HSC Science Class 12 - Tamil Nadu Board of Secondary Education Question Bank Solutions for Mathematics

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

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If the line `(x  - )/3 = (y - 1)/(-5) = (x + 2)/2` lies in the plane x + 3y – αz + ß = 0 then (α + ß) is

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The angle between the line `vec"r" = (hat"i" + 2hat"j" - 3hat"k") + "t"(2hat"i" + hat"j" - 2hat"k")` and the plane `vec"r"(hat"i" + hat"j") + 4` = 0 is

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Distance from the origin to the plane 3x – 6y + 2z + 7 = 0 is

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The distance between the planes x + 2y + 3z + 7 = 0 and 2x + 4y + 6z + 7 = 0 is

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If the distance of the point (1, 1, 1) from the origin is half of its distance from the plane x + y + z + k = 0, then the values of k are

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If the planes `vec"r"(2hat"i" - lambdahat"j" + hatk")` =  and `vec"r"(4hat"i" + hat"j" - muhat"k")` = 5 are parallel, then the value of λ and µ are

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If the length of the perpendicular from the origin to the plane 2x + 3y + λz = 1, λ > 0 is `1/5, then the value of λ is

Find two positive numbers whose sum is 12 and their product is maximum

Find two positive numbers whose product is 20 and their sum is minimum