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

Long Answer Type

41. ABCD is a quadrilateral and O is any point in its plane. Show that it

OA with rightwards arrow on top space plus space OB with rightwards arrow on top space plus space OC with rightwards arrow on top space plus space OD with rightwards arrow on top space equals space 0 with rightwards arrow on top

then O is the point of intersection of lines joining the middle points of the opposite sides of ABCD.

86 Views

42. For any two vectors $straight a with rightwards arrow on top space and space straight b with rightwards arrow on top$ , prove that
(i) $open vertical bar straight a with rightwards arrow on top space plus space straight b with rightwards arrow on top close vertical bar space less or equal than space open vertical bar straight a with rightwards arrow on top close vertical bar space plus space open vertical bar straight b with rightwards arrow on top close vertical bar$ (ii) $open vertical bar straight a with rightwards arrow on top space minus space straight b with rightwards arrow on top close vertical bar space less or equal than space open vertical bar straight a with rightwards arrow on top close vertical bar space plus space open vertical bar straight b with rightwards arrow on top close vertical bar$ (iii) $open vertical bar straight a with rightwards arrow on top space minus space straight b with rightwards arrow on top close vertical bar space space space greater-than or slanted equal to space space open vertical bar straight a with rightwards arrow on top close vertical bar space minus space open vertical bar straight b with rightwards arrow on top close vertical bar$

Case I,

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

are non-collinear vectors.
Let

OA with rightwards arrow on top space equals space straight a with rightwards arrow on top comma space space AB with rightwards arrow on top space equals straight b with rightwards arrow on top

therefore space space space space space space space OB with rightwards arrow on top space equals OA with rightwards arrow on top space plus space AB with rightwards arrow on top space equals space straight a with rightwards arrow on top space plus straight b with rightwards arrow on top Now space space space OA space equals space open vertical bar straight a with rightwards arrow on top close vertical bar comma space space AB space equals space open vertical bar straight b with rightwards arrow on top close vertical bar comma space space space space space space space space space space space space OB space equals space open vertical bar straight a with rightwards arrow on top space plus space straight b with rightwards arrow on top close vertical bar

We know that in a triangle sum of two sides of triangle is always > third side

therefore space space space space space space space OA plus AB greater than OB rightwards double arrow space space space space space space space open vertical bar straight a with rightwards arrow on top close vertical bar space plus space open vertical bar straight b with rightwards arrow on top close vertical bar space greater than space space space open vertical bar straight a with rightwards arrow on top space plus space straight b with rightwards arrow on top close vertical bar therefore space space space space space space space space open vertical bar straight a with rightwards arrow on top space plus space straight b with rightwards arrow on top close vertical bar space less than space space open vertical bar straight a with rightwards arrow on top close vertical bar space plus space open vertical bar straight b with rightwards arrow on top close vertical bar

Case II. $straight a with rightwards arrow on top space and space straight b with rightwards arrow on top$ are collinear vectors.
Let

OA with rightwards arrow on top space equals space straight a with rightwards arrow on top comma space space space AB with rightwards arrow on top space equals space straight b with rightwards arrow on top space then space OB with rightwards arrow on top space equals space OA with rightwards arrow on top space plus space AB with rightwards arrow on top space equals space straight a with rightwards arrow on top space plus space straight b with rightwards arrow on top

Also,

OA space equals space open vertical bar straight a with rightwards arrow on top close vertical bar comma space space space AB space equals open vertical bar straight b with rightwards arrow on top close vertical bar comma space space space OB space equals space open vertical bar straight a with rightwards arrow on top space plus space straight b with rightwards arrow on top close vertical bar

Now,

OB space equals space OA plus AB

therefore space space space space space space open vertical bar straight a with rightwards arrow on top space plus space straight b with rightwards arrow on top close vertical bar space equals space open vertical bar straight a with rightwards arrow on top close vertical bar space plus space open vertical bar straight b with rightwards arrow on top close vertical bar

Combining the results of Case I and II, we get,

open vertical bar straight a with rightwards arrow on top space plus space straight b with rightwards arrow on top close vertical bar space less or equal than space space open vertical bar straight a with rightwards arrow on top close vertical bar space plus space open vertical bar straight b with rightwards arrow on top close vertical bar

Combining the results of Case I and II, we get,

open vertical bar straight a with rightwards arrow on top space plus space straight b with rightwards arrow on top close vertical bar space less or equal than space open vertical bar straight a with rightwards arrow on top close vertical bar space plus space open vertical bar straight b with rightwards arrow on top close vertical bar

(ii)

open vertical bar straight a with rightwards arrow on top minus straight b with rightwards arrow on top close vertical bar space equals space open vertical bar straight a with rightwards arrow on top plus left parenthesis negative straight b with rightwards arrow on top right parenthesis close vertical bar space less or equal than space open vertical bar straight a with rightwards arrow on top close vertical bar space plus space open vertical bar negative straight b with rightwards arrow on top close vertical bar space equals space open vertical bar straight a with rightwards arrow on top close vertical bar space plus space open vertical bar straight b with rightwards arrow on top close vertical bar

therefore

open vertical bar straight a with rightwards arrow on top minus straight b with rightwards arrow on top close vertical bar space less or equal than open vertical bar straight a with rightwards arrow on top close vertical bar space plus space open vertical bar straight b with rightwards arrow on top close vertical bar

(iii)

open vertical bar straight a with rightwards arrow on top close vertical bar space equals space open vertical bar left parenthesis straight a with rightwards arrow on top space minus space straight b with rightwards arrow on top right parenthesis space plus space straight b with rightwards arrow on top close vertical bar space less or equal than space open vertical bar stack straight a space with rightwards arrow on top space minus space straight b with rightwards arrow on top close vertical bar space plus space open vertical bar straight b with rightwards arrow on top close vertical bar

therefore space space space space space space open vertical bar straight a with rightwards arrow on top close vertical bar space minus space open vertical bar straight b with rightwards arrow on top close vertical bar space less or equal than space space open vertical bar straight a with rightwards arrow on top space minus space straight b with rightwards arrow on top close vertical bar rightwards double arrow space space space space space space space open vertical bar straight a with rightwards arrow on top space minus space straight b with rightwards arrow on top close vertical bar space greater or equal than space space open vertical bar straight a with rightwards arrow on top close vertical bar space minus space open vertical bar straight b with rightwards arrow on top close vertical bar.

76 Views

43.

If $straight c with rightwards arrow on top space equals space 3 straight a with rightwards arrow on top space plus space 4 straight b with rightwards arrow on top$ and $2 straight c with rightwards arrow on top space equals space straight a with rightwards arrow on top space minus space 3 straight b with rightwards arrow on top$ , show that
(i) $straight c with rightwards arrow on top space and space straight a with rightwards arrow on top$ have the same direction and $open vertical bar straight c with rightwards arrow on top close vertical bar space greater than space open vertical bar straight a with rightwards arrow on top close vertical bar$
(ii) $straight c with rightwards arrow on top space and space straight b with rightwards arrow on top$ have opposite direction and

77 Views

Short Answer Type

44. Show that the line joining the middle points of the consecutive sides of a quadrilateral is a parallelogram.

80 Views

Long Answer Type

45. In the figure, M is the mid-point of AB and N is the mid-point of CD and O is the mid-point of MN. Prove that
(i)

OA with rightwards arrow on top space plus space OB with rightwards arrow on top space plus space OC with rightwards arrow on top space plus space OD with rightwards arrow on top space equals space straight O with rightwards arrow on top

(ii)

BC with rightwards arrow on top space plus space AD with rightwards arrow on top space equals space 2 space MN with rightwards arrow on top

87 Views

46. Prove by vector method that the line segment joining the mid-points of the diagonals of trapezium is parallel to the parallel sides and equal to help of there difference.

133 Views