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# CBSE Class 10 - Maths

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CBSE X - All India
MATHS PAPER 2010

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.

SECTION A

### Question 1

1. Write whether $\frac{2\sqrt{45}+3\sqrt{20}}{2\sqrt{5}}$ on simplification gives a rational or an irrational number.

### Question 2

2. If α, β are the zeroes of the polynomial ${2y}^{2}$ + 7y + 5, write the value of α + β + αβ.

### Question 3

3. If the sum of the first q terms of an A.P. is 2q + 3${q}^{2}$, what is its common difference?

### Question 4

4. In the given figure, CP and CQ are tangents from an external point C to a circle with centre O. AB is another tangent which touches the circle at R. If CP =11 cm and BR = 4 cm, find the length of BC.

### Question 5

5. In Figure, DE||BC in ΔABC such that BC = 8 cm, AB = 6 cm and DA = 1.5. Find DE.

### Question 6

6. If 5x = sec θ and $\frac{5}{x}$ = tan θ, find the value of 5$\left[{x}^{2}-\frac{1}{{x}^{2}}\right]$.

### Question 7

7. What is the distance between the points A(c, 0) and B(0, −c)?

### Question 8

8. In ΔABC, right- angled at C, AC = 6 cm and AB = 12 cm. Find ∠A.

### Question 9

9. The slant height of the frustum of a cone is 5 cm. If the difference between the radii of its two circular ends is 4 cm, find the height of the frustum.

### Question 10

10. A die is thrown once. What is the probability of getting a number greater than 4?

SECTION B

### Question 11

11. For what value of k, 3 is a zero of the polynomial 2${x}^{2}$ + x + k?

### Question 12

12. Find the value of m for which the pair of linear equations 2x + 3y − 7 = 0 and (m − 1) x + (m + 1) y = (3m − 1) has infinitely many solutions.

### Question 13

13. Find the common difference of an A.P. whose first term in 4, the last term is 49 and the sum of all its terms is 265.

### Question 14

14. In the given figure, there are two concentric circles with centre O and of radii 5 cm and 3 cm. From an external point P, tangents PA and PB are drawn to these circles. If AP = 12 cm, find the length of BP.

### Question 15

15. Without using trigonometric tables, evaluate the following:

$\frac{\left(\mathrm{cos}70°\right)}{\left(3\mathrm{sin}20°\right)}$+ $\frac{4\left(\mathrm{se}{c}^{2}59°-\mathrm{co}{t}^{2}31°\right)}{3}$ - $\frac{2}{3}$ sin 90°

OR

Find the value of sec 60° geometrically.

SECTION C

### Question 16

16. Prove that $\sqrt{3}$ is an irrational number.

### Question 17

17. Solve the following pair of liner equations for x and y :

$\frac{b}{a}$x + $\frac{a}{b}$y = ${a}^{2}$+ ${b}^{2}$

x + y = 2ab.

OR

The sum of the numerator and the denominator of a fraction is 4 more than twice the numerator. If 3 is added to each of the numerator and denominator, their ratio becomes 2:3. Find the fraction.

### Question 18

18. In an A.P., the sum of its first ten terms is −80 and the sum of its next ten terms is −280. Find the A.P.

### Question 19

19. In figure 4, ABC is an isosceles triangle in which AB = AC. E is a point on the side CB produced, such that FE ⊥ AC. If AD⊥CB, prove that AB × EF = AD × EC.

### Question 20

20. Prove the following:

(1 + cot A−cosec A) (1 + tan A + sec A) = 2

OR

Prove the following:

sin A (1 + tan A) + cos A (1 + cot A) = sec A + cosec A.

### Question 21

21. Construct a triangle ABC in which AB = 8 cm, BC = 10 cm and AC = 6 cm. Then construct another triangle whose sides are $\frac{4}{5}$ of the corresponding sides of ABC.

### Question 22

22. Point P divides the line segment joining the points A (−1, 3) and B (9, 8) such that $\frac{\mathrm{AP}}{\mathrm{PB}}$= $\frac{k}{1}$. If P lies on the line x− + 2 = 0, find the value of k.

### Question 23

23. If the points (p, q), (m, n) and (p − m, q − n) are collinear, show that pn = qm.

### Question 24

24.

The rain-water collected on the roof of a building, of dimensions 22 m × 20 m, is drained into a cylindrical vessel having base diameter 2 m and height 3.5 m. If the vessel is full up to the brim, find the height of rain-water on the roof. Use π = $\frac{22}{7}$.

OR

In figure 5, AB and CD are two perpendicular diameters of a circle with centre O. If OA = 7 cm, find the area of the shaded region. Use π = $\frac{22}{7}$.

### Question 25

25. A bag contains cards which are numbered from 2 to 90. A card is drawn at random from the bag. Find the probability that it bears

(i) a two digit number,

(ii) a number which is a perfect square.

SECTION D

### Question 26

26. A girl is twice as old as her sister. Four years hence, the product of their ages (in years) will be 160. Find their present ages.

### Question 27

27. In a triangle, if the square of one side is equal to the sum of the squares of the other two sides, then prove that the angle opposite the first side is a right angle.

Using the above, do the following:

In an isosceles triangle PQR, PQ = QR and ${\mathrm{PR}}^{2}$ = 2${\mathrm{PQ}}^{2}$. Prove that ∠Q is a right angle.

### Question 28

28. A man on the deck of a ship, 12 m above water level, observes that the angle of elevation of the top of a cliff is 60° and the angle of depression of the base of the cliff is 30°. Find the distance of the cliff from the ship and the height of the cliff. [Use $\sqrt{3}$√= 1.732]

OR

The angle of elevation of a cloud from a point 60 m above a lake is 30° and the angle of depression of the reflection of the cloud in the lake is 60°. Find the height of the cloud from the surface of the lake.

### Question 29

29. The surface area of a solid metallic sphere is 616 ${\mathrm{cm}}^{2}$. It is melted and recast into a cone of height 28 cm. Find the diameter of the base of the cone so formed. [Use π = $\frac{22}{7}$ ] .

OR

The difference between the outer and inner curved surface areas of a hollow right circular cylinder, 14 cm long, is 88 ${\mathrm{cm}}^{2}$. If the volume of metal used in making the cylinder is 176 ${\mathrm{cm}}^{3}$, find the outer and inner diameters of the cylinder. [Use π = $\frac{22}{7}$ ].

### Question 30

30. Draw ‘less than ogive’ and ‘more than ogive’ for the following distribution and hence find its median.

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