Vectors · NDA previous year questions

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2010 15 questions

Q432010 · 22 Aug 2010
If the position vector of a point $P$ with respect to origin $O$ is $\hat{i} + 3 \hat{j} - 2 \hat{k}$ and that of a point $Q$ is $3 \hat{i} + \hat{j} - 2 \hat{k}$ , then what is the position vector of the bisector of the angle $P O Q$ ?
1. A. $\hat{i} - \hat{j} - \hat{k}$
2. B. $\hat{i} + \hat{j} - \hat{k}$
3. C. $\hat{i} + \hat{j} + \hat{k}$
4. D.None of the above
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Answer: B
Q452010 · 22 Aug 2010
Let $a, b$ and $c$ be the distinct non-negative numbers. If the vectors $a \hat{i} + a \hat{j} + c \hat{k}, \hat{i} + \hat{k}, c \hat{i} + c \hat{j} + b \hat{k}$ lie on a plane, then which one of the following is correct?
1. A. $c$ is the arithmetic mean of $a$ and $b$
2. B. $c$ is the geometric mean of $a$ and $b$
3. C. $c$ is the harmonic mean of $a$ and $b$
4. D. $c$ is equal to zero
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Answer: B
Q502010 · 22 Aug 2010
What is the vector equally inclined to the vectors $\hat{i} + 3 \hat{j}$ and $3 \hat{i} + \hat{j}$ ?
1. A. $\hat{i} + \hat{j}$
2. B. $2 \hat{i} - \hat{j}$
3. C. $2 \hat{i} + \hat{j}$
4. D.None of the above
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Answer: A
Q512010 · 22 Aug 2010
$A B C D$ is a quadrilateral. Forces $A B, C B, C D$ and $D A$ act along its sides. What is their resultant?
1. A. $2 C D$
2. B. $2 D A$
3. C. $2 B C$
4. D. $2 C B$
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Answer: D
Q542010 · 22 Aug 2010
What is the value of $b$ such that the scalar product of the vector $\hat{i} + \hat{j} + \hat{k}$ with the unit vector parallel to the sum of the vectors $2 \hat{i} + 4 \hat{j} - 5 \hat{k}$ and $b \hat{i} + 2 \hat{j} + 3 \hat{k}$ is unity?
1. A. $- 2$
2. B. $- 1$
3. C.0
4. D.1
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Answer: D
Q42010 · 18 Apr 2010
Consider the following statements: 1. For any three vectors $a, b, c$ ; $a \cdot {(b + c) \times (a + b + c)} = 0$ 2. For any three coplanar unit vectors $d, e, f$ ; $(d \times e) \cdot f = 1$ 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: A
Q462010 · 18 Apr 2010
What is the value of $λ$ for which the vectors $\hat{i} - \hat{j} + \hat{k}$ , $2 \hat{i} + \hat{j} - \hat{k}$ , $λ \hat{i} - \hat{j} + λ \hat{k}$ are coplanar?
1. A.1
2. B.2
3. C.3
4. D.4
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Answer: A
Q862010 · 18 Apr 2010
A vector $b$ is collinear with the vector $a = (2, 1, - 1)$ and satisfies the condition $a \cdot b = 3$ . What is $b$ equal to?
1. A.(1, 1/2, -1/2)
2. B.(2/3, 1/3, -1/3)
3. C.(1/2, 1/4, -1/4)
4. D.(1, 1, 0)
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Answer: A
Q872010 · 18 Apr 2010
What is the least positive integer n for which $(\frac{1 + i}{1 - i})^{n} = 1$ ?
1. A.16
2. B.12
3. C.8
4. D.4
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Answer: D
Q882010 · 18 Apr 2010
The vectors $a = x \hat{i} + y \hat{j} + z \hat{k}$ , $b = \hat{k}$ , $c$ are such that they form a right-handed system. What is $c$ equal to?
1. A. $\hat{j}$
2. B. $y \hat{j} - x \hat{k}$
3. C. $y \hat{i} - x \hat{j}$
4. D. $x \hat{i} - y \hat{j}$
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Answer: C
Q892010 · 18 Apr 2010
If $x = t^{2}, y = t^{3}$ , then what is $\frac{d ^{2} y}{d x ^{2}}$ equal to?
1. A.1
2. B. $\frac{3}{2 t}$
3. C. $\frac{3}{4 t}$
4. D. $\frac{3}{2}$
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Answer: C
Q902010 · 18 Apr 2010
What is $\int_{- π /4}^{π /4} tan^{3} x d x$ equal to?
1. A. $3$
2. B. $\frac{1}{3}$
3. C. $\frac{1}{2}$
4. D.0
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Answer: D
Q952010 · 18 Apr 2010
If the angle between the vectors $a$ and $b$ is $\frac{π}{3}$ , what is the angle between $- 5 a$ and $6 b$ ?
1. A. $\frac{π}{6}$
2. B. $\frac{2 π}{3}$
3. C. $\frac{2 π}{5}$
4. D. $\frac{3 π}{7}$
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Answer: B
Q1192010 · 18 Apr 2010
Let $a$ and $b$ be two unit vectors and $α$ be the angle between them. If $(a + b)$ is also the unit vector, then what is the value of $α$ ?
1. A. $\frac{π}{4}$
2. B. $\frac{π}{3}$
3. C. $\frac{2 π}{3}$
4. D. $\frac{π}{2}$
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Answer: C
Q1202010 · 18 Apr 2010
What is the value of $\frac{( 0.101 ) _{2}^{(11)_{2}} + ( 0.011 ) _{2}^{(11)_{2}}}{( 0.101 ) _{2}^{(10)_{2}} - ( 0.101 ) _{2}^{(01)_{2}} ( 0.011 ) _{2}^{(01)_{2}} + ( 0.011 ) _{2}^{(10)_{2}}}$ ?
1. A. $(0.001)_{2}$
2. B. $(0.01)_{2}$
3. C. $(0.1)_{2}$
4. D. $(1)_{2}$
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Answer: D