ISC Maths 2011 Class-12 Solved Previous Year Question Papers

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


-: Select Your Topics :-

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:
ISC Class 12 Maths Previous Year Question Papers Solved 2011 Q1
ISC Class 12 Maths Previous Year Question Papers Solved 2011 Q1.1
ISC Class 12 Maths Previous Year Question Papers Solved 2011 Q1.2
ISC Class 12 Maths Previous Year Question Papers Solved 2011 Q1.3
ISC Class 12 Maths Previous Year Question Papers Solved 2011 Q1.4
ISC Class 12 Maths Previous Year Question Papers Solved 2011 Q1.5
ISC Class 12 Maths Previous Year Question Papers Solved 2011 Q1.6
ISC Class 12 Maths Previous Year Question Papers Solved 2011 Q1.7
ISC Class 12 Maths Previous Year Question Papers Solved 2011 Q1.8
ISC Class 12 Maths Previous Year Question Papers Solved 2011 Q1.9
ISC Class 12 Maths Previous Year Question Papers Solved 2011 Q1.10

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]
ISC Class 12 Maths Previous Year Question Papers Solved 2011 Q2
(b) Solve the following system of equations using matrix method: [5]
ISC Class 12 Maths Previous Year Question Papers Solved 2011 Q2.1
Solution 2:
ISC Class 12 Maths Previous Year Question Papers Solved 2011 Q2.2
ISC Class 12 Maths Previous Year Question Papers Solved 2011 Q2.3
ISC Class 12 Maths Previous Year Question Papers Solved 2011 Q2.4
ISC Class 12 Maths Previous Year Question Papers Solved 2011 Q2.5

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:
ISC Class 12 Maths Previous Year Question Papers Solved 2011 Q3
ISC Class 12 Maths Previous Year Question Papers Solved 2011 Q3.1
(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
ISC Class 12 Maths Previous Year Question Papers Solved 2011 Q3.2

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:
ISC Class 12 Maths Previous Year Question Papers Solved 2011 Q4
ISC Class 12 Maths Previous Year Question Papers Solved 2011 Q4.1
ISC Class 12 Maths Previous Year Question Papers Solved 2011 Q4.2

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:
ISC Class 12 Maths Previous Year Question Papers Solved 2011 Q5
ISC Class 12 Maths Previous Year Question Papers Solved 2011 Q5.1
ISC Class 12 Maths Previous Year Question Papers Solved 2011 Q5.2
ISC Class 12 Maths Previous Year Question Papers Solved 2011 Q5.3

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:

ISC Class 12 Maths Previous Year Question Papers Solved 2011 Q6
ISC Class 12 Maths Previous Year Question Papers Solved 2011 Q6.1
ISC Class 12 Maths Previous Year Question Papers Solved 2011 Q6.2

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]
ISC Class 12 Maths Previous Year Question Papers Solved 2011 Q7
Solution 7:
ISC Class 12 Maths Previous Year Question Papers Solved 2011 Q7.1
ISC Class 12 Maths Previous Year Question Papers Solved 2011 Q7.2
ISC Class 12 Maths Previous Year Question Papers Solved 2011 Q7.3

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.
ISC Class 12 Maths Previous Year Question Papers Solved 2011 Q8
ISC Class 12 Maths Previous Year Question Papers Solved 2011 Q8.1

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:

ISC Class 12 Maths Previous Year Question Papers Solved 2011 Q9
ISC Class 12 Maths Previous Year Question Papers Solved 2011 Q9.1
ISC Class 12 Maths Previous Year Question Papers Solved 2011 Q9.2


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:
ISC Class 12 Maths Previous Year Question Papers Solved 2011 Q10
ISC Class 12 Maths Previous Year Question Papers Solved 2011 Q10.1
ISC Class 12 Maths Previous Year Question Papers Solved 2011 Q10.2

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
ISC Class 12 Maths Previous Year Question Papers Solved 2011 Q11
(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
ISC Class 12 Maths Previous Year Question Papers Solved 2011 Q11.1

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
ISC Class 12 Maths Previous Year Question Papers Solved 2011 Q12
(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,
ISC Class 12 Maths Previous Year Question Papers Solved 2011 Q12.1


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:
ISC Class 12 Maths Previous Year Question Papers Solved 2011 Q13
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
ISC Class 12 Maths Previous Year Question Papers Solved 2011 Q13.1
ISC Class 12 Maths Previous Year Question Papers Solved 2011 Q13.2
ISC Class 12 Maths Previous Year Question Papers Solved 2011 Q13.3
ISC Class 12 Maths Previous Year Question Papers Solved 2011 Q13.4

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:
ISC Class 12 Maths Previous Year Question Papers Solved 2011 Q14
ISC Class 12 Maths Previous Year Question Papers Solved 2011 Q14.1
ISC Class 12 Maths Previous Year Question Papers Solved 2011 Q14.2

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.
ISC Class 12 Maths Previous Year Question Papers Solved 2011 Q15
(b) The profit Of a soft drink firm (in thousands of ₹) during each month of the year is as given below:
ISC Class 12 Maths Previous Year Question Papers Solved 2011 Q15.1
Calculate four monthly moving averages and plot these and the original data on a graph sheet.
Solution 15:
ISC Class 12 Maths Previous Year Question Papers Solved 2011 Q15.2
ISC Class 12 Maths Previous Year Question Papers Solved 2011 Q15.3
ISC Class 12 Maths Previous Year Question Papers Solved 2011 Q15.4

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