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Board Paper of Class 10 2019 Maths Abroad(Set 3) - Solutions

General Instructions:

(i) All questions are compulsory.

(ii) The question paper consists of 30 questions divided into four sections – A, B, C and D.

(iii) Section A comprises 6 questions of 1 mark each. Section B contains 6 questions of 2 marks each. Section C contains 10 questions of 3 marks each. Section D contains 8 questions of 4 marks each.

(iv) There is no overall choice. However, an internal choice has been provided in two questions of 1 mark, two questions of 2 marks, four questions of 3 marks each and three questions of 4 marks each. You have to attempt only one of the alternative in all such questions.

(v) Use of calculators is not permitted.

  • Question 1
    Which term of the A.P. −4, −1, 2, ... is 101? VIEW SOLUTION
  • Question 2
    Evaluate:
    tan 65°cot 25°
                                       OR

    Express (sin 67° + cos 75°) in terms of trigonometric ratios of the angle between 0° and 45°. VIEW SOLUTION
  • Question 3
    Find the value of k for which the quadratic equation kx (x − 2) + 6 = 0 has two equal roots. VIEW SOLUTION
  • Question 4
    Find a rational number between 2 and 7.
                                                 OR
    Write the number of zeroes in the end of a number whose prime factorization is 22 × 53 × 32 × 17. VIEW SOLUTION
  • Question 5
    Find the distance between the points (a, b) and (−a, −b). VIEW SOLUTION
  • Question 6
    Let ∆ ABC ∽ ∆ DEF and their areas be respectively, 64 cm2 and 121 cm2. If EF = 15⋅4 cm, find BC. VIEW SOLUTION
  • Question 7
    Find the solution of the pair of equation :
    3x+8y=-1;  1x-2y=2, x, y 0
                          OR
    Find the value(s) of k for which the pair of equations kx+2y=33x+6y=10 has a unique solution. VIEW SOLUTION
  • Question 8
    Use Euclid's division algorithm to find the HCF of 255 and 867. VIEW SOLUTION
  • Question 9
    The point R divides the line segment AB, where A(−4, 0) and B(0, 6) such that AR=34AB. Find the coordinates of R. VIEW SOLUTION
  • Question 10
    How many multiples of 4 lie between 10 and 205 ?
                                       OR
    Determine the A.P. whose third term is 16 and 7th term exceeds the 5th by 12. VIEW SOLUTION
  • Question 11
    Three different coins are tossed simultaneously. Find the probability of getting exactly one head. VIEW SOLUTION
  • Question 12
    A die is thrown once. Find the probability of getting.
    (a) a prime number.
    (b) an odd number VIEW SOLUTION
  • Question 13
    In Figure 1, BL and CM are medians of a ∆ABC right-angled at A. Prove that 4 (BL2 + CM2) = 5 BC2.
     
     
    OR                                               

    Prove that the sum of the squares of the sides of a rhombus is equal to the sum of the squares of its diagonals. VIEW SOLUTION
  • Question 14
    In Figure 2, two concentric circles with centre O, have radii 21 cm and 42 cm. If ∠AOB = 60°, find the area of the shaded region.
    VIEW SOLUTION
  • Question 15
    A cone of height 24 cm and radius of base 6 cm is made up of modelling clay. A child reshapes it in the form of a sphere. Find the radius of the sphere and hence find the surface area of this sphere.
                                                                                OR
    A farmer connects a pipe of internal diameter 20 cm from a canal into a cylindrical tank in his field which is 10 m in diameter and 2 m deep. If water flows through the pipe at the rate of 3 km/hr, how much time will the tank be filled ? VIEW SOLUTION
  • Question 16
    Calculate the mode of the following distribution :
     
    Class : 10 − 15 15 − 20 20 − 25 25 − 30 30 − 35
    Frequency : 4 7 20 8 1
    VIEW SOLUTION
  • Question 17
    Show that 2+327 is not a rational number, given that 2 is an irrational number. VIEW SOLUTION
  • Question 18
    Obtain all the zeroes of the polynomial 2x4 − 5x3 − 11x2 + 20x + 12 when 2 and − 2 are two zeroes of the above polynomial VIEW SOLUTION
  • Question 19
    A motorboat whose speed is 18 km/hr in still water takes on hour more to go 24 km upstream than to return downstream to the same spot. Find the speed of the stream. VIEW SOLUTION
  • Question 20
    Prove that:

    (sin θ + 1 + cos θ) (sin θ − 1 + cos θ) . sec θ cosec θ = 2

                                     OR

    Prove that :

    sec θ-1sec θ+1+sec θ+1sec θ-1=2cosec θ VIEW SOLUTION
  • Question 21
    In what ratio does the point P(−4, y) divide the line segment joining the points A(−6, 10) and B(3, −8) ? Hence find the value of y.

                                                            OR

    Find the value of p for which the points (−5, 1), (1, p) and (4, −2) are collinear. VIEW SOLUTION
  • Question 22
    ABC is a right triangle in which ∠B = 90°.  If AB = 8 cm and BC = 6 cm, find the diameter of the circle inscribed in the triangle. VIEW SOLUTION
  • Question 23
    In an A.P., the first term is −4, the last term is 29 and the sum of all its terms is 150. Find its common difference VIEW SOLUTION
  • Question 24
    Draw a circle of radius 4 cm. From a point 6 cm away from its centre, construct a pair of tangents to the circle and measure their lengths VIEW SOLUTION
  • Question 25
    Prove that :

    2(sin6 θ + cos6 θ) − 3 (sin4 θ + cos4 θ) + 1 = 0 VIEW SOLUTION
  • Question 26
    Solve for x :12a+b+2x=12a+1b+12x; x0, x-2a-b2, a, b0
     
                                                                       OR

    The sum of the areas of two squares is 640 m2. If the difference of their perimeters is 64 m, find the sides of the square. VIEW SOLUTION
  • Question 27
    In ∆ ABC (Figure 3), AD ⊥ BC. Prove that
    AC2 = AB2 +BC2 − 2BC × BD
    VIEW SOLUTION
  • Question 28
    A moving boat is observed from the top of a 150 m high cliff moving away from the cliff. The angle of depression of the boat changes from 60° to 45° in 2 minutes. Find the speed of the boat in m/min.

                                                           OR

    There are two poles, one each on either bank of a river just opposite to each other. One pole is 60 m high. From the top of this pole, the angle of depression of the top and foot of the other pole are 30° and 60° respectively. Find the width of the river and height of the other pole. VIEW SOLUTION
  • Question 29
    Calculate the mean of the following frequency distribution :
     
    Class : 10−30 30−50 50−70 70−90 90−110 110−130
    Frequency : 5 8 12 20 3 2

                                                    OR
    The following table gives production yield in kg per hectare of wheat of 100 farms of a village :
    Production yield
    (kg/hectare) :
    40−45 45−50 50−55 55−60 60−65 65−70
    Number of farms 4 6 16 20 30 24

    Change the distribution to a 'more than type' distribution, and draw its ogive. VIEW SOLUTION
  • Question 30
    A container opened at the top and made up of a metal sheet, is in the form of a frustum of a cone of height 16 cm with radii of its lower and upper ends as 8 cm and 20 cm respectively. Find the cost of milk which can completely fill the container, at the rate of ₹ 50 per litre. Also find the cost of metal sheet used to make the container, if it costs ₹ 10 per 100 cm2. (Take π = 3⋅14) VIEW SOLUTION
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