Q&A: practice questions with answers

Concept: Remaining share % → total = part $\times \frac{100}{\text{share}}$.

1. Blue = $100-25-35=40\%$
2. 40% of total = 40
3. Total = $\frac{40\times 100}{40}=100$

Trick: Blue count equals its percentage ($40=40\%$) → total is 100.

Answer: (A) 100

Concept: Area of rhombus = $\frac{1}{2}\times d_1\times d_2$.

1. Let $d_1=5k$, $d_2=6k$
2. $\frac{1}{2}\times 30k^2=135\Rightarrow k^2=9\Rightarrow k=3$
3. Diagonals = 15 cm and 18 cm

Trick: Only (C) is in ratio 5 : 6 (15 : 18); check $\frac{1}{2}\times 15\times 18=135$ ✓.

Answer: (C) 15 cm and 18 cm

Concept: Ring area = $\pi (R^2-r^2)=\pi (R+r)(R-r)$.

1. $R=12+3=15$, $r=12$
2. Area = $3.14\times (15+12)(15-12)=3.14\times 81$
3. $=254.34$ cm²

Trick: $(R+r)(R-r)=27\times 3=81$ – avoids squaring.

Answer: (B) 254.34 cm²

Concept: If $729<x<784$, then $829<x+100<884$.

1. Squares in 829–884: ${29}^2=841$ only.
2. $x+100=841\Rightarrow x=741$
3. 741 lies between 729 and 784 ✓

Trick: Add 100 to each option: 832, 841, 845 → only 841 is a square.

Answer: (C) 741

Concept: Circle touching all sides has diameter = side of square.

1. Side $=2r$ → square area $=4r^2$
2. Circle area $=\pi r^2$
3. % removed = $\frac{\pi }{4}\times 100=78.5\%$

Yaad rakhein: Inscribed circle always covers $\frac{\pi }{4}\approx 78.5\%$ of the square; remaining 21.5%.

Answer: (D) 78.5%

Concept: Complementary angles sum to 90°.

1. 2k + 7k = 90° → k = 10°
2. Angles = 20° and 70°

Trick: Only pairs adding to 90° in ratio 2 : 7 – 20 : 70 = 2 : 7 ✓.

Answer: (B) 20°, 70°

Concept: Distance = $h\cot \theta$; same side → subtract.

1. 45°: 60 m from tower.
2. 30°: $60\sqrt{3}$ m from tower.
3. Gap = $60(\sqrt{3}-1)=60\times 0.732=43.92$ m

Trick: $\sqrt{3}-1\approx 0.732$: multiply height by 0.732 directly.

Answer: (C) 43.92 m

Concept: $\tan \theta =\frac{P}{B}$; $\sec \theta =\frac{H}{B}$.

1. $P=5$, $B=12$
2. $H=\sqrt{25+144}=13$
3. $\sec \theta =\frac{13}{12}$

Trick: ${\sec }^2\theta =1+{\tan }^2\theta =1+\frac{25}{144}=\frac{169}{144}$ → $\frac{13}{12}$.

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

Concept: Put $x=0$ to get the y-intercept.

1. $5y=20$
2. $y=4$

Trick: Intercept form $\frac{x}{5}+\frac{y}{4}=1$ → y-intercept 4 (x-intercept 5).

Answer: (B) 4

Concept: Adjacent angles on a line = 180°; vertically opposite angles are equal.

1. $(4x-5)+(6x+25)=180\to 10x=160\to x=16$
2. $4\times 16-5={59}^{\circ }$
3. Opposite angle = 59°

Yaad rakhein: The adjacent angle is $6\times 16+25={121}^{\circ }$, and ${59}^{\circ }+{121}^{\circ }={180}^{\circ }$ ✓.

Answer: (A) 59°

Concept: Angle in a semicircle is 90°, so the hypotenuse is the circumcircle's diameter.

1. Circumcentre = midpoint of hypotenuse ✓
2. Centroid and incentre are always inside any triangle.
3. All centres coincide only in an equilateral triangle.

Yaad rakhein: In a right triangle the orthocentre lies at the right-angle vertex.

Answer: (B) The circumcentre is at the midpoint of the hypotenuse

Concept: A line parallel to one side divides the other two sides in the same ratio.

1. $\frac{KN}{NL}=\frac{KO}{OM}$
2. $\frac{2}{4}=\frac{3}{OM}$
3. OM = 6

Trick: KN : NL = 1 : 2, so $OM=2\times KO=6$.

Answer: (A) 6

Concept: Direct tangent${}^2=d^2-(r_1-r_2{)}^2$.

1. ${24}^2={25}^2-(10-r{)}^2$
2. $(10-r{)}^2=625-576=49$
3. $10-r=7\to r=3$ cm

Trick: 7-24-25 triplet: difference of radii = 7 → $10-7=3$.

Answer: (B) 3 cm

Concept: If $\sin A=\cos B$ (acute angles), then $A+B={90}^{\circ }$.

1. Angles must add to 90°: $2x+15+x-5=90$
2. So $3x=80$
3. $x=\frac{80}{3}\approx {26.67}^{\circ }$

Trick: $\frac{80}{3}$ is not whole → only the decimal option 26.67° can fit.

Answer: (D) 26.67°

Concept: Total work = workers × days (man-days).

1. Work = $12\times 20=240$ man-days
2. Done in 5 days = 60 → left 180
3. 9 workers: $180÷9=20$ days

Trick: Remaining 15 days $\times$ 12 workers $=$ 9 workers $\times D\to D=20$.

Answer: (A) 20 days

Concept: Idioms carry a figurative, not literal, meaning.

1. "Cut from the same cloth" = having a very similar character or nature.
2. e.g. The two brothers are cut from the same cloth – both are stubborn.
3. It is about nature, not dress, family line or job.

Trick: Literal-looking options (wear identical outfits) are almost always traps in idiom questions.

Answer: (B) Of the same nature

Concept: Break the word into parts: geo (Earth) + syn (same) + chron (time) + ous.

1. Correct: geosynchronous – "y" after s, and "ch" in chron.
2. Other options use i/v for y or drop the h.

Trick: "Chron" = time (chronology, chronic) – it always keeps "ch".

Answer: (A) geosynchronous

Concept: Taciturn = habitually silent, speaking very little.

1. Opposite: Loquacious = very talkative.
2. Reserved, Quiet, Uncommunicative are synonyms, not antonyms.

Trick: Three options mean the same thing – the odd one out is the antonym.

Answer: (B) Loquacious

Concept: Comes from secret ballots where a black ball meant a "no" vote.

1. To blackball = to keep someone out of a club/group, often by a secret vote.
2. e.g. He was blackballed by the golf club.

Yaad rakhein: Similar: to blacklist = to put on a list of people to be avoided or punished.

Answer: (D) To exclude someone from a group or society, often by secret vote

Concept: Medical compound words join roots with the connecting vowel "o".

1. cardio (heart) + thoraco (chest) + vascular (vessels) = Cardiothoracovascular.
2. Written as one word; "vascular" keeps its "s".

Trick: Spell each root separately – vas-cu-lar has an s, thorac-o not thoras-ic.

Answer: (D) Cardiothoracovascular