Vectors · NDA previous year questions

2023 12 questions

1. Q912023 · 3 Sept 2023

   What is the length of projection of the vector $\hat{i}+2\hat{j}+3\hat{k}$ on the vector $2\hat{i}+3\hat{j}-2\hat{k}$ ?

   1. A.$\frac{1}{\sqrt{17}}$
   2. B.$\frac{2}{\sqrt{17}}$
   3. C.$\frac{3}{\sqrt{17}}$
   4. D.$\frac{2}{\sqrt{14}}$
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   Answer: B

2. Q922023 · 3 Sept 2023

   If $(\vec{a}\times \vec{b}{)}^2+(\vec{a}\cdot \vec{b}{)}^2=144$ and $|\vec{b}|=4$, then what is the value of $|\vec{a}|$ ?

   1. A.3
   2. B.4
   3. C.5
   4. D.6
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   Answer: A

3. Q932023 · 3 Sept 2023

   If $\theta$ is the angle between vectors $\vec{a}$ and $\vec{b}$ such that $\vec{a}\cdot \vec{b}\ge 0$, then which one of the following is correct?

   1. A.$0\le \theta \le \pi$
   2. B.$\frac{\pi }{2}\le \theta \le \pi$
   3. C.$0\le \theta \le \frac{\pi }{2}$
   4. D.$0<\theta <\frac{\pi }{2}$
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   Answer: C

4. Q942023 · 3 Sept 2023

   The vectors $60\hat{i}+3\hat{j}$, $40\hat{i}-8\hat{j}$ and $\beta \hat{i}-52\hat{j}$ are collinear if :

   1. A.$\beta =20$
   2. B.$\beta =40$
   3. C.$\beta =-40$
   4. D.$\beta =26$
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   Answer: C

5. Q952023 · 3 Sept 2023

   Consider the following in respect of the vectors $\vec{a}=(0,1,1)$ and $\vec{b}=(1,0,1)$: 1. The number of unit vectors perpendicular to both $\vec{a}$ and $\vec{b}$ is only one. 2. The angle between the vectors is $\frac{\pi }{3}$. Which of the statements given above is/are correct?

   1. A.1 only
   2. B.2 only
   3. C.Both 1 and 2
   4. D.Neither 1 nor 2
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   Answer: B

6. Q672023 · 16 Apr 2023
   
   1. A.$\frac{1}{3}$
   2. B.$\frac{4}{9}$
   3. C.$\frac{5}{9}$
   4. D.$\frac{2}{3}$
   Show answer

   Answer: B

7. Q682023 · 16 Apr 2023

   Consider the following for the next two (02) items that follow : Let a vector $\vec{a}=4\hat{i}-8\hat{j}+\hat{k}$ make angles $\alpha ,\beta ,\gamma$ with the positive directions of x, y, z axes respectively. What is $\cos 2\beta +\cos 2\gamma$ equal to ?

   1. A.$-\frac{32}{81}$
   2. B.$-\frac{16}{81}$
   3. C.$\frac{16}{81}$
   4. D.$\frac{32}{81}$
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   Answer: A

8. Q692023 · 16 Apr 2023
   Passage

   Consider the following for the next two (02) items that follow : The position vectors of two points A and B are î - ĵ and ĵ + k̂ respectively. Consider the following points : 1. (-1, -3, 1) 2. (-1, 3, 2) 3. (-2, 5, 3)

   Which of the above points lie on the line joining A and B ?

   1. A.1 and 2 only
   2. B.2 and 3 only
   3. C.1 and 3 only
   4. D.1, 2 and 3
   Show answer

   Answer: B

9. Q702023 · 16 Apr 2023
   Passage

   Consider the following for the next two (02) items that follow : The position vectors of two points A and B are î - ĵ and ĵ + k̂ respectively. Consider the following points : 1. (-1, -3, 1) 2. (-1, 3, 2) 3. (-2, 5, 3)

   
   1. A.2
   2. B.3
   3. C.√6
   4. D.√3
   Show answer

   Answer: C

10. Q712023 · 16 Apr 2023

    Consider the following for the next three (03) items that follow : Let $f(x)=P{e}^x+Q{e}^{2x}+R{e}^{3x}$ where P, Q, R are real numbers. Further $f(0)=6$, $f^{\prime }(\ln 3)=282$ and ${\int }_0^{\ln 2}f(x)\mathrm{dx}=11$ ‘What is the value of Q ?

    1. A.1
    2. B.2
    3. C.3
    4. D.4
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    Answer: B

11. Q722023 · 16 Apr 2023

    What is the value of R ?

    1. A.1
    2. B.2
    3. C.3
    4. D.4
    Show answer

    Answer: C

12. Q732023 · 16 Apr 2023

    What is f'(0) equal to ?

    1. A.18
    2. B.16
    3. C.15
    4. D.14
    Show answer

    Answer: D