Maharashtra SSC Geometry (Mathematics Part II) Board Paper June 2026

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

Q.1 (A)(ii) What is the measure of semicircular arc of a circle ?

Q.1 (A)(iii) What is the equation of Y-axis ?

Q.1 (A)(iv) Radius of a sector of a circle is 35 cm and length of its arc is 22 cm. What is the area of the sector ?

Q.1 (B)(i) Find the diagonal of square whose side is 5 cm.

Q.1 (B)(ii) In the following figure, seg AE \(\perp\) seg BC, seg DF \(\perp\) line BC, AE = 4, DF = 6, then find \(\dfrac{A(\Delta ABC)}{A(\Delta DBC)}\).

[Figure: B, E, C, F are points in that order on a line; A is above E with AE \(\perp\) BC and D is above F with DF \(\perp\) line BC; segments AB, AC, DB, DC are drawn; right-angle marks at E and F.]

Q.1 (B)(iii) In the following figure, ray PQ touches the circle at point Q, PQ = 12, PR = 9. Find PS.

[Figure: P is a point outside the circle; ray PQ is a tangent touching the circle at Q; a secant from P cuts the circle first at R and then at S, so P – R – S; the line through P, R, S carries an arrow beyond S.]

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

Q.2 (A)(i) In the following figure, chord EN and chord FS intersect each other at point M in the exterior of the circle. If \(m(\text{arc } NS) = 125^\circ\), \(m(\text{arc } EF) = 37^\circ\), then complete the following activity to find \(\angle NMS\).

[Figure: M is a point outside the circle; one secant from M cuts the circle at E and then N (M – E – N) and the other cuts it at F and then S (M – F – S); both secants are extended with arrows beyond N and S.]

Activity : Chord EN and chord FS intersect each other at point M in the exterior of the circle.

\(m\angle NMS = \dfrac{1}{2}\left[\text{____} - m(\text{arc } EF)\right]\)
= \(\dfrac{1}{2}\left[\text{____} - 37\right]\)
= \(\dfrac{1}{2} \times \text{____}\)
\(\therefore m\angle NMS = \text{____}\)

Q.2 (A)(ii) If \(\sin\theta = \dfrac{20}{29}\), then complete the following activity to find \(\cos\theta\).

Activity : \(\sin^2\theta + \cos^2\theta = 1\)
\(\left(\text{____}\right)^2 + \cos^2\theta = 1\)
\(\dfrac{400}{841} + \cos^2\theta = 1\)
\(\therefore \cos^2\theta = 1 - \text{____}\)
\(\cos^2\theta = \text{____}\)
Taking square roots of both sides
\(\therefore \cos\theta = \text{____}\)

Q.2 (A)(iii) In the following figure, point O is the centre of the circle, \(\angle AOB = 100^\circ\). Complete the following activity with the help of given figure.

[Figure: circle with centre O; X and Y are the points where the vertical diameter meets the circle, Y at the top and X at the bottom; A is on the lower-left of the circle and B on the lower-right, with \(\angle AOB = 100^\circ\) marked at O; A and B are joined to O.]

Activity :

Q.2 (B)(i) In the following figure, in \(\Delta PQR\), seg RS bisects \(\angle R\). If PR = 15, QR = 20, PS = 12, then find QS.

[Figure: \(\Delta PQR\) with P at the lower left, Q at the lower right and R at the top; S is on side PQ; seg RS is drawn and the two equal parts of \(\angle PRQ\) are marked.]

Q.2 (B)(ii) Find the length of diagonal of a rectangle whose length is 35 cm and breadth is 12 cm.

Q.2 (B)(iii) In the following figure, point G, D, E, F are concyclic points of a circle with centre C. \(\angle ECF = 70^\circ\), \(m(\text{arc } DGF) = 200^\circ\). Find \(m(\text{arc } DE)\).

[Figure: circle with centre C; D in the upper left, G in the upper right, E in the lower left and F in the lower right on the circle; segments CD, CE and CF are drawn, with \(\angle ECF\) marked at C.]

Q.2 (B)(iv) Find the distance between the points P (–1, 1) and Q (5, –7).

Q.2 (B)(v) A person is standing at a distance of 80 m from a Church looking at its top. The angle of elevation is of 45º. Find the height of the Church.

[Figure: right triangle with the church as the vertical side, the distance 80 m along the ground and the angle of elevation \(45^\circ\) at the observer.]

Q.3 (A)(i) In the following figure, in \(\Delta ABC\) ray BD bisects \(\angle ABC\), A – D – C, seg ED \(\parallel\) side BC, A – E – B, then to prove that \(\dfrac{AB}{BC} = \dfrac{AE}{EB}\), complete the following activity.

[Figure: \(\Delta ABC\) with A at the top, B at the lower left and C at the lower right; D is on side AC and ray BD is drawn and extended beyond D with an arrow; E is on side AB; seg ED is drawn parallel to BC; the two equal parts of \(\angle ABC\) are marked at B.]

Activity : In \(\Delta ABC\), ray BD bisects \(\angle B\).

\(\therefore \dfrac{AB}{BC} = \dfrac{\text{____}}{\text{____}}\) .............. (I) (Angle bisector theorem)

In \(\Delta ABC\), seg ED \(\parallel\) side BC.

\(\therefore \dfrac{AE}{EB} = \dfrac{\text{____}}{DC}\) .............. (II) ____

\(\therefore \dfrac{AB}{\text{____}} = \dfrac{\text{____}}{EB}\) .............. from (I) and (II)

Q.3 (A)(ii) If P is the midpoint of segment joining the points A (–4, 2) and B (6, 2), then complete the following activity to find the co-ordinates of point P.

Activity : In the given example, suppose

[Figure: A (–4, 2), P (x, y) and B (6, 2) are shown as three points in that order on a line segment; AP and PB are marked with double tick marks.]

\((x_1, y_1) = (-4, 2)\), \((x_2, y_2) = (6, 2)\) and
Let co-ordinates of P are \((x, y)\).
According to midpoint formula,

\(x = \dfrac{x_1 + x_2}{2} = \dfrac{\text{____} + \text{____}}{2} = \dfrac{2}{2}\)
\(x = \text{____}\)
\(y = \dfrac{y_1 + y_2}{2} = \dfrac{\text{____} + \text{____}}{2} = \dfrac{4}{2}\)
\(y = \text{____}\)
\(\therefore\) Co-ordinates of midpoint are (1, 2).

Q.3 (B)(i) In \(\Delta PQR\), point S is the midpoint of side QR. If PQ = 11, PR = 17, PS = 13, find QR.

Q.3 (B)(ii) Prove that, opposite angles of a cyclic quadrilateral are supplementary.

Q.3 (B)(iii) Draw a circle of radius 3.3 cm. Draw a chord PQ of length 6.6 cm. Draw tangents to the circle at point P and Q. Write your observation about the tangents.

Q.3 (B)(iv) In the following figure if \(A(\text{P -- ABC}) = 154\) cm², radius of the circle is 14 cm, find : (a) \(m\angle APC\), (b) \(l(\text{arc } ABC)\).

[Figure: circle with centre P; A at the bottom of the circle, C at the right of the circle, B on the arc between A and C (lower right); seg PA and seg PC are drawn with a right angle marked at P; the sector P–ABC is the region bounded by PA, PC and arc ABC.]

Q.4 (i) In the following figure, line \(l\) intersects side AB and side AC of \(\Delta ABC\) in points P and Q respectively. Show that \(\dfrac{A(\Delta APQ)}{A(\Delta ABC)} = \dfrac{AP \times AQ}{AB \times AC}\).

[Figure: \(\Delta ABC\) with A at the top, B at the lower left and C at the lower right; a line \(l\) cuts side AB at P and side AC at Q and is drawn with arrows on both sides; seg PQ is drawn.]

Q.4 (iii) Find the number of coins of 2.2 cm diameter and 0.2 cm thick, can be made by melting a metallic right circular cylinder of height 0.2 m and diameter 8.8 cm.

Q.5 (i) AB is a chord of a circle with centre O. AC is a diameter of a circle. Line AT is a tangent at A. Write the answers of the following questions :

(a) Draw the figure using the given information.
(b) Find the measure of \(\angle CAT\).
(c) Find the measure of \(\angle ABC\).

Q.5 (ii) \(\alpha\) is acute angle. If \(3\sin\alpha - 4\cos\alpha = 0\), then find the values of \(\tan\alpha\) and \(\sec\alpha\).