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Maharashtra SSC Class 10 Geometry March 2024 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) (1)1 mark
    Out of the dates given below, which date constitutes a Pythagorean triplet?

    Tap an option to check your answer.

  2. Q.1 (A) (2)1 mark
    sin⁡θ×cosec⁡θ=\sin\theta \times \operatorname{cosec}\theta = ?

    Tap an option to check your answer.

  3. Q.1 (A) (3)1 mark
    Slope of the X-axis is ____.

    Tap an option to check your answer.

  4. Q.1 (A) (4)1 mark
    A circle has radius 3 cm. Then the length of its largest chord is ____.

    Tap an option to check your answer.

Q.1 (B)

1 mark each

  1. Q.1 (B) (1)1 mark
    If △ABC∼△PQR\triangle ABC \sim \triangle PQR and AB:PQ=2:3AB : PQ = 2 : 3, then find the value of A(△ABC)A(△PQR)\dfrac{A(\triangle ABC)}{A(\triangle PQR)}.
  2. Q.1 (B) (2)1 mark
    Two circles of radii 5 cm and 3 cm touch each other externally. Find the distance between their centres.
  3. Q.1 (B) (3)1 mark
    Find the side of a square whose diagonal is 10210\sqrt{2} cm.
  4. Q.1 (B) (4)1 mark
    Angle made by a line with the positive direction of the X-axis is 45∘45^\circ. Find the slope of that line.

Q.2 (A)

2 marks each · attempt any 2

  1. Q.2 (A) (1)2 marks
    In the figure, ∠ABC\angle ABC is inscribed in arc ABC. If ∠ABC=60∘\angle ABC = 60^\circ, complete the following activity to find m∠AOCm\angle AOC.
  2. Q.2 (A) (2)2 marks
    In right-angled △ABC\triangle ABC, ∠ABC=90∘\angle ABC = 90^\circ and ∠C=θ\angle C = \theta. Complete the following activity to find the value of sin⁡2θ+cos⁡2θ\sin^2\theta + \cos^2\theta.
  3. Q.2 (A) (3)2 marks
    In the figure, □ABCD\square ABCD is a square and a circle is inscribed in it so that all sides of the square touch the circle. If AB=14AB = 14 cm, complete the following activity to find the area of the shaded region.

Q.2 (B)

2 marks each · attempt any 4

  1. Q.2 (B) (1)2 marks
    Radius of a sector of a circle is 3.5 cm and the length of its arc is 2.2 cm. Find the area of the sector.
  2. Q.2 (B) (2)2 marks
    Find the length of the hypotenuse of a right-angled triangle if its remaining sides are 9 cm and 12 cm.
  3. Q.2 (B) (3)2 marks
    In the figure, m(arc NS)=125∘m(\text{arc } NS) = 125^\circ and m(arc EF)=37∘m(\text{arc } EF) = 37^\circ. Find the measure of ∠NMS\angle NMS.
  4. Q.2 (B) (4)2 marks
    Find the slope of the line passing through the points A(2,3)A(2, 3) and B(4,7)B(4, 7).
  5. Q.2 (B) (5)2 marks
    Find the surface area of a sphere of radius 7 cm. (π=227)\left(\pi = \dfrac{22}{7}\right)

Q.3 (A)

3 marks each · attempt any 1

  1. Q.3 (A) (1)3 marks
    In △ABC\triangle ABC, ray BD bisects ∠ABC\angle ABC with A−D−CA - D - C, and seg DE∥DE \parallel side BC with A−E−BA - E - B. Complete the following activity to show that ABBC=AEEB\dfrac{AB}{BC} = \dfrac{AE}{EB}.
  2. Q.3 (A) (2)3 marks
    Chords AB and CD of a circle with centre P intersect at point E. Draw seg AC and seg BD, then complete the following activity to prove that AE×EB=CE×EDAE \times EB = CE \times ED.

Q.3 (B)

3 marks each · attempt any 2

  1. Q.3 (B) (1)3 marks
    Determine whether the points A(1,−3)A(1, -3), B(2,−5)B(2, -5), C(−4,7)C(-4, 7) are collinear.
  2. Q.3 (B) (2)3 marks
    △ABC∼△LMN\triangle ABC \sim \triangle LMN. In △ABC\triangle ABC, AB=5.5AB = 5.5 cm, BC=6BC = 6 cm, CA=4.5CA = 4.5 cm. Construct △ABC\triangle ABC and △LMN\triangle LMN such that BCMN=54\dfrac{BC}{MN} = \dfrac{5}{4}.
  3. Q.3 (B) (3)3 marks
    Seg PM is a median of △PQR\triangle PQR, PM=9PM = 9 and PQ2+PR2=290PQ^2 + PR^2 = 290. Then find QR.
  4. Q.3 (B) (4)3 marks
    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 remaining sides in the same proportion.'

Q.4

4 marks each · attempt any 2

  1. Q.4 (1)4 marks
    If 1sin⁡2θ−1cos⁡2θ−1tan⁡2θ−1cot⁡2θ−1sec⁡2θ−1cosec⁡2θ=−3\dfrac{1}{\sin^2\theta} - \dfrac{1}{\cos^2\theta} - \dfrac{1}{\tan^2\theta} - \dfrac{1}{\cot^2\theta} - \dfrac{1}{\sec^2\theta} - \dfrac{1}{\operatorname{cosec}^2\theta} = -3, then find the value of θ\theta.
  2. Q.4 (2)4 marks
    A cylinder of radius 12 cm contains water up to a height of 20 cm. A spherical iron ball is dropped into the cylinder and the water level rises by 6.75 cm. What is the radius of the iron ball?
  3. Q.4 (3)4 marks
    Draw a circle with centre O and radius 3 cm. Draw tangent segments PA and PB through a point P outside the circle such that ∠APB=70∘\angle APB = 70^\circ.

Q.5

3 marks each · attempt any 1

  1. Q.5 (1)3 marks
    □ABCD\square ABCD is a trapezium with AB∥CDAB \parallel CD. The diagonals of the trapezium intersect at point P. (a) Draw the figure using the given information. (b) Write any one pair of alternate angles and one pair of opposite (vertically opposite) angles. (c) Write the names of the similar triangles with the test of similarity.
  2. Q.5 (2)3 marks
    AB is a chord of a circle with centre O. AOC is a diameter of the circle and AT is a tangent at A. (a) Draw the figure using the given information. (b) Find the measures of ∠CAT\angle CAT and ∠ABC\angle ABC with reasons. (c) Are ∠CAT\angle CAT and ∠ABC\angle ABC congruent? Justify your answer.