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

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.

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

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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

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

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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].

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

`f(x) = x^2 - 3x - 1, x ∈ [(-11)/7 , 13/7]`.

Verify Lagrange’s mean value theorem for the following functions : f(x) = 2x – x², x ∈ [0, 1].

Verify Lagrange’s mean value theorem for the following functions : f(x) = `(x - 1)/(x - 3)` on [4, 5].

A die is thrown 6 times. If ‘getting an odd number’ is a success, find the probability of 5 successes. 

A die is thrown 6 times. If ‘getting an odd number’ is a success, find the probability of at least 5 successes.

A die is thrown 6 times. If ‘getting an odd number’ is a success, find the probability of at most 5 successes.

A pair of dice is thrown 4 times. If getting a doublet is considered a success, find the probability of two successes.

Five cards are drawn successively with replacement from a well-shuffled deck of 52 cards; find the probability that all the five cards are spades.

Five cards are drawn successively with replacement from a well-shuffled deck of 52 cards, find the probability that only 3 cards are spades