Mathematics Official 2022-2023 ISC (Commerce) Class 12 Question Paper Solution

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

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

2|A|

4|A|

8|A|

6|A|

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

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 cm³/sec

750 cm³/sec

7500 cm³/sec

1250 cm³/sec

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}

For the curve y² = 2x³ – 7, the slope of the normal at (2, 3) is ______.

4

– 4

`1/4`

`(-1)/4`

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

2log(x² + 1) + c

`1/2`log(x² + 1) + c

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

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

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

`1/x`

`(-1)/x^3`

`(-1)/x`

– x

The intevral in which the function f(x) = 5 + 36x – 3x² increases will be ______.

(– ∞, 6)

(6, ∞)

(– 6, 6)

(0, – 6)

Evaluate: `int_-1^1 x^17.cos^4x  dx`

`oo`

1

– 1

0

Solve the differential equation: 

`dy/dx` = cosec y

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

Evaluate:

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

Evaluate:

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

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. 

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

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.

Evaluate the following determinant without expanding:

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

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.

Solve for x:

5tan^–1x + 3cot^–1x = 2π

Evaluate:

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

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

If `tan^-1 ((x - 1)/(x + 1)) + tan^-1 ((2x - 1)/(2x + 1)) = tan^-1 (23/36)` = then prove that 24x² – 23x – 12 = 0

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

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

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.

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.

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.

Solve the differential equation:

(1 + y²) dx = (tan⁻¹ y − x) dy

Solve the following differential equation:

(x² – y²)dx + 2xy dy = 0

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.

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

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.

Evaluate:

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

Evaluate: 

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

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.

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`

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)`

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

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

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

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")`.

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

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.

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.

Find the area of the region bounded by the curve x² = 4y and the line x = 4y – 2.

If the demand function is given by p = 1500 – 2x – x² then find the marginal revenue when x = 10.

1160

1600

1100

1200

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

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

The cost function C(x) = 3x² – 6x + 5. Find the average cost when x = 2.

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.

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.

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

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

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

Calculate:

1. the regression coefficient b_xy and b_yx
2. the probable value of y when x = 20

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