Vectors · NDA previous year questions

2021 11 questions

1. Q662021 · 14 Nov 2021

   Let $\vec{a}$, $\vec{b}$ and $\vec{c}$ be unit vectors such that $\vec{a}\times \vec{b}$ is perpendicular to $\vec{c}$. If $\theta$ is the angle between $\vec{a}$ and $\vec{b}$, then which of the following is/are correct? 1. $\vec{a}\times \vec{b}=\sin \theta \vec{c}$ 2. $\vec{a}\cdot (\vec{b}\times \vec{c})=0$ Select the correct answer using the code given below.

   1. A.1 only
   2. B.2 only
   3. C.Both 1 and 2
   4. D.Neither 1 nor 2
   Show answer

   Answer: B

2. Q672021 · 14 Nov 2021

   If $\vec{a}+3\vec{b}=3\hat{i}-\hat{j}$ and $2\vec{a}+\vec{b}=\hat{i}-2\hat{j}$, then what is the angle between $\vec{a}$ and $\vec{b}$

   1. A.0
   2. B.$\frac{\pi }{6}$
   3. C.$\frac{\pi }{3}$
   4. D.$\frac{\pi }{2}$
   Show answer

   Answer: D

3. Q682021 · 14 Nov 2021

   If $(\vec{a}+\vec{b})$ is perpendicular to $\vec{a}$ and magnitude of $\vec{b}$ is twice that of $\vec{a}$, then what is the value of $(4\vec{a}+\vec{b})\cdot \vec{b}$ equal to?

   1. A.0
   2. B.1
   3. C.$8|\vec{a}{|}^2$
   4. D.$8|\vec{b}{|}^2$
   Show answer

   Answer: A

4. Q692021 · 14 Nov 2021

   Let $\vec{a},\vec{b}$ and $\vec{c}$ be three vectors such that $\vec{a},\vec{b}$ and $\vec{c}$ are co-planar. Which of the following is/are correct? 1. $(\vec{a}\times \vec{b})\times \vec{c}$ is co-planar with $\vec{a}$ and $\vec{b}$ 2. $(\vec{a}\times \vec{b})\times \vec{c}$ is perpendicular to $\vec{a}$ and $\vec{b}$ Select the correct answer using the code given below.

   1. A.1 only
   2. B.2 only
   3. C.Both 1 and 2
   4. D.Neither 1 nor 2
   Show answer

   Answer: A

5. Q702021 · 14 Nov 2021

   If the position vectors of A and B are $(\sqrt{2}-1)\hat{i}-\hat{j}$ and $\hat{i}+(\sqrt{2}+1)\hat{j}$ respectively, then what is the magnitude of $\overrightarrow{AB}$?

   1. A.$2\sqrt{2}$
   2. B.$3\sqrt{2}$
   3. C.$2\sqrt{3}$
   4. D.$3\sqrt{3}$
   Show answer

   Answer: C

6. Q712021 · 14 Nov 2021

   If $y=(1+x)(1+x^2)(1+x^4)(1+x^8)(1+x^{16})$ then what is $\frac{\mathrm{d}y}{\mathrm{d}x}$ at $x=0$ equal to?

   1. A.0
   2. B.1
   3. C.2
   4. D.4
   Show answer

   Answer: B

7. Q1162021 · 18 Apr 2021

   A vector $\vec{r}=a\hat{i}+b\hat{j}$ is equally inclined to both x and y axes. If the magnitude of the vector is 2 units, then what are the values of a and b respectively?

   1. A.$\frac{1}{2}$, $\frac{1}{2}$
   2. B.$\frac{1}{\sqrt{2}}$, $\frac{1}{\sqrt{2}}$
   3. C.$\sqrt{2}$, $\sqrt{2}$
   4. D.2, 2
   Show answer

   Answer: C

8. Q1172021 · 18 Apr 2021

   Consider the following statements in respect of a vector $\vec{c}=\vec{a}+\vec{b}$, where $|\vec{a}|=|\vec{b}|\ne 0$: 1. $\vec{c}$ is perpendicular to $(\vec{a}-\vec{b})$. 2. $\vec{c}$ is perpendicular to $\vec{a}\times \vec{b}$. Which of the above statement is/are correct?

   1. A.1 only
   2. B.2 only
   3. C.Both 1 and 2
   4. D.Neither 1 nor 2
   Show answer

   Answer: C

9. Q1182021 · 18 Apr 2021

   If $\vec{a}$ and $\vec{b}$ are two vectors such that $|\vec{a}+\vec{b}|=|\vec{a}-\vec{b}|=4$, then which one of the following is correct?

   1. A.$\vec{a}$ and $\vec{b}$ must be unit vectors.
   2. B.$\vec{a}$ must be parallel to $\vec{b}$.
   3. C.$\vec{a}$ must be perpendicular to $\vec{b}$.
   4. D.$\vec{a}$ must be equal to $\vec{b}$.
   Show answer

   Answer: C

10. Q1192021 · 18 Apr 2021

    If $\vec{a},\vec{b}$ and $\vec{c}$ are coplanar, then what is $(2\vec{a}\times 3\vec{b})\cdot 4\vec{c}+(5\vec{b}\times 3\vec{c})\cdot 6\vec{a}$ equal to?

    1. A.114
    2. B.66
    3. C.0
    4. D.-66
    Show answer

    Answer: C

11. Q1202021 · 18 Apr 2021

    Consider the following statements: 1. The cross product of two unit vectors is always a unit vector. 2. The dot product of two unit vectors is always unity. 3. The magnitude of sum of two unit vectors is always greater than the magnitude of their difference. Which of the above statements are not correct?

    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: D