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Day 64: NDA Maths

5 questions (6 parts). Try each one first, then open its answer.

  1. Question 1

    Vectors. This type asked 2 times: NDA 1 2023, NDA 1 2025. This one: NDA 1 2025

    66
    A line makes angles α,β\alpha, \beta and γ\gamma with the positive directions of the coordinate axes. If a⃗=sin⁡2α i^+sin⁡2β j^+sin⁡2γ k^\vec{a} = \sin^2\alpha\,\hat{i} + \sin^2\beta\,\hat{j} + \sin^2\gamma\,\hat{k} and b⃗=i^+j^+k^\vec{b} = \hat{i}+\hat{j}+\hat{k}, then what is a⃗⋅b⃗\vec{a}\cdot\vec{b} equal to?

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  2. Question 2

    Vectors. This type asked 2 times: NDA 1 2018, NDA 1 2025. This one: NDA 1 2025

    69
    The position vectors of three points A, B and C are a⃗,b⃗\vec{a}, \vec{b} and c⃗\vec{c} respectively, where c⃗=cos⁡2θ a⃗+sin⁡2θ b⃗\vec{c} = \cos^2\theta\,\vec{a} + \sin^2\theta\,\vec{b}. What is (a⃗×b⃗)+(b⃗×c⃗)+(c⃗×a⃗)(\vec{a}\times\vec{b})+(\vec{b}\times\vec{c})+(\vec{c}\times\vec{a}) equal to?

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  3. Question 3

    Vectors. This type asked 2 times: NDA 2 2021, NDA 1 2025. This one: NDA 1 2025

    67
    Consider the following statements in respect of a vector d⃗=(a⃗×b⃗)×c⃗\vec{d} = (\vec{a}\times\vec{b})\times\vec{c}: (I). d⃗\vec{d} is coplanar with a⃗\vec{a} and b⃗\vec{b}. (II). d⃗\vec{d} is perpendicular to c⃗\vec{c}. Which of the statements given above is/are correct?

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  4. Question 4

    3D Geometry. This type asked 2 times: NDA 1 2024, NDA 2 2024. This one: NDA 2 2024

    Let SS be the line of intersection of two planes x+y+z=1x+y+z=1 and 2x+3y−4z=82x+3y-4z=8 with direction ratios ⟨−7,6,1⟩\langle -7,6,1\rangle. If (l,m,n)(l,m,n) are direction cosines of SS, then what is the value of 43l2−m2−n243l^2-m^2-n^2?
    67
    Which of the following are the direction ratios of SS?

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  5. Question 5 (case study, 2 parts)

    3D Geometry. This type asked 2 times: NDA 1 2024, NDA 2 2024. This one: NDA 2 2024

    Let L:x+y+z+4=0=2x−y−z+8L:x+y+z+4=0=2x-y-z+8 be a line and P:x+2y+3z+1=0P: x+2y+3z+1=0 be a plane.
    69
    What are the direction ratios of the line?

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    70
    What is the point of intersection of LL and PP?

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