Mathematics 65/2/3 2025-2026 Commerce (English Medium) Class 12 Question Paper Solution

Direction cosines of line `(1-x)/0=y=z` are:

1, 1, 1

`0,(-1)/sqrt2,(-1)/sqrt2`

1, 0, 0

0, –1, –1

In a linear programming problem, the linear function which has to be maximized or minimized is called ______.

a feasible function 

an objective function

an optimal function 

a constraint

For the feasible region shown below, the non-trivial constraints of the linear programming problem are:

x + y ≤ 5, x + 3y ≤ 9 

x + y ≤ 5, x + 3y ≥ 9 

x + y ≥ 5, x + 3y ≤ 9 

x + y ≥ 5, 3x + y ≤ 9 

For two events A and B such that P(A) ≠ 0 and P(B) ≠ 1, P(A'/B') =

1 – P(A/B)

1 – P(A'/B)

`(1-P(A∩B))/(P(B'))`

`(1-P(A∪B))/(P(B'))`

A relation R on set A = {1, 2, 3} is defined as R = {(1, 3), (3, 3), (1, 1), (2, 2), (3, 1} is ______.

only reflexive and symmetric

reflexive only

only reflexive and transitive

reflexive, symmetric and transitive

If A and B are square matrices of the same order, then which of the following statements is/are always true?

1. (A + B) (A – B) = A² – B²
2. AB = BA
3. (A + B)² = A² + AB + BA + B²
4. AB = 0 ⇒ A = 0 or B = 0

Only (i) and (iii)

Only (ii) and (iii)

Only (iii)

Only (iii) and (iv)

If A = `[(1,a,b),(-1,2,c),(0,5,3)]` is a symmetric matrix, then the value of 3a + b + c is ______.

2

6

4

0

If A = `[(tanx,cotx),(-cotx,tanx)]` and A + A' = 2I, then value of x ∈ `[0,pi/2]` is ______.

0

`pi/3`

`pi/4`

`pi/2`

For a square matrix A, (3A)^–1 = ______. 

3A^–1

9A^–1

`1/3`A^–1

`1/9`A^–1

If `[(–1,–2,5),(–2,a,–1),(0,4,2a)]` = –86, then the sum of all possible values of a is ______. 

4

5

–4

9

If x + y = xy, then `dy/dx` is ______.

`y/(x-1)`

`1/(x-1)`

`(y-1)/(x-1)`

`(1-y)/(x-1)`

`intdx/(secx+tanx)` is equal to ______.

log |secx + tanx| + C

log |secx – tanx| + C

log |1 + cosx| + C

log |1 + sinx| + C

For f(x) = x + `1/x` (x ≠ 0)

local maximum value is 2

local minimum value is –2

local maximum value is –2

local minimum value < local maximum value

Which of the following expressions will give the area of region the curve y = x² and line y = 16?

`int_0^4x^2dx`

`2int_0^4x^2dx`

`int_0^16sqrty  dy`

`2int_0^16sqrty  dy`

The general solution of the differential equation: x²dy + y²dx = 0 is ______. 

x³ + y³ = k

`1/y-1/x=k`

`1/y+1/x=k`

log y² + log x² = k

The integrating factor of the differential equation 2x`dy/dx` – y = 3 is ______.

`sqrtx`

`sqrt1/x`

e^x

e^–x

If `veca` = 5 and –2 ≤ λ ≤ 1, then the sum of greatest and the smallest value of `|λveca|` is ______.

–5

5

10

15

Vector of magnitude 3 making equal angles with x and y Perpendicular to z axis is ______.

`hati+2sqrt2hatj`

`3hatk`

`(3sqrt2)/2hati+(3sqrt2)/2hatj`

`sqrt3hati+sqrt3hatj+sqrt3hatk`

For two vectors `veca` and `vecb`

Assertion (A): `|vecaxxvecb|^2+(veca*vecb)^2=|veca|^2|vecb|^2`

Reason (R): `|vecaxxvecb|=(veca*vecb)tantheta,(theta≠pi/2)`

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

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

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

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

Assertion (A): A line can have direction cosines < 1, 1, 1>

Reason (R): cos θ = 1 is possible for θ = 0.

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

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

Find the co-ordinates of the point on line `x=(y-1)/2=(z-2)/3` whose y co-ordinate is 3 times the x co-ordinate.

Check whether f: R – {3} → R defined as f(x) = `(x-2)/(x-3)` is onto or not.

Check whether f: Z × Z → Z × Z (where Z is the set of integers) defined as f(x, y) = (2y, 3x) is injective or not.

If `x = e^(sin^-1t),y=e^(cos^-1t)`

find `dy/dx  "at"  "t"=1/sqrt2`

Find the absolute maximum value of f(x) = cos x + sin² x, x ∈ [0, π].

If the volume of a solid hemisphere increases at a uniform rate, prove that its surface area varies inversely as its radius.

If `vec(AB)=hatj+hatk` and `vec(AC)=3hati-hatj+4hatk` represent the two vectors along the sides AB and AC of ΔABC, prove that the median `vec(AD)=(vec(AB)+vec(AC))/2`, where D is midpoint of BC. Hence, find the length of median AD.

The Probability of hitting the target by a trained sniper is three times the Probability of not hitting the target on a stormy day due to high wind speed.

The sniper fired two shots on the target on a stormy day when wind speed was very high. Find the probability that

1. target is hit
2. atleast one shot misses the target.

Determine P(E|F).

Mother, father and son line up at random for a family picture

E: son on one end, F: father in middle

Find: `int(3x-1)/(sqrt(x^2-4x))dx`

Evaluate: `int_(pi/12)^((5pi)/12)dx/(1+sqrtcotx)`

Evaluate: `int_((-pi)/6)^(pi/2)(sin|x|+cos|x|)dx`

If `d/dx(F(x))=1/(e^x+1)`, then find F(x) given that F(0) = log `1/2`.

Solve the following differential equation:

`xdy/dx=y-xsin^2(y/x)`, given that y(1) = `pi/6`

Find the general solution of the differential equation: `y log y dx/dy+x=2/y`.

Solve the following linear programming problem graphically:

Maximize Z = 4500x + 5000y

Subject to constraints

x + y ≤ 250

25x + 40y ≤ 7000

x ≥ 0, y ≥ 0 

Find the sub-interval of `(0,pi/2)` in which f(x) = log (sin x + cos x) is increasing and decreasing.

A rectangle of perimeter 36 cm is revolved around one of its sides to sweep out a cylinder of maximum volume.

Find the dimensions of the rectangle.

Find the domain of q(x) = cos^–1 (4x² – 3). Hence, find the value of x for which q(x) = 0. Also, write the range of 3q(x) – π.

A line passing through the points A(1, 2, 3) and B(5, 8, 11) intersects the line `vecr=4hati+hatj+lambda(5hati+2hatj+hatk)`. Find the co-ordinates of the point of intersection. Hence, write the equation of a line passing through the point of intersection and perpendicular to both the lines.

If P = `[(1,-1,0),(2,3,4),(0,1,2)]` and Q = `[(2,2,-4),(-4,2,-4),(2,-1,5)]` find (QP) and hence solve the following system of equations using matrices:

x – y = 3, 2x + 3y + 4z = 17, y + 2z = 7

Obtain the value of Δ = `|(1+x,1,1),(1,1+y,1),(1,1,1+z)|` in terms of x, y and z. 

Further, if Δ = 0 and x, y, z are non-zero real numbers, prove that x^–1 + y^–1 + z^–1 = –1. 

Sports car racing is a form of motorsport which uses sports car prototypes. The competition is held on special tracks designed in various shapes.

The equation of one such track is given as follows:

`f(x) = {{:((x^4-4x^2+4)",",0 ≤ x < 3), (x^2+40",",x≥3):}`

Based on given information, answer the following questions:

1. Find f'(x) for 0 < x < 3. 1
2. Find f'(4) 1
3. 1. Test for continuity of f(x) at x = 3. 2
      OR
   2. Test for differentiability of f(x) at x = 3. 2

Smoking increases the risk of lung problems.

A study revealed that 170 in 1000 males who smoke develop lung related complications, while 120 out of 1000 females who smoke develop lung related problems. In a colony, 50 people were found to be smokers of which 30 are males.

A person is selected at random from these 50 people and tested for lung related problems.

Based on the given information, answer the following questions:

1. What is the probability that selected person is a female? 1
2. If a male person is selected, what is the probability that he will not be suffering from lung problems? 1
3. 1. A person selected at random is detected with lung complications. Find the probability that selected person is a female. 2
      OR
   2. A person selected at random is not having lung problems, find the probability that the person is a male. 2

A racing track is build around an elliptical ground whose equation is given by 9x² + 16y² = 144. The width of the track is 3 m as shown below:

Based on given information, answer the following questions:

1. Express y as a function of x from the given equation of ellipse. 1
2. Integrate the function obtained in (i) with respect to x. 1
3. 1. Find the area of the region enclosed within the elliptical excluding the track using integration. 2
      OR
   2. Write the co-ordinates of the points P and Q where the outer edge of the track cuts x axis and y axis in first quadrant and find the area of the triangle formed by points P, O, Q using integration. 2