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CBSE Class 10 Mathematics Solved Question Paper 2013

Long Answer Type

31.

A bucket open at the top, and made up of a metal sheet is in the form ofa frustum of a cone. The depth of the bucket is 24 cm and the diameters of its upper and lower circular ends are 30 cm and 10 cm respectively.Find the cost of metal sheet used in it at the rate of Rs 10 per 100 cm2.
$(Use π = 3.14)$

32.
In fig., l and m are two parallel tangents to a circle with centre O,touching the circle at A and B respectively. Another tangent at C intersects the line l at D and m at E. Prove that $∠$ DOE = 90⁰

$Given : l and m are two parallel tangents to the circle with centre O touching the circle at A and B respectively . DE is a tangent at the point C, which intersect l at D and m at E . To prove : ∠ DOE = 90 ° Contruction : Join OC . Proof :$

$In ∆ ODA and ∆ ODC, OA = OC (Radii of the same circle) AD = DC (Length of tangents drawn from an external point to a circle are equal) DO = OD (Commmon side) ∴ ∆ ODA ≅ ∆ ODC, (SSS congruence criterion) ∴ ∠ DOA = ∠ COD . . . . . . . . . . (1) Similarly, ∆ OEB ≅ ∆ OEC ∴ ∠ EOB = ∠ COE . . . . . . . . . . . (2) Now, AOB is a diameter of the circle . Hence, it is a straight line . ∠ DOA + ∠ COD + ∠ COE + ∠ EOB = 180 ° From (1) and (2), we have : \Rightarrow 2 ∠ COD + 2 ∠ COE = 180 ° \Rightarrow ∠ COD + ∠ COE = 90 ° \Rightarrow ∠ DOE = 90 ° Hence, proved .$

33.

Prove that the tangent at any point of a circle is perpendicular to theradius through the point of contact.

34.

Two poles of equal heights are standing opposite each other on either sideof the roads, which is 80 m wide. From a point between them on theroad, the angles of elevation of the top of the poles are 60o and 30orespectively. Find the height of the poles and the distances of the pointfrom the poles.