Quantitative Aptitude

From the figure, and are opposite corners of the bottom side and and are opposite corners of the right side; both side lengths are m. Hence the square has side m.
Manhattan distance : sum of axis-aligned displacements m.
Pythagorean (Euclidean) distance : diagonal of a square m (to one decimal).

Key facts about a parallelogram.
Opposite sides are parallel: and .
The area of a parallelogram is ``base corresponding altitude’’ and is independent of which side you pick as the base.
The altitude to a base is the perpendicular distance between the two lines containing a pair of opposite sides that are parallel to that base.
What the given perpendiculars represent.
with . Since , the perpendicular distance from any point on to the line equals the altitude corresponding to the base . Thus, is the altitude to base .
with . Since , the perpendicular distance from any point on to the line equals the altitude corresponding to the base . Thus, is the altitude to base .
Compute the area two ways. Using as the base: Equate the two area expressions: (Alternative coordinate/vector check.)
Place on the –axis with and (since and ). Let so that . Then ; the side has direction vector and length The area of the parallelogram is also (vertical separation between the parallels and ) .
Using base , the perpendicular distance to its opposite side is , hence area . Therefore , independent of , confirming the earlier result.

In least-squares adjustment, the (global model) test checks the stochastic model by comparing the observed variance factor (from residuals) with the a priori variance factor via If the statistic lies outside the acceptance region, the model is rejected—typically indicating gross errors/blunders or an incorrect noise model. Hence, it is used to test for the presence of gross errors.

A shape tiles the plane if copies meet edge-to-edge with interior angles around every vertex summing to .
- Circle: cannot tile because curved boundaries leave gaps unless stretched/overlapped. No.
- Regular octagon: interior angle ; is not an integer, so identical regular octagons alone cannot fill the plane (they need squares). No.
- Regular pentagon: interior angle ; not an integer; identical regular pentagons cannot tile the plane. No.
- Rhombus (any parallelogram with equal sides): opposite angles are equal and adjacent angles sum to ; two copies meet to make around each vertex via translations. All parallelograms, hence rhombi, tile the plane. Yes.

Standard error (SE) of the mean: If each observation has standard deviation , then for i.i.d.\ observations, the SE of the sample mean is .
Here the single‐observation standard error is (acts as ). We require . Hence Therefore, the surveyor must take 25 observations.

Let unit costs be . Quantities mixed are . Cost share (quantity) (unit cost). Hence Total cost is 1000, so cement’s part is

Why (C) is certain.
The first sentence says the Bank “does not plan to offer new financing … until the country has an adequate macroeconomic policy framework in place.” This implies the required framework is not yet in place. Therefore it follows with certainty that, according to the World Bank, Sri Lanka does not yet have an adequate macroeconomic policy framework.
Why the others are not entailed.
(A) The passage distinguishes between root causes and their effects: it says the crisis “has starved it of foreign exchange…”. So, lack of foreign exchange is presented as a consequence, not the root cause. Hence (A) overstates.
(B) The Bank “said Sri Lanka needed to adopt structural reforms” (a recommendation), but it never states it will advise the government (an explicit future action/engagement). Not guaranteed.
(D) The Bank is repurposing resources under existing loans—not providing additional (new) funds. Thus (D) contradicts the stated “does not plan to offer new financing.”

Expand each cubic: , .
Let with , and with .
In the product , the coefficient of is .
Compute: , , .
Sum: .