Q&A: practice questions with answers

Concept: For equal distances, average speed = $\frac{3}{\frac{1}{a}+\frac{1}{b}+\frac{1}{c}}$.

1. $\frac{1}{2}+\frac{1}{3}+\frac{1}{5}=\frac{31}{30}$.
2. Average = $3\times \frac{30}{31}=\frac{90}{31}$.
3. Average : lowest = $\frac{90}{31}:2=45:31$.

Trick: Take distance = LCM 30: times 15, 10, 6 → total 31 for 90 → $\frac{90}{31}$.

Answer: (C) 45 : 31

Concept: 20% of the money saved buys the extra 5 kg at the new price.

1. Saving = 20% of ₹100 = ₹20.
2. New price = $\frac{20}{5}=$ ₹4 per kg.
3. Original = $4÷0.8=$ ₹5 per kg.

Trick: Reduced price = $\frac{r\%\times \text{amount}}{\text{extra qty}}$ – then scale back up.

Answer: (C) ₹5

Concept: New average = $\frac{\text{old total}+\text{new value}}{\text{new count}}$.

1. Mon–Fri total = $5\times 30=150$.
2. With Saturday: $150+55=205$.
3. Average = $\frac{205}{6}\approx {34.17}^{\circ }$C.

Trick: Saturday is 25 above average; spread over 6 days → $30+\frac{25}{6}\approx 34.17$.

Answer: (C) 34.17°C

Concept: Discount % = $\frac{\text{discount}}{\text{marked price}}\times 100$.

1. $\frac{180}{3600}\times 100$.
2. $=\frac{1}{20}\times 100=5\%$.

Trick: 10% of 3600 = 360; 180 is half → 5%.

Answer: (A) 5%

Concept: SP = MP − discount; CP = SP − profit.

1. SP = 360 − 72 = ₹288.
2. CP = 288 − 18 = ₹270.

Trick: CP = MP − discount − profit = 360 − 90 = 270 in one go.

Answer: (B) ₹270

Concept: Equal distances: average speed = $\frac{2xy}{x+y}$.

1. $\frac{2\times 50\times 25}{50+25}=\frac{2500}{75}$.
2. = 33.33 km/h.
3. (Check: 400 km in $4+8=12$ h.)

Trick: Average speed is never the simple mean 37.5 – that's the trap option.

Answer: (A) 33.33 km/h

Concept: $\frac{M_1{D}_1}{W_1}=\frac{M_2{D}_2}{W_2}$.

1. $\frac{77\times 89}{1}=\frac{7\times D}{1/11}$.
2. $7D=\frac{77\times 89}{11}=7\times 89$.
3. $D=89$ days.

Trick: Women became $\frac{1}{11}$ and work became $\frac{1}{11}$ – they cancel, so days stay 89.

Answer: (C) 89

Concept: Perimeter = $2(l+b)$, $l=4b$.

1. $2(4b+b)=10b=200\Rightarrow b=20$ cm.
2. $l=80$ cm.
3. Area = $80\times 20=1600$ cm².

Trick: $l+b=100$ split 4 : 1 → 80 and 20.

Answer: (D) 1600 cm²

Concept: Build a right triangle: perpendicular 2x, hypotenuse $1+x^2$.

1. ${\text{Base}}^2$ $=(1+x^2{)}^2-(2x{)}^2=(1-x^2{)}^2$.
2. Base = $1-x^2$.
3. $\cot A$ = $\frac{\text{base}}{\text{perpendicular}}$ = $\frac{1-x^2}{2x}$.

Trick: Remember the triplet $2x,1-x^2,1+x^2$ (like $\frac{2\tan \theta }{1+{\tan }^2\theta }$). Check $x=\frac{1}{2}$: $\sin =\frac{4}{5}$, $\cot =\frac{3}{4}=\frac{1-\frac{1}{4}}{1}$ ✓.

Answer: (D) $\frac{1-x\mathrm{²}}{2x}$

Concept: Alcohol after = alcohol left + alcohol added.

1. Start: 30% of 150 = 45 L.
2. Remove 20 L mixture → lose 6 L alcohol → 39 L.
3. Add 20 L at 40% → +8 L → 47 L.
4. $\frac{47}{150}\times 100\approx 31.33\%$.

Trick: Net change = $20\text{L}\times (40-30)\%=+2\text{L}$ → $45+2=47$.

Answer: (A) 31.33%

Concept: Radian = degree $\times \frac{\pi }{180}$.

1. $160\times \frac{\pi }{180}$.
2. $=\frac{16\pi }{18}=\frac{8\pi }{9}$.

Trick: ${20}^{\circ }=\frac{\pi }{9}$, and ${160}^{\circ }=8\times {20}^{\circ }\Rightarrow \frac{8\pi }{9}$.

Answer: (A) $\frac{8\pi }{9}$

Concept: tan θ = cot(90° − θ).

1. tan x = cot y = tan(90° − y).
2. So x = 90° − y.
3. x + y = 90°.

Trick: Whenever tan of one = cot of the other (acute angles), they add to 90°.

Answer: (C) 90°

Concept: Area = $r\times s$, where s = semi-perimeter.

1. s = $\frac{40}{2}=20$ cm.
2. Area = $2\times 20=40$ cm².

Trick: Area = $r\times \frac{\text{perimeter}}{2}$ – works for every triangle.

Answer: (D) 40 cm²

Concept: Perpendicular from centre bisects the chord: $r^2=(\text{half chord}{)}^2+d^2$.

1. Half chord = 9 cm.
2. $r^2=9^2+{13}^2=81+169=250$.
3. $r=\sqrt{250}\approx 15.8\approx 16$ cm.

Trick: r must exceed 13 (distance) and 9 (half chord) but be less than $13+9$ – only 16 and 18 fit; ${16}^2=256\approx 250$.

Answer: (C) 16 cm

Concept: Tangent at any point is perpendicular to the radius at that point.

1. Radius to the point of contact ⟂ tangent.
2. Angle = 90°, whatever the radius (8 cm is extra data).

Yaad rakhein: Two tangents from an external point are equal in length.

Answer: (C) 90°

Concept: Desiccation = the process of drying out completely.

1. Correct pattern: de-SI-CC-ation → one 's', double 'c'.
2. dessication (ss, one c), desication (one c), desiccasion (-sion) are wrong.

Trick: Think 'one Sun dries, two Clouds don't come' → 1 S, 2 C.

Answer: (C) desiccation

Concept: 'Do not + V' becomes 'told/instructed + object + not to + V'.

1. Add the listener as object: instructed them.
2. 'Do not reshelve' → not to reshelve.
3. Reflexive changes: yourselves → themselves.

Trick: Reject any option with 'do not' or 'yourselves' still inside, and any without an object after 'instructed'.

Answer: (D) The librarian instructed them not to reshelve the books themselves.

Concept: Exacerbate = make a problem worse.

1. Opposite: alleviate = make less severe, ease.
2. Worsen, aggravate, intensify all mean the same as exacerbate.

Trick: When three options are synonyms of each other, the odd one out is usually the antonym.

Answer: (C) Alleviate

Concept: Come clean = admit the truth, especially about a wrongdoing.

1. Example: He finally came clean about taking the money.
2. (A), (B) are literal meanings; (C) fits 'come out unscathed'.

Trick: Idioms rarely mean the literal action – drop 'tidy up' and 'bath' first.

Answer: (D) To confess something, especially something wrong

Concept: Simple past active → was/were + V3; the full object phrase becomes the subject.

1. Object = 'the proposal that the survey results should be re-evaluated before publication' – keep it together.
2. rejected → was rejected; add 'by the panel'.
3. Inner clause stays unchanged.

Trick: Don't touch the tense of the inner clause; (C) and (D) alter tenses, (B) breaks the noun phrase.

Answer: (A) The proposal that the survey results should be re-evaluated before publication was rejected by the panel.