Q&A: practice questions with answers

Concept: Sleep is a low-power state: open documents and programs are kept in RAM, the display and most hardware are switched off, and the PC resumes within seconds exactly where you left it.

1. Shut down closes everything and powers off; Restart reboots; Disk Cleanup deletes temporary files.
2. Hibernate is similar but saves the session to the hard disk and uses no power.

Trick: Sleep = RAM (fast, little power); Hibernate = Disk (zero power).

Answer: (C) To save all open documents and programs while reducing power consumption

Concept: In an isosceles triangle the two base angles are equal, and the three angles add up to ${180}^{\circ }$.

1. Let each base angle be $x$. Then ${132}^{\circ }+x+x={180}^{\circ }$.
2. $2x={180}^{\circ }-{132}^{\circ }={48}^{\circ }\Rightarrow x={24}^{\circ }$.
3. Options A, C, D give unequal base angles, impossible for an isosceles triangle.

Trick: Base angle $=\frac{{180}^{\circ }-\text{vertex angle}}{2}$.

Answer: (B) ${24}^{\circ }$ and ${24}^{\circ }$

Concept: The distance of a point $(x,y)$ from the x-axis is $|y|$ and from the y-axis is $|x|$.

1. For $P(2,3)$: $x=2$, $y=3$.
2. Distance from the x-axis $=|y|=3$.
3. Option D (2) is the distance from the y-axis – the usual trap.

Trick: "From x-axis → take y"; "from y-axis → take x" (the opposite coordinate).

Answer: (B) 3

Concept: An open drum has a curved inner wall and a circular base (no lid). Area to paint $=\text{CSA}+\text{base}=2\pi rh+\pi r^2=\pi r(2h+r)$.

1. $r=\frac{78}{2}=39\text{m}$, $h=79\text{m}$.
2. $2h+r=158+39=197$.
3. Area $=\frac{22}{7}\times 39\times 197=\frac{169026}{7}\approx 24146.57\approx 24146.6{\text{m}}^2$.

Trick: "Open" → add only one circle; "closed" → add two. Factor out $\pi r$ to cut the arithmetic.

Answer: (D) 24146.6

Concept: In a percentage pie chart, amount of a sector $=\frac{\text{sector }\%}{100}\times \text{total}$.

1. From the chart, Food $=25\%$ (Education 27%, Rent 16%, Bills 32%).
2. Expenditure on Food $=\frac{25}{100}\times 10000=$ ₹$2500$.
3. ₹2000 would be 20% and ₹1700 is 17% – neither is the Food sector.

Trick: $25\%=\frac{1}{4}$ → simply divide the total by 4.

Answer: (C) 2500

Concept: Evaluate each square root of a perfect square separately, then add/subtract.

1. $\sqrt{144}=12$, $\sqrt{0.04}=\sqrt{\frac{4}{100}}=0.2$, $\sqrt{9}=3$.
2. $12+0.2-3=9.2$.

Trick: For decimals, $\sqrt{0.04}=0.2$ (half the decimal places), not $0.02$.

Answer: (C) 9.2

Concept: Two consecutive positive integers are $x$ and $x+1$; translate "product is 277" into an equation.

1. $x(x+1)=277$.
2. $x^2+x=277\Rightarrow x^2+x-277=0$.
3. Option C would come from $x(x-1)$ and A/B have the wrong sign of 277.

Trick: "Product = k" → move $k$ to the left as $-k$.

Answer: (D) $x^2+x-277=0$

Concept: In an A.P., $a_n=a+(n-1)d$; the difference between two terms is $(\text{gap in positions})\times d$.

1. $a_{20}-a_{16}=4d=441-385=56\Rightarrow d=14$.
2. $a_{90}=a_{20}+70d=441+70\times 14=441+980=1421$.

Trick: No need to find $a$: jump directly from a known term, $a_m=a_n+(m-n)d$.

Answer: (C) 1421

Concept: For two numbers, $\text{HCF}\times \text{LCM}=\text{product of the numbers}$.

1. Let the numbers be $\frac{4}{9}b$ and $b$. Then $\frac{4}{9}b\times b=12\times 300=3600$.
2. $b^2=3600\times \frac{9}{4}=8100\Rightarrow b=90$.
3. Smaller number $=\frac{4}{9}\times 90=40$. The pair satisfying the ratio and the product is $40$ and $90$, whose smaller member is $40$.

Trick: Ratio $4:9$ → numbers $4k,9k$; $36k^2=3600\Rightarrow k=10$ → $40,90$.

Answer: (C) 40

Concept: The tower, its shadow and the line to the sun form a right triangle; $\tan \theta =\frac{\text{height}}{\text{shadow}}$.

1. Let shadow $=x$; then height $=\sqrt{3}x$.
2. $\tan \theta =\frac{\sqrt{3}x}{x}=\sqrt{3}$.
3. $\tan {60}^{\circ }=\sqrt{3}$, so $\theta ={60}^{\circ }$.

Trick: height $=\sqrt{3}\times$ shadow → ${60}^{\circ }$; shadow $=\sqrt{3}\times$ height → ${30}^{\circ }$; equal → ${45}^{\circ }$.

Answer: (C) ${60}^{\circ }$

Concept: ${\sigma }^2=\frac{\sum x^2}{n}-{\overline{x}}^2$. Recover $\sum x$ and $\sum x^2$, replace the wrong value, and recompute.

1. Wrong data: $\sum x=25\times 42=1050$; $\sum x^2=n({\sigma }^2+{\overline{x}}^2)=25(25+1764)=44725$.
2. Corrected: $\sum x=1050-10+210=1250\Rightarrow \overline{x}=50$; $\sum x^2=44725-{10}^2+{210}^2=44725-100+44100=88725$.
3. ${\sigma }^2=\frac{88725}{25}-{50}^2=3549-2500=1049\Rightarrow \sigma =\sqrt{1049}$.

Trick: Only $\sum x$ and $\sum x^2$ change: subtract the wrong value (and its square), add the correct one.

Answer: (A) $\sqrt{1049}$

Concept: A quadratic with roots $\alpha ,\beta$ is $x^2-(\alpha +\beta )x+\alpha \beta =0$.

1. Sum of roots $=9+11=20$; product $=9\times 11=99$.
2. Equation: $x^2-20x+99=0$.
3. Check: $(x-9)(x-11)=x^2-20x+99$ ✓.

Trick: "minus sum, plus product" – signs are the usual trap (options B, D).

Answer: (A) $x^2-20x+99=0$

Concept: Simplify the denominator first, then divide by a fraction by multiplying with its reciprocal.

1. $\frac{5}{6}+\frac{2}{4}=\frac{5}{6}+\frac{1}{2}=\frac{5+3}{6}=\frac{8}{6}=\frac{4}{3}$
2. $\frac{1}{4/3}=\frac{3}{4}$
3. $\frac{3}{4}÷\frac{4}{29}=\frac{3}{4}\times \frac{29}{4}=\frac{87}{16}=5\frac{7}{16}$

Trick: $\frac{1}{a/b}=\frac{b}{a}$ — flipping a fraction is just swapping numerator and denominator.

Answer: (B) $5\frac{7}{16}$

Concept: Three points are collinear if the slope between any two pairs is equal.

1. Slope of $(-5,1)$ and $(4,-2)$: $\frac{-2-1}{4-(-5)}=\frac{-3}{9}=-\frac{1}{3}$
2. Slope of $(-5,1)$ and $(1,k)$: $\frac{k-1}{1-(-5)}=\frac{k-1}{6}$
3. $\frac{k-1}{6}=-\frac{1}{3}\Rightarrow k-1=-2\Rightarrow k=-1$

Trick: Or use area $=0$: $x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)=0$.

Answer: (A) $-1$

Concept: $B^{\prime }=U-B$ = elements of $U$ that are not in $B$; $n(B^{\prime })=n(U)-n(B)$.

1. $n(U)=9$, $n(B)=4$
2. $B^{\prime }=\{1,3,5,7,9\}$, so $n(B^{\prime })=9-4=5$

Trick: Complement count = total − given.

Answer: (A) 5

Concept: $P(A\cup B)=P(A)+P(B)-P(A\cap B)$, and for independent events $P(A\cap B)=P(A)\times P(B)$.

1. $P(A\cap B)=0.4\times 0.6=0.24$
2. $P(A\text{ or }B)=0.4+0.6-0.24=0.76$

Trick: $P(A\cup B)=1-P(A^{\prime })P(B^{\prime })=1-0.6\times 0.4=0.76$.

Answer: (D) 0.76

Concept: When first term $a$ and last term $l$ are known: $S_n=\frac{n}{2}(a+l)$.

1. $236=\frac{n}{2}(28+31)=\frac{59n}{2}$
2. $n=\frac{236\times 2}{59}=\frac{472}{59}=8$

Trick: Sum = number of terms × average of first and last term: $\frac{236}{29.5}=8$.

Answer: (A) 8

Concept: $n(M\cup P)=n(M)+n(P)-n(M\cap P)$.

1. $26=14+n(P)-6\Rightarrow n(P)=18$ teach physics in all.
2. Teachers who teach only physics $=26-14=12$ (everyone outside the mathematics circle).
3. Only mathematics $=14-6=8$; check: $8+6+12=26$ ✓

Trick: Draw the Venn diagram: 8 | 6 | 12. 'Physics' (total) = 18, 'only physics' = 12 — read the question carefully.

Answer: (B) 12 — the number who teach physics but not mathematics (the intended reading of the question).

Concept: In similar triangles, corresponding sides and perimeters are in the same ratio.

1. $YZ$ corresponds to $BC$: ratio $=\frac{YZ}{BC}=\frac{8}{4}=2$
2. Perimeter of $△ABC=5+4+4.5=13.5\text{cm}$
3. Perimeter of $△XYZ=2\times 13.5=27\text{cm}$

Trick: $\frac{{\text{Perimeter}}_1}{{\text{Perimeter}}_2}=\frac{{\text{side}}_1}{{\text{side}}_2}$ — no need to find each side.

Answer: (B) 27 cm

Concept: A force is conservative if the work done by it depends only on the initial and final positions (zero over a closed path); only then is mechanical energy $KE+PE$ conserved.

1. Gravitational, spring (elastic) and electrostatic forces are conservative – each has a potential energy function.
2. Friction always opposes motion, so its work over a round trip is negative, not zero. It converts mechanical energy into heat.
3. Hence mechanical energy is not conserved when friction acts.

Yaad rakhein: Friction, air drag, viscous force = non-conservative (energy "leaks" as heat).

Answer: (B) Frictional force