Maharashtra State Board

Class X Mathematics – Geometry – Part 2

Board Paper 2023
Time: 2 Hours. Maximum Marks: 40

Note:
(i) All questions are compulsory.
(ii) Use of calculator is not allowed.
(iii) The numbers to the right of the questions indicate full marks.
(iv) In case of MCQs IQ. No. 1(A)) only the first attempt will be evaluated and will be given
credit.
(v) For every MCQ, the correct alternative (A), (B), (C) or (D)) with sub-question number
is to be written as an answer.
(vi) Draw the proper figures for answers wherever necessary.
(vii) The marks of construction should be clear and distinct. Do not erase them.
(viii) Diagram is essential for writing the proof of the theorem.

1. (A) Four alternative answer are given for every sub question. Select the correct
alternative and write the alphabet of that answer. 4
(1) If a, b, c are sides of a triangle and a + b = c , name the type of triangle:
2 2 2

(A) Obtuse angled triangle

(B) Acute angled triangle
(C) Right angled triangle
(D) Equilateral triangle

(2) Chords AB and CD of a circle intersect inside the circle at point E. If AE = 4, EB =

10, CE = 8, then find ED:
(A) 7
(B) 5
(C) 8
(D) 9

(3) Co-ordinates of origin are ………………

(A) (0,0)
(B) (0,1)
(C) (1,0)
(D) (1,1)

(4) If radius of the base of cone is 7 cm and height is 24 cm, then find its slant height:
(A) 23 cm
(B) 26 cm
(C) 31 cm
(D) 25 cm

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 (B) Solve the following sub-questions: 4
A(ABC) 16
(1) If ΔABC  ΔPQR and  , then find AB:PQ.
A(PQR) 25

(2) In ΔRST, ∠S = 90°, ∠T = 30°, RT = 12 cm, then find RS.

(3) If radius of a circle is 5 cm, then find the length of longest chord of a circle.

(4) Find the distance between the point O(0, 0) and P(3, 4).

2. (A) Complete the following activities (Any two): 4

(1)

In the above figure, ∠L =35°, find

(i) m(are MN)
(ii) m(are MLN)
Solution:
1
(i) L m arc MN  ……(By Inscribed Angle Theorem)
2
1
  m arc MN 
2
235  m arc MN 
 m arc MN  
(ii)
m arc MLN    m arc MN  ……(Definition of measure of arc)
 360 70
m arc MLN  

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 (2) Show that: cot  + tan  = cosec  × sec 
Solution:
L.H.S.  cot   tan 
cos  sin 
 
sin  cos 

 .
sin  cos 
1
 ......
sin  cos 
1 1
 
sin 
 cosec  sec 
L.H.S. = R.H.S.
 cot  + tan  = cosec  × sec 

(3) Find the surface area of a sphere of radius 7 cm.

Solution:
Surface area of sphere = 4r2
22 2
 4 
7
22
 4 
7
 7
 Surface area of sphere  sq.cm

(B) Solve the following sub-questions (Any four): 8

(1)

In trapezium ABCD side AB || side PQ || side DC. AP = 15, PD = 12, QC = 14, find BQ.

(2) Find the length of the diagonal of a rectangle whose length is 35 cm and breadth is
12 cm.

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 (3)

In the given figure point G, D, E, F are point of a circle with centre C, ECF = 70°, m
(arc DGF) = 200°.
Find:
(i) m (arc DE)
(ii) m (arc DEF).

(4) Show that points A(–1, –1), B(0, 1), C(1, 3) are collinear.

(5) A person is standing at a distance of 50 m from a temple looking at its top. The angle
of elevation is of 45°. Find the height of the temple.

3. (A) Complete the following activities (Any one):

(1)

In ΔPQR, seg PM is median. Angle bisectors of ∠PMQ and ∠PMR intersect side PQ
and side PR in points X and Y respectively.
Prove that XY || QR.
Complete the proof by filling in the boxes.
Solution:
In PMQ,
Ray MX is the bisector of ∠PMQ.
MP
 ...... I   Theorem of angle bisector
MQ
Similarly, in PMR, Ray MY is bisector of ∠PMR.
MP
 ...... II  Theorem of angle bisector
MR

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 MP MP
But  ...... III  As M is the midpoint of QR
MQ MR
Hence MQ = MR
PX
  ......  From  I  , II  and  III 
YR
 XY QR

(2) Find the co-ordinates of point P where P is the midpoint of a line segment AB with
A(–4, 2) and B(6, 2)
Solution:

Suppose (–4, 2) = (x1, y1) and (6, 2) = (x2, y2) and co-ordinates of P are (x, y).
 According to midpoint theorem,
x1  x2 6
x   
2 2 2
y y 2 4
y 1 2   
2 2 2
 Co-ordinates of midpoint P are .

(B) Solve the following sub-questions (any two): 6

(1) In ΔABC, seg AP is a median. If BC = 18, AB 2 + AC2 = 260, find AP.

(2) Prove that, “Angles inscribed in the same arc are congruent.”

(3) Draw a circle of radius 3.3 cm. Draw a chord PQ of length 6.6 cm. Draw tangents to
the circle at points P and Q.

(4) The radii of circular ends of a frustum are 14 cm and 6 cm respectively and its
height is 6 cm. Find its curved surface area. (π = 3.14)

4. Solve the following sub-questions (Any two): 8

(1) In ΔABC, seg DE || side BC, If 2A(ΔADE) = A(▭DBCE), find AB:AD and show that BC
= 3 DE.

SH 3
(2) ΔSHR – ΔSVU, In ΔSHR, SH = 4.5 cm, HR = 5.2 cm, SR = 5.8 cm and  ,
SV 5
construct ΔSVU.

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 (3) An ice-cream pot has a right circular cylindrical shape. The radius of the base is 12
cm and height is 7 cm. This pot is completely filled with ice-cream. The entire ice-
cream is given to the students in the form of right circular ice-cream cones, having
diameter 4 cm and height is 3.5 cm. If each student is given one cone, how many
students can be served?

5. Solve the following sub-question (Any one): 3

(1)

A circle touches side BC at point P of the ΔABC, from out-side of the triangle. Further
extended lines AC and AB are tangents to the circle at N and M respectively. Prove
that:
1
AM  (Perimeter ofABC)
2

(2) Eliminate θ if x = r cos θ and y = r sin θ

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