Geometry Mathematics 2 Model set 3 by shaalaa.com 2021-2022 SSC (English Medium) 10th Standard Board Exam Question Paper Solution

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Geometry Mathematics 2 [Model set 3 by shaalaa.com]
Marks: 40 Maharashtra State Board
SSC (English Medium)
SSC (Marathi Semi-English)

Academic Year: 2021-2022
Date: March 2022
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Note:

  1. All questions are compulsory.
  2. Use of calculator is not allowed.
  3. Figures to the right of questions indicates full marks.
  4. Draw proper figures for answers wherever necessary.
  5. The marks of construction should be clear and distinct. Do not erase them.
  6. While writing any proof, drawing relevant figure is necessary. Also the proof should be consistent with the figure.

[8]1
[4]1.A | Choose the correct alternative :
[1]1.A.i

If the length of the segment joining point L(x, 7) and point M(1, 15) is 10 cm, then the value of x is ______

7

7 or – 5

–1

1

Concept: Distance Formula
Chapter: [0.05] Co-ordinate Geometry
[1]1.A.ii

Choose the correct alternative:

If length of sides of a triangle are a, b, c and a2 + b2 = c2, then which type of triangle it is?

Obtuse angled triangle

Acute angled triangle

Equilateral triangle

Right angled triangle

Concept: Right-angled Triangles and Pythagoras Property
Chapter: [0.02] Pythagoras Theorem
[1]1.A.iii

In fig, seg DE || sec BC, identify the correct statement.

`"AD"/"DB" = "AE"/"AC"`

`"AD"/"DB" = "AB"/"AC"`

`"AD"/"DB" = "EC"/"AC"`

`"AD"/"DB" = "AE"/"EC"`

Concept: Basic Proportionality Theorem (Thales Theorem)
Chapter: [0.01] Similarity
[1]1.A.iv

Choose the correct alternative answer for the following question.

Find the side of a cube of volume 1 m3.

 1 cm

10 cm 

100 cm 

1000 cm

Concept: Circumference of a Circle
Chapter: [0.07] Mensuration
[4]1.B | Solve the following questions :
[1]1.B.i

Choose the correct alternative:

∆ABC ∼ ∆AQR. `"AB"/"AQ" = 7/5`, then which of the following option is true?

A–Q–B

A–B–Q

A-C–B

A–R–B

Concept: Division of a Line Segment
Chapter: [0.04] Geometric Constructions [0.05] Co-ordinate Geometry
[1]1.B.ii

The radius of a circle is 10 cm. The measure of an arc of the circle is 54°. Find the area of the sector associated with the arc. (\[\pi\]= 3.14 )

Concept: Length of an Arc
Chapter: [0.07] Mensuration
[1]1.B.iii

ΔPQR ~ ΔSUV. Write pairs of congruent angles

Concept: Similarity of Triangles
Chapter: [0.01] Similarity
[1]1.B.iv

Draw seg AB of length 4.5 cm and draw its perpendicular bisector

Concept: Basic Geometric Constructions
Chapter: [0.04] Geometric Constructions
[12]2
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[4]2.A | Complete the following activities : (Attempt Any TWO)
[2]2.A.i

In figure, chord EF || chord GH. Prove that, chord EG ≅ chord FH. Fill in the blanks and write the proof.

Proof: Draw seg GF.


∠EFG = ∠FGH     ......`square`    .....(I)

∠EFG = `square`   ......[inscribed angle theorem] (II)

∠FGH = `square`   ......[inscribed angle theorem] (III)

∴ m(arc EG) = `square`  ......[By (I), (II), and (III)]

chord EG ≅ chord FH   ........[corresponding chords of congruent arcs]

Concept: Inscribed Angle Theorem
Chapter: [0.03] Circle
[2]2.A.ii

From the given figure, in ∆ABC, if AD ⊥ BC, ∠C = 45°, AC = `8sqrt(2)` , BD = 5, then for finding value of AD and BC, complete the following activity.

Activity: In ∆ADC, if ∠ADC = 90°, ∠C = 45°    ......[Given]

∴ ∠DAC = `square`   .....[Remaining angle of ∆ADC]

By theorem of 45° – 45° – 90° triangle,

∴ `square = 1/sqrt(2)` AC and `square = 1/sqrt(2)` AC

∴ AD =`1/sqrt(2) xx square` and DC = `1/sqrt(2) xx 8sqrt(2)`

∴ AD = 8 and DC = 8

∴ BC = BD +DC

= 5 + 8

= 13

Concept: Property of 30°- 60°- 90° Triangle Theorem
Chapter: [0.02] Pythagoras Theorem
[2]2.A.iii

Find distance between point Q(3, – 7) and point R(3, 3)

Solution: Suppose Q(x1, y1) and point R(x2, y2)

x1 = 3, y1 = – 7 and x2 = 3, y2 = 3

Using distance formula,

d(Q, R) = `sqrt(square)`

∴ d(Q, R) = `sqrt(square - 100)`

∴ d(Q, R) =  `sqrt(square)`

∴ d(Q, R) = `square`

Concept: Distance Formula
Chapter: [0.05] Co-ordinate Geometry
[8]2.B | Solve the following questions : (Attempt Any FOUR)
[2]2.B.i

In the adjoining figure, circle with center D touches the sides of ∠ACB at A and B. If ∠ACB = 52°, find measure of ∠ADB.

Concept: Converse of Tangent Theorem
Chapter: [0.03] Circle
[2]2.B.ii

In the given figure, if A(P-ABC) = 154 cm2  radius of the circle is 14 cm, find

(1) `∠APC` 

(2) l ( arc ABC) .

Concept: Circumference of a Circle
Chapter: [0.07] Mensuration
[2]2.B.iii

ΔABP ~ ΔDEF and A(ΔABP) : A(ΔDEF) = 144:81, then AB : DE = ?

Concept: Similarity of Triangles
Chapter: [0.01] Similarity
[2]2.B.iv

Draw a circle of radius 3.4 cm, take any point P on it. Draw tangent to the circle from point P

Concept: Construction of a Tangent to the Circle at a Point on the Circle
Chapter: [0.04] Geometric Constructions
[2]2.B.v

In the given figure, m(arc NS) = 125°, m(arc EF) = 37°, find the measure ∠NMS.

Concept: Touching Circles
Chapter: [0.03] Circle
[9]3
[3]3.A | Complete the following activities : (Attempt Any ONE)
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[3]3.A.i

From fig., seg PQ || side BC, AP = x + 3, PB = x – 3, AQ = x + 5, QC = x – 2, then complete the activity to find the value of x.

In ΔPQB, PQ || side BC

`"AP"/"PB" = "AQ"/(["______"])`    ...[______]

`(x + 3)/(x - 3) = (x + 5)/(["______"])`

(x + 3) [______] = (x + 5)(x – 3)

x2 + x – [______] = x2 + 2x – 15

x = [______]

Concept: Basic Proportionality Theorem (Thales Theorem)
Chapter: [0.01] Similarity
[3]3.A.ii

Complete the following activity to draw tangents to the circle.

  1. Draw a circle with radius 3.3 cm and center O. Draw chord PQ of length 6.6 cm. Draw ray OP and ray OQ.
  2. Draw a line perpendicular to the ray OP from P.
  3. Draw a line perpendicular to the ray OQ from Q.
Concept: Construction of a Tangent to the Circle at a Point on the Circle
Chapter: [0.04] Geometric Constructions
[6]3.B | Solve the following questions : (Attempt Any TWO)
[3]3.B.i

Some plastic balls of radius 1 cm were melted and cast into a tube. The thickness, length and outer radius of the tube were 2 cm, 90 cm, and 30 cm respectively. How many balls were melted to make the tube?

Concept: Concept of Surface Area, Volume, and Capacity
Chapter: [0.07] Mensuration
[3]3.B.ii

If sec θ = `41/40`, then find values of sin θ, cot θ, cosec θ

Concept: Trigonometric Identities
Chapter: [0.06] Trigonometry
[3]3.B.iii

In Quadrilateral ABCD, side AD || BC, diagonal AC and BD intersect in point P, then prove that `"AP"/"PD" = "PC"/"BP"`

Concept: Similarity of Triangles
Chapter: [0.01] Similarity
[3]3.B.iv

Prove the following theorems:

Opposite angles of a cyclic quadrilateral are supplementary.

Concept: Theorem: Opposite angles of a cyclic quadrilateral are supplementary.
Chapter: [0.03] Circle
[8]4 | Solve the following questions : (Attempt Any TWO)
[4]4.A

As shown in figure, LK = `6sqrt(2)` then

(i) MK = ?

(ii) ML = ?

(iii) MN = ?

Concept: Property of 30°- 60°- 90° Triangle Theorem
Chapter: [0.02] Pythagoras Theorem
[4]4.B

Show that points A(– 4, –7), B(–1, 2), C(8, 5) and D(5, – 4) are the vertices of a parallelogram ABCD

Concept: Slope of a Line
Chapter: [0.05] Co-ordinate Geometry
[4]4.C

Prove that `"cot A"/(1 - tan "A") + "tan A"/(1 - cot"A")` = 1 + tan A + cot A = sec A . cosec A + 1

Concept: Trigonometric Identities
Chapter: [0.06] Trigonometry
[3]5 | Solve the following questions : (Attempt Any ONE)
[3]5.A

If AB and CD are the common tangents in the circles of two unequal (different) radii, then show that seg AB ≅ seg CD.

Concept: Tangent Segment Theorem
Chapter: [0.03] Circle
[3]5.B

In the given figure, square ABCD is inscribed in the sector A - PCQ. The radius of sector C - BXD is 20 cm. Complete the following activity to find the area of shaded region 

Concept: Length of an Arc
Chapter: [0.07] Mensuration

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