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

Q.1 (A)(1) If \(\Delta ABC \sim \Delta DEF\), \(m\angle B = 60^\circ\), then \(m\angle E\) = ....................

Q.1 (A)(2) Two circles of radii 5.5 cm and 4.2 cm touch each other externally then distance between their centres is ....................

Q.1 (A)(3) A line makes an angle of 45º with the positive direction of X-axis. So the slope of the line is ....................

Q.1 (A)(4) The volume of a cube of side 2 cm is ....................

Q.1 (B)(1) Find the diagonal of square whose side is 10 cm.

Q.1 (B)(2) The ratio of corresponding sides of similar triangles is 3 : 5, then find the ratio of their areas.

Q.1 (B)(3) Find the slope of the line passing through the points A(2, 3) and B(4, 7).

Q.1 (B)(4) If \(\sin\theta = \dfrac{7}{25}\), then find the value of \(\operatorname{cosec}\theta\).

Q.2 (A)(1) In the given figure, AR ⊥ BC, AR ⊥ PQ, then complete the activity for finding \(\dfrac{A(\Delta ABC)}{A(\Delta APQ)}\).

[Figure: Δ ABC with A at the top; B, P, R, Q, C lie on the base line in that order; seg AR is drawn from A to the base with a right-angle mark at R on BC.]

Activity :

\(\dfrac{A(\Delta ABC)}{A(\Delta APQ)} = \dfrac{____ \times AR}{PQ \times ____}\)

\(\therefore \dfrac{A(\Delta ABC)}{A(\Delta APQ)} = \dfrac{____}{____}\)

Q.2 (A)(2) In the following figure, seg PS is a tangent segment, line PR is a secant. If PQ = 3.6, QR = 6.4, then find PS by completing the following activity.

[Figure: circle with an external point P; the secant from P cuts the circle at Q (nearer) and R (farther); the tangent from P touches the circle at S.]

Activity :

\(PS^2 = PQ \times ____\) ......... (tangent secant segments theorem)

\(\therefore PS^2 = PQ \times (PQ + ____)\)

\(\therefore PS^2 = 3.6 \times (3.6 + ____)\)

\(\therefore PS^2 = 3.6 \times 10\)

\(\therefore PS^2 = 36\)

\(\therefore PS = ____\)

Q.2 (A)(3) Measure of an arc of a circle is 90º and its radius is 14 cm. Complete the following activity to find the length of an arc.

Activity :

Length of an arc \(= \dfrac{\theta}{360} \times ____\) ......... (Formula)

\(= \dfrac{90}{360} \times 2 \times \dfrac{22}{7} \times ____\)

\(= \dfrac{1}{4} \times ____\)

Length of an arc \(= ____\) cm

Q.2 (B)(1) In the following figure, in \(\Delta LMN\), ray MT bisects \(\angle LMN\).

If LM = 6, MN = 10, TN = 8, then find LT.

[Figure: Δ LMN with M at the lower left; ray MT from M meets side LN at T and is produced beyond T; LM = 6, MN = 10, TN = 8, with equal angles marked at M.]

Q.2 (B)(2) Find surface area of sphere of radius 7 cm.

Q.2 (B)(3) In the following figure, \(m(\text{arc NS}) = 125^\circ\), \(m(\text{arc EF}) = 37^\circ\). Find \(m\angle NMS\).

[Figure: circle with chords NS and EF intersecting at the point M inside the circle; N, F on the upper part of the circle, E and S on the lower part.]

Q.2 (B)(4) Find the co-ordinates of midpoint of the segment joining the points P(22, 20) and Q(0, 16).

Q.2 (B)(5) Find the volume of a cone if radius of its base is 7 cm and its perpendicular height is 15 cm.

Q.3 (A)(1) If \(\tan\theta = 1\), then find the value of \(\dfrac{\sin\theta + \cos\theta}{\sec\theta + \operatorname{cosec}\theta}\) by completing the following activity.

Activity :

\(\tan\theta = 1\) ......... (given)

but \(\tan ____ = 1\)

\(\therefore \theta = ____\)

\(\therefore \dfrac{\sin\theta + \cos\theta}{\sec\theta + \operatorname{cosec}\theta} = \dfrac{\sin 45^\circ + \cos 45^\circ}{\sec 45^\circ + \operatorname{cosec} 45^\circ}\)

\(= \dfrac{\dfrac{1}{____} + \dfrac{1}{\sqrt2}}{\sqrt2 + ____}\)

\(= \dfrac{\dfrac{2}{\sqrt2}}{____}\)

\(\dfrac{\sin\theta + \cos\theta}{\sec\theta + \operatorname{cosec}\theta} = \dfrac{1}{____}\)

Q.3 (A)(2) In the following figure, point O is the centre of the circle and length of chord AB is equal to the radius of the circle. Find the measures of :

(i) ∠AOB
(ii) arc AB
(iii) ∠ACB

[Figure: circle with centre O; chord AB below the centre, C on the major arc AB above the centre; seg OA, seg OB, seg CA and seg CB drawn.]

by completing the activity.

Activity :

In \(\Delta AOB\),

AO = OB = AB

\(\therefore \Delta AOB\) is an ____ triangle.

\(\therefore m\angle AOB = ____\)

\(m\angle AOB = m(\text{arc AB}) = ____\) ......... (definition of measure of an arc)

\(m\angle ACB = \dfrac{1}{2} \times ____\) ......... (inscribed angle theorem)

\(= \dfrac{1}{2} \times 60^\circ\)

\(m\angle ACB = ____\)

Q.3 (B)(1) Find co-ordinates of point P, if P divides the line segment joining the points A(–1, 7) and B(4, –3) in the ratio 2 : 3.

Q.3 (B)(2) Draw a circle with centre O of 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.

Q.3 (B)(3) A storm broke a tree and the treetop rested 20 m from the base of the tree, making an angle of 60º with the horizontal. Find the height of the tree.

[Figure: vertical broken tree AB with base A; the broken top rests on the ground at C, 20 m from A, with ∠BCA = 60º, where B is the point where the tree broke.]

Q.3 (B)(4) Prove that, ‘In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of remaining two sides’.

Q.4 (1) Δ ABC has sides of length 4 cm, 5 cm and 6 cm while Δ PQR has perimeter of 90 cm. If Δ ABC is similar to Δ PQR, then find the length of corresponding sides of Δ PQR.

Q.4 (2) \(\Delta ABC \sim \Delta PBR\), BC = 8 cm, AC = 10 cm, ∠B = 90º, \(\dfrac{BC}{BR} = \dfrac{5}{4}\), then construct Δ PBR.

Q.4 (3) In the following figure Δ ABC is an isosceles triangle with perimeter 44 cm. The base BC is of length 12 cm. Side AB and AC are congruent. A circle touches the three sides of triangle as shown.

Find the length of tangent segment from A to circle.

[Figure: isosceles Δ ABC with A at the top and base BC; the incircle touches AB at P, AC at Q and BC at R; centre of the circle marked inside.]

Q.5 (1) Draw right-angled \(\Delta ABC\) of lengths of sides are 3 cm, 4 cm and 5 cm. Draw median on the hypotenuse of \(\Delta ABC\). Then :

(i) Measure the length of median and write it.
(ii) By observing lengths of median and hypotenuse write your observations.

Q.5 (2) Observe the given figure and answer the following questions :

[Figure: a solid cone with apex P and centre of the base O; AB is a diameter of the circular base, so A and B are on the base circle; seg PO is drawn to O with a right-angle mark at O, and PA, PB are the slant edges.]

(i) How many surfaces does a solid cone have ?
(ii) What are the names of slant height and perpendicular height in the given figure ?
(iii) If slant height of solid cone is 10 cm and perpendicular height is 8 cm, then find diameter of base of solid cone ?