Class 12th

Let f : [0,∞) → be a function defined by f(x) = { max{sint: 0≤ t ≤x}, 0≤x≤π, 2 + cos x, x > π
Then which of the following is true?

Let y = y(x) be the solution of the differential equation (x - x³) dy = (y + yx² – 3x? )dx, x > 2. If y(3) = 3 then y(4) is equal to

Let a,b and c be three vectors such that a = b*(bxc). If magnitudes of the vectors a,b and c are √2, 1 and 2 respectively and the angle between b and c is θ(0<

Let a = i - αj + βk, b = 3i + βj - αk and c = -αi - 2j + k, where α and β are integers. If a . b = -1and b . c = 10, then (a * b) . c is equal to ……… equal to ………

Let y = y(x) be the solution of the differential equation dy = e^(αx+y)dx; α ∈ N . If y(logₑ2) = -logₑ2 and y(0) = -logₑα, then the value of α is equal to………

If A = [1 1 1; 0 1 1; 0 0 1] and M = A + A² + A³ + . + A²?, then the sum of all the elements of the matrix M is equal to………

If ∫₀^(π/2) (sin³x)e⁻sin²ˣdx = α - (β/e)∫₀¹ √t eᵗdt, then α + β is equal to ………

Let f:(a,b) →R be twice differentiable such that f(x) = ∫ₐˣ g(t)dt for a differentiable function g(x). If f(x) = 0 has exactly five distinct roots in (a, b), then g(x)g'(x) = 0 has at least :

A student appeared in an examination consisting of 8 true – false type questions. The student guesses the answers with equal probability. The smallest value of n, so that the probability of guessing at least 'n' correct answers is less than 1/2, is :

Let A and B be two 3 x 3 real matrices such that (A² - B²) is invertible matrix. If A⁵ = B⁵ and A³B² = A²B³, then the value of the determinant of the matrix A³ + B³ is equal to :