Biology GS TIFR- Tata Institute of Fundamental Research Questions 6-10_ Solved

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6. Where would the graph of y=x^2 and y=x^3 intersect?

(i) At x = -1

(ii) At all x>1

(iii) At x=0 and x=1

(iv) At all points between x=0 and x=1

Answer: To find the points of intersection between the graphs of y = x^2 and y = x^3, we need to solve the equation x^2 = x^3.

We can start by factoring out x^2 from both sides of the equation:

x^2(1 - x) = 0

This equation is satisfied when either x^2 = 0 or 1 - x = 0.

If x^2 = 0, then x = 0 is a solution.

If 1 - x = 0, then x = 1 is a solution.

Therefore, the graphs of y = x^2 and y = x^3 intersect at the points (0, 0) and (1, 1).

7. Avni has 8 favorite paintings but only 2 wall hooks where she can hang them. In how many different ways can she hang the paintings?

(i)32 (ii)56

(iii)16

(iv)64

Answer: Avni can hang the first painting on either of the two hooks and the second painting on the remaining hook. For the third painting, she has two options: she can either hang it on one of the hooks that already has a painting or on the remaining hook. Similarly, for each subsequent painting, she has two options.

Therefore, the total number of different ways Avni can hang the paintings is:

2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 = 256

So there are 256 different ways that Avni can hang her 8 favorite paintings on 2 wall hooks.

8. Consider the following data as a sample.15, 15, 33, 17, 30, 30, 20, 60, 45, 15. If each data value is increased by 5, what is the effect on the mean and standard deviation?

(i) Both the mean and the standard deviation increase by 5.

(ii) The mean remains the same, but the standard deviation increases by 5.

(iii) The standard deviation remains the same, but the mean increases by 5.

(iv) Both the mean and the standard deviation remain the same

Answer: To find the effect of increasing each data value by 5 on the mean and standard deviation, we need to calculate the new mean and standard deviation after adding 5 to each data value.

New Data: 20, 20, 38, 22, 35, 35, 25, 65, 50, 20

To find the mean, we sum up all the data values and divide by the total number of values:

Old Mean = (15 + 15 + 33 + 17 + 30 + 30 + 20 + 60 + 45 + 15) / 10 = 28

New Mean = (20 + 20 + 38 + 22 + 35 + 35 + 25 + 65 + 50 + 20) / 10 = 32

So, adding 5 to each data value increases the mean from 28 to 32.

To find the standard deviation, we first need to find the variance. The variance is the average of the squared differences between each data value and the mean:

Old Variance = ((15-28)^2 + (15-28)^2 + (33-28)^2 + (17-28)^2 + (30-28)^2 + (30-28)^2 + (20-28)^2 + (60-28)^2 + (45-28)^2 + (15-28)^2) / 10

= 324.2

New Variance = ((20-32)^2 + (20-32)^2 + (38-32)^2 + (22-32)^2 + (35-32)^2 + (35-32)^2 + (25-32)^2 + (65-32)^2 + (50-32)^2 + (20-32)^2) / 10

= 347.6

So, the old standard deviation is the square root of the old variance:

Old Standard Deviation = sqrt(324.2) = 18

And the new standard deviation is the square root of the new variance:

New Standard Deviation = sqrt(347.6) = 18.64

So, adding 5 to each data value increases the standard deviation from 18 to 18.64.

Therefore, adding 5 to each data value increases the mean from 28 to 32 and the standard deviation from 18 to 18.64.

9. Electric cars are considered more eco-friendly because:

(i) They do not cause air pollution

(ii) They are built from bio-degradable parts

(iii) They can carry more people

(iv) They are cheaper

Answer: Electric cars are considered more eco-friendly than traditional gasoline cars for several reasons:

Zero Emissions: Electric cars produce zero emissions at the tailpipe, which means they don't emit any harmful pollutants into the air, making them a cleaner and greener alternative to traditional gasoline cars.

Renewable Energy: Electric cars can be powered by electricity from renewable energy sources like wind, solar, and hydropower. This means that they can be fueled by sustainable and clean energy sources.

Energy Efficiency: Electric cars are much more energy-efficient than gasoline cars, which means they require less energy to travel the same distance. This translates into lower overall energy consumption, which can help reduce greenhouse gas emissions.

Noise Pollution Reduction: Electric cars are also quieter than traditional gasoline cars, which can help reduce noise pollution in cities and urban areas.

Overall, electric cars are considered more eco-friendly than traditional gasoline cars because they produce zero emissions, can be powered by renewable energy sources, are more energy-efficient, and produce less noise pollution.

10. A large sealed drum with a radius of 1 m and a height of 20 m, needs to be painted all over. Assuming 1 liter of paint covers 6 square meters, how many liters of paint will be required to paint the drum?

(i) 132 litres

(ii) 22 litres

(iii) 20 litres

(iv) 125 litres

Answer: The total surface area of the drum can be found by adding the area of the top and bottom circles (which are the same) and the lateral surface area.

The area of a circle is given by the formula A = πr^2, where r is the radius of the circle. Therefore, the area of each circular end of the drum is:

A1 = π(1m)^2 = π m^2

The lateral surface area of a cylinder is given by the formula A2 = 2πrh, where r is the radius of the cylinder and h is its height. Therefore, the lateral surface area of the drum is:

A2 = 2π(1m)(20m) = 40π m^2

So, the total surface area of the drum is:

A = 2A1 + A2 = 2π m^2 + 40π m^2 = 42π m^2

Now, we can calculate the amount of paint required. If 1 liter of paint covers 6 square meters, then the amount of paint required to cover 42π square meters is:

paint required = (42π m^2) / (6 m^2/litre) = 7π litres

Therefore, approximately 22.05 liters of paint will be required to paint the drum.

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