Find the directional derivative - a real root of equation

Question

Q1. The following system of

x + y + 2z = 3
x + 2y + 3z = 6
x + 4y + kz = 12
has unique solution. The only possible value(s) of k is/are -

Q2. The number of linearly independent Eigen vectors of

[a   b]

[0   a]

is/are

Q3. Probability that the divisors of 6¹⁰ is a multiple of 6⁶ is

Q4. What would be the expectation of the number of successes preceeding the first failure in an infinite series of independent trials with constant probability of success p = 0.4.

Q5. A Variate X has the following distribution

     X :        0     1      2     3

P (X = x):  1/3    1/6  1/3   1/6

The E [2X + 3)²] is

Q6. What will be the value of y(0.04), from the differential equation dy/dx +y = 0 with y(0) = 1, h = 0.04, by Runge-Kutta Method.

Q7. A real root of equation cos x = 4x - 1 correct to seven decimal places by method of successive approximation is

Q8. (Repeat Q2) The number of linearly independent Eigen vectors of

[a   b]

[0   a]

is/are

Q9. The solution of differential equation

dy/dx + y/x = x² with y(1) =1 is

Q10. The general solution of

(xe^xy + 2y) dy + ye^xydx = 0

Q11. If

f(x) = -x², x ≤ 0

= 5x - 4, 0 < x ≤ 1

= 4x² - 3x, 1 < x < 2

= 3x + 4, x ≥ 2

then f(x) is discontinuous at x =

Q12. Find the directional derivative of

f(x, y, x) = x²yz + 4xz²

at (1, -2, -1) along 2i^ - j^ - 2k^

Q13. ∫√(x² - a²)dx is

Q14. I = ₀∫^Π₀∫^acosθ rdrdθ is

Q15. Find c of Rolle's Theorem for

f(x) = e^x sin x in [0, Π] .