Mathematics 65/1/1 2025-2026 Commerce (English Medium) Class 12 Question Paper Solution

If 2 cos⁻¹x = y, then ______.

0 ≤ y ≤ π

−π ≤ y ≤ π

0 ≤ y ≤ 2π

−π ≤ y ≤ 0

Which of the following cannot be the order of a row-matrix?

2 × 1

1 × 2

1 × 1

1 × n

Which of the following properties is/are true for two matrices of suitable orders?

1. (A + B)′ = A′ + B′
2. (A − B)′ = B′ − А′
3. (AB)′ = A′B′
4. (kAB)′ = kB′A′ (k is a scalar)

(i) only

(i), (ii) and (iii)

(i) and (ii)

(i) and (iv)

If `Δ_1 = |(1  0  0), (0  2  0), (0  0  3)| and Δ_2 = |(0  2  0), (1  0  0), (0  0  6)|`, then ______.

Δ₁ = 2Δ₂

Δ₂ = −2Δ₁

Δ₁ = Δ₂

Δ₂ = −Δ₁

One of the values of x for which `|(cos x, sin x), (- cos x, sin x)|` = 1 is ______.

0

`π/4`

`π/3`

`π/2`

If A and B are skew-symmetric matrices of same order, then which of the following matrices is also skew-symmetric?

AB

AB + BA

(A + B)²

A − B

The least value of f(x) = x³ − 12x, ∈ [0, 3] is ______.

−16

−9

0

16

If `int (3ax)/(b^2 + c^2x^2) = A log |b^2 + c^2x^2| + K`, then the value of A is ______.

3a

`(3a)/(2b^2)`

`(3a)/(b^2c^2)`

`(3a)/(2c^2)`

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

0

log 2

2 log 2

`1/2` log 2

The area bounded by the curve y = x|x|, x-axis and the ordinates x = −1 and x = 1 is given by ______.

0

`1/3`

`2/3`

`4/3`

The integrating factor of differential equation `R(dx)/(dy) + Px = Q`, where P, Q, R are functions of y, is ______.

`e^(intP/Qdy)`

`e^(intPdy)`

`e^(intP/Rdy)`

`e^(intP/Rdx)`

The order and degree of the differential equation `d/(dx)(e^y)` = 0, respectively, are ______.

0, 1

1, 1

2, 1

1, not defined

The value of p for which vectors `hati + 2hatj + 3hatk and 2hati - phatj + hatk` are perpendicular to each other is ______.

0

1

`5/2`

`-5/2`

The value of m for which the points with position vectors `-hati -hatj + 2hatk, 2hati + mhatj + 5hatk and 3hati + 11hatj + 6hatk` are collinear, is ______.

8

−8

2

`5/2`

If `|vec"a"| = 8, |vec"b"| = 3 and |vec"a" xx vec"b"|` = 12, then value of `vec"a" * vec"b"` is ______.

`6sqrt(3)`

`8sqrt(3)`

`12sqrt(3)`

None of these

`3sqrt(12)`

The length of perpendicular drawn from point (2, 5, 7) on line `x/1 = y/0 = z/0` is ______.

2

5

`sqrt74`

`sqrt78`

The feasible region of a linear programming problem with objective function Z = 5x + 7y is shown below:

The maximum value of Z − minimum value of Z is ______.

8

29

35

43

The degree of an objective function of a linear programming problem is ______.

0

1

2

Any natural number

Assertion (A): In an experiment of throwing an unbiased die, the probability of getting a prime number given that number appearing on the die being odd is `2/3`.

Reason (R): For any two events A and B, P(A|B) = `(P(A ∪ B))/(P(B))`

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

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

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

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

Assertion (A): Lines given by x = py + q, z = ry + s and x = p′y + q′, z = r′y + s′ are perpendicular to each other when pp′ + rr′ = 1.

Reason (R): Two lines `vecr = veca_1 + lambda vecb_1 and vecr = veca_2 + mu vecb_2` are perpendicular to each other if `vecb_1 * vecb_2 = 0`.

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

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

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

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

Check whether the function f(x) is defined as:

`f(x) = {((|x - 3|)/(2(x - 3))","  x < 3), ((x - 6)/6","  x ≥ 3):}` is continuous at x = 3 or not?

If `sqrt3(x^2 + y^2) = 4xy`, then find `(dy)/(dx)  at (1/2, sqrt3/2)`.

A room freshener bottle in the shape of an inverted cone sprays the perfume at regular intervals, such that the volume of perfume in the bottle decreases at a steady rate of 1 mm³/min. Find the rate at which the level of perfume is dropping at an instant when the level of perfume in the bottle is 10 mm, if the semi-vertical angle of the conical bottle is `π/6`.

Find the vector of magnitude 14 in the direction of `vecQP`, where P and Q are the points (1, 3, 2) and (−1, 0, 8) respectively.

Vectors `veca = 3hati - 2hatj + 2hatk and vecb = hati + 2hatk` represent the two adjacent sides of a parallelogram. Find the vectors representing its diagonals and hence find their lengths.

Simplify:

`tan^-1((cos 2x - sin 2x)/(cos 2x + sin 2x)), 0 < x < π/4`.

Evaluate:

`tan (sin^-1 - cos^-1(1/2))`

Evaluate:

`int_0^1 x tan^-1x*dx`

Find `int sqrt((x + 2)/(x - 2))` dx

Find: `int x^2/((x^2 + 9) (x^2 + 16))` dx

If `I_1 = int_(-π/4)^(π/4) (dx)/(1 + cos 2x) and I_2 = int_(-1/2)^(1/2) |x|dx`, then show that I₁ − 4I₂ = 0.

Find the general solution of the following differential equation:

`x^2(dy)/(dx) = x^2 + xy + y^2`

Find the particular solution of the differential equation:

`xy(dy)/(dx) = (x + 2) (y + 2)`, given that y(1) = −1.

Solve the following linear programming problem graphically:

Minimize Z = 13x − 15y

Subject to constraints:

x + y ≤ 7,

2x − 3y + 6 ≥ 0,

x ≥ 0, y ≥ 0

Out of two bags, bag I contains 3 red and 4 white balls and bag II contains 8 red and 6 white balls. A die is thrown. If it shows а number less than 3, then a ball is drawn at random from bag I, otherwise a ball is drawn at random from bag II. Find the probability that the ball drawn from one of the bags is a red ball.

The probability of the simultaneous occurrence of at least one of the two events X and Y is a. If the probability that exactly one of the events X and Y occurs is b, prove that P(X′) + P(Y′) = 2 − 2a + b.

A relation R is defined on Z, the set of integers, as,

R = {(x, y): |x − y| is divisible by a prime number ‘p’, x, y ∈ Z}.

Check whether R is an equivalence relation or not.

A function f: `R - {3/5} -> R - {3/5}` is defined as `f(x) = (3x + 2)/(5x - 3)`. Show that f is one-one and onto.

If `A = [(0, 2, 1), (-2, -1, -2), (1, -1, 0)]`, find A⁻¹ and use it to solve the following system of equations:

−2y + z = 7, 2x − y − z = 8, x − 2y = 10

If `[(3, -1, sin3x), (-7, 4, cos2x), (-11, 7, 2)]` is a singular matrix, then find all values of x, where x ∈ `[0, π/2]`.

If x = cos t, y = cos mt, prove that `(1 - x^2) (d^2y)/(dx^2) - x (dy)/(dx) + m^2y = 0`.

Check whether the lines given by `(x - 1)/2 = (y - 2)/3 = (z - 3)/4 and (x - 4)/5 = (y - 1)/2 = z` are parallel or not. If parallel, find the distance between them, otherwise find their point of intersection, if the lines are intersecting.

An online delivery company in a city has 5,000 subscribers and collects annual subscription fees of ₹ 300 per subscriber for unlimited free deliveries.

The company wishes to increase the annual subscription fee. It is predicted that, for every increase of ₹ 1, ten subscribers will discontinue. Assume that the company increased the annual fee by ₹ x.

Based on the given information, answer the following questions:

1. How many subscribers will discontinue after an increase of ₹ x in annual fee?
2. If R(x) denotes the total revenue collected after the increase of ₹ x in subscription fee, express R(x) as a function of x.
3. 1. Find the value of x for which R(x) is maximum.
      OR
   2. Find the sub-intervals of (0, 5000) in which R(x) is increasing and decreasing.

In an online jackpot, there is one first prize of ₹ 3,00,000, two second prizes of ₹ 2,00,000 each and three third prizes of ₹ 50,000 each.

A total of 1,00,000 jackpot tickets, each costing ₹ 100, were sold there by raising a fund of ₹ 1,00,00,000.

Rohan bought one ticket.

Based on given information, answer the following questions:

1. What are the possible amounts the person can win?
2. 1. What is the probability that the person wins atleast ₹ 2,00,000?
      OR
   2. What is the probability that the person does not win any amount?
3. In another jackpot, Rohan also bought a ticket having a prize money of ₹ 5,00,000. The chances of winning the jackpot are 1 in 1,00,000. Find the probability that on exactly one of tickets he wins the jackpot.

Roundabouts are often made on busy roads to ease the traffic and avoid red lights.

One such round-about is made such that equation representing its boundary is given by C₁ ;  x² + y² = 64.

There is a circular pond with a fountain in the middle of the roundabout whose equation is given by C₂ ;  x² + y² = 64.

Based on the given information, answer the following questions:

1. Represent the given equations C₁ and C₂ with the help of a diagram.
2. Express y as a function of x, (y = f(x)), for both C₁ and C₂.
3. 1. Using integration find the area of region covered by the roundabout.
      OR
   2. Using integration, find the area of region covered by circular pond.