HSC Arts (English Medium) 12th Standard Board Exam - Maharashtra State Board Question Bank Solutions

Prove the following: 

`2tan^-1(1/3) = tan^-1(3/4)`

Prove the following:

`tan^-1["cosθ + sinθ"/"cosθ - sinθ"] = pi/(4) + θ, if θ ∈ (- pi/4, pi/4)`

Prove the following:

`tan^-1[sqrt((1 - cosθ)/(1 + cosθ))] = θ/(2)`, if θ ∈ (– π, π).

A table of values of f, g, f' and g' is given :

If r(x) =f [g(x)] find r' (2).

A table of values of f, g, f' and g' is given :

If R(x) =g[3 + f(x)] find R'(4).

A table of values of f, g, f' and g' is given:

If s(x) = f[9 − f (x)] find s'(4).

A table of values of f, g, f' and g' is given :

If S(x) =g [g(x)] find S'(6).

Assume that `f'(3) = -1,"g"'(2) = 5, "g"(2) = 3 and y = f["g"(x)], "then" ["dy"/"dx"]_(x = 2) = ?`

If h(x) = `sqrt(4f(x) + 3"g"(x)), f(1) = 4, "g"(1) = 3, f'(1) = 3, "g"'(1) = 4, "find h"'(1)`.

Find the x co-ordinates of all the points on the curve y = sin 2x − 2 sin x, 0 ≤ x < 2π, where `"dy"/"dx"` = 0.

Select the appropriate hint from the hint basket and fill up the blank spaces in the following paragraph. [Activity]:

"Let f(x) = x² + 5 and g (x) = e^x + 3 then
f[g(x)] = .......... and g[f(x)] =...........
Now f'(x) = .......... and g'(x) = ..........
The derivative of f[g(x)] w. r. t. x in terms of f and g is ..........

Therefore `"d"/"dx"[f["g"(x)]]` = .......... and

`["d"/"dx"[f["g"(x)]]]_(x  =  0)` = ..........
The derivative of g[f(x)] w. r. t. x in terms of f and g is

Therefore `"d"/"dx"["g"[f(x)]]` = .......... and

`["d"/"dx"["g"[f(x)]]]_(x  = -1)` = .........."

Hint basket : `{f'["g"(x)]·"g"'(x), 2e^(2x) + 6e^x, 8, "g"' [ f (x)]· f'(x),2xe^(x^2+5),  − 2e^6,e^(2x) + 6e^x + 14, e^(x^2+5) + 3, 2x, e^x}`

Find the shortest distance between the lines `barr = (4hati - hatj) + λ(hati + 2hatj - 3hatk)` and `barr = (hati - hatj + 2hatk) + μ(hati + 4hatj - 5hatk)`

Find the shortest distance between the lines `(x + 1)/(7) = (y + 1)/(-6) = (z + 1)/(1) and (x - 3)/(1) = (y - 5)/(-2) = (z - 7)/(1)`

By computing the shortest distance, determine whether following lines intersect each other.

`bar"r" = (hat"i" - hat"j") + lambda(2hat"i" + hat"k") and bar"r" = (2hat"i" - hat"j") + mu(hat"i" + hat"j" - hat"k")`

By computing the shortest distance, determine whether following lines intersect each other.

`(x - 5)/(4) = (y -7)/(-5) = (z + 3)/(-5) and (x - 8)/(7) = (y - 7)/(1) = (z - 5)/(3)`

By computing the shortest distance determine whether following lines intersect each other : `bar"r" = (hat"i" + hat"j" - hat"k") + lambda(2hat"i"  - hat"j" + hat"k") and bar"r" (2hat"i" + 2hat"j" - 3hat"k") + mu(hat"i" + hat"j" - 2hat"k")`

By computing the shortest distance determine whether the following lines intersect each other: `(x -5)/(4) = (y - 7)/(5) = (z + 3)/(5)` and x – 6 = y – 8 = z + 2.

Find the direction cosines of the lines `bar"r" = (-2hat"i" + 5/2hat"j" - hat"k") + lambda(2hat"i" + 3hat"j")`.

Choose correct alternatives :

The shortest distance between the lines `vecr = (hati + 2hatj + hatk) + lambda(hati - hatj + hatk) and vecr = (2hati - hatj - hatk) + μ(2hati + hatj + 2hatk)` is ______.

The direction ratios of the line 3x + 1 = 6y – 2 = 1 – z are ______.