Q. No.The figure shows the orientation of two vectors \(u\) and \(v\) in the XY plane.
If \(u=a\hat{i}+b\hat{j}\) and \(v=p\hat{i}+q\hat{j}\).
Which of the following is correct?
| 1. | \(a\) and \(p\) are positive while \(b\) and \(q\) are negative. |
| 2. | \(a,\) \(p\) and \(b\) are positive while \(q\) is negative. |
| 3. | \(a,\) \(q\) and \(b\) are positive while \(p\) is negative. |
| 4. | \(a,\) \(b,\) \(p\) and \(q\) are all positive. |
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The component of a vector \(\vec{r}\) along the X-axis will have maximum value if:
| 1. | \(\vec{r}\) is along the positive Y-axis. |
| 2. | \(\vec{r}\) is along the positive X-axis. |
| 3. | \(\vec{r}\) makes an angle of \(45^\circ\) with the X-axis. |
| 4. | \(\vec{r}\) is along the negative Y-axis. |
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Consider the quantities of pressure, power, energy, impulse, gravitational potential, electric charge, temperature, and area. Out of these, the only vector quantities are:
| 1. | impulse, pressure, and area |
| 2. | impulse and area |
| 3. | area and gravitational potential |
| 4. | impulse and pressure |
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| 1. | vector \((A\times B)\times C\) is not zero unless vectors \(B\) and \(C\) are parallel. |
| 2. | vector \((A\times B).C\) is not zero unless vectors \(B\) and \(C\) are parallel. |
| 3. | if vectors \(A,B\) and \(C\) define a plane, \((A\times B)\times C\) is in that plane. |
| 4. | \((A\times B). C= |A||B||C|\rightarrow C^2= A^2+B^2\) |
The incorrect statement/s is/are:
1. (b), (d)
2. (a), (c)
3. (b), (c), (d)
4. (a), (b)
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It is found that \(|\vec{A}+\vec{B}|=|\vec{A}|\). This necessarily implies:
| 1. | \(\vec{B}=0\) |
| 2. | \(\vec{A},\) \(\vec{B}\) are antiparallel |
| 3. | \(\vec{A}\) and \(\vec{B}\) are perpendicular |
| 4. | \(\vec{A}.\vec{B}\leq0\) |
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Given below in Column-I are the relations between vectors \(a,\) \(b,\) and \(c\) and in Column-II are the orientations of \(a,\) \(b,\) and \(c\) in the \(xy\)-plane. Match the relation in Column-I to the correct orientations in Column-II.
| Column-I | Column-II | ||
| (a) | \(a + b = c\) | (i) |  |
| (b) | \(a- c = b\) | (ii) |  |
| (c) | \(b - a = c\) | (iii) |  |
| (d) | \(a + b + c = 0\) | (iv) |  |
Choose the correct option from the given table.
| 1. | a-(ii), b-(iv), c-(iii), d-(i) |
| 2. | a-(i), b-(iii), c-(iv), d-(ii) |
| 3. | a-(iv), b-(iii), c-(i), d-(ii) |
| 4. | a-(iii), b-(iv), c-(i), d-(ii) |
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If \(|\vec{A}|=2\) and \(|\vec{B}|=4\), then match the relations in Column I with the angle \(\theta\) between \(\vec{A}\) and \(\vec{B}\) in Column II.
| Column I | Column II | ||
| (a) | \(\vec{A}.\vec{B}=0\) | (i) | \(\theta=0^{\circ}\) |
| (b) | \(\vec{A}.\vec{B}=8\) | (ii) | \(\theta=90^{\circ}\) |
| (c) | \(\vec{A}.\vec{B}=4\) | (iii) | \(\theta=180^{\circ}\) |
| (d) | \(\vec{A}.\vec{B}=-8\) | (iv) | \(\theta=60^{\circ}\) |
Choose the correct answer from the options given below:
| 1. | (a)–(iii), (b)-(ii), (c)-(i), (d)-(iv) |
| 2. | (a)–(ii), (b)-(i), (c)-(iv), (d)-(iii) |
| 3. | (a)–(ii), (b)-(iv), (c)-(iii), (d)-(i) |
| 4. | (a)–(iii), (b)-(i), (c)-(ii), (d)-(iv) |
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If \(\left| \vec{A}\right|\) = \(2\) and \(\left| \vec{B}\right|\) = \(4,\) then match the relations in column-I with the angle \(\theta\) between \(\vec{A}\) and \(\vec{B}\) in column-II.
| Column-I | Column-II |
| (A) \(\left| \vec{A}\times \vec{B}\right|\) \(=0\) | (p) \(\theta=30^\circ\) |
| (B)\(\left| \vec{A}\times \vec{B}\right|\)\(=8\) | (q) \(\theta=45^\circ\) |
| (C) \(\left| \vec{A}\times \vec{B}\right|\) \(=4\) | (r) \(\theta=90^\circ\) |
| (D) \(\left| \vec{A}\times \vec{B}\right|\) \(=4\sqrt2\) | (s) \(\theta=0^\circ\) |
| 1. | \(\mathrm{A(s), B(r), C(q), D(p)}\) |
| 2. | \(\mathrm{A(s), B(p), C(r), D(q)}\) |
| 3. | \(\mathrm{A(s), B(p), C(q), D(r)}\) |
| 4. | \(\mathrm{A(s), B(r), C(p), D(q)}\) |
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For two vectors \(\vec A\) and \(\vec B\), |\(\vec A\)+\(\vec B\)|=|\(\vec A\) - \(\vec B\)| is always true when:
| (a) | |\(\vec A\)| = |\(\vec B\)| ≠ \(0\) |
| (b) | \(\vec A\perp\vec B\) |
| (c) | |\(\vec A\)| = |\(\vec B\)| ≠ \(0\) and \(\vec A\) and \(\vec B\) are parallel or antiparallel. |
| (d) | when either |\(\vec A\)| or |\(\vec B\)| is zero. |
Choose the correct option from the given ones:
| 1. | (a), (d) |
| 2. | (b), (c) |
| 3. | (b), (d) |
| 4. | (a), (b) |
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1. \(45^{\circ} \)
2. \(90^{\circ} \)
3. \(-45^{\circ} \)
4. \(180^{\circ}\)
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