Vectors · NDA previous year questions

2011 20 questions

1. Q292011 · 21 Aug 2011

   If $\vec{a}$ and $\vec{b}$ are two vectors such that $\vec{a}\cdot \vec{b}=0$ and $\vec{a}\times \vec{b}=\vec{0}$, then which one of the following is correct?

   1. A.$\vec{a}$ is parallel to $\vec{b}$.
   2. B.$\vec{a}$ is perpendicular to $\vec{b}$.
   3. C.Either $\vec{a}$ or $\vec{b}$ is a null vector.
   4. D.None of the above
   Show answer

   Answer: C

2. Q302011 · 21 Aug 2011

   If $y=\ln \sqrt{\tan x}$, then what is the value of $\frac{dy}{dx}$ at $x=\frac{\pi }{4}$?

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

   Answer: D

3. Q502011 · 21 Aug 2011

   If the vectors $-\hat{i}-2x\hat{j}-3y\hat{k}$ and $\hat{i}-3x\hat{j}-2y\hat{k}$ are orthogonal to each other, then what is the locus of the point $(x,y)$ ?

   1. A.a straight line
   2. B.an ellipse
   3. C.a parabola
   4. D.a circle
   Show answer

   Answer: D

4. Q582011 · 21 Aug 2011

   If $\vec{c}$ is the unit vector perpendicular to both the vectors $\vec{a}$ and $\vec{b}$, then what is another unit vector perpendicular to both the vectors $\vec{a}$ and $\vec{b}$ ?

   1. A.$\vec{c}\times \vec{a}$
   2. B.$\vec{c}\times \vec{b}$
   3. C.$-\frac{(\vec{a}\times \vec{b})}{|\vec{a}\times \vec{b}|}$
   4. D.$\frac{(\vec{a}\times \vec{b})}{|\vec{a}\times \vec{b}|}$
   Show answer

   Answer: C

5. Q662011 · 21 Aug 2011

   For what value of m are the points with position vectors $10\hat{i}+3\hat{j}$, $12\hat{i}-5\hat{j}$ and $m\hat{i}+11\hat{j}$ collinear ?

   1. A.$-8$
   2. B.4
   3. C.8
   4. D.12
   Show answer

   Answer: C

6. Q712011 · 21 Aug 2011

   For what value of m are the vectors $2\hat{i}-3\hat{j}+4\hat{k}$, $\hat{i}+2\hat{j}-\hat{k}$ and $m\hat{i}-\hat{j}+2\hat{k}$ coplanar?

   1. A.0
   2. B.5/3
   3. C.1
   4. D.8/5
   Show answer

   Answer: D

7. Q92011 · 17 Apr 2011

   $A,B,C,D$ and $E$ are coplanar points and three of them lie in a straight line. What is the maximum number of triangles that can be drawn with these points as their vertices?

   1. A.5
   2. B.9
   3. C.10
   4. D.12
   Show answer

   Answer: B

8. Q102011 · 17 Apr 2011

   The vector $\vec{a}$ lies in the plane of vectors $\vec{b}$ and $\vec{c}$. Which one of the following is correct?

   1. A.$\vec{a}\cdot (\vec{b}\times \vec{c})=0$
   2. B.$\vec{a}\cdot (\vec{b}\times \vec{c})=1$
   3. C.$\vec{a}\cdot (\vec{b}\times \vec{c})=-1$
   4. D.$\vec{a}\cdot (\vec{b}\times \vec{c})=3$
   Show answer

   Answer: A

9. Q112011 · 17 Apr 2011

   What is the projection of the vector $\hat{i}-2\hat{j}+\hat{k}$ on the vector $4\hat{i}-4\hat{j}+7\hat{k}$?

   1. A.$\frac{\sqrt{5}}{2}$
   2. B.$\frac{19}{9}$
   3. C.$\frac{\sqrt{5}}{4}$
   4. D.$\frac{11}{3}$
   Show answer

   Answer: B

10. Q522011 · 17 Apr 2011

    The points with position vectors $10\hat{i}+3\hat{j},12\hat{i}-5\hat{j},a\hat{i}+11\hat{j}$ are collinear, if the value of $a$ is

    1. A.$-8$
    2. B.4
    3. C.8
    4. D.12
    Show answer

    Answer: C

11. Q532011 · 17 Apr 2011

    What is the sine of angle between the vectors $\hat{i}+2\hat{j}+3\hat{k}$ and $-\hat{i}+2\hat{j}+3\hat{k}$?

    1. A.$\sqrt{\frac{13}{7}}$
    2. B.$\frac{\sqrt{13}}{7}$
    3. C.$\frac{13}{\sqrt{7}}$
    4. D.None of the above
    Show answer

    Answer: B

12. Q542011 · 17 Apr 2011

    If $\sin \left({\sin }^{-1}\frac{1}{5}+{\cos }^{-1}x\right)=1$ then what is $x$ equal to?

    1. A.0
    2. B.1
    3. C.$\frac{4}{5}$
    4. D.$\frac{1}{5}$
    Show answer

    Answer: D

13. Q552011 · 17 Apr 2011

    What is $\log \left(a+\sqrt{a^2+1}\right)+\log \left(\frac{1}{a+\sqrt{a^2+1}}\right)$ equal to?

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

    Answer: B

14. Q872011 · 17 Apr 2011

    Which one of the following is the unit vector perpendicular to the vectors $4\hat{i}+2\hat{j}$ and $-3\hat{i}+2\hat{j}$?

    1. A.$\frac{\hat{i}+\hat{j}}{\sqrt{2}}$
    2. B.$\frac{\hat{i}-\hat{j}}{\sqrt{2}}$
    3. C.$\hat{k}$
    4. D.$\frac{\hat{i}+\hat{j}+\hat{k}}{\sqrt{3}}$
    Show answer

    Answer: C

15. Q882011 · 17 Apr 2011

    Consider the following statements in respect of the vectors ${\vec{u}}_1=(1,2,3),{\vec{u}}_2=(2,3,1),{\vec{u}}_3=(1,3,2)$ and ${\vec{u}}_4=(4,6,2)$ 1. ${\vec{u}}_1$ is parallel to ${\vec{u}}_4$ 2. ${\vec{u}}_2$ is parallel to ${\vec{u}}_4$ 3. ${\vec{u}}_2$ is parallel to ${\vec{u}}_3$ Which of the statements given above is/are correct?

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

    Answer: B

16. Q892011 · 17 Apr 2011

    The line $mx+ny=1$ passes through the points $(1,2)$ and $(2,1)$. What is the value of $m$?

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

    Answer: D

17. Q1042011 · 17 Apr 2011

    Consider the following relations among the angles $\alpha ,\beta ,\gamma$ made by a vector with the coordinate axes : 1. $\cos 2\alpha +\cos 2\beta +\cos 2\gamma =-1$ 2. ${\sin }^2\alpha +{\sin }^2\beta +{\sin }^2\gamma =1$ Which of the above 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: A

18. Q1112011 · 17 Apr 2011

    Let $p,q,r,s$ be respectively the magnitudes of the vectors $3\hat{i}-2\hat{j}$, $2\hat{i}+2\hat{j}+\hat{k}$, $4\hat{i}-\hat{j}+\hat{k}$, $2\hat{i}+2\hat{j}+3\hat{k}$. Which one of the following is correct?

    1. A.$r>s>q>p$
    2. B.$s>r>p>q$
    3. C.$r>s>p>q$
    4. D.$s>r>q>p$
    Show answer

    Answer: C

19. Q1122011 · 17 Apr 2011

    In a triangle $ABC$, if $c=2$, $A=120^{\circ }$, $a=\sqrt{6}$, then what is $C$ equal to ?

    1. A.$30^{\circ }$
    2. B.$45^{\circ }$
    3. C.$60^{\circ }$
    4. D.$75^{\circ }$
    Show answer

    Answer: B

20. Q1202011 · 17 Apr 2011

    If $x\hat{i}+y\hat{j}+z\hat{k}$ is a unit vector and $x:y:z=\sqrt{3}:2:3$, then what is the value of $z$ ?

    1. A.$\frac{3}{16}$
    2. B.3
    3. C.$\frac{3}{4}$
    4. D.2
    Show answer

    Answer: C