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Class 10 Mathematics Chapter 9: Tangents and Angles of a Circle | Model Paper 2027
Class 10 Math Unit 9 Board Model Paper 2027
Class 10 Mathematics: Tangents and Angles of a Circle – Full Unit Test Model Paper is a useful practice paper for students preparing for the Punjab Board 2027 Model Papers. This Class 10 Mathematics Unit 9 Test covers important concepts including tangents, angles of a circle, arcs, sectors, radii, chords, central angles and related geometry problems. Students can use this Class 10 Tangents and Angles of a Circle Model Paper for revision, self-assessment and preparation for their upcoming Punjab Board examinations.
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Full Unit Test
Unit 9: Tangents and Angles of a Circle
T-4
Objective Type
Time Allowed: 20 Min.
Max. Marks: 15
Note: Four possible answer A, B, C and D to each question are given.
The choice which you think is correct, fill that circle front of that
question with Marker or ink pen in the answer book. Cutting filling
two or more circles will result in zero mark in that question.
(i) Angle between tangent line and radial line segment joining point of contact to the origin is:
(ii) Number of tangents drawn to a circle from a point outside the circle is:
(iii) Two circles of radius 3 and 4 touches each other externally, then distance between their centers is:
(iv) Two circles of radii 3 and 4 touches each other internally then distance between their centers is:
(v) Consider the accompanying figure. If m∠C = 30° then m∠D =
(vi) From the figure m∠ACB =
(vii) From the figure, if m∠ACB = 30° then m∠ADB =
(viii) A round table has a decorative quadrilateral pattern inscribed on its surface.
If one angle of the quadrilateral is 105°, what is the measure of the opposite angle.
(ix) Length ℓ of an arc which subtends an angle of θ radius of a circle of radius r is calculated by the formula ℓ =
(x) If r = 4 and ℓ = 14 then angle subtends by the arc in radian measure is θ =
(xi) If r = 6 and θ = 60° then length of the arc is ℓ =
(xii) If r = 6 and θ = 60°, when area of the sector is A =
(xiii) Angles x and y in the accompanying figure are:
(xiv) An angle in a segment greater than a semi-circle is __________ angle.
(xv) Number of tangents can be drawn to a circle from center of the circle is:
Here is Subjective Part of Paper.
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Subjective Type
Time Allowed:
2:10 hours
2:10 hours
(Part-I)
Max. Marks: 60
2. Write short answers to any six (06) questions:
(i) The Radii of the two circles are 4cm and 6cm.
What is the distance between their centres if they touch externally?
(ii) An inner decorative circle touches the outer clock face internally.
If the outer circle has radius 12cm and the inner has radius 7.5cm,
find the distance between their centres.
(iii) A curved running track forms an arc that subtends
60° at the centre of the circular ground. What angle does it subtend
at a point on the boundary?
(iv) A radar antenna scans an arc subtending 90° at the centre.
What is the angle subtended by the arc at the edge of its circular path?
(v) Find the value of x.
(vi) Find the angle ABC.
(vii) D, E and F are points on the circumference of a circle
with centre at O and ∠DOF = 130°.
Find ∠DEF.
(viii) X, Y and Z are points on the circumference of a circle
with centre at O and ∠XYZ = 85°. Find x.
(ix) In a historical monument, a circular fountain with a radius
of 3m is built. A flagpole is erected 7m away from the centre of the
fountain. Two ropes from the pole are tied to the edge of the fountain,
just touching it. Find the length of each rope.
3. Write short answers to any six (06) questions:
(i) An inner holder touches the outer cylindrical container internally.
If the outer container has radius 8cm and the inner holder has radius 6cm,
find the distance between their centres.
(ii) A small sensor lies inside a satellite dish and touches its wall
internally. If the dish has radius 15cm and the sensor has radius 2.5cm,
find the distance between their centres.
(iii) Two circular gears touch each other externally for proper rotation
in a machine. The radii of the two circular gears are 5cm and 7cm.
What is the distance between their centres if they touch externally?
(iv) In a large circular stained-glass window of a library, two decorative
beams are drawn from the same chord AB to two points
P and Q on the arc opposite to
AB such that ∠APB and
∠AQB are formed. If ∠APB is measured
to be 40°, what is ∠AQB?
(v) A projector casts an image over an arc on a circular wall.
Two observers on the arc segment report one observing a 50° angle.
What is the other angle in the same segment?
(vi) In the diagram P, Q and R are points on the circumference
of a circle and the line segment PQ is the diameter. Find the angles
x and y.
(vii) S, T, U and V are points on the circumference of a circle.
Find the angles x and y.
(viii) A round table has a decorative quadrilateral pattern inscribed
on its surface. If one angle of the quadrilateral is 105°, what is the
measure of the opposite angles?
(ix) Prove that
Area of sector = ½r²θ,
where θ is in radians.
4. Write short answers to any six (06) questions:
(i) Prove that
ℓ = rθ,
where θ is in radian.
(ii) Find the area of sector of a circle having central angle
60° and radius 7cm. Also find the length of the arc.
(iii) The length of an arc and area of sector of a circle are
5cm and 25cm² respectively. Find radius and the central angle of sector.
(iv) If ℓ is the arch length and
r is the radius of the circle where
ℓ = 14m, r = 4m, then find the central angle.
Also find the area of the sector of the circle.
(v) A, B, C and D are points on circumference of a circle with
centre at O as shown in figure. Find the angle x.
(vi) Write the formula of area of a sector.
(vii) Define circum angle of a circle.
(viii) Find the angles x and
y in the following figures.
(ix) In a circular arch over a monument entrance, two spotlight
fixtures are placed such that they each shine from two different points
on the arch to the same chord PQ on the base of the arch. If the angle
formed at one light is 55°, what is the angle at the second light on the
same side of chord PQ?
(Part-II) Note: Attempt any two (02) questions.
5:(a) QR is a tangent to a circle with centre P at point Q
of radius 4cm. It meets the line segment PR such that
PR = 8 cm. What is the length of
QR?
(b) An architect designs a circular dome on a building.
A lighting rod is placed 12m away from the centre of the dome which
has a radius of 5m. Support cables are attached from the rod to the dome,
touching the dome at two points. What is the length of each cable?
6:(a) A, B and C are points on the circumference of a circle with
centre at O. AC is the diameter of the circle and
DE is the tangent to the circle at the point C and
∠BCD = 62°. Find ∠BCA.
(b) A pyramid-shaped sculpture is placed in the center of a circular
plaza with a radius of 10m. A decorative pole stands 26m from the center
of the circle. Two guide wires are attached from the pole to the plaza
edge, just touching the circle. Find the length of each wire.
7:(a) In the adjoining figure
m∠BDC = 74°, find
m∠BAC,
m∠BOC and
m∠OBC.
(b) Find the angle x in the given figure.
(Part-III) Note: Attempt any one (01) question.
8:(a) In a circular arch over a monument entrance, two spotlight
fixtures are placed such that they each shine from two different points
on the arch to the same chord PQ on the base of the arch. If the angle
formed at one light is 55°, what is the angle at the second light on the
same side of chord PQ?
(b) Uzma cut a pizza of radius 14cm into 8 equal slices.
What is the area of one slice (sector)?
9:(a) The perimeter and area of a sector are 14cm and 10cm²
respectively. Find the radius of the circle and the central angle of
the sector.
(b) Find ℓ and area of sector, when:
(i) r = 1.7cm, θ = 0.25 radian
(ii) r = 3c, θ = 45°
