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Mathematics Test 213
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Mathematics Test 213

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  • Question 1/10
    4 / -1

    The function f(x) = |2sgn(2x)| + 2 is (where sgn(x) means signum x)

    Solutions

    Given function f(x) = |2sgn(2x)| + 2

    At x = 0, we have

    L.H.L.= R.H.L.= 4 and f(0) = 2

    Hence,f(x) is discontinuous at x = 0

     

  • Question 2/10
    4 / -1

    If G.M. and H.M. of two numbers are 10 and 8 respectively. The numbers a

    Solutions

    Let the numbers are a and b

    ∴ √ab = 10

    ⇒ a + b = 25 …(2)

    On solving equations (1) & (2), we get

    a = 5, b = 20 or a = 20, b = 5

     

  • Question 3/10
    4 / -1

    Out of 3n consecutive natural numbers, 3 natural numbers are chosen at random without replacement. The probability that the sum of the chosen numbers is divisible by 3, is

    Solutions

    In 3n consecutive natural numbers, either

    (i) n numbers are of from 3P

    (ii) n numbers are of from 3P + 1

    (iii) n numbers are of from 3P + 2

    Here favourable number of cases = Either we can select three numbers from any of the set or we can select one from each set

     

     

  • Question 4/10
    4 / -1

    The coefficient of x5 in the expansion of (1 + x)21 + (1 + x)22 +...+(1 + x)30 is

    Solutions

     

  • Question 5/10
    4 / -1

    If e[sin2⁡α + sin4⁡α + sin6⁡α +... ∞] loge⁡2       is a root of equation x2 - 9x + 8=0, where 0 < α < π/2, find the value of 

    Solutions

     

  • Question 6/10
    4 / -1

    Which of the following function is periodic?

    Solutions

    Clearly, f(x) = x−[x] = {x}

    which has period 1.

    Let sin(1/x) be periodic with period T.

    Then,

    Now for a variable x and constant T, the given relation cannot hold ∀ allowable x. Hence, sin1/x is not periodic.

    Similarly, for f(x) = xcosx, let T be the period.

    Hence, (x + T) cos(x + T) = xcosx

    Note that LHS is a constant while RHS varies as xx varies for allowable values of x. Hence, no such T is possible, so xcosx is also non-periodic.

     

  • Question 7/10
    4 / -1

    Solutions

    Let (x, y)x, y is the set of points equidistant from point (2,3) and the line 3x + 4y − 2 = 0.

    So the given equation represents a parabola.

     

  • Question 8/10
    4 / -1

    The sum of an infinite geometric series is 2 and the sum of geometric series made from the cube of this infinite series is 24. Then the series is

    Solutions

     

  • Question 9/10
    4 / -1

    The area (in sq. units) of the region bounded by the curve y = 2x - x2 and the line y = x is

    Solutions

     

  • Question 10/10
    4 / -1

    If the lines x + 2ay + a = 0, x + 3by + b = 0, x + 4cy + c = 0 are concurrent, then a, b, c are in

    Solutions

     

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