ISC (Commerce)

ISC (Arts)

ISC (Science)

Academic Year: 2022-2023

Date & Time: 20th February 2023, 2:00 pm

Duration: 3h

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- Candidates are allowed an additional 15 minutes for only reading the paper.
- They must NOT start writing during this time.
- The Question Paper consists of three 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.
**Section A**: Internal choice has been provided in two questions of two marks each, two questions of four marks each and two questions of six marks each.**Section B**: Internal choice has been provided in one question of two marks and one question of four marks.**Section C**: Internal choice has been provided in one question of two marks and one question of four marks.- 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.**

A relation R on (1, 2, 3) is given by R = {(1, 1), (2, 2), (1, 2), (3, 3), (2, 3)}. Then the relation R is ______.

Reflexive

Symmetric

Transitive

Symmetric and Transitive

Chapter: [0.01] Relations and Functions (Section A)

If A is a square matrix of order 3, then |2A| is equal to ______.

2|A|

4|A|

8|A|

6|A|

Chapter: [0.021] Matrices and Determinants

If the following function is continuous at x = 2 then the value of k will be ______.

f(x) = `{{:(2x + 1",", if x < 2),( k",", if x = 2),(3x - 1",", if x > 2):}`

2

3

5

– 1

Chapter: [0.031] Continuity, Differentiability and Differentiation

An edge of a variable cube is increasing at the rate of 10 cm/sec. How fast will the volume of the cube increase if the edge is 5 cm long?

75 cm^{3}/sec

750 cm^{3}/sec

7500 cm^{3}/sec

1250 cm^{3}/sec

Chapter: [0.032] Applications of Derivatives

Let \[f\left(x\right) = x^3\] be a function with domain {0, 1, 2, 3}. Then domain of \[f^{-1}\] is ______.

{3, 2, 1, 0}

{0, −1, −2, −3}

{0, 1, 8, 27}

{0, −1, −8, −27}

Chapter: [0.01] Relations and Functions (Section A)

For the curve y^{2} = 2x^{3} – 7, the slope of the normal at (2, 3) is ______.

4

– 4

`1/4`

`(-1)/4`

Chapter: [0.032] Applications of Derivatives

**Evaluate: **`int x/(x^2 + 1)"d"x`

2log(x^{2} + 1) + c

`1/2`log(x^{2} + 1) + c

`"e"^(x^2 + 1) + "c"`

`logx + x^2/2 + "c"`

Chapter: [0.033] Integrals

The derivative of log x with respect to `1/x` is ______.

`1/x`

`(-1)/x^3`

`(-1)/x`

– x

Chapter: [0.031] Continuity, Differentiability and Differentiation

The intevral in which the function f(x) = 5 + 36x – 3x^{2} increases will be ______.

(– ∞, 6)

(6, ∞)

(– 6, 6)

(0, – 6)

Chapter: [0.032] Applications of Derivatives

**Evaluate:** `int_-1^1 x^17.cos^4x dx`

`oo`

1

– 1

0

Chapter: [0.033] Integrals

**Solve the differential equation: **

`dy/dx` = cosec y

Chapter: [0.034] Differential Equations

For what value of k the matrix `[(0, k),(-6, 0)]` is a skew symmetric matrix?

Chapter: [0.021] Matrices and Determinants

**Evaluate: **

`int_0^1 |2x + 1|dx`

Chapter: [0.033] Integrals

**Evaluate: **

`int (1 + cosx)/(sin^2x)dx`

Chapter: [0.033] Integrals

A bag contains 19 tickets, numbered from 1 to 19. Two tickets are drawn randomly in succession with replacement. Find the probability that both the tickets drawn are even numbers.

Chapter: [0.04] Probability (Section A)

If f(x) = [4 – (x – 7)^{3}]^{1/5} is a real invertible function, then find f^{–1}(x).

Chapter: [0.01] Relations and Functions (Section A)

Let A = R – {2} and B = R – {1}. If f: A `→` B is a function defined by f(x) = `(x - 1)/(x - 2)` then show that f is a one-one and an onto function.

Chapter: [0.01] Relations and Functions (Section A)

**Evaluate the following determinant without expanding:**

`|(5, 5, 5),(a, b, c),(b + c, c + a, a + b)|`

Chapter: [0.021] Matrices and Determinants

The probability of the event A occurring is `1/3` and of the event B occurring is `1/2`. If A and B are independent events, then find the probability of neither A nor B occurring.

Chapter: [0.04] Probability (Section A)

**Solve for x:**

5tan^{–1}x + 3cot^{–1}x = 2π

Chapter: [0.01] Relations and Functions (Section A)

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**Evaluate:**

\[\int \cos^{-1} \left(\sin x \right) \text{dx}\]

Chapter: [0.033] Integrals

If `int x^5 cos (x^6)"d"x = "k" sin (x^6) + "C"`, find the value of k.

Chapter: [0.033] Integrals

If `tan^-1 ((x - 1)/(x + 1)) + tan^-1 ((2x - 1)/(2x + 1)) = tan^-1 (23/36)` = then prove that 24x^{2} – 23x – 12 = 0

Chapter: [0.01] Relations and Functions (Section A)

If y = e^{ax}. cos bx, then prove that

`(d^2y)/(dx^2) - 2ady/dx + (a^2 + b^2)y` = 0

Chapter: [0.031] Continuity, Differentiability and Differentiation

In a company, 15% of the employees are graduates and 85% of the employees are non-graduates. As per the annual report of the company, 80% of the graduate employees and 10% of the non-graduate employees are in the Administrative positions. Find the probability that an employee selected at random from those working in administrative positions will be a graduate.

Chapter: [0.04] Probability (Section A)

A Problem in Mathematics is given to the three students A, B and C. Their chances of solving the problem are `1/2, 1/3` and `1/4` respectively. Find the probability that exactly two students will solve the problem.

Chapter: [0.04] Probability (Section A)

A Problem in Mathematics is given to the three students A, B and C. Their chances of solving the problem are `1/2, 1/3` and `1/4` respectively. Find the probability that at least two of them will solve the problem.

Chapter: [0.04] Probability (Section A)

**Solve the differential equation: **

(1 + y^{2}) dx = (tan^{−1 }y − x) dy

Chapter: [0.034] Differential Equations

**Solve the following differential equation:**

(x^{2} – y^{2})dx + 2xy dy = 0

Chapter: [0.034] Differential Equations

**Using the matrix method, solve the following system of linear equations:**

`2/x + 3/y + 10/z` = 4, `4/x - 6/y + 5/z` = 1, `6/x + 9/y - 20/z` = 2.

Chapter: [0.021] Matrices and Determinants

Prove that the semi-vertical angle of the right circular cone of given volume and least curved surface is \[\cot^{- 1} \left( \sqrt{2} \right)\] .

Chapter: [0.032] Applications of Derivatives

A running track of 440 m is to be laid out enclosing a football field. The football field is in the shape of a rectangle with a semi-circle at each end. If the area of the rectangular portion is to be maximum,then find the length of its sides. Also calculate the area of the football field.

Chapter: [0.032] Applications of Derivatives

**Evaluate:**

`int (3"e"^(2x) - 2"e"^x)/("e"^(2x) + 2"e"^x - 8)"d"x`

Chapter: [0.033] Integrals

**Evaluate: **

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

Chapter: [0.033] Integrals

A box contains 30 fruits, out of which 10 are rotten. Two fruits are selected at random one by one without replacement from the box. Find the probability distribution of the number of unspoiled fruits. Also find the mean of the probability distribution.

Chapter: [0.04] Probability (Section A)

If `|veca| = 3, |vecb| = sqrt(2)/3` and `veca xx vecb` is a unit vector then the angle between `veca` and `vecb` will be ______.

`π/6`

`π/4`

`π/3`

`π/2`

Chapter: [0.05] Vectors (Section B)

The distance of the point `2hati + hatj - hatk` from the plane `vecr.(hati - 2hatj + 4hatk)` = 9 will be ______.

13

`13/sqrt(21)`

21

`21/sqrt(13)`

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

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Find the area of the parallelogram whose diagonals are `hati - 3hatj + hatk` and `hati + hatj + hatk`.

Chapter: [0.05] Vectors (Section B)

Find the equation of the plane passing through the point (2, 4, 6) and making equal intercepts on the coordinate axes.

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

If the two vectors `3hati + αhatj + hatk` and `2hati - hatj + 8hatk` are perpendicular to each other, then find the value of α.

Chapter: [0.05] Vectors (Section B)

If A(1, 2, – 3) and B(– 1, – 2, 1) are the end points of a vector `vec("AB")` then find the unit vector in the direction of `vec("AB")`.

Chapter: [0.05] Vectors (Section B)

If `hata` is unit vector and `(2vecx - 3hata)*(2vecx + 3hata)` = 91, find the value of `|vecx|`.

Chapter: [0.05] Vectors (Section B)

Find the equation of the plane passing through the point (1, 1, –1) and perpendicular to the planes x + 2y + 3z = 7 and 2x – 3y + 4z = 0.

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

A line passes through the point (2, – 1, 3) and is perpendicular to the lines `vecr = (hati + hatj - hatk) + λ(2hati - 2hatj + hatk)` and `vecr = (2hati - hatj - 3hatk) + μ(hati + 2hatj + 2hatk)` obtain its equation.

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

Find the area of the region bounded by the curve x^{2} = 4y and the line x = 4y – 2.

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

If the demand function is given by p = 1500 – 2x – x^{2} then find the marginal revenue when x = 10.

1160

1600

1100

1200

Chapter: [0.08] Application of Calculus (Section C)

If the two regression coefficients are 0.8 and 0.2, then the value of coefficient of correlation r will be ______.

± 0.4

± 0.16

0.4

0.16

Chapter: [0.09] Linear Regression (Section C)

Out of the two regression lines x + 2y – 5 = 0 and 2x + 3y = 8, find the line of regression of y on x.

Chapter: [0.09] Linear Regression (Section C)

The cost function C(x) = 3x^{2} – 6x + 5. Find the average cost when x = 2.

Chapter: [0.08] Application of Calculus (Section C)

The fixed cost of a product is ₹ 30,000 and its variable cost per unit is ₹ 800. If the demand function is p(x) = 4500 – 100x. Find the break-even values.

Chapter: [0.08] Application of Calculus (Section C)

The total cost function for x units is given by C(x) = `sqrt(6x + 5) + 2500`. Show that the marginal cost decreases as the output x increases.

Chapter: [0.08] Application of Calculus (Section C)

The average revenue function is given by AR = `25 - x/4`. Find total revenue function and marginal revenue function.

Chapter: [0.08] Application of Calculus (Section C)

Solve the following Linear Programming Problem graphically.

Maximise Z = 5x + 2y subject to:

x – 2y ≤ 2,

3x + 2y ≤ 12,

– 3x + 2y ≤ 3,

x ≥ 0, y ≥ 0

Chapter: [0.1] Linear Programming (Section C)

The following table shows the Mean, the Standard Deviation and the coefficient of correlation of two variables x and y.

Series | x | y |

Mean | 8 | 6 |

Standard deviation | 12 | 4 |

Coefficient of correlation | 0.6 |

**Calculate:**

- the regression coefficient b
_{xy}and b_{yx} - the probable value of y when x = 20

Chapter: [0.09] Linear Regression (Section C)

An analyst analysed 102 trips of a travel company. He studied the relation between travel expenses (y) and the duration (x) of these trips. He found that the relation between x and y was linear. Given the following data, find the regression equation of y on x.

`sumx` = 510, `sumy` = 7140, `sumx^2` = 4150, `sumy^2` = 740200, `sumxy` = 54900

Chapter: [0.09] Linear Regression (Section C)

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