CBSE X
Delhi
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. The decimal expansion of the rational number $\frac{43}{{2}^{4}{5}^{3}}$ will terminate after how many places of decimals?
2. For what value of k , −4 is a zero of the polynomial${x}^{2}$ − x − (2k + 2)?
3. For what value of p, 2p − 1, 7 and 3p are three consecutive terms of an A.P.?
4. In the figure CP and CQ are tangents to a circle with centre O. ARB is another tangent touching the circle at R. If CP = 11 cm, and BC = 7 cm, then find the length of BR.
5. In Fig. 2, ∠M = ∠N = 46°. Express x in terms of a, b and c where a, b and c are lengths, of LM, MN and NK respectively.
6. If sin θ =$\frac{1}{3}$ , then find the value of (2${\mathrm{cot}}^{2}$θ + 2).
7. Find the value of a so that the point (3, a) lies on the line represented by 2x − 3y = 5.
8. A cylinder and a cone are of same base radius and of same height. Find the ratio of the volume of cylinder to that of the cone.
9. Find the distance between the points (-$\frac{8}{5}$, 2) and ($\frac{2}{5}$, 2).
10. Write the median class of the following distribution :
11. If the polynomial 6${x}^{4}$+ 8${x}^{3}$+ 17${x}^{2}$+ 21x + 7 is divided by another polynomial 3${x}^{2}$ + 4x + 1 , the remainder comes out to be (ax+b), find a and b.
12. Find the value(s) of k for which the pair of linear equations kx + 3y = k − 2 and 12x + ky = k has no solution.
13. If ${S}_{n}$is the sum of first n terms of an A.P. is given by ${S}_{n}$= 3${n}^{2}$- 4n, then find its ${n}^{\mathrm{th}}$ term.
14. Two tangents PA and PB are drawn to a circle with centre O from an external point P. Prove that ∠APB = 2 ∠OAB.
OR
Prove that the parallelogram circumscribing a circle is a rhombus.
15. Simplify: $\frac{{\mathrm{sin}}^{3}\theta +{\mathrm{cos}}^{3}\theta}{\mathrm{sin}\theta +\mathrm{cos}\theta}$ + sinθ cosθ
16. Prove that √5 is an irrational number.
17. Solve the following pair of equations:
$\frac{5}{(x-1)}+\frac{1}{(y-2)}$= 2
$\frac{6}{(x-1)}-\frac{3}{(y-2)}=1$
18. The sum of 4th and 8th terms of an A.P. is 24 and sum of 6th and 10th terms is 44. Find the A.P.
19. Construct a ΔABC in which BC = 6.5 cm, AB = 4.5 cm and ∠ABC = 60°. Construct a triangle similar to this triangle whose sides are $\frac{3}{4}$ of the corresponding sides of the triangle ABC.
20. In Fig. 4, ΔABC is right angled at C and DE ⊥ AB. Prove that ΔABC ∼ ΔADE and hence find the lengths of AE and DE.
OR
In Fig, 5, DEFG is a square and ∠BAC = 90°. Show that ${\mathrm{DE}}^{2}$ = BD × EC.
21. Find the value of sin 30° geometrically.
OR
Without using trigonometrical tables, evaluate:
$\frac{\mathrm{cos}58\xb0}{\mathrm{sin}32\xb0}$+ $\frac{\mathrm{sin}22\xb0}{\mathrm{cos}68\xb0}$ - $\frac{\mathrm{cos}38\xb0\mathrm{cos}\mathrm{ec}52\xb0}{\mathrm{tan}18\xb0\mathrm{tan}35\xb0\mathrm{tan}60\xb0\mathrm{an}72\xb0\mathrm{tan}55\xb0}$
22. Find the point on y - axis which is equidistant from the points (5, −2) and (−3, 2)
OR
The line segment joining the points A (2, 1) and B (5, −8) is trisected at the points P and Q such that P is nearer to A. If P also lies on the line given by
2x − y + k = 0, find the value of k.
23. If P (x, y) is any point on the line joining the points A (a, 0) and B (0, b), then show that
$\frac{x}{a}$+ $\frac{y}{b}$= 1.
24. In Fig. 6, PQ = 24 cm, PR = 7 cm and O is the centre of the circle. Find the area of shaded region (take π = 3.14)
25. The king, queen and jack of clubs are removed from a deck of 52 playing cards and the remaining cards are shuffled. A card is drawn from the remaining cards. Find the probability of getting a card of (i) heart (ii) queen (iii) clubs.
26. The sum of the squares of two consecutive odd numbers is 394. Find the numbers.
OR
Places A and B are 100 km apart on a highway. One car starts from A and another from B at the same time. If the cars travel in the same direction at different speeds, they meet in 5 hours. If they travel towards each other, they meet in 1 hour. What are the speeds of the two cars?
27. Prove that, if a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio.
Using the above result, do the following:
In Fig. 7, DE || BC and BD = CE. Prove that ΔABC is an isosceles triangle.
28. A straight highway leads to the foot of a tower. A man standing at the top of the tower observes a car at an angle of depression of 30°, which is approaching the foot of the tower with a uniform speed. Six seconds later the angle of depression of the car is found to be 60°. Find the time taken by the car to reach the foot of the tower from this point.
29. From a solid cylinder whose height is 8 cm and radius 6 cm, a conical cavity of height 8 cm and of base radius 6 cm, is hollowed out. Find the volume of the remaining solid correct to two places of decimals. Also find the total surface area of the remaining solid. (take π = 3.1416)
OR
In Fig. 8, ABC is a right triangle right angled at A. Find the area of shaded region if AB = 6 cm, BC = 10 cm and O is the centre of the incircle of ΔABC.
(take π = 3.14)
30. The following table gives the daily income of 50 workers of a factory:
Find the Mean, Mode and Median of the above data.