Class - X Maths Sample Paper

SECTION - A

Question numbers 1 to 10 carry 1 mark each. For each of the questions 1 to 10 four alternative choices have been provided of which only one is correct. You have to select the correct choice.
1. Which of the following is not a quadratic equation :
(A) (x-2)² +1 = 2x-3       (B) x(x+1) + 8 = (x+2)(x-2)
(C) x(2x +3) = x²+1        (D) (x+2)  = x -4

2. For what value of p are 2p+1, 13, 5p-3, three consecutive terms of an A.P. ?
(A) 2       (B) - 2        (C) 4          (D) -4

3. A tangent PA is drawn from an external point P to a circle of radius 3√2  cm such that the distance of the point P from O is 6 cm as shown in figure.  The value of ∠APO  is

(A)    30°           (B)    45°          (C)    60°        (D)     50°

4. The maximum number of common tangents that can be drawn to two circles intersecting at two distinct points is
(A)    1         (B)    2          (C)    3             (D)    4

5. In the given figure, O is the centre of the circle.  If PA and PB are tangents from an external point P to the circle, then ∠AQB is equal to

(A) 100°          (B) 80°        (C) 70°          (D) 50°

6. The probability  that a leap year has 53 Sundays is 
(A)          (B)          (C)         (D) 
7. If the angle of depression of an object from a  5 m high tower is 30°, then the distance of the object from the base of tower is
(A) 25√3  m     (B) 50√3  m      (C) 75√3  m     (D) 150m

 8. The perimeter of a quadrant of a circle of radius   cm is
(A) 3.5 cm    (B) 5.5 cm     (C) .5 cm     (D) 12.5 cm

9. If a solid right circular cone of height 24 cm and base radius 6 cm is melted and recast in the shape of a sphere, then the radius of the sphere is
(A) 6 cm (B) 4 cm (C) 8 cm (D) 12 cm

10. In the given figure, AA₁=A₁A₂=A₂A =A₃B.  If B₁A₁||CB, then A₁  divides AB in the ratio
 
(A) 1 : 2      (B) 1 : 3       (C) 1 : 4      (D) 1 : 1

SECTION - B

Question numbers 11 to 18 carry 2 marks each.
11. The sum of circumferences of two circles is 132 cm. If the radius of one circle is 14 cm, find the radius of the second circle.

12. For what value(s) of k, the equation x² - 2kx - k= a will  have equal roots ?
13. Find the 10th term from the end of the A.P. 4, 9, 14, ........ 254.

14. A point P is at a distance of √10  from the point (2, 3). Find the coordinates of the point P if its y coordinate is twice its x coordinate.

OR

Find the coordinates of the point B, if the point P(-4, 1) divides the line segment joining the points A(2, -2) and B in the ratio 3 : 5.

15. If the points A (4, 3) and B (x, 5) are on the circle with  centre O(2, 3) ; find  the value of x.

16. A bag contains cards numbered from 2 to 26.  0ne  card is drawn  from the bag at random.  Find the probability  that it has a number divisible by both 2 and 3.
17. How many lead shots, each 0.3 cm in diameter, can be made from a cuboid of dimensions 9 cm x 11 cm x 12 cm.

18. A quadrilateral ABCD is drawn to circumscribe a circle. Prove that AB+CD=AD+BC. 

SECTION - C

Question numbers 19 to  28 carry 3 marks each.
19. Solve for x :
      ;x  - 1, -2, -4

OR 

Solve for x :
9x² - 3(a+b)x + ab = a

20. Determine the ratio in which the line 3x+y-9=a divides the line segment joining the points (1, 3) and (2, 7)

21. The area of a triangle whose vertices are (-2, -2), (-1, -3) and (x, a) is 3 square units.  Find the value of x.

22. How many terms of the A.P.  8,  1, 64, ...... are needed to give the sum 465 ?  Also find the last term of this A.P.

23.In the given figure, diameter AB is 12 cm long.  AB is trisected at points P and Q. Find the area of the shaded region.

OR 

A chord of a circle of radius 12 cm subtends an angle of 120°at the centre.  Find the  area of the corresponding minor segment of the circle.  (Use π=3.14)
24. Water is flowing  at the rate of 5 km/hour through  a pipe of diameter 14 cm into a rectangular tank, which is 50 m long and 44 m wide.  Determine the time in which the level of water in the tank will  rise by 7cm.

25. In the figure, OP is equal to the diameter of the circle. Prove that ABP is an equilateral triangle.

OR

In the figure, AB and CD are two parallel tangents to a circle with  centre O.  ST is the tangent segment between two parallel tangents touching the circle at Q.  Show that ∠SOT=90°.

26. The shadow of a tower standing on a level ground is found to be 4a m longer when the sun's altitude is 30° than when it is 60°.  Find the height of the tower.

27. Draw two tangents to a circle of radius 3.5 cm from a point P at a distance of 6 cm from its centre O.

28. All the three face cards of spades are removed from a well - shuffled pack of 52 cards.A card is then drawn  at random from the remaining pack.  Find the probability  of getting
(a) a black face card
(b) a queen
(c) a black card 

SECTION - D

Question numbers  29 to 34  carry 4 marks each.
29. Prove that the lengths of tangents drawn from an external point to a circle are equal.

30. A person on tour has Rs. 360 for his daily expenses. If he extends his tour for four days, he has to cut down his daily expenses by Rs. 3. Find the original duration of the tour.

31. 200 logs are stacked in the following manner : 20 logs in the bottom row, 19 in the next row, 18 in the row next to it and so on. In how many rows are the 200 logs placed and how many logs are in the top row ?

32. An aircraft is flying  at a constant height with  a speed of 360 km/hour.  From a point on the ground, the angle of elevation at an instant was observed to be 45°.   After 20 seconds, the angle of elevation was observed to be 30°.   Determine the height at which the aircraft is flying.   (use  √3 = 1.732 )

OR

A boy 2 m tall is standing at some distance from a 30 m tall building. The angle of elevation from his eyes of the top of the building  increases from 30° to 60° as he walks towards the building.   Find the distance he walked towards the building.

33. The area of an equilateral triangle ABC is 17320.5 cm². With each vertex of the triangle as centre, a circle is drawn with radius equal to half the length of the side of the triangle.Find the area of the shaded region.  (use π= 3.14 and √3 = 1.73205)
 
34. The internal radii of the ends of a bucket, full of milk and of internal height 16 cm, are14 cm and 7cm. If this milk is poured into a hemispherical vessel, the vessel is completely filled.  Find the internal diameter of the hemispherical vessel.

OR

A right  triangle, whose sides are 6 cm and 8 cm (other than hypotenuse) is made to revolve about its hypotenuse. Find the volume and surface area of the double cone so formed.