IPMAT Rohtak 2019 · Question Paper & Solutions

Reading the charts. Both pies are printed with the title "downstream", but the questions only work if the 4800 km pie is upstream and the 2400 km pie is downstream (Q2 confirms this).

• Upstream (4800 km): Sun 960, Mon 768, Tue 864, Wed 576, Thu 480, Fri 672, Sat 480
• Downstream (2400 km): Sun 288, Mon 336, Tue 432, Wed 264, Thu 240, Fri 408, Sat 432
• Stream speed: Mon 5, Tue 2, Wed 6, Fri 1, Sun 3 (Thu, Sat not given)

Monday downstream: $t=\frac{336}{15+5}=16.8$ h.

Wednesday upstream time $=\frac{6}{7}\times 16.8=14.4$ h, so upstream speed $=\frac{576}{14.4}=40$ km/h.

Still-water speed on Wednesday $=40+6=46$ km/h, which is not among 52, 62, 42. (Reading the pies the other way round gives about 14 km/h, also not an option.)

Answer: None of these.

Reading the charts. Both pies are printed with the title "downstream", but the questions only work if the 4800 km pie is upstream and the 2400 km pie is downstream (Q2 confirms this).

• Upstream (4800 km): Sun 960, Mon 768, Tue 864, Wed 576, Thu 480, Fri 672, Sat 480
• Downstream (2400 km): Sun 288, Mon 336, Tue 432, Wed 264, Thu 240, Fri 408, Sat 432
• Stream speed: Mon 5, Tue 2, Wed 6, Fri 1, Sun 3 (Thu, Sat not given)

Let still-water speed be $b$. With stream 5 km/h:
$$\frac{768}{b-5}-\frac{336}{b+5}=27.2$$

Try $b=25$: $\frac{768}{20}-\frac{336}{30}=38.4-11.2=27.2$ ✓.

The left side decreases as $b$ grows, so the root is unique (check $b=20$: $51.2-13.44=37.76\ne 27.2$).

Answer: 25 km/h.

Reading the charts. Both pies are printed with the title "downstream", but the questions only work if the 4800 km pie is upstream and the 2400 km pie is downstream (Q2 confirms this).

• Upstream (4800 km): Sun 960, Mon 768, Tue 864, Wed 576, Thu 480, Fri 672, Sat 480
• Downstream (2400 km): Sun 288, Mon 336, Tue 432, Wed 264, Thu 240, Fri 408, Sat 432
• Stream speed: Mon 5, Tue 2, Wed 6, Fri 1, Sun 3 (Thu, Sat not given)

Wednesday still-water speed $=27\times \frac{5}{3}=45$ km/h, so upstream speed $=45-6=39$ km/h.

Wednesday upstream time $=\frac{576}{39}=\frac{192}{13}$ h.

As printed (ratio $16\frac{1}{3}=\frac{49}{3}$): Saturday downstream time would be $\frac{192}{13}\times \frac{3}{49}\approx 0.90$ h, i.e. a downstream speed of about 478 km/h. That is impossible, and none of the options work.

Intended ratio $\frac{16}{13}$ (the fraction seems to have been misprinted as a mixed number):
$$t_{\text{down, Sat}}=\frac{192}{13}\times \frac{13}{16}=12\text{ h}\Rightarrow 27+s=\frac{432}{12}=36\Rightarrow s=9$$

Answer: 9 km/h.

Reading the charts. Both pies are printed with the title "downstream", but the questions only work if the 4800 km pie is upstream and the 2400 km pie is downstream (Q2 confirms this).

• Upstream (4800 km): Sun 960, Mon 768, Tue 864, Wed 576, Thu 480, Fri 672, Sat 480
• Downstream (2400 km): Sun 288, Mon 336, Tue 432, Wed 264, Thu 240, Fri 408, Sat 432
• Stream speed: Mon 5, Tue 2, Wed 6, Fri 1, Sun 3 (Thu, Sat not given)

Sunday still-water speed $=21\times \frac{9}{7}=27$ km/h, so downstream speed $=27+3=30$ km/h.

Sunday downstream time $=\frac{288}{30}=9.6$ h. Saturday upstream time $=2.5\times 9.6=24$ h.

Upstream speed on Saturday $=\frac{480}{24}=20$ km/h (so the stream is 1 km/h).

Time for 125 km $=\frac{125}{20}=6.25$ h $=6$ h $15$ min.

Answer: 6 hrs 15 min.

Reading the charts. Both pies are printed with the title "downstream", but the questions only work if the 4800 km pie is upstream and the 2400 km pie is downstream (Q2 confirms this).

• Upstream (4800 km): Sun 960, Mon 768, Tue 864, Wed 576, Thu 480, Fri 672, Sat 480
• Downstream (2400 km): Sun 288, Mon 336, Tue 432, Wed 264, Thu 240, Fri 408, Sat 432
• Stream speed: Mon 5, Tue 2, Wed 6, Fri 1, Sun 3 (Thu, Sat not given)

Friday upstream speed $=17-1=16$ km/h, so time $=\frac{672}{16}=42$ h.

Wednesday downstream time $=42-30=12$ h, so downstream speed $=\frac{264}{12}=22$ km/h.

Wednesday still-water speed $=22-6=16$, so upstream speed $=16-6=10$ km/h. That is not among 27, 22, 20.

Answer: None of these.

Water is only ever swapped in for milk, so the total stays $X$ litres. Water $=X-(X-122)=122$ litres.

Check (find $X$): $X{\left(1-\frac{50}{X}\right)}^3=X-122\Rightarrow (X-50{)}^3=X^2(X-122)$
$$\Rightarrow 28X^2-7500X+125000=0\Rightarrow X=250$$
(the other root, about 17.9, is less than 50 and so invalid).

Milk left $=250\times (0.8{)}^3=128$, so water $=250-128=122$ ✓.

Answer: 122 litres.

Veena: $P\left(1+0.06\times \frac{5}{12}\right)=1.025P=2460\Rightarrow P=2400$.

Gita: interest $=60=2400\times 0.075\times \frac{t}{12}=15t\Rightarrow t=4$ months.

Answer: 4 months.

Each small circle has diameter 15, so radius $7.5$. Circle K's diameter is $15+15=30$, so its radius is $15$.

• Area of K $=225\pi$
• Area of L $=$ area of M $=56.25\pi$

Shaded region (K minus L and M) $=225\pi -112.5\pi =112.5\pi$.

(The two small circles together also make $112.5\pi$, so the answer is the same whichever part is shaded.)

Answer: $112.5\pi$ sq ft.

The radius of M ($7.5$) is half the radius of K ($15$). Area scales with the square of the radius:
$$\frac{\text{M}}{\text{K}}={\left(\frac{7.5}{15}\right)}^2=\frac{1}{4}$$

Directly: $56.25\pi :225\pi =1:4$.

Answer: 1 : 4.

$\frac{xy}{100}=z$.

Percentage of $z$ that $x$ is:
$$\frac{x}{z}\times 100=\frac{x\cdot 100}{xy/100}=\frac{100^2}{y}$$

Check: $x=10,y=200\Rightarrow z=20$, and $10$ is $50\%$ of $20$. Also $\frac{10000}{200}=50$ ✓.

Answer: $\frac{100^2}{y}$.

$$\frac{450}{v_A}-\frac{450}{v_B}=\frac{3}{4}\frac{450}{\frac{3}{4}{v}_B}-\frac{450}{v_A}=\frac{1}{2}$$

The second equation is $\frac{600}{v_B}-\frac{450}{v_A}=\frac{1}{2}$. Adding the two equations: $\frac{150}{v_B}=\frac{5}{4}\Rightarrow v_B=120$.

Then $\frac{450}{v_A}=\frac{450}{120}+0.75=4.5\Rightarrow v_A=100$ km/h.

Check: A takes 4.5 h and B 3.75 h (45 min apart). Slowed B runs at 90 km/h and takes 5 h, 30 min more than A ✓.

Answer: 100 km/h.

Let the selling price be $S$.

• Profit 20% on cost: cost $=\frac{S}{1.2}$, profit $=\frac{S}{6}$
• Profit 20% on SP: profit $=0.2S=\frac{S}{5}$

Difference: $\frac{S}{5}-\frac{S}{6}=\frac{S}{30}=85\Rightarrow S=2550$.

Answer: Rs 2550.

Let the old price be $p$ per egg. Then $\frac{24}{p}-\frac{24}{1.2p}=2$.

Simplifying: $\frac{24}{p}\cdot \frac{1}{6}=2\Rightarrow p=2$. The new price is $2.4$ per egg.

Per dozen: $12\times 2.4=28.8$.

Check: Rs 24 bought 12 eggs before and 10 now ✓.

Answer: Rs 28.8/dozen.

$$\sin \frac{13\pi }{6}=\sin \left(2\pi +\frac{\pi }{6}\right)=\sin \frac{\pi }{6}=\frac{1}{2}$$

Answer: $\frac{1}{2}$.

9 boys eat $\frac{3}{5}$ in 2 hours, so one boy eats $\frac{3/5}{9\times 2}=\frac{1}{30}$ of the fruit per hour.

Left: $\frac{2}{5}$ of the fruit, to be eaten by 3 boys at $\frac{3}{30}=\frac{1}{10}$ per hour.

Time $=\frac{2/5}{1/10}=4$ hours. (The Rs 350 is irrelevant.)

Answer: 4 hours.

$f(x)=(x+b{)}^2+2c^2-b^2$, so $\min f=2c^2-b^2$.

$g(x)=-(x+c{)}^2+c^2+b^2$, so $\max g=c^2+b^2$.

Requirement: $2c^2-b^2>c^2+b^2\Rightarrow c^2>2b^2\Rightarrow |c|>\sqrt{2}|b|$.

Answer: $|c|>\sqrt{2}|b|$.

Case $x\ge -2$: $x^2-(x+2)+x=x^2-2>0\Rightarrow x>\sqrt{2}$ or $-2\le x<-\sqrt{2}$.

Case $x<-2$: $x^2+(x+2)+x=(x+1{)}^2+1>0$, which always holds.

Union: $(-\infty ,-\sqrt{2})\cup (\sqrt{2},\infty )$.

Check: $x=0$ gives $-2$ (fails), $x=-1.5$ gives $-1.25$ (fails), $x=1.5$ gives $0.25$ (works) ✓.

Answer: $(-\infty ,-\sqrt{2})\cup (\sqrt{2},\infty )$.

Successive divisors are $÷12,÷10,÷8,÷6,÷4$, so the next step is $÷2$:
$$2÷2=1$$

Answer: 1.

Base + roof $=2\times 15\times 6=180$.

Four walls $=2h(15+6)=42h$.

$42h=180\Rightarrow h=\frac{30}{7}\approx 4.29$.

Answer: 4.29 cm.

$a_3+a_9=2a+10d=2a_6=8\Rightarrow a_6=4$.

$S_{11}=\frac{11}{2}(a_1+a_{11})=11a_6=44$.

Answer: 44.

$B$ is a geometric series: $B=2^0+2^1+\cdots +2^{64}=2^{65}-1$.

So $A-B=1$.

Check on a small case: $2^3=8$ and $1+2+4=7$ ✓.

Answer: A is larger than B by 1.

A segment from the centre to any point on the circle is, by definition, a radius (option B) — the correct answer.

A diameter is a chord through the centre, joining two points on the circle, and is twice the radius. Area is not a line, and midpoint is a point.

The official key gives A (diameter), which contradicts the definition.

Total ways: $\left(\genfrac{}{}{0ex}{}{15}{2}\right)=105$.

Exactly one white: $\left(\genfrac{}{}{0ex}{}{5}{1}\right)\left(\genfrac{}{}{0ex}{}{10}{1}\right)=50$.

$P=\frac{50}{105}=\frac{10}{21}$.

Check by the complement: $P(\text{0 white})=\frac{45}{105}$ and $P(\text{2 white})=\frac{10}{105}$, so $1-\frac{55}{105}=\frac{50}{105}$ ✓.

Answer: $\frac{10}{21}$.

All three together: $\frac{1}{20}+\frac{1}{25}-\frac{1}{30}=\frac{15+12-10}{300}=\frac{17}{300}$ per hour.

After 10 h the tank is $\frac{170}{300}$ full, leaving $\frac{130}{300}$.

P and Q together: $\frac{27}{300}$ per hour, so they need $\frac{130}{27}$ h more.

Total time $=10+\frac{130}{27}=\frac{400}{27}$ h (the key counts the total time).

Answer: $\frac{400}{27}$ hrs.

Make $b$ common (LCM of 3 and 5 is 15): $a:b=10:15$ and $b:c=15:6$.

So $a:b:c=10:15:6$. The ratio $c:d$ isn't needed.

Answer: 10 : 15 : 6.

Let the counts be $k,8k,16k$.

Value: $k+0.5(8k)+0.25(16k)=k+4k+4k=9k=495\Rightarrow k=55$.

50 paise coins $=8\times 55=440$.

Answer: 440.

$1$ second $=\frac{1}{3600}$ hour $=0.000277\dots$

• Rounded to five decimal places this is $0.00028$ (option A) — the correct answer.
• Truncated it is $0.00027$ (option B), which is what the official key gives.

The exact value, $0.0002\overline{7}$, is closer to $0.00028$, so A is right.

$$0.0056=\frac{56}{10000}=\frac{7}{1250}$$

Check: $7÷1250=0.0056$ ✓.

Answer: $\frac{7}{1250}$.

Let the numbers be $x$ and $y$. From $\frac{x}{5}=\frac{5y}{8}$ we get $x=\frac{25y}{8}$.

$x+35=4y\Rightarrow \frac{25y}{8}+35=4y\Rightarrow \frac{7y}{8}=35\Rightarrow y=40$.

Check: $x=125$. Then $125/5=25=\frac{5}{8}(40)$, and $125+35=160=4(40)$ ✓.

Answer: 40.

After swapping the symbols: $43+561\times 500÷100-10$.

Working through: $561\times 500=280500$, then $÷100=2805$, so $43+2805-10=2838$.

Answer: 2838.

Triangle area $=\frac{\sqrt{3}}{4}(24{)}^2=144\sqrt{3}$.

Inradius $r=\frac{a}{2\sqrt{3}}=\frac{24}{2\sqrt{3}}=4\sqrt{3}$, so circle area $=\pi (48)=48\pi$.

Remaining area $=144\sqrt{3}-48\pi$ cm².

Answer: $144\sqrt{3}-48\pi$ cm².

4 strikes have 3 gaps, so each gap $=\frac{9}{3}=3$ s.

12 strikes have 11 gaps: $11\times 3=33$ s.

Answer: 33 seconds.

Raman is $14$ now, so he will be $24$ in 10 years.

Father in 10 years $=48$, so father now $=38$ and two years ago $=36$.

John $=\frac{36}{4}=9$.

Answer: 9 years.

Let Monty stay in for $12-m$ months.

$\frac{76000\times 12}{57000\times (12-m)}=2\Rightarrow 912000=114000(12-m)\Rightarrow 12-m=8\Rightarrow m=4$.

Answer: 4 months.

Let R's rate be $r$. Then Q $=3r$ and P $=6r$, so together $10r=\frac{1}{30}\Rightarrow r=\frac{1}{300}$.

P's rate $=\frac{6}{300}=\frac{1}{50}$, so P alone takes 50 days.

Check: $\frac{1}{50}+\frac{1}{100}+\frac{1}{300}=\frac{6+3+1}{300}=\frac{1}{30}$ ✓.

Upstream speed $=2$ km/h. Downstream speed $=1$ km in $\frac{1}{6}$ h $=6$ km/h.

Still-water speed $=\frac{2+6}{2}=4$ km/h.

Time for 5 km $=\frac{5}{4}$ h $=1$ h $15$ min.

Answer: 1 hr 15 min.

$25^{2.7}=5^{5.4}$, so the expression is $5^{5.4}\times 5^{4.2}÷5^{5.4}=5^{5.4+4.2-5.4}=5^{4.2}$.

Answer: $5^{4.2}$.

Work left to right: $18800÷470=40$, then $40÷20=2$.

Answer: 2.

First reduce to lowest terms: $\frac{2}{3},\frac{2}{3},\frac{8}{27}$.

$$\text{HCF}=\frac{\text{HCF}(2,2,8)}{\text{LCM}(3,3,27)}=\frac{2}{27}$$

Check: $\frac{2}{3}÷\frac{2}{27}=9$ and $\frac{8}{27}÷\frac{2}{27}=4$, both whole numbers. The next candidate up, $\frac{4}{27}$, fails because $\frac{2}{3}÷\frac{4}{27}=4.5$.

Answer: $\frac{2}{27}$.

For an AP, $2\log (2^x-1)=\log 2+\log (2^x+3)$, so $(2^x-1{)}^2=2(2^x+3)$.

Put $t=2^x$: $t^2-2t+1=2t+6\Rightarrow t^2-4t-5=0\Rightarrow t=5$ or $t=-1$.

Since $2^x>0$, $t=5$ and $x={\log }_{2}5$.

Check: the terms are $\log 2,\log 4,\log 8$, which are in AP ✓.

Answer: ${\log }_{2}5$.

Reading the net. The cross has five squares: a centre square and four side panels. Each side panel ends in a triangular flap: top flap black, right flap black, bottom flap white, left flap white. The bottom panel carries a dot. When folded, the four flaps meet to form the sixth face, so the black flaps belong to two neighbouring panels (top and right).

Testing the views (net drawn on the outside):

• (3): the triangular face is entirely white, but two of its four triangles must be black ✗.
• (4): the whole triangular face is black, but only two of its triangles are black ✗.
• (2): left half black, right half white, dot on the right wall. This fits: right wall = dot panel (bottom), left wall = right panel. Their flaps give black (right + top) on one side and white (bottom + left) on the other ✓.
• (1): shows a black triangle on a side wall, but no side panel in the net has shading ✗.

On this analysis only (2) matches, which isn't one of the choices. The key gives "1 and 2 only".

• The son of Manju's only brother is the brother of Raju's wife.
• So Raju's wife is a child of Manju's brother, which makes Manju's brother Raju's father-in-law.
• Manju is therefore the sister of Raju's father-in-law.

Mother-in-law would be the brother's wife, not his sister.

Answer: Sister of father-in-law.

A quiver is the container where arrows are kept. Likewise, money is kept in a bank.

• A coin is *made* at a mint, not stored there.
• Fear causes trembling (cause and effect).
• Music is made of sound (component).

Answer: Money : Bank.

*Humiliate* is a stronger degree of *embarrass*. Likewise, *exasperate* is a stronger degree of *annoy*.

• Enquire : Ask are near-synonyms of the same degree.
• Embezzle : Peculate are exact synonyms.
• Gamble : Investment are different activities, not a difference in degree.

Answer: Annoy : Exasperate.

• I: fruits that are trees are trees, and no tree is a flower. So I follows.
• II: fruits that are *not* trees could still be flowers. Nothing rules that out, so II does not follow.

Answer: Only conclusion I follows.

Chain: Cauliflower > Lady's finger > Cabbage, and Peas ≥ Cabbage.

• The chain directly gives Cauliflower > Cabbage.
• Peas are only known to be at least as tasty as cabbage. Their place relative to lady's finger or cauliflower is unknown, so A, B and C cannot be concluded.

Answer: Cauliflower is tastier than cabbage.

• II: "Some A are B" converts directly to "Some B are A", so II follows.
• I: the C's that are A need not be the A's that are B. Two particular premises give no conclusion, so I does not follow.

Answer: Only conclusion II follows.

Arrangement that matches the key. Seats 1–8 run west to east. A north-facing person's left is west; a south-facing person's left is east.

• Seat 1: R, faces south, White
• Seat 2: U, faces north, Red
• Seat 3: P, faces south, Yellow
• Seat 4: Q, faces south, Orange
• Seat 5: V, faces south, Black
• Seat 6: vacant
• Seat 7: T, faces north, Purple
• Seat 8: S, faces north, Blue

Checks:
- T (7) and P (3) have 3 seats between them, and neither is at an end.
- Persons between U and Q: 1 (P). Between Q and S: 2 (V, T).
- V's only neighbour is Q (seat 4), which is 3rd to the left of R (seat 1, facing south).
- U and T face north; Q faces south.
- Orange (Q) is 3rd to the left of T. T likes Purple.
- The ends, R and S, face opposite ways.
- Seats between Blue (8) and Yellow (3): 4. Persons between Red (2) and Black (5): 2. So $4=2\times 2$.
- White (R) is not next to the vacant seat, and Red (U) sits beside White.
- Yellow (P) faces south, opposite to T.
- The people beside the vacant seat, V and T, face the same way as their other neighbours, Q and S.

Caution: a systematic check shows the printed conditions also allow many other arrangements, so the puzzle does not force this one.

Solving. Purple is T (seat 7, facing north), so T's left is toward seat 1. Second to the left of T is seat 5, V.

V faces south, so V's left is toward seat 8. Third to the left of V is seat 8, S, who likes Blue.

Arrangement that matches the key. Seats 1–8 run west to east. A north-facing person's left is west; a south-facing person's left is east.

• Seat 1: R, faces south, White
• Seat 2: U, faces north, Red
• Seat 3: P, faces south, Yellow
• Seat 4: Q, faces south, Orange
• Seat 5: V, faces south, Black
• Seat 6: vacant
• Seat 7: T, faces north, Purple
• Seat 8: S, faces north, Blue

Checks:
- T (7) and P (3) have 3 seats between them, and neither is at an end.
- Persons between U and Q: 1 (P). Between Q and S: 2 (V, T).
- V's only neighbour is Q (seat 4), which is 3rd to the left of R (seat 1, facing south).
- U and T face north; Q faces south.
- Orange (Q) is 3rd to the left of T. T likes Purple.
- The ends, R and S, face opposite ways.
- Seats between Blue (8) and Yellow (3): 4. Persons between Red (2) and Black (5): 2. So $4=2\times 2$.
- White (R) is not next to the vacant seat, and Red (U) sits beside White.
- Yellow (P) faces south, opposite to T.
- The people beside the vacant seat, V and T, face the same way as their other neighbours, Q and S.

Caution: a systematic check shows the printed conditions also allow many other arrangements, so the puzzle does not force this one.

Solving. Red is U (seat 2) and the vacant seat is 6. Between them sit P, Q and V: three persons.

Arrangement that matches the key. Seats 1–8 run west to east. A north-facing person's left is west; a south-facing person's left is east.

• Seat 1: R, faces south, White
• Seat 2: U, faces north, Red
• Seat 3: P, faces south, Yellow
• Seat 4: Q, faces south, Orange
• Seat 5: V, faces south, Black
• Seat 6: vacant
• Seat 7: T, faces north, Purple
• Seat 8: S, faces north, Blue

Checks:
- T (7) and P (3) have 3 seats between them, and neither is at an end.
- Persons between U and Q: 1 (P). Between Q and S: 2 (V, T).
- V's only neighbour is Q (seat 4), which is 3rd to the left of R (seat 1, facing south).
- U and T face north; Q faces south.
- Orange (Q) is 3rd to the left of T. T likes Purple.
- The ends, R and S, face opposite ways.
- Seats between Blue (8) and Yellow (3): 4. Persons between Red (2) and Black (5): 2. So $4=2\times 2$.
- White (R) is not next to the vacant seat, and Red (U) sits beside White.
- Yellow (P) faces south, opposite to T.
- The people beside the vacant seat, V and T, face the same way as their other neighbours, Q and S.

Caution: a systematic check shows the printed conditions also allow many other arrangements, so the puzzle does not force this one.

Solving. The vacant seat (6) sits between V (5) and T (7). P (3) sits between U (2) and Q (4).

By the same relation, U (2) sits between R (1) and P (3).

Arrangement that matches the key. Seats 1–8 run west to east. A north-facing person's left is west; a south-facing person's left is east.

• Seat 1: R, faces south, White
• Seat 2: U, faces north, Red
• Seat 3: P, faces south, Yellow
• Seat 4: Q, faces south, Orange
• Seat 5: V, faces south, Black
• Seat 6: vacant
• Seat 7: T, faces north, Purple
• Seat 8: S, faces north, Blue

Checks:
- T (7) and P (3) have 3 seats between them, and neither is at an end.
- Persons between U and Q: 1 (P). Between Q and S: 2 (V, T).
- V's only neighbour is Q (seat 4), which is 3rd to the left of R (seat 1, facing south).
- U and T face north; Q faces south.
- Orange (Q) is 3rd to the left of T. T likes Purple.
- The ends, R and S, face opposite ways.
- Seats between Blue (8) and Yellow (3): 4. Persons between Red (2) and Black (5): 2. So $4=2\times 2$.
- White (R) is not next to the vacant seat, and Red (U) sits beside White.
- Yellow (P) faces south, opposite to T.
- The people beside the vacant seat, V and T, face the same way as their other neighbours, Q and S.

Caution: a systematic check shows the printed conditions also allow many other arrangements, so the puzzle does not force this one.

Solving. The ends are seat 1, R (White), and seat 8, S (Blue).

Arrangement that matches the key. Seats 1–8 run west to east. A north-facing person's left is west; a south-facing person's left is east.

• Seat 1: R, faces south, White
• Seat 2: U, faces north, Red
• Seat 3: P, faces south, Yellow
• Seat 4: Q, faces south, Orange
• Seat 5: V, faces south, Black
• Seat 6: vacant
• Seat 7: T, faces north, Purple
• Seat 8: S, faces north, Blue

Checks:
- T (7) and P (3) have 3 seats between them, and neither is at an end.
- Persons between U and Q: 1 (P). Between Q and S: 2 (V, T).
- V's only neighbour is Q (seat 4), which is 3rd to the left of R (seat 1, facing south).
- U and T face north; Q faces south.
- Orange (Q) is 3rd to the left of T. T likes Purple.
- The ends, R and S, face opposite ways.
- Seats between Blue (8) and Yellow (3): 4. Persons between Red (2) and Black (5): 2. So $4=2\times 2$.
- White (R) is not next to the vacant seat, and Red (U) sits beside White.
- Yellow (P) faces south, opposite to T.
- The people beside the vacant seat, V and T, face the same way as their other neighbours, Q and S.

Caution: a systematic check shows the printed conditions also allow many other arrangements, so the puzzle does not force this one.

Solving. Q is at seat 4 facing south, so Q's left runs toward seat 8. Second to the left of Q is seat 6, the vacant seat, now taken by N.

A mirror beside the text reverses the order of the characters and flips each one left to right. The result reads A 4 1 0 1 N I A R A T from left to right, with N, R, 4 and 1 drawn reversed.

• (4) has exactly that order with the reversed N and R ✓.
• (2) keeps the digits as 1014, so their order is not reversed ✗.
• (1) ends in a wrong letter shape in place of T ✗.
• (3) jumbles the letter order ✗.

Answer: (4).

Fixing the order. The blocks are EC (E immediately before C) and DB (D immediately before B), plus A. Neither D nor E may be 1st or 5th.

• If E or D starts a block that is placed first, the rule is broken. So A is first, followed by EC and DB in some order.
• A, E, C, D, B: A (1) and B (5) have three plays between them ✗.
• A, D, B, E, C: D is 2nd and E is 4th ✓, and exactly one play (D) sits between A and B ✓.

The order is A D B E C.

The first play is A.

Fixing the order. The blocks are EC (E immediately before C) and DB (D immediately before B), plus A. Neither D nor E may be 1st or 5th.

• If E or D starts a block that is placed first, the rule is broken. So A is first, followed by EC and DB in some order.
• A, E, C, D, B: A (1) and B (5) have three plays between them ✗.
• A, D, B, E, C: D is 2nd and E is 4th ✓, and exactly one play (D) sits between A and B ✓.

The order is A D B E C.

Answer: ADBEC.

Fixing the order. The blocks are EC (E immediately before C) and DB (D immediately before B), plus A. Neither D nor E may be 1st or 5th.

• If E or D starts a block that is placed first, the rule is broken. So A is first, followed by EC and DB in some order.
• A, E, C, D, B: A (1) and B (5) have three plays between them ✗.
• A, D, B, E, C: D is 2nd and E is 4th ✓, and exactly one play (D) sits between A and B ✓.

The order is A D B E C.

Wednesday is the 3rd day, so the play is B.

In a four-handed card game, partners sit opposite each other.

• I alone: D is opposite A, so A and D are partners and C's partner is B. Sufficient.
• II alone: B is on A's right and on D's left, so A and D are B's two neighbours. That puts A opposite D, and C must be opposite B, making them partners. Also sufficient.

Since each statement alone is sufficient, the logical choice is C (either I or II alone). The key gives A (I alone).

• I alone: the month's length (28, 29, 30 or 31 days) isn't given, so the 14th can't be fixed. Not sufficient.
• II alone: the 17th is a Saturday, so the 14th, three days earlier, is a Wednesday. Sufficient.

The logical answer is B (II alone). The key gives A.

Rule: write the word backwards, then move each letter forward by one.

SEQUENCE reversed is E C N E U Q E S. Adding one to each letter gives F D O F V R F T ✓.

CHILDREN reversed is N E R D L I H C. Adding one gives O F S E M J I D.

Answer: OFSEMJID.

The king of the jungle is the lion. In this code, Lion is called Goat.

Answer: Goat.

• I: calling age 5 *desirable* presumes the child is developmentally ready then. Implicit.
• II: the statement is about what's desirable, not about admission rules. Nothing suggests schools refuse older children. Not implicit.

Answer: Only I is implicit.

• I contradicts the speaker, who expects the allowance to *bring* punctuality. Not implicit.
• II: the speaker pairs a reward (the allowance) with a disciplinary goal (punctuality). This assumes the two work together. Implicit.

Answer: Only II is implicit.

Put the start at $(0,0)$.

• 5 km south takes him to $(0,-5)$.
• Turning right while facing south means facing west. 3 km takes him to $(-3,-5)$.
• Turning left while facing west means facing south. 5 km takes him to $(-3,-10)$.

He ends west and south of the start: South-West.

The series is two interleaved sequences:

• Odd positions: 2, 10, 18, 26 (adding 8 each time)
• Even positions: 7, 22, 37, 52 (adding 15 each time)

Answer: 52.

Each term is 35 less than the one before. $439-35=404$.

Answer: 404.

Every pair of subjects overlaps, and some students take all three. That needs three mutually overlapping circles with a common central region, which is diagram (A).

• (B) is a chain, so the outer circles never meet.
• (C) has separate subsets.
• (D) has nested circles.

Answer: A.

• I: spinners scoring 80% of the runs says nothing about how many players are spinners. Doesn't follow.
• II: nothing is said about the batting order. Doesn't follow.

Logically, neither conclusion follows (option D). The key gives A.

• I: the statement is the classic expression that change is inevitable, the way of nature. Follows.
• II: the old gave way to the new, but nothing says old ideas should be thrown out *because* they are old. Doesn't follow.

Answer: Only I follows.

Step VI: 95, 84, 78, 76, 67, 38, 13, 18.

Between the 1st (95) and the 3rd (78) is 84.

Step IV: 95, 84, 78, 76, 13, 67, 18, 38.

The number after 18 is 38.

Step VII: 95, 84, 78, 76, 67, 38, 18, 13.

Between the 3rd (78) and the 5th (67) is 76.

Let the present ages be $T$ and $G$.

• 16 years ago: $G-16=8(T-16)$, so $G=8T-112$.
• In 8 years: $G+8=3(T+8)$, so $8T-104=3T+24$, giving $T=25.6$ and $G=92.8$.

8 years ago: $17.6:84.8=176:848=11:53$.

Check: $11:53$ scaled by $1.6$ is $17.6:84.8$ ✓.

Answer: 11 : 53.

2008 is a leap year, so February has 29 days.

From 2 p.m. Feb 25 to 2 p.m. Mar 3 is 7 days. Stopping at 9 a.m. instead gives 6 days 19 hours $=163$ h $=9780$ min shown on the clock.

The clock shows $1436$ min for every $1440$ real minutes, so:
$$\text{real time}=9780\times \frac{1440}{1436}=9780\times \frac{360}{359}$$

Minutes to add $=9780\left(\frac{360}{359}-1\right)=\frac{9780}{359}$.

Answer: 9780/359.

From $U\ge R\ge S$ we get $U\ge S$. So I ($U>S$) is not definite.

$R\ge S$ means exactly one of $R=S$ or $R>S$ holds, and neither alone is certain. II and III form a complementary pair.

Answer: Either II or III is true.

• I: $U<S$ and $S\ge V$ give no relation between U and V. Not definite.
• II: $B<A<P=C$, so $B<C$ ✓.
• III: $Q>C\ge D$, so $Q>D$ ✓.

Answer: Only II and III are true.

• I: the facts say *some* chairs have beans, not which ones. Children's chairs are never mentioned.
• II: *most* stuffed toys have beans, not all, so not all tigers necessarily do.
• III: monkeys are never mentioned.

Answer: None of the statements is a known fact.

Boys in the line $=12+4-1=15$.

Boys to add $=28-15=13$.

Answer: 13.

Read "between" as immediately between.

• I gives the block E B C (or C B E).
• III gives the block A D E (or E D A).
• The blocks share E, which must end both of them. The row is A D E B C or its reverse, and E is in the middle either way.

So I and III together already answer the question (option C). II only fixes the direction, which isn't needed for the middle. If "between" meant not necessarily adjacent, even all three wouldn't be enough (for example E B D C A). So option D is never the minimal set.

Chain the three statements:

• II: Gopal's brother was born in 1982.
• III: Gopal's brother is eight years younger than Gopal, so Gopal was born in 1974.
• I: Sanjay is six years older than Gopal, so Sanjay was born in 1968.

Take away any one statement and the chain breaks, so all three are needed.

Answer: I, II and III together.

Compare positions: I M T I T J U becomes T M I I U J T.

Rule: swap letters 1 and 3, and swap letters 5 and 7. Letters 2, 4 and 6 stay put.

T E M R E M P becomes M E T R P M E.

Answer: METRPME.

*Abominate* means to loathe or detest. Its opposite is to hold in high regard: esteem.

• *Gratify* means to please.
• *Console* means to comfort.
• *Support* means to back up.

None of these three is the opposite of hating.

Answer: esteem.

*Abhor* means to hate intensely. The exact opposite is greatly admire.

*Inspire*, *credit* and *improve* have nothing to do with hating or liking.

Answer: greatly admire.

*Enormous* means extremely large. Its exact opposite is extremely small: tiny (option C).

*Average* is only a middle point between large and small, not the opposite. *Soft* and *weak* describe texture and strength, not size.

The key gives *Average*, but the true antonym is *Tiny*.

*Impromptu* means done without preparation, on the spur of the moment. Offhand (unrehearsed) has the same meaning.

*Unimportant*, *unreal* and *effective* don't mean unprepared.

Answer: Offhand.

*Wretched* means miserable, pitiable, in a poor or unhappy state. Poor, in the sense of unfortunate or pitiable, is the closest match.

• *Foolish* and *insane* concern the mind.
• *Strained* means tense.

Answer: Poor.

*Erroneous* means containing errors, i.e. incorrect.

• *Digressive* means wandering off topic.
• *Confused* means unclear.
• *Impenetrable* means impossible to get through or understand.

Answer: Incorrect.

The passage says that in his dream Harold found himself in a land where slug-like creatures lived on people's bodies. The setting is a strange land, not his village, house or office. The office is only mentioned as his real-life workplace.

Answer: in a different land.

The creatures are first described as small slug-like animals with slimy tentacles, so those two phrases define them.

• *Security bug* is the later metaphor for what they stood for, not a description of them.
• *Fearful dream* describes the dream, not the creatures.

Answer: Slimy tentacles and slug-like animals.

The "security bug" stands for Harold's reliance on a secure job, letting society feed on him in exchange for support in retirement. Throwing it off, and then taking up freelance work, means he overcame his need for security.

• He killed no real bugs.
• He didn't stop valuing dreams; the dream is what changed him.
• He stopped tolerating the arrangement, rather than starting to.

Answer: Harold succeeded in overcoming the need for security.

The fear came from the dream's content: loathsome slimy creatures living and feeding on people's bodies, and then discovering that he was covered with them too. That is what made him wake up screaming.

• A: the confrontation with reality is what the dream *meant* on reflection, not why it was frightening.
• B: the creatures were slug-like, not snakes.
• C: the elephants were the harmless transport system.

Answer: D.

*But* signals a contrast with the stern appearance. Only a heart full of compassion gives that contrast.

• "Looks very serious" and "people are afraid of him" continue the same idea.
• "Uncompromising on tasks" also goes along with an unsmiling manner rather than against it.

Answer: A.

The purpose is to *raise profit*. Only extra work for no extra pay does that.

• Paid leave and new welfare schemes add costs.
• Overtime "with lucrative compensation" also adds costs.

Answer: D.

*Manners and morals* is a natural pairing about behaviour and values. A subject that keeps coming back in literature is a recurrent theme.

• *Style, motif*: "manners and style" is redundant.
• *Morals, story*: "a recurrent story" is unidiomatic.
• *Wealth, theme*: "manners and wealth" is an odd pairing, since the nouveau riche are already defined by their wealth.

Answer: morals, theme.

• S1 → Q: "The future beckons" leads to the question of where we go and what our endeavour will be.
• Q → S: S answers it: *It* (the endeavour) will be to bring freedom and opportunity.
• S → R: R adds a further task with "also": fight poverty, ignorance and disease.
• R → P: P sums up, "In fact we have hard work ahead".
• P → S6: hard work leads naturally to "no resting … till we redeem our pledge".

Answer: QSRP.

• S1 → S: having halted on the road, he describes where the elephant was, 8 yards away.
• S → P: seeing the elephant, he knew he should not shoot it.
• P → Q: Q gives the reason: shooting a working elephant is a serious matter.
• Q → R: R adds that the elephant's *must* was already passing off, so it was no longer dangerous.
• R → S6: so he decided to watch it for a while and go home.

Answer: SPQR.

• S1 → Q: the noise was meant to show everyone that something was wrong ("That" refers to the noise).
• Q → P: *But* people did not take it seriously.
• P → R: ignoring it was a dangerous thing to do.
• R → S: "For, within minutes the ship began to sink" explains why it was dangerous.
• S → S6: nearly 200 lives were lost.

Answer: QPRS.

The decimal system is the number system with base ten. The binary system is the one with base two.

• A septet and a quartet are *groups* of seven and four performers, not number systems.
• A quince is a fruit.

Answer: two : binary.

A cat's foot is a paw; a horse's foot is a hoof.

• A lion also has paws.
• An elephant's foot is not called a hoof.
• The horse is the standard example of a hoofed animal.

Answer: Horse.

To *make a clean breast of* something is to confess it fully and frankly, holding nothing back.

The other options (gaining prominence, self-praise, destroying something early) have no link to the idiom.

Answer: To confess without reserve.

Someone with *an axe to grind* has a private motive or selfish interest behind what they do or say.

It doesn't mean failing to arouse interest, having no result, or working for both sides.

Answer: A private end to serve.

The subject *I* takes was in the past tense: "When I was a newspaper journalist". *Were* after "I" is only used in hypotheticals ("If I were…").

The rest of the sentence is correct.

Answer: A.

The key finds the error in part A. The reasoning is tense agreement: the clause "I know about my friend" is in the present, so the main verb should be "I can count on one hand".

However, *could* is also used in natural English as a hypothetical ("I could count them on one hand if asked"). On that reading the sentence is acceptable and "no error" (D) is defensible.

The statement is only about both spouses working, not about who earns more. So "Wives earn more than their husbands" cannot be drawn.

• B, C and D all follow: working spouses were less common before, family income used to be lower, and more couples work now.

Answer: A.

• Argument 1 ties directly to the question's premise: the weaker nation threatens world peace, and force is justified to keep that peace. It is relevant and forceful.
• Argument 2 is an absolute ("never"). Such sweeping claims are treated as weak because they ignore the specific situation.

Answer: Only 1.

Capital is the chief city, or seat of government, of a country.

• *Capitol* is a legislative building, such as the US Capitol.
• *Capetol* is not a word.
• *Capitalisation* is unrelated.

Answer: Capital.

Farther is used for physical distance ("the farther shore").

• *Further* is preferred for figurative or additional degree ("further discussion").
• *Father* is a parent.
• *Ferther* is not a word.

Answer: Farther.

*Room* here means available space, and "no more room for you" is idiomatic and correct.

• "No more seat" is ungrammatical (it would need "no more seats").
• "Space" and "accommodation" are no better.

Answer: No improvement.

A third conditional (an unreal past) uses had + past participle in the *if*-clause: "If he had tried, he would have succeeded." *Would have* must not appear in the *if*-clause.

"Is tried" and "was tried" are passive and wrong in meaning.

Answer: had tried.

*Conflate* means to combine two or more things into one.

• (I) merges two processes into one ✓.
• (II) uses it as an adjective meaning "separate", which is wrong.
• (III) uses it to mean "separate the dogs", which is the opposite of its meaning.

Answer: Only (I).

*Rare* means uncommon.

• (I) a rare edition ✓.
• (II) a rare form of bone cancer ✓.
• (III) confuses *rare* with rear (the back of the line) ✗.

Answer: Both (I) and (II).

The passage describes the NFL revising its safety rules: sideline concussion assessments, stricter penalties for helmet-to-helmet hits, and whistling the ball dead when a helmet comes off. It even calls these efforts laudable. So "NFL officials have done little" is false.

The other options are supported:
- Proper tackling form is urged.
- Autopsies found CTE in every player examined who had committed suicide.
- Glorifying brutal hits rewards and encourages such behaviour.

Answer: C.

• I is supported: professional rules "cannot protect amateur players", and youth leagues should adopt stricter rules. The standards are not applied consistently across levels.
• II is supported: young athletes "should be educated" about the dangers of head injuries.
• III is also stated in the final paragraph: when the media glorifies violent hits, amateurs become more likely to imitate their favourite NFL athletes.

The passage supports all three, but no option includes all three. The key picks "I and II only".

The author says the NFL's efforts are *laudable* and that "we should commend every attempt". *Laudable* means deserving praise: praiseworthy.

• *Ineffectual* means useless.
• *Memorable* means worth remembering.
• *Audacious* means bold.

None of these fits.

Answer: praiseworthy.

The last paragraph is full of firm demands: the media "should engage in a collective effort to cease glorifying brutal plays", fans "should stop expecting…", and players "must not be encouraged…". That is an insistent tone.

It isn't remorseful (no regret), hopeless (it proposes remedies) or perplexed (it isn't confused).

Answer: insistent.

*Exalts* here means glorifies or plays up dangerous behaviour. The opposite idea is to play it down or lessen it, and mitigates (lessens) is the closest.

• *Venerates* is a synonym of exalts.
• *Mollifies* means soothes a person.
• *Expedites* means speeds up.

Answer: mitigates.

The author says the most important factor "lies … with the media and fans" and calls on media professionals to act together to stop glorifying brutal plays. The emphasis is on the media's responsibility.

• Entertainment value is what the author criticises, not what they stress.
• "Senselessness" and "danger" aren't how the media itself is characterised.

Answer: responsibility.

The highlight shows with weekly countdowns of the hardest hits are listed with the replays and admiring commentary. They are examples of how the media glorifies violence (I).

II (amateur imitation) and III (professionals being rewarded) are *consequences* the author draws in the next sentence. They are not what the highlight shows exemplify.

Answer: I only.

The towns were *overthrown* by the storm, meaning knocked down or destroyed: devastated.

*Surrendered*, *capitulated* and *yielded* are the political sense of *overthrown* (giving up power), which doesn't fit towns hit by a storm.

Answer: devastated.

Whole districts *levelled* by waterspouts means flattened to the ground: razed.

*Marked* and *spread* don't fit, and *raised* is the opposite of flattened.

Answer: razed.

The destruction listed (towns overthrown, forests uprooted, thousands killed) is the mark of the storm's violent anger: its fury.

• *Velocity* is merely speed.
• *Reward* and *benevolence* contradict the destruction.

Answer: Fury.