HSC Science (General) 12th Standard Board Exam - Maharashtra State Board Question Bank Solutions for Mathematics and Statistics

Use quantifiers to convert the given open sentence defined on N into a true statement

3x – 4 < 9

Use quantifiers to convert the given open sentence defined on N into a true statement

Y + 4 > 6

If x² + y² = 1, then `(d^2x)/(dy^2)` = ______.

If x² + y² = t + `1/"t"` and x⁴ + y⁴ = t² + `1/"t"^2` then `("d"y)/("d"x)` = ______

If x = a t⁴ y = 2a t² then `("d"y)/("d"x)` = ______

If y = `sqrt(tansqrt(x)`, find `("d"y)/("d"x)`.

If x = sin θ, y = tan θ, then find `("d"y)/("d"x)`.

Let f(x) = x³ − 6x² + 9𝑥 + 18, then f(x) is strictly decreasing in ______

Prove that function f(x) = `x - 1/x`, x ∈ R and x ≠ 0 is increasing function

Show that f(x) = x – cos x is increasing for all x.

Show that the function f(x) = x³ + 10x + 7 for x ∈ R is strictly increasing

Test whether the function f(x) = x³ + 6x² + 12x − 5 is increasing or decreasing for all x ∈ R

Test whether the following function f(x) = 2 – 3x + 3x² – x³, x ∈ R is increasing or decreasing

Find the values of x for which the function f(x) = 2x³ – 6x² + 6x + 24 is strictly increasing

Find the values of x for which the function f(x) = x³ – 6x² – 36x + 7 is strictly increasing

Find the values of x, for which the function f(x) = x³ + 12x² + 36𝑥 + 6 is monotonically decreasing

Find the values of x for which f(x) = 2x³ – 15x² – 144x – 7 is

1. Strictly increasing
2. strictly decreasing

Show that function f(x) = tan x is increasing in `(0, π/2)`.

If y = `e^(m tan^-1x)` then show that `(1 + x^2) (d^2y)/(dx^2) + (2x - m) (dy)/(dx)` = 0

Lines `overliner = (hati + hatj - hatk) + λ(2hati - 2hatj + hatk)` and `overliner = (4hati - 3hatj + 2hatk) + μ(hati - 2hatj + 2hatk)` are coplanar. Find the equation of the plane determined by them.