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Day 59: NDA Maths

5 questions (7 parts). Try each one first, then open its answer.

  1. Question 1 (case study, 2 parts)

    Vectors. This type asked 2 times: NDA 1 2017, NDA 1 2026. This one: NDA 1 2026

    For the next two (02) items that follow: Let a⃗×b⃗=c⃗\vec{a}\times\vec{b}=\vec{c} and b⃗×c⃗=a⃗\vec{b}\times\vec{c}=\vec{a}.
    67
    Consider the following statements: I. a⃗\vec{a}, b⃗\vec{b}, c⃗\vec{c} are orthogonal in pairs. II. a⃗\vec{a}, b⃗\vec{b}, c⃗\vec{c} are unit vectors. Which of the statements given above is/are correct?

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    68
    Consider the following statements: I. (a⃗×b⃗)⋅c⃗+(b⃗×c⃗)⋅a⃗=(c⃗×a⃗)⋅b⃗(\vec{a}\times\vec{b})\cdot\vec{c}+(\vec{b}\times\vec{c})\cdot\vec{a}=(\vec{c}\times\vec{a})\cdot\vec{b}. II. {(a⃗×b⃗)×(b⃗×c⃗)}⋅b⃗=1\{(\vec{a}\times\vec{b})\times(\vec{b}\times\vec{c})\}\cdot\vec{b}=1. Which of the statements given above is/are correct?

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  2. Question 2

    Binary Numbers. This type asked 2 times: NDA 1 2020, NDA 2 2025. This one: NDA 2 2025

    41
    What is the sum of the binary numbers (101101101)2(101101101)_2 and (100011)2(100011)_2?

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  3. Question 3

    Complex Numbers. This type asked 2 times: NDA 2 2024, NDA 2 2025. This one: NDA 2 2025

    8
    If α=−1+−32\alpha = \dfrac{-1+\sqrt{-3}}{2}, then what is the value of (1+α19−α35)100−(1−3α25+α38)50(1+\alpha^{19}-\alpha^{35})^{100}-(1-3\alpha^{25}+\alpha^{38})^{50}?

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  4. Question 4

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

    59
    Consider the following statements in respect of the equation x2+3y=0x^2+3y=0: I. The equation represents the equation to parabola that opens upwards. II. The axis of the parabola is x=0x=0. III. The equation of the latus rectum is 4y−3=04y-3=0. How many of the statements given above are correct?

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  5. Question 5 (case study, 2 parts)

    Functions. This type asked 2 times: NDA 1 2017, NDA 2 2025. This one: NDA 2 2025

    For the following two (02) items: Consider the function f(x)=x1−xf(x)=\dfrac{x}{1-x} (x>0, x≠1)(x>0,\,x\neq1).
    76
    What is f(x)f(x+1)\dfrac{f(x)}{f(x+1)} equal to?

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    77
    What is (1−x)f(x)+xf(x+1)(1-x)f(\sqrt{x})+xf(\sqrt{x}+1) equal to?

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