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

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

If x = `(2bt)/(1 + t^2), y = a((1 - t^2)/(1 + t^2)), "show that" "dx"/"dy" = -(b^2y)/(a^2x)`.

Differentiate 3^x w.r.t. log_x3.

Find the second order derivatives of the following : x³.logx

Find the second order derivatives of the following : log(logx)

If y = `log(x + sqrt(x^2 + a^2))^m`, show that `(x^2 + a^2)(d^2y)/(dx^2) + x "d"/"dx"` = 0.

If y = log (log 2x), show that xy₂ + y₁ (1 + xy₁) = 0.

Find the n^th derivative of the following: log (ax + b)

Find the n^th derivative of the following : log (2x + 3)

Choose the correct option from the given alternatives :

If x^y = y^x, then `"dy"/"dx"` = ..........

If y = A cos (log x) + B sin (log x), show that x²y² + xy₁ + y = 0.

Find the angle between planes `bar"r".(hat"i" + hat"j" + 2hat"k") = 13 and bar"r"(2hat"i" + hat"j" + hat"k")` = 31.

Find the acute angle between the line `barr = (hati + 2hatj + 2hatk) + lambda(2hati + 3hatj - 6hatk)` and the plane `barr*(2hati - hatj + hatk)` = 0

Find the value of λ so that the lines `(1 - x)/(3) = (7y - 14)/(λ) = (z - 3)/(2) and (7 - 7x)/(3λ) = (y - 5)/(1) = (6 - z)/(5)` are at right angles.