Mathematics 65/5/1 2025-2026 Commerce (English Medium) Class 12 Question Paper Solution

If matrix A = `[(-p, q),(r, p)]` is such that A² = I, then ______.

1 + p² + qr = 0

1 − p² − qr = 0

1 − p² + qr = 0

1 + p² − qr = 0

If A is a square matrix such that A² = A, then (A − I)³ − A is equal to ______.

I

−I

A

A²

For the inverse trigonometric functions, which of the following Principal Value Branch is not correctly defined?

tan⁻¹: R → `[(π/2, π/2)]`

Sec⁻¹: R − (−1, 1) → [0, π] − `[π/2]`

Cot⁻¹: R → (0, π)

cosec⁻¹: R − (−1, 1) → `[−π/2,π/2]`

Let A = `[(0, -3, 4),(1, 0, 2)]` and B = `[(-3, 0, 1),(2, 4, 0)]`. If A + B + C = 0, then matrix C is ______.

`[(-3, -3, 5),(3, 4, 2)]`

`[(3, 3, 5),(-3, -4, -2)]`

`[(3, 3, -5),(-3, -4, -2)]`

`[(-3, -3, -5),(3, 4, 2)]`

If A is a non-singular matrix, then which of the following is not true?

adj A is singular

(adj A)⁻¹ = (adj A⁻¹)

|A| ≠ 0

A⁻¹ exists

If f(x) = `{{:((x^2 - 4x - 5)/(x + 1)",", x ≠ -1), (k",",x = -1):}`

is continuous at x = −1, then the value of k is:

Any real value

6

−1

−6

If the area of Δ ABC with vertices A(3, 1), B(−2, 1) and C(0, k) is 5 sq. units, then values of k are ______.

3, 1

−1, 3

−1, 2

0, 2

Derivative of cos⁻¹ `((sin x + cos x)/(sqrt2)), -pi/4 < x < pi/4` with respect to x is ______.

−1

1

`pi/4`

`-pi/4`

Absolute minimum value of f(x) = (x − 2)² + 5 in the interval [−3, 2] is ______.

−3

2

5

30

`int1/(sqrt(1 + cos 2x)) dx` is equal to ______.

log cos x + C

`1/sqrt2` log |sec x + tan x| + C

`1/sqrt2` log |sec x − tan x| + C

log sin 2x + C

The value of `int_-5^-1 1/x dx` is equal to ______.

−log 5

x⁶

log (−5)

x⁻⁶

An ant is observed crawling on a sheet of paper along a straight line given by equation y = 2x − 4. Area of the surface covered by the ant bounded by y-axis, x-axis and x = 1 is ______.

1 sq. unit

3 sq. unit

2 sq. unit

4 sq. unit

The order and degree of the differential equation 1 + `((d^3y)/dx^3)^3` = λ `(d^2y)/dx^2` is ______.

Order = 3, Degree = 3

Order = 2, Degree = 2

Order = 3, Degree = 1

Order = 2, Degree = 1

The general solution for the differential equation `dy/dx = e^(3x - y)` is ______.

3e^y = e^3x + C

log (3x − y) = C

`e^(3x - y) = C`

−e^y + 3e^3x = C

The corner points of the feasible region determined by the system of linear constraints are (0, 0), (0, 40), (20, 40), (60, 20) and (60, 0). If the objective function of an LPP is Z = 4x + 3y, then the maximum value is ______.

200

300

240

120

If position vector `vecP` of a point (24, n) is such that `vec|P|` = 25, then the value of n is ______.

±49

±5

±1

±7

If vectors `vec a = 3 hati + 2 hatj + λ hatk and vecb = 2 hati - 4 hatj + 5 hatk`, represent the two strips of the Red Cross sign placed outside a doctor’s clinic, then the value of λ is ______.

1

`5/2`

`2/5`

0

If 3P(A) = P(B) = `3/5` and P(A | B) = `2/3`, then P(A∪B) is ______.

`3/5`

`1/5`

`2/15`

`2/5`

Assertion (A): A relation R on the set {1, 2, 3} defined as R = {(1, 1), (1, 2), (2, 1), (2, 2), (3, 3)} is an equivalence relation.

Reason (R): A relation that is reflexive, symmetric and transitive is an equivalence relation.

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

Both Assertion (A) and Reason (R) are a true, but 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): Consider a Linear Programming Problem with minimise Z = x + 2y subject to constraints 2x + y ≥ 3, x + 2y ≥ 6, x, y ≥ 0 which gives minimum Z at infinitely many points. The corner points of feasible region are (0, 3) and (6, 0).

Reason (R): If two corner points produce the same minimum value of the objective function, then every point on the line segment joining the points will give the same minimum value.

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

Both Assertion (A) and Reason (R) are a true, but 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.

Evaluate sin `[tan^-1 tan((3pi)/4)]`.

Differentiate x^x with respect to x log x.

If y = P cos ux + Q sin ux, show that `(d^2y)/(dx^2) + u^2y = 0`.

Determine the values of x for which f(x) = `(x - 3)/(x + 1)`, x ≠ −1 is an increasing function.

Three honey bees were found flying along the vectors `vec a = 2 hat i - 3 hat j + hat k, vec b = 4 hat j - 2 hat k and vec c = 3 hat i + 2 hat k` respectively. Find the value of λ such that the path for `vec a + λ vec b` is perpendicular to `vec c`.

If A, B and C be three non-collinear points such that `vec ("AB") = hat i + 2 hat j - hat k and vec ("AC") = 2 hat i - 3 hat j`, then find the area of ΔABC.

Find the angle between the following pair of lines:

`(x - 2)/3 = (y + 5)/2 = (1 - z)/-6 and (x - 7)/1 = y/2 = (6 - z)/-2`

A spherical balloon loses its volume due to escape of air from it in such a way that decrease of volume at any instant is proportional to its surface area. Show that the radius is decreasing at a constant rate.

Find:

`int (x - sin x)/(1 - cos x) dx`

Evaluate:

`int_0^2 1/sqrt(x^2 + 2x + 3) dx`

Solve the differential equation (x + 2y³) dy = y dx.

Solve the following Linear Programming Problem graphically:

Maximize Z = `(2x)/5 + (3y)/10`

Subject to constraints

2x + y ≤ 1000

x + y ≤ 800

x, y ≥ 0

Let three toys A, B and C be placed in the same straight line. If the position vectors of A, B and C are `55 hat i - 2 hat j, 5 hat i + 8 hat j and a hat i - 52 hat j` respectively, find the value of ‘a’.

If `vec a, vec b and vec c` are unit vectors, then prove that `|vec a - vec b|^2 + |vec b - vec c|^2 + |vec c - vec a|^2` ≤ 9.

A die is rolled. Consider events:

A = {1, 2, 5}, B = {3, 5}, C = {2, 3, 4, 5}

and hence find:

1. P(A|C) and P(C|A)
2. P(A ∩ B|C) and P(A ∪ B|C)