Maharashtra SSC Geometry (Mathematics Part II) Board Paper March 2024

Q.1 (A)(1) Out of the dates given below which date constitutes a Pythagorean triplet ?

Q.1 (A)(2) \(\sin\theta \times \operatorname{cosec}\theta = \,?\)

Q.1 (A)(3) Slope of X-axis is .................... .

Q.1 (A)(4) A circle having radius 3 cm, then the length of its largest chord is .................... .

Q.1 (B)(1) If \(\Delta ABC \sim \Delta PQR\) and \(AB : PQ = 2 : 3\), then find the value of \(\dfrac{A(\Delta ABC)}{A(\Delta PQR)}\).

Q.1 (B)(2) Two circles of radii 5 cm and 3 cm touch each other externally. Find the distance between their centres.

Q.1 (B)(3) Find the side of a square whose diagonal is \(10\sqrt{2}\) cm.

Q.1 (B)(4) Angle made by the line with the positive direction of X-axis is 45º. Find the slope of that line.

Q.2 (A)(1)

[Figure: a circle with centre O. Points A, B, C lie on the circle, B at the top and A, C lower down; chords AB and BC are drawn; X is a point on the arc AXC below, and the rays from A and from C through X are produced beyond X, outside the circle.]

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

If ∠ABC = 60º, find m∠AOC.

Solution :

∠ABC = \(\dfrac{1}{2}\) m(arc AXC) ............ ____

60º = \(\dfrac{1}{2}\) m(arc AXC)

____ = m(arc AXC)

But m∠AOC = m(arc .....) ............ (Property of central angle)

∴ m∠AOC = ____

Q.2 (A)(2) Find the value of \(\sin^2\theta + \cos^2\theta\).

[Figure: right-angled Δ ABC, ∠ABC = 90º, ∠C = θ, with AB the side opposite θ and BC the side adjacent to θ.]

Solution :

In Δ ABC, ∠ABC = 90º, ∠C = θ.

AB² + BC² = ____ ............ (Pythagoras theorem)

Divide both sides by AC²

\(\dfrac{AB^2}{AC^2} + \dfrac{BC^2}{AC^2} = \dfrac{AC^2}{AC^2}\)

∴ \(\left(\dfrac{AB}{AC}\right)^2 + \left(\dfrac{BC}{AC}\right)^2 = 1\)

But \(\dfrac{AB}{AC}\) = ____ and \(\dfrac{BC}{AC}\) = ____

∴ \(\sin^2\theta + \cos^2\theta\) = ____

Q.2 (A)(3)

[Figure: square ABCD with a circle inscribed in it; the four corner regions between the circle and the sides of the square are shaded.]

In the figure given above, □ 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 = (____)² ............ (Formula)

= 14²

= ____ cm²

Area of circle = ____ ............ (Formula)

= \(\dfrac{22}{7} \times 7 \times 7\)

= 154 cm²

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

= 196 – 154

= ____ cm²

Q.2 (B)(1) 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.

Q.2 (B)(2) Find the length of the hypotenuse of a right-angled triangle if remaining sides are 9 cm and 12 cm.

Q.2 (B)(3)

[Figure: a circle; two secants from an external point M cut the circle at E, N and at F, S respectively, so that M – E – N and M – F – S.]

In the above figure, m(arc NS) = 125º, m(arc EF) = 37º. Find the measure of ∠NMS.

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

Q.2 (B)(5) Find the surface area of a sphere of radius 7 cm.

Q.3 (A)(1)

[Figure: Δ ABC with D on side AC (A – D – C); ray BD bisects ∠ABC; E is on side AB (A – E – B) and seg DE || side BC.]

In Δ ABC, ray BD bisects ∠ABC, A – D – C, seg DE || side BC, A – E – B, then for showing \(\dfrac{AB}{BC} = \dfrac{AE}{EB}\), complete the following activity :

Proof :

In Δ ABC, ray BD bisects ∠B

∴ \(\dfrac{____}{BC} = \dfrac{AD}{DC}\) ............ (I) (____)

In Δ ABC, DE || BC

∴ \(\dfrac{____}{EB} = \dfrac{AD}{DC}\) ............ (II) (____)

\(\dfrac{AB}{____} = \dfrac{____}{EB}\) ............ [from (I) and (II)]

Q.3 (A)(2)

[Figure: a circle with centre P; chords AB and CD intersect at E inside the circle; seg AC and seg BD are drawn (dashed).]

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 ............ ____

____ ≅ ∠BDE (angles inscribed in the same arc)

∴ Δ CAE ~ Δ BDE ............ ____

∴ \(\dfrac{____}{DE} = \dfrac{CE}{____}\) ............ ____

∴ AE × EB = CE × ED.

Q.3 (B)(1) Determine whether the points are collinear.

A(1, –3), B(2, –5), C(–4, 7)

Q.3 (B)(2) \(\Delta ABC \sim \Delta LMN\). In \(\Delta ABC\), AB = 5.5 cm, BC = 6 cm, CA = 4.5 cm. Construct \(\Delta ABC\) and \(\Delta LMN\) such that \(\dfrac{BC}{MN} = \dfrac{5}{4}\).

[Figure: none; construct both triangles.]

Q.3 (B)(3) Seg PM is a median of \(\Delta PQR\), PM = 9 and PQ² + PR² = 290, then find QR.

Q.3 (B)(4) Prove that, ‘If a line parallel to a side of a triangle intersects the remaining sides in two distinct points, then the line divides the side in the same proportion’.

Q.4 (1) \(\dfrac{1}{\sin^2\theta} - \dfrac{1}{\cos^2\theta} - \dfrac{1}{\tan^2\theta} - \dfrac{1}{\cot^2\theta} - \dfrac{1}{\sec^2\theta} - \dfrac{1}{\operatorname{cosec}^2\theta} = -3\), then find the value of θ.

Q.4 (2) 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 ?

Q.4 (3) 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 = 70º.

Q.5 (1) □ ABCD is a trapezium, AB || CD, diagonals of trapezium intersects in point P.

Write the answers of the following questions :

(a) Draw the figure using given information.

(b) Write any one pair of alternate angles and opposite angles.

(c) Write the names of similar triangles with test of similarity.

Q.5 (2) 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 :

(a) Draw the figure using given information.

(b) Find the measures of ∠CAT and ∠ABC with reasons.

(c) Whether ∠CAT and ∠ABC are congruent ? Justify your answer.