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Class X: Math, Board Paper 2015, Set-1

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

    Find the 25th term of the A.P. -5, -52, 0, 52, ... 


  • Q2

    A pole casts a shadow of length 23 m on the ground, when the sun's elevation is 60°. Find the height of the pole. 


  • Q3

    A game of chance consists of spinning an arrow which comes to rest pointing at one of the numbers 1, 2, 3, 4, 5, 6, 7, 8 and these are equally likely outcomes. Find the probability that the arrow will point at any factor of 8. 


  • Q4

    Two concentric circles of radii a and b (a > b) are given. Find the length of the chord of the larger circle which touches the smaller circle. 


  • Q5

    In figure 1, O is the centre of a circle. PT and PQ are tangents to the circle from an external point P. If ∠TPQ = 70°, find ∠TRQ.



  • Q6

    In Figure 2, PQ is a chord of length 8 cm of a circle of radius 5 cm. The tangents at P and Q intersect at a point T. Find the lengths of TP and TQ.



  • Q7

    Solve for x :

    x2-3+1 x+3=0 


  • Q8

    The fourth term of an A.P. is 11. The sum of the fifth and seventh terms of the A.P. is 34. Find its common difference. 


  • Q9

    Show that the points (a, a), (–a, –a) and -3a, 3a are the vertices of an equilateral triangle. 


  • Q10

    For what values of k are the points (8, 1), (3, –2k) and (k, –5) collinear ? 


  • Q11

    Point A lies on the line segment PQ joining P(6, –6) and Q(–4, –1) in such a way that PAPQ=25. If point P also lies on the line 3x + k (y + 1) = 0, find the value of k. 


  • Q12

    Solve for x :
    x2 + 5x − (a2 + a − 6) = 0 


  • Q13

    In an A.P., if the 12th term is −13 and the sum of its first four terms is 24, find the sum of its first ten terms. 


  • Q14

    A bag contains 18 balls out of which x balls are red.

    (i) If one ball is drawn at random from the bag, what is the probability that it is not red?
    (ii) If 2 more red balls are put in the bag, the probability of drawing a red ball will be 98 times the probability of drawing a red ball in the first case. Find the value of x. 


  • Q15

    From the top of a tower of height 50 m, the angles of depression of the top and bottom of a pole are 30° and 45° respectively. Find
    (i) how far the pole is from the bottom of a tower,
    (ii) the height of the pole. (Use 3=1.732) 


  • Q16

    The long and short hands of a clock are 6 cm and 4 cm long respectively. Find the sum of the distances travelled by their tips in 24 hours. (Use π = 3.14) 


  • Q17

    Two spheres of same metal weigh 1 kg and 7 kg. The radius of the smaller sphere is 3 cm. The two spheres are melted to form a single big sphere. Find the diameter of the new sphere. 


  • Q18

    A metallic cylinder has radius 3 cm and height 5 cm. To reduce its weight, a conical hole is drilled in the cylinder. The conical hole has a radius of 32 cm and its depth is 89 cm. Calculate the ratio of the volume of metal left in the cylinder to the volume of metal taken out in conical shape. 


  • Q19

    In Figure 3, ABCD is a trapezium with AB || DC, AB = 18 cm, DC = 32 cm and the distance between AB and DC is 14 cm. If arcs of equal radii 7 cm have been drawn, with centres A,B, C  and D, then find the area of the shaded region.

                  Figure 3 


  • Q20

    A solid right-circular cone of height 60 cm and radius 30 cm is dropped in a right-circular cylinder full of water of height 180 cm and radius 60 cm. Find the volume of water left in the cylinder, in cubic metres.

    Use π=227



  • Q21

    If x = −2 is a root of the equation 3x2 + 7x + p = 0, find the values of k so that the roots of the equation x2 + k(4x + k − 1) + p = 0 are equal. 


  • Q22

    Find the middle term of the sequence formed by all three-digit numbers which leave a remainder 3, when divided by 4. Also find the sum of all numbers on both sides of the middle term separately. 


  • Q23

    The total cost of a certain length of a piece of cloth is Rs 200. If the piece was 5 m longer and each metre of cloth costs Rs 2 less, the cost of the piece would have remained unchanged. How long is the piece and what is its original rate per metre ? 


  • Q24

    Prove that the tangent at any point of a circle is perpendicular to the radius through the point of contact. 


  • Q25

    In Figure 4, O is the centre of the circle and TP is the tangent to the circle from an external point T. If ∠PBT = 30°, prove that BA : AT = 2 : 1.

    Figure 4


  • Q26

    Draw a circle of radius 3 cm. From a point P, 7 cm away from its centre draw two tangents to the circle. Measure the length of each tangent. 


  • Q27

    Two poles of equal heights are standing opposite to each other on either side of the road which is 80 m wide. From a point P between them on the road, the angle of elevation of the top of a pole is 60° and the angle of depression from the top of another pole at point P is 30°. Find the heights of the poles and the distances of the point P from the poles. 


  • Q28

    A box contains cards bearing numbers from 6 to 70. If one card is drawn at random from the box, find the probability that it bears

    (i) a one digit number.

    (ii) a number divisible by 5.

    (iii) an odd number less than 30.

    (iv) a composite number between 50 and 70.



  • Q29

    The base BC of an equilateral triangle ABC lies on y-axis. The coordinates of point C are (0, −3). The origin is the mid-point of the base. Find the coordinates of the points A and B. Also find the coordinates of another point D such that BACD is a rhombus. 


  • Q30

    A vessel full of water is in the form of an inverted cone of height 8 cm and the radius of its top, which is open, is 5 cm. 100 spherical lead balls are dropped into the vessel. One-fourth of the water flows out of the vessel. Find the radius of a spherical ball. 


  • Q31

    Milk in a container, which is in the form of a frustum of a cone of height 30 cm and the radii of whose lower and upper circular ends are 20 cm and 40 cm respectively, is to be distributed in a camp for flood victims. If this milk is available at the rate of Rs 35 per litre and 880 litres of milk is needed daily for a camp, find how many such containers of milk are needed for a camp and what cost will it put on the donor agency for this. What value is indicated through this by the donor agency ? 


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