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

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.

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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].

In a certain culture of bacteria, the rate of increase is proportional to the number present. If it is found that the number doubles in 4 hours, find the number of times the bacteria are increased in 12 hours.

If the population of a country doubles in 60 years, in how many years will it be triple (treble) under the assumption that the rate of increase is proportional to the number of inhabitants?
(Given log 2 = 0.6912, log 3 = 1.0986)