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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.

The given AP is 8, 10, 12, ....

So,

First term =a = 8

Common difference = d = 10-8 =2

We know that n^th term of an AP, a_n = a + (n - 1)

60^th term of the given AP = a₆₀ = 8 +( 60-1) x 2 = 8 + 59 x 2 = 8 + 118 = 126

Therefore, the 60^th term of the given AP is 126

It is given that the AP has a total of 60 terms. So, in order to find sum of last n terms. we take

First term, A = 126
Common difference, D = -2
Now,
$Sum space of space straight n space term space of space AP space from space the space end space equals space straight n over 2 left square bracket 2 straight A space plus left parenthesis straight n space minus space 1 right parenthesis space straight D right square bracket therefore space Sum space of space last space 10 space terms space of space the space given space AP equals space 10 over 2 left square bracket space 2 space straight x space 126 space plus space left parenthesis 10 space minus space 1 space right parenthesis space straight x space left parenthesis negative 2 right parenthesis right square bracket equals space 5 space left square bracket 252 space plus 9 space straight x space left parenthesis negative 2 right parenthesis right square bracket equals space 2 space left parenthesis 252 space minus 18 right parenthesis equals space 5 space straight x space 234 equals space 1170 hence space comma space the space sum space of space last space 10 space terms space of space the space given space AP space is space 1170$

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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).