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

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.

Find the acute angle between the lines `(x - 1)/(1) = (y - 2)/(-1) = (z - 3)/(2) and (x - 1)/(2) = (y - 2)/(1) = (z - 3)/(1)`.

Find the acute angle between the lines x = y, z = 0 and x = 0, z = 0.

Find the acute angle between the lines x = –y, z = 0 and x = 0, z = 0.

Choose correct alternatives :

The angle between the lines 2x = 3y = – z and 6x = – y = – 4z is

Choose correct alternatives :

The angle between the planes `bar"r".(hat"i" - 2hat"j" + 3hat"k") + 4 = 0 and bar"r".(2hat"i" + hat"j" - 3hat"k") + 7 = 0` is

Choose correct alternatives :

Measure of angle between the plane 5x – 2y + 3z – 7 = 0 and 15x – 6y + 9z + 5 = 0 is

Choose correct alternatives:

If the line `(x + 1)/(2) = (y - m)/(3) = (z - 4)/(6)` lies in the plane 3x – 14y + 6z + 49 = 0, then the value of m is ______.

Solve the following :

Find the angle between the planes `bar"r".(-2hat"i" + hat"j" + 2hat"k")` = 17 and `bar"r".(2hat"i" + 2hat"j" + hat"k")` = 71.

Find the acute angle between the line `bar r = lambda (hat i - hat j + hat k)` and the plane `bar r * (2hat i - hat j + hat k)` = 23.

Verify Lagrange’s mean value theorem for the following function:

f(x) = log x, on [1, e]

Verify Lagrange’s mean value theorem for the following functions : f(x) = (x – 1)(x – 2)(x – 3) on [0, 4].