**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**In fig. 1, S and T are points on the sides PQ and PR, respectively of ΔPQR, such that PT = 2 cm, TR = 4 cm and ST is parallel to QR. Find the ratio of the areas of ΔPST and ΔPQR.

VIEW SOLUTION**Q3**In the given figure, ΔAHK is similar to ΔABC. If AK = 10 cm, BC = 3.5 cm and HK = 7 cm, find AC.

VIEW SOLUTION**Q4**If

VIEW SOLUTION*α*,*β*are the zeroes of a polynomial, such that*α*+*β*= 6 and*αβ*= 4, then write the polynomial.**Q5****Q6****Q7**A card is drawn at random from a well shuffled pack of 52 playing cards. Find the probability of getting a red face card.

VIEW SOLUTION**Q8**The slant height of a frustum of a cone is 4 cm and the perimeters (circumferences) of its circular ends are 18 cm and 6 cm. Find the curved surface area of the frustum.

VIEW SOLUTION**Q9**If A(1, 2), B(4, 3) and C(6, 6) are the three vertices of a parallelogram ABCD, find the coordinates of the fourth vertex D.

VIEW SOLUTION**Q10**If P(2,

VIEW SOLUTION*p*) is the mid-point of the line segment joining the points A(6, − 5) and B(− 2, 11), find the value of*p*.**Q11**If and are two zeroes of the polynomial

VIEW SOLUTION*x*^{3}+ 3*x*^{2}− 5*x*− 15, find its third zero.**Q12**If all the sides of a parallelogram touch a circle, show that the parallelogram is a rhombus.

VIEW SOLUTION**Q13****Q14**Find the value of

*k*for which the following pair of linear equation have infinitely many solutions:2

VIEW SOLUTION*x*+ 3*y*= 7; (*k*− 1)*x*+ (*k*+ 2)*y*= 3*k***Q15**In an A.P., first term is 2, the last term is 29 and sum of the terms is 155. Find the common difference of the A.P.

VIEW SOLUTION**Q16****Q17**In figure 3, ABC is a right triangle, right angled at C and D is the mid-point of BC. Prove that AB

VIEW SOLUTION^{2}= 4AD^{2}− 3AC^{2}.**Q18****Q19**In an A.P., the sum of first ten terms is −150 and the sum of its next ten terms is −550. Find the A.P.

VIEW SOLUTION**Q20**The sum of numerator and denominator of a fraction is 3 less than twice the denominator. If each of the numerator and denominator is decreased by 1, the fraction becomes. Find the fraction.

**OR**Solve the following pair of equations:

VIEW SOLUTION**Q21**Construct a triangle PQR in which QR = 6 cm, ∠Q = 60° and ∠R = 45°. Construct another triangle similar to ΔPQR such that its side are of the corresponding sides of ΔPQR.

VIEW SOLUTION**Q22**Cards bearing numbers 1, 3, 5, ..., 35 are kept in a bag. A card is drawn at random from the bag. Find the probability of getting a card bearing

**(i)**a prime number less than 15.

VIEW SOLUTION**(ii)**a number divisible by 3 and 5.**Q23**If the point P (

VIEW SOLUTION*m*, 3) lies on the line segment joining the points and B (2, 8), find the value of*m*.**Q24**Point P divides the line segment joining the points A (2, 1) and B (5, −8) such that. If P lies on the line 2

VIEW SOLUTION*x*−*y*+*k*= 0, find the value of*k*.**Q25**In figure 4, the boundary of shaded region consists of four semicircular arcs, two smallest being equal. If diameter of the largest is 14 cm and that of the smallest is 3.5 cm, calculate the area of the shaded region. [Use π =]

**OR**Find the area of shaded region in figure 5, if AC = 24 cm, BC = 10 cm and O is the centre of the circle. [Use π = 3.14]

VIEW SOLUTION**Q26**A milk container is made of metal sheet in the shape of frustum of a cone whose volume is . The radii of its lower and upper circular ends are 8 cm and 20 cm respectively. Find the cost of metal sheet used in making the container at the rate of Rs. 1.40 per square centimetre.

**Or**A toy is in the form of a hemisphere surmounted by a right circular cone of the same base radius as that of the hemisphere. If the radius of base of the cone is 21 cm and its volume is of the volume of the hemisphere, calculate the height of the cone and the surface area of the toy. .

VIEW SOLUTION**Q27**Prove that, in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

Using the above, prove the following:

Point D is the mid-point of the side BC of a right triangle ABC, right angled at C. Prove that 4AD

VIEW SOLUTION^{2}= 4AC^{2}+ BC^{2}**Q28**Three consecutive positive integers are such that the sum of the square of the first and the product of the other two is 46, find the integers.

**Or**The difference of squares of two numbers is 88. If the larger number is 5 less than twice the smaller number, then find the two numbers.

VIEW SOLUTION**Q29**From the top of a 7 m high building, the angle of elevation of the top of a tower is 60° and the angle of depression of the foot of the tower is 30°. Find the height of the tower.

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

**Class**:0 − 10

10 − 20

20 − 30

30 − 40

40 − 50

50 − 60

60 − 70

**Frequency**:8

7

15

20

12

8

10

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## Class X: Math, Board Paper 2010, Set-3