# Geometry Shaalaa.com Model Set 1 2019-2020 SSC (Marathi Semi-English) 10th Standard [इयत्ता १० वी] Question Paper Solution

Geometry [Shaalaa.com Model Set 1]
Date: March 2020
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: [0.01] Similarity
[1] 1.A.ii

Some question and their alternative answer are given.

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?

15

13

12

Concept: Apollonius Theorem
Chapter: [0.02] Pythagoras Theorem
[1] 1.A.iii

Some question and their alternative answer are given.

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?

15

13

12

Concept: Apollonius Theorem
Chapter: [0.02] Pythagoras Theorem
[1] 1.A.iv

Choose the correct alternative answer for the following question.

Find the volume of a cube of side 0.01 cm.

1 cm3

0.001 cm

0.0001 cm3

0.000001 cm3

Concept: Circumference of a Circle
Chapter: [0.07] Mensuration [0.07] Mensuration
[4] 1.B
[1] 1.B.i

Find the distance between the following pairs of point.

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

Concept: Distance Formula
Chapter: [0.04] Co-ordinate Geometry
[1] 1.B.ii

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

Concept: Application of Trigonometry
Chapter: [0.06] 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: [0.04] Co-ordinate Geometry [0.05] 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 with the help of activity.

Proof: In ∆RMO and ∆RNO,

∠RMO ≅ ∠RNO = 90°   ......[square]

hypt OR ≅ hypt OR    ......[square]

seg OM ≅ seg square    ......[Radii of the same circle]

∴ ∆RMO ≅ ∆RNO      ......[square]

∠MOR ≅ ∠NOR

Similairy ∠MRO ≅ square    ......[square]

Concept: Tangent Segment Theorem
Chapter: [0.03] 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: [0.01] Similarity [0.02] 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: [0.07] Mensuration
[2] 2.A.iii

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

Concept: Application of Trigonometry
Chapter: [0.06] 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: [0.01] Similarity
[2] 2.B.ii

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

Concept: Right-angled Triangles and Pythagoras Property
Chapter: [0.02] 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: [0.06] 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: [0.06] Trigonometry
[2] 2.B.v

Solve the following example.

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

Concept: Apollonius Theorem
Chapter: [0.02] 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: [0.01] 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: [0.03] 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: [0.05] 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: [0.04] Co-ordinate Geometry [0.05] Geometric Constructions
[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: [0.03] 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: [0.07] 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: [0.01] 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: [0.03] Circle [0.03] 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 π =22/7)

Concept: Surface Area of a Combination of Solids
Chapter: [0.07] 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: [0.05] 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: [0.04] Co-ordinate Geometry [0.05] Geometric Constructions

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