Q&A: practice questions with answers

Concept: In a quadrilateral, if bisectors of $\angle C$ and $\angle D$ meet at O, then $\angle DOC=\frac{\angle A+\angle B}{2}$.

1. $\angle C+\angle D={360}^{\circ }-({53}^{\circ }+{46}^{\circ })={261}^{\circ }$.
2. In $△DOC$: $\angle DOC={180}^{\circ }-\frac{{261}^{\circ }}{2}={180}^{\circ }-{130.5}^{\circ }$.
3. $\angle DOC={49.5}^{\circ }$.

Trick: Directly: $\frac{{53}^{\circ }+{46}^{\circ }}{2}={49.5}^{\circ }$.

Answer: (C) ${49.5}^{\circ }$

Concept: If girls : boys = 1 : 5, then boys = 5 × girls.

1. $\frac{\text{Girls}}{\text{Boys}}=\frac{1}{5}$
2. $\text{Boys}=5\times 201=1005$

Trick: 1 part = 201, so 5 parts = 1005.

Answer: (C) 1005

Concept: Median = middle value after arranging data in order; for $n$ odd it is the ${\left(\frac{n+1}{2}\right)}^{\text{th}}$ value.

1. Ascending order: 11, 12, 38, 38, 49, 53, 55, 60, 70, 93, 98.
2. $n=11$, so median = ${\left(\frac{11+1}{2}\right)}^{\text{th}}=6^{\text{th}}$ value = 53.

Trick: Always sort first – the unsorted middle (98) is a trap.

Answer: (B) 53

Concept: Production ∝ (manpower × hours). If one factor falls, the other must rise to keep the product constant.

1. Manpower effect: production becomes $80\%$, i.e. factor $=\frac{4}{5}$.
2. To restore, hours must be multiplied by $\frac{5}{4}$.
3. Increase $=\frac{5}{4}-1=\frac{1}{4}=25\%$.

Trick: Decrease $x\%$ → required increase $=\frac{x}{100-x}\times 100=\frac{20}{80}\times 100=25\%$.

Answer: (C) $25\%$

Concept: Centroid of a triangle (meeting point of medians) $G=(\frac{x_1+x_2+x_3}{3},\frac{y_1+y_2+y_3}{3})$.

1. $x=\frac{1+7+10}{3}=\frac{18}{3}=6$
2. $y=\frac{4+8+12}{3}=\frac{24}{3}=8$
3. $G=(6,8)$

Trick: Centroid = simple average of the three x's and the three y's.

Answer: (B) $(6,8)$

Concept: Ladder, wall and ground form a right triangle; the ladder is the hypotenuse, height is opposite to the 30° angle.

1. $\sin {30}^{\circ }=\frac{h}{8}$
2. $h=8\times \frac{1}{2}=4\text{m}$

Trick: In a 30°-60°-90° triangle, side opposite 30° = half of the hypotenuse.

Answer: (A) $4\text{m}$

Concept: Volume of a hemisphere $=\frac{2}{3}\pi r^3$.

1. $r^3={24}^3=13824$
2. $V=\frac{2}{3}\pi \times 13824=2\pi \times 4608=9216\pi {\text{cm}}^3$

Trick: Divide by 3 first: $\frac{13824}{3}=4608$, then double.

Answer: (C) $9216\pi$

Concept: Opposite directions: relative speed = sum of speeds; distance covered while crossing = sum of the lengths.

1. Relative speed $=99+72=171\text{km/h}=171\times \frac{5}{18}=47.5\text{m/s}$.
2. $x+1.5x=47.5\times 46=2185\Rightarrow x=874\text{m}$; length of B $=1.5\times 874=1311\text{m}$.
3. Speed of B $=72\times \frac{5}{18}=20\text{m/s}$; in 82 s it covers $20\times 82=1640\text{m}$.
4. Bridge $=1640-1311=329\text{m}$.

Trick: Train crossing a bridge covers (train + bridge) length.

Answer: (D) 329

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

1. $(5z-9y{)}^2=25z^2-90yz+81y^2$
2. $(9z+5y{)}^2=81z^2+90yz+25y^2$
3. Sum $=106z^2+106y^2$ (the $yz$ terms cancel).
4. $106z^2+106y^2-81z^2=25z^2+106y^2$

Trick: $(ax-by{)}^2+(bx+ay{)}^2=(a^2+b^2)(x^2+y^2)=106(z^2+y^2)$.

Answer: (C) $25z^2+106y^2$

Concept: For $a{x}^2+bx+c=0$: $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$.

1. $a=9,b=-9,c=1$
2. $D=b^2-4ac=81-36=45$
3. $x=\frac{9\pm \sqrt{45}}{2\times 9}=\frac{9\pm \sqrt{45}}{18}$

Trick: Check: sum of roots $=\frac{18}{18}=1=-\frac{b}{a}$ ✓; options with denominator 9 or 4.5 fail this.

Answer: (A) $\frac{9+\sqrt{45}}{18}$ & $\frac{9-\sqrt{45}}{18}$

Concept: A number is divisible by 4 if the number formed by its last two digits is divisible by 4.

1. In 32043p88, the last two digits are 88.
2. $88÷4=22$ ✓ – divisible for every value of $p$.
3. So $p$ can be any digit 0–9; maximum $p=9$.

Trick: If $p$ is not among the last two digits, it is free – take the largest digit 9.

Answer: (D) 9

Concept: Average speed $=\frac{\text{Total distance}}{\text{Total time}}$ (never the average of speeds).

1. Times: $\frac{204}{68}=3\text{h}$, $\frac{424}{53}=8\text{h}$, $\frac{366}{61}=6\text{h}$; total $=17\text{h}$.
2. Distance $=204+424+366=994\text{km}$.
3. Average speed $=\frac{994}{17}\approx 58.47\text{km/h}$.

Trick: All divisions are exact (3, 8, 6) – a sign you are on the right track.

Answer: (C) 58.47

Concept: With 30% gain, SP $=\frac{130}{100}\times$ CP, so profit $=\frac{30}{130}\times$ SP.

1. CP = ₹ $\frac{2938\times 100}{130}=2260$
2. Profit = ₹ $2938-2260=678$

Trick: Profit $=\frac{3}{13}\times 2938=3\times 226=678$.

Answer: (C) 678

Concept: Successive discounts multiply the remaining fractions: net price $=P(1-d_1)(1-d_2)(1-d_3)$.

1. Remaining fraction $=0.71\times 0.72\times 0.87$.
2. $0.71\times 0.72=0.5112$; $0.5112\times 0.87=0.444744$.
3. Single discount $=(1-0.444744)\times 100=55.5256\%\approx 55.53\%$.

Trick: Two at a time: $29\%,28\%\to 29+28-\frac{29\times 28}{100}=48.88\%$; then with $13\%$: $48.88+13-\frac{48.88\times 13}{100}=55.53\%$.

Answer: (C) $55.53\%$

Concept: Find the change in each letter and in the number.

1. MP 14 → LS −1: M−1=L, P+3=S, $14-15=-1$.
2. NK 5 → MN −10: N−1=M, K+3=N, $5-15=-10$ ✓.
3. SW −1: S−1=R, W+3=Z, $-1-15=-16$ → RZ −16.

Trick: Pattern: first letter −1, second letter +3, number −15.

Answer: (D) RZ −16

Concept: Find the fixed chain of operations in both given sets.

1. $5\to 6$ (+1), $6\to 10$ (+4), $10\to 20$ (×2).
2. $12\to 13$ (+1), $13\to 17$ (+4), $17\to 34$ (×2) ✓.
3. (A): $17\to 18$ (+1), $18\to 22$ (+4), $22\to 44$ (×2) ✓. Others fail at the last step (e.g. $20\times 2\ne 30$).

Trick: Check the last step (×2) first — it eliminates options fastest.

Answer: (A) 17 - 18 - 22 - 44

Concept: Check each condition for the group; if none applies, code directly from the table.

1. Group: # 8 9 $. Direct codes: # = Z, 8 = B, 9 = Y, $ = U.
2. Condition 1: first is a symbol but last ($) is not an odd number ✗. Condition 2: first is not an odd number ✗.
3. Condition 3: 8 is a perfect cube but 9 is not ✗.
4. So code directly: Z B Y U.

Trick: Test the conditions in order; when all fail, just read the table.

Answer: (D) ZBYU

Concept: Two interleaved series.

1. Odd positions: 61, 61, 61, 61 (constant).
2. Even positions: 54, 47, 40, ? → each −7.
3. $40-7=33$.

Trick: When a number keeps repeating, separate the alternate terms.

Answer: (B) 33

Concept: Decode each symbol and draw the family tree.

1. M × N: M is the wife of N.
2. N ÷ O: N is the father of O → M is the mother of O.
3. O – P: O is the brother of P; P + Q: P is the sister of Q.
4. So O, P, Q are siblings and M is the mother of Q.

Trick: Wife of the father = mother; siblings share the same mother.

Answer: (B) Mother

Concept: Compare the shift of each letter from the first cluster to the second.

1. UO-ZG: U+5=Z, O−8=G.
2. BV-GN: B+5=G, V−8=N. WQ-BI: W+5=B, Q−8=I.
3. JD-OW: J+5=O ✓ but D−8=V, not W ✗.

Trick: Use positions: D=4, $4-8+26=22$ = V.

Answer: (C) JD-OW