(1) The probability of selecting a red marble can be calculated as the number of favorable outcomes (selecting a red marble) divided by the total number of possible outcomes.

In this case, there are 3 red marbles and 4 green marbles, so the total number of possible outcomes is 3 + 4 = 7. Since there are 3 red marbles, the number of favorable outcomes is 3. Therefore, the probability of selecting a red marble is 3/7.

(2) To solve the system of equations:

2x - y = 5

3x + y = 1

We can use the method of substitution or elimination. Let's use the elimination method:

Multiply the first equation by 3 and the second equation by 2 to create opposite coefficients for y:

6x - 3y = 15

6x + 2y = 2

Subtract the second equation from the first equation:

(6x - 3y) - (6x + 2y) = 15 - 2

-3y - 2y = 13

-5y = 13

Divide both sides by -5:

y = -13/5

Now substitute the value of y back into one of the original equations:

2x - (-13/5) = 5

2x + 13/5 = 5

Multiply through by 5 to eliminate the denominator:

10x + 13 = 25

10x = 25 - 13

10x = 12

Divide both sides by 10:

x = 12/10 = 6/5

Therefore, the solution to the system of equations is x = 6/5 and y = -13/5.

(3) Let's denote the amount of chestnuts Mary picked as M, the amount Peter picked as P, and the amount Lucy picked as L.

According to the given information, we have the following equations:

M = 2P (Mary picked twice as much chestnuts as Peter)
L = P + 2 (Lucy picked 2 kg more than Peter)
M + P + L =26 (Together, they picked 26 kg of chestnuts)

Substituting the first two equations into the third equation, we get:

2P + P + (P + 2) = 26

Combining like terms:

4P + 2 = 26

Subtracting 2 from both sides:

4P = 24

Dividing both sides by 4:

P = 6

Now substitute this value of P back into the first two equations:

M = 2P
M = 2(6)
M = 12

L = P + 2
L = 6 + 2
L = 8

Therefore, Mary picked 12 kg, Peter picked 6 kg, and Lucy picked 8 kg of chestnuts.

(4) Let's denote the unknown number as x.

According to the given information, we can write the equation as:

2x - 138 = 102

Adding 138 to both sides of the equation:

2x = 240

Dividing both sides by 2:

x = 120

Therefore, the number the student chose was 120.

(5) To find the integral of x^2 dx, we can use the power rule of integration.

The power rule states that the integral of x^n dx, where n is any real number except -1, is (1/(n+1)) * x^(n+1) + C, where C is the constant of integration.

Applying the power rule, we have:

∫ x^2 dx = (1/3) * x^(2+1) + C

Simplifying:

∫ x^2 dx = (1/3) * x^3 + C

Therefore, the integral of x^2 dx is (1/3) * x^3 + C.

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