The ISC Class 12 Maths – Paper was held on 26th February 2018. The exam comprises 100 marks and 3 hours of time duration. It started at 2 PM. Here, we have provided the ISC Class 12 Maths Question Paper Solution 2018 so that students can easily get the answers to all questions. Going through the solution pdf will help them in analysing how marks are awarded in steps for each answer. They can also be able to verify their answers with the solutions provided by us in the pdf. To download the ISC Class 12 Maths Question Paper and Solution pdf 2018, click on the links below.

## ISC Class 12 Maths Question Paper 2018

## ISC Class 12 Maths Question Paper Solution 2018 PDF

Students can also access the Solved ISC Class 12 Maths Previous Year Question Papers compiled at one place. They can have a look at the ISC Class 12 Maths Question Paper Solution 2018 below.

### Difficult Topics of ISC Class 12 Maths Paper 2018

Topics which students found difficult while solving the Maths 2018 paper are listed below:

- Application of derivatives including Maxima and Minima.
- Integrals, and curve sketching.
- Vectors, interchange of vector equation to Cartesian equation and vice-versa.
- Probability and Probability distribution.
- Inverse circular functions.

### Confusing ISC Class 12 Maths Questions 2018

Maths concepts in which students got confused during the exam are:

- Open and closed interval of mean value theorems.
- Product and sum rule of probability and dependent and independent events.
- Dot and Cross product of vectors, Projection of a vector.
- Properties of definite integrals.

## ISC Class 12 Maths Question Paper Solution 2018

### Question 1:

(i) The binary operation ∗ : R × R → R is defined as a ∗ b = 2a + b.

Find (2 ∗ 3) ∗ 4.

(iii) Solve: 3tan^{-1}x + cot^{-1}x = 𝜋

(vi) Find the approximate change in the volume ‘𝑉’ of a cube of side x metres caused by decreasing the side by 1%.

(vii)

(viii) Find the differential equation of the family of concentric circles x^{2 }+ y^{2} = a^{2}

(ix) If A and B are events such that P(A) = 1/2 , P(B) = 1/3 and P(A∩B) = 1/4 , then find:

- P (𝐴⁄𝐵)
- P (𝐵⁄𝐴)

(x) In a race, the probabilities of A and B winning the race are 1/3 and 1/6 respectively.

Find the probability of neither of them winning the race.

**Answer:**

(i) a∗b = 2a + b

2∗3 = 2.2 + 3 = 7

(2∗3)∗4 = 7∗4

= 2·7+4 = 18

### Question 2: If the function f(x) = √2𝑥 − 3 is invertible then find its inverse. Hence prove that(𝑓 o 𝑓^{−1}) (𝑥) = 𝑥.

**Answer:**

### Question 3: If tan^{−1} 𝑎 + tan^{−1} 𝑏 + tan^{−1} 𝑐 = 𝜋, prove that 𝑎 + 𝑏 + 𝑐 = 𝑎𝑏𝑐.

**Answer:**

### Question 4: Use properties of determinants to solve for x:

**Answer:**

### Question 5:

(a)

**OR**

(b) Verify Rolle’s theorem for the following function: 𝑓(𝑥) = 𝑒^{−𝑥} 𝑠i𝑛 𝑥 on [0, 𝜋]

**Answer: (a)**

(b) f(x) = e^{-x} sin x is continuous in [0, 𝜋]

f^{1}(x) = e^{-x} (cos x − sin x) exists in (0, 𝜋)

f (0) = f (𝜋) = 0

Rolle’s theorem conditions are satisfied.

∴ there exists at least one value of x = c

such that f^{1}(c) = 0

∴ f^{1}(c) = e^{−𝑐}(cos c – sin c) = 0

⇒ cos c − sin c = 0

⇒ c = 𝜋/4

⇒ c = 𝜋/4 ∈ (0, 𝜋)

### Question 6:

**Answer:**

### Question 7: Evaluate

**Answer:**

### Question 8:

(a) Find the points on the curve 𝑦 = 4𝑥^{3} – 3𝑥 + 5 at which the equation of the tangent is parallel to the x-axis.

OR

(b) Water is dripping out from a conical funnel of semi-vertical angle 𝜋/4 at the uniform rate of 2 cm^{2}/sec in the surface, through a tiny hole at the vertex of the bottom. When the slant height of the water level is 4 cm, find the rate of decrease of the slant height of the water.

**Answer: (a)**

**OR**

**(b)**

### Question 9:

(a)

**OR**

(b) The population of a town grows at the rate of 10% per year. Using a differential equation, find how long it will take for the population to grow 4 times.

**Answer: (a)**

### Question 10:

(a) Using matrices, solve the following system of equations:

2𝑥 – 3𝑦 + 5𝑧 = 11

3𝑥 + 2𝑦 – 4𝑧 = −5

𝑥 + 𝑦 – 2𝑧 = – 3

OR

(b) Using elementary transformation, find the inverse of the matrix:

**Answer: **Solve the following system of equations:

2𝑥 – 3𝑦 + 5𝑧 = 11

3𝑥 + 2𝑦 – 4𝑧 = −5

𝑥 + 𝑦 – 2𝑧 = – 3

OR

(b)

### Question 11: A speaks truth in 60% of the cases, while B in 40% of the cases. In what percent of cases are they likely to contradict each other in stating the same fact?

**Answer:**

### Question 12: A cone is inscribed in a sphere of radius 12 cm. If the volume of the cone is maximum, find its height.

**Answer: **

### Question 13:

**Answer: **

### Question 14:

From a lot of 6 items containing 2 defective items, a sample of 4 items are drawn at random. Let the random variable X denote the number of defective items in the sample. If the sample is drawn without replacement, find:

- The probability distribution of X
- Mean of X
- Variance of X

**Answer: **

### Question 15:

(a)

(b) The Cartesian equation of a line is: 2𝑥 – 3 = 3𝑦 + 1 = 5 – 6𝑧. Find the vector equation of a line passing through (7, −5, 0) and parallel to the given line.

(c) Find the equation of the plane through the intersection of the planes

**Answer: **

### Question 16:

**Answer: **

### Question 17:

(a) Draw a rough sketch of the curve and find the area of the region bounded by curve

𝑦^{2} = 8𝑥 and the line 𝑥 = 2.

OR

(b) Sketch the graph of 𝑦 = |𝑥 + 4|. Using integration, find the area of the region bounded by the curve 𝑦 = |𝑥 + 4| and 𝑥 = −6 𝑎𝑛𝑑 𝑥 = 0.

**Answer: (a)**

OR

**(b) **𝑦 = |𝑥 + 4|

x + 4 > 0 ⇒ x > − 4

x + 4 < 0 ⇒ x <− 4

### Question 18:

**Answer: **

(3λ+3, -λ-2, 4λ+1)

3 (3λ+3) -1(-λ-2) +4(4λ+1) = 2

26λ + 15 = 2

26λ = -13

λ = -13/26 = -1/2

Let (x_{1}, y_{1}, z_{1}) be the image of the point with reference to the given plane

∴ foot of the perpendicular

⇒ x_{1}= 0, y_{1}= -1, z_{1}= -3

(0, -1, -3)

### Question 19:

(a) Given the total cost function for x units of a commodity as:

𝐶(𝑥) = 1/3𝑥^{3} + 3𝑥^{2} − 16𝑥 + 2.

Find:

(i) Marginal cost function

(ii) Average cost function

(b) Find the coefficient of correlation from the regression lines:

𝑥 – 2𝑦 + 3 = 0 and 4𝑥 – 5𝑦 + 1 = 0.

(c) The average cost function associated with producing and marketing x units of an item is given by 𝐴𝐶 = 2𝑥 – 11 + 50/𝑥 . Find the range of values of the output x, for which AC is increasing.

**Answer: **

### Question 20:

(a) Find the line of regression of y on x from the following table.

x |
1 |
2 |
3 |
4 |
5 |

y |
7 |
6 |
5 |
4 |
3 |

Hence, estimate the value of y when x = 6.

(b) From the given data:

Variable |
x |
y |

Mean |
6 |
8 |

Standard Deviation |
4 |
6 |

and correlation coefficient: 2/3 . Find:

(i) Regression coefficients b_{yx} and b_{xy}

(ii) Regression line x on y

(iii) Most likely value of x when y = 14

**Answer:**

### Question 21:

(a) A product can be manufactured at a total cost 𝐶(𝑥) = 𝑥^{2}/100 + 100𝑥 + 40, where x is the number of units produced. The price at which each unit can be sold is given by

P = (200 − 𝑥/400).

Determine the production level x at which the profit is maximum. What is the price per unit and total profit at the level of production?

OR

(b) A manufacturer’s marginal cost function is 500/√2𝑥+25 . Find the cost involved to increase production from 100 units to 300 units.

**Answer: **

### Question 22:

A manufacturing company makes two types of teaching aids A and B of Mathematics for Class X. Each type of A requires 9 labour hours for fabricating and 1 labour hour for finishing. Each type of B requires 12 labour hours for fabricating and 3 labour hours for finishing. For fabricating and finishing, the maximum labour hours available per week are 180 and 30 respectively. The company makes a profit of ₹ 80 on each piece of type A and ₹ 120 on each piece of type B. How many pieces of type A and type B should be manufactured per week to get a maximum profit? Formulate this as a Linear Programming Problem and solve it. Identify the feasible region from the rough sketch.

**Answer: **

We hope students must have found ISC Class 12 Maths Question Paper Solution 2018 helpful for their exam preparation. We have also compiled ISC Class 12 Previous Year Question Papers for Physics, Maths, Maths and Biology at one place for students convenience. Happy Learning and stay tuned to BYJU’S for the latest update on ICSE/CBSE/State Boards/Competitive exams. Also, don’t forget to download the BYJU’S App.

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