Geometry Mathematics 2 Official 2022-2023 SSC (English Medium) 10th Standard Board Exam Question Paper Solution

Some question and their alternative answer are given. Select the correct alternative.

If a, b, and c are sides of a triangle and a² + b² = c², name the type of triangle.

Obtuse angled triangle

Acute angled triangle 

Right-angled triangle

Equilateral triangle

Chords AB and CD of a circle intersect inside the circle at point E. If AE = 4, EB = 10, CE = 8, then find ED.

7

5

8

9

Co-ordinates of origin are ______.

(0, 0)

(0, 1)

(1, 0)

(1, 1)

If radius of the base of cone is 7 cm and height is 24 cm, then find its slant height.

23 cm

26 cm

31 cm

25 cm

If ΔABC ∼ ΔPQR and `(A(ΔABC))/(A(ΔPQR)) = 16/25`, then find AB : PQ.

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

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

In the above figure, ∠L = 35°, find :

1. m(arc MN)
2. m(arc MLN)

Solution :

1. ∠L = `1/2` m(arc MN) ............(By inscribed angle theorem)
   ∴ `square = 1/2` m(arc MN)
   ∴ 2 × 35 = m(arc MN)
   ∴ m(arc MN) = `square`
2. m(arc MLN) = `square` – m(arc MN) ...........[Definition of measure of arc]
   = 360° – 70°
   ∴ m(arc MLN) = `square`

Show that, cotθ + tanθ = cosecθ × secθ

Solution :

L.H.S. = cotθ + tanθ

= `cosθ/sinθ + sinθ/cosθ`

= `(square + square)/(sinθ xx cosθ)`

= `1/(sinθ xx cosθ)` ............... `square`

= `1/sinθ xx 1/square`

= cosecθ × secθ

L.H.S. = R.H.S

∴ cotθ + tanθ = cosecθ × secθ

Find the surface area of a sphere of radius 7 cm.

Solution :

The surface area of the sphere = 4πr²

= `4 xx 22/7 xx square^2`

= `4 xx 22/7 xx square`

= `square xx 7`

∴ The surface area of the sphere = `square` sq.cm.

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

In the given figure, points G, D, E, and F are concyclic points of a circle with centre C. ∠ECF = 70°, m(arc DGF) = 200°. Find m(arc DE) and m(arc DEF).

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

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

In ∆PQR seg PM is a 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")/("MQ") = square/square` .............(I) [Theorem of angle bisector]

Similarly, in ∆PMR, Ray MY is the bisector of ∠PMR.

∴ `("MP")/("MR") = square/square` .............(II) [Theorem of angle bisector]

But `("MP")/("MQ") = ("MP")/("MR")`  .............(III) [As M is the midpoint of QR.] 

Hence MQ = MR

∴ `("PX")/square = square/("YR")`  .............[From (I), (II) and (III)]

∴ XY || QR   .............[Converse of basic proportionality theorem]

Find the coordinates 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 the midpoint theorem,

x = `(x_1 + x_2)/2 = (square + 6)/2 = square/2 = square`

y = `(y_1 + y_2)/2 = (2 + square)/2 = 4/2 = square`

∴  Co-ordinates of midpoint P are `square`.

Prove the following theorem:

Angles inscribed in the same arc are congruent.

Draw a circle with a radius of 3.3 cm. Draw a chord PQ of length 6.6 cm. Draw tangents to the circle at points P and Q. Write your observation about the tangents.

The radii of the ends of a frustum are 14 cm and 6 cm respectively and its height is 6 cm. Find its curved surface area.

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

ΔSHR ~ ΔSVU. In ΔSHR, SH = 4.5 cm, HR = 5.2 cm, SR = 5.8 cm and `"SH"/("SV")=3/5`. Construct ΔSVU.

An ice cream pot has a right circular cylindrical shape. The radius of the base is 12 cm and the 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 a diameter of 4 cm and a height is 3.5 cm. If each student is given one cone, how many students can be served?

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

Eliminate θ if x = r cosθ and y = r sinθ.