ISC Class 12 Maths Question Paper Solution 2016

The ISC Class 12 Maths was conducted on 15th March 2016. The exam started at 2 PM. Exam was of 3 hours and 100 Marks. Here, we have provided the completely solved ISC Class 12 Maths Question Paper 2016. Along with the solutions, students will also find the marking scheme and step-wise marks allocated to each answer. So, it’s recommended to the students to go through the solution pdf after solving the previous years ISC Class 12 Maths Question Paper. They can download the ISC Class 12 Maths Question Paper Solution 2016 PDF from the link provided below.

For students convenience, we have also compiled the solved ISC Class 12 Previous Year Question Papers of Maths, Physics, Chemistry and Biology subjects at one place. Students can download and access them for free. They can have a look at the ISC Class 12 Maths Question Paper Solution 2016 below.

Difficult Topics of ISC Class 12 Maths Paper 2016

Topics which students found difficult while solving the Maths 2016 paper are mentioned below:

− Indefinite Integrals (use of substitution or integration by parts)

− Definite Integrals – use of properties.

− Inverse Circular Functions (formulae and relations)

− Differential Equations (solving Homogeneous and Linear Differential Equations)

− Vectors – in general

− Annuity (Deferred annuities)

− Conics in general

− Probability – use of sum and product laws and identifying all cases.

− Maxima and Minima

Confusing ISC Class 12 Maths Questions 2016

Maths concepts in which students got confused during the exam are mentioned below.

− Regression lines: y on x and x on y

− Sum and product laws of probability

− 3 – D: Image of a given point and perpendicular distance

− Conditional probability property in Baye’s theorem

− Price Index by aggregate and Price Relative methods

− Differences between and usage of formulae for BD, TD, BG, DV, etc.

ISC Class 12 Maths Question Paper Solution 2016

Question 1:

(i) Find the matrix X for which:

(ii) Solve for x, if:

(iii) Prove that the line 2x – 3y = 9 touches the conics y² = – 8x. Also, find the point of contact.

(iv) Using L’Hospital’s Rule, evaluate:

(vii) The two lines of regressions are x + 2y – 5 = 0 and 2x + 3y – 8 = 0 and the variance of x is 12. Find the variance of y and the coefficient of correlation.

(viii) Express (2+𝑖) / (1+ 𝑖) (1−2𝑖) in the form of a + ib. Find its modulus and argument.

(ix) A pair of dice is thrown. What is the probability of getting an even number on the first die or a total of 8?

(x) Solve the differential equation:

Answer:

(iii) Line 3y = 2x – 9

y = 2/3𝑥 − 3

m = 2/3, c = −3

y² = − 8x

a = −2

The condition: a = mc

⇒ −2 = ⅔ × −3

⇒ −2= −2

∴ the line touches the parabola

(ix) {(2,1),(2,2),(2,3),(2,4),(2,5),(2,6),(4,1),(4,2),(4,3),(4,4),(4,5),(4,6),(6,1),(6,2),(6,3),(6,4),(6,5),(6,6),(3,5),(5,3)}

P(E) = 20/36 = 59

Or P(A) + P(B) – P(A∩B) = 18/36 + 5/36 − 3/36 = 20/36

(x)

Question 2:

(a) Using properties of determinants, prove that:

(b) Solve the following system of linear equations using matrix method:

3x + y + z = 1, 2x + 2z = 0, 5x + y + 2z = 2

Answer: (a)

Question 3:

(a) If sin^-1x + tan^-1x = 𝜋/2 , prove that:

2x² + 1 = √5

(b) Write the Boolean function corresponding to the switching circuit given below:

A, B and C represent switches in ‘on’ position and Aʹ, Bʹ and Cʹ represent them in ‘off’ position. Using Boolean algebra, simplify the function and construct an equivalent switching circuit.

Answer:

(a)

(b) F (A, B, C) = A(Aʹ + B) + AʹB+ (A+Bʹ)C

= AAʹ + AB + AʹB + AC + BʹC

= O + B(A+Aʹ) + AC + BʹC

= B + BʹC + AC

= (B +Bʹ) (B+C) + AC

= B + C + AC = B + C

Question 4:

(a) Verify the conditions of Rolle’s Theorem for the following function:

f(x) = log(x² + 2) – log 3 on [-1,1]

Find a point in the interval, where the tangent to the curve is parallel to x-axis.

(b) Find the equation of the standard ellipse, taking its axes as the coordinate axes, whose minor axis is equal to the distance between the foci and whose length of latus rectum is 10. Also, find its eccentricity.

Answer: (a) f(x) = log(x²+2) – log 3 is continuous [-1, 1]

f ʹ(x) = 2𝑥 / (𝑥²+2)

f ʹ(x) exists in (-1, 1)

f( -1) = f (1) = 0

All the conditions of Rolle’s theorem are satisfied then

there exists ‘c’ in ( -1, 1) such that f ʹ (c) = 0

2c / (1+𝑐²) = 0

c = 0 lies between -1 and 1. Hence, Rolle’s theorem is verified.

The point where the tangent is parallel to x axis is (0, log 2/3)

(b)

Question 5:

(a) If log y = tan^-1x, prove that:

(b) A rectangle is inscribed in a semicircle of radius r with one of its sides on the diameter of the semicircle. Find the dimensions of the rectangle to get maximum area. Also, find the maximum area.

Answer:

(b)

Question 6:

(a) Evaluate:

(b) Find the area of the region bounded by the curves y = 6x – x² and y = x² – 2x.

Answer: (a)

(b) The curve y = 6x – x²

y = −(x −3)² + 9

represents a parabola with vertex at (3, 9) and it opens downward.

The curve y = x² – 2 x

y = (x −1)²−1

represents a parabola with vertex at (1 -1) and it opens upward.

Both the curves pass through origin and intersect in the first quadrant at (4, 8)

Question 7:

(a) Calculate Karl Pearson’s coefficient of correlation between x and y for the following data and interpret the result:

(1, 6), (2, 5), (3,7), (4, 9), (5, 8), (6, 10), (7, 11), (8, 13), (9, 12)

(b) The marks obtained by 10 candidates in English and Mathematics are given below:

Estimate the probable score for Mathematics if the marks obtained in English are 24.

Answer: (a)

(b)

Question 8:

(a) A committee of 4 persons has to be chosen from 8 boys and 6 girls, consisting of at least one girl. Find the probability that the committee consists of more girls than boys.

(b) An urn contains 10 white and 3 black balls while another urn contains 3 white and 5 black balls. Two balls are drawn from the first urn and put into the second urn and then a ball is drawn from the second urn. Find the probability that the ball drawn from the second urn is a white ball.

Answer: (a) Number of ways the committee can be selected:

= ¹⁴C₄−⁸C₄ = 1001 – 70 = 931 or (⁶C₁.⁸C₃)+(⁶C₂.⁸C₂)+(⁶C₃.⁸C₁)+⁶C₄

No. of Committees consists of more girls than boys= ⁶C₄+ ⁶C₃× ⁸C₁

P(E) = ⁶C₄+ ⁶C₃× ⁸C₁ / 931

= 15 +160 / 931 = 175 / 931 = 0·188

(b) P (transferring 2 white balls to urn 2 and drawing a white ball from urn 2)

= ¹⁰C₂×⁵C₁ / ¹³C₂×¹⁰C₁

= 45 / 13×12

p (transferring 2 black balls to urn 2 land drawing a white from urn 2)

= ¹³C₂׳C₁ / ¹³C₂×¹⁰C₁

= 9 / 13×12×5

P(transferring1white and a black ball to urn 2 and drawing a white ball from urn 2)

= ¹⁰C₁׳C₁×⁴C₁ / ¹³C₂×¹⁰C₁

= 24 / 13×12

Required probability =

Question 9:

(a) Find the locus of a complex number, z = x + iy, satisfying the relation

Illustrate the locus of z in the Argand plane.

(b) Solve the following differential equation:

x² dy + (xy + y²) dx = 0, when x = 1 and y =1.

Answer:

Question 10:

Answer:

Question 11:

Answer: (a)

Question 12:

(a) In an automobile factory, certain parts are to be fixed into the chassis in a section before it moves into another section. On a given day, one of the three persons A, B and C carries out this task. A has 45% chance, B has 35% chance and C has 20% chance of doing the task. The probability that A, B and C will take more than the allotted time is 1/6,1/10, 1/20 respectively. If it is found that the time taken is more than the allotted time, what is the probability that A has done the task?

(b) The difference between mean and variance of a binomial distribution is 1 and the difference of their squares is 11. Find the distribution.

Answer: (a) Let E₁, E₂, E₃, denote the events of carrying out the task by A, B and C respectively.

Let H denote the event of taking more time.

Then P(E₁) = 0.45

P(E₂) = 0.35

P(E₃) = 0.20

P(H/E₁) = 1/6

P(H/E₂) = 1/10

P(H/E₃) = 1/20

(b) np – npq = 1, np(1 – q) = 1 ………………..(i)

(np)² – (npq)² = 11, (np)² (1 – q²) = 11 …………(ii)

dividing (ii) by (i), we get (1 + q) / (1−1) = 11

1 + q = 11 – 11q

12q = 10

q = 5/6, p = 1/6

we get n = 36

The distribution is given by (1/6 + 5/6)³⁶ or x ~ B (36, 1/6)

Question 13:

(a) A man borrows 20,000 at 12% per annum, compounded semi-annually and agrees to pay it in 10 equal semi-annual installments. Find the value of each installment, if the first payment is due at the end of two years

(b) A company manufactures two types of products A and B. Each unit of A requires 3 grams of nickel and r grams of chromium, while each unit of B requires 1 gram of nickel and 2 grams of chromium. The firm can produce 9 grams of nickel and 8 grams of chromium. The profit is Rs 40 on each unit of product qf type A and Rs 50 on each unit of type B. How many units of each type should the company manufacture so as to earn maximum profit? Use linear programming to find the solution.

Answer: (a)

(b) Let x units of product A and y units of product B.

Maximize Z = 40x + 50y

Subject to constraints

3x + y ≤ 9

x + 2y ≤ 8

x ≥ 0, y ≥ 0

Solving, we get A(0, 4), B(3, 0), C(2, 3)

At A, z = 40×0 + 50× 4 = ₹ 200

B, z = 40×3 + 50×0 = ₹ 120

C, z = 40×2 + 50×3 = ₹ 230

Maximum profit is ₹ 230, when 2 units of type A and 3 units of type B are produced.

Question 14:

(a) The demand function is x = (24−2p) / 3 where x is the number of units demanded and p is the price per unit. Find:

(i) The revenue function R in terms of p.

(ii) The price and the number of units demanded for which the revenue is maximum.

(b) A bill of ₹ 1,800 drawn on 10th September, 2010 at 6 months was discounted for ₹1,782 at a bank. If the rate of interest was 5% per annum, on what date was the bill discounted?

Answer:

(b) A = 1800; i= 5% p.a.

BD = 1800 – 1782 = 18

BD = Ani

⇒18 = 1800 × n × 5/100

⇒ n = 1/5 year = 73 days

Date of expiry: March 13, 2011

Date of discounting: December 30, 2010.

Question 15:

(a) The index number by the method of aggregates for the year 2010, taking 2000 as the base year, was found to be 116. If sum of the prices in the year 2000 is Rs 300, find the values of x and y in the data given below:

(b) From the details given below, calculate the five yearly moving averages of the number of students who have studied in a school. Also, plot these and original data on the same graph paper.

Answer: (a)

(b)

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