Mathematics Set 1 2021-2022 ISC (Commerce) Class 12 Question Paper Solution

If `int (log "x")^2/"x" "dx" = (log "x")^"k"/"k" + "c"`, then the value of k is:

3

2

1

None of the above options

`int_0^(2"a") "f"("x") "dx" = int_0^"a" "f"("x") "dx" + int_0^"a" "f"("k" - "x") "dx"`, then the value of k is:

a

2a

Independent of a

None of the above options

The degree of the differential equation `("d"^2"y")/("dx"^2) + 3("dy"/"dx")^2 = "x"^2 (("d"^2"y")/("dx"^2))^2` is:

1

2

3

4

Given `int "e"^"x" (("x" - 1)/("x"^2)) "dx" = "e"^"x" "f"("x") + "c"`. Then f(x) satisfying the equation is:

x

x²

`1/"x"`

None of the above options

Two cards are drawn out randomly from a pack of 52 cards one after the other, without replacement. The probability of first card being a king and second card not being a king is:

`48/663`

`24/663`

`12/663`

`4/663`

If two balls are drawn from a bag containing 3 white, 4 black and 5 red balls. Then, the probability that the drawn balls are of different colours is:

`1/66`

`3/66`

`19/66`

`47/66`

Find the following integrals:

`int (x^3 - x^2 + x - 1)/(x - 1) dx`

Evaluate:

`∫ log_10 "x dx"`

Solve the differential equation:

cosec³ x dy − cosec y dx = 0

Solve the differential equation:

`"dy"/"dx" = 2^(-"y")`

Evaluate:

`int_2^8 (sqrt(10 - "x"))/(sqrt"x" + sqrt(10 - "x")) "dx"`

A bag contains 6 red and 5 blue balls and another bag contains 5 red and 8 blue balls. A ball is drawn from the first bag and without noticing its colour is placed in the second bag. If a ball is drawn from the second bag, then find the probability that the drawn ball is red in colour.

A bag contains 3 red and 4 white balls and another bag contains 2 red and 3 white balls. If one ball is drawn from the first bag and 2 balls are drawn from the second bag, then find the probability that all three balls are of the same colour.

Evaluate:

`int (1 + sin "x")/(1 - sin "x") "dx"`

In a bolt factory, machines X, Y and Z manufacture 20%, 35% and 45% respectively of the total output. Of their output 8%, 6% and 5% respectively are defective bolts. One bolt is drawn at random from the product and is found to be defective. What is the probability that it was manufactured in machine Y?

Evaluate:

`int "dx"/(sin "x" + sin 2"x")`

By using the properties of the definite integral, evaluate the integral:

`int_0^(pi/4) log (1+ tan x) dx`

The equation of the plane which is parallel to 2x − 3y + z = 0 and which passes through (1, −1, 2) is:

2x − 3y + z − 7 = 0

2x − 3y + z + 7 = 0

2x − 3y + z − 8 = 0

2x − 3y + z + 6 = 0

The intercepts made on the coordinate axes by the plane 2x + y − 2z = 3 are:

`(-3)/2, -3, (-3)/2`

`3/2, 3, (-3)/2`

`3/2, -3, (-3)/2`

`3/2, 3, 3/2`

Find the equation of the plane passing through the point (1, 1, 1) and is perpendicular to the line `("x" - 1)/3 = ("y" - 2)/0 = ("z" - 3)/4`. Also, find the distance of this plane from the origin.

Using integration, find the area of the region bounded between the line x = 4 and the parabola y² = 16x.

If the regression line of x on y is, 9x + 3y − 46 = 0 and y on x is, 3x + 12y − 7 = 0, then the correlation coefficient ‘r’ is equal to:

`(-1)/12`

`1/12`

`(-1)/(2sqrt3)`

`1/(2sqrt3)`

If `bar"X"` = 40, `bar"Y"` = 6, σ_x = 10, σ_y = 1.5 and r = 0.9 for the two sets of data X and Y, then the regression line of X on Y will be:

x − 6y − 4 = 0

x + 6y − 4 = 0

x − 6y + 4 = 0

x + 6y + 4 = 0

For 5 observations of pairs (x, y) of variables X and Y, the following results are obtained:

∑x = 15, ∑y = 25, ∑x² = 55, ∑y² = 135, ∑xy = 83.

Calculate the value of b_xy and b_yx.

A manufacturer wishes to produce two commodities A and B. The number of units of material, labour and equipment needed to produce one unit of each commodity is shown in the table given below. Also shown is the available number of units of each item, material, labour, and equipment.

Find the maximum profit if each unit of commodity A earns a profit of ₹ 2 and each unit of B earns a profit of ₹ 3.