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Maharashtra SSC Class 10 Geometry March 2022 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

  1. Q.1 (A) (i)1 mark
    If △ABC∼△DEF\triangle ABC \sim \triangle DEF and ∠A=48∘\angle A = 48^\circ, then ∠D=\angle D = ______.

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  2. Q.1 (A) (ii)1 mark
    AP is a tangent at A drawn to the circle with centre O from an external point P. OP=12OP = 12 cm and ∠OPA=30∘\angle OPA = 30^\circ, then the radius of the circle is ______.

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  3. Q.1 (A) (iii)1 mark
    Seg AB is parallel to X-axis and co-ordinates of the point A are (1,3)(1, 3), then the co-ordinates of the point B can be ______.

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  4. Q.1 (A) (iv)1 mark
    The value of 2tan⁡45∘−2sin⁡30∘2\tan 45^\circ - 2\sin 30^\circ is ______.

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Q.1 (B)

1 mark each

  1. Q.1 (B) (i)1 mark
    In △ABC\triangle ABC, ∠ABC=90∘\angle ABC = 90^\circ, ∠BAC=∠BCA=45∘\angle BAC = \angle BCA = 45^\circ. If AC=92AC = 9\sqrt{2}, then find the value of AB.
  2. Q.1 (B) (ii)1 mark
    Chord AB and chord CD of a circle with centre O are congruent. If m(arc AB)=120∘m(\text{arc } AB) = 120^\circ, then find the m(arc CD)m(\text{arc } CD).
  3. Q.1 (B) (iii)1 mark
    Find the Y-co-ordinate of the centroid of a triangle whose vertices are (4,−3)(4, -3), (7,5)(7, 5) and (−2,1)(-2, 1).
  4. Q.1 (B) (iv)1 mark
    If sin⁡θ=cos⁡θ\sin\theta = \cos\theta, then what will be the measure of angle θ\theta?

Q.2 (A)

2 marks each · attempt any 2

  1. Q.2 (A) (i)2 marks
    In the figure, seg AC and seg BD intersect each other at point P. If APCP=BPDP\dfrac{AP}{CP} = \dfrac{BP}{DP}, then complete the following activity to prove △ABP∼△CDP\triangle ABP \sim \triangle CDP.
  2. Q.2 (A) (ii)2 marks
    In the figure, □ABCD\square ABCD is a rectangle. If AB=5AB = 5, AC=13AC = 13, then complete the following activity to find BC.
  3. Q.2 (A) (iii)2 marks
    Complete the following activity to prove: cot⁡θ+tan⁡θ=cosec⁡θ×sec⁡θ\cot\theta + \tan\theta = \operatorname{cosec}\theta \times \sec\theta.

Q.2 (B)

2 marks each · attempt any 4

  1. Q.2 (B) (i)2 marks
    If △ABC∼△PQR\triangle ABC \sim \triangle PQR, AB:PQ=4:5AB : PQ = 4 : 5 and A(△PQR)=125A(\triangle PQR) = 125 cm2^2, then find A(△ABC)A(\triangle ABC).
  2. Q.2 (B) (ii)2 marks
    In the figure, m(arc DXE)=105∘m(\text{arc } DXE) = 105^\circ, m(arc AYC)=47∘m(\text{arc } AYC) = 47^\circ, then find the measure of ∠DBE\angle DBE.
  3. Q.2 (B) (iii)2 marks
    Draw a circle of radius 3.2 cm and centre 'O'. Take any point P on it. Draw a tangent to the circle through point P using the centre of the circle.
  4. Q.2 (B) (iv)2 marks
    If sin⁡θ=1161\sin\theta = \dfrac{11}{61}, then find the value of cos⁡θ\cos\theta using trigonometric identity.
  5. Q.2 (B) (v)2 marks
    In △ABC\triangle ABC, AB=9AB = 9 cm, BC=40BC = 40 cm, AC=41AC = 41 cm. State whether △ABC\triangle ABC is a right-angled triangle or not? Write reason.

Q.3 (A)

3 marks each · attempt any 1

  1. Q.3 (A) (i)3 marks
    In the figure, chord PQ and chord RS intersect each other at point T. If ∠STQ=58∘\angle STQ = 58^\circ and ∠PSR=24∘\angle PSR = 24^\circ, then complete the following activity to verify: ∠STQ=12[m(arc PR)+m(arc SQ)]\angle STQ = \dfrac{1}{2}\left[m(\text{arc } PR) + m(\text{arc } SQ)\right].
  2. Q.3 (A) (ii)3 marks
    Complete the following activity to find the co-ordinates of point P which divides seg AB in the ratio 3:13 : 1 where A(4,−3)A(4, -3) and B(8,5)B(8, 5).

Q.3 (B)

3 marks each · attempt any 2

  1. Q.3 (B) (i)3 marks
    In △ABC\triangle ABC, seg XY ∥\parallel side AC. If 2AX=3BX2AX = 3BX and XY=9XY = 9, then find the value of AC.
  2. Q.3 (B) (ii)3 marks
    Prove that, "Opposite angles of a cyclic quadrilateral are supplementary".
  3. Q.3 (B) (iii)3 marks
    △ABC∼△PQR\triangle ABC \sim \triangle PQR. In △ABC\triangle ABC, AB=5.4AB = 5.4 cm, BC=4.2BC = 4.2 cm, AC=6.0AC = 6.0 cm, AB:PQ=3:2AB : PQ = 3 : 2, then construct △ABC\triangle ABC and △PQR\triangle PQR.
  4. Q.3 (B) (iv)3 marks
    Show that: tan⁡A(1+tan⁡2A)2+cot⁡A(1+cot⁡2A)2=sin⁡A×cos⁡A.\dfrac{\tan A}{\left(1 + \tan^2 A\right)^2} + \dfrac{\cot A}{\left(1 + \cot^2 A\right)^2} = \sin A \times \cos A.

Q.4

4 marks each · attempt any 2

  1. Q.4 (i)4 marks
    □ABCD\square ABCD is a parallelogram. Point P is the midpoint of side CD. Seg BP intersects diagonal AC at point X, then prove that: 3AX=2AC3AX = 2AC.
  2. Q.4 (ii)4 marks
    In the figure, seg AB and seg AD are tangent segments drawn to a circle with centre C from exterior point A, then prove that: ∠A=12[m(arc BYD)−m(arc BXD)]\angle A = \dfrac{1}{2}\left[m(\text{arc } BYD) - m(\text{arc } BXD)\right].
  3. Q.4 (iii)4 marks
    Find the co-ordinates of centroid of a triangle if points D(−7,6)D(-7, 6), E(8,5)E(8, 5) and F(2,−2)F(2, -2) are the mid-points of the sides of that triangle.

Q.5

3 marks each · attempt any 1

  1. Q.5 (i)3 marks
    If a and b are natural numbers and a>ba > b. If (a2+b2)(a^2 + b^2), (a2−b2)(a^2 - b^2) and 2ab2ab are the sides of the triangle, then prove that the triangle is right angled. Find out two Pythagorean triplets by taking suitable values of a and b.
  2. Q.5 (ii)3 marks
    Construct two concentric circles with centre O with radii 3 cm and 5 cm. Construct a tangent to the smaller circle from any point A on the larger circle. Measure and write the length of the tangent segment. Calculate the length of the tangent segment using Pythagoras theorem.