Maharashtra SSC Geometry (Mathematics Part II) Board Paper March 2025

Q.1 (A)(1) Out of the following which is a Pythagorean triplet ?

Q.1 (A)(2) ∠ACB is inscribed angle in a circle with centre O. If m∠ACB = 65º, then what is measure of its intercepted arc AXB ?

Q.1 (A)(3) Distance of point (3, 4) from the origin is .................... .

Q.1 (A)(4) If radius of cone is 5 cm and its perpendicular height is 12 cm, then the slant height is ..................... .

Q.1 (B)(1) In the following figure Δ ABC, B – D – C and BD = 7, BC = 20, then find \(\dfrac{A(\Delta ABD)}{A(\Delta ABC)}\).

[Figure: Δ ABC with point D on side BC; segment AD drawn.]

Q.1 (B)(2) In the following figure ∠MNP = 90º, seg NQ ⊥ seg MP, MQ = 9, QP = 4, find NQ.

[Figure: right-angled Δ MNP with the right angle at N; Q on hypotenuse MP with NQ ⊥ MP.]

Q.1 (B)(3) Angle made by a line with the positive direction of X-axis is 30º. Find slope of that line.

Q.1 (B)(4) In cyclic quadrilateral ABCD m∠A = 100º, then find m∠C.

Q.2 (A)(1) The radius of a circle with centre ‘P’ is 10 cm. If chord AB of the circle subtends a right angle at P, find area of minor sector by using the following activity. (π = 3.14)

[Figure: circle with centre P, chord AB, ∠APB = 90º, X on minor arc AB.]

Activity :

r = 10 cm, θ = 90º, π = 3.14.
A(P–AXB) = \(\dfrac{\theta}{360}\) × ____
= \(\dfrac{\text{____}}{360}\) × 3.14 × 10²
= \(\dfrac{1}{4}\) × ____
A(P–AXB) = ____ sq. cm.

Q.2 (A)(2) In the following figure chord MN and chord RS intersect at point D. If RD = 15, DS = 4, MD = 8, find DN by completing the following activity :

[Figure: circle with chords MN and RS intersecting at D inside the circle.]

Activity :

∴ MD × DN = ____ × DS .............. (Theorem of internal division of chords)
∴ ____ × DN = 15 × 4
∴ DN = ____ / 8
∴ DN = ____

Q.2 (A)(3) An observer at a distance of 10 m from tree looks at the top of the tree, the angle of elevation is 60º. To find the height of tree complete the activity. (\(\sqrt3\) = 1.73)

[Figure: right-angled Δ ABC, right angle at B; AB is the tree, BC = 10 m, ∠BCA = 60º.]

Activity :

In the figure given above, AB = h = height of tree, BC = 10 m, distance of the observer from the tree.
Angle of elevation (θ) = ∠BCA = 60º
tan θ = ____ / BC .................. (I)
tan 60º = ____ .................. (II)
\(\dfrac{AB}{BC} = \sqrt3\) (From (I) and (II))
AB = BC × \(\sqrt3\) = \(10\sqrt3\)
AB = 10 × 1.73 = ____
∴ height of the tree is ____ m.

Q.2 (B)(1) In Δ ABC, DE || BC. If DB = 5.4 cm, AD = 1.8 cm, EC = 7.2 cm, then find AE.

[Figure: Δ ABC with D on AB and E on AC, DE parallel to BC.]

Q.2 (B)(2) In the figure given below, find RS and PS using the information given in Δ PSR.

[Figure: Δ PSR with ∠S = 90º, ∠P = 30º and hypotenuse RP = 12.]

Q.2 (B)(3) In the following figure, circle with centre D touches the sides of ∠ACB at A and B. If ∠ACB = 52º, find measure of ∠ADB.

[Figure: circle with centre D; CA and CB are tangents touching the circle at A and B; radii DA and DB drawn.]

Q.2 (B)(4) Verify, whether points, A(1, –3), B(2, –5) and C(–4, 7) are collinear or not.

Q.2 (B)(5) If \(\sin\theta = \dfrac{11}{61}\), find the values of cos θ using trigonometric identity.

Q.3 (A)(1) In the following figure, XY || seg AC. If 2AX = 3BX and XY = 9. Complete the activity to find the value of AC.

[Figure: Δ ABC with X on side AB and Y on side BC, XY parallel to AC.]

Activity :

2AX = 3BX (Given)
∴ \(\dfrac{AX}{BX} = \dfrac{3}{\text{____}}\)
\(\dfrac{AX + BX}{BX} = \dfrac{3 + 2}{2}\) .......... (by componendo)
\(\dfrac{\text{____}}{BX} = \dfrac{5}{2}\) ................ (I)
Now Δ BCA ~ Δ BYX ............. (____ test of similarity)
∴ \(\dfrac{BA}{BX} = \dfrac{AC}{XY}\) .............. (corresponding sides of similar triangles)
∴ \(\dfrac{\text{____}}{\text{____}} = \dfrac{AC}{9}\) .............. from (I)
∴ AC = ____

Q.3 (A)(2) Complete the following activity to prove that the sum of squares of diagonals of a rhombus is equal to the sum of the squares of the sides.

[Figure: rhombus PQRS with diagonals PR and QS intersecting at T.]

Given : PQRS is a rhombus. Diagonals PR and SQ intersect each other at point T.
To prove : PS² + SR² + QR² + PQ² = PR² + QS²

Activity :

Diagonals of a rhombus bisect each other.
In Δ PQS, PT is the median and in Δ QRS, RT is the median.
∴ by Apollonius theorem,
PQ² + PS² = ____ + 2QT² ................. (I)
QR² + SR² = ____ + 2QT² ................. (II)
adding (I) and (II),
PQ² + PS² + QR² + SR² = 2 (PT² + ____) + 4QT²
= 2 (PT² + ____) + 4QT² .............. (RT = PT)
= 4PT² + 4QT²
= (____)² + (2QT)²
∴ PQ² + PS² + QR² + SR² = PR² + ____ .

Q.3 (B)(1) Show that points P(1, –2), Q(5, 2), R(3, –1), S(–1, –5) are the vertices of a parallelogram.

Q.3 (B)(2) Prove that tangent segments drawn from an external point to a circle are congruent.

Q.3 (B)(3) Draw a circle with radius 4.1 cm. Construct tangents to the circle from a point at a distance 7.3 cm from the centre.

Q.3 (B)(4) How many solid cylinders of radius 10 cm and height 6 cm can be made by melting a solid sphere of radius 30 cm ?

Q.4 (1) In the following figure DE || BC, then :

[Figure: Δ ABC with D on side AB and E on side AC, DE parallel to BC.]

(i) If DE = 4 cm, BC = 8 cm, A(Δ ADE) = 25 cm², find A(Δ ABC).
(ii) If DE : BC = 3 : 5, then find A(Δ ADE) : A(□ DBCE).

Q.4 (2) Δ ABC ~ Δ PQR. In Δ ABC, AB = 3.6 cm, BC = 4 cm and AC = 4.2 cm. The corresponding sides of Δ ABC and Δ PQR are in the ratio 2 : 3, construct Δ ABC and Δ PQR.

Q.4 (3) The radii of the circular ends of a frustum of a cone are 14 cm and 8 cm. If the height of the frustum is 8 cm, find : (π = 3.14)
(i) Curved surface area of frustum.
(ii) Total surface area of the frustum.
(iii) Volume of the frustum.

Q.5 (1) ABCD is a rectangle. Taking AD as a diameter, a semicircle AXD is drawn which intersects the diagonal BD at X. If AB = 12 cm, AD = 9 cm, then find the values of BD and BX.

Q.5 (2) Taking θ = 30º to verify the following Trigonometric identities :
(i) \(\sin^2\theta + \cos^2\theta = 1\)
(ii) \(1 + \tan^2\theta = \sec^2\theta\)
(iii) \(1 + \cot^2\theta = \operatorname{cosec}^2\theta\).