Chapter 13 — Verbal and Analytical Reasoning

Concepts, worked examples and 45 CSIR NET-JRF Part A questions (December 2016 – December 2025) with full solutions. Click Show solution under any question.

Concepts

13.1 Orderings and arrangements

A large class of reasoning questions asks us to place a set of objects in order, by position in a row, by height, by seniority or by place in a race, from a list of partial statements. Each statement is a relation between two objects, and the relations “taller than”, “senior to” and “ahead of” are transitive: if \(X > Y\) and \(Y > Z\), then \(X > Z\). The working method is therefore to translate every statement into a short chain such as \(F > D > B\), to merge chains that share a member, and to see which positions remain undetermined. When two people are equal in the ordering, as batchmates are in seniority, they occupy a single place in the chain, and a strict statement about one of them applies equally to the other.

Words must be read exactly. “R is just ahead of T” makes R and T adjacent, whereas “R is ahead of T” does not; “K is the only one between T and V” fixes the block T K V or V K T; “C is four places to the left of F” means that F occupies the position four seats to the right of C, so that if the seats are numbered \(1,2,\dots,n\) from the left, \(p_F = p_C + 4\) and in particular \(p_F \geq 5\). A person “at the centre” of a row of \(2m+1\) seats has \(m\) people on each side. These numerical translations frequently settle a question without the whole arrangement being found.

When several arrangements satisfy the conditions, the question usually asks what is certain, what is possible or what is impossible. A statement is certain only if it holds in every admissible arrangement, and it is shown to be false by a single counter-example. It is therefore efficient to list all admissible arrangements, which are seldom more than four, and to test each option against the list. A row read from the other end is again an arrangement, so a configuration and its mirror image are admissible together unless a statement refers to a particular end.

Matching problems, in which each person is paired with exactly one object, are handled in the same spirit. A grid with people along one side and objects along the other is filled with ticks and crosses; a tick in a cell places crosses in the rest of its row and column, and a row or column with a single cell left must take a tick. Conditions of the type “exactly two initials match” are applied last, when only a few cells remain open.

Some arrangements are governed by parity. If boys and girls sit alternately in a row which begins and ends with a boy, the row reads \(BGBG\cdots GB\); it contains \(k+1\) boys and \(k\) girls, and so an odd number \(2k+1\) of seats. A sum of three odd numbers is odd, so a class seated in three such rows has an odd number of students.

13.2 Blood relations

Questions on blood relations are solved by drawing a family tree. Each generation is placed on its own level, a vertical line joins a parent to a child, a horizontal line joins siblings, and a double line joins a married couple; the sex of each person is recorded as soon as it is known, since many relations (“brother”, “aunt”, “niece”) depend on it. A relation is then read off by moving along the tree. Thus “my father’s son”, spoken by a man with no brothers, can only be the speaker himself, and “the daughter of my father’s son”, spoken by the same man, is his own daughter. The relations most often needed are summarised below:

Relation	Meaning
uncle, aunt	brother or sister of a parent, or the spouse of one
nephew, niece	son or daughter of a brother or sister (or of a spouse’s sibling)
first cousin	child of an uncle or aunt
father-in-law	father of one’s husband or wife
grandson, granddaughter	son or daughter of one’s son or daughter

Counting questions rest on the fact that siblings share their brothers and sisters. If each of several sisters “has one brother”, it is the same brother, so seven sisters and one brother make eight children; if each of four sons “has exactly two sisters”, there are two daughters in all. Coded relations, in which a symbol such as \(*\) or \(\Delta\) stands for “is the father of” or “is the husband of”, are decoded link by link from left to right, and each link is drawn on the tree before the next is read.

A useful structural fact concerns parent–child pairs. Every person has exactly one father, and nobody can be his own ancestor, so in any group at least one man (the eldest, for instance) has no father in the group. Among \(n\) men, therefore, at most \(n-1\) have their fathers present, and

\[\boxed{\text{number of father–son pairs among } n \text{ men} \leq n-1},\]

with equality, for example, when one man is the father of all the others, or when the men form a single line of descent. A group of \(n\) people contains \(\binom{n}{2} = n(n-1)/2\) pairs in all, a count that is needed when the question asks for the largest possible number of pairs of some kind.

13.3 Direction sense

A direction problem is a sequence of straight walks and turns, and it is best treated as vector addition. Take east as the positive \(x\)-axis and north as the positive \(y\)-axis, so that the eight compass points are

\[\text{E} = (1,0),\quad \text{N} = (0,1),\quad \text{NE} = \left(\tfrac{1}{\sqrt2},\tfrac{1}{\sqrt2}\right),\quad \text{SE} = \left(\tfrac{1}{\sqrt2},-\tfrac{1}{\sqrt2}\right),\]

and so on. A turn to the right is a clockwise rotation of the heading and a turn to the left an anticlockwise one, as seen from above: a walker facing north who turns right faces east, and one facing west who turns left faces south. A right turn through \(45^\circ\) from north gives north-east, and a further right turn through \(90^\circ\) gives south-east. Each leg of length \(\ell\) in the direction \((\cos\theta,\sin\theta)\) contributes \((\ell\cos\theta,\ell\sin\theta)\), the legs are added, and the final position \((x,y)\) gives

\[\boxed{d = \sqrt{x^2+y^2}},\]

the shortest distance from the start, together with its direction. In most questions the legs are along the four cardinal directions and the sums are immediate; diagonal legs bring in the factor \(1/\sqrt2 \approx 0.707\).

The direction of a point Q as seen from a point P is the direction of the vector \(\mathbf{r}_Q - \mathbf{r}_P\), and not of either position vector alone. When the answer must be one of a few compass points, we choose the one nearest to this vector. The same rule governs instructions given to a moving body that carries its own sense of left and right, such as an ant or a robot: each turn is made relative to the current heading, and the sequence of moving forward a distance \(a\) and turning right through \(90^\circ\), repeated four times, traces a closed square and returns the body to its starting point and heading.

13.4 Coding, analogies and series

In a coding question a rule converts one word or number into another, and the rule is inferred from a single example and then applied to a new word. The common rules are a reversal of the letters (NOODLES becomes SELDOON), a shift of each letter by a fixed number of places in the alphabet, a substitution of a digit for each letter, and an exchange of arithmetic operations. In a letter-to-digit substitution each letter keeps its digit wherever it occurs, so repeated letters in the example fix the code of the repeated letters in the new word, and an option that gives two different digits to the same letter is excluded. When the symbols of arithmetic are exchanged, the new expression is evaluated by the usual precedence of brackets, then multiplication and division, then addition and subtraction.

Series are completed by finding the rule that generates successive terms. For numbers we examine the differences, the ratios and the alternate terms in turn. For words or letter strings we examine how each term is built from the previous one; a string may, for instance, be extended by appending a copy of itself in which every letter is replaced by its complement, as in \(\text{A} \to \text{AB} \to \text{ABBA} \to \text{ABBABAAB}\), where the length doubles at each step. Positional patterns, such as the order in which books are placed on a shelf, are found in the same way by writing down the differences between successive entries.

An analogy \(A : B :: C : D\) asserts that \(C\) stands to \(D\) in the same relation as \(A\) to \(B\). The relation must be stated precisely before the options are examined: a script is the document from which a director works, a city may be the largest industrial centre of a state that is not its capital, and so on. Of several options that bear some relation to \(C\) or \(D\), only the one that reproduces the same relation is correct.

13.5 Statements, syllogisms and Venn diagrams

A categorical statement relates two classes, a subject \(S\) and a predicate \(P\), in one of four forms: “All \(S\) are \(P\)” (\(S \subseteq P\)), “No \(S\) is \(P\)” (\(S \cap P = \varnothing\)), “Some \(S\) are \(P\)” (\(S \cap P \neq \varnothing\)) and “Some \(S\) are not \(P\)”. A syllogism draws a conclusion from two such statements sharing a middle term. The conclusion is valid only if it holds in every Venn diagram consistent with the premises, so the reliable test is to draw the most unfavourable diagram and see whether the conclusion survives. From “All B are M” and “All M are C” it follows that all B are C, because inclusion is transitive, but neither “All M are B” nor “All C are B” follows. Similarly “Some A are T” together with “All T are P” gives “Some A are P”, because the members common to A and T lie inside P. A universal statement does not by itself guarantee that any member exists, and its converse is not in general true.

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Chapter 13 Verbal and Analytical Reasoning