SAMPLE PAPER - 2008
Class - X
SUBJECT - MATHEMATICS
Marks: 80 Time: 3Hrs
SECTION-A
- Show that any positive odd integer is of the form 4q + 1 or 4q + 3 where q is some integer.
2) Consider the number 7n where n is a natural number. Check whether there is any value nЄ N for which 7n ends with the digit 0. Why? (No)
3) For a triangle ABC show that sin (b + c) = cos (A/2) where A, B, C are inte
2
rior angles of ∆ ABC.
4) Find the value of 9 sec² A – 9tan²A. (9)
5) Prove that sin4 θ – cos4 θ = sin² θ - cos² θ.
6) Show that the quadrilateral with vertices (3, 2), (0, 5), (-3, 2) and (0, -1) is a square.
7) The area of a circle is 78.5cm2.Calculate the circumference of the circle.
(31.4cm)
8) Find the area of a sector which subtends an angle of 120º, at the centre, given that the radius of the circle is 21cm. (462cm²)
9) Three coins are tossed. Find the probability of getting one head.
(3/4)
10) If the zeroes of the polynomial x³ - 3x² + x + 1 are (a – b), a, (a + b). Find a and b. (1, ± √ 2)
SECTION-B
11) For what values of a and b, the following system of linear equations have infinite number of solutions?
2x + 3 y = 7, (a – b) x + (a + b) y = (3a + b – 2) (5, 1)
12) Solve for x: x + x -1 = 4 1 + √3 1 - √3
x - 1 x 2 , 2
13) In an AP show that tp + tp +2q = 2tp + q.
14) Which term of the sequence 17, 16 1/5, 15 2/5, 14 3/5 , is the 1st negative term?
(23)
15) Prove that cot A + cosec A - 1 = 1 +cos A
Cot A – cosec A + 1 sin A
SECTION-C
16) In figure A, B, C are points on OP, QR, and OR respectively, such that
AB ║ PQ and AC ║ PR. Show that BC ║ QR.
P
A
B
C
O
Q
R
17) ABC is an isosceles triangle in which AB = AC, is circumscribed about a circle, show that BC is bisected at the point of contact.
18) Two concentric circles are of radii 5cm and 3cm. Find the length of the larger circle which touches the smaller circle. (
19) 20 tickets are numbered from 1 to 20. One of them is drawn at random. Find the number on it is divisible by 3 or 5. (9/20)
20) Savitha tosses 2 different coins simultaneously. What is the probability that she gets at least one head? (3/4)
21) The area enclosed between 2 concentric circles is 770m². If the radius of outer circle is 21cm, find the radius of inner circle. (14cm)
22) Find the co ordinates of the point where the diagonals of the parallelogram formed by joining the points (-2, -1), (1, 0), (4, 3), (1, 2) meet. (1, 1)
23) If in a triangle AB = AC and AD is a median, show that AD ┴ BC.
24) If 2 liquids are mixed in the ratio 3:2, a mixture is obtained weighing 1.04gm/cc, while if they are mixing in the ratio 5:3, the resulting mixture weighs 1.05gm/cc. Find the weight of a cc of each of the original liquids. (1.2gm, 0.8gm)
25) A kite is flying at a height of 60m above the ground. The inclination of the string to the ground is 60º on either side of it. Find the length of a string assuming that there is no black in the string. (69.28m)
SECTION-D
26) An aero plane at an altitude of 200m observes the angles of depression of opposite points on the 2 banks of a river to be 45º and 60º. Find the width of the river in meters. (315.5m)
27) Prove that the tangent at any point of a circle is ┴ to the radius through the point of contact. From a point P, 2 tangents PA and PB are drawn to a circle with centre O. If OP is equal to the diameter of the circle, prove that ∆ PAB is equilateral using above theorem.
28) State and prove converse of Pythagoras theorem, and hence show that in an isosceles ∆ ABC, with AC = BC, and AB² = 2AC², prove that ∟ACB = 90º.
29) Draw a circle of diameter 6cm. From a point 5cm from its centre, construct the pair of tangents to the circle.
30) A toy is in the form of a cone mounted on a hemisphere. The diameter of the base of the cone is 18cm and its height is 12cm. Calculate the surface area of the toy.
(932.58cm)
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