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

If the two lines `(x - 1)/2 = (y + 1)/3 = (z - 1)/4` and `(x - 3)/1 = (y - "m")/2` = z intersect at a point, find the value of m

Show that the lines `(x - 3)/3 = (y - 3)/(-1), z - 1` = 0 and `(x - 6)/2 = (z - 1)/3, y - 2` = 0 intersect. Aslo find the point of intersection

Show that the straight lines x + 1 = 2y = – 12z and x = y + 2 = 6z – 6 are skew and hence find the shortest distance between them

Find the parametric form of vector equation of the straight line passing through (−1, 2, 1) and parallel to the straight line `vec"r" = (2hat"i" + 3hat"j" - hat"k") + "t"(hat"i" - 2hat"j" + hat"k")` and hence find the shortest distance between the lines

Find the foot of the perpendicular drawn: from the point (5, 4, 2) to the line `(x + 1)/2 = (y - 3)/3 = (z - 1)/(-1)`. Also, find the equation of the perpendicular

Choose the correct alternative:

If `[vec"a", vec"b", vec"c"]` = 1, then the value of `(vec"a"*(vec"b" xx vec"c"))/((vec"c" xx vec"a")*vec"b") + (vec"b"*(vec"c" xx vec"a"))/((vec"a" xx vec"b")*vec"c") + (vec"c"*(vec"a" xx vec"b"))/((vec"c" xx vec"b")*vec"a")` is

Choose the correct alternative:

If `vec"a", vec"b", vec"c"` are non-coplanar, non-zero vectors `[vec"a", vec"b", vec"c"]` = 3, then `{[[vec"a" xx vec"b", vec"b" xx vec"c", vec"c" xx vec"a"]]}^2` is equal to

Choose the correct alternative:

I`vec"a" xx  (vec"b" xx vec"c") = (vec"a" xx vec"b") xx vec"c"`, where `vec"a", vec"b", vec"c"` are any three vectors such that `vec"b"*vec"c" ≠ 0` and `vec"a"*vec"b" ≠ 0`, then `vec"a"` and `vec"c"` are

Choose the correct alternative:

The vector equation `vec"r" = (hat"i" - hat"j" - hat"k") + "t"(6hat"i" - hat"k")` represents a straight line passing through the points

Find intervals of concavity and points of inflexion for the following functions:

f(x) = x(x – 4)³ 

Find intervals of concavity and points of inflection for the following functions:

f(x) = sin x + cos x, 0 < x < 2π

Find intervals of concavity and points of inflection for the following functions:

f(x) = `1/2 ("e"^x - "e"^-x)`

Find the local extrema for the following functions using second derivative test:

f(x) = – 3x⁵ + 5x³

Find the local extrema for the following functions using second derivative test:

f(x) = x log x

Find the local extrema for the following functions using second derivative test:

f(x) = x² e^–2x 

For the function f(x) = 4x³ + 3x² – 6x + 1 find the intervals of monotonicity, local extrema, intervals of concavity and points of inflection

Choose the correct alternative:

The curve y = ax⁴ + bx² with ab > 0

Choose the correct alternative:

The point of inflection of the curve y = (x – 1)³ is

Evaluate the following:

`int_0^oo x^5 "e"^(-3x)  "d"x`

Evaluate the following:

`int_0^(pi/2) ("e"^(-tanx))/(cos^6x)  "d"x`