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

Choose correct alternatives:

If the equation 4x² + hxy + y² = 0 represents two coincident lines, then h = _______

If the lines represented by kx² − 3xy + 6y² = 0 are perpendicular to each other, then

Choose correct alternatives:

The difference between the slopes of the lines represented by 3x² - 4xy + y² = 0 is 2

If A = {1, 2, 3, 4, 5, 6, 7, 8, 9}, determine the truth value of the following statement:

∃ x ∈ A such that x + 8 = 15

If A = {1, 2, 3, 4, 5, 6, 7, 8, 9}, determine the truth value of the following statement:

∀ x ∈ A, x + 5 < 12.

If A = {1, 2, 3, 4, 5, 6, 7, 8, 9}, determine the truth value of the following statement:

∃ x ∈ A, such that x + 7 ≥ 11.

If A = {1, 2, 3, 4, 5, 6, 7, 8, 9}, determine the truth value of the following statement:

∀ x ∈ A, 3x ≤ 25.

If log (x + y) = log(xy) + p, where p is a constant, then prove that `"dy"/"dx" = (-y^2)/(x^2)`.

If `log_10((x^3 - y^3)/(x^3 + y^3))` = 2, show that `dy/dx = -(99x^2)/(101y^2)`.

If `log_5((x^4 + y^4)/(x^4 - y^4)) = 2, "show that""dy"/"dx" = (12x^3)/(13y^3)`.

If x^y = e^x–y, then show that `"dy"/"dx" = logx/(1 + logx)^2`.

`"If"  y = sqrt(logx + sqrt(log x + sqrt(log x + ... ∞))), "then show that"  dy/dx = (1)/(x(2y - 1).`

If y = `x^(x^(x^(.^(.^.∞))`, then show that `"dy"/"dx" = y^2/(x(1 - logy).`.

If e^y = y^x, then show that `"dy"/"dx" = (logy)^2/(log y - 1)`.

If x = `asqrt(secθ - tanθ), y = asqrt(secθ + tanθ), "then show that" "dy"/"dx" = -y/x`.

If x = e^sin3t, y = e^cos3t, then show that `dy/dx = -(ylogx)/(xlogy)`.

If x = a cos³t, y = a sin³t, show that `"dy"/"dx" = -(y/x)^(1/3)`.

If x = 2cos⁴(t + 3), y = 3sin⁴⁽t + 3), show that `"dy"/"dx" = -sqrt((3y)/(2x)`.

If x = log(1 + t²), y = t – tan^–1t,show that `"dy"/"dx" = sqrt(e^x - 1)/(2)`.

If x = sin^–1(e^t), y = `sqrt(1 - e^(2t)), "show that"  sin x + dy/dx` = 0