Date: April 2022

Duration: 1h30m

**Note :**

- Candidates are al1owed an additional 10 minutes for only reading the paper.
- They must NOT start writing during this time.
- This question paper is divided into 3 Sections A, B and C
- Candidates are required to attempt all questions from Section A and all questions EITHER from Section B OR Section C
- All working, including rough work, should be done on the same sheet as and adjacent to the rest of the answer.
- The intended marks for questions or parts of questions are given in brackets [ ].
- Mathematical tables and graph papers are provided.

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

Chapter: [0.033] Integrals

`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

Chapter: [0.033] Integrals

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

Chapter: [0.034] Differential Equations

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

x

x^{2}

`1/"x"`

None of the above options

Chapter: [0.033] Integrals

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`

Chapter: [0.04] Probability (Section A)

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`

Chapter: [0.04] Probability (Section A)

Find the following integrals `int (x^3 - x^2 + x - 1)/(x - 1) dx`

Chapter: [0.033] Integrals

**Evaluate:**

`∫ log_10 "x dx"`

Chapter: [0.033] Integrals

**Solve the differential equation:**

cosec^{3} x dy − cosec y dx = 0

Chapter: [0.034] Differential Equations

**Solve the differential equation:**

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

Chapter: [0.034] Differential Equations

**Evaluate:**

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

Chapter: [0.033] Integrals

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.

Chapter: [0.04] Probability (Section A)

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.

Chapter: [0.04] Probability (Section A)

**Evaluate:**

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

Chapter: [0.033] Integrals

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?

Chapter: [0.04] Probability (Section A)

**Evaluate:**

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

Chapter: [0.033] Integrals

By using the properties of definite integrals, evaluate the integrals

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

Chapter: [0.033] Integrals

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

Chapter: [0.06] Three - Dimensional Geometry (Section B)

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`

Chapter: [0.06] Three - Dimensional Geometry (Section B)

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.

Chapter: [0.06] Three - Dimensional Geometry (Section B)

Using integration, find the area of the region bounded between the line x = 4 and the parabola y^{2} = 16x.

Chapter: [0.07] Application of Integrals (Section B)

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)`

Chapter: [0.09] Linear Regression (Section C)

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

Chapter: [0.09] Linear Regression (Section C)

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

∑x = 15, ∑y = 25, ∑x^{2} = 55, ∑y^{2} = 135, ∑xy = 83.

Calculate the value of b_{xy} and b_{yx}.

Chapter: [0.09] Linear Regression (Section C)

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.

Items |
Commodity A |
Commodity B |
Available no. of Units |

Material | 1 | 2 | 8 |

Labour | 3 | 2 | 12 |

Equipment | 1 | 1 | 10 |

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.

Chapter: [0.1] Linear Programming (Section C)

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