Unified Tertiary Matriculation Examination · Mathematics

Geometry, Trigonometry and Coordinate Geometry

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The interior angles of a pentagon are $x^{\circ}, (x + 20)^{\circ}, (x + 40)^{\circ}, (x + 60)^{\circ}$ , and $(x + 80)^{\circ}$ . Find the value of $x$ .
An arc subtends an angle of $13 0^{\circ}$ at the center of a circle. What is the angle subtended by the same arc at any point on the major circumference?
In a cyclic quadrilateral $A B C D$ , opposite angles are $∠ A = (2 x + 10)^{\circ}$ and $∠ C = (3 x - 15)^{\circ}$ . Find the value of $x$ .
A chord of length 16 cm is at a distance of 6 cm from the center of a circle. What is the radius of the circle?
A vertical communication mast casts a horizontal shadow of $153 m$ when the angle of elevation of the sun is $3 0^{\circ}$ . Find the height of the mast.
From the top of a vertical coastal cliff 60 m high, the angle of depression of a stationary fishing canoe is $4 5^{\circ}$ . What is the horizontal distance from the base of the cliff to the canoe?
A patrol vessel navigates on a three-figure bearing of $12 0^{\circ}$ . What is the back bearing to its port of departure?
A surveyor walks 8 km due North and then 6 km due East. What is his three-figure bearing from the starting point to the nearest degree?
Calculate the straight-line distance between the points $P (2, - 3)$ and $Q (- 4, 5)$ in the Cartesian plane.
Find the coordinates of the midpoint of the line segment joining $A (- 3, 7)$ and $B (5, - 1)$ .
Determine the gradient (slope) of the straight line passing through the points $M (3, 4)$ and $N (- 1, - 4)$ .
Find the gradient of a line that is perpendicular to the line $3 x - 4 y + 7 = 0$ .
Find the point of intersection of the linear equations: $x + 2 y = 8$ and $3 x - y = 3$ .
Calculate the perpendicular distance from the point $(1, 2)$ to the straight line $3 x + 4 y - 1 = 0$ .
Evaluate the exact numerical value of: $sin 6 0^{\circ} cos 3 0^{\circ} + cos 6 0^{\circ} sin 3 0^{\circ}$ .
If $tan θ = \frac{3}{4}$ and $θ$ is an acute angle, evaluate the value of $cos θ - sin θ$ .
Simplify the trigonometric expression: $\frac{1 - c o s ^{2} θ}{s i n θ c o s θ}$ .
Simplify the algebraic trigonometric expression: $(sin θ + cos θ)^{2} - 2 sin θ cos θ$ .
In triangle $A B C$ , $∠ A = 3 0^{\circ}, ∠ B = 4 5^{\circ}$ , and side $a = 6 cm$ . Find the exact length of side $b$ .
In triangle $A B C$ , side $b = 5 cm$ , side $c = 8 cm$ , and $∠ A = 6 0^{\circ}$ . Find the length of side $a$ .
Calculate the area of triangle $P QR$ where $p = 10 cm$ , $q = 12 cm$ , and included angle $∠ R = 3 0^{\circ}$ .
Find the area of a sector of a circle of radius 6 cm that subtends an angle of $6 0^{\circ}$ at the center, in terms of $π$ .
An arc subtends an angle of $7 0^{\circ}$ at the center of a circle of radius 9 cm. Calculate the length of the arc in terms of $π$ .
A right circular metal cylinder has a base radius of 7 cm and a height of 10 cm. Find its volume taking $π = \frac{22}{7}$ .
A solid right cone has a base radius of 5 cm and a vertical height of 12 cm. Find its curved surface area in terms of $π$ .
Find the volume of a solid sphere of radius 3 cm in terms of $π$ .
Solve for $θ \in [0^{\circ}, 9 0^{\circ}]$ in the equation: $2 sin θ - 1 = 0$ .
Solve for $θ \in [0^{\circ}, 18 0^{\circ}]$ in the trigonometric equation: $2 cos^{2} θ - cos θ - 1 = 0$ .
Two tangents $P A$ and $P B$ are drawn from an external point $P$ to a circle centered at $O$ . If $∠ A P B = 4 0^{\circ}$ , calculate the central angle $∠ A O B$ .
Find the equation of the line passing through $(3, 5)$ that is parallel to the line $2 x - y + 4 = 0$ .
If $sin θ = - \frac{3}{5}$ and $θ$ lies in the third quadrant ( $18 0^{\circ} < θ < 27 0^{\circ}$ ), find the value of $tan θ$ .
Each exterior angle of a regular polygon measures $2 4^{\circ}$ . How many sides does the polygon have?
The diagonals of a rhombus measure 12 cm and 16 cm. What is the perimeter of the rhombus?
A tangent $T A T^{'}$ touches a circle at $A$ . Chord $A B$ makes an angle of $5 2^{\circ}$ with the tangent. Find the angle subtended by chord $A B$ at the center of the circle.
Find the equation of the straight line with gradient -3 passing through the point (2, -1).

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