Mathematics Outside Delhi Set - 2 2023-2024 Commerce (English Medium) Class 12 Question Paper Solution

The lines `(1-x)/2=(y-1)/3=z/1` and `(2x-3)/(2p)=(y)/-1=(z-4)/7` are perpendicular to each other for p equal to ______.

`-1/2`

`1/2`

2

3

The maximum value of Z = 4x + y for a L.P.P. whose feasible region is given below is ______.

50

110

120

170

The probability distribution of a random variable X is:

where k is some unknown constant.

The probability that the random variable X takes the value 2 is ______.

`1/5`

`2/5`

`4/5`

1

If A = [a_ij] = `[(2,-1,5),(1,3,2),(5,0,4)]` and c_ij is the cofactor of element a_ij, then the value of a₂₁ × c₁₁ + a₂₂ × c₁₂ + a₂₃ × c₁₃ is ______.

−57

0

9

57

If A `= [(1,3),(3,4)]` and A² − kA − 5I = 0, then the value of k is ______.

5

3

7

9

If `e^(x^2y)` = c, then `dy/dx` is ______.

`(xe^(x^2y))/(2y)`

`(-2y)/x`

`(2y)/x`

`x/(2y)`

The value of constant c that makes the function f defined by 

`f(x) = {{:(x^2-c^2; if x < 4),(cx+20  ; if x > 4):}`

continuous for all real numbers is ______.

−2

−1

0

2

The value of `int_-1^1|x|` dx is ______.

–2

–1

1

2

The number of arbitrary constants in the particular solution of the differential equation `log (dy/dx) = 3x+4y; y(0) = 0` is/are ______.

2

1

0

3

If `[(a,c,0),(b,d,0),(0,0,5)]` is a scalar matrix, then the value of a + 2b + 3c + 4d is ______.

0

5

10

25

If A = `[(2,1),(-4,-2)]`, then the value of I – A + A² – A³ + ... is ______.

`[(-1,-1),(4,3)]`

`[(3,1),(-4,-1)]`

`[(0,0),(0,0)]`

`[(1,0),(0,1)]`

Given that A^-1= `1/7 [(2,1),(-3,2)]`, matrix A is ______.

`7 [(2,-1),(3,2)]`

`[(2,-1),(3,2)]`

`1/7 [(2,-1),(3,2)]`

`1/49 [(2,-1),(3,2)]`

The integrating factor of the differential equation `(x+2y^2) dy/dx = y  (y>0)` is ______.

`1/x`

x

y

`1/y`

A vector perpendicular to the line `vecr=hati+hatj-hatk+λ(3hati-hatj)` is ______.

`5hati+hatj+6k`

`hati+3hatj+5hatk`

`2hati-2hatj`

`9hati-3hatj`

The vectors `\veca= 2hati - hatj + hatk, \vecb = hati - 3hatj - 5hatk and \vecc = -3hati + 4hatj +4hatk` represents the sides of ______.

an equilateral triangle

an obtuse-angled triangle

an isosceles triangle

a right-angled triangle

Let `veca` be any vector such that `|veca| = a`. The value of `|vecaxxhati|^2 + |vecaxxhatj|^2 + |vecaxxhatk|^2` is ______.

a²

2a²

3a²

0

If `veca`  and  `vecb`  are two vectors such that `|vec{a}| = 1, |vec{b}| = 2 and vec{a}.vec{b} = root  3,` then the angle between `2vec{a} and vec{-b}` is ______.

`pi/6`

`pi/3`

`(5pi)/6`

`(11pi)/6`

The function f(x) = kx – sin x is strictly increasing for ______.

k > 1

k < 1

k > –1

k < –1

Assertion (A): The corner points of the bounded feasible region of a L.P.P. are shown below. The maximum value of Z = x + 2y occurs at infinite points.

Reason (R): The optimal solution of a LPP having bounded feasible region must occur at corner points.

Both Assertion (A) and Reason (R) are true and the Reason (R) is the correct explanation of Assertion (A).

Both Assertion (A) and Reason (R) are true and Reason (R) is not the correct explanation of the Assertion (A).

Assertion (A) is true, but Reason (R) is false.

Assertion (A) is false, but Reason (R) is true.

Assertion(A): The relation R = {(x,y) : (x +y) is a prime number and x, y ∈ N} is not a reflexive relation.

Reason (R): The number ‘2n’ is composite for all natural numbers n.

Both Assertion (A) and Reason (R) are true and the Reason (R) is the correct explanation of Assertion (A).

Both Assertion (A) and Reason (R) are true and Reason (R) is not the correct explanation of the Assertion (A).

Assertion (A) is true, but Reason (R) is false.

Assertion (A) is false, but Reason (R) is true.

The volume of a cube is increasing at the rate of 6 cm³/s. How fast is the surface area of cube increasing, when the length of an edge is 8 cm?

Express `tan^-1(cos x/(1-sin x)), "where" (-pi)/2 <x<pi/2` in the simplest form.

Find the principal value of `tan^-1(1)+cos^-1(-1/2)+sin^-1(-1/root  2)`.

Show that `f(x) =(4 sinx)/(2+cosx) -x` is an increasing function of x in `[0, pi/2]`

If y = cos³ (sec² 2t), find `dy/dt`

If `x^y = e^(x-y), "prove that" dy/dx = logx/(1 + log x)^2`.

Evaluate: 

`int_(-1/2)^(1/2) cosx.log ((1+x)/(1-x))dx`

Given that x^y + y^x = a^b, where a and b are positive constants, find `dy/dx`.

Find the particular solution of the differential equation `dy/dx = y cot 2x`, given that `y(pi/4) = 2.`

Find the particular solution of the differential equation `(xe^(y/x) +y) dx = x dy,` given that y = 1 when x = 1.

Find: 

`int (2x+3)/(x^2(x+3))  dx`

A card from a well shuffled deck of 52 playing cards is lost. From the  remaining cards of the pack, a card is drawn at random and is found to be a King. Find the probability of the lost card being a King.

A biased die is twice as likely to show an even number as an odd number. If such a die is thrown twice, find the probability distribution of the number of sixes. Also, find the mean of the distribution.

Solve the following L.P.P. graphically:

Maximise Z = x + 3y
subject to the constraints:

x + 2y ≤ 200
x + y ≤ 150
y ≤ 75
x, y ≥ 0

Evaluate: 

`int_0^(pi/4) (x dx)/(1+cos2x+sin2x)`

Find:

`inte^x [1/(1+x^2)^(3/2) + x/(sqrt (1+x^2))] dx`

Let A = R – {5} and B = R – {1}. Consider the function f : A → B, defined by `f(x) = (x-3)/(x-5)`. Show that f is one-one and onto.

Check whether the relation S in the set of real numbers R defined by S = {(a, b): where a – b + `sqrt2` is an irrational number} is reflexive, symmetric or transitive.

Find the distance between the line `x/2 = (2y-6)/4 = (1-z)/-1` and another line parallel to it passing through the point (4, 0, −5).

If the lines `(x-1)/-3=(y-2)/(2k)=(z-3)/2 and (x-1)/(3k)=(y-1)/1=(z-6)/-7` are perpendicular  to each other, find the value of k and hence write the vector equation of a line perpendicular to these two lines and passing through the point (3, –4, 7).

Use the product of matrices `[(1,2,-3),(3,2,-2),(2,-1,1)][(0,1,2),(-7,7,-7),(-7,5,-4)]` to solve the following system of equations:

x + 2y – 3z = 6

3x + 2y – 2z = 3

2x  –  y + z = 2

Sketch the graph of y = x |x| and hence find the area bounded by this curve, X-axis and the ordinates x = –2 and x = 2, using integration.

Using integration, find the area bounded by the ellipse 9x² + 25y² = 225, the lines x = –2, x = 2, and the X-axis.

Based on the above information, answer the following questions:

(i) How far is the star V from star A?  (1)

(ii) Find a unit vector in the direction of `vec(DA)`.  (1)

(iii) Find the measure of ∠VDA.  (2)

OR

(iii) What is the projection of vector `vec(DV)  "on vector"  vec(DA)`  (2)

Based on the above information, answer the following questions:

(i) What is the probability that at least one of them is selected?  (1)

(ii) Find `P(G|barH)` where G is the event of Jaspreet’s selection and H denotes the event that Rohit is not selected.  (1)

(iii) Find the probability that exactly one of them is selected.  (2)

OR

(iii) Find the probability that exactly two of them are selected.  (2)

Based on the above information, answer the following questions:

1. Determine the number of units (x) that should be sold to maximise the revenue R(x) = x p(x). Also, verify the result.
2. What rebate in price of calculator should the store give to maximise the revenue?