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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.

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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).

Area of the square lawn PQRS = 42 m x 42 m

Let OP = OS = xm
So, x² + x² = (42)²

⇒ 2x² = 42 x 42

⇒ x² = 21 x 42

Now,
area of sector POS=

$90 over 360 space x space πx squared space equals space 1 fourth space straight x space left parenthesis πx right parenthesis squared space.... space left parenthesis space straight i space right parenthesis equals 1 fourth space space straight x space 22 over 7 space straight x space 21 space straight x space 42 space straight m squared space.... space left parenthesis ii right parenthesis Also comma space Area space of space increment POS space equals space 1 fourth space straight x space Area space of space square space lawn space PQRS space equals space 1 fourth space straight x space left parenthesis space 42 space straight x space 42 right parenthesis squared space straight m squared space space space space space left parenthesis angle POQ space equals space 90 degree right parenthesis space.. space left parenthesis iii right parenthesis So comma space Area space of space flower space bed space PSP space equals space Area space of space Sector space POS space minus space Area space of space increment POS space equals space 1 fourth space straight x space 22 over 7 space straight x space 21 space straight x 42 space minus space 1 fourth space straight x space 42 space straight x space 42 space space left square bracket frome space left parenthesis ii right parenthesis space and space left parenthesis iii right parenthesis right square bracket space equals space 1 fourth space straight x space 21 space straight x space 42 space straight x open parentheses 22 over 7 space minus space 2 close parentheses equals space 1 fourth space straight x space 21 space straight x space 42 space straight x space open parentheses 8 over 7 close parentheses straight m squared Therefore space area space of space the space two space flower space beds space equals space 2 space straight x space 1 fourth space straight x space 21 space straight x space 42 space straight x open parentheses space 8 over 7 close parentheses space space equals space 504 space straight m squared Hence comma space the space total space area space of space the space two space flower space beds space 504 space straight m squared$