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Day 205: NDA Maths

5 questions. Try each one first, then open its answer.

  1. Question 1

    Complex Numbers. Asked once: NDA 1 2021 (same type asked 2 times)

    39
    Consider the following in respect of a complex number ZZ: 1. (Z−1)‾=(Zˉ)−1\overline{(Z^{-1})} = (\bar{Z})^{-1}. 2. ZZ−1=∣Z∣2ZZ^{-1} = |Z|^2. Which of the above is/are correct?

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  2. Question 2

    Matrices & Determinants. Asked once: NDA 1 2017 (same type asked 2 times)

    30
    If A=(cos⁡αsin⁡α−sin⁡αcos⁡α)A = \begin{pmatrix} \cos\alpha & \sin\alpha \\ -\sin\alpha & \cos\alpha \end{pmatrix}, then what is AATAA^T equal to (where ATA^T is the transpose of AA)?

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  3. Question 3

    Matrices & Determinants. Asked once: NDA 2 2022 (same type asked 2 times)

    5
    Consider the determinant Δ=∣a11a12a13a21a22a23a31a32a33∣\Delta = \begin{vmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{vmatrix}. If a13=yza_{13} = yz, a23=zxa_{23} = zx, a33=xya_{33} = xy and the minors of a13,a23,a33a_{13}, a_{23}, a_{33} are respectively (z−y)(z-y), (z−x)(z-x), (y−x)(y-x), then what is the value of Δ\Delta?

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  4. Question 4

    3D Geometry. Asked once: NDA 1 2024 (same type asked 2 times)

    57
    If the direction cosines ⟨l,m,n⟩\langle l,m,n\rangle of a line are connected by relation l+2m+n=0l+2m+n=0, 2l−2m+3n=02l-2m+3n=0, then what is the value of l2+m2−n2l^2+m^2-n^2?

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  5. Question 5

    Matrices & Determinants. Asked once: NDA 2 2023 (same type asked 2 times)

    8
    If A=(034−305−4−50)A = \begin{pmatrix}0&3&4\\-3&0&5\\-4&-5&0\end{pmatrix}, then which one of the following statements is correct?

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