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Mathematics Test 229
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Mathematics Test 229
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  • Question 1/10
    4 / -1

    Solutions


     

  • Question 2/10
    4 / -1

    The range of the function 

    Solutions

    Calculation:


     

  • Question 3/10
    4 / -1

    Solutions

    Concept:

    The points where the f'(x) = 0 are known as critical values.

    At critical point, if f ''(x) > 0 then function has minima.

    If f ''(x) < 0 then function has maxima.

    Calculation:

    For x ≥ 0

    f(x) = − 3sin x

    ⇒ f '(x) = − 3cos x

    ⇒ f ′(0) = − 3

    For x < 0, f (x) = x3 + x2 + 10x

    ⇒ f ′(x) = 3x2 + 2x + 10

    ⇒ f ′(0) = 10

    ⇒ f ′(x) > 0 for x < 0 and f ′(x) < 0 for x ≥ 0

    ⇒ f (x) has maxima at x = 0.

    ∴ At x = 0, there is a point of maximum.

    The correct answer is Option 1.

     

  • Question 4/10
    4 / -1

     where α ∈ Q is equal to  

    Solutions


  • Question 5/10
    4 / -1

    Four numbers are multiplied together. Then the probability that the product will be divisible by 5 or 10 is

    Solutions

    Calculation:

    The divisibility of the product of four numbers depends upon the value of the last digit of each number.

    The last digit of a number can be any of the 10 digits

    ⇒ The total number of ways of selecting last digits of four numbers is 10 × 10 × 10 × 10 = 104 

    Now, if the product of the 4 numbers is not divisible by 5 or 10

    ⇒ Then the number of choices for the last digit of each number is 8 (excluding 0 or 5).

    ⇒ Favourable number of ways = 8 × 8 × 8 × 8 = 84 

    ∴ P(product is divisible by 5 or 10)

    = 1 - P(product is divisible by 5 or 10)

     

  • Question 6/10
    4 / -1

    The differential equation of all circles which pass through the origin and whose centres lie on y-axis is

    Solutions

    Calculation:

    Given, the circles pass through the origin.

    They have their centres at (0, a) The circles have radius a.

    ∴ The equation of the family of circles in given by x2 + (y - a)2 = a2 

    ⇒ x2 + y2 + a2 - 2ya = a2 

    ⇒ x2 + y2 = 2ay ⋯ (i)

    Differentiating wrt x, we get:

    ∴ The differential equation of all circles which pass through the origin and whose centres lie on y-axis is 

    The correct answer is Option 1.

     

  • Question 7/10
    4 / -1

    The coefficient of x49 in the product (x – 1) (x – 3) ... (x – 99) is

    Solutions

    Concept:

    The sum of first n terms of an AP with first term and common difference d is given by:

    Calculation:

    We can observe, (x − a)(x − b) = x− (a + b)x + ab ​

    (x − a)(x − b)(x − c) = x3 − (a + b + c)x2 + (ab + bc + ca) x − abc

    (x − a)(x − b)(x − c)(x − d) = x4 − (a + b + c + d)x3 + (ab + bc + cd + da)x2 − (abc + bcd + cda + abd)x + abcd

    Now, (x − 1)(x − 3)(x − 5) ⋯ (x − 99)

    = x50 − (1 + 3 + 5 + ⋯ + 99)x49 +(1⋅3 + 1⋅5 + 1⋅7 + ⋯ + 1⋅99 + 3⋅5 + 3⋅7 + ⋯ + 3⋅99 + 5⋅7 + ⋯ + 97⋅99)x48 + ⋯

    ∴ Coefficient of x48 =

    = − [1 + 3 + 5 + ⋯ + 99]

    = − 25[2 + 49 × 2]

    = − 25 × 100

    = − 2500

    ∴ The coefficient of x49 in the product (x – 1) (x – 3) ... (x – 99) is − 2500.

    The correct answer is Option 3.

     

  • Question 8/10
    4 / -1

    Area bounded by the curves y = |x| - 2 and y = 1 - |x - 1| is equal to

    Solutions

    Calculation:

    ∴ Required area = ar(ABCD)

    = ar(ΔOAB) + ar(ΔODC) + ar(ΔOBC)

    = 1 + 1 + 2

    = 4 sq. units

    ∴ The area bounded by the curves y = |x| - 2 and y = 1 - |x - 1| is equal to 4 sq. units.

    The correct answer is Option 1.

     

  • Question 9/10
    4 / -1

    If two events A and B are such that P(A') = 0.3, P(B) = 0.4 and P (A ∩ B') = 0.5, then 

    Solutions


     

  • Question 10/10
    4 / -1

    If the point (1, a) lines in between the line x + y = 1 and 2(x + y) = 3, then a lies in

    Solutions

     

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