{"id":27401,"date":"2020-06-30T14:40:56","date_gmt":"2020-06-30T09:10:56","guid":{"rendered":"https:\/\/kseebsolutions.guru\/?p=27401"},"modified":"2021-07-02T15:07:10","modified_gmt":"2021-07-02T09:37:10","slug":"2nd-puc-maths-previous-year-question-paper-march-2017","status":"publish","type":"post","link":"https:\/\/kseebsolutions.guru\/2nd-puc-maths-previous-year-question-paper-march-2017\/","title":{"rendered":"2nd PUC Maths Previous Year Question Paper March 2017"},"content":{"rendered":"<p>Students can Download 2nd PUC Maths Previous Year Question Paper March 2017, Karnataka <a href=\"https:\/\/kseebsolutions.guru\/2nd-puc-maths-model-question-papers-with-answers\/\">2nd PUC Maths Model Question Papers with Answers<\/a> helps you to revise the complete Karnataka State Board Syllabus and score more marks in your examinations.<\/p>\n<h2>Karnataka 2nd PUC Maths Previous Year Question Paper March 2017<\/h2>\n<p>Time: 3 Hrs 15 Min<br \/>\nMax. Marks: 100<\/p>\n<p>Instructions<\/p>\n<ul>\n<li>The question paper has five parts namely A, B, C, D, and E. Answer all the parts.<\/li>\n<li>Use the graph sheet for the question on Linear programming in Part &#8211; E<\/li>\n<\/ul>\n<p style=\"text-align: center;\"><span style=\"color: #0000ff;\">Part &#8211; A<\/span><\/p>\n<p><span style=\"color: #0000ff;\">Answer ALL the following questions: (10 \u00d7 1 = 10)<\/span><\/p>\n<p>Question 1.<br \/>\nLet * be the binary operation on N, given by a * b = LCM of a and b. Find 20 * 16.<br \/>\nSolution:<br \/>\n20 * 16 = LCM of 20 and 16 = 80<\/p>\n<p>Question 2.<br \/>\nFind the principal value of cosec<sup>-1<\/sup> (-\u221a2)<br \/>\nSolution:<br \/>\ncosec<sup>-1<\/sup> (-\u221a2) = -cosec \u221a2 = \\(-\\pi \/ 4\\)<\/p>\n<p>Question 3.<br \/>\nConstruct a 2 \u00d7 2 matrix, A = [a<sub>ij<\/sub>], where elements are given by, a<sub>ij<\/sub> = \\(\\frac{i}{j}\\)<br \/>\nSolution:<br \/>\na<sub>11<\/sub> = \\(\\frac{1}{1}\\) = 1<br \/>\na<sub>12<\/sub> = \\(\\frac{1}{2}\\)<br \/>\na<sub>21<\/sub> = \\(\\frac{2}{1}\\) = 2<br \/>\na<sub>22<\/sub> = \\(\\frac{2}{2}\\) = 1<br \/>\n\u2234 A = \\(\\left[\\begin{array}{ll}1 &amp; 2 \\\\1 \/ 2 &amp; 1\\end{array}\\right]\\)<\/p>\n<p><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/11\/KSEEB-Solutions.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2017\" width=\"172\" height=\"16\" \/><\/p>\n<p>Question 4.<br \/>\nIf A is a square matrix with |A| = 8 then find the value of |AA&#8217;|.<br \/>\nSolution:<br \/>\n|AA&#8217;| = |A| |A&#8217;| = (8) (8) = 64<\/p>\n<p>Question 5.<br \/>\nIf y = cos\u221ax, find \\(\\frac{d y}{d x}\\)<br \/>\nSolution:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63521\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q5.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2017 Q5\" width=\"165\" height=\"57\" \/><\/p>\n<p>Question 6.<br \/>\nFind \\(\\int\\left(\\sqrt{x}+\\frac{1}{\\sqrt{x}}\\right) d x\\)<br \/>\nSolution:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63522\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q6.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2017 Q6\" width=\"466\" height=\"96\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q6.png 466w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q6-300x62.png 300w\" sizes=\"(max-width: 466px) 100vw, 466px\" \/><\/p>\n<p>Question 7.<br \/>\nDefine collinear vectors.<br \/>\nSolution:<br \/>\nCollinear vectors Two or moe non-zero vectors are said to be collinear if they are parallel to the same line. Two vectors a and b are collinear if a = \u03bbb for some scalar \u03bb.<\/p>\n<p><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/11\/KSEEB-Solutions.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2017\" width=\"172\" height=\"16\" \/><\/p>\n<p>Question 8.<br \/>\nFind the direction cosines of a line which makes equal angles with the positive co-ordinates axes.<br \/>\nSolution:<br \/>\n\u03b1 = \u03b2 = r<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63523\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q8.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2017 Q8\" width=\"252\" height=\"202\" \/><\/p>\n<p>Question 9.<br \/>\nDefine a feasible region in a linear programming problem.<br \/>\nSolution:<br \/>\nThe region containing the set of points satisfying all the constraints of an LPP is called the feasible region.<\/p>\n<p>Question 10.<br \/>\nIf A and B are independent events, P(A) = \\(\\frac{3}{5}\\) and P(B) = \\(\\frac{1}{5}\\) then find P(A\u2229B).<br \/>\nSolution:<br \/>\nP(A\u2229B) = P(A) . P(B) = \\(\\frac{3}{5} \\cdot \\frac{1}{5}=\\frac{3}{25}\\)<\/p>\n<p style=\"text-align: center;\"><span style=\"color: #0000ff;\">Part &#8211; B<\/span><\/p>\n<p><span style=\"color: #0000ff;\">Answer any TEN questions: (10 \u00d7 2 = 20)<\/span><\/p>\n<p>Question 11.<br \/>\nIf f : R \u2192 R defined by f(x) = 1 + x<sup>2<\/sup>, then show that f is neither 1-1 nor onto.<br \/>\nSolution:<br \/>\nf(1) = 1 + 1<sup>2<\/sup> = 2<br \/>\nf(-1) = 1 + (-1)<sup>2<\/sup> = 2<br \/>\n\u2234 f(1) = f(-1) But 1 \u2260 -1<br \/>\n\u2234 f is not one-one<br \/>\nLet y \u2208 R \u2203 x \u2208 R such that<br \/>\nf(x) = y<br \/>\n\u21d2 1 + x<sup>2<\/sup> = y<br \/>\n\u21d2 x<sup>2<\/sup> = y &#8211; 1<br \/>\n\u21d2 x = \\(\\sqrt{y-1}\\)<br \/>\nif y = 0 then x = \\(\\sqrt{-1} \\notin \\mathrm{R}\\)<br \/>\n\u2234 0 has no pre-image<br \/>\n\u2234 f is not onto.<\/p>\n<p><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/11\/KSEEB-Solutions.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2017\" width=\"172\" height=\"16\" \/><\/p>\n<p>Question 12.<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63524\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q12.1.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2017 Q12.1\" width=\"398\" height=\"55\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q12.1.png 398w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q12.1-300x41.png 300w\" sizes=\"(max-width: 398px) 100vw, 398px\" \/><br \/>\nSolution:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63525\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q12.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2017 Q12\" width=\"411\" height=\"181\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q12.png 411w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q12-300x132.png 300w\" sizes=\"(max-width: 411px) 100vw, 411px\" \/><\/p>\n<p>Question 13.<br \/>\nSolve the equaton \\(\\tan ^{-1}\\left(\\frac{1-x}{1+x}\\right)=\\frac{1}{2} \\tan ^{-1} x,(x&gt;0)\\)<br \/>\nSolution:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63526\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q13.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2017 Q13\" width=\"255\" height=\"376\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q13.png 255w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q13-203x300.png 203w\" sizes=\"(max-width: 255px) 100vw, 255px\" \/><\/p>\n<p>Question 14.<br \/>\nFind the values of k, if the area of the triangle is 4 sq. units and vertices are (k, 0), (4, 0) and (0, 2) using determinant.<br \/>\nSolution:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63527\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q14.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2017 Q14\" width=\"461\" height=\"196\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q14.png 461w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q14-300x128.png 300w\" sizes=\"(max-width: 461px) 100vw, 461px\" \/><br \/>\n\u21d2 -2k + 8 = 8<br \/>\n\u21d2 2k =<br \/>\n\u21d2 k = 0<br \/>\non taking -ve sign we get -2k + 8 = -8<br \/>\n\u21d2 2k = 16<br \/>\n\u21d2 k = 8<br \/>\n\u2234 k = 0, 8.<\/p>\n<p>Question 15.<br \/>\nIf ax + by<sup>2<\/sup> = cos y, find \\(\\frac{d y}{d x}\\)<br \/>\nSolution:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63528\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q15.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2017 Q15\" width=\"193\" height=\"170\" \/><\/p>\n<p>Question 16.<br \/>\nVerify Rolle\u2019s theorem for the function f(x) = x<sup>2<\/sup> + 2x &#8211; 8, x \u2208 [-4, 2]<br \/>\nSolution:<br \/>\nf(x) is a polynomial in x. Flence it is continuous over {-4, 2} and differentiable over (-4, 2).<br \/>\nf(-4) = (-4)<sup>2<\/sup> + 2(-4) &#8211; 8 = 16 &#8211; 8 &#8211; 8 = 0<br \/>\nf(2) = 4 + 4 &#8211; 8 = 0<br \/>\n\u2234 f(-4) = f(2)<br \/>\n\u2234 All the conditions of the Rolle\u2019s theorem are satisfied.<br \/>\n\u2234 there exists a c \u2208 [-4, 2] such that<br \/>\nf'(c) = 0<br \/>\nf'(x) = 2x + 2<br \/>\nf'(c) = 2c + 2<br \/>\nf'(c) = 0<br \/>\n\u21d2 2c + 2 = 0<br \/>\n\u21d22c = -2<br \/>\n\u21d2 c = -1 \u2208 [-4, 2]<br \/>\n\u2234 Rolle\u2019s theorem is verified.<\/p>\n<p><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/11\/KSEEB-Solutions.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2017\" width=\"172\" height=\"16\" \/><\/p>\n<p>Question 17.<br \/>\nFind the approximate change in the volume of a cube of side x metres caused by increasing the side by 3%.<br \/>\nSolution:<br \/>\nv = x<sup>3<\/sup><br \/>\n\u2206x = 3% of x = (0.03)x<br \/>\n\\(\\frac{d v}{d x}\\) = 3x<sup>2<\/sup><br \/>\n\u2206v = \\(\\frac{d v}{d x}\\) . \u2206x<br \/>\n= (3x<sup>2<\/sup>) (0.03) x<br \/>\n= 0.09 x<sup>3<\/sup> m<sup>3<\/sup><\/p>\n<p>Question 18.<br \/>\nIntegrate \\(\\frac{\\tan ^{4} \\sqrt{x} \\sec ^{2} \\sqrt{x}}{\\sqrt{x}}\\) with respect to x.<br \/>\nSolution:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63529\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q18.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2017 Q18\" width=\"329\" height=\"152\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q18.png 329w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q18-300x139.png 300w\" sizes=\"(max-width: 329px) 100vw, 329px\" \/><\/p>\n<p>Question 19.<br \/>\nEvaluate \\(\\int_{0}^{2 \/ 3} \\frac{d x}{4+9 x^{2}}\\)<br \/>\nSolution:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63530\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q19.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2017 Q19\" width=\"495\" height=\"85\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q19.png 495w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q19-300x52.png 300w\" sizes=\"(max-width: 495px) 100vw, 495px\" \/><\/p>\n<p>Question 20.<br \/>\nFind the order and degree of the differential equation \\(\\left( \\frac { dy }{ dx } \\right) ^{ 2 }+\\frac { dy }{ dx } { sin }^{ 2 }y=0\\)<br \/>\nSolution:<br \/>\nOrder = 1, Degree = 2.<\/p>\n<p><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/11\/KSEEB-Solutions.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2017\" width=\"172\" height=\"16\" \/><\/p>\n<p>Question 21.<br \/>\nFind the position vector of a point R which divides the lipe joining two points P and Q whose position vectors are \\(\\hat{i}+2 \\hat{j}-\\hat{k}\\) and \\(-\\hat{i}+\\hat{j}+\\hat{k}\\) respectively in theratio 2 : 1<br \/>\n(i) Internally.<br \/>\n(ii) Externally.<br \/>\nSolution:<br \/>\nThe position vector of a point R divided the line segment joining two points P and Q in the ratio m : n is given by<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63533\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q21.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2017 Q21\" width=\"430\" height=\"269\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q21.png 430w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q21-300x188.png 300w\" sizes=\"(max-width: 430px) 100vw, 430px\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63532\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q21.1.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2017 Q21.1\" width=\"593\" height=\"386\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q21.1.png 593w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q21.1-300x195.png 300w\" sizes=\"(max-width: 593px) 100vw, 593px\" \/><\/p>\n<p>Question 22.<br \/>\nFind the area of the parallelogram whose adjacent sides are determined by the vector \\(\\vec{a}=\\hat{i}-\\hat{j}+3 \\hat{k}\\) and \\(\\vec{b}=2 \\hat{i}-7 \\hat{j}+\\hat{k}\\)<br \/>\nSolution:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63534\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q22.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2017 Q22\" width=\"515\" height=\"151\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q22.png 515w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q22-300x88.png 300w\" sizes=\"(max-width: 515px) 100vw, 515px\" \/><br \/>\nArea of the parallelogram = \\(|\\vec{a} \\times \\vec{b}|\\) = \u221a450 sq units.<\/p>\n<p>Question 23.<br \/>\nFind the vector and Cartesian equation of the line that passes through the points (3, -2, 5) and (3, -2, 6).<br \/>\nSolution:<br \/>\nLet a and b be the position vectors of points (3, -2, -5), and (3, -2, 6) respectively.<br \/>\n\\(\\vec{a}=3 \\hat{i}-2 \\hat{j}-5 \\hat{k}\\) and \\(b=3 \\hat{i}-2 \\hat{j}+6 \\hat{k}\\)<br \/>\nWe know that the vector equation of a line passing through the points having position vectors a and b is \\(r=\\vec{a}+\\lambda(\\vec{b}-\\vec{a})\\)<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63535\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q23.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2017 Q23\" width=\"685\" height=\"366\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q23.png 685w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q23-300x160.png 300w\" sizes=\"(max-width: 685px) 100vw, 685px\" \/><\/p>\n<p><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/11\/KSEEB-Solutions.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2017\" width=\"172\" height=\"16\" \/><\/p>\n<p>Question 24.<br \/>\nFind the probability distribution of the number of heads in two tosses of a coin.<br \/>\nSolution:<br \/>\nWhen one coin is tossed twice, the sample space is<br \/>\nS = {HH, HT, TH, TT}.<br \/>\nLet X denotes, the number of heads in any outcome in S,<br \/>\nX(HH) = 2, X(HT) = 1, X(TH) = 1 and X(TT) = 0<br \/>\nTherefore, X can take the value of 0, 1 or 2. It is known that<br \/>\nP(HH) = P(HT) = P(TH) = P(TT) = \\(\\frac{1}{4}\\)<br \/>\nP(X = 0) = P (tail occurs on both tosses) = P({TT}) = \\(\\frac{1}{4}\\)<br \/>\nP(X = 1) = P(one head and one tail occurs) = P({TH, HT}) = \\(\\frac{2}{4}\\) = \\(\\frac{1}{2}\\)<br \/>\nand P(X = 2) = P (head occurs on both tosses) = P({HH}) = \\(\\frac{1}{4}\\)<br \/>\nThus, the required probability distribution is as follows.<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63536\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q24.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2017 Q24\" width=\"497\" height=\"89\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q24.png 497w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q24-300x54.png 300w\" sizes=\"(max-width: 497px) 100vw, 497px\" \/><\/p>\n<p style=\"text-align: center;\"><span style=\"color: #0000ff;\">Part &#8211; C<\/span><\/p>\n<p><span style=\"color: #0000ff;\">Anstver any TEN questions: (10 \u00d7 3 = 30)<\/span><\/p>\n<p>Question 25.<br \/>\nShow that the relation R in R (set of real numbers) is defined as R = {(a, b) : a \u2264 b} is reflexive and transitive but not symmetric.<br \/>\nSolution:<br \/>\na \u2264 a is always true<br \/>\n\u2234 (a, a) R a R<br \/>\n\u2234 R is reflexive<br \/>\nLet (a, b) \u2208 R \u21d2 a \u2264 b which does not imply b \u2264 a<br \/>\n\u2234 (b, a) \u2209 R<br \/>\n\u2234 R is not symmetric<br \/>\nLet (a, b) \u2208 R and (b, c) \u2208 R<br \/>\n\u21d2 a \u2264 b and b \u2264 c<br \/>\n\u21d2 a \u2264 c<br \/>\n\u21d2 (a, c) \u2208 R<br \/>\n\u2234 R is transitive<\/p>\n<p><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/11\/KSEEB-Solutions.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2017\" width=\"172\" height=\"16\" \/><\/p>\n<p>Question 26.<br \/>\nWrite \\(\\tan ^{-1}\\left(\\frac{\\sqrt{1+x^{2}}-1}{x}\\right)\\), x \u2260 0 in the simplest form.<br \/>\nSolution:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63537\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q26.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2017 Q26\" width=\"568\" height=\"468\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q26.png 568w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q26-300x247.png 300w\" sizes=\"(max-width: 568px) 100vw, 568px\" \/><\/p>\n<p>Question 27.<br \/>\nIf A and B are symmetric matrices of the same order, then show that AB is symmetric if and only if AB = BA.<br \/>\nSolution:<br \/>\nLet A and B are symmetric<br \/>\n\u2234 A&#8217; = A and B&#8217; = B<br \/>\nLet AB is symmetric<br \/>\n\u2234 (AB)&#8217; = AB, B&#8217;A&#8217; = AB, BA = AB<br \/>\nConversely, Let AB = BA<br \/>\n(AB)&#8217; = B&#8217;A&#8217; = BA = AB<br \/>\n\u2234 AB is symmetric<\/p>\n<p>Question 28.<br \/>\nDifferentiate (log x) cos x with respect to x.<br \/>\nSolution:<br \/>\nLet cos x<br \/>\ny = (log x) cos x<br \/>\nlog y = log (log x)<br \/>\nlog y = cos x (log x)<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63538\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q28.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2017 Q28\" width=\"383\" height=\"175\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q28.png 383w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q28-300x137.png 300w\" sizes=\"(max-width: 383px) 100vw, 383px\" \/><\/p>\n<p>Question 29.<br \/>\nDifferentiate sin<sup>2<\/sup> x with respect to e<sup>cos x<\/sup>.<br \/>\nSolution:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63539\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q29.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2017 Q29\" width=\"564\" height=\"119\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q29.png 564w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q29-300x63.png 300w\" sizes=\"(max-width: 564px) 100vw, 564px\" \/><\/p>\n<p>Question 30.<br \/>\nFind two positive numbers x and y such that x + y = 60 and xy<sup>3<\/sup> is maximum.<br \/>\nSolution:<br \/>\nLet the two numbers be x, y and P = xy<sup>3<\/sup><br \/>\nGiven x + y = 60 \u21d2 x = 60 &#8211; y<br \/>\nOn putting this value in P = xy<sup>3<\/sup>, we get<br \/>\nP = (60 &#8211; y) y<sup>3<\/sup> \u21d2 P = 60y<sup>3<\/sup> &#8211; y<sup>4<\/sup><br \/>\nOn differentiating twice w.r.t. y, we get<br \/>\n\\(\\frac{d P}{d y}\\) = 180y<sup>2<\/sup> &#8211; 4y<sup>3<\/sup><br \/>\nand \\(\\frac{d^{2} P}{d y^{2}}\\) = 360y &#8211; 12y<sup>2<\/sup><br \/>\nFor maxima, we must have \\(\\frac{d P}{d y}\\) = 0<br \/>\n\u21d2 180y<sup>2<\/sup> &#8211; 4y<sup>3<\/sup> = 0<br \/>\n\u21d2 4y<sup>2<\/sup>(45 &#8211; y) = 0<br \/>\n\u21d2 y = 0, 45<br \/>\nBut y \u2260 0, so y = 45<br \/>\nAt y = 45, \\(\\left(\\frac{d^{2} P}{d y^{2}}\\right)_{y=45}\\) = 360 \u00d7 45 &#8211; 12 \u00d7 (45)<sup>2<\/sup><br \/>\n=16200 &#8211; 24300<br \/>\n= -8100 &lt; 0<br \/>\n\u21d2 P has afocal maxima at y = 45<br \/>\n\u2234 By second derivative test, x = 45 is a point of local maxima of P. Thus, the function xy<sup>3<\/sup> is maximum when y = 45 and x = 60 &#8211; 45 = 15.<br \/>\nHence, the required numbers are 15 and 45.<\/p>\n<p><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/11\/KSEEB-Solutions.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2017\" width=\"172\" height=\"16\" \/><\/p>\n<p>Question 31.<br \/>\nEvaluate \\(\\int \\frac{2 x}{x^{2}+3 x+2} d x\\)<br \/>\nSolution:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63540\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q31.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2017 Q31\" width=\"524\" height=\"311\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q31.png 524w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q31-300x178.png 300w\" sizes=\"(max-width: 524px) 100vw, 524px\" \/><\/p>\n<p>Question 32.<br \/>\nEvaluate \u222be<sup>x<\/sup> sin x dx.<br \/>\nSolution:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63541\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q32.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2017 Q32\" width=\"618\" height=\"199\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q32.png 618w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q32-300x97.png 300w\" sizes=\"(max-width: 618px) 100vw, 618px\" \/><\/p>\n<p>Question 33.<br \/>\nFind the area of the region bounded by the curve y<sup>2<\/sup> = 4x, y-axis and the line y = 3.<br \/>\nSolution:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63543\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q33.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2017 Q33\" width=\"312\" height=\"234\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q33.png 312w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q33-300x225.png 300w\" sizes=\"(max-width: 312px) 100vw, 312px\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63542\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q33.1.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2017 Q33.1\" width=\"349\" height=\"104\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q33.1.png 349w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q33.1-300x89.png 300w\" sizes=\"(max-width: 349px) 100vw, 349px\" \/><\/p>\n<p>Question 34.<br \/>\nForm the differential equation of the family of circles having a centre on y-axis and radius 3 units.<br \/>\nSolution:<br \/>\nThe equation of the family circles having centre on y-axis and radius 3 unit is<br \/>\nx<sup>2<\/sup> + (y &#8211; b)<sup>2<\/sup> = 9 &#8230;&#8230;.(i)<br \/>\nOn differentiating Eq. (i) w.r.t x, we get<br \/>\n2x + 2(y &#8211; b) y&#8217; = 0<br \/>\n\u21d2 y &#8211; b = \\(-\\frac{x}{y^{\\prime}}\\) &#8230;&#8230;. (ii)<br \/>\nOn substituting this value of (y &#8211; b) in Eq. (i) we get<br \/>\n\\(x^{2}+\\left(-\\frac{x}{y^{\\prime}}\\right)^{2}=9\\)<br \/>\n\u21d2 x<sup>2<\/sup>[(y&#8217;)<sup>2<\/sup> + 1] = 9(y&#8217;)<sup>2<\/sup><br \/>\n\u21d2 (x<sup>2<\/sup> &#8211; 9)(y&#8217;)<sup>2<\/sup> + x<sup>2<\/sup> = 0<br \/>\nwhich is the required differential equation.<\/p>\n<p><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/11\/KSEEB-Solutions.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2017\" width=\"172\" height=\"16\" \/><\/p>\n<p>Question 35.<br \/>\nFind x, such that the four points A (3, 2, 1), B(4, x, 5), C(4, 2, -2) and D (6, 5, -1) are coplanar.<br \/>\nSolution:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63545\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q35.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2017 Q35\" width=\"272\" height=\"309\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q35.png 272w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q35-264x300.png 264w\" sizes=\"(max-width: 272px) 100vw, 272px\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63544\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q35.1.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2017 Q35.1\" width=\"314\" height=\"236\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q35.1.png 314w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q35.1-300x225.png 300w\" sizes=\"(max-width: 314px) 100vw, 314px\" \/><\/p>\n<p>Question 36.<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63548\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q36.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2017 Q36\" width=\"685\" height=\"68\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q36.png 685w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q36-300x30.png 300w\" sizes=\"(max-width: 685px) 100vw, 685px\" \/><br \/>\nSolution:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63546\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q36.1.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2017 Q36.1\" width=\"293\" height=\"116\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63547\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q36.2.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2017 Q36.2\" width=\"197\" height=\"196\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q36.2.png 197w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q36.2-150x150.png 150w\" sizes=\"(max-width: 197px) 100vw, 197px\" \/><\/p>\n<p>Question 37.<br \/>\nFind the shortest distance between the lines \\(\\vec{r}=\\hat{i}+2 \\hat{j}+\\hat{k}+\\lambda(\\hat{i}-\\hat{j}+\\hat{k})\\) and \\(\\vec{r}=2 \\hat{i}-\\hat{j}-\\hat{k}+\\mu(2 \\hat{i}+\\hat{j}+2 \\hat{k})\\)<br \/>\nSolution:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63550\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q37.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2017 Q37\" width=\"596\" height=\"424\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q37.png 596w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q37-300x213.png 300w\" sizes=\"(max-width: 596px) 100vw, 596px\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63549\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q37.1.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2017 Q37.1\" width=\"622\" height=\"266\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q37.1.png 622w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q37.1-300x128.png 300w\" sizes=\"(max-width: 622px) 100vw, 622px\" \/><\/p>\n<p>Question 38.<br \/>\nGiven that the two numbers appearing on throwing two dice are different. Find the probability of the event &#8216;the sum of numbers on the dice is 4&#8217;.<br \/>\nSolution:<br \/>\nWhen dice is thrown, number of observations in the sample space S = 6 \u00d7 6 = 36 (equally likely sample events)<br \/>\ni.e., n(S) = 36<br \/>\nLet E : set of numbers in which numbers appearing on the two dice are different<br \/>\nThen, E = {(1, 3), (2, 2), (3, 1)}<br \/>\n\u21d2 n(E) = 3<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63552\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q38.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2017 Q38\" width=\"369\" height=\"172\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q38.png 369w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q38-300x140.png 300w\" sizes=\"(max-width: 369px) 100vw, 369px\" \/><br \/>\nn(F) = 30<br \/>\nHere, F contains all points of S except {(1, 1), (2, 2), (3, 3),(4, 4), (5, 5), (6, 6)}<br \/>\nE \u2229 F = {(1, 3), (3, 1)}<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63551\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q38.1.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2017 Q38.1\" width=\"495\" height=\"232\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q38.1.png 495w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q38.1-300x141.png 300w\" sizes=\"(max-width: 495px) 100vw, 495px\" \/><\/p>\n<p style=\"text-align: center;\"><span style=\"color: #0000ff;\">Part &#8211; D<\/span><\/p>\n<p><span style=\"color: #0000ff;\">Answer any SIX questions: (6 \u00d7 5 = 30)<\/span><\/p>\n<p>Question 39.<br \/>\nLet f : N \u2192 R be a function defined as f(x) = 4x<sup>2<\/sup> + 12x + 15. Show that f : N \u2192 S, where S is the range of f, is invertible. Find the inverse of f.<br \/>\nSolution:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63555\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q39.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2017 Q39\" width=\"433\" height=\"67\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q39.png 433w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q39-300x46.png 300w\" sizes=\"(max-width: 433px) 100vw, 433px\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63553\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q39.1.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2017 Q39.1\" width=\"557\" height=\"427\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q39.1.png 557w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q39.1-300x230.png 300w\" sizes=\"(max-width: 557px) 100vw, 557px\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63554\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q39.2.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2017 Q39.2\" width=\"511\" height=\"447\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q39.2.png 511w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q39.2-300x262.png 300w\" sizes=\"(max-width: 511px) 100vw, 511px\" \/><\/p>\n<p>Question 40.<br \/>\nIf A = \\(\\left[\\begin{array}{lll}1 &amp; 0 &amp; 2 \\\\0 &amp; 2 &amp; 1 \\\\2 &amp; 0 &amp; 3\\end{array}\\right]\\) prove that A<sup>3<\/sup> &#8211; 6A<sup>2<\/sup> + 7A + 2I = 0.<br \/>\nSolution:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63556\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q40.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2017 Q40\" width=\"680\" height=\"317\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q40.png 680w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q40-300x140.png 300w\" sizes=\"(max-width: 680px) 100vw, 680px\" \/><\/p>\n<p><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/11\/KSEEB-Solutions.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2017\" width=\"172\" height=\"16\" \/><\/p>\n<p>Question 41.<br \/>\nSolve the following system of linear equations by matrix method.<br \/>\nx &#8211; y + 2z = 1, 2y &#8211; 3z = 1 and 3x &#8211; 2y + 4z = 2.<br \/>\nSolution:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63558\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q41.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2017 Q41\" width=\"293\" height=\"194\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63557\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q41.1.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2017 Q41.1\" width=\"606\" height=\"459\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q41.1.png 606w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q41.1-300x227.png 300w\" sizes=\"(max-width: 606px) 100vw, 606px\" \/><\/p>\n<p>Question 42.<br \/>\nIf y = (tan<sup>-1<\/sup> x)<sup>2<\/sup>. Show that (x<sup>2<\/sup> + 1)<sup>2<\/sup> y<sub>2<\/sub> + 2x(x<sup>2<\/sup> + 1) y<sub>1<\/sub> = 2.<br \/>\nSolution:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63559\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q42.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2017 Q42\" width=\"677\" height=\"312\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q42.png 677w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q42-300x138.png 300w\" sizes=\"(max-width: 677px) 100vw, 677px\" \/><\/p>\n<p>Question 43.<br \/>\nThe length x of a rectangle is decreasing at the rate of 5 cm\/minute and the width y is increasing at the rate of 4 cm\/minute when x = 8 cm and y = 6 cm, find the rate of change of<br \/>\n(i) The perimeter and<br \/>\n(ii) The are of the rectangle.<br \/>\nSolution:<br \/>\nAt any instant of time t, let length, breadth, perimeter and area of the rectangle are x, y, P and A respectively, then<br \/>\nP = 2(x + y) and A = xy &#8230;&#8230; (i)<br \/>\nIt is given that \\(\\frac{d x}{d t}\\) = -5 cm\/mm and \\(\\frac{d y}{d t}\\) = -5 = 4 cm\/min<br \/>\n(-ve sign shows that the length is decreasing)<br \/>\n(i) Now, P = 2(x + y).<br \/>\nOn differentiating w.r.t. t, we get Rate of change of pert meter<br \/>\n\\(\\frac{d P}{d t}=2\\left(\\frac{d x}{d t}+\\frac{d y}{d t}\\right)\\)<br \/>\n= 2(-5 + 4) cm\/min<br \/>\n= -2 cm\/min [\u2234 \\(\\frac{d x}{d t}\\) = -5 and \\(\\frac{d y}{d t}\\) = 4]<br \/>\nHence, perimeter of the rectangle is decreasing (-ve sign) at the rate of 2 cm\/min.<br \/>\n(ii) Here, area of rectangle A = xy.<br \/>\nOn differentiating w.r.t. t, we get<br \/>\nRate of change \\(\\frac{d A}{d t}=x \\frac{d y}{d t}+y \\frac{d x}{d t}\\)<br \/>\n= 8 \u00d7 4 + 6 \u00d7 (5) [\u2234 \\(\\frac{d x}{d t}\\) = -5 and \\(\\frac{d y}{d t}\\) = 4]<br \/>\n= 32 &#8211; 30<br \/>\n= 2 cm<sup>2<\/sup>\/min<br \/>\nHence, area of the rectangle is increasing at the rate of 2 cm<sup>2<\/sup>\/min.<br \/>\nNote: It rate of change is increasing, we take positive sign and if the rate of change is decreasing, then we take a negative sign.<\/p>\n<p><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/11\/KSEEB-Solutions.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2017\" width=\"172\" height=\"16\" \/><\/p>\n<p>Question 44.<br \/>\nFind the integral of \\(\\sqrt{x^{2}-a^{2}}\\) with respect to x and hence evaluate \\(\\int \\sqrt{x^{2}-8 x+7}\\) dx<br \/>\nSolution:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63563\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q44.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2017 Q44\" width=\"286\" height=\"206\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63560\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q44.1.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2017 Q44.1\" width=\"351\" height=\"276\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q44.1.png 351w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q44.1-300x236.png 300w\" sizes=\"(max-width: 351px) 100vw, 351px\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63561\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q44.2.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2017 Q44.2\" width=\"425\" height=\"308\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q44.2.png 425w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q44.2-300x217.png 300w\" sizes=\"(max-width: 425px) 100vw, 425px\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63562\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q44.3.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2017 Q44.3\" width=\"559\" height=\"104\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q44.3.png 559w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q44.3-300x56.png 300w\" sizes=\"(max-width: 559px) 100vw, 559px\" \/><\/p>\n<p>Question 45.<br \/>\nUsing integration find the area of the triangular region whose sides have the equations y = 2x + 1, y = 3x + 1 and x = 4.<br \/>\nSolution:<br \/>\nGiven equation of sides of the triangle are y = 2x + 1, y = 3x + 1 and x = 4.<br \/>\nOn solving these equations, we obtain the vertices of the triangle as A(0, 1), B(4, 13) and C(4, 9).<br \/>\nRequired area (shown in the shaded region)<br \/>\n= Area (OLBAO) &#8211; Area (OLCAO)<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63565\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q45.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2017 Q45\" width=\"506\" height=\"118\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q45.png 506w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q45-300x70.png 300w\" sizes=\"(max-width: 506px) 100vw, 506px\" \/><br \/>\n= 8 sq.units<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63564\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q45.1.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2017 Q45.1\" width=\"271\" height=\"235\" \/><\/p>\n<p><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/11\/KSEEB-Solutions.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2017\" width=\"172\" height=\"16\" \/><\/p>\n<p>Question 46.<br \/>\nSolve the differential equation \\(\\cos ^{2} x \\frac{d y}{d x}+y=\\tan x\\left(0 \\leq x&lt;\\frac{\\pi}{2}\\right)\\)<br \/>\nSolution:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63566\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q46.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2017 Q46\" width=\"248\" height=\"494\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q46.png 248w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q46-151x300.png 151w\" sizes=\"(max-width: 248px) 100vw, 248px\" \/><\/p>\n<p>Question 47.<br \/>\nDerive the equation of a plane perpendicular to a given vector and passing through a given point both in vector and Cartesian form.<br \/>\nSolution:<br \/>\nVector Form<br \/>\nLet a plane pass through a point A with position vector \\(\\overrightarrow{\\mathrm{a}}\\) and perpendicular to the vector \\(\\overrightarrow{\\mathrm{N}}\\)<br \/>\nLet \\(\\overrightarrow{\\mathrm{r}}\\) be the position vector of any point P(x, y, z) in the plane.<br \/>\nThen the point P lies in the plane if and only if \\(\\overline{\\mathrm{AP}}\\) is perpendicular to \\(\\overrightarrow{\\mathrm{N}}\\)<br \/>\ni.e. \\(\\overline{\\mathrm{AP}} \\cdot \\overline{\\mathrm{N}}=0\\)<br \/>\nBut \\(\\overline{\\mathrm{AP}}=\\overrightarrow{\\mathrm{r}}-\\overrightarrow{\\mathrm{a}}\\)<br \/>\nTherefore \\((\\overrightarrow{\\mathbf{r}}-\\overrightarrow{\\mathrm{a}}) \\cdot \\overrightarrow{\\mathrm{N}}=0\\) &#8230;&#8230;.. (1)<br \/>\nThis is the vector equation of the plane.<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63568\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q47.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2017 Q47\" width=\"275\" height=\"245\" \/><br \/>\nCartesian form<br \/>\nLet the given point A be (x<sub>1<\/sub>, y<sub>1<\/sub>, z<sub>1<\/sub>), P be (x, y, z) and direction ratios of<br \/>\n\\(\\overline{\\mathrm{N}}\\) are A, B and C. Then<br \/>\n\\(\\overrightarrow{\\mathrm{a}}=x_{1} \\hat{\\mathrm{i}}+\\mathrm{y}_{1} \\hat{\\mathrm{j}}+\\mathrm{z}_{1} \\hat{\\mathrm{k}}, \\quad \\overrightarrow{\\mathrm{r}}=x \\hat{\\mathrm{i}}+\\mathrm{y} \\hat{\\mathrm{j}}+\\mathrm{z} \\hat{\\mathrm{k}}\\) and \\(\\overline{\\mathrm{N}}=\\mathrm{A} \\hat{\\mathrm{i}}+\\mathrm{B} \\hat{\\mathrm{j}}+\\mathrm{C} \\hat{\\mathrm{k}}\\)<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63567\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q47.1.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2017 Q47.1\" width=\"475\" height=\"150\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q47.1.png 475w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q47.1-300x95.png 300w\" sizes=\"(max-width: 475px) 100vw, 475px\" \/><\/p>\n<p>Question 48.<br \/>\nThe probability that a bulb produced by a factory will fuse after 150 days of use is 0.05. Find the probability that out of 5 such bulbs.<br \/>\n(i) none<br \/>\n(ii) not more than one<br \/>\n(iii) more than one will fuse after 150 days of use.<br \/>\nSolution:<br \/>\nLet X represents the number of bulbs that will fuse after 150 days of use in an experiment of 5 trials. The trials are Bernoulli trials, n = 5p = p (success) = 0.05, q = 1 &#8211; p = 1 &#8211; 0.05 = 0.95<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63569\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q48.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2017 Q48\" width=\"549\" height=\"111\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q48.png 549w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q48-300x61.png 300w\" sizes=\"(max-width: 549px) 100vw, 549px\" \/><\/p>\n<p style=\"text-align: center;\"><span style=\"color: #0000ff;\">Part &#8211; E<\/span><\/p>\n<p><span style=\"color: #0000ff;\">Answer ary ONE question: (1 \u00d7 10 = 10)<\/span><\/p>\n<p>Question 49(a).<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63571\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q49a.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2017 Q49(a)\" width=\"488\" height=\"122\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q49a.png 488w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q49a-300x75.png 300w\" sizes=\"(max-width: 488px) 100vw, 488px\" \/><br \/>\nSolution:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63570\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q49a.1.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2017 Q49(a).1\" width=\"474\" height=\"346\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q49a.1.png 474w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q49a.1-300x219.png 300w\" sizes=\"(max-width: 474px) 100vw, 474px\" \/><\/p>\n<p><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/11\/KSEEB-Solutions.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2017\" width=\"172\" height=\"16\" \/><\/p>\n<p>Question 49(b).<br \/>\nShow that \\(\\left|\\begin{array}{ccc}x &amp; x^{2} &amp; y z \\\\y &amp; y^{2} &amp; z x \\\\z &amp; z^{2} &amp; x y\\end{array}\\right|\\) = (x &#8211; y) (y &#8211; z) (z &#8211; x) (xy + yz + zx)<br \/>\nSolution:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63573\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q49b.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2017 Q49(b)\" width=\"662\" height=\"93\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q49b.png 662w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q49b-300x42.png 300w\" sizes=\"(max-width: 662px) 100vw, 662px\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63572\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q49b.1.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2017 Q49(b).1\" width=\"513\" height=\"306\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q49b.1.png 513w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q49b.1-300x179.png 300w\" sizes=\"(max-width: 513px) 100vw, 513px\" \/><br \/>\n= (y + x)(y &#8211; x)(z &#8211; x)(z<sup>3<\/sup> + x<sup>2<\/sup> + xz) &#8211; (z + x)(z &#8211; x)(y &#8211; x)(y<sup>2<\/sup> + x<sup>2<\/sup> + xy)<br \/>\n= (y &#8211; x)(z &#8211; x)[(y &#8211; x)(z<sup>3<\/sup> + x<sup>2<\/sup> + xz) &#8211; (z + x)(y<sup>2<\/sup> + x<sup>2<\/sup> + xy)]<br \/>\n= (y &#8211; x)(z &#8211; x) [(yz<sup>2<\/sup> + yx<sup>2<\/sup> + xyz + xz<sup>2<\/sup> + x<sup>3<\/sup> + x<sup>2<\/sup>z &#8211; zy<sup>2<\/sup> &#8211; zx<sup>2<\/sup> &#8211; xyz &#8211; xy<sup>2<\/sup> &#8211; x<sup>3<\/sup> &#8211; x<sup>2<\/sup>y]<br \/>\n= (y &#8211; x)(z &#8211; x)[yz<sup>2<\/sup> + zy<sup>2<\/sup> + xz<sup>2<\/sup> &#8211; xy<sup>2<\/sup>]<br \/>\n= (y &#8211; x) (z &#8211; x) [yz (z &#8211; y) + x (z &#8211; y)(z + y)]<br \/>\n= (y &#8211; x) (z &#8211; x) [(z &#8211; y) (xy + yz + zx)]<br \/>\n= (x &#8211; y) (y &#8211; z) (z &#8211; x) (xy + yz + zx)<br \/>\n= RHS<br \/>\nHence proved.<\/p>\n<p>Question 50(a).<br \/>\nMinimize and maximize z = 600x + 400y<br \/>\nSubject to the constraints:<br \/>\nx + 2y \u2264 12<br \/>\n2x + y \u2264 12<br \/>\n4x + 5y \u2265 20 and x \u2265 0, y \u2265 0 by graphical method.<br \/>\nSolution:<br \/>\nx + 2y = 12<br \/>\ny = 0 \u21d2 x = 12 \u2234 P (12, 0)<br \/>\nx = 0 \u21d2 y = 6 \u2234 D (0, 6)<br \/>\n2x + 2y = 12<br \/>\ny = 0 \u21d2 x = 6 \u2234 B (6, 0)<br \/>\nx = 0 \u21d2 y = 12 \u2234 F (0, 12)<br \/>\n4x + 5y = 20<br \/>\ny = 0 \u21d2 x = 5 \u2234 A (5, 0)<br \/>\nx = 0 \u21d2 y = 4 \u2234 E (0, 4)<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63575\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q50a.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2017 Q50(a)\" width=\"417\" height=\"424\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q50a.png 417w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q50a-295x300.png 295w\" sizes=\"(max-width: 417px) 100vw, 417px\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63574\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q50a.1.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2017 Q50(a).1\" width=\"397\" height=\"197\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q50a.1.png 397w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q50a.1-300x149.png 300w\" sizes=\"(max-width: 397px) 100vw, 397px\" \/><\/p>\n<p><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/11\/KSEEB-Solutions.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2017\" width=\"172\" height=\"16\" \/><\/p>\n<p>Question 50(b).<br \/>\nFind the value of k, if<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63577\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q50b.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2017 Q50(b)\" width=\"196\" height=\"107\" \/><br \/>\nSolution:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63576\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q50b.1.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2017 Q50(b).1\" width=\"584\" height=\"427\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q50b.1.png 584w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2017-Q50b.1-300x219.png 300w\" sizes=\"(max-width: 584px) 100vw, 584px\" \/><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Students can Download 2nd PUC Maths Previous Year Question Paper March 2017, Karnataka 2nd PUC Maths Model Question Papers with Answers helps you to revise the complete Karnataka State Board Syllabus and score more marks in your examinations. Karnataka 2nd PUC Maths Previous Year Question Paper March 2017 Time: 3 Hrs 15 Min Max. Marks: &hellip;<\/p>\n","protected":false},"author":5,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[82],"tags":[],"jetpack_sharing_enabled":true,"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/kseebsolutions.guru\/wp-json\/wp\/v2\/posts\/27401"}],"collection":[{"href":"https:\/\/kseebsolutions.guru\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/kseebsolutions.guru\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/kseebsolutions.guru\/wp-json\/wp\/v2\/users\/5"}],"replies":[{"embeddable":true,"href":"https:\/\/kseebsolutions.guru\/wp-json\/wp\/v2\/comments?post=27401"}],"version-history":[{"count":0,"href":"https:\/\/kseebsolutions.guru\/wp-json\/wp\/v2\/posts\/27401\/revisions"}],"wp:attachment":[{"href":"https:\/\/kseebsolutions.guru\/wp-json\/wp\/v2\/media?parent=27401"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/kseebsolutions.guru\/wp-json\/wp\/v2\/categories?post=27401"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/kseebsolutions.guru\/wp-json\/wp\/v2\/tags?post=27401"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}