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.

Three Dimensional Geometry

Long Answer Type

31. The projection of a line on the co-ordinate axes are 6, 2, 3. Find the length of the line and its direction cosines. Also, find the projection of the line segment joining (2, – 3, 1) to (4, 2, 3) on this line.

112 Views

32. A (– 1, 2, – 3), B (5, 0, – 6), C (0, 4, – 1) are three points. Show that the direction cosines of the bisectors of are proportional to 25, 8, 5 and -11, $AB space equals space square root of left parenthesis 5 plus 1 right parenthesis squared plus left parenthesis 0 minus 2 right parenthesis squared plus left parenthesis negative 6 plus 3 right parenthesis squared end root space space space space space space equals space square root of 36 plus 4 plus 9 end root space equals space square root of 49 space equals space 7 AC space equals space square root of left parenthesis 0 plus 1 right parenthesis squared plus left parenthesis 4 minus 2 right parenthesis squared plus left parenthesis negative 1 plus 3 right parenthesis squared end root space space space space space space space space equals space square root of 1 plus 4 plus 4 end root space equals space square root of 9 space equals space 3$

A, B, C are the points (– 1, 2, – 3), (5, 0, – 6), (0, 4, – 1) respectively.

AB space equals space square root of left parenthesis 5 plus 1 right parenthesis squared plus left parenthesis 0 minus 2 right parenthesis squared plus left parenthesis negative 6 plus 3 right parenthesis squared end root space space space space space space space equals square root of 36 plus 4 plus 9 end root space equals space square root of 49 space equals space 7 AC space equals space square root of left parenthesis 0 plus 1 right parenthesis squared plus left parenthesis 4 minus 2 right parenthesis squared plus left parenthesis negative 1 plus 3 right parenthesis squared end root space space space space space space space equals square root of 1 plus 4 plus 4 end root space equals space square root of 9 space equals space 3

Syntax error from line 1 column 252 to line 1 column 355. Unexpected '<mlongdiv '.

because space space space BD over CD equals AB over AC equals 7 over 3

∴ D divides BC internally in the ratio 7 : 3 and E divides BC externally in the ratio 7 : 3

therefore space space space straight D space is space open parentheses fraction numerator left parenthesis 7 right parenthesis thin space left parenthesis 0 right parenthesis space plus space left parenthesis 3 right parenthesis thin space left parenthesis 5 right parenthesis over denominator 7 plus 3 end fraction comma space fraction numerator left parenthesis 7 right parenthesis thin space left parenthesis 4 right parenthesis space plus space left parenthesis 3 right parenthesis thin space left parenthesis 0 right parenthesis over denominator 7 plus 3 end fraction comma space fraction numerator left parenthesis 7 right parenthesis thin space left parenthesis negative 1 right parenthesis plus left parenthesis negative 3 right parenthesis thin space left parenthesis negative 6 right parenthesis over denominator 7 plus 3 end fraction close parentheses

i.e.,

open parentheses 15 over 10 comma space 28 over 10 comma space minus 25 over 10 close parentheses space space space straight i. straight e. comma space space space open parentheses 3 over 2 comma space 14 over 5 comma space minus 5 over 2 close parentheses

E is

open parentheses fraction numerator left parenthesis 7 right parenthesis thin space left parenthesis 0 right parenthesis space plus space left parenthesis negative 3 right parenthesis thin space left parenthesis 5 right parenthesis over denominator 7 minus 3 end fraction comma space fraction numerator left parenthesis 7 right parenthesis thin space left parenthesis 4 right parenthesis space plus space left parenthesis negative 3 right parenthesis thin space left parenthesis 0 right parenthesis over denominator 7 minus 3 end fraction comma space fraction numerator left parenthesis 7 right parenthesis thin space left parenthesis negative 1 right parenthesis space plus space left parenthesis negative 3 right parenthesis thin space left parenthesis negative 6 right parenthesis over denominator 7 minus 3 end fraction close parentheses

i.e,

open parentheses negative 15 over 4 comma space 28 over 4 comma space 11 over 4 close parentheses space space space space straight i. straight e. comma space space space space open parentheses fraction numerator negative 15 over denominator 4 end fraction comma space 7 comma space 11 over 4 close parentheses

Directionratios of AD are

3 over 2 plus 1 comma space space 14 over 5 minus 2 comma space space fraction numerator negative 5 over denominator 2 end fraction plus 3

i.e.,

5 over 2 comma space space 4 over 5 comma space 1 half space space space or space space space 25 comma space 8 comma space 5

Direction ratios of AE are

negative 15 over 4 plus 1 comma space space 7 minus 2 comma space space 11 over 4 plus 3

i.e.,

negative 11 over 4 comma space 5 comma space 23 over 4 space space or space space space minus 11 comma space 20 comma space 33

88 Views

Short Answer Type

33. The cartesian equations of a line are

fraction numerator straight x minus 5 over denominator 3 end fraction space equals space fraction numerator straight y plus 4 over denominator 7 end fraction space equals fraction numerator straight z minus 6 over denominator 2 end fraction.

Find a vector equation for the line.

90 Views

34.

The Cartesian equation of a line is $fraction numerator straight x plus 3 over denominator 2 end fraction space equals space fraction numerator straight y minus 5 over denominator 4 end fraction space equals fraction numerator straight z plus 6 over denominator 2 end fraction.$ Find the vector equation for the line.

80 Views

35.

Find the cartesian equation of the line which passes through the point (-2, 4, -5) and is parallel to the line given by $fraction numerator straight x plus 3 over denominator 3 end fraction space equals space fraction numerator straight y minus 4 over denominator 5 end fraction space equals fraction numerator straight z plus 8 over denominator 6 end fraction$

109 Views

36.

Find the equations of a line which is parallel to the line $fraction numerator straight x minus 2 over denominator 3 end fraction space equals fraction numerator straight y plus 1 over denominator 1 end fraction space equals space fraction numerator straight z minus 7 over denominator 9 end fraction$ and which passes through the point (3, 0, 5).

127 Views

Long Answer Type

37.

For the cartesian and vector equation of a line which passes through the point (1, 2, 3) and is parallel to the line $fraction numerator negative straight x minus 2 over denominator 1 end fraction space equals space fraction numerator straight y plus 3 over denominator 7 end fraction space equals space fraction numerator 2 straight z minus 6 over denominator 3 end fraction.$

99 Views

Short Answer Type

38.

Find the equation of line (vector and cartesian both) which is parallel to the vector $2 straight i with hat on top space minus straight j with hat on top plus 3 space straight k with hat on top$ and which passes through the point (5, -2, 4).

76 Views

39. Find the vector equation of the line passing through the point (2, –1, –1) which is parallel to the lines 6 x – 2 = 3 y + 1 = 2 z – 2.

95 Views

40.

The equation of a line is given by $fraction numerator 4 minus straight x over denominator 2 end fraction space equals space fraction numerator straight y plus 3 over denominator 3 end fraction space equals fraction numerator straight z plus 2 over denominator 6 end fraction$ . Write the direction cosines of a line parallel to the above line.

105 Views