List of practice Questions

Let f: R→ R be a function. Note: R denotes the set of real numbers.
\(f(x) = \left\{ \begin{array}{ll} -x, & \text{if } x < -2 \\ ax^2+bx+c, & \text{if }\ x \in[-2,2]\\ x, & \text{if } x>2 \end{array} \right.\)
Which ONE of the following choices gives the values of a, b, c that make the function f continuous and differentiable ?

Consider the following Python code :
def count (child_dict, i) :
if i not in child_dict.keys() :
return 1
ans = 1
for j in child_dict[i] :
ans += count (child_dict, j)
return ans
child_dict = dict()
child_dict [0] = [1, 2]
child_dict [1] = [3, 4, 5]
child_dict [2] = [6, 7, 8]
print (count(child_dict,0))
Which ONE of the following is the output of this code ?

Given the relational schema R = (U, V, W, X, Y, Z) and the set of functional dependencies :
{U → V,U → W, WX → Y,WX → Z,V → X}
Which of the following functional dependencies can be derived from the above set ?

Consider the function f: \(\R→ \R\) where \(\R\) is the set of all real numbers.
\(f(x)=\frac{x^4}{4}-\frac{2x^3}{3}-\frac{3x^2}{2}+1\)
Which of the following statements is/are TRUE ?

Consider the following figures representing datasets consisting of two-dimensional features with two classes denoted by circles and squares.

Which of the following is/are TRUE ?

Let X be a random variable exponentially distributed with parameter λ > 0. The probability density function of X is given by :
\(f(x) = \left\{ \begin{array}{ll} \lambda e^{-\lambda x,} & x \ge0 \\ 0 & \text{otherwise} \end{array} \right.\)
If 5E(X) = Var (X), where E (X) and Var (X) indicate the expectation and variance of X, respectively, the value of λ is _____
(rounded off to one decimal place).

Consider a joint probability density function of two random variables X and Y
\(f_{x,y}(x,y) = \left\{ \begin{array}{ll} 2xy, & 0\lt x \lt2 , & 0\lt y\lt x\\ 0 & \text{otherwise } \end{array} \right.\)
Then, E[Y | X = 1.5] is ______.

Evaluate the following limit :
\(\lim\limits_{x\rightarrow0}\frac{\text{ln}((x^2+1)\cos x)}{x^2}\) = ________.

Details of ten international cricket games between two teams "Green" and "Blue" are given in Table C. This table consists of matches played on different pitches, across formats along with their winners. The attribute Pitch can take one of two values: spin-friendly (represented as S) or pace-friendly (represented as F). The attribute Format can take one of two values: one-day match (represented as 0) or test match (represented as T).
A cricket organization would like to use the information given in Table C to develop a decision-tree model to predict outcomes of future games between these two teams.
To develop such a model, the computed InformationGain(C, Pitch) with respect to the Target is ______ (rounded off to two decimal places).
Table C

Match Number	Pitch	Format	Winner (Target)
1	S	T	Green
2	S	T	Blue
3	F	O	Blue
4	S	O	Blue
5	F	T	Green
6	F	O	Blue
7	S	O	Green
8	F	T	Blue
9	F	O	Blue
10	S	O	Green

Let \( P \) be a parabola with vertex \( (2, 3) \) and directrix \( 2x + y = 6 \). Let an ellipse \( E : \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), \( a > b \), of eccentricity \( \frac{1}{\sqrt{2}} \) pass through the focus of the parabola \( P \). Then the square of the length of the latus rectum of \( E \) is

An objective function $Z = ax + by$ is maximum at points $(8, 2)$ and $(4, 6)$. If $a \geq 0$ and $b \geq 0$ and $ab = 25$, then the maximum value of the function is:

A partial differential equation
\(\frac{∂^2T}{∂x^2}+\frac{∂^2T}{∂y^2}=0\)
is defined for the two-dimensional field T: T(x, y), inside a planar square domain of size 2 m x 2 m. Three boundary edges of the square domain are maintained at value T = 50, whereas the fourth boundary edge is maintained T = 100.
The value of T at the center of the domain is

The statements P and Q are related to matrices A and B, which are conformable for both addition and multiplication.
P : (A + B)^T = A^T + B^T
Q: (AB)^T = A^TB^T
Which one of the following options is CORRECT ?

The second derivative of a function f is computed using the fourth-order Central Divided Difference method with a step length h.
The CORRECT expression for the second derivative is

The function f(x) = x³ - 27x + 4, 1 ≤ x ≤ 6 has

Consider two Ordinary Differential Equations (ODEs) :
\(P:\frac{dy}{dx}=\frac{x^4+3x^2y^2+2y^4}{x^3y}\)
\(Q:\frac{dy}{dx}=\frac{-y^2}{x^2}\)
Which one of the following options is CORRECT ?

A 3 m long, horizontal, rigid, uniform beam PQ has negligible mass. The beam is subjected to a 3 kN concentrated vertically downward force at 1 m from P, as shown in the figure. The beam is resting on vertical linear springs at the ends P and Q. For the spring at the end P, the spring constant K_p 100 kN/m.

If the beam DOES NOT rotate under the application of the force and displaces only vertically, the value of the spring constant KQ (in kN/m) for the spring at the end Q is

Consider the statements P and Q.
P : In a Pure project organization, the project manager maintains complete authority and has maximum control over the project.
Q : A Matrix organization structure facilitates quick response to changes, conflicts, and project needs.
Which one of the following options is CORRECT ?

For a thin-walled section shown in the figure, points P, Q, and R are located on the major bending axis X - X of the section. Point Q is located on the web whereas point S is located at the intersection of the web and the top flange of the section.

Qualitatively, the shear center of the section lies at

Consider the following data for a project of 300 days duration.
Budgeted Cost of Work Scheduled (BCWS) = Rs. 200
Budgeted Cost of Work Performed (BCWP) = Rs. 150
Actual Cost of Work Performed (ACWP) = Rs. 190
The 'schedule variance' for the project is

A simply supported, uniformly loaded, two-way slab panel is torsionally unrestrained. The effective span lengths along the short span (x) and long span (y) directions of the panel are l_x and l_y, respectively. The design moments for the reinforcements along the x and y directions are M_ux and M_uy, respectively. By using Rankine-Grashoff method, the ratio M_ux/M_uy is proportional to

The structural design method that DOES NOT take into account the safety factors on the design loads is

The contact pressure distribution shown in the figure belongs to a

Which one of the following saturated fine-grained soils can attain a negative Skempton's pore pressure coefficient (A) ?