta \cos \theta}

.

Question 87Question →

A solid hemisphere of radius $6\text{ cm}$ has the same total surface area as a solid right circular cone with a base radius of $6\text{ cm}$ . What is the slant height of the cone?

6\text{ cm}

12\text{ cm}

18\text{ cm}

24\text{ cm}

Show answer & explanation

Answer: $12\text{ cm}$

Answer

The slant height of the cone is

12\text{ cm}

.

The correct answer is derived by setting the total surface area of the solid hemisphere (

3\pi R^2 = 108\pi\text{ cm}^2

) equal to the total surface area of the solid cone (

\pi r^2 + \pi r l = 36\pi + 6\pi l

). Solving

36\pi + 6\pi l = 108\pi

gives

l = 12\text{ cm}

.

Step-by-Step Solution

1

Calculate the total surface area of the solid hemisphere.

\text{TSA}_{\text{hemisphere}} = 3\pi R^2 = 3\pi(6)^2 = 108\pi\text{ cm}^2

A solid hemisphere consists of a curved surface (

2\pi R^2

) plus its flat circular base (

\pi R^2

).

2

Express the total surface area of the solid cone in terms of slant height

l

.

\text{TSA}_{\text{cone}} = \pi r^2 + \pi r l = \pi(6)^2 + \pi(6)l = 36\pi + 6\pi l

A solid cone has a base area of

\pi r^2

and a curved surface area of

\pi r l

.

3

Equate the two surface area expressions and solve for

l

.

36\pi + 6\pi l = 108\pi \implies 6\pi l = 72\pi \implies l = 12\text{ cm}

The question states that the total surface area of both solids is equal.

Key Concept

Total Surface Area of Composite 3D Solids (Hemisphere and Cone)

Question 88Question →

A solid right circular cylinder has a height of $10\text{ cm}$ and a total surface area of $192\pi\text{ cm}^2$ . What is the volume of the cylinder in $\text{cm}^3$ expressed in terms of $\pi$ ?

Show answer & explanation

Answer: 360

Answer

360

The total surface area of a closed cylinder is

A = 2\pi r^2 + 2\pi r h

. Given

A = 192\pi

and

h = 10\text{ cm}

, we have

2\pi r^2 + 20\pi r = 192\pi

. Dividing by

2\pi

yields

r^2 + 10r - 96 = 0

. Factoring the quadratic gives

(r + 16)(r - 6) = 0

, so

r = 6\text{ cm}

(since radius must be positive). Substituting

r = 6\text{ cm}

and

h = 10\text{ cm}

into the volume formula

V = \pi r^2 h

yields

V = \pi (6^2)(10) = 360\pi\text{ cm}^3

. The numerical coefficient of

\pi

is

360

.

Step-by-Step Solution

1

Set up the equation for the total surface area of the cylinder

2\pi r^2 + 2\pi r(10) = 192\pi

The total surface area of a solid cylinder comprises the areas of the top and bottom circular bases (

2\pi r^2

) plus the curved surface area (

2\pi r h

).

2

Simplify and solve the resulting quadratic equation for radius

r

r^2 + 10r - 96 = 0 \implies (r + 16)(r - 6) = 0 \implies r = 6\text{ cm}

Dividing the equation by

2\pi

yields

r^2 + 10r - 96 = 0

. Since radius must be positive,

r = 6\text{ cm}

.

3

Calculate the volume of the cylinder using the radius and height

V = \pi r^2 h = \pi (6)^2 (10) = 360\pi\text{ cm}^3

The volume of a cylinder is calculated using the formula

V = \pi r^2 h

.

Key Concept

Surface Area and Volume of 3D Solids

Question 89Question →

A piece of wire of length $88\text{ cm}$ is bent to form the perimeter of a sector of a circle of radius $21\text{ cm}$ . What is the area of the sector in $\text{cm}^2$ ?

Show answer & explanation

Answer: 483

Answer

The area of the sector is

483\text{ cm}^2

.

The total length of the wire forms the perimeter of the sector, which consists of two radii and the arc length (

P = 2r + l

). With

P = 88\text{ cm}

and

r = 21\text{ cm}

, the arc length is

l = 88 - 2(21) = 46\text{ cm}

. Using the sector area formula

A = \frac{1}{2} r l

, the area is

\frac{1}{2} \times 21 \times 46 = 483\text{ cm}^2

.

Step-by-Step Solution

1

Determine the arc length of the sector from the given perimeter

l = 46\text{ cm}

The total perimeter of a sector includes two straight radii and the curved arc length:

P = 2r + l

. Subtracting twice the radius (

2 \times 21 = 42\text{ cm}

) from the total wire length of

88\text{ cm}

gives an arc length of

46\text{ cm}

.

2

Calculate the area of the sector

A = 483\text{ cm}^2

The area of a sector can be computed using the formula

A = \frac{1}{2} r l

. Substituting

r = 21\text{ cm}

and

l = 46\text{ cm}

yields

A = \frac{1}{2} \times 21 \times 46 = 483\text{ cm}^2

.

Key Concept

Perimeter and Area of a Sector of a Circle

Question 90Question →

A solid wooden block is shaped as a frustum of a right circular cone with a top base radius of $4\text{ cm}$ , a bottom base radius of $12\text{ cm}$ , and a vertical height of $15\text{ cm}$ . A cylindrical hole of radius $4\text{ cm}$ is drilled vertically through the center of the frustum from the top base straight down to the bottom base. What is the volume of the remaining wooden solid in $\text{cm}^3$ ?

800\pi\text{ cm}^3

2880\pi\text{ cm}^3

560\pi\text{ cm}^3

80\pi\text{ cm}^3

Show answer & explanation

Answer: $800\pi\text{ cm}^3$

Answer

The volume of the remaining wooden solid is

800\pi\text{ cm}^3

.

The volume of a frustum of a cone with radii

R=12\text{ cm}

,

r=4\text{ cm}

, and height

h=15\text{ cm}

is

\frac{1}{3}\pi(15)(12^2 + 4^2 + 12 \times 4) = 1040\pi\text{ cm}^3

. Subtracting the volume of the central cylindrical hole of radius

4\text{ cm}

and height

15\text{ cm}

, which is

\pi(4^2)(15) = 240\pi\text{ cm}^3

, yields

1040\pi - 240\pi = 800\pi\text{ cm}^3

.

Step-by-Step Solution

1

Calculate the total volume of the conical frustum before drilling.

V_{\text{frustum}} = \frac{1}{3}\pi h (R^2 + r^2 + R r) = \frac{1}{3}\pi (15)(12^2 + 4^2 + 12 \times 4) = 5\pi(144 + 16 + 48) = 1040\pi\text{ cm}^3

.

The total volume of a frustum of a cone is determined by its vertical height and the radii of its top and bottom circular bases.

2

Calculate the volume of the cylindrical hole drilled through the solid.

V_{\text{cylinder}} = \pi r_{\text{hole}}^2 h = \pi (4^2)(15) = 240\pi\text{ cm}^3

.

The drilled hole forms a right circular cylinder of radius

4\text{ cm}

and height equal to the full height of the frustum (

15\text{ cm}

).

3

Subtract the cylinder's volume from the frustum's volume to find the remaining volume.

V_{\text{remaining}} = V_{\text{frustum}} - V_{\text{cylinder}} = 1040\pi - 240\pi = 800\pi\text{ cm}^3

.

Removing material by drilling decreases the overall volume of the original solid by the exact volume of the cylindrical bore.

Key Concept

Volume of Composite Solids and Conical Frustums

Estimated Time:2m 30s

Question 91Question →

Each exterior angle of a regular polygon measures $24^\circ$ . How many sides does this polygon have?

Show answer & explanation

Answer: 15

Answer

The polygon has 15 sides.

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

360^\circ

. Because the polygon is regular, all its exterior angles are equal. Dividing

360^\circ

by the measure of one exterior angle (

24^\circ

) yields exactly 15 sides.

Step-by-Step Solution

1

Identify the formula relating exterior angle and number of sides

The sum of exterior angles of any convex polygon is

360^\circ

. For a regular polygon with

n

sides, each exterior angle equals

\frac{360^\circ}{n}

.

All exterior angles in a regular polygon are equal.

2

Rearrange the formula to solve for the number of sides

n

n = \frac{360^\circ}{\text{Exterior Angle}} = \frac{360^\circ}{24^\circ} = 15

.

Dividing the total sum of exterior angles by the measure of one exterior angle yields the total number of sides.

Key Concept

Exterior angle property of regular polygons

Question 92Question →

The ratio of the measure of each interior angle to each exterior angle of a regular convex polygon is $7 : 2$ . What is the total number of diagonals of this polygon?

27

9

36

54

Show answer & explanation

Answer: 27

Answer

The total number of diagonals of the polygon is 27.

Since interior and exterior angles are supplementary, their ratio

7 : 2

divides

180^\circ

into 9 equal parts of

20^\circ

. The exterior angle is

2 \times 20^\circ = 40^\circ

. The number of sides

n

is

360^\circ / 40^\circ = 9

. Using the formula for diagonals

D = \frac{n(n-3)}{2}

, we obtain

D = \frac{9 \times 6}{2} = 27

.

Step-by-Step Solution

1

Calculate the measure of the exterior angle

Exterior angle

= \frac{2}{7 + 2} \times 180^\circ = 40^\circ

Interior and exterior angles at a vertex are supplementary, summing to

180^\circ

.

2

Determine the number of sides

n

n = \frac{360^\circ}{40^\circ} = 9

The sum of exterior angles of any convex polygon is

360^\circ

, so

n = \frac{360^\circ}{\text{exterior angle}}

.

3

Calculate the number of diagonals using

D = \frac{n(n - 3)}{2}

D = \frac{9(9 - 3)}{2} = \frac{9 \times 6}{2} = 27

Each vertex connects to

n - 3

non-adjacent vertices, and dividing by 2 avoids double-counting.

Key Concept

Interior and exterior angles of regular polygons and the polygon diagonal count formula.

Estimated Time:1m 30s

Question 93Question →

A composite plane figure is formed from a rectangle $ABCD$ measuring $20\text{ cm}$ by $14\text{ cm}$ . A semicircle with diameter $BC = 14\text{ cm}$ is attached externally along side $BC$ . At the opposite end, an isosceles triangle $AED$ with base $AD = 14\text{ cm}$ and perpendicular height $6\text{ cm}$ is cut out from the interior of the rectangle. Taking $\pi = \frac{22}{7}$ , what is the total area of the figure in $\text{cm}^2$ ?

Show answer & explanation

Answer: 315

Answer

The total area of the composite figure is 315 cm².

The area of the complete shape is found by adding the area of the external semicircle (

77\text{ cm}^2

) to the area of the rectangle (

280\text{ cm}^2

) and subtracting the area of the removed triangle (

42\text{ cm}^2

), yielding a net area of

315\text{ cm}^2

.

Step-by-Step Solution

1

Calculate the area of the rectangular portion

ABCD

Area of rectangle

= 20\text{ cm} \times 14\text{ cm} = 280\text{ cm}^2

The main section of the figure before modifications is a rectangle with length

20\text{ cm}

and width

14\text{ cm}

.

2

Calculate the area of the externally attached semicircle

Area of semicircle

= \frac{1}{2} \times \frac{22}{7} \times 7^2 = 77\text{ cm}^2

The diameter of the semicircle is

BC = 14\text{ cm}

, giving a radius

r = 7\text{ cm}

. Area is given by

\frac{1}{2}\pi r^2

.

3

Calculate the area of the removed internal triangle

AED

Area of triangle

= \frac{1}{2} \times 14\text{ cm} \times 6\text{ cm} = 42\text{ cm}^2

The triangle has base

AD = 14\text{ cm}

and height

6\text{ cm}

. Area is calculated as

\frac{1}{2} \times \text{base} \times \text{height}

.

4

Combine the areas to find the total remaining area of the figure

Total Area

= 280\text{ cm}^2 + 77\text{ cm}^2 - 42\text{ cm}^2 = 315\text{ cm}^2

The external semicircle increases the total area while the internal cut-out triangle reduces it.

Key Concept

Area of Composite Plane Figures

Estimated Time:2m 0s

Question 94Question →

Two parallel lines, $L_1$ and $L_2$ , are situated in a plane with $L_1$ above $L_2$ . A regular polygon of $n$ sides has one of its sides, $AB$ , lying entirely on line $L_1$ . An adjacent side, $BC$ , extends downwards into the region between $L_1$ and $L_2$ . A line segment $BD$ is drawn from vertex $B$ perpendicular to line $L_2$ , meeting $L_2$ at point $D$ . The segment $BD$ lies inside the interior angle $\angle ABC$ of the polygon and divides $\angle ABC$ into two angles, $\angle ABD$ and $\angle DBC$ , such that $\angle ABD : \angle DBC = 3 : 2$ . What is the total number of diagonals of this regular polygon?

Show answer & explanation

Answer: 54

Answer

54

Because line L_1 is parallel to line L_2 and BD is perpendicular to L_2, BD is also perpendicular to L_1, making angle ABD equal to 90°. Based on the 3:2 ratio, angle DBC equals 60°, giving a total interior angle of 150°. Solving (n - 2) × 180° / n = 150° yields n = 12 sides. Substituting n = 12 into the diagonal formula n(n - 3) / 2 gives 54 diagonals.

Step-by-Step Solution

1

Determine the measure of angle ABD using parallel line properties

\angle ABD = 90^\circ

Because line L_1 is parallel to line L_2 and line segment BD is perpendicular to L_2, BD is also perpendicular to L_1.

2

Calculate the measure of angle DBC from the given angle ratio

\angle DBC = 60^\circ

With \angle ABD : \angle DBC = 3 : 2 and \angle ABD = 90^\circ, setting 3 ratio units equal to 90^\circ gives 1 unit = 30^\circ, so 2 units = 60^\circ.

3

Find the total interior angle measure of the regular polygon

\angle ABC = 150^\circ

The interior angle \angle ABC is the sum of \angle ABD and \angle DBC, which gives 90^\circ + 60^\circ = 150^\circ.

4

Determine the number of sides n of the regular polygon

n = 12

Using the regular polygon interior angle formula \frac{(n - 2) \times 180^\circ}{n} = 150^\circ, solving for n yields 180n - 360 = 150n \implies 30n = 360 \implies n = 12.

5

Calculate the total number of diagonals

54

Applying the polygon diagonal formula D = \frac{n(n - 3)}{2} with n = 12 gives D = \frac{12 \times (12 - 3)}{2} = \frac{12 \times 9}{2} = 54.

Key Concept

Combining parallel line perpendicularity properties, angle partitioning, regular polygon interior angle measure, and diagonal count calculation

Question 95Question →

A point $P(x, y)$ moves in the Cartesian plane such that its distance from the origin $O(0, 0)$ is always half of its distance from the fixed point $Q(6, 0)$ . Which of the following equations represents the locus of $P$ ?

x^2 + y^2 + 4x - 12 = 0

x^2 + y^2 - 4x - 12 = 0

x^2 + y^2 + 12x - 36 = 0

x^2 + y^2 - 16x + 48 = 0

Show answer & explanation

Answer: $x^2 + y^2 + 4x - 12 = 0$

Answer

The equation of the locus of

P

is

x^2 + y^2 + 4x - 12 = 0

.

The correct equation

x^2 + y^2 + 4x - 12 = 0

is derived by expressing the condition

PO = \frac{1}{2} PQ

as

2 \cdot PO = PQ

, squaring both sides to get

4(x^2 + y^2) = (x - 6)^2 + y^2

, and simplifying to standard circle form.

Step-by-Step Solution

1

Express the distance condition algebraically.

The distance from

P(x,y)

to

O(0,0)

is

PO = \sqrt{x^2 + y^2}

, and the distance from

P(x,y)

to

Q(6,0)

is

PQ = \sqrt{(x-6)^2 + y^2}

. Given

PO = \frac{1}{2} PQ

, we have

2 \cdot PO = PQ

.

Translate the geometric distance description into algebraic expressions.

2

Square both sides of the equation to eliminate square roots.

4(x^2 + y^2) = (x - 6)^2 + y^2

.

Squaring removes the radicals; note that

(2 \cdot PO)^2 = 4 \cdot PO^2

.

3

Expand and group like terms.

4x^2 + 4y^2 = x^2 - 12x + 36 + y^2 \implies 3x^2 + 12x + 3y^2 - 36 = 0

.

Expand

(x - 6)^2 = x^2 - 12x + 36

and collect terms on one side.

4

Divide the entire equation by 3 to standard form.

x^2 + y^2 + 4x - 12 = 0

.

Simplifying by the common factor of 3 gives the equation of a circle representing the Circle of Apollonius.

Key Concept

Locus of a point with a constant ratio of distances from two fixed points (Circle of Apollonius)

Estimated Time:2m 0s

Question 96Question →

Given that $\tan \theta + \cot \theta = 4$ for an acute angle $\theta$ , what is the exact decimal value of $\sin^4 \theta + \cos^4 \theta$ ?

Show answer & explanation

Answer: 0.875

Answer

The exact decimal value of

\sin^4 \theta + \cos^4 \theta

is 0.875.

By writing

\tan \theta + \cot \theta

as

\frac{\sin \theta}{\cos \theta} + \frac{\cos \theta}{\sin \theta} = \frac{\sin^2 \theta + \cos^2 \theta}{\sin \theta \cos \theta} = 4

, we find that

\sin \theta \cos \theta = \frac{1}{4}

. Squaring the fundamental identity

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

gives

\sin^4 \theta + 2\sin^2 \theta \cos^2 \theta + \cos^4 \theta = 1

. Isolating

\sin^4 \theta + \cos^4 \theta

yields

1 - 2(\sin \theta \cos \theta)^2 = 1 - 2\left(\frac{1}{16}\right) = 1 - \frac{1}{8} = \frac{7}{8} = 0.875

.

Step-by-Step Solution

1

Rewrite the expression

\tan \theta + \cot \theta = 4

using sine and cosine ratios

\frac{\sin \theta}{\cos \theta} + \frac{\cos \theta}{\sin \theta} = 4

Applies the fundamental quotient identities for tangent and cotangent.

2

Combine fractions over a common denominator and apply the Pythagorean identity

\frac{\sin^2 \theta + \cos^2 \theta}{\sin \theta \cos \theta} = \frac{1}{\sin \theta \cos \theta} = 4 \implies \sin \theta \cos \theta = \frac{1}{4}

Utilizes the identity

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

.

3

Relate

\sin^4 \theta + \cos^4 \theta

to

(\sin^2 \theta + \cos^2 \theta)^2

\sin^4 \theta + \cos^4 \theta = (\sin^2 \theta + \cos^2 \theta)^2 - 2\sin^2 \theta \cos^2 \theta = 1 - 2(\sin \theta \cos \theta)^2

Uses the algebraic identity

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

where

a = \sin^2 \theta

and

b = \cos^2 \theta

.

4

Substitute

\sin \theta \cos \theta = \frac{1}{4}

into the algebraic relation

1 - 2\left(\frac{1}{4}\right)^2 = 1 - 2\left(\frac{1}{16}\right) = 1 - \frac{1}{8} = \frac{7}{8} = 0.875

Evaluates the expression to obtain the final decimal result.

Key Concept

Trigonometric Identities and Algebraic Polynomial Expansion

Estimated Time:2m 0s

Question 97Question →

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 98Question →

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 99Question →

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 100Question →

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

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