**General Instructions:**

1. All questions are compulsory.

2. 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 2marks each, Section C comprises of ten questions of 3 marks each and Section D comprises of five questions of 6marks 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.

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.

4. In question on construction, the drawing should be neat and as per the given measurements.

5. Use of calculators is not permitted.

**Q1****Q2****Q3****Q4****Q5****Q6**The lengths of the diagonals of a rhombus are 30 cm and 40 cm. Find the side of the rhombus.

VIEW SOLUTION**Q7****Q8****Q9****Q10****Q11**Find all the zeroes of the polynomial

VIEW SOLUTION*x*^{4}+*x*^{3}− 34*x*^{2 }− 4*x*+ 120, if two of its zeros are 2 and −2.**Q12**A pair of dice is thrown once. Find the probability of getting the same number on each dice.

VIEW SOLUTION**Q13**If sec 4A = cosec (A − 20°) where 4A is an acute angle, then find the value of A.

**OR**In Δ ABC right-angled at C, if then find the value of sin A cos B + cos A sin B.

VIEW SOLUTION**Q14****Q15**E is a point on AD produced of a ||

VIEW SOLUTION^{gm }ABCD and BE intersects CD at F. Show that Δ ABE ∼ Δ CFB.**Q16**Use Euclid’s Division Lemma to show that the square of any positive integer is either of the form 3

VIEW SOLUTION*m*or (3*m*+ 1) for some integer*m*.**Q17**Represent the following pair of equations graphically and write the co-ordinates of points where the lines intersect the y-axis:

*x*+ 3*y*= 62

VIEW SOLUTION*x*− 3*y*= 12**Q18**For what value of

*n*are the*n*^{th}terms of two A.P.’s 63, 65, 67, … and 3, 10, 17, … equal?**OR**If

VIEW SOLUTION*m*times the*m*^{th}term of an A.P. is equal to*n*times its*n*^{th}term, then find the (*m*+*n*)^{th}term of the A.P.**Q19**In an A.P. the first term is 8, the

VIEW SOLUTION*n*^{th}term is 33, and sum of the first*n*terms is 123. Find*n*and*d*, the common difference.**Q20**Prove that:

(1+ cot A + tan A) (sin A − cos A) = sin A tan A − cot A cos A

**OR**Without using trigonometric tables, evaluate the following:

VIEW SOLUTION**Q21**If P divides the join of A(−2, −2) and B(2, −4) such that , find the co-ordinates of P.

VIEW SOLUTION**Q22**The mid-points of the sides of a triangle are (3, 4), (4, 6), and (5, 7). Find the co-ordinates of the vertices of the triangle.

VIEW SOLUTION**Q23**Draw a right triangle where the sides containing the right angle are 5 cm and 4 cm. Construct a similar triangle whose sides are times the sides of the above triangle.

VIEW SOLUTION**Q24**Prove that a parallelogram circumscribing a circle is a rhombus.

**OR**In Figure 2, AD ⊥ BC. Prove that AB

VIEW SOLUTION^{2}+ CD^{2}= BD^{2}+ AC^{2}.**Q25**In Figure 3, ABC is the quadrant of a circle of radius 14 cm and a semi-circle is drawn with BC as the diameter. Find the area of the shaded region.

**Q26**A peacock is sitting on the top of a pillar, which is 9 m high. From a point 27 m away from the bottom of the pillar, a snake is coming to its hole at the base of the pillar. Seeing the snake the peacock pounces on it. If their speeds are equal, at what distance from the hole is the snake caught?

**OR**The difference of two numbers is 4. If the difference of their reciprocals is, find the two numbers.

VIEW SOLUTION**Q27**The angle of elevation of an aeroplane from a point A on the ground is 60°. After a flight of 30 seconds, the angle of elevation changes to 30°. If the plane is flying at a constant height of m, find the speed in km/hour of the plane.

VIEW SOLUTION**Q28**If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, prove that the other two sides are divided in the same ratio.

Using the above, prove the following:

In Figure 4, AB || DE and BC || EF. Prove that AC || DF.

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

Using the above, prove the following:

ABC is an isosceles triangle in which AB = AC circumscribed about a circle, as shown in Figure 5. Prove that the base is bisected by the point of contact.

**Q29**If the radii of the circular ends of a conical bucket, which is 16 cm high, are 20 cm and 8 cm, find the capacity and the total surface area of the bucket. [Use π ]

VIEW SOLUTION**Q30**Find mean, median, and mode of the following data:

Classes

Frequency

0 − 20

6

20 − 40

8

40 − 60

10

60 − 80

12

80 − 100

6

100 − 120

5

120 − 140

3

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## Class X: Math, Board Paper 2008, Set-1