PYQs Based Test - 17 : Engineering Mathematics

Let $f\left(x\right)=e^{x+x^2}$ for real x. From among the following, choose the Taylor series approximation of f(x) around x = 0, which includes all powers of x less than or equal to 3.

Consider the following system of linear equations,

x₁ + 2x₂ = b₁

2x1 + 4x2 = b2

3x1 + 7x2 = b3

3x1 + 9x2 = b4

Which one of the following conditions ensures that a solution exists for the above system?

The maximum value of the determinant among all 2 × 2 real symmetric matrices with trace 24 is _______.

A box contains the following three coins.

I. A fair coin with head on one face and tail on the other face.

II. A coin with heads on both the faces.

III. A coin with tails on both the faces.

A coin is picked randomly from the box and tossed. Out of the two remaining coins in the box, one coin is then picked randomly and tossed. If the first toss results in a head, the probability of getting a head in the second toss is

The position of a particle (𝑡) is described by the differential equation:

$\frac{{\mathrm{d}}^2\mathrm{y}}{\mathrm{d}{\mathrm{t}}^2}=-\frac{\mathrm{dy}}{\mathrm{dt}}-\frac{5\mathrm{y}}{4}$

The initial conditions are y(0) = 1 and ${\frac{\mathrm{dy}}{\mathrm{dt}}|}_{\mathrm{t}=0}=0$. The position (accurate to two decimal places) of the particle at t = π is _______.