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Federal (FBISE) · Chapter test · Free

1st Year Mathematics Chapter 3 Test3. Vectors

30 marks · 40 minutes · 21 questions. Covers: 3. Vectors.

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1st Year Mathematics Chapter 3 Test

1st Year · Mathematics · Federal (FBISE)

Name: ____________Roll No: ________Class/Section: ________Date: ________Total marks: 30Time: 40 minutes

Q1. Multiple choice questions (Attempt all · 10 × 1 = 10)

  1. The projection of i‾−2j‾+ak‾ \underline{i} - 2\underline{j} + a\underline{k} along 2i‾−3j‾+23k‾ 2\underline{i} - 3\underline{j} + 2\sqrt{3}\underline{k} is 4 then α \alpha is :
    1. 3\sqrt{3}
    2. 232\sqrt{3}
    3. -3\sqrt{3}
    4. 333\sqrt{3}
  2. The line segment joining the midpoints of the diagonals of a trapezium is parallel to the parallel sides and its length is half the ____________ lengths of the parallel sides.
    1. Sum
    2. Difference
    3. Product
    4. Quotient
  3. If a‾=i‾−2k‾\underline{a} = \underline{i} - 2\underline{k} and b‾=3i‾−2j‾\underline{b} = 3\underline{i} - 2\underline{j}, then a‾×b‾=?\underline{a} \times \underline{b} = ?
    1. 2(i‾−3j‾+k‾)2\left( \underline{i} - 3\underline{j} +\underline{k}\right)
    2. −2(2i‾+3j‾+k‾)-2\left(2 \underline{i}+ 3\underline{j} +\underline{k}\right)
    3. 2(i‾+3j‾−k‾)2\left( \underline{i} + 3\underline{j} -\underline{k}\right)
    4. 2i‾+3j‾+4k‾2\underline{i} + 3\underline{j} +4\underline{k}
  4. If AB→=CD→\overrightarrow{AB}=\overrightarrow{CD}, then coordinates of point A is when B ( 1 , 2 ) , C ( - 2 , 5 ) and D ( 4 , 11 ) are.
    1. A ( 5 , 4 )
    2. A ( - 5 , 4 )
    3. A ( - 5 , - 4 )
    4. A ( 5 , - 4 )
  5. A vector having magnitude one is called :
    1. Null vector
    2. Unit vector
    3. Position vector
    4. None of these
  6. The vector from the point A to the origin where AB‾=4i‾−2j‾\overline{AB} =4\underline{i} - 2\underline{j} and B is the point ( - 2 , 5 ) is
    1. ( 6 , 7 )
    2. ( - 6 , 7 )
    3. ( 6 , - 7 )
    4. ( - 6 , - 7 )
  7. Which of the following is Lagrange , identity ?
    1. ∣a×b∣2=∣a∣2∣b∣2−a⋅b|a \times b|^2 = |a|^2 |b|^2 - a \cdot b
    2. ∣a×b∣2=∣a∣2∣b∣2+(a⋅b)2|a \times b|^2 = |a|^2 |b|^2 + (a \cdot b)^2
    3. ∣a×b∣2=∣a∣2∣b∣2−(a⋅b)2|a \times b|^2 = |a|^2 |b|^2 - (a \cdot b)^2
    4. ∣a×b∣2=∣a∣2∣b∣2+a⋅b|a \times b|^2 = |a|^2 |b|^2 + a \cdot b
  8. Which of the following is a vector quantity:
    1. Distance
    2. Volume
    3. Time
    4. Weight
  9. If a‾=b‾ \underline{a} = \underline{b} then a‾⋅b‾ \underline{a} \cdot \underline{b} is equal to
    1. ∣a‾∣|\underline{a}|
    2. ∣a‾∣2|\underline{a}|^2
    3. 2∣a‾∣2|\underline{a}|
    4. 12∣a‾∣\frac{1}{2}|\underline{a}|
  10. The moment of force 2i‾+3j‾+7k‾2\underline{i} + 3\underline{j} + 7\underline{k} about the point ( 1 , 2 , 3 ) when applied at the point ( - 1 , 2 , 0 )
    1. 9i‾+8j‾−6k‾9\underline{i} + 8\underline{j} - 6\underline{k}
    2. −9i‾−8j‾+6k‾-9\underline{i} - 8\underline{j} + 6\underline{k}
    3. 9i‾+8j‾+6k‾9\underline{i} + 8\underline{j} + 6\underline{k}
    4. 9i‾−8j‾−6k‾9\underline{i} - 8\underline{j} - 6\underline{k}

Q2. Short questions (Attempt any 6 of 8 · 6 × 2 = 12)

  1. Find the moment of the force F⃗=3i^=2j^−5k^\vec{F} = 3\hat{i} = 2\hat{j} - 5\hat{k} about the point ( 2 , 1 , -1 ) when it is applied at point ( 3 , 0 , 2 ).
  2. Find the angle between the vectors i^−2j^+k^\hat{i} -2\hat{j} + \hat{k} and 2i^−3j^+k^2\hat{i} -3\hat{j} + \hat{k} .
  3. If a⃗=2i^−3j^+k^\vec{a} = 2\hat{i} - 3\hat{j} + \hat{k} and b⃗=i^−3j^+5k^\vec{b} = \hat{i} - 3\hat{j} + 5\hat{k} then find: A vector of magnitude 37\frac{3}{7} in the direction 3a⃗+b⃗3\vec{a} + \vec{b}.
  4. If a⃗=2i^+5j^−2k^andb⃗=i^−j^+3k^ \vec{a} = 2\hat{i} + 5\hat{j} - 2\hat{k} \quad \text{and} \quad \vec{b} = \hat{i} - \hat{j} + 3\hat{k} are two adjacent sides of a parallelogram, then find its area.
  5. Three vertices of triangle are A ( 0, -1, -2 ) ; B ( 3, 1, 4 ) and C ( 5, 7, 1 ). Show that ABC is a right-angled triangle and find the other two angles.
  6. For the given vectors a⃗,b⃗\vec{a}, \vec{b} and c⃗\vec{c}; prove that a⃗⋅b⃗×c⃗=b⃗⋅c⃗×a⃗=c⃗⋅a⃗×b⃗.\vec{a} \cdot \vec{b} \times \vec{c} = \vec{b} \cdot \vec{c} \times \vec{a} = \vec{c} \cdot \vec{a} \times \vec{b}. a⃗=3i^−j^+2k^;b⃗=2i^+3j^−k^;c⃗=−i^+2j^−3k^.\vec{a} = 3\hat{i} - \hat{j} + 2\hat{k}; \quad \vec{b} = 2\hat{i} + 3\hat{j} - \hat{k}; \quad \vec{c} = -\hat{i} + 2\hat{j} - 3\hat{k}.
  7. Find the required vector in its component form. Give that P = ( 3 , -1 ) ; Q = ( -4 , -6 ) ; R = ( 1 , 4 ) and S = ( 2 , 5 ).2PR→+2PS→ 2\overrightarrow{PR}+2 \overrightarrow{PS}
  8. If a⃗=2i^−3j^+k^\vec{a} = 2\hat{i} - 3\hat{j} + \hat{k}; b⃗=i^−3j^+4k^\vec{b} = \hat{i} - 3\hat{j} + 4\hat{k} and c⃗=−i^−2j^+5k^\vec{c} = -\hat{i} - 2\hat{j} + 5\hat{k}, then evaluate the following. i^.a⃗+j^.b⃗+k^.c⃗\hat {i}.\vec{a} +\hat {j}.\vec{b}+\hat {k}.\vec{c}

Q3. Long questions (Attempt any 2 of 3 · 2 × 4 = 8)

  1. OPQR is a trapezium made from three equilateral triangles with OP→=r⃗\overrightarrow{OP} = \vec{r}, OS→=s⃗\overrightarrow{OS} = \vec{s} and M is the mid point of QR‾\overline{QR}. (1) Write PS→\overrightarrow{PS} in terms of r⃗\vec{r} and s⃗\vec{s}. (2) Show that OQ→\overrightarrow{OQ} is parallel to SM→\overrightarrow{SM}.
  2. Find the volume of the tetrahedron with given vertices. A ( 0, 1, 0) ; B (2, 0, 1) ; C (3, 1, 2) ; D (5, 6, -1)
  3. The scalar product of i⃗+j⃗−k⃗ \vec{i} + \vec{j} - \vec{k} with the unit vector along the sum of the vectors 2i⃗−3j⃗+k⃗2\vec{i} - 3\vec{j} + \vec{k} and λi⃗−2j⃗+3k⃗\lambda\vec{i} - 2\vec{j} + 3\vec{k} is 1. Find the value of λ\lambda.