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Q1
mcq
1 mark
The common difference of the AP 1/p, 1-p/p, 1-2p/p, ... is:
Correct Answer: C —
Solution
The common difference (d) is calculated by subtracting the first term from the second: d = (1-p)/p - 1/p = (1 - p - 1)/p = -p/p = -1. Thus, option C is correct.
Q2
mcq
1 mark
In Fig. 1, PA and PB are two tangents drawn from an external point P to a circle with centre C and radius 4 cm. If PA ⊥ PB, then the length of each tangent is:
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A.
3 cm
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B.
4 cm
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C.
5 cm
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D.
6 cm
Correct Answer: B —
Solution
The radius CA is perpendicular to the tangent PA, so ∠PAC = 90°. Since PA ⊥ PB, ∠APB = 90°. Also, CA = CB = 4 cm (radii). In the quadrilateral PACB, three angles are 90°, making it a square. Therefore, PA = AC = 4 cm.
Q3
mcq
1 mark
In Fig. 2, a circle with centre O is inscribed in a quadrilateral ABCD such that, it touches the sides BC, AB, AD and CD at points P, Q, R and S respectively. If AB=29 cm, AD=23 cm, ∠B=90° and DS = 5 cm, then the radius of the circle (in cm.) is
Correct Answer: A —
Solution
Tangents from a point are equal: DS = DR = 5 cm. Given AD = 23 cm, AR = AD - DS = 23 - 5 = 18 cm. Since AR = AQ = 18 cm, and AB = 29 cm, then QB = AB - AQ = 29 - 18 = 11 cm. In quadrilateral OQBP, ∠B = 90°, and OQ = OP = r. Thus OQBP is a square, so radius r = QB = 11 cm.
Q4
mcq
1 mark
The angle of depression of a car, standing on the ground, from the top of a 75 m high tower, is 30°. The distance of the car from the base of the tower (in m.) is:
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A.
25√3
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B.
50√3
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C.
75√3
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D.
150
Correct Answer: C —
Solution
Let the tower height h = 75 m and distance = x. In the right triangle, tan(30°) = height / base = 75/x. Since tan(30°) = 1/√3, then 1/√3 = 75/x, which implies x = 75√3 m.
Q5
mcq
1 mark
The probability of getting an even number, when a die is thrown once, is
-
A.
1/2
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B.
1/3
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C.
1/6
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D.
5/6
Correct Answer: A —
Solution
The sample space of a die is {1, 2, 3, 4, 5, 6}. The even numbers are {2, 4, 6}. Number of favorable outcomes = 3. Total outcomes = 6. Probability = 3/6 = 1/2.
Q20
short answer
Find the ratio in which the y-axis divides the line segment joining the points (-4, -6) and (10, 12). Also find the coordinates of the point of division.
Q21
short answer
In Fig.5, AB and CD are two diameters of a circle with centre O, which are perpendicular to each other. OB is the diameter of the smaller circle. If OA = 7 cm, find the area of the shaded region. [Use π = 22/7]
