Gujarat Board Class 10 Math Sample Papers 2005

Subject – mathematics

Class - x

1. If cosθ + sinθ = √2 cosθ , then show that cosθ − sinθ = √2 sinθ.

2. Prove that 5 – √3 is an irrational number.

3. Find all the zeroes of x4 – 5x3 + 3x2 + 15x -18, if two of its zeroes are √ 3 and -√3.

4. D is any point on the side BC of ΔABC such that ∠ADC = ∠BAC. Prove that CA/CD = CB/CA.

5. A railway half ticket cost half the full fare & the reservation charge is the same on half ticket as on full ticket. One reserved first class ticket from Bombay to Ahmadabad costs Rs. 216 & one full & one half reserved first class tickets cost Rs. 327. What is the basic first class full fare & what is the reservation charge.

6. A point D is on the side BC of an equilateral triangle ABC such that CD = ¼ BC. Prove that AD2 = 13CD2.

7. The rain water from a roof 22m ´ 20 m drains into a cylindrical vessel having diameter of base 2m and height 3.5 m . If the vessel is just full, find the rainfall in cm.

8. Water flows at the rate of 10m per minute through a cylindrical pipe having its diameter as 5mm.How much time will it take to fill a conical vessel whose diameter of base is 40cm and depth 24cm ?

9. Solve the following pair of linear equation graphically: x-2y=4,   x-y =3.

10. Draw a right triangle ABC in which BC = 12 cm, AB = 5 cm and B = 90°. Construct a triangle similar to it and of scale factor 2/3. Is the new triangle also a right triangle?

11. In the centre of rectangular lawn of dimensions 50m X 40m , a rectangular pond is constructed so that ,the area of grass surrounding the pond is 1184 m2 Find the length and breadth of the pond .

12. Draw a circle of 3 .4 cm radius. Take a point P outside the circle. Draw two tangents to the circle from the point P without using the center .

13. A solid toy is in the form of a hemisphere surmounted by a right circular cone. The height of the cone is 4 cm and the diameter of the base is 8 cm.

14. Three cubes of a metal whose edges are in ratio 3:4:5 are melted and converted in to a single cube whose diagonal is 12√3 cm. Find the edges of these cubes.

15. An iron solid sphere of radius 3 cm is melted and recast into small spherical balls of radius 1cm each. Assuming that there is no wastage in the process, find the number of small spherical balls made from the given sphere.

16. In ∆OPQ, right angled at P, OP = 7cm and OQ – PQ = 1cm.Determine the values of sinQ and cosQ.

17. From a balloon vertically above a straight road, the angles of depression of two cars at an instant are found to be 45° and 60°. If the cars are 100 m apart, find the height of the balloon.

18. A train covers a distance of 90 Km at uniform speed .Had the speed been 15 Km /hr more . it would have taken 30 minutes less for the journey find the original speed of the train

19. A ladder 15m long just reaches the top of vertical wall if ladder makes an angle of 600 with the wall, find the height of the wall.

20. A man travels 600km partly by train and partly by car. If he covers 400km by train and the rest by car, it takes him 6 hours and 30 minutes. But if he travels 200 km by train and rest by car, he takes half an hour longer. Fine the speed of the train and that of the car.

21. The mid-points D, E, F of the sides of a triangle ABC are (3, 4), (8, 9) and (6, 7). Find the coordinates of the vertices of the triangle.

22. Find the zeroes of the polynomial x2 – 3 and verify the relationship between the zeroes and the coefficients.

23. Use Euclid's algorithm to find the HCF of 56, 96 and 404.