Total Marks: 40 Time: 2 Hrs
Q. 1 Four alternative answers are given for every sub question select the correct alternative and write the alphabet of that answer. ——————–(4)
i) If a, b, c are sides of a triangle and a2 + b2 = c2, name the type of the triangle.
A) Obtuse angled triangle B) Acute angled triangle
C) Right angle triangle D) equilateral triangle
ii) AB is tangent to the circle with centre O at point A, OB = 20cm and ∠OBA = 300, then radius of the circle is…….
A) 10cm B) 20cm
C) 10 D) 20
cm
iii) Distance of point (-5, 12) from the origin is…….
A) 13 B) 7
C) 17 D) 0
iv) Find the side of a cube of volume 1 m3.
A) 1 cm B) 10cm
C) 100 cm D) 1000cm
Q. 1 A) Solve the following sub-questions. ——————–(4)
i) ∆ABC ∆PQR and ∆ABC: ∆PQR = 36: 81, then find AB: PQ.
ii) Two circles of radii 12cm and 8cm touch each other externally. Find the distance between their centres.
iii) In ∆ LMN, l = 5, m = 13, n = 12. State whether ∆ LMN is a right-angled triangle or not.
iv) Angle made by the line with the positive direction of X- axis is given, find the slope of the line. 600.
Q.2 A) Complete the following activities. (Any 2) ——————–(4)
i) In the adjoining figure circles with centres X and Y touch each other at point Z. A secant passing through Z intersects the circles at points A and B respectively. Prove that,
radius XA || radius YB.
Fill in the blanks and complete the proof.
Construction: Draw segments XZ and ……….
Proof: By theorem of touching circles, points X, Z, Y are……..
∠ XZA
…………. Opposite angles
Let ∠ XZA = ∠ BZY = a —————(I)
Now, seg XA seg XZ ————(……….)
∠ XAZ = ………. = a ————(Isosceles triangle) —-(II)
Similarly, seg YB …… ————(…………..)
∠ BZY = ………. = a ————(———–) ————(III)
From I, II and III,
∠ XAZ = ……
radius XA || radius YB ————(……………)
ii) If 5sin – 12cos = 0, find the value of sec.
Solution:
5sin – 12cos = 0
5sin = 12cos
=
=
We have,
1 + tan2 = sec2
1 +
= sec2
sec2 =
sec =
iii) The circumference of circular forces of a frustrum are 132cm and 88cm and its height is 24cm. To find the curved surface area of the frustum complete the following activity.
Circumference1 = 2r1 = 132
r1 = =
Circumference2 = 2r2 = 88
r2 = =
Slant height of frustrum = l =
=
=
Curved surface area of the frustrum =
l
=
= sq.cm
B) Solve the following sub-questions. (Any 4) ——————–(8)
i) In ∆ABC, DE || BC, If DB = 5.4cm, AD = 1.8cm, EC = 7.2 cm then find AE.
ii) In ∆PQR, ∠ Q = 900, ∠ P = 300, then find PQ and QR.
iii) In a circle with centre O, length of chord AB is equal to the radius of the circle. Find measure of each of the following
- ∠ AOB
- Arc AB
iv) Find k, if R(1, -1), S(-2, k) and slope of line RS is -2.
v) Person is standing at a distance of 50m from a building looking at its top. The angle of elevation is 600. Find the height of building.
Q.3 A) Complete the following activities. (Any 1) ——————–(3)
i) In the figure, X is any point in the interior of triangle. Point X is joined to vertices of triangle. Seg PQ || seg DE, seg QR || seg EF. Fill in the blanks to prove that,
seg PR || seg DF.
Proof: In ∆XDE,
PQ || DE ——————-
=
———– (I) (Basic proportionality theorem)
In ∆XEF,
QR || EF ——————-
=
———– (II) (
)
=
———–(From I and II)
seg PR || seg DE ——-(converse of basic proportionality theorem)
ii) If point P(-4, 6) divides the line segment AB with A(-6, -10) and B(a, b) in the ratio 2:1, find the co-ordinate of B.
Solution:
Suppose, P(-4, 6) = (x, y)
By section formula,
x =
-4 =
-4 =
-12 = 2r – 6
2r =
r =
And,
y =
6 =
6 =
18 = 2s + 10
2s =
s =
Co-ordinate of point B are ( ,
).
Q. 3 B) Solve the following sub-questions. (Any 2) ——————–(6)
i) Find the length of the side and perimeter of an equilateral triangle whose height is cm.
ii) Prove that: Opposite angles of a cyclic quadrilateral are supplementary.
iii) Draw a circle with radius 4.1cm, Construct tangents to the circle from a point at a distance 7.3cm from the centre.
iv) In radius of circle is 3.4cm and perimeter of sector P-ABC is 12.8cm. Find A(P-ABC).
Q.4 Solve the following sub-questions. (Any 2) ——————–(8)
i) From the information given in the figure, prove that PM = PN = a
ii) ∆ABC ∆PQR, in ∆ABC, AB = 5.4cm, BC = 4.2cm, AC = 6cm. AB: PQ = 3:2. Construct ∆ABC and ∆PQR.
iii) While landing at an airport, a pilot made an angle of depression of 200. Average speed of the plane was 200 km/hr. The plane reached the ground after 54 seconds. Find the height at which the plane was when it started landing. ( sin 200 = 0.342).
Q.5 Solve the following sub-questions. (Any 1) ——————–(3)
i) The circles with centres P and Q touch each other at R. A line passing through R meets the circles at A and B respectively. Prove that,
a) seg AP || seg BQ
b) ∆APR ∆RQB
c) Find ∠RQB, if ∠PAR = 350.
ii) Prove that: = sec2 + tan