Vardaan Learning Institute
Created by Team Vardaan
Answer Key / Marking Scheme
SECTION A: MCQs (1 Mark each)
1. (d) 7
$392 = 2^3 \times 7^2$. Need one 7 to make pair of 3.
$392 = 2^3 \times 7^2$. Need one 7 to make pair of 3.
2. (c) 0.948
$\sqrt{0.9000...} \approx 0.948$
$\sqrt{0.9000...} \approx 0.948$
3. (b) $10^{100}$
4. (c) 24
$15 \times 48 = x \times 30 \Rightarrow x = 24$
$15 \times 48 = x \times 30 \Rightarrow x = 24$
5. (b) 4%
Disc = 600. $\frac{600}{15000} \times 100 = 4\%$
Disc = 600. $\frac{600}{15000} \times 100 = 4\%$
6. (c) $P(1 +
R/200)^{2n}$
7. (b) $4ab$
8. (c) 3
9. (b) $80^\circ,
100^\circ$
$4x+5x=180 \Rightarrow x=20$
$4x+5x=180 \Rightarrow x=20$
10. (c) 2
11. (c) Square
12. (b) Two adjacent
sides
13. (c) $64 \text{
cm}^3$
$6a^2=96 \Rightarrow a=4$
$6a^2=96 \Rightarrow a=4$
14. (d) Negative
y-axis
15. (b) Class Interval
16. (a) $2/5$
Red=4, Total=10. $4/10 = 2/5$
Red=4, Total=10. $4/10 = 2/5$
17. (a) $F + V - E =
2$
18. (b) 6
19. (b) $4x - 5y$
20. (c) 4 times
SECTION B (2 Marks each)
21.
12.5
Pairing: $\overline{1} \overline{56} . \overline{25}$
Step 1: $1 \times 1 = 1$. Remainder 0. Bring down 56.
Step 2: Divisor 22. $22 \times 2 = 44$. Rem 12. Bring down 25.
Step 3: Divisor 245. $245 \times 5 = 1225$. Rem 0.
Step 1: $1 \times 1 = 1$. Remainder 0. Bring down 56.
Step 2: Divisor 22. $22 \times 2 = 44$. Rem 12. Bring down 25.
Step 3: Divisor 245. $245 \times 5 = 1225$. Rem 0.
22.
-1
$\{(1/3)^{-1} - (1/4)^{-1}\}^{-1} = \{3 - 4\}^{-1} = \{-1\}^{-1} = \frac{1}{-1} = -1$
23.
$(5a - 2b + 7c)(5a + 2b - 7c)$
$= 25a^2 - (4b^2 - 28bc + 49c^2) = (5a)^2 - (2b - 7c)^2$
Apply $x^2 - y^2 = (x-y)(x+y)$.
Apply $x^2 - y^2 = (x-y)(x+y)$.
24.
7 cm
Area = $\frac{1}{2}(a+b)h \Rightarrow 34 = \frac{1}{2}(10+b) \times 4$
$34 = 2(10+b) \Rightarrow 17 = 10+b \Rightarrow b = 7$.
$34 = 2(10+b) \Rightarrow 17 = 10+b \Rightarrow b = 7$.
25.
$x = 5$
$5^{x+1} \times 5^3 = 5^9 \Rightarrow 5^{x+1+3} = 5^9$
$x+4 = 9 \Rightarrow x = 5$.
$x+4 = 9 \Rightarrow x = 5$.
26.
$2/5$
Total tosses = 200. Tails = 80.
Probability = $80/200 = 8/20 = 2/5$.
Probability = $80/200 = 8/20 = 2/5$.
SECTION C (3 Marks each)
27.
Number: 1024, Square Root: 32
Smallest 4-digit number is 1000.
$31^2 = 961$ (3 digits). $32^2 = 1024$ (4 digits).
$31^2 = 961$ (3 digits). $32^2 = 1024$ (4 digits).
28.
Number: 85 (or 58)
Let digits be $x, y$. Number $10x+y$. $x-y=3$.
Sum of numbers: $(10x+y) + (10y+x) = 143 \Rightarrow 11(x+y)=143 \Rightarrow x+y=13$.
Solving $x+y=13, x-y=3$ gives $x=8, y=5$.
Sum of numbers: $(10x+y) + (10y+x) = 143 \Rightarrow 11(x+y)=143 \Rightarrow x+y=13$.
Solving $x+y=13, x-y=3$ gives $x=8, y=5$.
29.
$\angle 1 = 83^\circ, \angle 2 = 97^\circ$
Consecutive interior angles sum to $180^\circ$.
$(2x+15) + (3x-5) = 180 \Rightarrow 5x + 10 = 180 \Rightarrow 5x=170 \Rightarrow x=34$.
$\angle 1 = 2(34)+15 = 83^\circ$. $\angle 2 = 3(34)-5 = 97^\circ$.
$(2x+15) + (3x-5) = 180 \Rightarrow 5x + 10 = 180 \Rightarrow 5x=170 \Rightarrow x=34$.
$\angle 1 = 2(34)+15 = 83^\circ$. $\angle 2 = 3(34)-5 = 97^\circ$.
30.
(i) 8, (ii) $2\sqrt{15}$
$(x+1/x)^2 = x^2+1/x^2+2 = 62+2=64 \Rightarrow x+1/x = 8$.
$(x-1/x)^2 = x^2+1/x^2-2 = 62-2=60 \Rightarrow x-1/x = \sqrt{60} = 2\sqrt{15}$.
$(x-1/x)^2 = x^2+1/x^2-2 = 62-2=60 \Rightarrow x-1/x = \sqrt{60} = 2\sqrt{15}$.
31.
10 days
Inverse Variation. Men $\times$ Days = Constant.
$125 \times 16 = (125+75) \times D_2$
$2000 = 200 \times D_2 \Rightarrow D_2 = 10$.
$125 \times 16 = (125+75) \times D_2$
$2000 = 200 \times D_2 \Rightarrow D_2 = 10$.
32.
CP = ₹600
MP = 800. Discount 10% = ₹80. SP = ₹720.
Profit 20%. $SP = CP(1 + \frac{20}{100}) \Rightarrow 720 = CP(1.2)$
$CP = 720 / 1.2 = 600$.
Profit 20%. $SP = CP(1 + \frac{20}{100}) \Rightarrow 720 = CP(1.2)$
$CP = 720 / 1.2 = 600$.
33.
Construction Step Check
1. Draw QR = 6cm. 2. From Q, arc 4cm. From R, arc 7cm. Intersection P.
3. From P, arc 5.5cm. From R, arc 5cm. Intersection S. 4. Join all.
3. From P, arc 5.5cm. From R, arc 5cm. Intersection S. 4. Join all.
SECTION D (4 Marks each)
35.
Sum = ₹8000
$SI = \frac{P \times 5 \times 3}{100} = 0.15P$.
$CI = P(1.05)^3 - P = P(1.157625 - 1) = 0.157625P$.
Difference = $0.007625P = 61$.
$P = 61 / 0.007625 = 8000$.
$CI = P(1.05)^3 - P = P(1.157625 - 1) = 0.157625P$.
Difference = $0.007625P = 61$.
$P = 61 / 0.007625 = 8000$.
36.
(ii) $\angle DAO = 55^\circ$
(i) SAS Congruence (Diagonals bisect, Vertically opp angles).
(ii) Vertically opp $\angle DOC = 110^\circ$. $\Delta DOC$ is isosceles ($OD=OC$).
$\angle ODC = (180-110)/2 = 35^\circ$. Alt interior $\angle OBA = 35^\circ$.
In right $\Delta DAB$ ($\angle A=90$), $\angle DAO = 90 - \angle OAB = 90 - 35 = 55^\circ$.
(ii) Vertically opp $\angle DOC = 110^\circ$. $\Delta DOC$ is isosceles ($OD=OC$).
$\angle ODC = (180-110)/2 = 35^\circ$. Alt interior $\angle OBA = 35^\circ$.
In right $\Delta DAB$ ($\angle A=90$), $\angle DAO = 90 - \angle OAB = 90 - 35 = 55^\circ$.
37.
Quotient: $2x^2 - x - 9$, Remainder: $-80x - 28$
Perform polynomial long division.
Verify: Dividend = Divisor $\times$ Quotient + Remainder.
38.
1980 $m^2$
$r = 42 \text{ cm} = 0.42 \text{ m}$. $h = 1 \text{ m}$.
CSA of one rev = $2\pi rh = 2 \times \frac{22}{7} \times 0.42 \times 1 = 2.64 \text{ m}^2$.
Total Area = $2.64 \times 750 = 1980 \text{ m}^2$.
CSA of one rev = $2\pi rh = 2 \times \frac{22}{7} \times 0.42 \times 1 = 2.64 \text{ m}^2$.
Total Area = $2.64 \times 750 = 1980 \text{ m}^2$.
39.
(ii) ₹200, (iii) ₹3500
Relationship is linear ($SI = \frac{P \times R \times T}{100}$). Rate is 8%.
(ii) For 2500, SI = 200. (iii) For 280, Deposit = 3500.
(ii) For 2500, SI = 200. (iii) For 280, Deposit = 3500.
40.
8.4 seconds
Relative Speed = $60 + 90 = 150 \text{ km/h}$.
Convert to m/s: $150 \times \frac{5}{18} = \frac{125}{3} \text{ m/s}$.
Total Distance = $150 + 200 = 350 \text{ m}$.
Time = $\frac{\text{Distance}}{\text{Speed}} = 350 \div \frac{125}{3} = 350 \times \frac{3}{125} = 8.4 \text{ s}$.
Convert to m/s: $150 \times \frac{5}{18} = \frac{125}{3} \text{ m/s}$.
Total Distance = $150 + 200 = 350 \text{ m}$.
Time = $\frac{\text{Distance}}{\text{Speed}} = 350 \div \frac{125}{3} = 350 \times \frac{3}{125} = 8.4 \text{ s}$.
*** END OF ANSWER KEY ***