Maharashtra SSC Geometry (Mathematics Part 2) Practice Paper 1 with Solutions | Class 10

Q.1 (A)(1) If Δ ABC ~ Δ PQR and AB : PQ = 2 : 3, then A(Δ ABC) : A(Δ PQR) = .................... .

Q.1 (A)(2) \(1 + \tan^2\theta =\) .................... .

Q.1 (A)(3) The measure of an angle inscribed in a semicircle is .................... .

Q.1 (A)(4) The distance of the point (−6, 8) from the origin is .................... .

Q.1 (B)(1) Is (9, 40, 41) a Pythagorean triplet? Give reason.

Q.1 (B)(2) Find the curved surface area of a cylinder whose radius is 7 cm and height is 10 cm. \(\left(\pi = \dfrac{22}{7}\right)\)

Q.1 (B)(3) Find the slope of the line passing through the points A(2, 5) and B(4, 11).

Q.1 (B)(4) Δ ABC ~ Δ DEF. If ∠A = 50° and ∠E = 60°, find ∠C.

Q.2 (A)(1) In the figure, A, B and C are points on a circle with centre O. Point C lies on the major arc AB, and ∠ACB = 40°. Seg OA and seg OB are drawn. Complete the following activity to find m(arc AB) and ∠AOB.

[Figure: a circle with centre O; chord AB near the bottom of the circle; point C on the upper (major) arc; seg CA and seg CB form the inscribed angle ∠ACB = 40°; radii OA and OB form the central angle ∠AOB.]

Solution :

∠ACB is an inscribed angle intercepting arc AB.
∴ m∠ACB = ½ × m(arc ____) .................... (inscribed angle theorem)
∴ m(arc AB) = 2 × ____ = ____
∠AOB is the central angle of arc AB.
∴ m∠AOB = m(arc AB) = ____

Q.2 (A)(2) If \(\sin\theta = \dfrac{5}{13}\) and \(\theta\) is an acute angle, complete the following activity to find \(\cos\theta\).

Solution :

\(\sin^2\theta + \cos^2\theta =\) ____
\(\therefore \left(\dfrac{5}{13}\right)^2 + \cos^2\theta = 1\)
\(\therefore \cos^2\theta = 1 -\) ____
\(= \dfrac{\text{____}}{169}\)
\(\therefore \cos\theta =\) ____

Q.2 (A)(3) The radius of a circle is 14 cm and the measure of the central angle of a sector is 45°. Complete the following activity to find the area of the sector and the length of its arc. \(\left(\pi = \dfrac{22}{7}\right)\)

Solution :

Area of sector \(= \dfrac{\theta}{360} \times\) ____
\(= \dfrac{45}{360} \times \dfrac{22}{7} \times 14 \times 14\)
\(=\) ____ cm²
Length of arc \(= \dfrac{\theta}{360} \times\) ____
\(= \dfrac{45}{360} \times 2 \times \dfrac{22}{7} \times 14\)
\(=\) ____ cm

Q.2 (B)(1) In Δ ABC, point D lies on side AB and point E lies on side AC (A–D–B and A–E–C) such that seg DE ∥ side BC. If AD = 6, DB = 9 and AE = 8, find EC.

Q.2 (B)(2) In Δ ABC, seg AD is a median (D is the midpoint of side BC). If \(AB^2 + AC^2 = 122\) and BC = 10, find AD.

Q.2 (B)(3) □ ABCD is a cyclic quadrilateral. If ∠A = (2x + 10)° and ∠C = (x + 20)°, find x, ∠A and ∠C.

Q.2 (B)(4) Using slopes, show that the points A(1, −1), B(2, 1) and C(4, 5) are collinear.

Q.2 (B)(5) Prove that : \(\cot\theta + \tan\theta = \text{cosec}\,\theta \sec\theta\).

Q.3 (A)(1) Δ ABC ~ Δ PQR. A(Δ ABC) = 80 cm², A(Δ PQR) = 125 cm² and AB = 12 cm. Complete the following activity to find PQ.

Solution :

Δ ABC ~ Δ PQR .................... (given)
\(\therefore \dfrac{A(\Delta ABC)}{A(\Delta PQR)} = \dfrac{\text{____}}{PQ^2}\) .................... (theorem of areas of similar triangles)
\(\therefore \dfrac{80}{125} = \dfrac{12^2}{PQ^2}\)
\(\therefore \dfrac{16}{25} = \dfrac{12^2}{PQ^2}\)
\(\therefore \dfrac{\text{____}}{5} = \dfrac{12}{PQ}\) .................... (taking square roots)
\(\therefore PQ =\) ____ cm

Q.3 (A)(2) The vertices of Δ ABC are A(−7, 6), B(2, −2) and C(8, 5). Complete the following activity to find the co-ordinates of the centroid G of Δ ABC.

Solution :

Let \(A(x_1, y_1) = (-7, 6)\), \(B(x_2, y_2) = (2, -2)\), \(C(x_3, y_3) = (8, 5)\).
By the centroid formula,
\(x = \dfrac{x_1 + x_2 + x_3}{3} = \dfrac{-7 + 2 + 8}{3} = \dfrac{\text{____}}{3} =\) ____
\(y = \dfrac{\text{____}}{3} = \dfrac{6 - 2 + 5}{3} = \dfrac{\text{____}}{3} =\) ____
\(\therefore\) G = ( ____ , ____ )

Q.3 (B)(1) 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 sides in the same proportion.

[Figure: Δ ABC with A at the top, B at the lower left and C at the lower right; a line parallel to side BC cuts side AB at P and side AC at Q, so A–P–B and A–Q–C; seg PC and seg BQ are drawn.]

Q.3 (B)(2) Draw a circle of radius 3.6 cm. Take any point M on the circle. Draw a tangent to the circle at point M without using the centre of the circle.

Q.3 (B)(3) A solid metallic sphere of radius 6 cm is melted and recast into a solid cylinder of radius 4 cm. Find the height of the cylinder.

Q.3 (B)(4) From the top of a lighthouse 50 m high, the angle of depression of a boat in the sea is 30°. Find the distance of the boat from the foot of the lighthouse. \((\sqrt{3} = 1.73)\)

Q.4 (1) Δ ABC ~ Δ PQR. In Δ ABC, AB = 5.4 cm, BC = 4.2 cm, AC = 6.0 cm and \(\dfrac{AB}{PQ} = \dfrac{3}{2}\). Construct Δ ABC and Δ PQR.

Q.4 (2) In Δ ABC, ∠ABC = 90° and seg BD ⊥ hypotenuse AC, with A–D–C. If AD = 9 cm and DC = 16 cm, find BD, AB and BC.

[Figure: right-angled Δ ABC with the right angle at B; the hypotenuse AC is horizontal at the bottom with A at the left and C at the right; B is above AC; the perpendicular BD from B meets AC at D.]

Q.4 (3) A chord of a circle of radius 14 cm subtends a right angle at the centre. Find the area of the minor segment and the area of the major segment. \(\left(\pi = \dfrac{22}{7}\right)\)

Q.5 (1) Show that the points A(1, 1), B(5, 4), C(2, 8) and D(−2, 5) are the vertices of a square.

Q.5 (2) Line XY touches the circle with centre O at point A, with X and Y on opposite sides of A. Seg AB is a chord of the circle and ∠BAY = 65°. Point C lies on the major arc AB. Seg OA, seg OB, seg CA and seg CB are drawn.

[Figure: a circle with centre O; the tangent XY touches it at A at the bottom; chord AB goes up to the right, so the minor arc AB lies inside ∠BAY on the Y side; point C is on the far (major) arc; radii OA and OB are drawn.]

(a) Find m(minor arc AB) and m(major arc AB).
(b) Find m∠ACB.
(c) Find m∠AOB and m∠OAB.