The ISC Class 12 Maths Paper was conducted on 5th March 2019. The exam started at 2 PM, and students were allotted 3 hours of time duration to finish the paper. Here, we have provided the ISC Class 12 Maths Question Paper Solution 2019 to help students with their exam preparation. They can download the ISC Class 12 Maths Question Paper and Solution pdf for the year 2019 by clicking on the link below.

## ISC Class 12 Maths Question Paper 2019

## ISC Class 12 Maths Question Paper Solution 2019 PDF

Students can access the solved Maths papers of all the years from 2015 to 2019 by visiting the ISC Class 12 Maths Previous Year Question Papers page. They can have a look at the ISC Class 12 Maths Question Paper Solution 2019 below.

### Difficult Topics of ISC Class 12 Maths Paper 2019

Topics which students found difficult during the Maths 2019 exam are listed below:

- Increasing and decreasing functions.
- In proving the function “onto”.
- Applying formulae of inverse trigonometric functions.
- Finding the area of a triangle using determinant.
- Proving differentiability.
- Application of derivatives, in particular, rate measurers, Maxima and Minima.
- Conditional Probability and Bayes’ theorem.
- Family of planes and the equation of straight line of 3D Geometry.
- Area under the curve.
- Application of calculus in Commerce and Economics.

### Confusing Questions of ISC Class 12 Maths Paper 2019

Questions based on concepts on which students got confused while solving them during the exam are as below.

- Between the composite functions f o g and g o f.
- Between open and closed intervals in Mean Value theorem.
- Between mutually exclusive and independent events.
- Identifying the type of differential equation.
- Definite integrals and their properties.
- Between Marginal cost and average cost.
- Between the regression coefficients b
_{yx}and b_{xy}and regression lines*y*on*x*and*x*on*y*. - Between Dot and Cross product of vectors.

## ISC Class 12 Maths Question Paper Solution 2019

### Question 1:

(i) If f : R→R, *f(x)= x ^{3}* and

*g*: R→ R, g(x) = 2

*x*

^{2}+1, and R is the set of real numbers, then find

*fog (x)*and

*gof (x)*.

(ii) Solve: Sin (2 tan^{-1}x) = 1

(iii) Using determinants, find the values of k, if the area of triangle with vertices

(−2, 0), (0, 4) and (0, k) is 4 square units.

(vi) Prove that the function f(x) = x^{3} − 6x^{2} + 12x + 5 is increasing on R.

(ix) Two balls are drawn from an urn containing 3 white, 5 red and 2 black balls, one by one without replacement. What is the probability that at least one ball is red?

(x) If events A and B are independent, such that P(A) = 3/5, P(B) = 2/3, find P(A∪B) .

**Answer: (i)**

(ii)

(ix) 3W, 5R, 2B

Event A: not drawing a red ball in 1st instance.

Event B: not drawing a red ball in 2nd instance.

P(A) = 5/10, or 𝑃(𝐵) = 4/9

Probability of not drawing red ball in 1st and 2nd instances:

(x)

### Question 2:

**Answer:**

### Question 3:

**Answer:**

**(a)**

**OR**

**(b)**

### Question 4:

**Answer:**

### Question 5: (a) Show that the function f(x) = |x – 4| , x∈R is continuous, but not differentiable at x = 4.

OR

(b) Verify the Lagrange’s mean value theorem for the function:

f(x) = x + 1/x in the interval [1, 3].

**Answer: (a)**

**(b)**

### Question 6:

**Answer: **

### Question 7: A 13 m long ladder is leaning against a wall, touching the wall at a certain height from the ground level. The bottom of the ladder is pulled away from the wall, along the ground, at the rate of 2 m/s. How fast is the height on the wall decreasing when the foot of the ladder is 5 m away from the wall?

**Answer: **

### Question 8:

**Answer: (a)**

**OR**

**(b) **

### Question 9:

**Answer:**

### Question 10: Bag A contains 4 white balls and 3 black balls, while Bag B contains 3 white balls and 5 black balls. Two balls are drawn from Bag A and placed in Bag B. Then, what is the probability of drawing a white ball from Bag B?

**Answer:**

### Question 11:

**Answer:**

### Question 12: (a) The volume of a closed rectangular metal box with a square base is 4096 cm^{3}. The cost of polishing the outer surface of the box is ₹ 4 per cm2. Find the dimensions of the box for the minimum cost of polishing it.

OR

(b) Find the point on the straight line 2x + 3y = 6, which is closest to the origin.

**Answer: (a)**

**OR**

**(b) **

### Question 13:

**Answer:**

### Question 14:

(a) Given three identical Boxes A, B and C, Box A contains 2 gold and 1 silver coin, Box B contains 1 gold and 2 silver coins and Box C contains 3 silver coins. A person chooses a Box at random and takes out a coin. If the coin drawn is of silver, find the probability that it has been drawn from the Box which has the remaining two coins also of silver.

OR

(b) Determine the binomial distribution where mean is 9 and standard deviation is 3/2. Also, find the probability of obtaining at most one success.

**Answer: (a)**

**OR**

**(b)**

### Question 15:

**Answer:**

**(c) **

### Question 16:

**Answer:**

### Question 17:

**Answer: (a)**

**OR**

**(b) **

### Question 18: Draw a rough sketch and find the area bounded by the curve x^{2} = y and x + y = 2.

**Answer:**

### Question 19:

(a) A company produces a commodity with ₹ 24,000 as fixed cost. The variable cost, estimated to be 25% of the total revenue received on selling the product, is at the rate of ₹ 8 per unit. Find the break-even point.

**Answer: (a) **Given fixed cost = Rs 24000

Let number of units be: x

Selling price per unit p(x) = Rs.8

∴ Revenue function R(x) = 8x

Cost function = C (x) = 24000 + 25% of 8x

= 24000 + 2x

For break even points: R(x) = C(x)

8x = 24000 + 2x

6x = 24000

x = 4000 units

### Question 20:

(a) The following results were obtained with respect to two variables x and y:

(i) Find the regression coefficient b_{xy}.

(ii) Find the regression equation of x on y.

OR

(b) Find the equation of the regression line of y on x, if the observations (x, y) are as follows:

(1, 4), (2, 8), (3, 2), (4, 12), (5, 10), (6, 14), (7, 16), (8, 6), (9, 18)

Also, find the estimated value of y when x = 14.

**Answer: (a)**

**(b)**

### Question 21:

(a)

(b) The marginal cost function of x units of a product is given by MC = 3x^{2} −10x + 3. The cost of producing one unit is ₹ 7. Find the total cost function and average cost function.

**Answer: (a)**

**OR**

**(b)**

### Question 22: A carpenter has 90, 80 and 50 running feet respectively of teak wood, plywood and rosewood which is used to produce product A and product B. Each unit of product A requires 2, 1 and 1 running feet and each unit of product B requires 1, 2 and 1 running feet of teak wood, plywood and rosewood respectively. If product A is sold for ₹ 48 per unit and product B is sold for ₹ 40 per unit, how many units of product A and product B should be produced and sold by the carpenter, in order to obtain the maximum gross income?

Formulate the above as a Linear Programming Problem and solve it, indicating clearly the feasible region in the graph.

**Answer:**

The ISC Class 12 Maths Question Paper Solution 2019 must have helped students with their exam preparation. Apart from the 2019 paper solution, students can also find the answers to other papers of ISC Class 12 Previous Years Questions by clicking here. 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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