The area of cross-section ($x$) of four rock samples and the respective applied loads ($y$) at failure under uniaxial loading are given below:

If the best fit line $y = 4.88x$ represents the above data, the coefficient of determination ($R^2$) of the best fit line is ______.

The block grade model of an ore deposit is shown in the figure below. The relationship between block value per tonne ($B_v$) in rupees and the block grade in percentage ($x$) is given below:

If each square block contains 1000 tonne of material and the overall pit slope angle is $45^\circ$, the total value of the pit determined by the floating cone algorithm in Lakhs of rupees is ______.

A wheel of radius 0.5 m rotates under a moment of 2000 N-m as shown in the figure. A block brake is used to stop the wheel. If the coefficient of static friction between the wheel and the block brake is 0.3, the smallest force of P in N required to stop the wheel is ______.

Consider the following sentences:

All benches are beds. No bed is a bulb. Some bulbs are lamps.

Which of the following can be inferred?

i. Some beds are lamps.
ii. Some lamps are beds.

The bar graph below shows the output of five carpenters over one month, each of whom made different items of furniture: chairs, tables, and beds.

Consider the following statements.

i. The number of beds made by carpenter C2 is exactly the same as the number of tables made by carpenter C3.
ii. The total number of chairs made by all carpenters is less than the total number of tables.

Which one of the following is true?

Match the following for coal mining operation:

A double ended ranging drum shearer operates in a longwall retreating panel. The following data are provided:

• Face length: $200\ \text{m}$
• Average cutting speed: $1.5\ \text{m/min}$
• Web depth: $0.6\ \text{m}$
• Average flitting speed: $4.0\ \text{m/min}$
• Cutting height: $2.5\ \text{m}$
• Number of shifts/day: $3$
• Density of coal: $1.4\ \text{tonne/m}^3$

In the linear programming problem,

Maximize $Z = 48X_1 + 36X_2$

Subject to:

• $X_1 \leq 5$
• $X_2 \leq 8$
• $2X_1 + \frac{4}{3}X_2 \leq 16$
• $X_1 + X_2 = 10$
• $X_1 \geq 4$
• $X_1 \geq 0, X_2 \geq 0$

The value of $Z$ is ______.

A rock block of mass 100 kg is to be lifted by a horizontal force P as shown in the figure below. Smooth rollers are placed between the wedges. The coefficient of static friction between wedge A and surface C and between wedge B and surface D is 0.3. Ignoring the weight of the wedges and the friction between the roller and the wedges, the minimum force P in kg required to lift the block (round off to one decimal place) is ______

Out of the support categories given for an underground coal mine, identify the ‘active support’.