Class 12 ISC Maths Specimen 2023

In subparts (i) to (x) choose the correct options and in subparts (xi) to (xv), answer the questions as instructed.

(i) Which of the following is NOT an equivalence relation on Z?

1. 𝑎𝑅𝑏 ⇔ 𝑎 + 𝑏 𝑖𝑠 𝑎𝑛 𝑒𝑣𝑒𝑛 𝑖𝑛𝑡𝑒𝑔𝑒r
2. 𝑎𝑅𝑏 ⇔ 𝑎 − 𝑏 𝑖𝑠 𝑎𝑛 𝑒𝑣𝑒𝑛 𝑖𝑛𝑡𝑒𝑔𝑒r
3. 𝑎𝑅𝑏 ⇔ 𝑎 < b
4. 𝑎𝑅𝑏 ⇔ 𝑎 = b

Solution

(ii) Let A be the set of all 50 cards numbered from 1 to 50. Let 𝑓: 𝐴 → 𝑁 be a function defined by 𝑓(𝑥) = card number of the card ‘𝑥’. Then the function ‘𝑓’ is:

1. one to one but not onto.
2. onto but not one to one.
3. neither one to one nor onto
4. one to one and onto.

Solution

(iii) The value of tan−1 √3 − sec^-1 (−2) is equal to

1. π / 3
2. 2π / 3
3. - π / 3
4. π / 4

Solution

(iv) If

Solution

(v) If 𝐴 is a 3 × 3 matrix such that

then the value of |A^T| will be

1. 27
2. 9
3. 3
4. -3

Solution

(vi) If
, then the value of |𝐵| is equal to

1. 𝑎 + 2𝑏 + 3c
2. 𝑎 − 𝑏 + 2c
3. 0
4. 3a

Solution

(vii) If sin^-1 𝑥 + sin^-1 𝑦 =

π / 2

, then

dy / dx

is equal to

1. x / y
2. -
   x / y
3. y / x
4. -
   y / x

Solution

(viii) If f(x)=

4-x² / 4x-x³

then the function is:

1. discontinuous at only one point
2. discontinuous at exactly two points.
3. discontinuous at exactly three points
4. discontinuous at exactly four points

Solution

(ix) The degree of the differential equation is

1. 2
2. 1
3. 3
4. not defined

Solution

(x) If A and B are two events such that 𝑃(𝐴) > 0 and 𝑃(𝐵) ≠ 1 , then P (𝐴̅/𝐵̅) is

1. 1 − 𝑃 (𝐴̅/𝐵 )
2. 1 − 𝑃(𝐴/𝐵)
3. 1 − P(𝐴 ∪ 𝐵) / 𝑃(𝐵̅)
4. 𝑃(𝐴̅) / 𝑃(𝐵)

Solution

(xi) Write the smallest equivalence relation on the set 𝐴 = {𝑎, 𝑏, 𝑐}

Solution

(xii) If

3	5
7	9

=X+Y,

where 𝑋 is skew- symmetric matrix and 𝑌 is symmetric matrix. Find |𝑋|.

Solution

(xiii) If ∫ log 2𝑥 𝑑𝑥 = 𝑥log2𝑥 − 𝑘 + 𝑐 where 𝑘 is a function of 𝑥, then find k

Solution

(xiv) 50 tickets in a box are numbered 00, 01, 02 , .. .. , 49. One ticket is drawn randomly from the box. Find the probability of the ticket having the product of its digits 7, given that the sum of the digits is 8?

Solution

(xv) A speaks truth in 60% of cases and B speaks truth in 90% of the cases. In what percentage of cases are they likely to contradict each other in stating the same fact?

Solution

(a) If 𝑦 = √sin 𝑥 + 𝑦 , then find

dy / dx

OR

(b) Find the point on the curve y² = 4𝑥 + 8 for which the abscissa and ordinate changes at the same rate

Solution

Let 𝑓: 𝑅 → 𝑅 defined as 𝑓(𝑥) = 2𝑥 − 3. Find

(i) 𝑓^-1(𝑥)
(ii) domain and range of 𝑓^-1(𝑥)

Solution

The function 𝑓 is defined for all x ∈ R . The line with equation 𝑦 = 6𝑥 − 1 is the tangent to the graph of 𝑓 at 𝑥 = 4

(i) Write down the value of 𝑓′(4).
(ii) Find 𝑓(4).

Solution

Evaluate: ∫[sin(log 𝑥) + cos(log 𝑥)] 𝑑x

OR
Evaluate:

Solution

Solve the differential equation: (1 + 𝑦²)(1 + log 𝑥)𝑑𝑥 + 𝑥𝑑𝑦 = 0

Solution

Solution

If y=(x+√1+x²)ⁿ,then prove that

Solution

Evaluate :


OR

Evaluate: ∫ √cosec 𝑥 − 1 𝑑x

Solution

(i) A student answers a multiple choice question with 5 alternatives, of which exactly one is correct. The probability that he knows the correct answer is 1/5. If he does not know the correct answer, he randomly ticks one answer. Given that he has answered the question correctly, find the probability that he did not tick the answer randomly.

(ii) A candidate takes three tests in succession and the probability of passing the first test is 1/2 . The probability of passing each succeeding test is 1/2 or 1/4 depending on whether he passes or fails in the preceding one. The candidate is selected, if he passes at least two tests. Find the probability that the candidate is selected.

Solution

Find 𝐴^-1, if

Using 𝐴^-1, solve the system of linear equations:
x-2𝑦 = 10, 2𝑥 − 𝑦 − 𝑧 = 8, −2𝑦 + 𝑧 = 7

Solution

(i) Find the particular solution of the differential equation given below
(1 + 𝑥²)𝑑𝑦 = (tan^-1 𝑥 − 𝑦)𝑑𝑥, given that 𝑦 = 1 𝑤ℎ𝑒𝑛 𝑥 = 0

OR
Evaluate: ∫

1 / sin⁴x+cos⁴x

dx

Solution

(i) Find the maximum volume of the cylinder which can be inscribed in a sphere of radius 3√3 𝑐𝑚. (find the answer in terms of 𝜋)

OR

(ii) Find the coordinates of a point on the curve 𝑦 = 𝑥 2 + 7𝑥 + 2 which is closest to the straight line 𝑦 = 3𝑥 − 3.

Solution

A biased four-sided die with faces labelled 1, 2,3 and 4 is rolled and recorded. Let 𝑋 be the result obtained when the die is rolled. The probability distribution for 𝑋 is given in the following table where p and 𝑞 are constants.

For the probability distribution, it is known that 𝐸(𝑋) = 2. Find 𝑝 and 𝑞.
Also, find 𝑃(𝑋 > 2).

Ajay plays a game with this four-sided die. In this game he is allowed a maximum of five rolls. His score is calculated by adding the results of each roll. He wins the game if his score is at least 10. After 3 rolls, Ajay has score of four points. Assuming that rolls of the die are independent, find the probability that Ajay wins the game.

Solution

In subparts (i) and (ii) choose the correct options and in subparts (iii) to (v), answer the questions as instructed.
(i) If 𝑎⃗ = 𝑖̂+ 2𝑗̂+ 3𝑘̂ and 𝑏⃗⃗ = -𝑖̂+ 2𝑗̂+ 𝑘̂ and 𝑐⃗ = 3𝑖̂+ 𝑗̂, find 𝑡 such that 𝑎⃗ + 𝑡𝑏⃗⃗ is perpendicular to 𝑐⃗ is

1. 0
2. 5
3. 4
4. 2

(ii) The planes 2x – y + 4z = 5 and 5x – 2·5y + 10z = 6 are

1. parallel
2. intersect on y axis
3. perpendicular
4. pass through (0, 0, 5/4)

(iii) Find a vector of magnitude of 10 units and parallel to the vector 2𝑖̂+ 3𝑗̂− 𝑘̂.

(iv) Find the position vector of a point R which divides the line joining the two-points 𝑃 and 𝑄 with position vectors 2𝑖̂+𝑗̂and 𝑖̂− 2𝑗̂ respectively in the ratio of 2:1 externally.

(v) Find the equation of the plane with intercept 3 on the 𝑦 − axis and parallel to 𝑥𝑧 −plane.

Solution

(i) Find the area of the triangle whose adjacent sides are 𝑖̂+ 4𝑗̂– 𝑘̂ and 𝑖̂+ 𝑗̂+ 2𝑘̂

OR
(ii) For any two non-zero vectors 𝑎⃗ and 𝑏⃗⃗, if | 𝑎⃗.𝑏⃗⃗ |=|𝑎⃗ ×𝑏⃗⃗|, then find the angle between them.

Solution

Show that

intersect each other. Also find out the point of intersection.

OR

(ii) Find the equation of the plane passing through the points (0,4, −3) and (6, −4,3) if the sum of their intercepts on three axes is 0.

Solution

If the area is bounded by the parabola 𝑦² = 16𝑥 and the line 𝑦 = 4𝑚𝑥 is 1/12 sq units, then using integration, find the value of 𝑚. (𝑚 > 0).

Solution

In subparts (i) and (ii) choose the correct options and in subparts (iii) to (v), answer the questions as instructed.

(i) Which relation is correct for breakeven point where 𝑅(𝑋) is revenue function and 𝐶(𝑥) is cost function?

1. 𝑅(𝑋) > 𝐶(𝑋)
2. 𝑅(𝑋) < 𝐶(𝑋)
3. 𝑅(𝑋) = 𝐶(𝑋)
4. 𝑅(𝑋) = 2𝐶(𝑋)

(ii) If correlation coefficient 𝑟 = 0, then regression lines are

1. parallel to each other.
2. not mutually perpendicular.
3. parallel to coordinate axis.
4. overlapping lines.

(iii) Average revenue of a commodity is given by 𝐴𝑅(𝑥) = 𝑎 +𝑏/𝑥. Find the demand function when marginal revenue is zero.

(iv) If 𝑅(𝑋) = 36𝑥 + 3𝑥² + 5 then find actual revenue from selling 50th item.

(v) For a given bi-variant distribution, the mean of variable 𝑥 = 4 and the mean of variable 𝑦 = 6. Find the point of intersection of two regression lines.

Solution

(i) The total revenue function R(x) = x + 2x³– 3·5x² . Find the point where marginal revenue curve cuts the co-ordinate axis.

OR

(ii) For the revenue function 𝑅(𝑥) =

bx / a+x

, show that the marginal revenue function is increasing for all 𝑏 < 0 and 𝑎> 0.

Solution

The line of regression of marks in Maths (𝑋) and marks in English (𝑌) for a class of 50 students is 3𝑌 − 5𝑋 + 180 = 0. The average score in English is 44 and variance of marks in Maths is 9/16 𝑡ℎ of the variance of marks in English. Find the average score in Maths. Also, find out the coefficient of correlation between marks in Maths and English.

Solution

(i) Solve the following linear programming problem graphically and interpret your result of 𝑍 = 2𝑥 – 5𝑦 subject to the constraints 𝑥 + 𝑦 ≥ 2, 𝑥 − 𝑦 ≥ 0, 𝑥 ≤ 1, 𝑥 ≥0, 𝑦 ≥ 0.

OR

(ii) The standard weight of a special purpose brick is 5 kg and it must contain two basic ingredients B₁ and B₂ . B₁ costs ₹ 5 per kg and B₂ costs ₹ 8 per kg. Strength considerations dictate that the brick should not contain more than 4 kg of B₁ and minimum 2 kg of B₂ .Since the demand for the product is likely to be related to the price of the bricks, find the minimum cost of the brick satisfying the above conditions. Formulate this situation as an L.P.P. and solve it graphically.

Solution