Geometry Mathematics 2 Official 2023-2024 SSC (English Medium) 10th Standard Question Paper Solution

Four alternative answers for the following question are given. Choose the correct alternative and write its alphabet:

Out of the dates given below which date constitutes a Pythagorean triplet?

15/8/17 

16/8/16 

3/5/17

4/9/15

Four alternative answers for the following question are given. Choose the correct alternative and write its alphabet:

sin θ × cosec θ = ______

1

0

`1/2`

`sqrt2`

Four alternative answers for the following question are given. Choose the correct alternative and write its alphabet:

Slope of X-axis is ______.

1

−1

0

Cannot be determined

Four alternative answers for the following question are given. Choose the correct alternative and write its alphabet:

A circle having radius 3 cm, then the length of its largest chord is ______.

1.5 cm

3 cm

6 cm

9 cm

If ΔABC ∼ ∆PQR and AB : PQ = 2 : 3, then find the value of `(A(triangleABC))/(A(trianglePQR))`.

Two circles of radii 5 cm and 3 cm touch each other externally. Find the distance between their centres.

Find the side of a square whose diagonal is `10sqrt2` cm.

Angle made by the line with the positive direction of X-axis is given. Find the slope of the line.

 45° 

In the above figure, ∠ABC is inscribed in arc ABC.

If ∠ABC = 60°. find m ∠AOC.

Solution:

∠ABC = `1/2` m(arc AXC)   ......`square`

60° = `1/2` m(arc AXC) 

`square` = m(arc AXC) 

But m ∠AOC = \[\boxed{m(arc ....)}\]   ......(Property of central angle)

∴ m ∠AOC = `square`

Find the value of sin²θ  + cos²θ

Solution:

In Δ ABC, ∠ABC = 90°, ∠C = θ°

AB² + BC² = `square`   .....(Pythagoras theorem)

Divide both sides by AC²

`"AB"^2/"AC"^2 + "BC"^2/"AC"^2 = "AC"^2/"AC"^2`

∴ `("AB"^2/"AC"^2) + ("BC"^2/"AC"^2) = 1`

But `"AB"/"AC" = square and "BC"/"AC" = square`

∴ `sin^2 theta  + cos^2 theta = square` 

In the figure given above, `square`ABCD is a square and a circle is inscribed in it. All sides of a square touch the circle. If AB = 14 cm, find the area of shaded region.

Solution:

Area of square = `(square)^2`   ......(Formula)

= 14²

= `square  "cm"^2`

Area of circle = `square`    ......(Formula)

= `22/7 xx 7 xx 7`

= 154 cm²

(Area of shaded portion) = (Area of square) - (Area of circle)

= 196 − 154

= `square  "cm"^2`

The radius of a sector of a circle is 3.5 cm and length of its arc is 2.2 cm. Find the area of the sector.

Find the length of the hypotenuse of a right angled triangle if remaining sides are 9 cm and 12 cm.

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

Find the slope of the line passing through the points A(2, 3) and B(4, 7).

In ΔABC, ray BD bisects ∠ABC, A – D – C, seg DE || side BC, A – E – B, then for showing `("AB")/("BC") = ("AE")/("EB")`, complete the following activity:

Proof :

In ΔABC, ray BD bisects ∠B.

∴ `square/("BC") = ("AD")/("DC")`   ...(I) (`square`)

ΔABC, DE || BC

∴ `(square)/("EB") = ("AD")/("DC")`   ...(II) (`square`)

∴ `("AB")/square = square/("EB")`   ...[from (I) and (II)]

Given:

Chords AB and CD of a circle with centre P intersect at point E.

To prove:

AE × EB = CE × ED

Construction:

Draw seg AC and seg BD.

Fill in the blanks and complete the proof.

Proof:

In Δ CAE and Δ BDE,

∠AEC ≅ ∠DEB    ...`square`

`square` ≅ ∠BDE    ...(angles inscribed in the same arc)

∴ ΔCAE ~ ΔBDE    ...`square`

∴ `square/ ("DE") = ("CE")/square`    ...`square` 

∴ AE × EB = CE × ED.

Determine whether the points are collinear.

A(1, −3), B(2, −5), C(−4, 7)

∆ABC ~ ∆LMN. In ∆ABC, AB = 5.5 cm, BC = 6 cm, CA = 4.5 cm. Construct ∆ABC and ∆LMN such that `"BC"/"MN" = 5/4`.

In ΔPQR, seg PM is a median, PM = 9 and PQ² + PR²  = 290. Find the length of QR. 

Prove that, if a line parallel to a side of a triangle intersects the other sides in two district points, then the line divides those sides in proportion.

`1/sin^2θ - 1/cos^2θ - 1/tan^2θ - 1/cot^2θ - 1/sec^2θ - 1/("cosec"^2θ) = -3`, then find the value of θ.

A cylinder of radius 12 cm contains water up to the height 20 cm. A spherical iron ball is dropped into the cylinder and thus water level raised by 6.75 cm. What is the radius of iron ball?

Draw a circle with centre O having radius 3 cm. Draw tangent segments PA and PB through the point P outside the circle such that ∠APB = 70°.

`square`ABCD is trapezium, AB || CD diagonals of trapezium intersects in point P.

Write the answers of the following questions:

1. Draw the figure using the given information.
2. Write any one pair of alternate angles and opposite angles.
3. Write the names of similar triangles with the test of similarity.

AB is a chord of a circle with centre O. AOC is diameter of circle, AT is a tangent at A.

Write answers of the following questions:

1. Draw the figure using the given information.
2. Find the measures of ∠CAT and ∠ABC with reasons.
3. Whether ∠CAT and ∠ABC are congruent? Justify your answer.