Mathematics Specimen Paper 2024-2025 ISC (Commerce) Class 12 Question Paper Solution

A matrix which is both symmetric and skew symmetric matrix is a ______.

triangular matrix

identity matrix

diagonal matrix

null matrix

The value of `inta^x.e^x dx` equals

`(a^x.log_ea)e^x + c`

`(a^x.e^x)/(log_e(ae)) + c`

`(a^x.e^x)/(log_(ae)e) + c`

`log_e(ae)(ae)^x + c`

The trigonometric equation tan^–1x = 3tan^–1 a has solution for ______.

`|a| ≤ 1/sqrt(3)`

`|a| > 1/sqrt(3)`

`|a| < 1/sqrt(3)`

all real value of a.

Assertion: Degree of the differential equation: `a(dy/dx)^2 + bdx/dy = c`, is 3

Reason: If each term involving derivatives of a differential equation is a polynomial (or can be expressed as polynomial) then highest exponent of the highest order derivative is called the degree of the differential equation.

Which of the following is correct?

Both Assertion and Reason are true and Reason is the correct explanation for Assertion.

Both Assertion and Reason are true but Reason is not the correct explanation for Assertion.

Assertion is true and Reason is false.

Assertion is false and Reason is true.

Five numbers x₁, x₂, x₃, x₄, x₅ are randomly selected from the numbers 1, 2, 3, ......., 18 and are arranged in the increasing order such that x₁ < x₂ < x₃ < x₄ < x₅. What is the probability that x₂ = 7 and x₄ = 11?

`26/51`

`3/104`

`1/68`

`1/34`

if `|(a, b, c),(m, n, p),(x, y, z)| = k`, then what is the value of `|(6a, 2b, 2c),(3m, n, p),(3x, y, z)|`?

`k/6`

2k

3k

6k

Consider the graph `y = x^(1/3)`

Statement 1: The above graph is continuous at x = 0

Statement 2: The above graph is differentiable at x = 0

Which of the following is correct?

Statement 1 is true and Statement 2 is false.

Statement 2 is true and Statement 1 is false.

Both the statements are true.

Both the statements are false.

The value of `dy/dx` if y = |x – 1| + |x – 4| at x = 3 is ______.

–2

0

2

4

Statement 1: The intersection of two equivalence relations is always an equivalence relation.

Statement 2: The Union of two equivalence relations is always an equivalence relation.

Which one of the following is correct?

Statement 1 implies Statement 2.

Statement 2 implies Statement 1.

Statement 1 is true only if Statement 2 is true.

Statement 1 and 2 are independent of each other.

In a third order matrix a_ij denotes the element of the i^th row and the j^th column.

A = `a_(ij) = {(0",", for, i = j),(1",", f or, i > j),(-1",", f or, i < j):}`

Assertion: Matrix ‘A’ is not invertible.

Reason: Determinant A = 0

Which of the following is correct?

Both Assertion and Reason are true and Reason is the correct explanation for Assertion.

Both Assertion and Reason are true but Reason is not the correct explanation for Assertion.

Assertion is true and Reason is false.

Assertion is false and Reason is true.

Given two events A and B such that (A/B) = 0.25 and P(A ∩ B) = 0.12. The value P(A ∩ B') is ______.

0.36

0.48

0.88

0.036

The value of the determinant of a matrix A of order 3 is 3. If C is the matrix of cofactors of the matrix A, then what is the value of determinant of C²?

If a relation R on the set {a, b, c} defined by R = {(b, b)}, then classify the relation.

The given function f : R → R is not ‘onto’ function. Give reason.

There are three machines and 2 of them are faulty. They are tested one by one in a random order till both the faulty machines are identified. What is the probability that only two tests are needed to identify the faulty machines?

If x^y = y^x, then find `dy/dx`

Find the interval in which the function f(x) = x²e^–x is strictly increasing or decreasing.

Evaluate:

`int_0^sqrt(2)[x^2]dx`

Find the equation to the tangent at (0, 0) on the curve y = 4x² – 2x³

Evaluate:

`int(2x^3 - 1)/(x^4 + x)dx`

Evaluate:

`inte^x "cosec"  x(1 - cot x)dx`

Find the value of `tan^-1(x/y) + tan^-1((y - x)/(y + x))`

Solve:

sin^–1(x) + sin^–1(1 – x) = cos^–1x.

Evaluate:

`intsqrt(sec  x/2 - 1)dx`

A kite is being pulled down by a string that goes through a ring on the ground 8 meters away from the person pulling it. If the string is pulled in at 1 meter per second, how fast is the kite coming down when it is 15 meters high?

If `y = (x + sqrt(a^2 + x^2))^m`, prove that `(a^2 + x^2)(d^2y)/(dx^2) + xdy/dx - m^2y = 0`

Three friends go to a restaurant to have pizza. They decide who will pay for the pizza by tossing a coin. It is decided that each one of them will toss a coin and if one person gets a different result (heads or tails) than the other two, that person would pay. If all three get the same result (all heads or all tails), they will toss again until they get a different result.

1. What is the probability that all three friends will get the same result (all heads or all tails) in one round of tossing?
2. What is the probability that they will get a different result in one round of tossing?
3. What is the probability that they will need exactly four rounds of tossing to determine who would pay?

Students of under graduation submitted a case study on “Understanding the Probability of Left-Handedness in Children Based on Parental Handedness”. Following Recent studies suggest that roughly 12% of the world population is left-handed. Depending on the parents’ handedness, the chances of having a left-handed child are as follows:

Scenario A: Both parents are left-handed, with a 24% chance of the child being left-handed.

Scenario B: The fathers is right-handed and the mothers left-handed, with a 22% chance of child being left-handed.

Scenario C: The fathers left-handed and the mother is right-handed, with a 17% chance of child being left-handed.

Scenario D: Both parents are right-handed, with a 9% chance of having a left-handed child.

Assuming that scenarios A, B, C and D are equally likely and L denotes the event that the child is left-handed, answer the following questions.

1. What is the overall probability that a randomly selected child is left-handed?
2. Given that exactly one parent is left-handed, what is the probability that a randomly selected child is left-handed?
3. If a child is left-handed, what is the probability that both parents are left-handed?

Answer the following question:

1. Translate the problem into a system of equations.
2. Solve the system of equation by using matrix method.
3. Hence, find the cost of one paper bag, one scrap book and one pastel sheet.

Solve the differential equation:

`(xdy - ydx)  ysin(y/x) = (ydx + xdy)  xcos(y/x)`.

Find the particular solution satisfying the condition that y = π when x = 1.

Evaluate:

`int_0^π(sin^4x + cos^4x)dx`

Hence evaluate:

`int_(-2π)^(2π) (sin^4x + cos^4x)/(1 + e^x)dx`

Show that the cone of the greatest volume which can be inscribed in a given sphere has an altitude equal to \[ \frac{2}{3} \] of the diameter of the sphere.

A given quantity of metal is to be cast into a half cylinder with a rectangular base and semicircular ends. Show that in order that the total surface area may be minimum the ratio of the length of the cylinder to the diameter of its semi-circular ends is \[\pi : (\pi + 2)\].

Kiran plays a game of throwing a fair die 3 times but to quit as and when she gets a six. Kiran gets +1 point for a six and –1 for any other number.

1. If X denotes the random variable “points earned” then what are the possible values X can take?
2. Find the probability distribution of this random variable X.
3. Find the expected value of the points she gets.

Consider the following statements and choose the correct option:

Statement 1: If `veca` and `vecb` represents two adjacent sides of a parallelogram then the diagonals are represented by `veca + vecb` and `veca - vecb`.

Statement 2: If `veca` and `vecb` represents two diagonals of a parallelogram then the adjacent sides are represented by `2(veca + vecb)` and `2(veca - vecb)`.

Which of the following is correct?

Only Statement 1

Only Statement 2

Both Statements 1 and 2

Neither Statement 1 nor Statement 2

If a plane passes through the point (1, 1, 1) and is perpendicular to the line \[\frac{x - 1}{3} = \frac{y - 1}{0} = \frac{z - 1}{4}\] then its perpendicular distance from the origin is ______.

`3/4`

`4/3`

`7/5`

If the direction cosines of a line are `(1/c, 1/c, 1/c)` then ______.

0 < c < 1

c = ± 3

c > 2

c > 0

c = `±sqrt(3)`

If `veca` is a unit vector perpendicular to `vecb` and `(veca + 2vecb).(3veca - vecb) = -5`, find `|vecb|`.

Shown below is a cuboid. Find `vec(BA).vec(BC)`

Find a vector of magnitude 9 units and perpendicular to the vectors.

`veca = 4hati - hatj + hatk` and `vecb = -2hati + hatj - 2hatk`

What are the values of x for which the angle between the vectors? `2x^2hati + 3xhatj + hatk` and `hati - 2hatj + x^2hatk` is obtuse?

Show that the line whose vector equation is `vecr = (2hati - 2hatj + 3hatk) + λ(hati - hatj + 4hatk)` is parallel to the plane whose vector equation is `vecr.(hati + 5hatj + hatk) = 5`. Also find the distance between them.

Find the equation of the plane containing the line `x/(-2) = (y - 1)/3 = (1 - z)/1` and the point (–1, 0, 2).

Sketch the region enclosed bounded by the curve, y = x |x| and the ordinates x = −1 and x = 1.

Evaluate:

`int_0^1x^2dx`

Hence find the area bounded by the curve, y = x |x| and the coordinates x = −1 and x = 1.

Which condition is true if Average Cost (AC) is constant at all levels of output?

MC > AC

MC = AC

MC < AC

MC = `1/2` AC

Read the following statements and choose the correct option:

1. If r = 0, then regression lines are not defined.
2. If r = 0, then regression lines are parallel.
3. If r = 0, then regression lines are perpendicular.
4. If r = ±1, then regression lines coincide.

Which of the following is correct?

Only IV is correct.

Only I and II are correct.

Only I and IV are correct.

Only III and IV are correct.

Mean of x = 53, mean of y = 28 regression co-efficient y on x = −1.2, regression co-efficient x on y = −0.3. Find coefficient of correlation (r).

The total revenue received from the sale of x unit of a product is given by R(x) = 3x² + 36x + 5. Find the marginal revenue when x = 5.

A manufacturing company finds that the daily cost of producing x item of product is given by C(x) = 210x + 7000. Find the minimum number that must be produced and sold daily, if each item is sold for ₹ 280.

The cost function of a commodity is `C(x) = 200 + 20x - 1/2x^2` (in rupees). Find the range where AC falls.

The demand function of a monopoly is given by x = 100 − 4p. Find the quantity at which the MR will be zero.

A survey of 50 families to study the relationships between expenditure on accommodation in (₹ x) and expenditure on food and entertainment (₹ y) gave the following results: `sumx = 8500, sumy = 9600, σ_x = 60, σ_y = 20, r = 0.6`

Estimate the expenditure on food and entertainment when expenditure on accommodation is ₹ 200.

The random variables have regression lines 3x + 2y − 26 = 0 and 6x + y − 31 = 0. Calculate mean value of x and y.

The random variables have regression lines 3x + 2y − 26 = 0 and 6x + y − 31 = 0. Calculate co-efficient of correlations.

A linear programming problem is given by Z = px + qy where p, q > 0 subject to the constraints: x + y ≤ 60, 5x + y ≤ 100, x ≥ 0 and y ≥ 0

1. Solve graphically to find the corner points of the feasible region.
2. If Z = px + qy is maximum at (0, 60) and (10, 50), find the relation of p and q. Also mention the number of optimal solution(s) in this case.