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NDA I 2025 Mathematics Test - 6
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NDA I 2025 Mathematics Test - 6

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  • Question 1/10
    2.5 / -0.83

    Solutions

  • Question 2/10
    2.5 / -0.83

    Solutions

  • Question 3/10
    2.5 / -0.83

    Let C be the circle with centre at (1, 1) and radius = 1. If T is the circle centred at (0, y) passing through origin and touching the circle C externally, then the radius of T is equal to

    Solutions

    Calculation:

    Given, C is the circle with centre at (1, 1) and radius = 1.

    T is the circle centred at (0, y) passing through origin and touching the circle C externally.

  • Question 4/10
    2.5 / -0.83

    Solutions

    Concept:

    Formulae

    Calculation:

    Statement I:

    We have, d(x2 + 1) = 2x dx

  • Question 5/10
    2.5 / -0.83

     where a1, a2,⋯ , an ∈ R arnd ω is the complex cube root of unity. Then the value of 

    Solutions

    Concept:

    The roots of equation x3 = 1 are given by the cube roots of unity.

    = i - i

    = 0

    ∴ The required answer is 0.

    The correct answer is Option 1.

  • Question 6/10
    2.5 / -0.83

    If the system of equations 

    11x + y + λz = –5

    2x + 3y + 5z = 3

    8x – 19y – 39z = µ

    has infinitely many solutions, then λ4 – µ is equal to :

    Solutions

    Calculation:

    Given,

    11x + y + λz = –5

    2x + 3y + 5z = 3

    8x – 19y – 39z = µ

    Now, for infinite sol D = 0

    ⇒ -5(-117 + 95) -1(-117 - 5μ) -2(-57 - 3μ) = 0

    ⇒ -5(-22) + 117 + 5μ + 114 + 6μ = 0

    ⇒ 11μ = -110 - 231 = -341

    ⇒ μ = - 31

    ∴ λ4 - μ = (- 2)4 - (- 31)

    = 16 + 31 = 47

    ∴ The value of λ4 – µ is equal to 49.

    The correct answer is Option 3.

  • Question 7/10
    2.5 / -0.83

    If p and q are respectively the coefficient of x-3 and x-5 in the expansion of 

    Solutions

    Concept:

    General term of expansion (a + b)n is:

    Tr + 1 =  nCr an-r ̇br

    Calculation:

    Given p is the coefficient of x-3 

    ⇒ (21 -2r)/3  = - 3

    ⇒ 21 - 2r = - 9

    ⇒ 2r = 30

    ⇒ r = 15

    Substituting r = 15 in equation (1)

  • Question 8/10
    2.5 / -0.83

    If in a triangle ABC, sin3 A + sin3 B + sin3 C = 3 sin A sin B sin C, then what is the value of the determinant  where a, b, c are sides of the triangle?

    Solutions

    Concept:

    (i) sine rule: Let ABC be a triangle with angles A, B, C and corresponding sides a, b, c then

  • Question 9/10
    2.5 / -0.83

    Let A and B be two 3 × 3 non singular matrices, such that det (ATBA) = 27 and det (AB-1) = 8. Then det (BTA-1B) =

    Solutions

    Concept:

    |AB| = |A| × |B|

    |AT| = |A|

    Calculation:

    Given det (ATBA) = 27 and det (AB-1) = 8.

    ⇒ |AT| × |B| × |A| = 27

    ∵ |AT| = |A| ,

    |A|2 |B| = 27 ...(1)

    Similarly det (AB-1) = 8

    ⇒ |A| × |B-1| = 8

    ⇒ |A| = 8 × |B| ...(2)

    Substitute |A| in eq (1)

    ⇒ 82 |B|2|B| = 27

  • Question 10/10
    2.5 / -0.83

    Consider the following statements:

    Which of the above statement is/are correct?

    Solutions

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