Given:

pq + qr + rp = 0        ----(1)

Calculation:

Now, We have to find the value of 

[p² / (p² - qr)] + [q² / (q² - rp)] + [r² / (r² - pq)]

From equation (i), we get 

- qr = rp + pq      ----(ii)

- rp = pq + qr      ----(iii)

- pq = qr + rp      ----(iv) 

Now, From equation (ii), (iii) and (iv), we get 

⇒ [p² / (p² + pq + rp)] + [q² / (q² + pq + qr)] + [r² / (r² + qr + rp)]  

⇒ [p² / p (p + q + r)] + [q² / q (q + p + r)] + [r² / r (r + q +p)]

⇒ [p / (p + q + r)] + [q / (p + q + r)] + [r / (p + q + r)]

⇒ (p + q + r) / (p + q + r)

⇒ 1

∴ The required value of the expression is 1.

Shortcut Trick

⇒ 1 × 2

⇒ 2

From option 1.

⇒ 2 (x² + y²)

⇒ 2 × (1 + 0)

⇒ 2 × 1

⇒ 2 (satisfied)

Alternate Method

(A) If N = (2378)² + 2378 + 2379, then the value of √N is 2379.

⇒ N = 2378² + 2378 + 2378 + 1

⇒ N = 2378² + (2 × 2378) + 1

⇒ N = 2378² + (2 × 2378) + 1²

⇒ N = (2378 + 1)²

⇒ √N = 2379

∴ (A) is correct statement.

(B) The unit digit of 349 × 468 × 637 × 698 × 436 is 6.

⇒ 9 × 8 × 7 × 8 × 6

⇒ (72) × (56) × 6

⇒ (2 × 6) × 6

⇒ 72

⇒ 2

∴ The unit digit of the given expression is 2.

So, (B) is the wrong statement.

(C) The total number of positive factors of 3024 is 40.

⇒ 3024 = 2⁴ × 3³ × 7¹

Let there be a composite number N and its prime factors be a, b, c, d, … etc. and p, q, r, s, … etc. be the indices (or powers) of the a, b, c, d, … etc respectively i.e., if N can be expressed as –

N = a^p ×  b^q × c^r × d^s…..

⇒ The number of positive number of factors = (p + 1) (q + 1) (r + 1) (s + 1)

∴ The total number of positive factors of 3024 is (4 + 1) (3 + 1) (1 + 1) = 40

∴ (C) is the correct statement.

∴ A and C both are correct.

GIVEN:

When 6 is added to both numerator and denominator, then it becomes 7/9 and when 4 is subtracted from both numerator and denominator, then it becomes 1/2.

CALCULATION:

According to the question:

(p + 6)/(q + 6) = 7/9

⇒ 9p + 54 = 7q + 42

⇒ 7q - 9p = 12  ---(1)

And

(p - 4)/(q - 4) = 1/2

⇒ 2p - 8 = q - 4

⇒ 2p - q = 4       ---(2)

Multiplying equation (2) by 7 and adding in equation (1):

(7q - 9p) + (14p - 7q) = 12 + 28

⇒ 5p = 40

⇒ p = 8 and q = 12

∴ Simplified value of (p/q) = 8/12 = 2/3

Given:

If (x² - 4) is a factor of x⁴ - 5x³ + ax² + bx + c

Calculation:

If (x² - 4) is a factor of x⁴ - 5x³ + ax² + bx + c

We know, x = +2, -2

Put the value of x and polynomial is equal to 0

P(2) = 16 - 40 + 4a + 2b + c = 0

⇒ 4a + 2b + c = 24 ---(1)

P(-2) = 16 + 40 + 4a -2b + c = 0

⇒ 4a - 2b + c = -56

Adding (1) and (2)

⇒ 8a + 2c = -32

⇒ 4a + c + 16 = 0

∴ The required answer is 4a + c + 16 = 0

∴ (x³ + y³ + z³) = 3xyz

Given:

Statement I base is 7 times height of triangular field

Rolling cost Rs 1400 Rs 100 per hectare

Statement II base is 3 times height of triangular field

Rolling cost Rs 1350 Rs 100 per hectare

Formula used:

Area of triangle = 1/2 × b × h (b = base and h = height)

Calculation:

Statement-I

⇒ b = 7h

⇒ Area of triangular field (A) = 1400/100 hectare = 14 hectare

⇒ A = 14 × 10000 = 140000 m²  (1 hectare = 10000 m²)

Now,

⇒ A = 1/2 × 7h × h

⇒ 140000 m² =(7/2) × h²

⇒ 140000 m² × (2/7) = h²

⇒ h = √ 40000 m² = 200 m

Statement-II

⇒ b = 3h

⇒ Area of triangular field (A) = 1350/100 hectare = 13.5 hectare

⇒ A = 13.5 × 10000 = 135000 m2  (1 hectare = 10000 m2)

Now,

⇒ A = 1/2 × 3h × h

⇒ 135000 m² =(3/2) × h²

⇒ 135000 m² × (2/3) = h²

⇒ h = √ 90000 m² = 300 m

∴ The correct answer is option 1.

Calculation: 

For Statement I

Total surface Area of Cuboid = 2(lb + bh + hl)

Let length be 2y, breadth = 3y & height = y 

⇒ 550 = 2(2y × 3y + 3y × y + y × 2y)

⇒ 550 = 2 × 11y²

⇒ 25 = y²

⇒ y = 5

So, length = 10 cm, breadth = 15 cm, height = 5 cm

Volume = lbh

⇒ 10 × 15 × 5 = 750 cm³

Statement I alone is not sufficient to answer the question.

For Statement II

Total surface Area  of Cube = 6a²

⇒ 384 = 6a²

⇒ a = 8 cm

The volume of the Cube = a³

⇒ 83 = 512 cm³

Statement II alone is not sufficient to answer the question.

For Statement III

Height of the cuboid = y cm

Breadth = 3y cm

Length = 2y cm

Difference = 2y-y= y cm

⇒ y = 5cm

∴ Length = 10 cm, Breadth = 15 cm, Height = 5 cm

Volume = lbh

 ⇒10X 15X 5 = 750 cm³

Statement III alone is not sufficient to answer the question.

But Either statement II and  III or statement I and II are sufficient to answer the question.

∴ Correct Answer is option (4)

Given:

x + y + z = 0

xyz = -1/2

Formula used:

(a + b)² = a² + b² + 2ab

Calculation:

x + y + z = 0

⇒ x + y = -z

Squaring both side

⇒ (x + y)² = z²

⇒ x² + y² + 2xy = z²

⇒ x² + y² – z² = -2xy    ----(1)

Similarly,

⇒ x² + z² – y² = -2xz     -----(2)

⇒ y² + z² – x² = -2yz     -----(3)

Hence, required value,