Class – X (Outside Delhi)
Subject : Mathematics
Time allowed : 3 hours Maximum Marks : 80
General Instructions :
(i) All questions are compulsory.
(ii) The question paper consists of 30 questions divided into four sections - A, B, C and D. Section A comprises of ten questions of 1 mark each, Section B comprises of five questions of 2 marks each, Section C comprises of ten questions of 3 marks each and Section D comprises of five questions of 6 marks each
(iii) All questions in Section A are to be answered in one word, one sentence or as per the exact requirement of the question.
(iv) There is no overall choice. However, an internal choice has been provided in one question of 2 marks each, three questions of 3 marks each and two questions of 6 marks each. You have to attempt only one of the alternatives in all such questions.
(v) In question on construction, the drawings should be neat and exactly as per the given measurement.
(vi) Use of calculators is not permitted.
SECTION A
Questions number 1 to 10 carry 1 mark each.
1. Find the [ HCF x LCM ] for the numbers 100 and 190.
2. If 1 is a zero of the polynomial then find p(x) = ax2 – 3(a-1)x -1, then find the value of a.
3. In ∆LMN, angle L = 500 and angle N = 600. If ∆LMN ~ ∆PQR, then find angle Q.
4. If sec2θ (1 + sin θ) (1 – sin θ) = k, then find the value of k.
5. If the diameter of a semicircular protractor is 14 cm, then find its perimeter.
6. Find the number of solutions of the following pair of linear equations:
X + 2y – 8 =0
2x + 4y = 16
7. Find the discriminate of the quadratic equation
3√3x2 + 10x + √3 = 0
8. If 4/5 , a, 2 are three consecutive terms of an A.P., then find the value of a.
9. In Figure 1, ∆ABC is circumscribing a circle. Find the length of BC.
10. Two coins are tossed simultaneously. Find the probability of getting exactly one head.
SECTION B
Questions number 11 to 15 carry 2 marks each.
11. Find all the zeroes of the polynomial x3 + 3x2 – 2x – 6, if two of its zeroes are -√2 and √2.
12. Which term of the A.P. 3, 15, 27, 39, ….. will be 120 more than its 21St term ?
13. In Figure 2, ∆ABD is a right triangle, right-angled at A and AC perpendicular BD. Prove that AB2 = BC.BD
14. If cot θ = 15/8, then evaluate [(2 + 2sin θ) (1 – sin θ)]/[(1 + cos θ)(2 – 2cos θ)].
OR
Find the value of tan 600, geometrically.
15. If the points A (4, 3) and B (x, 5) are on the circle with the centre O (2, 3), find the value of x.
Section – C
Question number 16 to 25 carry 3 marks each.
16. Prove that 3 + √2 is an irrational number.
17. Solve for x and y:
(ax)/b – (by)/a = a + b
ax – by =2ab
OR
The sum of two numbers is 8. Determine the numbers if the sum of their reciprocals is 8/ 15 .
18. The sum of first six terms of an arithmetic progression is 42. The ratio of its 10th term to its 30th term is 1 : 3. Calculate the first and the thirteenth term of the A.P.
19. Evaluate:
2/3 cosec2 580 – 2/3cot580 tan320 – 5/3 tan130 tan370 tan450 tan530 tan770
20. Draw a right triangle in which sides (other than hypotenuse) are of lengths 8 cm and 6 cm. Then construct another triangle whose sides are ¾ times the corresponding sides of the first triangle.
21. In figure, AD perpendicular BC and BD = 1/3CD. Prove that
2 CA2 = 2 AB2 + BC2 .
OR
In Figure 4, M is mid-point of side CD of a parallelogram ABCD. The line BM is drawn intersecting AC at L and AD produced at E. Prove that EL = 2 BL.
22. Find the ratio in which the point (2, y) divides the line segment joining the points A (-2, 2) and B (3, 7). Also find the value of y.
23. Find the area of the quadrilateral ABCD whose vertices are A (- 4, - 2), B (- 3, - 5), C (3, - 2) and D (2, 3).
24. The area of an equilateral triangle is 49√3 cm2. Taking each angular point as centre, circles are drawn with radius equal to half the length of the side of the triangle. Find the area of triangle not included in the circles. [Take √3 = 1.73]
OR
Figure 5 shows a decorative block which is made of two solids - a cube and a hemisphere. The base of the block is a cube with edge 5 cm and the hemisphere, fixed on the top, has a diameter of 4.2 cm. Find the total surface area of the block. [Take π=72]
25. Two dice are thrown simultaneously. What is the probability that
(i) 5 will not come up on either of them ?
(ii) 5 will come up on at least one ?
(iii) 5 will come up at both dice ?
SECTION D
Questions number 26 to 30 carry 6 marks each.
26. Solve the following equation for x :
9x2 – 9(a + b)x – (2a2 + 5ab 2b2) = 0
If (- 5) is a root of the quadratic equation 2x2 + px - 15 = 0 and the quadratic equation p (x2 + x) + k = 0 has equal roots, then find the values of p and k.
27. Prove that the lengths of the tangents drawn from an external point to a circle are equal.
Using the above theorem prove that If quadrilateral ABCD is circumscribing a circle, then
AB + CD = AD + BC.
28. An aeroplane when flying at a height of 3125 m from the ground passes vertically below another plane at an instant when the angles of elevation of the two planes from the same point on the ground are 30° and 60° respectively. Find the distance between the two planes at that instant.
29. A juice seller serves his customers using a glass as shown in Figure 6. The inner diameter of the cylindrical glass is 5 cm, but the bottom of the glass has a hemispherical portion raised which reduces the capacity of the glass. If the height of the glass is 10 cm, find the apparent capacity of the glass and its actual capacity. (Use π = 3.14)
A cylindrical vessel with internal diameter 10 cm and height 10.5 cm is full of water. A solid cone of base diameter 7 cm and height 6 cm is completely immersed in water. Find the volume of
(i) water displaced out of the cylindrical vessel.
(ii) water left in the cylindrical vessel. [Take π =22/7]
30. During the medical check-up of 35 students of a class their weights were recorded as follows :
| Weight (in kg) | Number of students |
| 38-40 | 3 |
| 40-42 | 2 |
| 42-44 | 4 |
| 44-46 | 5 |
| 46-48 | 14 |
| 48-50 | 4 |
| 50-52 | 3 |
Draw a less than type and a more than type ogive from the given data. Hence obtain the median weight from the graph.