Mathematics Board Sample Paper 2023-2024 Commerce (English Medium) Class 12 Question Paper Solution

If A = [a_ij] is a square matrix of order 2 such that a_ij = ${\displaystyle \{\begin{array}{l}1\text{,} \text{when i}\ne \text{j}\\ 0\text{,} \text{when} \text{i}=\text{j}\end{array},}$ then A² is ______.

${\displaystyle \left[\begin{array}{cc}1& 0\\ 1& 0\end{array}\right]}$

${\displaystyle \left|\begin{array}{cc}1& 1\\ 0& 0\end{array}\right|}$

${\displaystyle \left|\begin{array}{cc}1& 1\\ 1& 0\end{array}\right|}$

${\displaystyle \left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]}$

If A and B are invertible square matrices of the same order, then which of the following is not correct?

|AB^–1| = ${\displaystyle \frac{\left|A\right|}{\left|B\right|}}$

|(AB)^–1| = ${\displaystyle \frac{1}{\left|A\right|\left|B\right|}}$

(AB)^–1 = B^–1A^–1

(A + B)^–1 = B^–1 + A^–1

The area of a triangle with vertices (–3, 0), (3, 0) and (0, k) is 9 sq.units. The value of k will be ______.

9

±3

– 9

6

If f(x) = ${\displaystyle \{\begin{array}{l}\frac{kx}{\left|x\right|}\text{,}ifx<0\\ 3\text{,} ifx\ge 0\end{array}}$ is continuous at x = 0, then the value of k is ______.

–3

0

3

any real number

The lines ${\displaystyle \overrightarrow{r}=\hat{i}+\hat{j}-\hat{k}+\lambda (2\hat{i}+3\hat{j}-6\hat{k})}$ and ${\displaystyle \overrightarrow{r}=2\hat{i}-\hat{j}-\hat{k}+\mu (6\hat{i}+9\hat{j}-18\hat{k})}$; (where λ and μ are scalars) are ______.

coincident

skew

intersecting

parallel

The degree of the differential equation ${\displaystyle {[1+{\left(\frac{dy}{dx}\right)}^2]}^3={\left(\frac{d^{2}y}{{dx}^2}\right)}^2}$ is ______.

4

${\displaystyle \frac{3}{2}}$

2

Not defined

Corner points of the feasible region determined by the system of linear constraints are (0, 3), (1, 1) and (3, 0). Let Z = px + qy, where p, q > 0. Condition on p and q so that the minimum of Z occurs at (3, 0) and (1, 1) is ______.

p = 2q

p = ${\displaystyle \frac{\text{q}}{2}}$

p = 3q

p = q

ABCD is a rhombus whose diagonals intersect at E . Then ${\displaystyle \overrightarrow{EA}+\overrightarrow{EB}+\overrightarrow{EC}+\overrightarrow{ED}}$ equals to ______.

${\displaystyle \overrightarrow{0}}$

${\displaystyle \overrightarrow{AD}}$

${\displaystyle 2\overrightarrow{BD}}$

${\displaystyle 2\overrightarrow{AD}}$

For any integer n, the value of ${\displaystyle {\int }_{-\pi }^{\pi }{e}^{{\cos }^{2}x}{\sin }^3(2n+1)x dx}$ is ______.

–1

0

1

2

The value of |A|, if A = ${\displaystyle \left[\begin{array}{ccc}0& 2x-1& \sqrt{x}\\ 1-2x& 0& 2\sqrt{x}\\ -\sqrt{x}& -2\sqrt{x}& 0\end{array}\right]}$, where x ∈ R⁺, is ______.

(2x + 1)²

0

(2x + 1)³

(2x – 1)²

The feasible region corresponding to the linear constraints of a Linear Programming Problem is given below.

Which of the following is not a constraint to the given Linear Programming Problem?

x + y ≥ 2

x + 2y ≤ 10

x – y ≥ 1

x – y ≤ 1

If ${\displaystyle \overrightarrow{a}=4\hat{i}+6\hat{j}}$ and ${\displaystyle \overrightarrow{b}=3\hat{j}+4\hat{k}}$, then the vector form of the component of ${\displaystyle \overrightarrow{a}}$ along ${\displaystyle \overrightarrow{b}}$ is ______.

${\displaystyle \frac{18}{5}(3\hat{i}+4\hat{k})}$

${\displaystyle \frac{18}{25}(3\hat{j}+4\hat{k})}$

${\displaystyle \frac{18}{5}(3\hat{i}+4\hat{k})}$

${\displaystyle \frac{18}{25}(4\hat{i}+6\hat{j})}$

Given that A is a square matrix of order 3 and |A| = –2, then |adj(2A)| is equal to ______.

–2⁶

+4

–2⁸

2⁸

A problem in Mathematics is given to three students whose chances of solving it are ${\displaystyle \frac{1}{2},\frac{1}{3},\frac{1}{4}}$ respectively. If the events of their solving the problem are independent then the probability that the problem will be solved, is ______.

${\displaystyle \frac{1}{4}}$

${\displaystyle \frac{1}{3}}$

${\displaystyle \frac{1}{2}}$

${\displaystyle \frac{3}{4}}$

The general solution of the differential equation ydx – xdy = 0; (Given x, y > 0), is of the form

(Where 'c' is an arbitrary positive constant of integration)

xy = c

x = cy²

y = cx

y = cx²

The value of λ for which the two vectors ${\displaystyle 2\hat{i}-\hat{j}+2\hat{k}}$ and ${\displaystyle 3\hat{i}+\lambda \hat{j}+\hat{k}}$ are perpendicular is ______.

2

4

6

8

The set of all points where the function f(x) = x + |x| is differentiable, is ______.

(0, ∞)

(–∞, 0)

(–∞, 0) ∪ (0, ∞)

(–∞, ∞)

If the direction cosines of a line are ${\displaystyle (\frac{1}{c},\frac{1}{c},\frac{1}{c})}$ then ______.

0 < c < 1

c = ± 3

c > 2

c > 0

c = ${\displaystyle \pm \sqrt{3}}$

Let f(x) be a polynomial function of degree 6 such that ${\displaystyle \frac{d}{dx}\left(f\left(x\right)\right)}$ = (x – 1)³ (x – 3)², then

Assertion (A): f(x) has a minimum at x = 1.

Reason (R): When ${\displaystyle \frac{d}{dx}\left(f\left(x\right)\right)<0,\forall x\in (a-h,a)}$ and ${\displaystyle \frac{d}{dx}\left(f\left(x\right)\right)>0,\forall x\in (a,a+h)}$; where 'h' is an infinitesimally small positive quantity, then f(x) has a minimum at x = a, provided f(x) is continuous at x = a.

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 relation f : {1, 2, 3, 4} ${\displaystyle \to }$ {x, y, z, p} defined by f = {(1, x), (2, y), (3, z)} is a bijective function.

REASON (R): The function f : {1, 2, 3} ${\displaystyle \to }$ {x, y, z, p} such that f = {(1, x), (2, y), (3, z)} is one-one.

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 ${\displaystyle {\sin }^{-1}\left(\cos \left(\frac{33\pi }{5}\right)\right)}$.

Find the domain of sin^–1 (x² – 4).

Find the interval/s in which the function f : R ${\displaystyle \to }$ R defined by f(x) = xe^x, is increasing.

If f(x) = ${\displaystyle \frac{1}{4x^2+2x+1};x\in R}$, then find the maximum value of f(x).

Find the maximum profit that a company can make, if the profit function is given by P(x) = 72 + 42x – x², where x is the number of units and P is the profit in rupees.

Evaluate : ${\displaystyle {\int }_{-1}^1\log \left(\frac{2-x}{2+x}\right)dx}$.

Check whether the function f : R ${\displaystyle \to }$ R defined by f(x) = x³ + x, has any critical point/s or not ? If yes, then find the point/s.

Find : ${\displaystyle \int \frac{2x^2+3}{x^2(x^2+9)}dx;x\ne 0}$.

The random variable X has probability distribution P(X) of the following form, where k is some number:

${\displaystyle P(X=x)\{\begin{array}{ll}k& ifx=0\\ 2k& ifx=1\\ 3k& ifx=2\\ 0& \text{otherwise}\end{array}}$

1. Determine the value of 'k'.
2. Find P(X < 2), P(X ≥ 2), P(X ≤ 2).

Find : ${\displaystyle \int \sqrt{\frac{x}{1-x^3}}dx;x\in (0,1)}$.

Evaluate: ${\displaystyle {\int }_0^{\frac{\pi }{4}}\log (1+\tan x)dx}$.

Solve the differential equation  ${\displaystyle y{e}^{\frac{x}{y}}dx=(x{e}^{\frac{x}{y}}+y^2)dy,(y\ne 0)}$

Solve the differential equation ${\displaystyle {\cos }^{2}x\frac{dy}{dx}}$ + y = tan x

Solve the following Linear Programming Problem graphically:

Minimize: z = x + 2y,

Subject to the constraints: x + 2y ≥ 100, 2x – y ≤ 0, 2x + y ≤ 200, x, y ≥ 0.

Solve the following Linear Programming Problem graphically:

Maximize: z = – x + 2y,

Subject to the constraints: x ≥ 3, x + y ≥ 5, x + 2y ≥ 6, y ≥ 0.

If ${\displaystyle (a+bx)e^{\frac{y}{x}}}$ = x then prove that ${\displaystyle x\frac{d^{2}y}{{dx}^2}={\left(\frac{a}{a+bx}\right)}^2}$.

Make a rough sketch of the region {(x, y) : 0 ≤ y ≤ x² + 1, 0 ≤ y ≤ x + 1, 0 ≤ x ≤ 2} and find the area of the region, using the method of integration.

Let N be the set of all natural numbers and R be a relation on N × N defined by (a, b) R (c, d) ${\displaystyle \iff }$ ad = bc for all (a, b), (c, d) ∈ N × N. Show that R is an equivalence relation on N × N. Also, find the equivalence class of (2, 6), i.e., [(2, 6)].

Show that function f: R ${\displaystyle \to }$ {x ∈ R : −1 < x < 1} defined by f(x) = ${\displaystyle \frac{x}{1+\left|x\right|}}$, x ∈ R is one-one and onto function.

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

${\displaystyle \frac{2}{x}+\frac{3}{y}+\frac{10}{z}}$ = 4, ${\displaystyle \frac{4}{x}-\frac{6}{y}+\frac{5}{z}}$ = 1, ${\displaystyle \frac{6}{x}+\frac{9}{y}-\frac{20}{z}}$ = 2.

Find the coordinates of the image of the point (1, 6, 3) with respect to the line ${\displaystyle \overrightarrow{r}=(\hat{j}+2\hat{k})+\lambda (\hat{i}+2\hat{j}+3\hat{k})}$; where 'λ' is a scalar. Also, find the distance of the image from the y – axis.

An aeroplane is flying along the line ${\displaystyle \overrightarrow{r}=\lambda (\hat{i}-\hat{j}+\hat{k})}$; where 'λ' is a scalar and another aeroplane is flying along the line ${\displaystyle \overrightarrow{r}=\hat{i}-\hat{j}+\mu (-2\hat{j}+\hat{k})}$; where 'μ' is a scalar. At what points on the lines should they reach, so that the distance between them is the shortest? Find the shortest possible distance between them.

Read the following passage and answer the questions given below:

In an Office three employees Jayant, Sonia and Oliver process incoming copies of a certain form. Jayant processes 50% of the forms, Sonia processes 20% and Oliver the remaining 30% of the forms. Jayant has an error rate of 0.06, Sonia has an error rate of 0.04 and Oliver has an error rate of 0.03.

Based on the above information, answer the following questions.

1. Find the probability that Sonia processed the form and committed an error.
2. Find the total probability of committing an error in processing the form.
3. The manager of the Company wants to do a quality check. During inspection, he selects a form at random from the days output of processed form. If the form selected at random has an error, find the probability that the form is not processed by Jayant.
   OR
   Let E be the event of committing an error in processing the form and let E₁, E₂ and E₃ be the events that Jayant, Sonia and Oliver processed the form. Find the value of ${\displaystyle \sum _{i=1}^{3}P\left(E_i\mid E\right)}$.

Read the following passage and answer the questions given below:

1. What is the magnitude of the force of Team A ?
2. Which team will win the game?
3. Find the magnitude of the resultant force exerted by the teams.
   OR
   In what direction is the ring getting pulled?

Read the following passage and answer the questions given below:

1. Find the rate of growth of the plant with respect to the number of days exposed to the sunlight.
2. Does the rate of growth of the plant increase or decrease in the first three days? What will be the height of the plant after 2 days?