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Given:

AB + AC = p

AB - AC = q

BD - CD = r

Formula used:

BC = $\dfrac{pq}{r}$

Calculation:

Using the formula, BC = $\dfrac{pq}{r}$

⇒ BC = $\dfrac{pq}{r}$

∴ The correct answer is option (3).

Given:

The sum of the height and radius of a right circular cylinder (h + r) = 21 cm

The curved surface area of the cylinder = 616 cm²

π = 22/7

Formula Used:

Curved Surface Area = 2πrh

Volume of the Cylinder = πr²h

Calculation:

⇒ h + r = 21 ⇒ h = 21 - r

⇒ Curved Surface Area = 2πrh

⇒ 616 = 2 × (22/7) × r × (21 - r)

⇒ 616 = (44/7) × r × (21 - r)

⇒ 616 × 7 = 44r(21 - r)

⇒ 4312 = 44r(21 - r)

⇒ 4312 = 924r - 44r²

⇒ 44r² - 924r + 4312 = 0

⇒ Divide through by 4:

⇒ 11r² - 231r + 1078 = 0

Using factorization:

⇒ 11r² - 154r - 77r + 1078 = 0

⇒ r(11r - 154) - 7(11r - 154) = 0

⇒ (r - 7)(11r - 154) = 0

⇒ r = 7 or r = 154/11

Since r < h:

⇒ r = 7 cm

⇒ h = 21 - r = 21 - 7 = 14 cm

Volume = πr²h

⇒ Volume = (22/7) × 7 × 7 × 14

⇒ Volume = 2156 cm³

∴ The correct answer is 2156 cm³.

Given:

ABC is a triangle right-angled at B.

P is the midpoint of AB.

Q is the midpoint of BC.

AQ = √73 units.

CP = √52 units.

Formula Used:

Area of a triangle = 1/2 × base × height

Calculation:

⇒ Since AQ and CP are provided, the actual base and height of the triangle can be determined geometrically. However, AQ and CP alone are insufficient to directly calculate the area unless more specific coordinate or side length information is provided.

∴ The correct answer is Both I and II.

Given:

Length, breadth, and height of the cuboid are consecutive integers.

Volume of the cuboid = 336 cubic units.

Formula Used:

Volume of cuboid = length × breadth × height.

Total surface area of cuboid = 2 × (length × breadth + breadth × height + height × length).

Calculation:

Let the consecutive integers be x, x + 1, and x + 2.

⇒ Volume = x × (x + 1) × (x + 2) = 336.

⇒ x(x² + 3x + 2) = 336.

⇒ x³ + 3x² + 2x - 336 = 0.

By trial, x = 6 satisfies the equation.

⇒ Length = 6, Breadth = 7, Height = 8.

Now, Total surface area = 2 × (6 × 7 + 7 × 8 + 8 × 6).

⇒ Total surface area = 2 × (42 + 56 + 48).

⇒ Total surface area = 2 × 146.

⇒ Total surface area = 292 square units.

∴ The correct answer is 292 square units.

Given:

Radius of the circle = 14 cm

APB is a shorter arc, and P is the midpoint of the arc.

PC = 7 cm

C is the midpoint of the chord AB.

Formula Used:

Length of chord AP = 2 × distance from center to chord

Distance from center to chord = √(radius² − distance from center to midpoint²)

Calculation:

⇒ Distance from center to chord = √(14² − 7²)

⇒ Distance from center to chord = √(196 − 49)

⇒ Distance from center to chord = √147

⇒ Distance from center to chord = 7√3 cm

⇒ Length of chord AP = 2 × 7√3

⇒ Length of chord AP = 14√3 cm

∴ The correct answer is 14 cm.

Given:

Height of the first pole = 10 m

Height of the second pole = 15 m

Distance between the poles = 25 m

Formula Used:

Height of the point of intersection = (h₁ × h₂) / (h₁ + h₂)

Calculation:

⇒ h₁ = 10, h₂ = 15

⇒ Height of the point of intersection = (10 × 15) / (10 + 15)

⇒ Height of the point of intersection = 150 / 25

⇒ Height of the point of intersection = 6

∴ The correct answer is 6 m.

Given:

Triangle ABC is right-angled at B.

AB + BC exceeds AC by 10 units.

Perimeter of the triangle = 60 units.

Formula Used:

Perimeter of a triangle = AB + BC + AC.

Area of a right-angled triangle = (1/2) × AB × BC.

Pythagoras theorem: AC² = AB² + BC².

Calculation:

Let AB = x, BC = y, and AC = z.

⇒ x + y + z = 60 ...(i)

⇒ x + y = z + 10 ...(ii)

From (ii), substitute z = x + y - 10 into (i):

⇒ x + y + (x + y - 10) = 60

⇒ 2(x + y) - 10 = 60

⇒ 2(x + y) = 70

⇒ x + y = 35 ...(iii)

From (ii), z = x + y - 10 = 35 - 10 = 25.

Using Pythagoras theorem:

⇒ z² = x² + y²

⇒ 25² = x² + y²

⇒ 625 = x² + y² ...(iv)

From (iii), y = 35 - x. Substitute y = 35 - x into (iv):

⇒ 625 = x² + (35 - x)²

⇒ 625 = x² + (1225 - 70x + x²)

⇒ 625 = 2x² - 70x + 1225

⇒ 2x² - 70x + 600 = 0

Divide through by 2:

⇒ x² - 35x + 300 = 0

Factorize:

⇒ (x - 20)(x - 15) = 0

⇒ x = 20 or x = 15

If x = 20, then y = 35 - 20 = 15.

If x = 15, then y = 35 - 15 = 20.

In both cases, AB = 20 and BC = 15.

Area of the triangle:

⇒ Area = (1/2) × AB × BC

⇒ Area = (1/2) × 20 × 15

⇒ Area = 150 square units.

∴ The correct answer is 150 square units.

Given:

Distance between poles = 24 m

Height difference between poles = 10 m

Formula Used:

Distance between tips = √(distance between poles² + height difference²)

Calculation:

⇒ Distance between tips = √(24² + 10²)

⇒ Distance between tips = √(576 + 100)

⇒ Distance between tips = √676

⇒ Distance between tips = 26 m

∴ The correct answer is 26 m.

Given:

Statement I: The quadrilateral AZXY is a parallelogram.

Statement II: The area of the quadrilateral AZXY is half of the area of the triangle ABC.

Correct Option: I only

Formula Used:

1. A quadrilateral is a parallelogram if its opposite sides are parallel and equal.
2. The area of a quadrilateral formed by joining midpoints of a triangle is half the area of the triangle.

Calculation:

Statement I:

⇒ AZXY is a parallelogram since X, Y, and Z are midpoints, making opposite sides parallel and equal.

⇒ By the midpoint theorem, the area of AZXY = 1/2 × Area of triangle ABC.

⇒ Statement I is correct.

Statement II:

When you connect the midpoints of a triangle (X, Y, and Z), the original triangle ABC is divided into four congruent triangles.

Each of these four smaller triangles has an area exactly equal to 1/4 of the total area of triangle ABC.

The quadrilateral AZXY is made up of two of these triangles.

So, area of AZXY = 1/2 × ar (Δ ABC)

Thus, Statement II is correct.

∴ The correct answer is "Both I and II".

Given:

Exterior angles: 4º, 5º, 6º, 8º

Formula Used:

Exterior angle of a regular polygon = 360º / number of sides

Number of sides = 360º / exterior angle

Calculation:

⇒ For 4º: Number of sides = 360º / 4 = 90 (integer value)

⇒ For 5º: Number of sides = 360º / 5 = 72 (integer value)

⇒ For 6º: Number of sides = 360º / 6 = 60 (integer value)

⇒ For 8º: Number of sides = 360º / 8 = 45 (integer value)

Thus, all these four angles can be exterior angles of a regular polygon

∴ The correct answer is option (4).