VARDAAN LEARNING INSTITUTE

(a) 1.275 × 10^-5
Decimal moves 5 places to the right to get 1.275.

(c) 4% decrease
Net change = x + y + (xy/100) = 20 - 20 + (20*-20/100) = -4%.

(b) ₹ 210
A = 1000(1 + 10/100)² = 1000(1.1)² = 1000(1.21) = 1210. CI = 1210 - 1000 = 210.

(c) 1
x^(a-b)(a+b) × x^(b-c)(b+c) × x^(c-a)(c+a) = x^{a²-b² + b²-c² + c²-a²} = x⁰ = 1.

(a) 6
No. of diagonals = n(n-3)/2 = 9. => n(n-3) = 18. Since 6(3) = 18, n = 6.

(a) 35
Adjacent angles sum to 180°. (3x - 10) + (2x + 15) = 180 => 5x + 5 = 180 => 5x = 175 => x = 35.

(b) 1/9
Pairs summing to 9: (3,6), (4,5), (5,4), (6,3). Total outcomes = 36. Prob = 4/36 = 1/9.

(b) 14
x² + 1/x² = (x + 1/x)² - 2 = (4)² - 2 = 16 - 2 = 14.

(c) 4
Highest sum of powers of variables in a term: 3xy³ => 1+3 = 4.

(c) (-3, 0)
A point on the x-axis has y-coordinate 0.

(a) 486 cm²
a³ = 729 => a = 9. TSA = 6a² = 6(81) = 486.

(c) 16
Inverse variation: M1*D1 = M2*D2 => 12*20 = M2*15 => M2 = 240/15 = 16.

(b) (2x - 3)²
(2x)² - 2(2x)(3) + (3)² = (2x - 3)².

(d) 7
392 = 2 × 2 × 2 × 7 × 7. To complete the triplet for 7, we need one more 7.

(a) 2.4
y = 1.6 × 1.5 = 2.40.

(a) 40, 55
x + (x+15) = 95 => 2x = 80 => x = 40. Numbers are 40 and 55.

(b) 68 cm
Diagonals bisect at 90°. Half-diagonals are 8 and 15. Side = √(8²+15²) = √(64+225) = √289 = 17. Perimeter = 4 × 17 = 68.

(b) Work done and Number of workers

(b) 7 cm
Area = 1/2 × (sum of parallel sides) × h. 34 = 1/2 × (10 + x) × 4. 17 = 10 + x => x = 7.

(d) (0, 0)
The origin is the intersection of axes at (0,0).

LCM of 8, 15, 20 = 120. Prime factorization of 120 = 2 × 2 × 2 × 3 × 5. To make it a square, we need pairs. Unpaired factors are 2, 3, 5. Multiply 120 by (2 × 3 × 5) = 30. Smallest square number = 120 × 30 = 3600.

5^2x+1 ÷ 5² = 5³ 5^{(2x+1) - 2} = 5³ 2x - 1 = 3 => 2x = 4 => x = 2.

(-3)^m+1+5 = (-3)⁷ m + 6 = 7 m = 1.

Let angles be 2x and 3x. Adjacent angles sum to 180°. 2x + 3x = 180 => 5x = 180 => x = 36°. Angles are 72° and 108°. Opposite angles are equal, so angles are 72°, 108°, 72°, 108°.

Direct variation. Let bottles be x. 840/6 = x/5 140 = x/5 => x = 700. 700 bottles.

Cube root of smaller number = 3. So, smaller number = 3³ = 27. Let larger number be x. x - 27 = 189. x = 189 + 27 = 216. Cube root of 216 = 6. Cube root of larger number is 6.

Fabina (SI): P=12500, R=12%, T=3. SI = (12500×12×3)/100 = ₹ 4500. Radha (CI): P=12500, R=10%, T=3. A = 12500(1 + 10/100)³ = 12500(1.331) = ₹ 16637.5. CI = 16637.5 - 12500 = ₹ 4137.5. Difference = 4500 - 4137.5 = 362.5. Fabina pays more interest by ₹ 362.5.

Division: z(5z² - 80) = z × 5(z² - 16) = 5z(z - 4)(z + 4). Divide by 5z(z + 4) => [5z(z-4)(z+4)] / [5z(z+4)] = z - 4.
OR (Factorise): p² + 6p - 16. Split middle term: p² + 8p - 2p - 16. p(p + 8) - 2(p + 8) = (p - 2)(p + 8).

Area of 1 tile = 1/2 × d1 × d2 = 1/2 × 45 × 30 = 675 cm². Total Area = 3000 × 675 = 20,25,000 cm². Convert to m²: 2025000 / 10000 = 202.5 m². Cost = 202.5 × 4 = ₹ 810. Total cost is ₹ 810.

LCM of 4 and 3 is 12. Multiply entire equation by 12. 3(3t - 2) - 4(2t + 3) = 12(2/3) - 12t 9t - 6 - 8t - 12 = 8 - 12t t - 18 = 8 - 12t 13t = 26 => t = 2.

Let numerator = x. Denominator = x + 8. (x + 17) / (x + 8 - 1) = 3/2 (x + 17) / (x + 7) = 3/2 2(x + 17) = 3(x + 7) => 2x + 34 = 3x + 21. x = 13. Denominator = 21. Rational number is 13/21.

(i) (100+2)(100+3) = 10000 + 500 + 6 = 10506. (ii) (100-1)² = 10000 - 200 + 1 = 9801. (iii) (300-3)(300+3) = 300² - 3² = 90000 - 9 = 89991.
OR: Let x = a+b. (x+c)(x-c) = x² - c². (a+b)² - c² = a² + 2ab + b² - c².

Case 1 (Height=4cm): Circumference = 11 cm. 2πr = 11 => r = 11×7/44 = 1.75 cm. V1 = πr²h = (22/7)×(1.75)²×4 = 38.5 cm³. Case 2 (Height=11cm): Circumference = 4 cm. 2πr = 4 => r = 4×7/44 = 0.636 cm. V2 = πr²h = (22/7)×(7/11)²×11 = (22/7)×(49/121)×11 = 14 cm³. Ratio V1:V2 = 38.5 : 14 = 385 : 140 = 11 : 4.

Speed = 75 km/h. (i) Distance = Speed × Time = 75 × (20/60) = 75 × (1/3) = 25 km. (ii) Time = Distance / Speed = 250 / 75 = 10/3 hours = 3 hours 20 minutes.

(a) All x-coordinates are 4. This is a vertical line parallel to the y-axis. Yes, they lie on a line. (b) y = x for all points. Yes, they lie on a line passing through origin. (c) Points (2,3), (5,3), (5,5), (2,5) form a rectangle, not a single straight line. No.

(i) Area(ABC) = 1/2 × base × height = 1/2 × 48 × 15 = 360 m². (ii) Area(ADC) = 1/2 × 48 × 25 = 600 m². (iii) Total Area = 360 + 600 = 960 m².

(i) Total MP = 1450 + (2 × 850) = 1450 + 1700 = ₹ 3150. (ii) Discount = 10% of 3150 = ₹ 315. (iii) Net amount = 3150 - 315 = ₹ 2835.

Frequency Table Construction:

Class Interval (Wages in ₹)	Tally Marks	Frequency (No. of Workers)
800-810	|||	3
810-820	||	2
820-830	|	1
830-840	|||| ||||	9
840-850	||||	5
850-860	|	1
860-870	|||	3
870-880	|	1
880-890	|	1
890-900	||||	4
Total		30

(i) 830-840 has the maximum workers (9). (ii) ₹ 850 and more = 1+3+1+1+4 = 10 workers. (iii) Less than ₹ 830 = 3+2+1 = 6 workers.