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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.

Let the shorter side of the rectangular field be x m.

Then, diagonal of the rectangular field = ( x+ 16) m

Also, longer side of the rectangular field = (x +14) m

In right ΔABC

left parenthesis AB right parenthesis squared space plus left parenthesis BC right parenthesis squared space equals space left parenthesis AC right parenthesis squared rightwards double arrow space left parenthesis straight x space plus 14 right parenthesis squared space plus space straight x squared space equals space left parenthesis space straight x space plus space 16 right parenthesis squared rightwards double arrow space straight x squared space plus 196 space plus 28 straight x space plus space straight x squared space equals space straight x squared space plus 256 space plus space 32 straight x rightwards double arrow space straight x squared space minus 4 straight x space minus 60 space equals space 0 rightwards double arrow space straight x squared space minus 10 straight x space plus space 6 straight x minus 60 space equals space 0 rightwards double arrow straight x left parenthesis straight x space minus space 10 right parenthesis space plus space 6 space left parenthesis straight x space minus 10 right parenthesis space equals space 0 rightwards double arrow space straight x space plus space 6 space equals space 0 space or space straight x minus 10 space equals 0 rightwards double arrow space straight x space equals negative 6 space or space space straight x space equals space 10

Since length cannot be negative, so x = 10

therefore, the length of the shorter side = 10 m

Length of the diagonal = 10 +16 = 26m

Length of the longer side = 10 +14 = 24

Hence, the length of the sides of the rectangular field is 10 m and 24 m.

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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.

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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).