HSC Arts (English Medium) १२ वीं कक्षा - Maharashtra State Board Question Bank Solutions for Mathematics and Statistics

Choose correct alternatives:

If acute angle between lines ax² + 2hxy + by² = 0 is, `pi/4`, then 4h² = ______.

Find the joint equation of the pair of lines which bisect angles between the lines given by x² + 3xy + 2y² = 0 

Show that the lines x² − 4xy + y² = 0 and x + y = 10 contain the sides of an equilateral triangle. Find the area of the triangle. 

If the slope of one of the lines given by ax² + 2hxy + by² = 0 is three times the other, prove that 3h² = 4ab.

Show that the line 3x + 4y + 5 = 0 and the lines (3x + 4y)² - 3(4x - 3y)² = 0 form the sides of an equilateral triangle.

Show that the lines x² - 4xy + y² = 0 and the line x + y = `sqrt6` form an equilateral triangle. Find its area and perimeter.

If the slope of one of the lines given by ax² + 2hxy + by² = 0 is square of the slope of the other line, show that a²b + ab² + 8h³ = 6abh.

Prove that the product of length of perpendiculars drawn from P(x₁, y₁) to the lines represented by ax² + 2hxy + by² = 0 is `|("ax"_1^2 + "2hx"_1"y"_1 + "by"_1^2)/(sqrt("a - b")^2 + "4h"^2)|`

Show that the difference between the slopes of the lines given by (tan²θ + cos²θ)x² − 2xy tan θ + (sin²θ)y² = 0 is two.

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")`