Vectors · NDA previous year questions

2025 12 questions

1. Q672025 · 14 Sept 2025

   How many of the following can be a vector perpendicular to both the vectors $2\hat{i}-\hat{j}+\hat{k}$ and $\hat{i}+\hat{j}+3\hat{k}$ ? I. $4\hat{i}+5\hat{j}-3\hat{k}$ II. $-8\hat{i}-10\hat{j}+6\hat{k}$ III. $\frac{1}{50}(-4\hat{i}-5\hat{j}+3\hat{k})$ Select the correct answer.

   1. A.None
   2. B.One
   3. C.Two
   4. D.All three
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   Answer: D

2. Q682025 · 14 Sept 2025

   What is the area of the parallelogram whose sides are represented by the vectors $\hat{i}+2\hat{j}+3\hat{k}$ and $2\hat{i}+\hat{j}+2\hat{k}$ ?

   1. A.$\frac{1}{2}\sqrt{26}$ square units
   2. B.$\frac{1}{2}\sqrt{27}$ square units
   3. C.$\sqrt{26}$ square units
   4. D.$\sqrt{27}$ square units
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   Answer: C

3. Q692025 · 14 Sept 2025

   The position vectors of the vertices A, B, Cand D of a quadrilateral ABCD are given by $3\hat{i}+4\hat{j}-2\hat{k}$ , $4\hat{i}-4\hat{j}-3\hat{k},2\hat{i}-3\hat{j}+2\hat{k}$ and $6\hat{i}-2\hat{j}+\hat{k}$ respectively. What is the angle between the diagonals AC and BD of the quadrilateral?

   1. A.$90^{\circ }$
   2. B.$75^{\circ }$
   3. C.$60^{\circ }$
   4. D.$45^{\circ }$
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   Answer: A

4. Q702025 · 14 Sept 2025

   A force $\vec{F}=2\hat{i}-\lambda \hat{j}+5\hat{k}$ is applied at the point A(1, 2, 5) If its moment about the point B(- 1, - 2, 3) is $16\hat{i}-6\hat{j}+2\lambda \hat{k}$, then what is the value of $\lambda$ ?

   1. A.-2
   2. B.0
   3. C.1
   4. D.2
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   Answer: A

5. Q712025 · 14 Sept 2025

   For the following two (02) items: Let $$f(x)=\{\begin{array}{ll}\frac{1-\cos 2x}{x^2}& ,x<0\\ 9& ,x=0\\ \frac{\sqrt{x}}{\sqrt{16+\sqrt{x}}-4}& ,x>0\end{array}$$ What is ${\lim }_{x\to 0^{-}}f(x)$ equal to?

   1. A.2
   2. B.4
   3. C.6
   4. D.8
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   Answer: A

6. Q722025 · 14 Sept 2025

   For the following two (02) items: Let $$f(x)=\{\begin{array}{ll}\frac{1-\cos 2x}{x^2}& ,x<0\\ 9& ,x=0\\ \frac{\sqrt{x}}{\sqrt{16+\sqrt{x}}-4}& ,x>0\end{array}$$ What is ${\lim }_{x\to 0^{+}}f(x)$ equal to?

   1. A.6
   2. B.7
   3. C.8
   4. D.9
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   Answer: C

7. Q672025 · 13 Apr 2025

   Consider the following statements with respect to a vector d = (a × b) × c: I. d is coplanar with a and b. II. d is perpendicular to c. Which of the statements given above is/are correct?

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

8. Q682025 · 13 Apr 2025

   The position vectors of three points A, B and C are $\vec{a}$, $\vec{b}$ and $\vec{c}$ respectively such that $3\vec{a}-4\vec{b}+\vec{c}=\vec{0}$. What is AB:BC equal to?

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

9. Q692025 · 13 Apr 2025

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

   1. A.$\vec{0}$
   2. B.$2\vec{c}$
   3. C.$3\vec{c}$
   4. D.Unit vector
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   Answer: A

10. Q702025 · 13 Apr 2025

    Let $\vec{a},\vec{b},(\vec{a}\times \vec{b})$ be unit vectors. What is $\vec{a}.\vec{b}$ equal to?

    1. A.0
    2. B.1/2
    3. C.1
    4. D.3
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    Answer: A

11. Q712025 · 13 Apr 2025

    Let $x=\sec \theta -\cos \theta$ and $y={\sec }^4\theta -{\cos }^4\theta$ What is ${\left(\frac{dy}{dx}\right)}^2$ equal to?

    1. A.$\frac{4(y^2+4)}{(x^2+4)}$
    2. B.$\frac{4(y^2-4)}{(x^2-4)}$
    3. C.$\frac{16(y^2+4)}{(x^2+4)}$
    4. D.$\frac{16(y^2-4)}{(x^2-4)}$
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    Answer: C

12. Q722025 · 13 Apr 2025

    Let $x=\sec \theta -\cos \theta$ and $y={\sec }^4\theta -{\cos }^4\theta$ What $\left[\frac{x^2+4}{y^2+4}\frac{dy}{dx}\left(x^2+4\frac{d^{2}y}{d{x}^2}-16y\right)\right]$ equal to?

    1. A.$16x$
    2. B.$16y$
    3. C.$-16x$
    4. D.$-16y$
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    Answer: C