Previous Year Papers

Download Solved Question Papers Free for Offline Practice and view Solutions Online.

Test Series

Take Zigya Full and Sectional Test Series. Time it out for real assessment and get your results instantly.

Test Yourself

Practice and master your preparation for a specific topic or chapter. Check you scores at the end of the test.

Vector Algebra

Short Answer Type

71. For given vectors,

straight a with rightwards arrow on top space equals space 2 space straight i with hat on top space minus space straight j with hat on top space plus space 2 space straight k with hat on top space and space straight b with rightwards arrow on top space equals space minus straight i with hat on top space plus space straight j with hat on top space minus space straight k with hat on top comma space

find the unit vector in the direction of the vector

straight a with rightwards arrow on top space plus space straight b with rightwards arrow on top

77 Views

72.

If $straight a with rightwards arrow on top space equals straight i with hat on top space plus space straight j with hat on top space plus space straight k with hat on top$ , $straight b with rightwards arrow on top space equals space 2 space straight i with hat on top space minus space straight j with hat on top space plus space 3 space straight k with hat on top$ and $straight c with rightwards arrow on top space equals space straight i with hat on top space minus space 2 space straight j with hat on top space plus space straight k with hat on top$ , find a vector parallel to the vector $2 space straight a with rightwards arrow on top space minus space straight b with rightwards arrow on top space plus space 3 space straight c with rightwards arrow on top$ .

73 Views

73. Find a vector in the direction of vector

5 space straight i with hat on top space minus space straight j with hat on top space plus space 2 space straight k with hat on top

which has magnitude 8 units.

78 Views

74. Find a vector of magnitude 5 units, and parallel to the resultant of the vectors

straight a with rightwards arrow on top space equals space 2 straight i with hat on top space plus space 3 straight j with hat on top space minus space straight k with hat on top space and space straight b with rightwards arrow on top space equals space straight i with hat on top space minus space 2 space straight j with hat on top space plus space straight k with hat on top.

81 Views

75. Show that the vectors

2 space straight i with hat on top minus 3 space straight j with hat on top space plus 4 space straight k with hat on top

and

negative 4 space straight i with hat on top space plus space 6 space straight j with hat on top space minus space 8 space straight k with hat on top

are collinear.

91 Views

76. Find the condition that the vectors

straight a with rightwards arrow on top space equals space straight k space straight i with hat on top space plus space straight l space straight j with hat on top space and space straight b with rightwards arrow on top space equals space straight l space straight i with hat on top space plus space straight k space straight j with hat on top comma space space left parenthesis straight k comma space straight l space not equal to 0 right parenthesis space are space parallel

74 Views

Long Answer Type

77. Prove that if $straight u with rightwards arrow on top space equals space straight u subscript 1 space straight i with hat on top space plus space straight u subscript 2 space straight j with hat on top space and space straight v with rightwards arrow on top space equals space straight v subscript 1 space straight i with hat on top space plus space straight v subscript 2 space straight j with hat on top$ are non-zero vectors, then they are parallel if and only if u₂v₂ – u₂v₁ = 0.

Let $straight u with rightwards arrow on top space equals space straight u subscript 1 straight i with hat on top space plus space straight u subscript 2 straight j with hat on top comma space space straight v with rightwards arrow on top space equals space straight v subscript 1 straight i with hat on top space plus space straight v subscript 2 straight j with hat on top$ be parallel vectors.
$therefore space space space there space exists space straight a space non minus zero space scalar space straight k space such space that space straight u with rightwards arrow on top space equals space straight k space straight v with rightwards arrow on top therefore space space space space straight u subscript 1 straight i with hat on top space plus space straight u subscript 2 straight j with hat on top space equals space straight k left parenthesis straight v straight i with hat on top space plus space straight v subscript 2 straight j with hat on top right parenthesis or space space left parenthesis straight u subscript 1 space space minus space kv subscript 1 right parenthesis space equals space 0 comma space space space straight u subscript 2 space minus kv subscript 2 space equals space 0 therefore space space space space straight k space equals space straight u subscript 1 over straight v subscript 1 comma space space space straight k space equals space straight u subscript 2 over straight v subscript 2 Equating space values space of space straight k comma space we space get comma space space space space space space space space space space space space space space space space space space space space straight u subscript 1 over straight v subscript 1 space equals space straight u subscript 2 over straight v subscript 2 therefore space space space space space space space space straight u subscript 1 straight v subscript 2 space equals space straight u subscript 2 straight v subscript 1 space space space space space space space space space rightwards double arrow space space space space space straight u subscript 1 straight v subscript 2 space minus space straight u subscript 2 space straight v subscript 1 space equals space 0$
Similarly, if $straight u subscript 1 space straight v subscript 2 space equals space straight u subscript 2 space straight v subscript 1 space space space space rightwards double arrow space space space space space straight u subscript 1 space straight v subscript 2 space minus space straight u subscript 2 space straight v subscript 1 space equals space 0$
i.e., $straight u with rightwards arrow on top comma space space straight v with rightwards arrow on top$ are parallel
Hence the result.