Section – A<\/strong> Question 1. Question 2. Question 3. Question 4. Question 5. Question 6. Question 7. Question 8. Question 9. Section – B<\/strong><\/p>\n Question 10 Question 11. Question 12. Section – C<\/strong><\/p>\n Question 13. Question 14. Question 15. ISC Maths Previous Year Question Paper 2011 Solved for Class 12 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 […]<\/p>\n","protected":false},"author":5,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_genesis_hide_title":false,"_genesis_hide_breadcrumbs":false,"_genesis_hide_singular_image":false,"_genesis_hide_footer_widgets":false,"_genesis_custom_body_class":"","_genesis_custom_post_class":"","_genesis_layout":"","footnotes":""},"categories":[41556],"tags":[],"yoast_head":"\n
\n(All questions are compulsory in this part)<\/strong><\/p>\n
\n(i) If \\(A=\\left[\\begin{array}{ll}{3} & {-2} \\\\ {4} & {-2}\\end{array}\\right]\\), find x such that A2<\/sup> = xA – 2I. Hence find A-1<\/sup>. [3]
\n(ii) Find the values of k, if the equation 8x2<\/sup> – 16xy + ky2<\/sup> – 22x + 34y = 12 represents an elhpse. [3]
\n(iii) Solve for x: sin (2 tan-1<\/sup>x) = 1 [3]
\n(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]
\n(v) Evaluate: [3]
\n\\(\\int \\frac{\\csc x}{\\log \\tan \\left(\\frac{x}{2}\\right)} d x\\)
\n(vi) Evaluate: [3]
\n\\(\\lim _{y \\rightarrow 0} \\frac{y-\\tan ^{-1} y}{y-\\sin y}\\)
\n(vii) Evaluate: [3]
\n\\(\\int_{0}^{1} \\frac{x e^{x}}{(1+x)^{2}} d x\\)
\n(viii) Find the modulus and argument of the complex number \\(\\frac{2+i}{4 i+(1+i)^{2}}\\) [3]
\n(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]
\n(x) Solve the differential equation: [3]
\n\\(\\frac{d y}{d x}=e^{x+y}+x^{2} e^{y}\\)
\nAnswer:
\n![\"ISC](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2020\/12\/ISC-Maths-Question-Paper-2011-Solved-for-Class-12-image-1.png\")
\n![\"ISC](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2020\/12\/ISC-Maths-Question-Paper-2011-Solved-for-Class-12-image-2.png\")
\n![\"ISC](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2020\/12\/ISC-Maths-Question-Paper-2011-Solved-for-Class-12-image-3.png\")
\n![\"ISC](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2020\/12\/ISC-Maths-Question-Paper-2011-Solved-for-Class-12-image-4.png\")
\n![\"ISC](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2020\/12\/ISC-Maths-Question-Paper-2011-Solved-for-Class-12-image-5.png\")
\n![\"ISC](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2020\/12\/ISC-Maths-Question-Paper-2011-Solved-for-Class-12-image-6.png\")
\n![\"ISC](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2020\/12\/ISC-Maths-Question-Paper-2011-Solved-for-Class-12-image-7.png\")
\n![\"ISC](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2020\/12\/ISC-Maths-Question-Paper-2011-Solved-for-Class-12-image-8.png\")
\n![\"ISC](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2020\/12\/ISC-Maths-Question-Paper-2011-Solved-for-Class-12-image-9.png\")
\n![\"ISC](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2020\/12\/ISC-Maths-Question-Paper-2011-Solved-for-Class-12-image-10.png\")
\n![\"ISC](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2020\/12\/ISC-Maths-Question-Paper-2011-Solved-for-Class-12-image-11.png\")
<\/p>\n
\n(a) Using properties of determinants, show that p\u03b12<\/sup> + 2q\u03b1 + r = 0, given that p, q and r are not in GP and [5]
\n![\"ISC](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2020\/12\/ISC-Maths-Question-Paper-2011-Solved-for-Class-12-image-12.png\")
\n(b) Solve the following system of equations using matrix method: [5]
\n![\"ISC](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2020\/12\/ISC-Maths-Question-Paper-2011-Solved-for-Class-12-image-13.png\")
\nAnswer:
\n![\"ISC](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2020\/12\/ISC-Maths-Question-Paper-2011-Solved-for-Class-12-image-14.png\")
\n![\"ISC](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2020\/12\/ISC-Maths-Question-Paper-2011-Solved-for-Class-12-image-15.png\")
\n![\"ISC](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2020\/12\/ISC-Maths-Question-Paper-2011-Solved-for-Class-12-image-16.png\")
\n![\"ISC](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2020\/12\/ISC-Maths-Question-Paper-2011-Solved-for-Class-12-image-17.png\")
<\/p>\n
\n(a) Prove that: [5]
\n\\(2 \\tan ^{-1} \\frac{1}{5}+\\cos ^{-1} \\frac{7}{5 \\sqrt{2}}+2 \\tan ^{-1} \\frac{1}{8}=\\frac{\\pi}{4}\\)
\n(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]
\nP(P + Q)Q(Q + R’)
\nUse 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.
\nAnswer:
\n![\"ISC](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2020\/12\/ISC-Maths-Question-Paper-2011-Solved-for-Class-12-image-18.png\")
\n![\"ISC](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2020\/12\/ISC-Maths-Question-Paper-2011-Solved-for-Class-12-image-19.png\")
\n(b) P, Q, R represent switches in ON position and P’, Q’, R’ represent in OFF position.
\nGiven polynomial is
\nP(P + Q) Q (Q + R’) = (PP + PQ) (QQ + QR’)
\n= (P + PQ) (Q + QR’)
\n= P(1 + Q)Q(1 + R’)
\n= P.1.Q.1
\n= PQ
\n![\"ISC](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2020\/12\/ISC-Maths-Question-Paper-2011-Solved-for-Class-12-image-20.png\")
<\/p>\n
\n(a) Verify Lagrange’s mean value theorem for the function f(x) = sin x – sin 2x in the interval [0, \u03c0]. [5]
\n(b) Find the equation of the hyperbola whose foci are (0, \u00b113) and the length of the conjugate axis is 20. [5]
\nAnswer:
\n![\"ISC](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2020\/12\/ISC-Maths-Question-Paper-2011-Solved-for-Class-12-image-21.png\")
\n![\"ISC](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2020\/12\/ISC-Maths-Question-Paper-2011-Solved-for-Class-12-image-22.png\")
\n![\"ISC](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2020\/12\/ISC-Maths-Question-Paper-2011-Solved-for-Class-12-image-23.png\")
<\/p>\n
\n(a) Evaluate: [5]
\n\\(\\int \\frac{x^{2}-5 x-1}{x^{4}+x^{2}+1} d x\\)
\n(b) Draw a rough sketch of the curves y = (x – 1)2<\/sup> and y = |x – 1|. Hence, find the area of the region bounded by these curves.
\nAnswer:
\n![\"ISC](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2020\/12\/ISC-Maths-Question-Paper-2011-Solved-for-Class-12-image-24.png\")
\n![\"ISC](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2020\/12\/ISC-Maths-Question-Paper-2011-Solved-for-Class-12-image-25.png\")
\n![\"ISC](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2020\/12\/ISC-Maths-Question-Paper-2011-Solved-for-Class-12-image-26.png\")
\n![\"ISC](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2020\/12\/ISC-Maths-Question-Paper-2011-Solved-for-Class-12-image-27.png\")
<\/p>\n
\n(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]
\n(b) If y = xx<\/sup>, prove that: [5]
\n\\(\\frac{d^{2} y}{d x^{2}}-\\frac{1}{y}\\left(\\frac{d y}{d x}\\right)^{2}-\\frac{y}{x}=0\\)
\nAnswer:
\n![\"ISC](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2020\/12\/ISC-Maths-Question-Paper-2011-Solved-for-Class-12-image-28.png\")
\n![\"ISC](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2020\/12\/ISC-Maths-Question-Paper-2011-Solved-for-Class-12-image-29.png\")
\n![\"ISC](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2020\/12\/ISC-Maths-Question-Paper-2011-Solved-for-Class-12-image-30.png\")
<\/p>\n
\n(a) The following observations are given:
\n(1, 4), (2, 8), (3, 2), (4, 12) (5, 10), (6, 14), (7, 16), (8, 6), (9, 18)
\nEstimate 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]
\n(b) Compute Karl Pearson’s Coefficient of Correlation between sales and expenditures of a firm for six months. [5]
\n![\"ISC](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2020\/12\/ISC-Maths-Question-Paper-2011-Solved-for-Class-12-image-31.png\")
\nAnswer:
\n![\"ISC](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2020\/12\/ISC-Maths-Question-Paper-2011-Solved-for-Class-12-image-32.png\")
\n![\"ISC](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2020\/12\/ISC-Maths-Question-Paper-2011-Solved-for-Class-12-image-33.png\")
\n![\"ISC](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2020\/12\/ISC-Maths-Question-Paper-2011-Solved-for-Class-12-image-34.png\")
<\/p>\n
\n(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]
\n(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]
\nAnswer:
\n(a) Let E1<\/sub> = selecting the first purse, E2<\/sub> selecting the second purse and A = coin drawn is silver.
\n![\"ISC](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2020\/12\/ISC-Maths-Question-Paper-2011-Solved-for-Class-12-image-35.png\")
\n![\"ISC](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2020\/12\/ISC-Maths-Question-Paper-2011-Solved-for-Class-12-image-36.png\")
<\/p>\n
\n(a) Using De Moivre’s theorem, find the value of: [5]
\n\\((1+i \\sqrt{3})^{6}+(1-i \\sqrt{3})^{6}\\)
\n(b) Solve the following differential equation for a particular solution: [5]
\n\\(y-x \\frac{d y}{d x}=x+y \\frac{d y}{d x}, \\text { when } y=0 \\text { and } x=1\\)
\nAnswer:
\n![\"ISC](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2020\/12\/ISC-Maths-Question-Paper-2011-Solved-for-Class-12-image-37.png\")
\n![\"ISC](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2020\/12\/ISC-Maths-Question-Paper-2011-Solved-for-Class-12-image-38.png\")
\n![\"ISC](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2020\/12\/ISC-Maths-Question-Paper-2011-Solved-for-Class-12-image-39.png\")
<\/p>\n
\n(a) Prove that: [5]
\n\\([\\vec{a}+\\vec{b} \\vec{b}+\\vec{c} \\vec{c}+\\vec{a}]=2[\\vec{a} \\vec{b} \\vec{c}]\\)
\n(b) If D, E, F are mid-points of the sides of a triangle ABC, prove by vector method that:
\nArea of \u2206DEF = \\(\\frac { 1 }{ 4 }\\) (Area of \u2206ABC). [5]
\nAnswer:
\n![\"ISC](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2020\/12\/ISC-Maths-Question-Paper-2011-Solved-for-Class-12-image-40.png\")
\n![\"ISC](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2020\/12\/ISC-Maths-Question-Paper-2011-Solved-for-Class-12-image-41.png\")
\n![\"ISC](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2020\/12\/ISC-Maths-Question-Paper-2011-Solved-for-Class-12-image-42.png\")
<\/p>\n
\n(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]
\n(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]
\nAnswer:
\n(a) \\(\\vec{r}=2 \\hat{i}+3 \\hat{j}-\\hat{k}+\\lambda(\\vec{i}-2 \\hat{j}+\\hat{k})\\) …(i)
\nThe 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}\\)
\nIt is given that the required line passes through the point (-1, 2, 1)
\nThe equation of the required line is
\n![\"ISC](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2020\/12\/ISC-Maths-Question-Paper-2011-Solved-for-Class-12-image-43.png\")
\n(b) Let the equation of the plane passing through the point A(2, 1, -3) be
\nA (x – 2) + B(y – 1) + C(z + 3) = 0 ….. (i)
\nPoints B (-3, -2,1) and C (2, 4, -1) lies on the plane.
\n\u21d2 A(-3 – 2) + B (-2 – 1) + C(1 + 3) = 0
\n\u21d2 -5A – 3B + 4C = 0 ……(ii)
\nAnd A(2 – 2) + B(4 – 1) + C(-1 + 3) = 0
\n\u21d2 A.0 + 3B + 2C = 0 ….(iii)
\nNow, eliminating A, B, C from (i), (ii) and (iii), we have
\n![\"ISC](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2020\/12\/ISC-Maths-Question-Paper-2011-Solved-for-Class-12-image-44.png\")
<\/p>\n
\n(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]
\n(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]
\nAnswer:
\n(a) The box contains 4 red and 5 black marbles
\n3 marbles are drawn.
\nProbability of one red marble drawn is
\n![\"ISC](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2020\/12\/ISC-Maths-Question-Paper-2011-Solved-for-Class-12-image-45.png\")
\n(b) Let E1<\/sub>, E2<\/sub> and E3<\/sub> the following events
\nE1<\/sub> = Bag A chosen; E2<\/sub> = Bag B chosen; E3<\/sub> = Bag C chosen.
\n\\(P\\left(E_{1}\\right)=P\\left(E_{2}\\right)=P\\left(E_{3}\\right)=\\frac{1}{3}\\)
\nNow, the probability of drawing a red and a black ball from bag A is,
\n![\"ISC](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2020\/12\/ISC-Maths-Question-Paper-2011-Solved-for-Class-12-image-46.png\")
<\/p>\n
\n(a) The price of a tape recorder is \u20b9 1,661. A person purchases it by making a cash payment of \u20b9 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]
\n(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:
\n![\"ISC](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2020\/12\/ISC-Maths-Question-Paper-2011-Solved-for-Class-12-image-47.png\")
\nEach machine is available for a maximum of six hours per day. If the profit on each cup of type A is \u20b9 1.50 and that on each cup of type B is \u20b9 1.00, find the number of cups of each type that should be manufactured in a day to get maximum profit. [5]
\nAnswer:
\n(a) The price of a tape recorder is \u20b9 1661.
\nThe man purchases it by \u20b9 400 cash down payment.
\nDue amount = 1661 – 400 = \u20b9 1261
\n![\"ISC](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2020\/12\/ISC-Maths-Question-Paper-2011-Solved-for-Class-12-image-48.png\")
\n![\"ISC](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2020\/12\/ISC-Maths-Question-Paper-2011-Solved-for-Class-12-image-49.png\")
\n![\"ISC](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2020\/12\/ISC-Maths-Question-Paper-2011-Solved-for-Class-12-image-50.png\")
\n![\"ISC](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2020\/12\/ISC-Maths-Question-Paper-2011-Solved-for-Class-12-image-51.png\")
<\/p>\n
\n(a) If the difference between Banker’s discount and True discount of a bill for 73 days at 5% per annum is \u20b9 10, find
\n(i) the amount of the bill
\n(ii) Banker’s discount. [5]
\n(b) Given that the total cost function for x units of a commodity is: [5]
\n\\(C(x)=\\frac{x^{3}}{3}+3 x^{2}-7 x+16\\)
\n(i) Find the Marginal Cost (MC)
\n(ii) Find the Average Cost (AC)
\n(iii) Prove that: Marginal Average Cost (MAC) = \\(\\frac{x(MC)-C(x)}{x^{2}}\\)
\nAnswer:
\n![\"ISC](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2020\/12\/ISC-Maths-Question-Paper-2011-Solved-for-Class-12-image-52.png\")
\n![\"ISC](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2020\/12\/ISC-Maths-Question-Paper-2011-Solved-for-Class-12-image-53.png\")
\n![\"ISC](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2020\/12\/ISC-Maths-Question-Paper-2011-Solved-for-Class-12-image-54.png\")
<\/p>\n
\n(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.
\n![\"ISC](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2020\/12\/ISC-Maths-Question-Paper-2011-Solved-for-Class-12-image-55.png\")
\n(b) The profit Of a soft drink firm (in thousands of \u20b9) during each month of the year is as given below:
\n![\"ISC](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2020\/12\/ISC-Maths-Question-Paper-2011-Solved-for-Class-12-image-56.png\")
\nCalculate four monthly moving averages and plot these and the original data on a graph sheet.
\nAnswer:
\n![\"ISC](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2020\/12\/ISC-Maths-Question-Paper-2011-Solved-for-Class-12-image-57.png\")
\n![\"ISC](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2020\/12\/ISC-Maths-Question-Paper-2011-Solved-for-Class-12-image-58.png\")
\n![\"ISC](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2020\/12\/ISC-Maths-Question-Paper-2011-Solved-for-Class-12-image-59.png\")
<\/p>\nISC Class 12 Maths Previous Year Question Papers<\/a><\/h4>\n","protected":false},"excerpt":{"rendered":"