**Sample Paper – 2009**

**Class – X**

**Subject – ****Mathematics**

Test Series III : Paper 1

Time: 180 min Mathematics Marks: 80

__SECTION A __{10 marks}

1. Find the zeroes of: t^{2} – 24

2. For what value of ‘m’, the given lines are parallel: 3*x* – 2*y* = 5; 4*y* = m*x* – 8

3. Find the probability that a number selected at random from the numbers 1, 2, 3, …., 35 is a

prime number less than 20.

4. If 16 cot A = 12, find the value of: __sin A + cos A__

sin A – cos A

5. The perimeters of two similar triangles ABC and PQR are 36 cm and 24 cm. If PQ = 10 cm,

find AB.

6. Given that LCM (96, 404) = 9696, find HCF (96, 404)

7. Write the AP whose n^{th} term is 4n – 5.

8. An arc of circle is of length 6 π cm and sector it bounds has an area 30 π cm^{2}. Find the radius.

9. Surface area of a sphere is 4 π cm^{2}. Find its diameter

10. Express 0.375 as a rational number in the simplest form.

__SECTION B__ {10 marks}

11. Find the LCM and HCF of 40, 36 and 126 by applying factorization method.

12. Find the ratio in which the point A (m, 6) divides the join of P (– 4, 3) and Q (2, 8). Also find

the value of m.

13. All cards of ace and jack are removed and a card is drawn from a deck of playing cards. Find the probability of drawing (a) black face card (b) none face card

14. Solve : __ 1 __ = __1 __ + __1 __ + __1 __ OR

a + b + *x *a b * x*

14. Divide 4*x*^{3 }+ 2*x*^{2} + 5*x* – 6 by 2*x*^{2} + 3*x* + 1 and find quotient and remainder.

15. Show that any positive even integer is of the form 8p, 8p + 2, 8p + 4 and 8p + 6, where p is

some integer

__SECTION C__ {30 marks}

16. Find the coordinates of the centre of circle passing through the points (0, 0), (– 2,1) and

(– 3, 2). Also find the radius OR

16. The line segment joining the points (3, – 4) and (1, 2) is trisected at points P and Q. Find the coordinates of P and Q

17. An AP consists of 50 terms of which 3^{rd} term is 12 and last term is 106. Find the 29^{th} term OR

17. The sum of three numbers in AP is 27 and their product is 405. Find the numbers.

18. Draw a circle of radius 5**.**2 cm. Draw tangents at the ends of any diameter of the circle.

20. An equilateral triangle is inscribed in a circle of radius 32 cm. Find the area of region outside the triangle, but inside the circle. OR

20. A square park has side 200 m. At each corner of the park, there is a flower bed in the form of a sector of radius 28 m. Find the area of remaining part of the park and cost of developing it at

the rate of 50 paise per m^{2}

21 Two circles touch externally. The sum of their areas is 130 π cm^{2} and the distance between their centres is 14 cm. Find the radii of the circles.

22. Prove: __ (1 + cot θ + tan θ )(sin θ – cos θ) __ = sin^{2 }θ cos^{2} θ .

sec^{3} θ – cosec^{3} θ

23. The sum of digits of a two digit number is 9. Also nine times this number is twice the number

obtained by reversing the digits. Find the number

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

25. The two opposite vertices of a square are (– 1, 2) and (3, 2). Find the other two vertices OR

25. In what ratio is line segment joining the points (– 2, – 3) and (3, 7) divided by the *y *– axis?

Also find the coordinates of the point of division.

__SECTION D__ {30 marks}

26. There is a vertical tower with a flag pole on top of the tower. At a point 45 m away from the foot of the tower, the angle of elevation of top and bottom of the flag pole are 60^{o} and 30^{o}. Find the heights of the tower and flag pole. OR

26. A boy standing on ground finds a bird flying at a distance of 100 m from him at an elevation of

30^{o}. A girl standing on the roof of a 20 m high building finds the angle of elevation of the same bird to be 45^{o}. Both the boy and girl are on opposite sides of the bird. Find the distance of the bird from the girl.

27. The height of a cone is 40 cm. A small cone is cut off at the top by plane parallel to the base. If the volume of the cut cone is 1/ 64 of the volume of the given cone, find at what height above the base is the section made. Also find ratio of their curved surface areas. OR

27. Water is flowing at the rate of 10 m per minute through a pipe having diameter 5 mm. How much time will it take to fill 10 conical vessels having diameter 40 cm and depth 30 cm?

28. Using Empirical formula, find the mode of the following data.

Class | 20 – 25 | 25 – 30 | 30 – 35 | 35 – 40 | 40 – 45 |

Frequency | 3 | 8 | 8 | 3 | 2 |

29. State and prove converse of Pythagoras theorem. Also prove that in an equilateral ∆ ABC, if D

is a point on side BC such that 3BD = BC, prove that 9AD^{2} = 7AB^{2}.

30. The speed of a boat in still water is 15 km/hr. It can go 30 km upstream and return downstream to the original point in 4 hours and 30 minutes. Find the speed of the stream

.

ALL THE BEST !