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Maharashtra SSC Class 10 Geometry March 2017 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

1 mark each · attempt any 5

  1. Q.1 (i)1 mark
    In the following figure, seg BE ⊥\perp seg AB and seg BA ⊥\perp seg AD. If BE = 6 and AD = 9, find A(△ABE)A(△BAD)\dfrac{A(\triangle ABE)}{A(\triangle BAD)}.
  2. Q.1 (ii)1 mark
    If two circles with radii 8 cm and 3 cm respectively touch internally, then find the distance between their centres.
  3. Q.1 (iii)1 mark
    Find the height of an equilateral triangle whose side is 6 units.
  4. Q.1 (iv)1 mark
    If the angle θ=−45∘\theta = -45^\circ, find the value of tan⁡θ\tan\theta.
  5. Q.1 (v)1 mark
    Find the slope and yy-intercept of the line y=3x−5y = 3x - 5.
  6. Q.1 (vi)1 mark
    Find the circumference of a circle whose radius is 7 cm.

Q.2

2 marks each · attempt any 4

  1. Q.2 (i)2 marks
    In △PQR\triangle PQR, seg RS is the bisector of ∠PRQ\angle PRQ. If PS = 6, SQ = 8 and PR = 12, find QR.
  2. Q.2 (ii)2 marks
    In the given figure, two chords AB and CD of a circle intersect at the point P. If PA = 10, PB = 2 and PC = 5, find PD.
  3. Q.2 (iii)2 marks
    Draw ∠ABC\angle ABC of measure 135∘135^\circ and bisect it.
  4. Q.2 (iv)2 marks
    Find the sine ratio of θ\theta in standard position whose terminal arm passes through (3,4)(3, 4).
  5. Q.2 (v)2 marks
    Find the slope of the line passing through the points G(4,5)G(4, 5) and H(−1,−2)H(-1, -2).
  6. Q.2 (vi)2 marks
    The dimensions of a cuboid in cm are 50×18×1050 \times 18 \times 10. Find its volume.

Q.3

3 marks each · attempt any 3

  1. Q.3 (i)3 marks
    Prove that: If the angles of a triangle are 45∘−45∘−90∘45^\circ - 45^\circ - 90^\circ, then each of the perpendicular sides is 12\dfrac{1}{\sqrt{2}} times the hypotenuse.
  2. Q.3 (ii)3 marks
    Find the angle between the two radii at the centre of the circle as shown in the figure. Lines PA and PB are tangents to the circle at the other ends A and B of the radii OA and OB, and ∠APR=140∘\angle APR = 140^\circ.
  3. Q.3 (iii)3 marks
    Construct tangents to a circle having radius 3.2 cm and centre 'C', from a point B which is at a distance 7.6 cm from the centre.
  4. Q.3 (iv)3 marks
    From the top of a lighthouse, an observer looks at a ship and finds the angle of depression to be 60∘60^\circ. If the height of the lighthouse is 90 metres, then find how far the ship is from the lighthouse. (3=1.73)(\sqrt{3} = 1.73)
  5. Q.3 (v)3 marks
    The volume of a cube is 343343 cm3^3. Find its total surface area.

Q.4

4 marks each · attempt any 2

  1. Q.4 (i)4 marks
    Prove that "The opposite angles of a cyclic quadrilateral are supplementary".
  2. Q.4 (ii)4 marks
    Eliminate θ\theta, if x=3csc⁡θ+4cot⁡θx = 3\csc\theta + 4\cot\theta and y=4csc⁡θ−3cot⁡θy = 4\csc\theta - 3\cot\theta.
  3. Q.4 (iii)4 marks
    A toy is a combination of a cylinder, hemisphere and a cone, each with radius 10 cm as shown in the figure. Height of the conical part is 10 cm and total height is 60 cm. Find the total surface area of the toy. (π=3.14, 2=1.41)(\pi = 3.14,\ \sqrt{2} = 1.41)

Q.5

5 marks each · attempt any 2

  1. Q.5 (i)5 marks
    In the given figure, AD is the bisector of the exterior ∠A\angle A of △ABC\triangle ABC. Seg AD intersects the side BC produced in D. Prove that BDCD=ABAC\dfrac{BD}{CD} = \dfrac{AB}{AC}.
  2. Q.5 (ii)5 marks
    Construct the circumcircle and incircle of an equilateral △XYZ\triangle XYZ with side 6.5 cm and centre O. Find the ratio of the radii of the incircle and the circumcircle.
  3. Q.5 (iii)5 marks
    A (5, 4), B (-3, -2) and C (1, -8) are the vertices of a triangle ABC. Find the equation of the median AD and the line parallel to AB passing through point C.