SSC (Marathi Semi-English) 10thMaharashtra State Board
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Geometry Shaalaa.com Model Set 1 2019-2020 SSC (Marathi Semi-English) 10th Question Paper Solution

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Geometry [Shaalaa.com Model Set 1]
Marks: 40Date: 2019-2020 March
Duration: 2h

[8]1
[4]1.A | MCQs
[1]1.A.i

Select the appropriate alternative.
In ∆ABC and ∆PQR, in a one to one correspondence \[\frac{AB}{QR} = \frac{BC}{PR} = \frac{CA}{PQ}\] 

∆PQR ~ ∆ABC 

∆PQR ~ ∆CAB 

∆CBA ~ ∆PQR 

∆BCA ~ ∆PQR

Concept: Similarity of Triangles
Chapter: [1] Similarity
[1]1.A.ii

In a right angled triangle, if sum of the squares of the sides making right angle is 169 then what is the length of the hypotenuse?

(A) 15 (B) 13 (C) 5 (D) 12

15

13

12

Concept: Apollonius Theorem
Chapter: [2] Pythagoras Theorem
[1]1.A.iii

In a right angled triangle, if sum of the squares of the sides making right angle is 169 then what is the length of the hypotenuse?

(A) 15 (B) 13 (C) 5 (D) 12

15

13

12

Concept: Apollonius Theorem
Chapter: [2] Pythagoras Theorem
[1]1.A.iv

Choose the correct alternative answer for each of the following questions

(7) Find the volume of a cube of side 0.01 cm.
 

(A) 1 cm3 

B) 0.001 cm

(C) 0.0001 cm3  

 (D) 0.000001 cm3

Concept: Problems Based on Areas and Perimeter Or Circumference of Circle, Sector and Segment of a Circle
Chapter: [7] Mensuration
[4]1.B
[1]1.B.i

Find the distance between the following pair of point.

\[W\left( \frac{- 7}{2} , 4 \right), X\left( 11, 4 \right)\]

Concept: Distance Formula
Chapter: [4] Co-ordinate Geometry
[1]1.B.ii

Prove that:
\[\frac{\sin^2 \theta}{\cos\theta} + \cos\theta = \sec\theta\]

Concept: Application of Trigonometry
Chapter: [6] Trigonometry
[1]1.B.iii

∆PQR ~ ∆LTR. In ∆PQR, PQ = 4.2 cm, QR = 5.4 cm, PR = 4.8 cm. Construct ∆PQR and ∆LTR, such that \[\frac{PQ}{LT} = \frac{3}{4} .\]

Concept: Division of a Line Segment
Chapter: [5] Geometric Constructions
[1]1.B.iv

Seg RM and seg RN are tangent segments of a circle with centre O. Prove that seg OR bisects ∠MRN as well as ∠MON.

Concept: Tangent Segment Theorem
Chapter: [3] Circle
[12]2
[4]2.A | Solve any 2 of the following
[2]2.A.i

In the given figure, ∠QPR = 90°, seg PM ⊥ seg QR and Q–M–R, PM = 10, QM = 8, find QR.

Concept: Application of Pythagoras Theorem in Acute Angle and Obtuse Angle
Chapter: [1] Similarity [2] Pythagoras Theorem
[2]2.A.ii

The dimensions of a cuboid are 44 cm, 21 cm, 12 cm. It is melted and a cone of height 24 cm is made. Find the radius of its base.

Concept: Surface Area of a Combination of Solids
Chapter: [7] Mensuration
[2]2.A.iii

Prove that:
\[\sec^4 \theta - \cos^4 \theta = 1 - 2 \cos^2 \theta\]

Concept: Application of Trigonometry
Chapter: [6] Trigonometry
[8]2.B | Solve any 4 of the following
[2]2.B.i

In adjoining figure PQ ⊥ BC, AD⊥ BC then find following ratios. 

(i) \[\frac{A\left( ∆ PQB \right)}{A\left( ∆ PBC \right)}\] 

(ii) \[\frac{A\left( ∆ PBC \right)}{A\left( ∆ ABC \right)}\] 

(iii) \[\frac{A\left( ∆ ABC \right)}{A\left( ∆ ADC \right)}\] 

(iv) \[\frac{A\left( ∆ ADC \right)}{A\left( ∆ PQC \right)}\] 

Concept: Similar Triangles
Chapter: [1] Similarity
[2]2.B.ii

In the given figure, M is the midpoint of QR. ∠PRQ = 90°. Prove that, PQ= 4PM– 3PR2

Concept: Pythagoras Theorem
Chapter: [2] Pythagoras Theorem
[2]2.B.iii

While landing at an airport, a pilot made an angle of depression of 20°. 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 20° = 0.342)

Concept: Heights and Distances
Chapter: [6] Trigonometry
[2]2.B.iv

From the top of a lighthouse, an observer looking at a ship makes angle of depression of 60°. If the height of the lighthouse is 90 metre, then find how far the ship is from the lighthouse.

\[\left( \sqrt{3} = 1 . 73 \right)\]
Concept: Heights and Distances
Chapter: [6] Trigonometry
[2]2.B.v

Solve the following example.

 Find the height of an equilateral triangle having side 2a.

Concept: Apollonius Theorem
Chapter: [2] Pythagoras Theorem
[9]3
[3]3.A | Solve any 1 of the following
[3]3.A.i

In ∆PQR, PM = 15, PQ = 25 PR = 20, NR = 8. State whether line NM is parallel to side RQ. Give reason. 

Concept: Property of three parallel lines and their transversals
Chapter: [1] Similarity
[3]3.A.ii

In the adjoining figure, O is the centre of the circle. From point R, seg RM and seg RN are tangent segments touching the circle at M and N. If (OR) = 10 cm and radius of the circle = 5 cm, then
(1) What is the length of each tangent segment ?
(2) What is the measure of ∠MRO ?
(3) What is the measure of ∠ MRN ?

Concept: Tangent Segment Theorem
Chapter: [3] Circle
[6]3.B | Solve any 2 of the following
[3]3.B.i

Draw a circle with radius 3.4 cm. Draw a chord MN of length 5.7 cm in it. construct tangents at point M and N to the circle.

Concept: Construction of Tangents to a Circle
Chapter: [5] Geometric Constructions
[3]3.B.ii

Find the co-ordinates of the points of trisection of the line segment AB with A(2, 7) and B(–4, –8).

Concept: Division of a Line Segment
Chapter: [4] Co-ordinate Geometry
[3]3.B.iii

In the given figure, 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 – (1) seg AP || seg BQ,
(2) ∆APR ~ ∆RQB, and
(3) Find ∠ RQB if ∠ PAR = 35°

Concept: Touching Circles
Chapter: [3] Circle
[3]3.B.iv

In the given figure shows a toy. Its lower part is a hemisphere and the upper part is a cone. Find the volume and surface area of the toy from the measures shown in the figure (\[\pi = 3 . 14\])

 

Concept: Surface Area of a Combination of Solids
Chapter: [7] Mensuration
[8]4 | Solve any 2 of the following
[4]4.A

In ΔABC, D and E are points on the sides AB and AC respectively such that DE || BC

If AD = 8x − 7, DB = 5x − 3, AE = 4x − 3 and EC = (3x − 1), find the value of x.

Concept: Basic Proportionality Theorem Or Thales Theorem
Chapter: [1] Similarity
[4]4.B

Prove that the tangents drawn at the ends of a diameter of a circle are parallel.

Concept: Number of Tangents from a Point on a Circle
Chapter: [3] Circle
[4]4.C

A solid is in the form of a right circular cylinder, with a hemisphere at one end and a cone at the other end. The radius of the common base is 3.5 cm and the heights of the cylindrical and conical portions are 10 cm. and 6 cm, respectively. Find the total surface area of the solid. (Use n =`22/7`)

Concept: Surface Area of a Combination of Solids
Chapter: [7] Mensuration
[3]5 | Solve any 1 of the following
[3]5.A

Draw a circle of radius 3.4 cm and centre E. Take a point F on the circle. Take another point A such that E-F-A and FA = 4.1 cm. Draw tangents to the circle from point A.

Concept: Construction of Tangents to a Circle
Chapter: [5] Geometric Constructions
[3]5.B

If A (–14, –10), B(6, –2) is given, find the coordinates of the points which divide segment AB into four equal parts.

Concept: Division of a Line Segment
Chapter: [4] Co-ordinate Geometry

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