JIPMAT 2025 · Question Paper & Solutions

Split each mixed number into its whole part and its fraction.

$$6\times 999+\frac{1+2+3+4+5+6}{7}=5994+\frac{21}{7}=5994+3=5997$$

They contradict each other when exactly one of them tells the truth.

$$P=0.75\times 0.20+0.25\times 0.80=0.15+0.20=0.35$$

So they contradict each other in 35% of cases.

For $x^2+bx+c=0$, the sum of roots is $-b$ and the product is $c$.

• Amit got only the constant wrong, so his $x$-coefficient is right: sum of roots $=4+3=7$.
• Alok got only the $x$-coefficient wrong, so his constant is right: product of roots $=3\times 2=6$.

The correct equation is $x^2-7x+6=0\Rightarrow (x-6)(x-1)=0$, so the roots are $6$ and $1$.

An even number must end in 2 or 4, so there are 2 choices for the units digit.

The hundreds and tens digits come from the remaining 4 digits in order: $4\times 3=12$ ways.

Total $=2\times 12=24$.

Cutting a 2 cm square from each corner removes 4 cm from each dimension. The flaps fold up to a height of 2 cm.

$$V=(25-4)(20-4)(2)=21\times 16\times 2=672{\text{ cm}}^3$$

Write every term as a power of 2: $4^{64}=2^{128}$, $8^{42}=2^{126}$.

$$2^{122}+2^{128}+2^{126}+2^{128}+2^{130}=2^{122}(1+2^6+2^4+2^6+2^8)=2^{122}(1+64+16+64+256)=2^{122}\times 401$$

$401$ is prime, so as printed the sum is not divisible by 3, 5, 7 or 11.

This is a well-known question whose second term is $4^{62}$, not $4^{64}$. With $4^{62}=2^{124}$:

$$2^{122}(1+2^2+2^4+2^6+2^8)=2^{122}(1+4+16+64+256)=2^{122}\times 341=2^{122}\times 11\times 31$$

So the intended expression is divisible by $11$, which is the official answer.

Equal fencing cost means equal perimeters.

$$\frac{\sqrt{3}}{4}{a}^2=16\sqrt{3}\Rightarrow a^2=64\Rightarrow a=8\text{ m}$$

Perimeter of the triangle $=3\times 8=24$ m, so the square has side $24/4=6$ m.

Diagonal $=6\sqrt{2}$ m.

Compounded half-yearly, the rate is 5% per period for 3 periods.

$$\text{CI}=P\left[(1.05{)}^3-1\right]=0.157625P,\text{SI}=P\times 0.10\times 1.5=0.15P$$

$$0.157625P-0.15P=0.007625P=183\Rightarrow P=\frac{183}{0.007625}=24000$$

A. Let $2^x=3^y=6^{-z}=k$. Then $2=k^{1/x}$, $3=k^{1/y}$ and $6=k^{-1/z}$. Since $2\times 3=6$:

$$k^{1/x+1/y}=k^{-1/z}\Rightarrow \frac{1}{x}+\frac{1}{y}+\frac{1}{z}=0$$

So A is correct.

B. $(243{)}^{0.16}=3^{5\times 0.16}=3^{0.8}$ and $9^{0.1}=3^{0.2}$. The product is $3^1=3$, not $0.3$. So B is false.

C. $0.00239+0.000715=0.003105=3.105\times 10^{-3}$, so $p=-3$. C is correct.

Answer: A and C only.

The same amount is spent each year, so the average price is the harmonic mean of the prices.

Litres bought $=\frac{6460}{17}+\frac{6460}{19}+\frac{6460}{20}=380+340+323=1043$.

$$\text{Average}=\frac{3\times 6460}{1043}=\frac{19380}{1043}\approx ₹18.58$$

Interest of ₹2,000 on ₹6,000 is $\frac{1}{3}$ of the principal, earned in 4 years.

For ₹525 to become ₹700, the interest is ₹175, which is also $\frac{175}{525}=\frac{1}{3}$ of the principal.

At the same rate it takes the same time: 4 years. (Check: rate $=\frac{2000\times 100}{6000\times 4}=8\frac{1}{3}\%$; $\frac{33\frac{1}{3}}{8\frac{1}{3}}=4$.)

Compute each new value and the increase.

• A: $\frac{2}{3}\to \frac{7}{8}$, change $=\frac{21-16}{24}=\frac{5}{24}\approx 0.208$
• B: $\frac{3}{4}\to \frac{8}{9}$, change $=\frac{32-27}{36}=\frac{5}{36}\approx 0.139$
• C: $\frac{4}{5}\to \frac{9}{10}$, change $=\frac{9-8}{10}=0.100$
• D: $\frac{5}{6}\to \frac{10}{11}$, change $=\frac{60-55}{66}=\frac{5}{66}\approx 0.076$

Ascending: $D<C<B<A$.

$$\text{CP}=\frac{1965}{0.75}=2620$$

At SP ₹3,013 the profit is $3013-2620=393$.

$$\text{Profit}\%=\frac{393}{2620}\times 100=15\%$$

A. Curved surface area of a cylinder is $2\pi rh$, which is proportional to $h$. Doubling $h$ doubles it. Correct.

B. Total surface area of a solid hemisphere is $3\pi r^2$. Doubling $r$ multiplies it by 4. Correct.

C. Equal bases and equal heights mean the cone has radius $r$ and height $r$ (the hemisphere's height is its radius). The cone's slant height is $l=\sqrt{r^2+r^2}=r\sqrt{2}$.

$$\frac{\text{CSA of hemisphere}}{\text{CSA of cone}}=\frac{2\pi r^2}{\pi r\cdot r\sqrt{2}}=\frac{2}{\sqrt{2}}=\sqrt{2}:1$$

This matches statement C, so by direct calculation C is also correct, which would make the answer "A, B and C" (option D).

The official key gives "A and B only". That is only possible if the ratio is read in the reverse order (cone to hemisphere, $1:\sqrt{2}$).

Let $a=4.53-3.07$, $b=3.07-2.15$, $c=2.15-4.53$. Then $a+b+c=0$.

The expression is

$$\frac{a^2}{bc}+\frac{b^2}{ca}+\frac{c^2}{ab}=\frac{a^3+b^3+c^3}{abc}$$

When $a+b+c=0$, $a^3+b^3+c^3=3abc$. So the value is $\frac{3abc}{abc}=3$.

Evaluate both lists.

• A: $0.60\times 264=158.4$
• B: $0.60\times 0.28\times 240=40.32$
• C: $0.85\times 485.5=412.675$
• D: $5.50\times 250=1375$

• I: $0.20\times 6875=1375$
• II: $0.50\times 825.35=412.675$
• III: $0.15\times 1056=158.4$
• IV: $0.084\times 480=40.32$

So A-III, B-IV, C-II, D-I.

Work from the right, using $\sqrt{2+t}\cdot \sqrt{2-t}=\sqrt{4-t^2}$.

Take the last two factors with $t=\sqrt{2+\sqrt{2+\sqrt{3}}}$:

$$\sqrt{4-\left(2+\sqrt{2+\sqrt{3}}\right)}=\sqrt{2-\sqrt{2+\sqrt{3}}}$$

Multiply by the second factor $\sqrt{2+\sqrt{2+\sqrt{3}}}$:

$$\sqrt{4-(2+\sqrt{3})}=\sqrt{2-\sqrt{3}}$$

Multiply by the first factor $\sqrt{2+\sqrt{3}}$:

$$\sqrt{4-3}=1$$

They change together again after the LCM of the intervals.

$48=2^4\cdot 3$, $72=2^3\cdot 3^2$, $108=2^2\cdot 3^3$, so $\text{LCM}=2^4\cdot 3^3=432$ s $=7$ min $12$ s.

$08:20:00+7:12=08:27:12$.

$$x^3+y^3=(x+y{)}^3-3xy(x+y)\Rightarrow 468=1728-36xy\Rightarrow xy=35$$

$$x^2+y^2=(x+y{)}^2-2xy=144-70=74$$

$$x^4+y^4=(x^2+y^2{)}^2-2(xy{)}^2=5476-2450=3026$$

Let $C=15k$. Then $B=\frac{2}{3}C=10k$ and $A=\frac{4}{5}B=8k$.

$$8k+10k+15k=33k=11550\Rightarrow k=350$$

$C-A=7k=7\times 350=₹2450$.

Take $B=100$. Then $A=60$ and $C=0.40\times 160=64$.

$$\frac{B-A}{C}\times 100=\frac{40}{64}\times 100=62.5\%$$

• A: side $=6$, perimeter $=24$ cm
• B: $3\times 9=27$ cm
• C: breadth $=40/10=4$, perimeter $=2(10+4)=28$ cm
• D: circumference $=2\pi \times 4=8\pi \approx 25.13$ cm

Descending: $C(28)>B(27)>D(25.13)>A(24)$.

Equal volumes give the rod's radius $r$:

$$\frac{4}{3}\pi (21{)}^3=\pi r^2(28)\Rightarrow r^2=\frac{4\times 9261}{3\times 28}=441\Rightarrow r=21\text{ cm}$$

$$\text{TSA of rod}=2\pi r(r+h)=2\pi (21)(49)=2058\pi ,\text{TSA of sphere}=4\pi (21{)}^2=1764\pi$$

$$2058:1764=7:6$$

Fractions: A is $\frac{3}{7}$ copper and $\frac{4}{7}$ zinc; B is $\frac{5}{14}$ copper and $\frac{9}{14}$ zinc.

Take 2 units of A and 3 units of B.

$$\text{Copper}=2\cdot \frac{3}{7}+3\cdot \frac{5}{14}=\frac{12+15}{14}=\frac{27}{14},\text{Zinc}=2\cdot \frac{4}{7}+3\cdot \frac{9}{14}=\frac{16+27}{14}=\frac{43}{14}$$

Ratio $=27:43$.

Sorted data (15 values): 8, 9, 13, 13, 14, 14, 14, 14, 15, 15, 15, 16, 16, 17, 17.

The median is the 8th value, so $x=14$.

The printed stem leaves out which number is replaced. In the standard version of this question it is 8. Replacing 8 with 18 gives:

9, 13, 13, 14, 14, 14, 14, 15, 15, 15, 16, 16, 17, 17, 18

Now the 8th value is 15, so $y=15$ and $x+y=29$.

(Replacing any value of 14 or less gives the same $y=15$. Replacing a value of 15 or more would leave the median at 14.)

Horizontal distances from the base:

$$d_1=\frac{100}{\tan 30^{\circ }}=100\sqrt{3}\approx 173.2\text{ m},d_2=\frac{100}{\tan 45^{\circ }}=100\text{ m}$$

The ships are on opposite sides, so the distance is $173.2+100\approx 273$ m.

Let CP of B $=b$, so CP of A $=b+200$.

$$0.90(b+200)+1.25b=1.04(2b+200)$$

$$2.15b+180=2.08b+208\Rightarrow 0.07b=28\Rightarrow b=400$$

Total work $=40\times 48=1920$ man-days.

After $x$ days there are $64+32=96$ men working for $\frac{50}{3}$ days.

$$64x+96\times \frac{50}{3}=1920\Rightarrow 64x+1600=1920\Rightarrow x=5$$

From $a+b=2c$: $a-c=\frac{a-b}{2}$ and $b-c=\frac{b-a}{2}$.

As printed:

$$\frac{a}{a-c}+\frac{b}{b-a}=\frac{2a}{a-b}-\frac{b}{a-b}=\frac{2a-b}{a-b}$$

This depends on $a$ and $b$, so it is not a constant. For example, $a=2,b=0$ gives 2, but $a=3,b=1$ gives 2.5.

Standard version (second denominator $b-c$):

$$\frac{a}{a-c}+\frac{b}{b-c}=\frac{2a}{a-b}+\frac{2b}{b-a}=\frac{2(a-b)}{a-b}=2$$

The official answer 2 matches this version.

The number is $\text{LCM}+5$.

$8=2^3$, $12=2^2\cdot 3$, $22=2\cdot 11$, $28=2^2\cdot 7$, $35=5\cdot 7$.

As printed: $\text{LCM}=2^3\cdot 3\cdot 5\cdot 7\cdot 11=9240$, so $x=9245$ with digit sum $9+2+4+5=20$. That is not among the options.

Without 35: $\text{LCM}(8,12,22,28)=2^3\cdot 3\cdot 7\cdot 11=1848$, so $x=1853$ with digit sum $1+8+5+3=17$. This matches the official key, which suggests the divisor 35 was not part of the intended question.

$$\sin \theta +\cos \theta =3\sin \theta -3\cos \theta \Rightarrow 2\sin \theta =4\cos \theta \Rightarrow \tan \theta =2$$

$${\sin }^4\theta -{\cos }^4\theta =({\sin }^2\theta -{\cos }^2\theta )({\sin }^2\theta +{\cos }^2\theta )={\sin }^2\theta -{\cos }^2\theta$$

With $\tan \theta =2$: ${\sin }^2\theta =\frac{4}{5}$ and ${\cos }^2\theta =\frac{1}{5}$. So the value is $\frac{4}{5}-\frac{1}{5}=\frac{3}{5}$.

Choose 2 of the 4 children and 2 of the 5 adults.

$$P=\frac{\left(\genfrac{}{}{0ex}{}{4}{2}\right)\left(\genfrac{}{}{0ex}{}{5}{2}\right)}{\left(\genfrac{}{}{0ex}{}{9}{4}\right)}=\frac{6\times 10}{126}=\frac{60}{126}=\frac{10}{21}$$

Let $Q=q$ and $S=2q$. Then $P=q+140$ and $R=2q+80$.

$$\frac{q+140}{2q+80}=\frac{2}{3}\Rightarrow 3q+420=4q+160\Rightarrow q=260$$

So $P=400$, $Q=260$, $R=600$, $S=520$.

Check: $400:600=2:3$ and $260:520=1:2$. Option A fails because $260:320\ne 2:3$.

The man's brother's father is the man's own father.

"The only son of my grandfather" is the woman's father, since he is her grandfather's only son.

So the man and the woman have the same father, which makes the woman his sister.

• Mother, aunt and grandmother are wrong because each would need a different generation, but they share the same father.

Count the upright and the tilted shapes separately.

Upright (axis-aligned) part. The central square is cut by a vertical and a horizontal midline into a $2\times 2$ grid.

• Rectangles in a $2\times 2$ grid: $\left(\genfrac{}{}{0ex}{}{3}{2}\right)\times \left(\genfrac{}{}{0ex}{}{3}{2}\right)=9$
• Of these, 5 are squares (4 small and 1 big) and 4 are non-square ($1\times 2$ or $2\times 1$).

Tilted part. The diamond joining the midpoints and the four corner tilted squares have collinear edges. Together they form a "plus" of 5 equal tilted cells: a line of 3 cells in each diagonal direction, sharing the diamond in the middle.

• A line of 3 cells has $3+2+1=6$ rectangles. Two such lines share the centre cell, so there are $6+6-1=11$ tilted rectangles.
• Of these, 5 are squares (the cells) and 6 are non-square.

Total $=9+11=20$ (10 squares and 10 non-square rectangles).

Genuine and authentic are synonyms.

A mirage is an optical illusion: something that appears real but is not. So "illusion" is its synonym.

• Reflection and image are related ideas, but they are real optical effects, not something false.
• Hideout is unrelated.

Each term equals the previous term minus that term's first three digits (the term divided by 10, dropping the decimal).

• $5824-582=5242$
• $5242-524=4718$
• $4718-471=4247$
• $4247-424=3823$

So the missing term is $4718$.

I. Cars are inside bottles, and bottles are inside windows, so all cars are windows. Follows.

II. Dolls and cars are both inside windows, but nothing links dolls to cars. They may not overlap. Does not follow.

III. "All cars are windows" converts to "Some windows are cars". Follows.

So only I and III follow.

Name the outer numbers top ($t$), left ($l$), right ($r$) and bottom ($b$).

B: centre $=t\cdot b+l\cdot r$. Check: $2\cdot 5+1\cdot 3=13$ and $3\cdot 4+1\cdot 5=17$. Missing: $4\cdot 6+1\cdot 5=29$, so B-III.

C: centre $=(l-r)(t-b)$. Check: $(15-8)(10-8)=14$ and $(9-5)(8-6)=8$. Missing: $(11-8)(6-4)=6$, so C-I.

D: centre $=l\cdot r-t\cdot b$. Check: $7\cdot 4-5\cdot 3=13$ and $8\cdot 4-9\cdot 3=5$. Missing: $9\cdot 4-8\cdot 3=12$, so D-II.

A: centre $=\frac{t\cdot b\cdot r}{2}$. Check: $\frac{3\cdot 2\cdot 4}{2}=12$ and $\frac{6\cdot 3\cdot 2}{2}=18$. Missing: $\frac{2\cdot 9\cdot 2}{2}=18$, so A-IV.

Option D is also the only option with C-I.

List the faces seen next to 3 in the views:

• View 1 (2, 3, 6): 3 is adjacent to 2 and 6.
• View 3 (6, 3, 1): 3 is adjacent to 6 and 1.
• View 4 (3, 5, 1): 3 is adjacent to 5 and 1.

So 3 is adjacent to 1, 2, 5 and 6. The only number left is 4, which must be opposite 3.

• A: First letters G, K, N, R follow $+4,+3,+4$, so the next is $+3$: U. Second letters J, M, Q, T follow $+3,+4,+3$, so the next is $+4$: X. Answer UX (III).
• B: First letters A, B, C give D. Second letters I, J, K give L. Answer DL (IV).
• C: First letters A, C, F follow $+2,+3$, so the next is $+4$: J. Second letters Z, X, U follow $-2,-3$, so the next is $-4$: Q. Answer JQ (II).
• D: First letters A, G, M follow $+6$: S. Second letters Z, T, N follow $-6$: H. Answer SH (I).

So A-III, B-IV, C-II, D-I.

Wheat, pea and mustard are all rabi crops, sown in winter and harvested in spring.

Rice is a kharif crop, sown with the monsoon. So rice is the odd one out.

Grouping by "cereal" does not give a single odd one: wheat and rice are both cereals, while pea is a pulse and mustard is an oilseed.

Replace each symbol and evaluate using BODMAS.

• A: $6\times 20÷12-7÷1=10-7=3\ne 38$
• B: $6÷20-12+7\times 1=0.3-12+7=-4.7\ne 57$
• C: $6\times 20÷12-7+1=10-7+1=4\ne 62$
• D: $6-20+12\times 7÷1=6-20+84=70$

The printed option D ends in "+ 70", where "= 70" is clearly meant. With that reading, D is the only correct equation.

The shifts alternate $-2,+2$: R→P ($-2$), O→Q ($+2$), A→Y ($-2$), S→U ($+2$), T→R ($-2$).

Apply the same pattern to SLOPPY:

S→Q ($-2$), L→N ($+2$), O→M ($-2$), P→R ($+2$), P→N ($-2$), Y→A ($+2$, wrapping round).

Code: QNMRNA.

• D is the father of A and the grandfather of F, so A is F's parent.
• B is the brother of E's husband. The only person who can be D's son and have a brother in the group is A, so A is E's husband and B is A's brother. That makes E the mother of C and F.
• There are two fathers (D and A) and one mother (E).
• There are three brothers: A and B are two. The third must be F, who is C's brother. So F is male, and C is his sister.
• Males: D, A, B and F, which makes four.

Checking the statements:

• A. E is the mother: true.
• B. A is E's husband: true.
• C. Four males: true.
• D. F is D's son: false, because F is D's grandson.

So A, B and C only.

Read the List II diagrams in order: I = one circle inside another plus a separate circle; II = three separate circles; III = two separate circles inside a big circle; IV = three nested circles.

• Hospital, Nurse, Patient: nurses and patients are both found in a hospital but are distinct groups, so III.
• Polygons ⊃ Quadrilaterals ⊃ Rhombus: fully nested, so IV.
• Engine, Piston, Wheel: a piston is part of an engine; a wheel is separate, so I.
• Colour, Cloth, Merchant: three unrelated categories, so II.

Answer: A-III, B-IV, C-I, D-II.

Take the start as $(0,0)$ with east as $+x$.

• 75 m east: $(75,0)$
• Left turn, now facing north, 25 m: $(75,25)$
• Left turn, now facing west, 40 m: $(35,25)$
• Left turn, now facing south, 25 m: $(35,0)$

He is 35 m east of the starting point.

When people face the centre, "left" means clockwise. Number the seats clockwise from 0.

• Put F at 0. C is to F's immediate left, so C is at 1. G is second to the left of C, so G is at 3.
• A is third to the left of E, so $A=E+3$. The free seats are 2, 4, 5 and 6.
• If $E=2$, then $A=5$. But E's neighbours would be C and G, so B cannot sit next to E. Rejected.
• $E=4$ gives $A=0$ and $E=5$ gives $A=1$, but both seats are taken.
• If $E=6$, then $A=2$. B sits between D and E, so $B=5$ and $D=4$. This works.

Clockwise order: F, C, A, G, D, B, E.

"Right" is anticlockwise, so the person immediately right of G (seat 3) is at seat 2: A.

• Ved's wife dislikes Sushma and Vimla, so she is Kusum or Poonam. Kusum does not care for Ved, so Ved's wife is Poonam.
• Prem's wife is friendly with Sushma, so she is not Sushma. Sushma does not like Raj, so she is not Raj's wife either. So Sushma is Ashok's wife.
• Prem's wife is friendly with Sushma, and Sushma's friend is Vimla, so Prem married Vimla. That leaves Raj with Kusum.

Checking the statements:

• A. Raj's wife is Kusum: true.
• B. Sushma's husband is Ashok: true.
• C. Poonam is Ved's wife: true.
• D. Poonam is friendly with Vimla: false, because Ved's wife (Poonam) does not like Vimla.

So A, B and C only.

Follow the process from raw material to finished product:

Jungle (source of wood), then Timber (cut wood), then Pulp (processed wood), then Paper (made from pulp), then Book (made of paper).

This gives D, C, B, E, A.

A mirror image swaps left and right; top and bottom stay the same.

In the original, the ball is top-left, the straight arm points left, the slanted arm goes down-right, the horizontal leg points left and the vertical leg is on the right.

After the swap, the image should have the ball top-right, the straight arm pointing right, the slanted arm going down-left, the horizontal leg pointing right and the vertical leg on the left. That matches D.

• A has both arms slanted.
• B moves only the ball.
• C is identical to the original.

One arrow flips from ">" to "<" in each step, working top to bottom: first the left column, then the right column.

• Fig 1: left > > >, right > >
• Fig 2: left < > >, right > >
• Fig 3: left < < >, right > >
• Fig 4: left < < <, right > >
• Fig 5: left < < <, right < >

The next step flips the second right-column arrow, giving left < < <, right < <. That is B.

• A shows the right column as > >.
• C and D do not have three arrows in the left column.

• A. Wrong. Diagnosis is what a doctor does; a court is where a judge works. The matching pair would be Judge : Judgment.
• B. Correct. A bouquet is made up of flowers, just as a sentence is made up of words.
• C. Wrong. A hygrometer measures humidity, not density.
• D. Correct. A sapling is a young tree and a foal is a young horse, so both are adult : young pairs.

So B and D only.

For $n=3$:

• Exactly one face painted: $6(n-2{)}^2=6$. A is true.
• No face painted: $(n-2{)}^3=1$. B is true.
• Exactly two faces painted: $12(n-2)=12$, not 6. C is false.
• Three faces painted: the 8 corner cubes. D is true.

So A, B and D only.

A water image flips top and bottom; left and right stay the same.

In the original:
- the arrow with the bar end runs from top-left (bar) to bottom-right (arrowhead);
- the arrow with the circle runs from top-left (arrowhead) to bottom-centre (circle).

After the flip:
- the bar end is at the bottom-left and the arrowhead at the top-right;
- the circle is at the top-centre and its arrowhead at the bottom-left.

Only C shows this. D is the original with the arrowheads and ends moved, and A and B are left-right mirror versions.

• "Sau-" comes before "Sav-", so Sausage is first.
• Among the "Sav-" words, compare the fourth letter: Savage, then Saviour, then Savour.

Order: Sausage, Savage, Saviour, Savour, which is C, B, A, D.

We need a region inside the circle but outside the square, the triangle and the rectangle.

• 10 is inside the circle only. Correct.
• 8 and 1 are inside the triangle only (musicians).
• 11 is inside the rectangle only (Indians).

So the answer is 10.

From 1 p.m. Tuesday to 1 p.m. Thursday is 48 hours. In that time the watch moves from $-1$ to $+2$ minutes, a gain of 3 minutes.

It shows the correct time once it has gained 1 minute:

$$\frac{1}{3}\times 48=16\text{ hours}$$

1 p.m. Tuesday plus 16 hours is 5:00 a.m. Wednesday.

Each code is twice the number of letters:

• DRIVER: 6 letters, code 12
• PEDESTRIAN: 10 letters, code 20
• ACCIDENT: 8 letters, code 16

CAR has 3 letters, so its code is $2\times 3=6$.

Read from the chart: Kolkata appeared 3250 and qualified 2250; Delhi appeared 2500 and qualified 2000.

Both averages divide by 2, so compare the totals directly:

$$\frac{2250+2000}{3250+2500}\times 100=\frac{4250}{5750}\times 100=73.91\%$$

Appeared: $3000+2250+3250+2500+3000=14000$, so the average is $2800$.

Qualified: $1750+1250+2250+2000+2000=9250$, so the average is $1850$.

Difference $=2800-1850=950$.

$$\frac{2500}{14000}\times 100\approx 17.86\%\approx 18\%$$

The amounts are proportional to the angles: Medical $39.6^{\circ }$ and Groceries $82.8^{\circ }$.

$$39.6:82.8=396:828=11:23$$

(Both numbers divide by 36.)

Entertainment $36^{\circ }$ plus Loan EMI $68.4^{\circ }$ gives $104.4^{\circ }$.

$$\frac{104.4}{360}\times 45800=0.29\times 45800=₹13,282$$

• Computer Science: $840\times 55\%=462$
• Mathematics: $220\times 65\%=143$
• Chemistry: $360\times 35\%=126$

Total $=462+143+126=731$.

• Physics: $900\times 23\%=207$
• Management: $450\times 44\%=198$

$$207:198=23:22$$

(Both numbers divide by 9.)

Standard narration rules:

• "said to me" becomes "told me that".
• "this" becomes "that".
• "we" stays "we", because the speaker (Pritha) and the listener (me) are both included.
• The reporting verb is in the past, so the present "have to" normally shifts back to "had to".

Applying all four gives *Pritha told me that we had to win that trophy* (option B).

The official key picks C, which keeps "have to". Backshift can be skipped when the reported statement is still true at the time of reporting. But option C already changes "this" to "that", which is a time/place shift, so leaving the tense unchanged is inconsistent.

Options A and D wrongly change "we" to "they".

• "Strip of" means to deprive (*stripped of his title*).
• "Strip off" means to remove clothing or a covering.
• "Strip away" means to remove layers.

All three are established phrasal verbs. "Strip against" is not an English collocation, so it is the odd one out.

A preposition must be followed by a noun or a gerund. "Criticised for + -ing" is the correct pattern: *criticised for taking into account...*

• A and C: "criticised to take" is wrong after "criticised".
• D: "by take" uses a bare verb after a preposition.

A sentence beginning "What a..." that expresses admiration is an exclamatory sentence, so it ends with "!".

It is not a question: the word order is "he is", not "is he". So a question mark is wrong, and a full stop or colon would lose the exclamation.

The sentence states a general truth in the present tense ("Happy is", "when he is young"), so the relative clause must also be simple present. The subject "the man" is singular, so the verb is "who acquires".

• C: "who acquire" is plural.
• B: "which" is not used for people.
• D: "has been acquired" is passive and makes no sense here.

Colossal means extremely large. Its opposite is tiny.

"Vast" and "epic" are near-synonyms, and "rust" is unrelated.

A resit (British usage) is a second attempt at an examination one has failed. Here that means *to take an exam again*.

It has nothing to do with rest, creative work or research.

Callous means showing a cruel disregard for others' feelings, so the synonym is insensitive.

• "Careless" means lacking care or attention, not lacking feeling.
• "Passive" and "cursed" are unrelated.

To put one's foot down means to be firm and refuse to allow something, i.e. to *take a firm stand*.

"Light" and "shaky" are the opposite idea, and "heavy stand" is not an idiom.

Embezzlement is the dishonest taking of money that has been entrusted to one's care, which is exactly misappropriation.

• Robbery involves force.
• Theft is stealing in general.
• Fraud is deception in general, not specifically misusing money one holds in trust.

"Instrument" starts with a vowel sound, so it takes an: *Education is an instrument*. The error is in part 1.

The other parts are grammatical.

To not mince matters (or words) means to speak bluntly and directly, without softening the message, i.e. *to speak out plainly*.

The other options describe being at ease, not confusing people or not interfering, none of which is the idiom's meaning.

A stoic endures pain and pleasure alike without showing feeling.

• A fatalist believes events are predetermined.
• An eccentric is unconventional.
• A philosopher studies fundamental questions.

An entreaty is an earnest, humble request or plea. Its opposite is a command, an order given with authority.

• "Request" is a synonym.
• "Immense" and "hatred" are unrelated.

"Truth" here means a specific truth (the truth of this matter), so it needs the definite article: *After knowing the truth*. The error is in part 1.

"Took the right decision" is accepted in British and Indian English, and parts 3 and 4 are correct.

Proscribe means to forbid or prohibit officially. Do not confuse it with "prescribe".

"Torture", "progressive" and "elemental" are unrelated.

"Will be made" is the simple future passive, so the active form is simple future: will make. The agent "the investment" becomes the subject.

• A and C change the tense to present perfect or present perfect continuous.
• B changes it to future continuous.

The correct spelling is L-i-e-u-t-e-n-a-n-t.

• "Lieutinant" has the wrong vowel.
• "Leutenant" drops the i.
• "Leftinant" copies the British pronunciation, not the spelling.

"Despite" is a preposition, so it must be followed by a noun or gerund, not a clause. Correct versions are *despite having received* or *although he had received*.

The error is in "despite he had received". The other parts are fine.

The second blank needs an adjective that takes "on" and means "depends on": dependent on. The first blank needs a noun for starting projects: initiation.

*The initiation of new projects... is dependent not only on low cost... but also on sustainability.*

• A: "vulnerable on" is not idiomatic (the correct form is "vulnerable to").
• B: "occurrence of new projects" is unnatural, since projects are started, not "occurring".
• D: "possible on" is ungrammatical.

The subject "The Indian Bill of Rights" is singular, and the passage describes a present, continuing fact. The next clause, "but it has also borrowed", confirms a present-time frame.

So the verb is the simple present singular: follows.

• "Follow" is plural.
• "Following" is not a finite verb.
• "Followed" clashes with the present-perfect "has also borrowed".

The blank modifies the verb "borrowed", so it needs an adverb: largely.

• "Large" and "largest" are adjectives.
• "Largly" is a misspelling.

The fixed phrase is on behalf of (meaning "as a representative of"): *demands were made on behalf of India*.

• "Behalf of" is missing "on".
• "On the behalf of" wrongly adds an article.
• "On behalf" alone is incomplete before a noun.

After the linking verb "was" we need a predicate adjective, and "___ to the Indian mind" needs the sense "able to be accepted". So the word is acceptable.

• "Accepted to" and "accepting to" are not idiomatic here.
• "Acceptably" is an adverb.

The comparison is between the situation in Britain and the Indian Constitution: *Unlike in Britain, the Indian Constitution has a Bill of Rights*.

• "The Britain" is wrong, because country names like Britain take no article.
• "On" and "with" do not fit the meaning.

• C introduces the topic: middle-class Indians using NGOs to reach poorer classes and spread ideas of rights. It is the only sentence without a back-reference.
• D ("this trend... such NGOs") comments on that development.
• A ("Such an intervention in political reforms") picks up D's "political reforms" and notes the Western inspiration.
• B ("These organisations") describes the NGOs' civil-society approach.
• E ("This bottom-up civil consciousness") concludes with the result.

Options B, C and D begin with A ("Such"), B ("These") or E ("This"), and each of those needs an earlier antecedent. So C-D-A-B-E is the only coherent order.

• D makes the general opening statement about India valuing spirituality.
• B ("People here...") adds that people accept religious and social change.
• C ("For instance...") gives an example: Dayanand Saraswati opposing idol worship.
• A ("Yet, he is remembered...") contrasts his challenge to tradition with the reverence he still receives.

"He" in A needs C before it, and "For instance" in C needs a general claim (B) before it.

• B "affirmative action is a policy" provides the subject and main verb.
• C "that tries to make up for past discrimination" is the relative clause.
• A "or unfair treatment" extends the object ("discrimination or unfair treatment").
• D "against certain groups of people" completes it.

*Affirmative action is a policy that tries to make up for past discrimination or unfair treatment against certain groups of people.*

• A. False. The writer says few authors have any claim to please or instruct, and that recent compilations only "distract choice". That is not a robust picture.
• B. True. The overall view of present-day publishing is bleak.
• C. True. The argument works by contrast: genuine teachers and entertainers against mere compilers, and useful compilations against careless ones.
• D. True. The passage is mainly in the present tense ("distinguish", "brings", "have") and mostly active, with occasional passives such as "is to be loved", "will be caressed" and "are not known".

So B, C and D only.

The writer's complaint is about authors who put two books in front of them and make a third "without any new materials of their own".

• Plagiarism (taking others' work as one's own) fits.
• Duplication (repeating what already exists) fits.
• Copyright laws are never discussed.
• Innovation is what the writer finds missing, so it does not describe the idea.

So B and C only.

• A. True. The writer criticises books compiled with no new material and little judgement.
• B. True. He praises the author who teaches (a "benefactor") and the one who amuses (a "pleasing companion"), yet worries about most current publications.
• C. False. He explicitly says he does not claim all compilations are useless, so he does not reject them outright.
• D. True. He says someone who collects scattered knowledge under proper heads is "laudably employed" and helps others make progress.

So A, B and D only.

• Be at cross purposes: two people misunderstand each other or work towards different aims, so II (not understanding each other).
• Wheels within wheels: hidden, secret influences at work behind a situation, so I (secret things).
• Wild goose chase: a hopeless, foolish pursuit, so IV (foolish effort).
• Off the cuff: said or done without preparation, so III.

Answer: A-II, B-I, C-IV, D-III.

• Alias: an assumed or second name, so IV.
• De jure: by law, as a matter of legal right, so III.
• In lieu of: instead of, in place of something, so II.
• Pro rata: in proportion, so I.

Answer: A-IV, B-III, C-II, D-I.

• "had won" is *had* + past participle, the past perfect (III).
• "won" is the simple past (I).
• "had been winning" is *had been* + -ing, the past perfect continuous (IV).
• "was winning" is *was* + -ing, the past continuous (II).

Answer: A-III, B-I, C-IV, D-II.