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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 calculated is not permitted.
Question 1
  • Q1

    If a tower 30 m high, casts a shadow 103 m long on the ground, then what is the angle of elevation of the sun? 

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  • Q2

    The probability of selecting a rotten apple randomly from a heap of 900 apples is 0.18. What is the number of rotten apples in the heap? 

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  • Q3

    What is the common difference of an A.P. in which a21 – a7 = 84? 

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  • Q4

    If the angle between two tangents drawn from an external point P to a circle of radius a and centre O, is 60°, then find the length of OP. 

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  • Q5

    A line intersects the y-axis and x-axis at the points P and Q respectively. If (2, –5) is the mid-point of PQ, then find the coordinates of P and Q. 

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  • Q6

    If the distances of P(x, y) from A(5, 1) and B(–1, 5) are equal, then prove that 3x = 2y. 

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  • Q7

    Find the value of p, for which one root of the quadratic equation px2 – 14x + 8 = 0 is 6 times the other.           

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  • Q8

    Prove that the tangents drawn at the end points of a chord of a circle make equal angles with the chord. 

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  • Q9

    A circle touches all the four sides of a quadrilateral ABCD. Prove that AB + CD = BC + DA. 

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  • Q10

    Which term of the A.P. 8, 14, 20, 26, ... will be 72 more than its 41st term? 

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  • Q11

    The dimensions of a solid iron cuboid are 4·4 m × 2·6 m × 1·0 m. It is melted and recast into a hollow cylindrical pipe of 30 cm inner radius and thickness 5 cm. Find the length of the pipe. 

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  • Q12

    In the given figure, two concentric circles with centre O have radii 21 cm and 42 cm. If ∠AOB = 60°, find the area of the shaded region. Use π=227

     

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  • Q13

    Water in a canal, 5·4 m wide and 1·8 m deep, is flowing with a speed of 25 km/hour. How much area can it irrigate in 40 minutes, if 10 cm of standing water is required for irrigation? 

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  • Q14

    In what ratio does the point 2411, y divide the line segment joining the points P(2, –2) and Q(3, 7)? Also find the value of y. 

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  • Q15

    On a straight line passing through the foot of a tower, two points C and D are at distances of 4 m and 16 m from the foot respectively. If the angles of elevation from C and D of the top of the tower are complementary, then find the height of the tower. 

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  • Q16

    A bag contains 15 white and some black balls. If the probability of drawing a black ball from the bag is thrice that of drawing a white ball, find the number of black balls in the bag. 

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  • Q17

    Three semicircles each of diameter 3 cm, a circle of diameter 4.5 cm and a semicircle of radius 4.5 cm are drawn in the given figure. Find the area of the shaded region.

     

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  • Q18

    From a solid right circular cylinder of height 2.4 cm and radius 0.7 cm, a right circular cone of same height and same radius is cut out. Find the total surface area of the remaining solid. 

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  • Q19

    If the 10th term of an A.P. is 52 and the 17th term is 20 more than the 13th term, find the A.P. 

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  • Q20

    If the roots of the equation (c2 – ab) x2 – 2 (a2 – bc) x + b2 – ac = 0 in x are equal, then show that either a = 0 or a3 + b3 + c3 = 3abc. 

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  • Q21

    If the points A(k + 1, 2k), B(3k, 2k + 3) and C(5k − 1, 5k) are collinear, then find the value of k. 

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  • Q22

    Two different dice are thrown together. Find the probability that the numbers obtained have

    (i) even sum, and
    (ii) even product. 

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  • Q23

    Construct a triangle ABC with side BC = 7 cm, ∠B = 45°, ∠A = 105°. Then construct another triangle whose sides are 34 times the corresponding sides of the ∆ABC. 

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  • Q24

    In a rain-water harvesting system, the rain-water from a roof of 22 m × 20 m drains into a cylindrical tank having diameter of base 2 m and height 3·5 m. If the tank is full, find the rainfall in cm. Write your views on water conservation. 

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  • Q25

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

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  • Q26

    In the given figure, XY and X'Y' are two parallel tangents to  circle with centre O and another tangent AB with point of contact C, is intersecting XY at A and X'Y' at B. Prove that ∠AOB = 90°.

     

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  • Q27

    If the ratio of the sum of the first n terms of two A.Ps is (7n + 1) : (4n + 27), then find the ratio of their 9th terms. 

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  • Q28

    Solve for x :

    12x-3+1x-5=119, x32, 5 

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  • Q29

    A train covers a distance of 300 km at a uniform speed. If the speed of the train is increased by 5 km/hour, it takes 2 hours less in the journey. Find the original speed of the train. 

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  • Q30

    A man observes a car from the top of a tower, which is moving towards the tower with a uniform speed. If the angle of depression of the car changes from 30° to 45° in 12 minutes, find the time taken by the car now to reach the tower. 

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  • Q31

    In the given figure, ∆ ABC is a right-angled triangle in which ∠ A is 90°. Semicircles are drawn on AB, AC and BC as diameters. Find the area of the shaded region.

     

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