Mathematics Sample 2022-2023 Commerce (English Medium) Class 12 Question Paper Solution

If A = [a_ij] is a skew-symmetric matrix of order n, then ______.

`a_(ij) = 1/(a_(ji)) ∀  i, j`

`a_(ij) ≠ 0  ∀  i, j`

`a_(ij) = 0, where  i = j`

`a_(ij) ≠ 0  where  i = j`

If A is a square matrix of order 3, |A′| = −3, then |AA′| = ______.

9

−9

3

−3

The area of a triangle with vertices A, B, C is given by ______. 

`|vec("AB") xx vec("AC")|`

`1/2|vec("AB") xx vec("AC")|`

`1/4|vec("AC") xx vec("AB")|`

`1/8|vec("AC") xx vec("AB")|`

The value of ‘k’ for which the function f(x) = `{{:((1 - cos4x)/(8x^2)",",  if x ≠ 0),(k",",  if x = 0):}` is continuous at x = 0 is ______.

0

–1

1

2

If f'(x) = `x + 1/x`, then f(x) is ______.

`x^2 + log |x| + C`

`x^2/2 + log |x| + C`

`x/2 + log |x| + C`

`x/2 - log |x| + C`

If m and n, respectively, are the order and the degree of the differential equation `d/(dx) [((dy)/(dx))]^4` = 0, then m + n = ______.

1

2

3

4

The solution set of the inequality 3x + 5y < 4 is ______.

an open half-plane not containing the origin.

an open half-plane containing the origin.

the whole XY-plane not containing the line 3x + 5y = 4.

a closed half-plane containing the origin.

The scalar projection of the vector `3hati - hatj - 2hatk` on the vector `hati + 2hatj - 3hatk` is ______.

`7/sqrt(14)`

`7/14`

`6/13`

`7/2`

The value of `int_2^3 x/(x^2 + 1)`dx is ______.

`log 4`

`log  3/2`

`1/2 log2`

`log  9/4`

If A, B are non-singular square matrices of the same order, then (AB^–1)^–1 = ______.

A^–1B

A^–1B^–1

BA^–1

AB

The corner points of the shaded unbounded feasible region of an LPP are (0, 4), (0.6, 1.6) and (3, 0) as shown in the figure. The minimum value of the objective function Z = 4x + 6y occurs at ______.

(0.6, 1.6) only

(3, 0) only

(0.6, 1.6) and (3, 0) only

at every point of the line-segment joining the points (0.6, 1.6) and (3, 0)

If `|(2, 4),(5, 1)| = |(2x, 4),(6, x)|`, then the possible value(s) of ‘x’ is/are ______.

3

`sqrt(3)`

`-sqrt(3)`

`sqrt(3), -sqrt(3)`

If A is a square matrix of order 3 and |A| = 5, then |adj A| = ______.

5

25

125

`1/5`

Given two independent events A and B such that P(A) = 0.3, P(B) = 0.6 and P(A' ∩ B') is ______.

0.9

0.18

0.28

0.1

The general solution of the differential equation y dx – x dy = 0 is ______.

xy = C

x = Cy²

y = Cx

y = Cx²

If y = sin^–1x, then (1 – x²)y₂ is equal to ______.

xy₁

xy

xy₂

x²

If two vectors `veca` and `vecb` are such that `|veca|` = 2, `|vecb|` = 3 and `veca.vecb` = 4, then `|veca - 2vecb|` is equal to ______.

`sqrt(2)`

`2sqrt(6)`

24

`2sqrt(2)`

P is a point on the line joining the points A(0, 5, −2) and B(3, −1, 2). If the x-coordinate of P is 6, then its z-coordinate is ______.

10

6

–6

–10

Assertion (A): The domain of the function sec^–12x is `(-∞, - 1/2] ∪ pi/2, ∞)`

Reason (R): sec^–1(–2) = `- pi/4`

Both A and R are true and R is the correct explanation of A.

Both A and R are true but R is not the correct explanation of A.

A is true but R is false.

A is false but R is true.

Assertion (A): The acute angle between the line `barr = hati + hatj + 2hatk  + λ(hati - hatj)` and the x-axis is `π/4`

Reason(R): The acute angle 𝜃 between the lines `barr = x_1hati + y_1hatj + z_1hatk  + λ(a_1hati + b_1hatj + c_1hatk)` and  `barr = x_2hati + y_2hatj + z_2hatk  + μ(a_2hati + b_2hatj + c_2hatk)` is given by cosθ = `(|a_1a_2 + b_1b_2 + c_1c_2|)/sqrt(a_1^2 + b_1^2 + c_1^2 sqrt(a_2^2 + b_2^2 + c_2^2)`

Both A and R are true and R is the correct explanation of A.

Both A and R are true but R is not the correct explanation of A.

A is true but R is false.

A is false but R is true.

Find the value of `sin^-1 [sin((13π)/7)]`

Prove that the function f is surjective, where f: N → N such that `f(n) = {{:((n + 1)/2",", if "n is odd"),(n/2",", if  "n is even"):}` Is the function injective? Justify your answer.

A man 1.6 m tall walks at the rate of 0.3 m/sec away from a street light that is 4 m above the ground. At what rate is the tip of his shadow moving? At what rate is his shadow lengthening?

If `veca = hati - hatj + 7hatk` and `vecb = 5hati - hatj + λhatk`, then find the value of λ so that the vectors `veca + vecb` and `veca - vecb` are orthogonal.

Find the direction ratio and direction cosines of a line parallel to the line whose equations are 6x − 12 = 3y + 9 = 2z − 2

If `ysqrt(1 - x^2) + xsqrt(1 - y^2)` = 1, then prove that `(dy)/(dx) = - sqrt((1 - y^2)/(1 - x^2))`

Find `|vecx|` if `(vecx - veca).(vecx + veca)` = 12, where `veca` is a unit vector.

Find: `int (dx)/sqrt(3 - 2x - x^2)`

Three friends go for coffee. They decide who will pay the bill, by each tossing a coin and then letting the “odd person” pay. There is no odd person if all three tosses produce the same result. If there is no odd person in the first round, they make a second round of tosses and they continue to do so until there is an odd person. What is the probability that exactly three rounds of tosses are made?

Find the mean number of defective items in a sample of two items drawn one-by-one without replacement from an urn containing 6 items, which include 2 defective items. Assume that the items are identical in shape and size.

Evaluate: `int_(pi/6)^(pi/3) (dx)/(1 + sqrt(tanx)`

Evaluate the following integral:

Solve the differential equation: y dx + (x – y²)dy = 0

Solve the differential equation: xdy – ydx = `sqrt(x^2 + y^2)dx`

Solve the following Linear Programming Problem graphically:

Maximize Z = 400x + 300y subject to x + y ≤ 200, x ≤ 40, x ≥ 20, y ≥ 0

Evaluate the following integral:

Make a rough sketch of the region {(x, y): 0 ≤ y ≤ x², 0 ≤ y ≤ x, 0 ≤ x ≤ 2} and find the area of the region using integration.

Define the relation R in the set N × N as follows:

For (a, b), (c, d) ∈ N × N, (a, b) R (c, d) if ad = bc. Prove that R is an equivalence relation in N × N.

Given a non-empty set X, define the relation R in P(X) as follows:

For A, B ∈ P(X), (4, B) ∈ R iff A ⊂ B. Prove that R is reflexive, transitive and not symmetric.

An insect is crawling along the line `barr = 6hati + 2hatj + 2hatk + λ(hati - 2hatj + 2hatk)` and another insect is crawling along the line `barr = - 4hati - hatk + μ(3hati - 2hatj - 2hatk)`. At what points on the lines should they reach so that the distance between them s the shortest? Find the shortest possible distance between them.

The equations of motion of a rocket are:
x = 2t,y = –4t, z = 4t, where the time t is given in seconds, and the coordinates of a ‘moving point in km. What is the path of the rocket? At what distances will the rocket be from the starting point O(0, 0, 0) and from the following line in 10 seconds? `vecr = 20hati - 10hatj + 40hatk + μ(10hati - 20hatj + 10hatk)`

If A = `[(2, -3, 5),(3, 2, -4),(1, 1, -2)]`, find A^–1. Use A^–1 to solve the following system of equations 2x − 3y + 5z = 11, 3x + 2y – 4z = –5, x + y – 2z = –3

Read the following passage and answer the questions given below.

1. Is the function differentiable in the interval (0, 12)? Justify your answer.
2. If 6 is the critical point of the function, then find the value of the constant m.
3. Find the intervals in which the function is strictly increasing/strictly decreasing.
   OR
   Find the points of local maximum/local minimum, if any, in the interval (0, 12) as well as the points of absolute maximum/absolute minimum in the interval [0, 12]. Also, find the corresponding local maximum/local minimum and the absolute ‘maximum/absolute minimum values of the function.

Read the following passage and answer the questions given below.

1. If the length and the breadth of the rectangular field be 2x and 2y respectively, then find the area function in terms of x.
2. Find the critical point of the function.
3. Use First derivative Test to find the length 2x and width 2y of the soccer field (in terms of a and b) that maximize its area.
   OR
   Use Second Derivative Test to find the length 2x and width 2y of the soccer field (in terms of a and b) that maximize its area.

Read the following passage and answer the questions given below.

There are two antiaircraft guns, named as A and B. The probabilities that the shell fired from them hits an airplane are 0.3 and 0.2 respectively. Both of them fired one shell at an airplane at the same time.

1. What is the probability that the shell fired from exactly one of them hit the plane?
2. If it is known that the shell fired from exactly one of them hit the plane, then what is the probability that it was fired from B?