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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

    A solid metallic cuboid of dimensions 9 m × 8 m × 2 m is melted and recast into solid cubes of edge 2 m. Find the number of cubes so formed. 

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  • Q2

    PQ is a tangent drawn from an external point P to a circle with centre O, QOR is the diameter of the circle. If ∠POR = 120°, what is the measure of ∠OPQ? 

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  • Q3

    A ladder 15 m long makes an angle of 60° with the wall. Find the height of the point where the ladder touches the wall. 

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  • Q4

    If one root of the quadratic equation 6x2x – k = 0 is 23, then find the value of k. 

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  • Q5

    If two adjacent vertices of a parallelogram are (3, 2) and (−1, 0) and the diagonals intersect at (2, −5), then find the coordinates of the other two vertices. 

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  • Q6

    In the given figure, if AB = AC, prove that BE = EC.

     

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  • Q7

    Find the probability that in a leap year there will be 53 Tuesdays. 

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  • Q8

    If seven times the 7th term of an A.P. is equal to eleven times the 11th term, then what will be its 18th term? 

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  • Q9

    Two different dice are thrown together. Find the probability that the product of the numbers appeared is less than 18. 

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  • Q10

    Solve for x:
    3x2-2x-83=0 

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  • Q11

    Show that ∆ ABC with vertices A (–2, 0), B (0, 2) and C (2, 0) is similar to ∆ DEF with vertices D (–4, 0), F (4, 0) and E (0, 4). 

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  • Q12

    In the given figure, ∆ABC is an equilateral triangle of side 3 units. Find the coordinates of the other two vertices.

     

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  • Q13

    In the given figure, ABCD is a trapezium with AB || DC, AB = 18 cm, DC = 32 cm and the distance between AB and AC is 14 cm. If arcs of equal radii 7 cm taking A, B, C and D as centres, have been drawn, then find the area of the shaded region.

     

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  • Q14

    Prove that the opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle. 

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  • Q15

    The shadow of a tower at a time is three times as long as its shadow when the angle of elevation of the sun is 60°. Find the angle of elevation of the sun at the time of the longer shadow. 

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  • Q16

    In the given figure, PA and PB are tangents to a circle from an external point P such that PA = 4 cm and ∠BAC = 135°. Find the length of chord AB.

     

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  • Q17

    Find the coordinates of the points of trisection of the line segment joining the points (3, –2) and (–3, –4). 

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  • Q18

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

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  • Q19

    In an A.P. of 50 terms, the sum of the first 10 terms is 210 and the sum of its last 15 terms is 2565. Find the A.P. 

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  • Q20

    A solid sphere of diameter 6 cm is dropped in a right circular cylindrical vessel partly filled with water. The diameter of the cylindrical vessel is 12 cm. If the sphere is completely submerged in water, by how much will the level of water rise in the cylindrical vessel? 

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  • Q21

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

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  • Q22

    A child puts one five-rupee coin of her saving in the piggy bank on the first day. She increases her saving by one five-rupee coin daily. If the piggy bank can hold 190 coins of five rupees in all, find the number of days she can continue to put the five-rupee coins into it and find the total money she saved.

    Write your views on the habit of saving. 

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  • Q23

    A park is of the shape of a circle of diameter 7 m. It is surrounded by a path of width of 0·7 m. Find the expenditure of cementing the path, if its cost is Rs 110 per sq. m. 

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  • Q24

    In a rectangular park of dimensions 50 m × 40 m, a rectangular pond is constructed so that the area of grass strip of uniform width surrounding the pond would be 1184 m2. Find the length and breadth of the pond. 

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  • Q25

    Two circles touch internally. The sum of their areas is 116 π cm2 and the distance between their centres is 6 cm. Find the radii of the circles. 

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  • Q26

    A box contains 90 discs which are numbered from 1 to 90. If one disc is drawn at random from the box, find the probability that it bears (i) a two-digit number, (ii) a number divisible by 5. 

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  • Q27

    A well of diameter 3 m is dug 14 m deep. The soil taken out of it is spread evenly all around it to a width of 5 m to form an embankment. Find the height of the embankment. 

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  • Q28

    Solve for x:

    4x2 + 4bx − (a2 − b2) = 0 

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  • Q29

    Draw a circle of radius of 3 cm. Take two points P and Q on one of its diameters extended on both sides, each at a distance of 7 cm on opposite sides of its centre. Draw tangents to the circle from these two points P and Q. 

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  • Q30

    From the top of a hill, the angles of depression of two consecutive kilometer stones due east are found to be 45° and 30° respectivly. Find the height of the hill. 

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  • Q31

    A right circular cone is divided into three parts by trisecting its height by two planes drawn parallel to the base. Show that the volumes of the three portions starting from the top are in the ratio 1 : 7 : 19. 

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