Q&A: practice questions with answers

Concept: $a^2-b^2=(a+b)(a-b)$ and ${\sin }^2\theta +{\cos }^2\theta =1$.

1. ${\sin }^4\theta -{\cos }^4\theta =({\sin }^2\theta +{\cos }^2\theta )({\sin }^2\theta -{\cos }^2\theta )={\sin }^2\theta -{\cos }^2\theta$
2. ${\sin }^2\theta -{\cos }^2\theta ={\sin }^2\theta -(1-{\sin }^2\theta )=2{\sin }^2\theta -1$
3. So $2{\sin }^2\theta -1=\frac{1}{2}$.

Trick: The asked expression equals the given one – no need to find $\theta$.

Answer: (B) $\frac{1}{2}$

Concept: Rolling along the length makes the length the circumference; the width becomes the height.

1. $2\pi r=32$ → $r=\frac{16}{\pi }$, $h=14$
2. $V=\pi r^{2}h=\pi \times \frac{256}{{\pi }^2}\times 14$
3. $=\frac{3584}{\pi }{\text{cm}}^3$

Trick: Options keep $\pi$ in the denominator – don't substitute $\frac{22}{7}$.

Answer: (D) $\frac{3584}{\pi }$

Concept: Translate the words into an equation.

1. $1200\%\text{ of }10=120$
2. $0.2k=120-10=110$
3. $k=\frac{110}{0.2}=550$

Trick: $20\%$ of $k$ = $\frac{k}{5}$, so $k=5\times 110$.

Answer: (B) 550

Concept: $\text{Mean}=\frac{\text{Sum of observations}}{\text{Number of observations}}$

1. Sum $=83+17+62+87+62+96+86+92+90=675$.
2. Number of observations $=9$.
3. Mean $=\frac{675}{9}=75$.

Trick: Pair to make round numbers: $83+17=100$, $62+87+...$ – add quickly.

Answer: (B) $75$

Concept: Compare ratios as fractions; a ratio greater than $\frac{1}{2}$ beats all those below $\frac{1}{2}$.

1. $\frac{38}{77}<\frac{1}{2}$ (since $38\times 2=76<77$).
2. $\frac{34}{51}=\frac{2}{3}>\frac{1}{2}$.
3. $\frac{27}{76}$ and $\frac{31}{72}$ are both $<\frac{1}{2}$.

Trick: Benchmark with $\frac{1}{2}$: only one fraction crosses it.

Answer: (B) $34:51$

Concept: A number is divisible by 8 if its last three digits are divisible by 8.

1. Last three digits: $p08$.
2. Try largest first: $908÷8=113.5$ ✗; $808÷8=101$ ✓.
3. So maximum $p=8$.

Trick: Start from 9 and come down – the first value that works is the maximum.

Answer: (C) $8$

Concept: For a fixed distance, time $\propto \frac{1}{\text{speed}}$.

1. Speed becomes $\frac{2}{7}$ of normal ⇒ time becomes $\frac{7}{2}$ of normal time $T$.
2. Extra time $=\frac{7}{2}T-T=\frac{5}{2}T=30$ min.
3. $T=30\times \frac{2}{5}=12$ min.

Trick: Speed ratio $2:7$ ⇒ time ratio $7:2$; difference 5 parts $=30$ ⇒ 1 part $=6$, normal $=2\times 6=12$.

Answer: (B) $12$

Concept: Half-yearly compounding: rate $=\frac{R}{2}$ per half-year, periods $=2n$; $A=P{(1+\frac{r}{100})}^t$.

1. Rate per half-year $=5\%$, periods $=2$.
2. $A=72000\times (1.05{)}^2=72000\times 1.1025$.
3. $A=79380$.

Trick: Two periods at $5\%$ ⇒ effective rate $10.25\%$: $72000+10.25\%$ of $72000=79380$.

Answer: (D) ₹$79,380$

Concept: Gain $={\text{SP}}_1-\text{CP}$, Loss $=\text{CP}-{\text{SP}}_2$.

1. Let loss $=L$; gain $=1.2L$.
2. $L+1.2L=8378-1756=6622$ ⇒ $L=3010$.
3. $\text{CP}=1756+3010=4766$.
4. For $50\%$ gain: $\text{SP}=1.5\times 4766=7149$.

Trick: Gain + Loss always equals the gap between the two selling prices.

Answer: (D) $7,149$

Concept: $a{x}^2+bx+c=0$ has equal real roots when $D=b^2-4ac=0$.

1. Here $a=4$, $c=16$.
2. $b^2-4\times 4\times 16=0$ ⇒ $b^2=256$.
3. $b=\pm 16$.

Trick: Equal roots ⇒ perfect square: $4x^2\pm 16x+16=4(x\pm 2{)}^2$.

Answer: (B) $16$ and $-16$

Concept: $\text{SI}=\frac{P\times R\times T}{100}$, Amount $=P+\text{SI}$.

1. $\text{SI}=\frac{1320\times 6\times 6}{100}=475.2$.
2. Amount $=1320+475.2=1795.2$.

Trick: Total rate $=6\times 6=36\%$ ⇒ Amount $=1.36\times 1320=1795.2$.

Answer: (B) $1,795.2$

Concept: $\text{Average price}=\frac{\text{Total cost}}{\text{Total books}}$

1. Cost $=51\times 1120+45\times 920=57120+41400=98520$.
2. Books $=51+45=96$.
3. Average $=\frac{98520}{96}=1026.25$.

Trick: Check by alligation: $(1120-1026.25):(1026.25-920)=93.75:106.25=45:51$ ✓.

Answer: (A) $1,026.25$

Concept: When a sheet is rolled into a cylinder, its whole area becomes the curved surface area.

1. Rolling along the breadth: length $12\text{m}$ becomes the circumference $2\pi r$; breadth $7.48\text{m}$ becomes the height.
2. $\text{CSA}=2\pi rh=12\times 7.48$.
3. $=89.76{\text{m}}^2$.

Trick: CSA of a rolled sheet = $l\times b$ – no need to use $\pi$.

Answer: (D) $89.76{\text{m}}^2$

Concept: Convert a ratio of fractions to whole numbers by multiplying by the LCM of denominators.

1. LCM of $14,13,14=182$: $\frac{5}{14}:\frac{16}{13}:\frac{20}{14}=65:224:260$.
2. Largest − smallest $=260-65=195$ parts $=30$.
3. 1 part $=\frac{30}{195}=\frac{2}{13}$; largest $=260\times \frac{2}{13}=40$.

Trick: Same denominators for 1st and 3rd: largest:smallest $=20:5$ ⇒ diff 15 parts of $\frac{1}{14}$ … largest $=\frac{20}{15}\times 30=40$.

Answer: (A) $40$

Concept: Solve brackets from innermost: ( ) → { } → [ ].

1. $24÷6=4$.
2. $\{72-4\}=68$; $68÷4=17$.
3. $[5+17]=22$; $57+22=79$.

Trick: Inside-out: small bracket first, then curly, then square.

Answer: (B) $79$

Concept: New member's value $=$ old average $+$ (new count $\times$ increase).

1. Old total $=7\times 54=378$.
2. New total $=8\times 56=448$.
3. New boy $=448-378=70\text{kg}$.

Trick: $54+8\times 2=70$ – the newcomer brings his share plus 2 kg for each of the 8 members.

Answer: (A) $70\text{kg}$

Concept: Check every number: the element just before it and just after it must both be symbols.

1. Numbers in order: 5, 6, 3, 4, 9, 6, 5, 8.
2. 5 (after @) is followed by 6 – no; 3 and 9 are preceded by numbers – no; 4 is followed by 9 – no.
3. 6 in '% 6 &' has symbols on both sides – yes; 5 in '© 5 8' is followed by 8 – no; 8 is the last element.

Trick: Scan only numbers that start a number-run of length 1 – a single number sandwiched between symbols.

Answer: (D) 1

Concept: Fix the sure clues first (product, 'only 2 above'), then place the pair clue.

1. B is on a prime floor and $B\times D=3$, so $B=3,D=1$.
2. Only 2 people above C → C is on floor 4.
3. F immediately below A: free floors are 2, 5, 6 → F = 5, A = 6; E = 2.
4. Between A (6) and C (4) only F lives → 1 person.

Trick: Product clues like $3=3\times 1$ fix two people at once – always start there.

Answer: (B) 1

Concept: Track the first letter, second letter and number separately.

1. 1st letters: G, I, K, M → +2 each → O.
2. 2nd letters: M, O, Q, S → +2 each → U.
3. Numbers: 17, 25, 33, 41 → +8 each → 49.

Trick: Three independent +2, +2, +8 patterns – check the number first to eliminate options fast.

Answer: (A) OU49

Concept: A 'Some' statement and an 'All' statement both give valid 'Some' conversions.

1. 'Some trees are flowers' converts to 'Some flowers are trees' → (I) follows.
2. 'All flowers are beautiful' → 'Some beautiful are flowers' → (II) follows.
3. The 'girls' statement is not needed.

Yaad rakhein: Some A are B ⇔ Some B are A; All A are B ⇒ Some B are A.

Answer: (C) Both conclusions (I) and (II) follow.