- Please check that this question paper contains 11 printed pages.

- Code number given on the right hand side of the question paper should be written on the title page of the answer-book by the candidate.

- Please check that this question paper contains 31 questions.

- Please write down the Serial Number of the question before attempting it.

- 15 minute time has been allotted to read this question paper. The question paper will be distributed at 10.15 a.m. From 10.15 a.m. to 10.30 a.m., the students will read the question paper only and will not write any answer on the answer-book during this period.

**SUMMATIVE ASSESSMENT – 2**

**MATHEMATICS**

**Time allowed:3 hours Maximum Marks : 90**

**General Instructions:**

**I. All questions are compulsory.**

**II. The question paper consists of 31 questions divided. into four sections - A, B, C and D.**

**III. Section A contains 4 questions of 1 mark each. Section B contains 6 questions of 2 marks each, Section C contains 10 questions of 3 marks each and Section D contains 11 questions of 4 marks each.**

**IV. Use of calculators is not permitted.**

**SECTION-A**

__Question. numbers 1 to 4 carry 1 mark each.__
1. If the quadratic equation px

^{2}– 2 √5 px + 15 = 0 has two equal roots, then find the value of p.
2. In Figure 1, a tower AB is 20 m high and BC, its shadow on the ground, is
20√3 m long. Find the Sun's altitude.

3. Two different dice are tossed together. Find the probability that the
product of the two numbers on the top of the dice is 6.

4. In Figure 2, PQ is a chord of a circle with centre O and PT is a tangent.
If ∠QPT = 60

^{0}, find ∠PRQ.**SECTION-B**

__Question numbers 5 to 10 carry 2 marks each.__
5. In Figure 3, two tangents RQ and RP are drawn from an external point R to
the circle with centre O. If ∠PRQ=120

^{0}, then prove that OR=PR+RQ.
6. In Figure 4, a triangle ABC is drawn to circumscribe a circle of radius 3
cm, such that the segments BD and DC are respectively of lengths 6 cm and 9 cm.
If the area of Δ ABC is 54 cm

^{2}, then find the lengths of sides AB and AC.
7. Solve the following quadratic equation for x : 4x

^{2}+ 4bx - (a^{2}-b^{2}) = 0
8. In an AP, if S

_{5}+ S_{7}= 167 and S_{10}- 235, then find the AP, where S_{n}denotes the sum of its first n terms.
9. The points A(4, 7), B(p, 3) and C(7, 3) are the vertices of a right
triangle, right-angled at B. Find the value of p.

10. Find the relation between x and y if the points A(x, y), B(- 5, 7) and
C(- 4,5) are collinear.

**SECTION-C**

__Question numbers 11 to 20 carry 3 marks each.__
11. The 14th term of an AP is twice its 8th term. If its 6th term is - 8,
then find the sum of its first 20 terms.

12. Solve for x: √3 x

^{2}- 2√2 x -2√3 =0
13. The angle of elevation of an aeroplane from a point A on the ground is
60°. After a flight of 15 seconds, the angle of elevation changes to 30°. If the
aeroplane is flying at a constant height of 1500√3

*m, find the speed of the plane in km/hr.*
14. If the coordinates of points A and Bare (- 2, - 2) and (2, - 4)
respectively, find the coordinates of P such that AP = 3/7 AB, where P lies on
the line segment AB.

15. The probability of selecting a red ball at random from a jar that
contains only red, blue and orange balls is ¼. The probability of selecting a
blue ball at random from the same jar is 1/3. If the jar contains 10 orange
balls, find the total number of balls in the jar.

16. Find the area of the minor segment of a circle of radius 14 cm, when its
central angle is 60°. Also find the area of the corresponding major segment [Use
Π

*= 22/7 ]*
17. Due to sudden floods, some welfare associations jointly requested the
government to get 100 tents fixed immediately and offered to contribute 50% of
the cost. If the lower part of each tent is of the form of a cylinder of
diameter 4.2 m and height 4 m with the conical upper part of same diameter but
of height 2.8 m, and the canvas to be used costs 100 rupees per sq. m, find the
amount, the associations will have to pay. What values are shown by these
associations? [Use Π

*= 22/7 ]*
18. A hemispherical bowl of internal diameter 36 cm contains liquid. This
liquid is filled into 72 cylindrical bottles of diameter 6 cm. Find the height
of the each bottle, if 10% liquid is wasted in this transfer.

19. A cubical block of side 10 cm is surmounted by a hemisphere. What is the
largest diameter that the hemisphere can have ? Find the cost of painting the
total surface area of the solid so formed, at the rate of 5 rupee per 100 sq.
cm. [Use Π

*= 3·14 ]*
20. 504 cones, each of diameter 3.5 cm and height 3 cm, are melted and recast
into a metallic sphere. Find the diameter of the sphere and hence find its
surface area. [Use Π

*= 22/7 ]***SECTION-D**

__Question numbers 21 to 31 carry 4 marks each.__
21. The diagonal of a rectangular field is 16 metres more than the shorter
side. If the longer side is 14 metres more than the shorter side, then find the
lengths of the sides of the field.

22. Find the 60th term of the AP 8,10, 12, ... , if it has a total of 60
terms and hence find the sum of its last 10 terms.

23. A train travels at a certain average speed for a distance of 54 km and
then travels a distance of 63 km at an average speed of 6 km/h more than the
first speed. if it takes 3 hours to complete the total journey, what is its
first speed?

24. Prove that the lengths of the tangents drawn from an external point to a
circle are equal.

25. Prove that the tangent drawn at the mid-point of an arc of a circle is
parallel to the chord joining the end points of the arc.

26. Construct a Δ ABC in which AB = 6 cm, ∠A = 30

^{0}and ∠B = 60^{0 }Construct another ΔAB'C' similar to ΔABC with base AB' = 8 cm.
27. At a point A, 20 metres above the level of water in a lake, the angle of
elevation of a cloud is 30°. The angle of depression of the reflection of the
cloud in the lake, at A is 60°. Find the distance of the cloud from A.

28. A card is drawn at random from a well-shuffled deck of playing cards.
Find the probability that the card drawn is

(i) a card of spade or an ace.

(ii) a black king.

(iii) neither a jack nor a king. (iv) either a king or a queen.

29. Find the values of k so that the area of the triangle with vertices (1, -
1), (- 4, 2k) and (- k, - 5) is 24 sq. units.

30. In Figure 5, PQRS is a square lawn with side PQ = 42 metres. Two circular
flower beds are there on the sides PS and QR with centre at 0, the intersection
of its diagonals. Find the total area of the two flower beds (shaded parts).

31. From each end of a solid metal cylinder, metal was scooped out in
hemispherical form of same diameter. The height of the cylinder is 10 cm and its
base is of radius 4.2 cm. The rest of the cylinder is melted and converted into
a cylindrical wire of 1.4 cm thickness. Find the length of the wire.[Use Π

*= 22/7 ]*
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