ISC Maths 2011 Class-12 Solved Previous Year Question Paper for practice. Step by step Solutions with section-A, B and C. Visit official website CISCE for detail information about ISC Board Class-12 Maths.

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## ISC Maths 2011 Class-12 Previous Year Question Papers Solved

Section-A

Section-B

Section-C

Time Allowed: 3 Hours
Maximum Marks: 100

(Candidates are allowed additional 15 minutes for only reading the paper. They must NOT start writing during this time.)

• The Question Paper consists of three sections A, B and C.
• Candidates are required to attempt all questions from Section A and all questions either from Section B or Section C.
• Section A: Internal choice has been provided in three questions of four marks each and two questions of six marks each.
• Section B: Internal choice has been provided in two questions of four marks each.
• Section C: Internal choice has been provided in two questions of four marks each.
• All working, including rough work, should be done on the same sheet as, and adjacent to the rest of the answer.
• The intended marks for questions or parts of questions are given in brackets [ ].
• Mathematical tables and graph papers are provided.

### Section – A (80 Marks)

ISC Maths 2011 Class-12 Previous Year Question Papers Solved

Que 1:
(i) If $A=\left[\begin{array}{ll}{3} & {-2} \\ {4} & {-2}\end{array}\right]$, find x such that A2 = xA – 2I. Hence find A-1. [3] (ii) Find the values of k, if the equation 8x2 – 16xy + ky2 – 22x + 34y = 12 represents an elhpse. [3] (iii) Solve for x: sin (2 tan-1x) = 1 [3]
(iv) Two regression lines are represented by 2x + 3y – 10 = 0 and 4x + y – 5 = 0. Find the line of regression of y on x. [3] (v) Evaluate: [3] $\int \frac{\csc x}{\log \tan \left(\frac{x}{2}\right)} d x$
(vi) Evaluate: [3]
$\lim _{y \rightarrow 0} \frac{y-\tan ^{-1} y}{y-\sin y}$
(vii) Evaluate: [3]
$\int_{0}^{1} \frac{x e^{x}}{(1+x)^{2}} d x$
(viii) Find the modulus and argument of the complex number $\frac{2+i}{4 i+(1+i)^{2}}$ [3] (ix) A word consists of 9 different alphabets, in which there are 4 consonants and 5 vowels. Three alphabets are chosen at random. What is the probability that more than one vowel will be selected? [3] (x) Solve the differential equation: [3] $\frac{d y}{d x}=e^{x+y}+x^{2} e^{y}$
Solution 1:

Que 2:
(a) Using properties of determinants, show that pα2 + 2qα + r = 0, given that p, q and r are not in GP and [5]

(b) Solve the following system of equations using matrix method: [5]

Solution 2:

Que 3:
(a) Prove that: [5]
$2 \tan ^{-1} \frac{1}{5}+\cos ^{-1} \frac{7}{5 \sqrt{2}}+2 \tan ^{-1} \frac{1}{8}=\frac{\pi}{4}$
(b) P, Q and R represent switches in ‘ON position’ and P’, Q’ and R’ represent switches in ‘OFF position’. Construct a switching circuit representing the polynomial: [5] P(P + Q)Q(Q + R’)
Use Boolean Algebra to show that the above circuit is equivalent to a switching circuit in which when P and Q are in ‘ON position’, the light is on.
Solution 3:

(b) P, Q, R represent switches in ON position and P’, Q’, R’ represent in OFF position.
Given polynomial is
P(P + Q) Q (Q + R’) = (PP + PQ) (QQ + QR’)
= (P + PQ) (Q + QR’)
= P(1 + Q)Q(1 + R’)
= P.1.Q.1
= PQ

Que 4:
(a) Verify Lagrange’s mean value theorem for the function f(x) = sin x – sin 2x in the interval [0, π]. [5] (b) Find the equation of the hyperbola whose foci are (0, ±13) and the length of the conjugate axis is 20. [5] Solution 4:

Que 5:
(a) Evaluate: [5] $\int \frac{x^{2}-5 x-1}{x^{4}+x^{2}+1} d x$
(b) Draw a rough sketch of the curves y = (x – 1)2 and y = |x – 1|. Hence, find the area of the region bounded by these curves.
Solution 5:

Que 6:
(a) If the sum of the lengths of the hypotenuse and a side of a right-angled triangle is given, show that the area of the triangle is maximum when the angle between them is $\frac{\pi}{3}$ [5]
(b) If y = xx, prove that: [5] $\frac{d^{2} y}{d x^{2}}-\frac{1}{y}\left(\frac{d y}{d x}\right)^{2}-\frac{y}{x}=0$
Solution 6:

Que 7:
(a) The following observations are given:
(1, 4), (2, 8), (3, 2), (4, 12) (5, 10), (6, 14), (7, 16), (8, 6), (9, 18)
Estimate the value of y when the value of x is 10 and also estimate the value of x when the value of y = 5. [5] (b) Compute Karl Pearson’s Coefficient of Correlation between sales and expenditures of a firm for six months. [5]

Solution 7:

Que 8:
(a) A purse contains 4 silver and 5 copper coins. A second purse contains 3 silver and 7 copper coins. If a coin is taken out at random from one of the purses, what is the probability that it is a copper coin? [5] (b) Aman arid Bhuvan throws a pair of dice alternately. In order to win, they have to get a sum of 8. Find their respective probabilities of winning if Aman starts the game. [5] Solution 8:
(a) Let E1 = selecting the first purse, E2 selecting the second purse and A = coin drawn is silver.

Que 9:
(a) Using De Moivre’s theorem, find the value of: [5]
$(1+i \sqrt{3})^{6}+(1-i \sqrt{3})^{6}$
(b) Solve the following differential equation for a particular solution: [5]
$y-x \frac{d y}{d x}=x+y \frac{d y}{d x}, \text { when } y=0 \text { and } x=1$
Solution 9:

Section – B (20 Marks)

### ISC Maths 2011 Class-12 Previous Year Question Papers Solved

Que 10:
(a) Prove that: [5] $[\vec{a}+\vec{b} \vec{b}+\vec{c} \vec{c}+\vec{a}]=2[\vec{a} \vec{b} \vec{c}]$
(b) If D, E, F are mid-points of the sides of a triangle ABC, prove by vector method that:
Area of ∆DEF = $\frac { 1 }{ 4 }$ (Area of ∆ABC). [5] Solution 10:

Que 11:
(a) Find the vector equation of the line passing through the point (-1, 2, 1) and parallel to the line $\vec{r}=2 \hat{i}+3 \hat{j}-\hat{k}+\lambda(\vec{i}-2 \hat{j}+\hat{k})$. Also, find the distance between these lines. [5] (b) Find the equation of the plane passing through the points A (2, 1, -3), B (-3, -2, 1) and C(2, 4, -1). [5] Solution 11:
(a) $\vec{r}=2 \hat{i}+3 \hat{j}-\hat{k}+\lambda(\vec{i}-2 \hat{j}+\hat{k})$ …(i)
The given fine is parallel to the vector $\hat{i}-2 \hat{j}+\hat{k}$ and the required line is parallel to given line So, required line is parallel to the vector $\hat{i}-2 \hat{j}+\hat{k}$
It is given that the required line passes through the point (-1, 2, 1)
The equation of the required line is

(b) Let the equation of the plane passing through the point A(2, 1, -3) be
A (x – 2) + B(y – 1) + C(z + 3) = 0 ….. (i)
Points B (-3, -2,1) and C (2, 4, -1) lies on the plane.
⇒ A(-3 – 2) + B (-2 – 1) + C(1 + 3) = 0
⇒ -5A – 3B + 4C = 0 ……(ii)
And A(2 – 2) + B(4 – 1) + C(-1 + 3) = 0
⇒ A.0 + 3B + 2C = 0 ….(iii)
Now, eliminating A, B, C from (i), (ii) and (iii), we have

Que 12:
(a) A box contains 4 red and 5 black marbles. Find the probability distribution of the red marbles in a random draw of three marbles. Also find the mean and standard deviation of the distribution. [5] (b) Bag A contains 2 white, 1 black and 3 red balls, Bag B contains 3 white, 2 black and 4 red balls and Bag C contains 4 white, 3 black and 2 red balls. One Bag is chosen at random and 2 balls are drawn at random from that Bag. If the randomly drawn balls happen to be red and black, what is the probability that both balls come from Bag B? [5] Solution 12:
(a) The box contains 4 red and 5 black marbles
3 marbles are drawn.
Probability of one red marble drawn is

(b) Let E1, E2 and E3 the following events
E1 = Bag A chosen; E2 = Bag B chosen; E3 = Bag C chosen.
$P\left(E_{1}\right)=P\left(E_{2}\right)=P\left(E_{3}\right)=\frac{1}{3}$
Now, the probability of drawing a red and a black ball from bag A is,

Section – C (20 Marks)

### ISC Maths 2011 Class-12 Previous Year Question Papers Solved

Que 13:
(a) The price of a tape recorder is ₹ 1,661. A person purchases it by making a cash payment of ₹ 400 and agrees to pay the balance with due interest in 3 half-yearly equal instalments. If the dealer charged interest at the rate of 10% per annum compounded half-yearly, find the value of the instalment. [5] (b) A manufacturer manufactures two types of tea-cups, A and B. Three machines are needed for manufacturing the tea-cups. The time in minutes required for manufacturing each cup on the machines is given below:

Each machine is available for a maximum of six hours per day. If the profit on each cup of type A is ₹ 1.50 and that on each cup of type B is ₹ 1.00, find the number of cups of each type that should be manufactured in a day to get maximum profit. [5] Solution 13:
(a) The price of a tape recorder is ₹ 1661.
The man purchases it by ₹ 400 cash down payment.
Due amount = 1661 – 400 = ₹ 1261

Que 14:
(a) If the difference between Banker’s discount and True discount of a bill for 73 days at 5% per annum is ₹ 10, find
(i) the amount of the bill
(ii) Banker’s discount. [5] (b) Given that the total cost function for x units of a commodity is: [5]
$C(x)=\frac{x^{3}}{3}+3 x^{2}-7 x+16$
(i) Find the Marginal Cost (MC)
(ii) Find the Average Cost (AC)
(iii) Prove that: Marginal Average Cost (MAC) = $\frac{x(MC)-C(x)}{x^{2}}$
Solution 14:

Que 15:
(a) The price quotations of four different commodities for 2001 and 2009 are as given below. Calculate the index number for 2009 with 2001 as the base year by using a weighted average of price relative method.

(b) The profit Of a soft drink firm (in thousands of ₹) during each month of the year is as given below:

Calculate four monthly moving averages and plot these and the original data on a graph sheet.
Solution 15:

-: End of ISC Maths 2011 Class-12 Solved Previous Year Question Paper :-

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