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CBSE Class 10 Mathematics Solved Question Paper 2015

Long Answer Type

21. The diagonal of a rectangular field is 16 metres more than the shorter side. If the longer side is 14 metres more than the shorter side, then find the lengths of the sides of the field.

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22.

Find the 60^th term of the AP 8, 10,12, ..., if it has a total of 60 terms and hence find the sum of its last 10 terms.

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23. A train travels at a certain average speed for a distance of 54 km and then travels a distance of 63 km at an average speed of 6 km/h more than the first speed. If it takes 3 hours to complete the total journey, what is its first speed?

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24.

Prove that the length of tangents drawn from an external point to a circle is equal.

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25. Prove that the tangent drawn at the midpoint of an arc of a circle is parallel to the chord joining the end points of the arc.

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26.

Construct a ΔABC in which AB = 6 cm, ∠A = 30^o and ∠B = 60^o. Construct another ΔAB'C' similar to ΔABC with base AB' = 8 cm

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27. At a point A, 20 metres above the level of water in a lake, the angle of elevation of a cloud is 30°. The angle of depression of the reflection of the cloud in the lake, at A is 60°. Find the distance of the cloud from A.

Let BE be the surface of the lake. Suppose C be the position of the cloud and C' be its reflection in the lake.
Then, AB = 20 m

Let CD = x m. Then,

CE =C'E = ( x + 20)

In ΔACD,

tan space 30 degree space equals space CD over AD rightwards double arrow space fraction numerator 1 over denominator square root of 3 space end fraction space equals space straight x over AD rightwards double arrow AD space equals space straight x square root of 3 space space.. space left parenthesis straight i right parenthesis Also comma sin space 30 degree space equals space CD over AC rightwards double arrow space 1 half space equals space straight x over AC rightwards double arrow space AC space equals 2 straight x space.... space left parenthesis ii right parenthesis Now comma space in space ΔADC apostrophe comma tan space 60 degree space equals space fraction numerator straight C apostrophe straight D over denominator AD end fraction rightwards double arrow space square root of 3 space equals space fraction numerator left parenthesis straight x space plus 20 right parenthesis space plus space 20 over denominator AD end fraction rightwards double arrow space AD space equals space fraction numerator left parenthesis straight x space plus 40 right parenthesis over denominator square root of 3 end fraction space... space left parenthesis iii right parenthesis From space left parenthesis straight i right parenthesis space and space left parenthesis iii right parenthesis comma space we space get straight x square root of 3 space equals space fraction numerator left parenthesis straight x space plus space 40 right parenthesis over denominator square root of 3 end fraction rightwards double arrow space 3 straight x space equals space straight x space plus 40 rightwards double arrow space 2 straight x space equals space 40 rightwards double arrow space straight x space equals space 20

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28. A card is drawn at random from a well-shuffled deck of playing cards.

(i) a card of a spade or an ace.
(ii) a black king
(iii) neither a jack nor a king
(iv) either a king or queen

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29.

Find the values of k so that the area of the triangle with vertices (1,-1), (-4, 2k) and (-k, 5) is 24 sq. units

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30. In Figure, PQRS is a square lawn with side PQ = 42 metres. Two circular flower beds are there on the sides PS and QR with centre at O, the intersection of its diagonals. Find the total area of the two flower beds (shaded parts).