Mathematics Delhi Set 3 2022-2023 Commerce (English Medium) Class 12 Question Paper Solution

Let R be the relation in the set N given by R = {(a, b) : a = b – 2, b > 6}, then ______.

(2, 4) ∈ R

(3, 8) ∈ R

(6, 8) ∈ R

(8, 7) ∈ R

If A = ${\displaystyle \left[\begin{array}{cc}5& x\\ y& 0\end{array}\right]}$ and A = A^T, where A^T is the transpose of the matrix A, then ______.

x = 0, y = 5

x = y

x + y = 5

x = 5, y = 0

If for a square matrix A, A² – A + I = 0, then A^–1 equals ______.

A

A + I

I – A

A – I

If A = ${\displaystyle \left[\begin{array}{cc}1& 0\\ 2& 1\end{array}\right]}$, B = ${\displaystyle \left[\begin{array}{cc}x& 0\\ 1& 1\end{array}\right]}$ and A = B², then x equals ______.

± 1

– 1

1

2

If ${\displaystyle \left|\begin{array}{ccc}\alpha & 3& 4\\ 1& 2& 1\\ 1& 4& 1\end{array}\right|}$ = 0, then the value of α is ______.

1

2

3

4

If f(x) = | cos x |, then ${\displaystyle f\left(\frac{3\pi }{4}\right)}$ is ______.

1

– 1

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

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

The function f(x) = [x], where [x] denotes the greatest integer less than or equal to x; is continuous at ______.

x = 1

x = 1.5

x = – 2

x = 1

If x = A cos 4t + B sin 4t, then ${\displaystyle \frac{d^{2}x}{{dt}^2}}$ is equal to ______.

x

– x

16x

– 16x

The function f(x) = x³ + 3x is increasing in interval ______.

(– ∞, 0)

(0, ∞)

R

(0, 1)

${\displaystyle \int \frac{\sec x}{\sec x-\tan x}dx}$ equals ______.

sec x – tan x + c

sec x + tan x + c

tan x + sec x + c

– (sec x + tan x) + c

${\displaystyle {\int }_{-1}^1\frac{|x-2|}{x-2}dx}$, x ≠ 2 is equal to ______.

1

– 1

2

– 2

The order and the degree of the differential equation ${\displaystyle {(1+3\frac{dy}{dx})}^2=4\frac{d^{3}y}{{dx}^3}}$ respectively are ______.

${\displaystyle 1,\frac{2}{3}}$

3, 1

3, 3

1, 2

If ${\displaystyle \overrightarrow{a}.\hat{i}=\overrightarrow{a}.(\hat{i}+\hat{j})=\overrightarrow{a}.(\hat{i}+\hat{j}+\hat{k})}$ = 1, then ${\displaystyle \overrightarrow{a}}$ is ______.

${\displaystyle \hat{k}}$

${\displaystyle \hat{i}}$

${\displaystyle \hat{j}}$

${\displaystyle \hat{i}+\hat{j}+\hat{k}}$

The magnitude of the vector ${\displaystyle 6\hat{i}-2\hat{j}+3\hat{k}}$ is ______.

1

5

7

12

If a line makes angles of 90°, 135° and 45° with the x, y and z axes respectively, then its direction cosines are ______.

${\displaystyle 0,-\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}}}$

${\displaystyle -\frac{1}{\sqrt{2}},0,\frac{1}{\sqrt{2}}}$

${\displaystyle \frac{1}{\sqrt{2}},0,-\frac{1}{\sqrt{2}}}$

${\displaystyle 0,\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}}}$

The angle between the lines 2x = 3y = – z and 6x = – y = – 4z is ______.

0°

30°

45°

90°

If for any two events A and B, P(A) = ${\displaystyle \frac{4}{5}}$ and P(A ∩ B) = ${\displaystyle \frac{7}{10}}$, then ${\displaystyle P\left(\frac{B}{A}\right)}$ is equal to ______.

${\displaystyle \frac{1}{10}}$

${\displaystyle \frac{1}{8}}$

${\displaystyle \frac{7}{8}}$

${\displaystyle \frac{17}{20}}$

Five fair coins are tossed simultaneously. The probability of the events that at least one head comes up is ______.

${\displaystyle \frac{27}{32}}$

${\displaystyle \frac{5}{32}}$

${\displaystyle \frac{31}{32}}$

${\displaystyle \frac{1}{32}}$

Assertion (A): Two coins are tossed simultaneously. The probability of getting two heads, if it is known that at least one head comes up, is ${\displaystyle \frac{1}{3}}$.

Reason (R): Let E and F be two events with a random experiment, then ${\displaystyle P\left(\frac{E}{F}\right)=\frac{P(E\cap F)}{P\left(E\right)}}$.

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): ${\displaystyle {\int }_2^8\frac{\sqrt{10-x}}{\sqrt{x}+\sqrt{10-x}}dx}$ = 3.

Reason (R): ${\displaystyle {\int }_a^{b}f\left(x\right)dx={\int }_a^{b}f(a+b-x)dx}$.

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 k for which the function f given as

f(x) =${\displaystyle \{\begin{array}{ll}\frac{1-\cos x}{2x^2}\text{,}& ifx\ne 0\\ k\text{,}& ifx=0\end{array}}$ 

is continuous at x = 0.

If x = a cos t and y = b sin t, then find ${\displaystyle \frac{d^{2}y}{{dx}^2}}$.

Find the value of ${\displaystyle {\tan }^{-1}\left[2\cos \left(2{\sin }^{-1} \frac{1}{2}\right)\right]+{\tan }^{-1}1}$.

Sketch the region bounded by the lines 2x + y = 8, y = 2, y = 4 and the Y-axis. Hence, obtain its area using integration.

If the vectors $$\overrightarrow{a}$$  and $$\overrightarrow{b}$$ are such that $$\left|\overrightarrow{a}\right|=3,\left|\overrightarrow{b}\right|=\frac{2}{3}$$ and $$\overrightarrow{a}\times \overrightarrow{b}$$ is a unit vector, then write the angle between $$\overrightarrow{a}$$ and $$\overrightarrow{b}$$ 

Find the area of a parallelogram whose adjacent sides are determined by the vectors ${\displaystyle \overrightarrow{a}=\hat{i}-\hat{j}+3\hat{k}}$ and ${\displaystyle \overrightarrow{b}=2\hat{i}-7\hat{j}+\hat{k}}$.

Find the vector equation of the line passing through the point A(1, 2, –1) and parallel to the line 5x – 25 = 14 – 7y = 35z.

Prove that the determinant ${\displaystyle \left|\begin{array}{ccc}x& \sin \theta & \cos \theta \\ -\sin \theta & -x& 1\\ \cos \theta & 1& x\end{array}\right|}$ is independent of θ.

Using integration, find the area of the region bounded by y = mx (m > 0), x = 1, x = 2 and the X-axis.

Find the coordinates of the foot of the perpendicular drawn from point (5, 7, 3) to the line ${\displaystyle \frac{x-15}{3}=\frac{y-29}{8}=\frac{z-5}{-5}}$.

If ${\displaystyle \overrightarrow{a}=\hat{i}+\hat{j}+\hat{k}}$ and ${\displaystyle \overrightarrow{b}=\hat{i}+2\hat{j}+3\hat{k}}$ then find a unit vector perpendicular to both ${\displaystyle \overrightarrow{a}+\overrightarrow{b}}$ and ${\displaystyle \overrightarrow{a}-\overrightarrow{b}}$.

Find the area of the following region using integration ((x, y) : y² ≤ 2x and y ≥ x – 4).

Find the co-ordinates of the foot of the perpendicular drawn from the point (0, 2, 3) to the line ${\displaystyle \frac{x+3}{5}=\frac{y-1}{2}=\frac{z+4}{3}}$.

Three vectors ${\displaystyle \overrightarrow{a},\overrightarrow{b}}$ and ${\displaystyle \overrightarrow{c}}$ satisfy the condition ${\displaystyle \overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}=\overrightarrow{0}}$. Evaluate the quantity μ = ${\displaystyle \overrightarrow{a}.\overrightarrow{b}+\overrightarrow{b}.\overrightarrow{c}+\overrightarrow{c}.\overrightarrow{a}}$, if ${\displaystyle \left|\overrightarrow{a}\right|}$ = 3, ${\displaystyle \left|\overrightarrow{b}\right|}$ = 4 and ${\displaystyle \left|\overrightarrow{c}\right|}$ = 2.

Find the distance between the lines:

${\displaystyle \overrightarrow{r}=(\hat{i}+2\hat{j}-4\hat{k})+\lambda (2\hat{i}+3\hat{j}+6\hat{k})}$;

${\displaystyle \overrightarrow{r}=(3\hat{i}+3\hat{j}-5\hat{k})+\mu (4\hat{i}+6\hat{j}+12\hat{k})}$

Solve the following Linear Programming Problem graphically:

Minimize: Z = 60x + 80y

Subject to constraints:

3x + 4y ≥ 8

5x + 2y ≥ 11

x, y ≥ 0

Evaluate: ${\displaystyle {\int }_0^{\frac{\pi }{2}}\sin 2x{\tan }^{-1}\left(\sin x\right)dx}$.

Solve the following Linear Programming Problem graphically:

Maximize: P = 70x + 40y

Subject to: 3x + 2y ≤ 9,

3x + y ≤ 9,

x ≥ 0,y ≥ 0.

In answering a question on a multiple choice test, a student either knows the answer or guesses. Let ${\displaystyle \frac{3}{5}}$ be the probability that he knows the answer and ${\displaystyle \frac{2}{5}}$ be the probability that he guesses. Assuming that a student who guesses at the answer will be correct with probability ${\displaystyle \frac{1}{3}}$. What is the probability that the student knows the answer, given that he answered it correctly?

A box contains 10 tickets, 2 of which carry a prize of ₹ 8 each. 5 of which carry a prize of ₹ 4 each and remaining 3 carry a prize of ₹ 2 each. If one ticket is drawn at random, find the mean value of the prize.

Read the following passage:

Based on the above information, answer the following questions:

1. How many relations are possible from B to G? (1)
2. Among all the possible relations from B to G, how many functions can be formed from B to G? (1)
3. Let R : B ${\displaystyle \to }$ B be defined by R = {(x, y) : x and y are students of the same sex}. Check if R is an equivalence relation. (2)
   OR
   A function f : B ${\displaystyle \to }$ G be defined by f = {(b₁, g₁), (b₂, g₂), (b₃, g₁)}. Check if f is bijective. Justify your answer. (2)

Read the following passage:

Based on the above information, answer the following questions:

1. Convert the given above situation into a matrix equation of the form AX = B. (1)
2. Find | A |. (1)
3. Find A^–1. (2)
   OR
   Determine P = A² – 5A. (2)

Read the following passage:

Based on the above, answer the following questions:

1. Show that (x² – y²) dx + 2xy dy = 0 is a differential equation of the type ${\displaystyle \frac{dy}{dx}=g\left(\frac{y}{x}\right)}$. (2)
2. Solve the above equation to find its general solution. (2)