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Maharashtra SSC Class 10 Geometry March 2019 question paper

Maximum marks 40 · Time 2 hours

Try each question first, then open its model answer. Where the paper offers a choice, both questions are shown with OR between them.

Q.1 (A)

1 mark each · attempt any 4

  1. Solve the following questions (Any four):
    Q.1 (A) (i)1 mark
    If △ABC∼△PQR\triangle ABC \sim \triangle PQR and ∠A=60∘\angle A = 60^\circ, then ∠P=?\angle P = ?
  2. Q.1 (A) (ii)1 mark
    In right-angled △ABC\triangle ABC, if ∠B=90∘\angle B = 90^\circ, AB=6AB = 6, BC=8BC = 8, then find AC.
  3. Q.1 (A) (iii)1 mark
    Write the length of the largest chord of a circle with radius 3.2 cm.
  4. Q.1 (A) (iv)1 mark
    From the given number line, find d(A, B).
  5. Q.1 (A) (v)1 mark
    Find the value of sin⁡30∘+cos⁡60∘\sin 30^\circ + \cos 60^\circ.
  6. Q.1 (A) (vi)1 mark
    Find the area of a circle of radius 7 cm.

Q.1 (B)

2 marks each · attempt any 2

  1. Solve the following questions (Any two):
    Q.1 (B) (i)2 marks
    Draw seg AB of length 5.7 cm and bisect it.
  2. Q.1 (B) (ii)2 marks
    In right-angled triangle PQR, if ∠P=60∘\angle P = 60^\circ, ∠R=30∘\angle R = 30^\circ and PR=12PR = 12, then find the values of PQ and QR.
  3. Q.1 (B) (iii)2 marks
    In a right circular cone, if perpendicular height is 12 cm and radius is 5 cm, then find its slant height.

Q.2 (A)

1 mark each

  1. Choose the correct alternative:
    Q.2 (A) (i)1 mark
    △ABC\triangle ABC and △DEF\triangle DEF are equilateral triangles. If A(△ABC):A(△DEF)=1:2A(\triangle ABC) : A(\triangle DEF) = 1 : 2 and AB=4AB = 4, then what is the length of DE?

    Tap an option to check your answer.

  2. Q.2 (A) (ii)1 mark
    Out of the following, which is a Pythagorean triplet?

    Tap an option to check your answer.

  3. Q.2 (A) (iii)1 mark
    ∠ACB\angle ACB is inscribed in arc ACB of a circle with centre O. If ∠ACB=65∘\angle ACB = 65^\circ, find m(arc ACB).

    Tap an option to check your answer.

  4. Q.2 (A) (iv)1 mark
    1+tan⁡2θ=?1 + \tan^2 \theta = ?

    Tap an option to check your answer.

Q.2 (B)

2 marks each · attempt any 2

  1. Solve the following questions (Any two):
    Q.2 (B) (i)2 marks
    Construct a tangent to a circle with centre A and radius 3.4 cm at any point P on it.
  2. Q.2 (B) (ii)2 marks
    Find the slope of a line passing through the points A(3, 1) and B(5, 3).
  3. Q.2 (B) (iii)2 marks
    Find the surface area of a sphere of radius 3.5 cm.

Q.3 (A)

2 marks each · attempt any 2

  1. Complete the following activities (Any two):
    Q.3 (A) (i)2 marks
    In △ABC\triangle ABC, ray BD bisects ∠ABC\angle ABC. If A-D-C, A-E-B and seg ED ∥\parallel side BC, then prove that ABBC=AEEB\frac{AB}{BC} = \frac{AE}{EB}.
  2. Q.3 (A) (ii)2 marks
    Prove that angles inscribed in the same arc are congruent. Given: ∠PQR\angle PQR and ∠PSR\angle PSR are inscribed in the same arc, and arc PXR is intercepted by the angles. To prove: ∠PQR≅∠PSR\angle PQR \cong \angle PSR.
  3. Q.3 (A) (iii)2 marks
    How many solid cylinders of radius 6 cm and height 12 cm can be made by melting a solid sphere of radius 18 cm?

Q.3 (B)

2 marks each · attempt any 2

  1. Solve the following questions (Any two):
    Q.3 (B) (i)2 marks
    In right-angled △ABC\triangle ABC, BD⊥ACBD \perp AC. If AD=4AD = 4 and DC=9DC = 9, then find BD.
  2. Q.3 (B) (ii)2 marks
    Verify whether the following points are collinear or not: A(1, -3), B(2, -5), C(-4, 7).
  3. Q.3 (B) (iii)2 marks
    If sec⁡θ=257\sec\theta = \frac{25}{7}, then find the value of tan⁡θ\tan\theta.

Q.4

3 marks each · attempt any 3

  1. Solve the following questions (Any three):
    Q.4 (i)3 marks
    In △PQR\triangle PQR, seg PM is a median, PM=9PM = 9 and PQ2+PR2=290PQ^2 + PR^2 = 290. Find the length of QR.
  2. Q.4 (ii)3 marks
    In the given figure, O is the centre of the circle. ∠QPR=70∘\angle QPR = 70^\circ and m(arc PYR)=160∘m(\text{arc PYR}) = 160^\circ, then find the value of each of the following: (a) m(arc QXR), (b) ∠QOR\angle QOR, (c) ∠PQR\angle PQR.
  3. Q.4 (iii)3 marks
    Draw a circle with radius 4.2 cm. Construct tangents to the circle from a point at a distance of 7 cm from the centre.
  4. Q.4 (iv)3 marks
    When an observer at a distance of 12 m from a tree looks at the top of the tree, the angle of elevation is 60∘60^\circ. What is the height of the tree? (3=1.73)(\sqrt{3} = 1.73)

Q.5

4 marks each · attempt any 1

  1. Solve the following questions (Any one):
    Q.5 (i)4 marks
    A circle with centre P is inscribed in △ABC\triangle ABC. Side AB, side BC and side AC touch the circle at points L, M and N respectively. The radius of the circle is r. Prove that A(△ABC)=12(AB+BC+AC)×rA(\triangle ABC) = \frac{1}{2}(AB + BC + AC) \times r.
  2. Q.5 (ii)4 marks
    In △ABC\triangle ABC, ∠ACB=90∘\angle ACB = 90^\circ. Seg CD ⊥\perp side AB and seg CE is the angle bisector of ∠ACB\angle ACB. Prove that ADBD=AE2BE2\frac{AD}{BD} = \frac{AE^2}{BE^2}.

Q.6

3 marks each · attempt any 1

  1. Solve the following questions (Any one):
    Q.6 (i)3 marks
    Show that the points (2, 0), (-2, 0) and (0, 2) are the vertices of a triangle. Also state with reason the type of the triangle.
  2. Q.6 (ii)3 marks
    In the figure, □XLMT\square XLMT is a rectangle with LM=21LM = 21 cm and XL=10.5XL = 10.5 cm. The diameter of the smaller semicircle is half the diameter of the larger semicircle. Find the area of the non-shaded region.