HSC Science कक्षा १२ - Tamil Nadu Board of Secondary Education Question Bank Solutions

If a plane meets the co-ordinate axes at A, B, C such that the centroid of the triangle ABC is the point (u, v, w), find the equation of the plane

Find the non-parametric form of vector equation and Cartesian equation of the plane passing through the point (2, 3, 6) and parallel to thestraight lines `(x - 1)/2 = (y + 1)/3 = (x - 3)/1` and `(x + 3)/2 = (y - 3)/(-5) = (z + 1)/(-3)`

Find the parametric form of vector equation, and Cartesian equations of the plane passing through the points (2, 2, 1), (9, 3, 6) and perpendicular to the plane 2x + 6y + 6z = 9

Find the parametric form of vector equation and Cartesian equations of the plane passing through the points (2, 2, 1), (1, – 2, 3) and parallel to the straight line passing through the points (2, 1, – 3) and (– 1, 5, – 8)

Find the non-parametric form of vector equation and cartesian equation of the plane passing through the point (1, − 2, 4) and perpendicular to the plane x + 2y − 3z = 11 and parallel to the line `(x + 7)/3 = (y + 3)/(-1) = z/1`

Find the parametric form of vector equation, and Cartesian equations of the plane containing the line `vec"r" = (hat"i" - hat"j" + 3hat"k") + "t"(2hat"i" - hat"j" + 4hat"k")` and perpendicular to plane `vec"r"*(hat"i" + 2hat"j" + hat"k")` = 8

Find the parametric vector, non-parametric vector and Cartesian form of the equation of the plane passing through the point (3, 6, – 2), (– 1, – 2, 6) and (6, 4, – 2)

Find the non-parametric form of vector equation and Cartesian equations of the plane `vec"r" = (6hat"i" - hat"j" + hat"k") + "s"(-hat"i" + 2hat"j" + hat"k") + "t"(-5hat"i" - 4hat"j" - 5hat"k")`

Show that the straight lines `vec"r" = (5hat"i" + 7hat"j" - 3hat"k") + "s"(4hat"i" + 4hat"j" - 5hat"k")` and `vec"r"(8hat"i" + 4hat"j" + 5hat"k") + "t"(7hat"i" + hat"j" + 3hat"k")` are coplanar. Find the vector equation of the plane in which they lie

Show that the lines `(x - 2)/1 = (y - 3)/1 = (z - 4)/3` and `(x - 1)/(-3) = (y - 4)/2 = (z - 5)/1` are coplanar. Also, find the plane containing these lines

If the straight lines `(x - 1)/1 - (y - 2)/2 = (z - 3)/"m"^2` and `(x - 3)/5 = (y - 2)/"m"^2 = (z - 1)/2` are coplanar, find the distinct real values of m

If the straight lines `(x - 1)/2 = (y + 1)/lambda = z/2` and `(x + 1)/5 = (y + 1)/2 = z/lambda` are coplanar, find λ and equations of the planes containing these two lines

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If `vec"a", vec"b", vec"c"` are three unit vectors such that `vec"a"` is perpendicular to `vec"b"`, and is parallel to `vec"c"` then `vec"a" xx (vec"b" xx vec"c")` is equal to

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The volume of the parallelepiped with its edges represented by the vectors `hat"i" + hat"j", hat"i" + 2hat"j", hat"i" + hat"j" + pihat"k"` is

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If `vec"a"` and `vec"b"` are unit vectors such that `[vec"a", vec"b", vec"a" xx vec"b"] = 1/4`, are unit vectors such that `vec"a"` nad `vec"b"` is

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If `vec"a", vec"b", vec"c"` are three non-coplanar vectors such that `vec"a" xx (vec"b" xx vec"c") = (vec"b" + vec"c")/sqrt(2)` then the angle between `vec"a"` and `vec"b"` is

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If the volume of the parallelepiped with `vec"a" xx vec"b", vec"b" xx vec"c", vec"c" xx vec"a"` as coterminous edges is 8 cubic units, then the volume of the parallelepiped with `(vec"a" xx vec"b") xx (vec"b" xx vec"c"), (vec"b" xx vec"c") xx (vec"c" xx vec"a")` and `(vec"c" xx vec"a") xx (vec"a" xx vec"b")` as coterminous edges is

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Consider the vectors  `vec"a", vec"b", vec"c", vec"d"` such that `(vec"a" xx vec"b") xx (vec"c" xx vec"d") = vec0`. Let P₁ and P₂ be the planes determined by the pairs of vectors `vec"a", vec"b"` and `vec'c", vec"d"` respectively. Then the angle between P₁ and P₂ is

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If `vec"a" = 2hat"i" + 3hat"j" - hat"k", vec"b" = hat"i" + 2hat"j" - 5hat"k", vec"c" = 3hat"i" + 5hat"j" - hat"k"`, then a vector perpendicular to `vec"a"` and lies in the plane containing `vec"b"` and `vec"c"` is 

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The angle between the lines `(x - 2)/3 = (y + 1)/(-2)`, z = 2 ad `(x - 1)/1 = (2y + 3)/3 = (z + 5)/2` is