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Ortho center = (14/5, 1/5)

Step-by-step explanation:

Given, the vertices of the triangle,

A = (5, -2)

B = (-1, 2)

C = (1, 4)

Orthogonal center is the cross section of altitudes of the triangle.

Slope of AB

= y₂−y/x₂−x₁

= 2+2/-1-5

= -2/3

Altitude from C to AB is perpendicular to AB.

= Perpendicular slope of AB

= −1/Slope of AB

= 3/2

The equation of CF is given as, (F is the point on AB)

y – y₁ = m(x – x₁)

y - 4 = 3/2(x – 1)

2y – 8 = 3x - 3

3x - 2y = -5 ——— (1)

Slope of BC

= y₂–y/x₂–x₁

= 4–2/1+1

= 1

Slope of AD (AD is altitude)

Perpendicular slope of BC

= −1/Slope of BC

= −1

The equation of AD is given as,

y – y₁ = m(x – x₁)

y + 2 = -1(x – 5)

x + y = 3 ——– (2)

Subtracting equation (1) and 3 x (2),

(3x  -  2y  =  -5) - (3x  + 3y =  9)

= -5y = -14

y = 14/5

Substituting the value of y in equation (2),

X = 3 – 14/5 = 1/5

Ortho center = (14/5, 1/5).

The smallest non-zero number is 1.

The factor of 1 = 1

1 has only one factor which is 1 itself.

The next smallest non-zero number is 2.

Factors of 2 = 1 and 2

It has only two factors, 1 and the number itself 2.

Therefore, 2 is the smallest prime number.

Let a be the first term and d be the common difference of the AP.

General term of an A.P.

⇒ a_n​=a+(n−1)d

We have, a₈​=31 and a₁₅​=16+a11​

⟹a+7d=31 and a+14d=16+a+10d

⟹a+7d=31 and 4d=16

⟹a+7d=31 and d=4

⟹a+7×4=31

⟹a+28=31

⟹a=3

Hence, the AP is a,a+d,a+2d,a+3d ...

i.e.,3, 7, 11, 15, 19, ...

S_n = 3n² - 4n

⇒ S_n-1 = 3(n - 1)² - 4(n - 1)

∴ S_n - S_n-1 = (3n²- 4n) - {3(n - 1)²- 4(n-1)}

⇒ a_n = (3n² - 4n)- {3(n²- 2n + 1) - 4n ± 4} = 3n² - 4n - {3n²- 6n + 3 - 4n + 4}

= 3n² - 4n - {3n² -10n + 7}

= 3n²- 4n - 3n² + 10n - 7

= 6n - 7

Hence, required n^th term is 6n - 7.

1. b) ₹ 72,000

2. d) ₹ 250

3. c) ₹ 350

4. d) ₹ 900.

i. (c) Debited to Partners capital A/c

ii. (c) Shown on Liability side of the Balance Sheet

iii. (a) Dr. Workmen’s Compensation Fund ₹ 10,000, Cr. Claim for Workmen’s Compensation ₹ 6,000; Cr. X’s Capital A/c ₹ 2,500; Cr. Y’s Capital A/c ₹ 1,500

iv. (a) Y’s Capital A/c Dr.3,000; X’s Capital A/c Cr.3,000

i. (b) ₹ 5000 Debited to Revaluation A/c

ii. (a) Dr. Investment Fluctuation Reserve A/c ₹ 30,000, Cr. Investment ₹ 10,000; Cr. A’s Capital A/c ₹ 10,000; Cr. B’s Capital A/c ₹ 6,000; Cr. C’s Capital A/c ₹ 4,000

iii. (a) B’s Capital A/c Dr.10,000; C ’s Capital A/c Dr.40,000; A’s Capital A/c Cr. 50,000

iv. (a) ₹ 17,000

1. a) 2/10

2. b) 4, 00,000

3. b) 80,000

4. c) Distributed among old partners in their old ratio.

1) (b) profit ₹1,20,000

2) (b) Land & Building

3) (c) ₹ 60,000

4) (d) old : old

1. c) ₹ 7,500

2. b) ₹ 10,000

3. c) ₹ 5,000

4. a) ₹ 2,500

Correct option is 

1 (b) Loss ₹ 750

2 (c) ₹1,50,000

3 (b) ₹ 30,750

4 (b) old, old

1. a) ₹ 1,500

2. b) ₹ 4,800

3. a) ₹ 12,000

4. b) ₹ 450

Correct option is 

1 (c) ₹ 19,500

2 (b) Profit ₹ 6,300

3 (c) 4:3

4 (b) sacrificing ratio

i. a. Dr. X Capital a/c ₹15,000; Dr. Y Capital a/c ₹10,000; Dr. Z Capital a/c ₹5,000; Cr. Deferred Advertisement Expenditure a/c ₹ 30,000

ii. a. Dr. Contingency Reserve a/c ₹9,000; Cr. X Capital a/c ₹4,500; Cr. Y Capital a/c ₹3,000; Cr. Z Capital a/c ₹ 1,500 b. Dr. Contingency Reserve a/c ₹9,00

iii. c. Dr. Z Capital a/c ₹ 16,000; Cr. Y Capital a/c ₹16,000

iv. c. X

1. (d)2:1

2. (d) Rs 2,80,000

3. (a) Rs 84,000

4. (b) C’s capital a/c

Correct option is 

1 (d) 2:1

2 (d) Rs 2,80,000

3 (a) Rs 84,000

4 (b) C’s capital a/c

Correct option is 

1 (b) Procuring additional capital

2 (d) ₹ 2,80,000

3 (d) None of the above

4 (c) ₹ 2,80,000

1. (b) Loss ₹ 750

2. (c) ₹1,50,000

3. (b) ₹ 30,750

4. (b) old, old

1. (d) Loss ₹ 20,000

2. (c) 1:1

3. (b) ₹ 3,20,000

4. (d) ₹ 34,000

1. b) ₹ 2,00,000

2. d) No salary will be given

3. a) ₹ 1,92,000

4. b) ₹ 96,000

1. a) ₹ 1,50,000

2. c) ₹ 4,76,000

3. b) ₹ 2,06,000

4. b) ₹ 1,03,000

1. (c) ₹ 19,500

2. (b) Profit ₹ 6,300

3. (c) 4:3

4. (b) sacrificing ratio

i. a. ₹ 33,000

ii. c. ₹ 80,000

iii. a. ₹ 44,000

iv. a. Dr. C’s capital a/c ₹25,000; Cr. A’s capital a/c ₹ 25,000

1. d) ₹ 4,60,000

2. c) ₹ 2,500

3. a) ₹ 1,00,000

4. b) ₹ 3,62,500

1. b. 50,000

2. a. 30,000

3. a. 1,13,000.

4. a. Share application a/c Dr to Bank

1. b. Issued capital

2. c. 25% of the issue price

3. c. Capital clause

4. a. Nothing

1. a) ₹(50,000)

2. c) Inventories

3. c) ₹6,38,000

4. a) ₹10,00,00,000

1. a. ₹19,400

2. a. ₹19,94,400

3. a. ₹23,95,000

4. a. Bank a/c Dr 5,93,900 To Share allotment a/c 5,93,900

Correct option is 

1 b. 50,000

2 a. 30,000

3 a. 1,13,000.

4 a. Share application a/c Dr to Bank

Correct option is 

1 b. Issued capital

2 c. 25% of the issue price

3 c. Capital clause

4 a. Nothing