CBSE X
All India
MATHS PAPER 2009
Time allowed: 180 minutes; Maximum Marks: 90
General Instructions: | |
1) | All questions are compulsory. |
2) | The question paper consists of thirty questions divided into 4 sections A, B, C and D. Section A comprises of ten questions of 01 mark each, Section B comprises of five questions of 02 marks each, Section C comprises ten questions of 03 marks each and Section D comprises of five questions of 06 marks each. |
3) | All questions in Section A are to be answered in one word, one sentence or as per the exact requirement of the question. |
4) | There is no overall choice. However, internal choice has been provided in one question of 02 marks each, three questions of 03 marks each and two questions of 06 marks each. You have to attempt only one of the alternatives in all such questions. |
5) | In question on construction, drawing should be near and exactly as per the given measurements. |
6) | Use of calculators is not permitted. |
1. Write whether the rational number $\frac{51}{1500}$ will have a terminating decimal expansion or a non-terminating repeating decimal expansion.
2. Write the polynomial, the product and sum of whose zeroes are $\frac{-9}{2}$ and$\frac{-3}{2}$ respectively.
3. Write whether the following pair of linear equations is consistent or not:
x + y = 14
x − y = 4
4. Write the nature of roots of quadratic equation 4${x}^{2}$+ 4$\sqrt{3}$x + 3 = 0.
5. For what value of k, are the numbers x, 2x + k and 3x + 6 three consecutive terms of an A.P.
6. In a ΔABC, DE || BC. If DE = $\frac{2}{3}$ BC and area of ΔABC = 81${\mathrm{cm}}^{2}$, find the area of ΔADE.
7. If sec A = $\frac{15}{7}$ and A + B = 90°, find the value of cosec B.
8. If the mid-point of the line segment joining the points P (6, b − 2) and Q (−2, 4) is (2, −3), find the value of b.
9. The length of the minute hand of a wall clock is 7 cm. How much area does it sweep in 20 minutes?
10. What is the lower limit of the modal class of the following frequency distribution?
11. Without drawing the graph, find out whether the lines representing the following pair of linear equations intersect at a point, are parallel or coincident:
9x −10y = 21, $\frac{3}{2}$x - $\frac{5}{3}$ y = $\frac{7}{2}$
12. The ${17}^{\mathrm{th}}$ term of an A.P. exceeds its ${10}^{\mathrm{th}}$term by 7. Find the common difference.
13. Without using trigonometric tables, evaluate: $\frac{7}{2}$× $\frac{\mathrm{cos}70\xb0}{\mathrm{sin}20\xb0}$ + $\frac{3}{2}$× $\frac{\mathrm{cos}55\xb0\mathrm{cos}\mathrm{ec}35\xb0}{\mathrm{tan}5\xb0\mathrm{tan}25\xb0\mathrm{tan}45\xb0\mathrm{tan}85\xb0\mathrm{tan}65\xb0}$.
14. Show that the points (−2, 5); (3, −4) and (7, 10) are the vertices of a right angled isosceles triangle.
OR
The centre of a circle is (2α − 1, 7) and it passes through the point (−3, −1). If the diameter of the circle is 20 units, then find the values(s) of α.
15. If C is a point lying on the line segment AB joining A (1, 1) and B (2, −3) such that 3AC = CB, then find the coordinates of C.
16. Show that the square of any positive odd integer is of the form 8m + 1, for some integer m.
OR
Prove that 7 + 3$\sqrt{2}$ is not a rational number.
17. If the polynomial 6${x}^{4}$+ 8${x}^{3}$−5${x}^{2}$+ ax + b is exactly divisible by the polynomial 2${x}^{2}$−5, the find the values of a and b.
18. If ${9}^{\mathrm{th}}$ term of an A.P. is zero, prove that its ${29}^{\mathrm{th}}$ term is double of its ${19}^{\mathrm{th}}$ term.
19. Draw a circle of radius 3 cm. From a point P, 6 cm away from its centre, construct a pair of tangents to the circle. Measure the lengths of the tangents.
20. In the given figure, two triangles ABC and DBC lie on the same side of base BC. P is a point on BC such that PQ || BA and PR || BD. Prove that QR || AD.
21. In the given figure, a triangle ABC is right angled at B. Side BC is trisected at points D and E. Prove that ${8\mathrm{AE}}^{2}=5{\mathrm{AD}}^{2}$.
OR
In the given figure, a circle is inscribed in a triangle ABC having side BC = 8 cm, AC = 10 cm and AB = 12 cm. Find AD, BE and CF.
22. Prove that ${\mathrm{sec}}^{2}\theta -\frac{{\mathrm{sin}}^{2}\theta -{2\mathrm{sin}}^{4}\theta}{{2\mathrm{cos}}^{4}\theta -{\mathrm{cos}}^{2}\theta}$ = 1.
23. Find a relation between x and y if the points (x, y), (1, 2) and (7, 0) are collinear.
24. In figure 4, the shape of the top of a table in a restaurant is that of a sector of a circle with centre O and ∠BOD = 90°. If BO = OD = 60 cm, find
(i) the area of the top of the table.
(ii) the perimeter of the table top.
(Take π = 3.14)
OR
In figure 5, ABCD is a square of side 14 cm and APD and BPC are semicircles. Find the area of the shaded region. (Take π = $\frac{22}{7}$)
25. A box has cards numbered 14 to 99. Cards are mixed thoroughly and a card is drawn from the bag at random. Find the probability that the number on the card, drawn from the box is an odd number, a perfect square number, a number divisible by 7.
26. A trader bought a number of articles for Rs 900. Five articles were found damaged. He sold each of the remaining articles at Rs. 2 more than what he paid for it. He got a profit of Rs. 80 on the whole transaction. Find the number of articles he bought.
OR
Two years ago the man’s age was three times the square of his son’s age. Three years hence his age will be four times his son’s age. Find their present ages.
27. Prove that the ratio of the areas of two similar triangles is equal to the ratio of the squares of their corresponding sides. Using the above theorem prove the following: The area of the equilateral triangle described on the side of a square is half the area of the equilateral triangle described on its diagonal.
28. The angle of elevation of the top of a building from the foot of a tower is 30° and the angle of elevation of the top of the tower from the foot of the building is 60°. If the tower is 50 m high, find the height of the building.
29. A spherical copper shell, of external diameter 18 cm, is melted and recast into a solid cone of base radius 14 cm and height 4 $\frac{3}{7}$ cm. Find the inner diameter of the shell.
OR
A bucket is in the form of a frustum of a cone with a capacity of 12308.8 ${\mathrm{cm}}^{3}$. The radii of the top and bottom circular ends of the bucket are 20 cm and 12 cm respectively. Find the height of the bucket and also the area of metal sheet used in making it.
30. Find the mode, median and mean for the following data: