Maharashtra SSC Geometry (Mathematics Part II) Board Paper July 2024

Q.1 (A)(1) In \(\Delta ABC\) and \(\Delta PQR\), in a one to one correspondence of vertices, \(\dfrac{AB}{QR} = \dfrac{BC}{PR} = \dfrac{CA}{PQ}\), then ....................... statement is true.

[Figure: a small triangle ABC with A at the top left, B below it and C to the right of B; a larger similarly shaped triangle with Q at the top, R below Q and P to the right of R, so that A corresponds to Q, B to R and C to P.]

Q.1 (A)(2) Two circles of radii 5.5 cm and 3.3 cm respectively touch each other externally. Then the distance between their centres is .............

Q.1 (A)(3) Seg AB is parallel to Y-axis and co-ordinates of point A are (1, 3), then the co-ordinates of point B are ........................

Q.1 (A)(4) The volume of a cube of side 10 cm is ..............................

Q.1 (B)(1) In \(\Delta ABC\), \(\angle B = 90^\circ\), \(\angle C = 30^\circ\), AC = 12 cm, then find AB.

[Figure: right-angled triangle ABC with the right angle at B, hypotenuse AC = 12 cm and \(\angle C = 30^\circ\).]

Q.1 (B)(2) In the above figure, \(m(\text{arc } MN) = 70^\circ\), find \(\angle MLN\).

[Figure: a circle with points L, M, N on it; chords LM and LN are drawn, so that \(\angle MLN\) is an inscribed angle standing on arc MN.]

Q.1 (B)(3) Find the value of \(\sin\theta \times \operatorname{cosec}\theta\).

Q.1 (B)(4) If radius of a circle is 4 cm and length of an arc is 10 cm, then find the area of the sector.

Q.2 (A)(1) In the above figure, chord PQ and chord RS intersect at point T. Then complete the following activity to prove, \(\angle STQ = \dfrac{1}{2}\left[m(\text{arc } SQ) + m(\text{arc } PR)\right]\)

[Figure: a circle with points P, S above and R, Q below on the circumference; chords PQ and RS intersect at T inside the circle; chord PS is also drawn.]

Activity :

\(\angle STQ = \angle SPQ +\) ____ ............ (Remote interior angles theorem of a triangle)
\(= \dfrac{1}{2}m(\text{arc } SQ) +\) ____ ........... (Inscribed angle theorem)
\(= \dfrac{1}{2}\left[\text{____} + \text{____}\right]\)

Q.2 (A)(2) If \(\sec\theta = \dfrac{25}{7}\), complete the following activity to find the value of \(\tan\theta\).

Activity :

We have, \(1 + \tan^2\theta = \sec^2\theta\)
\(\therefore 1 + \tan^2\theta =\) ____
\(\therefore \tan^2\theta = \dfrac{625}{49} -\) ____
\(\therefore \tan^2\theta = \dfrac{625 - 49}{49}\)
\(\therefore \tan^2\theta = \dfrac{\text{____}}{49}\)
\(\therefore \tan\theta =\) ____

Q.2 (A)(3) Complete the following activity to find the volume of a cone if the radius of its base is 7 cm and its perpendicular height is 6 cm.

Activity :

Here, radius (r) = 7 cm, height (h) = 6 cm

\(\therefore\) Volume of cone = ____ ....... (Formula)
\(= \dfrac{1}{3}\times\dfrac{22}{7}\times\) ____ \(\times 6\)
\(= \dfrac{1}{3}\times\dfrac{22}{7}\times\) ____ \(\times 6\)
\(\therefore\) Volume of cone = ____ cm³

Q.2 (B)(1) Construct a tangent to a circle with centre P and radius 3.2 cm at any point M on it.

Q.2 (B)(2) In \(\Delta PQR\), seg RS bisects \(\angle PRQ\). If PR = 15, RQ = 20, PS = 12, then find SQ.

[Figure: \(\Delta PQR\) with S on side PQ; seg RS is drawn and marks the two equal angles at R.]

Q.2 (B)(3) Find the surface area of a sphere of diameter 14 cm.

Q.2 (B)(4) In the above figure, \(\angle PQR = 90^\circ\), seg QN \(\perp\) seg PR, PN = 9, NR = 16, find QN.

[Figure: right-angled \(\Delta PQR\) with the right angle at Q; N lies on hypotenuse PR and seg QN is drawn perpendicular to PR, with PN = 9 and NR = 16.]

Q.2 (B)(5) Find the slope of the line passing through the points A(3, 3) and B(5, 7).

Q.3 (A)(1) In the given figure, AB || CD || EF. If AC = 12, CE = 9, BD = 8, then complete the following activity to find DF.

[Figure: three parallel lines with two transversals cutting them at A, C, E on one transversal and B, D, F on the other.]

Activity : AB || CD || EF

\(\therefore \dfrac{AC}{\text{____}} = \dfrac{\text{____}}{DF}\) ........... (Property of three parallel lines and their transversal)

\(\therefore \dfrac{12}{\text{____}} = \dfrac{\text{____}}{DF}\)

\(\therefore DF = \dfrac{8 \times 9}{\text{____}}\)

\(\therefore DF =\) ____

Q.3 (A)(2) A cylinder is having radius 7 cm and height 21 cm. Complete the following activity to calculate its volume and circumference of its base.

Activity :

Radius (r) = 7 cm, Height (h) = 21 cm

Volume of cylinder = ____ ........... (Formula)
\(= \dfrac{22}{7} \times 7 \times 7 \times\) ____
\(= 154 \times 21\)
\(=\) ____ cm³

Circumference of base = ____ ........ (Formula)
\(= 2 \times \dfrac{22}{7} \times\) ____
\(=\) ____ cm

Q.3 (B)(1) Prove that, ‘Opposite angles of a cyclic quadrilateral are supplementary’.

[Figure: a circle with centre O; points A, B, C, D on the circle in order, forming cyclic quadrilateral ABCD; diagonals AC and BD are drawn.]

Q.3 (B)(2) Draw a circle with centre P and radius 3.5 cm. Take point Q at a distance 8 cm from the centre. Construct tangents to the circle from point Q.

Q.3 (B)(3) Show that points P(–2, 3), Q(1, 2), R(4, 1) are collinear.

Q.3 (B)(4) If \(\Delta PQR \sim \Delta LMN\), \(9 \times A(\Delta PQR) = 16 \times A(\Delta LMN)\) and QR = 20, then find MN.

Q.4 (1) \(\Delta ABC\) is an equilateral triangle. Point D is on seg BC such that BD \(= \dfrac{1}{5}\) BC. Then prove that \(\dfrac{AD^2}{AB^2} = \dfrac{21}{25}\).

[Figure: equilateral triangle ABC with D on side BC, closer to B; seg AD is drawn, and AE is drawn perpendicular to BC with E the midpoint of BC.]

Q.4 (2) \(\Delta LMN \sim \Delta LQP\). In \(\Delta LMN\), LM = 3.6 cm, \(\angle L = 50^\circ\), LN = 4.2 cm and \(\dfrac{LM}{LQ} = \dfrac{4}{7}\). Construct \(\Delta LQP\).

Q.4 (3) To find the width of the river, a man observes the top of a tree, on the opposite bank making an angle of elevation of 60°. When he moves 24 meter backward from bank and observes the same top of the tree, his line of vision makes an angle of elevation of 30°. Find the height of the tree and width of the river. (\(\sqrt{3}\) = 1.73)

[Figure: a vertical tree of height h on the far bank; from the near bank the angle of elevation of its top is 60°, and from a point 24 m further back the angle of elevation is 30°; the width of the river is w.]

Q.5 (1) The diameter of circular garden is 13 meter. The distance between two gates of garden is 13 meter. An electric pole is to be erected on the circumference of the garden such that the difference between distance of the pole from each gate is 7 meter. Can such pole be erected ? If yes, find the distance of the pole from each gate.

[Figure: a circle of diameter 13 m with two gates A and B at the ends of a diameter; a pole P on the circumference with PA and PB drawn.]

Q.5 (2) The line \(x - 6y + 11 = 0\) bisects the segment joining the points (8, –1) and (0, k), then find the value of k.