Q&A: practice questions with answers

Concept: When working together, work done is in the ratio of efficiencies.

1. P is 20% more efficient ⇒ P : Q = $120:100=6:5$
2. P's share = $\frac{6}{11}$ of the work
3. $\frac{6}{11}\times 100=54.54\%$

Trick: The 18 days is extra data – the share depends only on the efficiency ratio.

Answer: (A) 54.54%

Concept: Cost price includes all expenses (transport, storage).

1. CP = $40,000+8,000+7,000=$ ₹55,000
2. Profit = $60,000-55,000=$ ₹5,000
3. Profit% = $\frac{5,000}{55,000}\times 100=\frac{1}{11}\times 100=9.09\%$

Trick: $\frac{1}{11}=9.09\%$ – memorise fraction-percent pairs ($\frac{1}{11}=9.09$, $\frac{1}{12}=8.33$).

Answer: (A) 9.09% profit

Concept: Use LCM as total work, then units per day.

1. Total = $\text{LCM}(12,15)=60$ units; Rahul 5, Pawan 4 units/day
2. 2 days together = $2\times 9=18$ units
3. Left = $\frac{42}{60}=\frac{7}{10}$

Trick: Fraction done = $2(\frac{1}{12}+\frac{1}{15})=2\times \frac{9}{60}=\frac{3}{10}\Rightarrow$ left $=\frac{7}{10}$.

Answer: (A) $\frac{7}{10}$

Concept: Average speed = total distance ÷ total time.

1. Take each part = $\text{LCM}(12,15,30)=60$ km; total 180 km
2. Times: 5 h + 4 h + 2 h = 11 h
3. Average = $\frac{180}{11}=16.36$ km/h

Trick: For 3 equal distances: $\frac{3abc}{ab+bc+ca}=\frac{3\times 12\times 15\times 30}{180+450+360}=\frac{16200}{990}=16.36$.

Answer: (B) 16.36 km/h

Concept: Each rectangular face = (side of base triangle) × prism height.

1. Hypotenuse = $\sqrt{8^2+{15}^2}=\sqrt{289}=17$ cm
2. Faces: $8\times 5$, $15\times 5$, $17\times 5$
3. Largest : smallest = $17\times 5:8\times 5=17:8$

Trick: Height is common, so the ratio = longest side : shortest side = 17 : 8 directly.

Answer: (B) 17 : 8

Concept: Juice stays constant; only water changes.

1. Juice = $60-20=40$ L
2. Need juice : water = 2 : 3 ⇒ water = $40\times \frac{3}{2}=60$ L
3. Water to add = $60-20=40$ L

Trick: Fix the unchanged quantity: 2 parts = 40 L ⇒ 1 part = 20 L ⇒ water 3 parts = 60 L.

Answer: (C) 40 L

Concept: sin = perpendicular/hypotenuse; cos = base/hypotenuse.

1. $\sin A=\frac{5}{13}$ ⇒ P = 5, H = 13
2. Base = $\sqrt{169-25}=12$
3. $\cos A=\frac{12}{13}$

Trick: 5-12-13 is a Pythagorean triplet – no calculation needed.

Answer: (A) $\frac{12}{13}$

Concept: Line: $y=mx+c$, where c is the y-intercept.

1. $m=6$; point $(4,-1)$
2. $-1=6\times 4+c$
3. $c=-1-24=-25$

Trick: $c=y_1-m·x_1$ directly: $-1-24=-25$.

Answer: (B) −25

Concept: Corresponding angles formed by a transversal on parallel lines are equal.

1. One corresponding angle = 100°.
2. Equal pair ⇒ other = 100°.

Yaad rakhein: Corresponding & alternate angles are equal; co-interior (same-side interior) angles add to 180°.

Answer: (C) 100°

Concept: In △PQR ∼ △MNO, vertices match in order: P↔M, Q↔N, R↔O.

1. ∠N corresponds to ∠Q.
2. ∠Q = 80° ⇒ ∠N = 80°

Trick: Read the letter positions: 2nd letter Q ↔ 2nd letter N. (∠R = ∠O = 30°.)

Answer: (B) 80°

Concept: $(a+b{)}^2=a^2+2ab+b^2$.

1. $x^2=(\sqrt{7}+3{)}^2=7+9+6\sqrt{7}=16+6\sqrt{7}$
2. $x^2-4\sqrt{7}=16+6\sqrt{7}-4\sqrt{7}$
3. $=16+2\sqrt{7}$

Trick: Rational part $7+9=16$ decides it – only one option starts with 16.

Answer: (A) 16 + 2$\sqrt{7}$

Concept: Equal chords of a circle are equidistant from the centre.

1. Both chords = 8 cm.
2. First chord is 2 cm from centre ⇒ second also 2 cm.

Yaad rakhein: Radius here = $\sqrt{4^2+2^2}=\sqrt{20}$ cm; distance $d=\sqrt{r^2-(\text{half-chord}{)}^2}$ is the same for equal chords.

Answer: (B) 2 cm

Concept: Radius ⟂ tangent at the point of contact.

1. ∠OMX = ∠ONX = 90°
2. In quadrilateral OMXN: 90° + 90° + α + ∠MXN = 360°
3. ∠MXN = 180° − α

Yaad rakhein: Angle between tangents and angle at centre are always supplementary.

Answer: (C) 180° − α

Concept: ${\sin }^{4}A+{\cos }^{4}A=1-2{\sin }^{2}A·{\cos }^{2}A$.

1. $\sin A=\frac{3}{5}\Rightarrow \cos A=\frac{4}{5}$
2. ${\sin }^{2}A·{\cos }^{2}A=\left(\frac{9}{25}\right)\left(\frac{16}{25}\right)=\frac{144}{625}$
3. $1-\frac{288}{625}=\frac{337}{625}$

Trick: Denominator must be $5^4=625$ – only option (a) has 625.

Answer: (A) $\frac{337}{625}$

Concept: Align decimal points before adding.

1. Write with 3 decimals: 8.650, 9.280, 0.067
2. Thousandths: 0+0+7 = 7; hundredths: 5+8+6 = 19 → 9 carry 1; tenths: 6+2+0+1 = 9; units 8+9 = 17
3. Sum = 17.997

Trick: Rough check: 8.65 + 9.28 ≈ 18 – only 17.997 is near 18.

Answer: (D) 17.997

Concept: Idiom meaning: repaying unkindness with kindness so the wrongdoer feels ashamed.

1. He was rude; she was kind in return.
2. Her goodness made him feel guilty and ashamed – no punishment or accusation involved.
3. So it means shaming an enemy through kindness.

Yaad rakhein: The phrase is biblical (Proverbs/Romans): return good for evil and the other person's conscience 'burns'.

Answer: (D) To shame an enemy through kindness

Concept: Accomplice = a partner in a crime.

1. Correct form: ac-com-plice (double c, single m, ends in -ice).
2. Accomplise (-ise), Acomplicee (single c, extra e), Acomplice (single c) are wrong.

Trick: Same family: accomplish, accompany – all start with 'acc'.

Answer: (A) Accomplice

Concept: Order: definition → how it works → uses → limitations.

1. 1 introduces VR with its full form – opener.
2. 3 explains how users access it; 2 gives its applications.
3. 4 states obstacles – conclusion.

Trick: A sentence with the full form 'Virtual reality (VR)' starts the paragraph; later sentences use only 'VR'.

Answer: (A) 1-3-2-4

Concept: Introduction → working → benefits → 'However' (problems).

1. 1 introduces EVs (full form given).
2. 3 describes how EVs work; 2 ('They reduce…') lists benefits.
3. 4 begins with 'However' – contrast, so it closes.

Trick: 'However' can never start a paragraph – it follows the positive points, so 4 goes last.

Answer: (A) 1-3-2-4

Concept: Amnesty = an official pardon, especially for political offences.

1. Correct form: am-nes-ty (single m, -nes-, ends in -ty).
2. Amnistl, Ammnisty, Amnistey – wrong vowels/letters.

Yaad rakhein: Amnesty International is the human-rights NGO – remembering the name fixes the spelling.

Answer: (C) Amnesty