PYQ Vault

Maharashtra SSC Class 10 Geometry March 2016 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
    △DEF∼△MNK\triangle DEF \sim \triangle MNK. If DE=5DE = 5, MN=6MN = 6, then find the value of A(△DEF)A(△MNK)\dfrac{A(\triangle DEF)}{A(\triangle MNK)}.
  2. Q.1 (ii)1 mark
    In △ABC\triangle ABC, ∠B=90∘\angle B = 90^\circ, ∠C=60∘\angle C = 60^\circ, ∠A=30∘\angle A = 30^\circ and AC=18AC = 18. Find BCBC.
  3. Q.1 (iii)1 mark
    In the figure, line SRSR is a tangent to the circle at QQ and m(arc PMQ)=130∘m(\text{arc } PMQ) = 130^\circ. Find ∠PQS\angle PQS.
  4. Q.1 (iv)1 mark
    If the angle θ=−60∘\theta = -60^\circ, find the value of cos⁡θ\cos\theta.
  5. Q.1 (v)1 mark
    Find the slope of the line with inclination 30∘30^\circ.
  6. Q.1 (vi)1 mark
    Using Euler's formula, find VV if E=30E = 30, F=12F = 12.

Q.2

2 marks each · attempt any 4

  1. Q.2 (i)2 marks
    In △PQR\triangle PQR, seg RSRS is the bisector of ∠PRQ\angle PRQ. If PS=6PS = 6, SQ=8SQ = 8, PR=15PR = 15, find QRQR.
  2. Q.2 (ii)2 marks
    In the figure, a tangent segment PAPA touching a circle at AA and a secant PBCPBC is shown. If AP=15AP = 15, BP=10BP = 10, find BCBC.
  3. Q.2 (iii)2 marks
    Draw an equilateral △ABC\triangle ABC with side 6.26.2 cm and construct its circumcircle.
  4. Q.2 (iv)2 marks
    For the angle in standard position, if the initial arm rotates 25∘25^\circ in the anticlockwise direction, then state the quadrant in which the terminal arm lies. (Draw the figure and write the answer.)
  5. Q.2 (v)2 marks
    Find the area of a sector whose arc length and radius are 1010 cm and 55 cm respectively.
  6. Q.2 (vi)2 marks
    Find the surface area of a sphere of radius 4.24.2 cm. (π=227)\left(\pi = \dfrac{22}{7}\right)

Q.3

3 marks each · attempt any 3

  1. Q.3 (i)3 marks
    Adjacent sides of a parallelogram are 1111 cm and 1717 cm. If the length of one of its diagonals is 2626 cm, find the length of the other.
  2. Q.3 (ii)3 marks
    In the figure, secants containing chords RSRS and PQPQ of a circle intersect each other at point AA in the exterior of the circle. If m(arc PCR)=26∘m(\text{arc } PCR) = 26^\circ and m(arc QDS)=48∘m(\text{arc } QDS) = 48^\circ, then find: (i) m∠PQRm\angle PQR (ii) m∠SPQm\angle SPQ (iii) m∠RAQm\angle RAQ.
  3. Q.3 (iii)3 marks
    Draw a circle of radius 3.53.5 cm. Take any point KK on it. Draw a tangent to the circle at KK without using the centre of the circle.
  4. Q.3 (iv)3 marks
    If sec⁡α=23\sec\alpha = \dfrac{2}{\sqrt{3}}, then find the value of 1−csc⁡α1+csc⁡α\dfrac{1 - \csc\alpha}{1 + \csc\alpha}, where α\alpha is in the IV quadrant.
  5. Q.3 (v)3 marks
    Write the equation of the line passing through the pair of points (2,3)(2, 3) and (4,7)(4, 7) in the form y=mx+cy = mx + c.

Q.4

4 marks each · attempt any 2

  1. Q.4 (i)4 marks
    Prove that "The lengths of the two tangent segments to a circle drawn from an external point are equal."
  2. Q.4 (ii)4 marks
    A person standing on the bank of a river observes that the angle of elevation of the top of a tree standing on the opposite bank is 60∘60^\circ. When he moves 4040 m away from the bank, he finds the angle of elevation to be 30∘30^\circ. Find the height of the tree and the width of the river. (3=1.73)(\sqrt{3} = 1.73)
  3. Q.4 (iii)4 marks
    A(5,4)A(5, 4), B(−3,−2)B(-3, -2) and C(1,−8)C(1, -8) are the vertices of △ABC\triangle ABC. Find the equations of median ADAD and the line parallel to ACAC passing through the point BB.

Q.5

5 marks each · attempt any 2

  1. Q.5 (i)5 marks
    In the figure, AE=EF=AF=BE=CF=aAE = EF = AF = BE = CF = a and AT⊥BCAT \perp BC. Show that AB=AC=3 aAB = AC = \sqrt{3}\,a.
  2. Q.5 (ii)5 marks
    △SHR∼△SVU\triangle SHR \sim \triangle SVU. In △SHR\triangle SHR, SH=4.5SH = 4.5 cm, HR=5.2HR = 5.2 cm, SR=5.8SR = 5.8 cm and SHSV=35\dfrac{SH}{SV} = \dfrac{3}{5}. Construct △SVU\triangle SVU.
  3. Q.5 (iii)5 marks
    Water flows at the rate of 1515 m per minute through a cylindrical pipe having diameter 2020 mm. How much time will it take to fill a conical vessel of base diameter 4040 cm and depth 4545 cm?