{"id":27317,"date":"2020-06-27T16:51:31","date_gmt":"2020-06-27T11:21:31","guid":{"rendered":"https:\/\/kseebsolutions.guru\/?p=27317"},"modified":"2021-07-02T15:19:13","modified_gmt":"2021-07-02T09:49:13","slug":"2nd-puc-maths-previous-year-question-paper-march-2020","status":"publish","type":"post","link":"https:\/\/kseebsolutions.guru\/2nd-puc-maths-previous-year-question-paper-march-2020\/","title":{"rendered":"2nd PUC Maths Previous Year Question Paper March 2020"},"content":{"rendered":"<p>Students can Download 2nd PUC Maths Previous Year Question Paper March 2020, 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 2020<\/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 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 5 * 7.<br \/>\nSolution:<br \/>\n5 * 7 = LCM of 5 and 7 = 35<\/p>\n<p>Question 2.<br \/>\nWrite the range of the function y = sec<sup>-1<\/sup> x.<br \/>\nSolution:<br \/>\n[0, \u03c0] &#8211; {\\(\\frac{\\pi}{2}\\)}<\/p>\n<p>Question 3.<br \/>\nIf a matrix has 5 elements, what are the possible orders it can have?<br \/>\nSolution:<br \/>\n1 \u00d7 5 and 5 \u00d7 1<\/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 2020\" width=\"172\" height=\"16\" \/><\/p>\n<p>Question 4.<br \/>\nFind the values of x for which \\(\\left|\\begin{array}{cc}x &amp; 2 \\\\18 &amp; x\\end{array}\\right|=\\left|\\begin{array}{cc}6 &amp; 2 \\\\18 &amp; 6\\end{array}\\right|\\)<br \/>\nSolution:<br \/>\nx<sup>2<\/sup> &#8211; 36 = 36 &#8211; 36<br \/>\n\u21d2 x<sup>2<\/sup> = 36<br \/>\n\u21d2 x = \u00b16<\/p>\n<p>Question 5.<br \/>\nIf y = tan(\u221ax), find \\(\\frac{d y}{d x}\\).<br \/>\nSolution:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63122\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q5.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2020 Q5\" width=\"156\" height=\"60\" \/><\/p>\n<p>Question 6.<br \/>\nFind \u222b(2x<sup>2<\/sup> + e<sup>x<\/sup>) dx.<br \/>\nSolution:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63123\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q6.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2020 Q6\" width=\"356\" height=\"55\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q6.png 356w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q6-300x46.png 300w\" sizes=\"(max-width: 356px) 100vw, 356px\" \/><\/p>\n<p>Question 7.<br \/>\nDefine the Negative of a vector.<br \/>\nSolution:<br \/>\nA vector whose magnitude is the same as that of a given vector but direction is opposite to that of it is called negative of a vector.<\/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 2020\" width=\"172\" height=\"16\" \/><\/p>\n<p>Question 8.<br \/>\nIf a line makes angles 90\u00b0, 135\u00b0 and 45\u00b0 with the X, Y and Z-axes respectively, find its direction cosines.<br \/>\nSolution:<br \/>\nLet direction cosines of the line be l, m and n with the X, Y, and Z-axes respectively, and given that<br \/>\n\u03b1 = 90\u00b0, \u03b2 = 135\u00b0 and \u03b3 = 45\u00b0<br \/>\nThen, l = cos \u03b1 = cos 90\u00b0 = 0<br \/>\nm = cos \u03b2 = cos 135\u00b0 = \\(-\\frac{1}{\\sqrt{2}}\\)<br \/>\nand n = cos \u03b3 = cos 45\u00b0 = \\(\\frac{1}{\\sqrt{2}}\\)<br \/>\nTherefore, the direction cosines of the line are 0, \\(-\\frac{1}{\\sqrt{2}}\\) and \\(\\frac{1}{\\sqrt{2}}\\).<\/p>\n<p>Question 9.<br \/>\nDefine optimal solution in Linear Programming Problem.<br \/>\nSolution:<br \/>\nAny feasible solution of LPP which maximizes or minimizes the objective function is called an optimal solution.<\/p>\n<p>Question 10.<br \/>\nIf P(A) = \\(\\frac{3}{5}\\) and P(B) = \\(\\frac{1}{5}\\). find P(A \u2229 B) if A and B are independent events.<br \/>\nSolution:<br \/>\nP(A \u2229 B) = P(A) \u00d7 P(B) = \\(\\frac{3}{5} \\times \\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 and g : R \u2192 R are given by f(x) = cos x and g(x) = 3x<sup>2<\/sup>. Find gof and fog.<br \/>\nSolution:<br \/>\ngof(x) = g[f(x)] = g[cosx] = 3cos<sup>2<\/sup> x<br \/>\nfog(x) = f[g(x)] = f[3x<sup>2<\/sup>] = cos 3x<sup>2<\/sup><\/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 2020\" width=\"172\" height=\"16\" \/><\/p>\n<p>Question 12.<br \/>\nProve that cot<sup>-1<\/sup>(-x) = \u03c0 &#8211; cot<sup>-1<\/sup> x, \u2200 x \u2208 R.<br \/>\nSolution:<br \/>\nLet cot<sup>-1<\/sup>(-x) = \u03b1<br \/>\n\u21d2 -cot \u03b1 = -x<br \/>\n\u21d2 -cot \u03b1 = x<br \/>\n\u21d2 cot(\u03c0 &#8211; \u03b1) = x<br \/>\n\u21d2 \u03c0 &#8211; \u03b1 = cot<sup>-1<\/sup> x<br \/>\n\u21d2 \u03b1 = \u03c0 &#8211; cot<sup>-1<\/sup> x<br \/>\n\u21d2 cot<sup>-1<\/sup> (-x) = \u03c0 &#8211; cot<sup>-1<\/sup> x<\/p>\n<p>Question 13.<br \/>\nFind the value of sin<sup>-1<\/sup>(sin \\(\\frac{3 \\pi}{5}\\))<br \/>\nSolution:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63124\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q13.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2020 Q13\" width=\"520\" height=\"63\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q13.png 520w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q13-300x36.png 300w\" sizes=\"(max-width: 520px) 100vw, 520px\" \/><\/p>\n<p>Question 14.<br \/>\nFind the area of the triangle whose vertices are (-2, -3), (3, 2) and (-1, -8) using the determinant method.<br \/>\nSolution:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63125\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q14.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2020 Q14\" width=\"379\" height=\"77\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q14.png 379w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q14-300x61.png 300w\" sizes=\"(max-width: 379px) 100vw, 379px\" \/><\/p>\n<p>Question 15.<br \/>\nFind \\(\\frac{d y}{d x}\\), if sin<sup>2<\/sup> x + cos<sup>2<\/sup> y = 1.<br \/>\nSolution:<br \/>\n2 sin x cos x + 2 cos y(-sin y) \\(\\frac{d y}{d x}\\) = 0<br \/>\n\u21d2 \\(\\frac{d y}{d x}=\\frac{2 \\sin x \\cos x}{2 \\sin y \\cos y}=\\frac{\\sin 2 x}{\\sin 2 y}\\)<\/p>\n<p>Question 16.<br \/>\nIf y = x<sup>x<\/sup>, find \\(\\frac{d y}{d x}\\).<br \/>\nSolution:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63126\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q16.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2020 Q16\" width=\"467\" height=\"163\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q16.png 467w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q16-300x105.png 300w\" sizes=\"(max-width: 467px) 100vw, 467px\" \/><\/p>\n<p>Question 17.<br \/>\nFind the interval in which the function f given by f(x) = x<sup>2<\/sup> &#8211; 4x + 6 is strictly decreasing.<br \/>\nSolution:<br \/>\nWe have f(x) = x<sup>2<\/sup> &#8211; 4x + 6<br \/>\n\u21d2 f'(x) = 2x &#8211; 4<br \/>\nTherefore, f'(x) = 0 gives x = 2.<br \/>\nNow the point x = 2 divides the real line into two disjoint intervals namely (-\u221e, 2) and (2, \u221e).<br \/>\nIn the interval (-\u221e, 2), f'(x) = 2x &#8211; 4 &lt; 0. Therefore, f is strictly decreasing in this interval. Also, in the interval (2, \u221e) f'(x) &gt; 0 and so the function f is strictly increasing in this interval.<\/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 2020\" width=\"172\" height=\"16\" \/><\/p>\n<p>Question 18.<br \/>\nFind \u222bcot x log(sin x) dx.<br \/>\nSolution:<br \/>\nPut log sin x = t<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63127\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q18.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2020 Q18\" width=\"285\" height=\"113\" \/><\/p>\n<p>Question 19.<br \/>\nFind \u222bx sec<sup>2<\/sup> x dx.<br \/>\nSolution:<br \/>\nx tan x &#8211; log |sec x| + c<\/p>\n<p>Question 20.<br \/>\nFind the order and degree (if defined) of the differential equation:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63128\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q20.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2020 Q20\" width=\"283\" height=\"61\" \/><br \/>\nSolution:<br \/>\nOrder = 2, Degree is not defined.<\/p>\n<p>Question 21.<br \/>\nFind the projection of the vector \\(\\vec{a}=\\hat{i}+3 \\hat{j}+7 \\hat{k}\\) on the vector \\(\\vec{b}=7 \\hat{i}-\\hat{j}+8 \\hat{k}\\).<br \/>\nSolution:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63129\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q21.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2020 Q21\" width=\"538\" height=\"267\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q21.png 538w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q21-300x149.png 300w\" sizes=\"(max-width: 538px) 100vw, 538px\" \/><\/p>\n<p>Question 22.<br \/>\nFind the area of the parallelogram whose adjacent sides are determined by the vectors \\(\\vec{a}=\\hat{i}-\\hat{j}+3 \\hat{k}\\) and \\(\\vec{b}=2 \\hat{i}-7 \\hat{j}+\\hat{k}\\)<br \/>\nSolution:<br \/>\nArea = \\(|\\vec{a} \\times \\vec{b}|\\) = 15\u221a2 sq.units<\/p>\n<p>Question 23.<br \/>\nFind the equation of the plane with intercepts 2, 3 and 4 on the X, Y and Z-axes respectively.<br \/>\nSolution:<br \/>\n\\(\\frac{x}{2}+\\frac{y}{3}+\\frac{z}{4}=1\\)<\/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 2020\" width=\"172\" height=\"16\" \/><\/p>\n<p>Question 24.<br \/>\nA random variable X has the following probability distribution:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63130\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q24.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2020 Q24\" width=\"370\" height=\"52\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q24.png 370w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q24-300x42.png 300w\" sizes=\"(max-width: 370px) 100vw, 370px\" \/><br \/>\nFind the value of K.<br \/>\nSolution:<br \/>\n\u03a3P(x) = 1<br \/>\n\u21d2 0.1 + K + 2K + 2K + K = 1<br \/>\n\u21d2 0.1 + 6K = 1<br \/>\n\u21d2 6K = 1 &#8211; 0.1 = 0.9<br \/>\n\u21d2 K = 0.9\/6 = 0.15<\/p>\n<p style=\"text-align: center;\"><span style=\"color: #0000ff;\">Part &#8211; C<\/span><\/p>\n<p><span style=\"color: #0000ff;\">Answer any TEN questions: (10 \u00d7 3 = 30)<\/span><\/p>\n<p>Question 25.<br \/>\nShow that the relation R defined in the set A of all triangles as R = {(T<sub>1<\/sub>, T<sub>2<\/sub>); T<sub>1<\/sub> is similar to T<sub>2<\/sub>} is an equivalence relation.<br \/>\nSolution:<br \/>\nT<sub>1<\/sub> is similar to T<sub>2<\/sub> \u21d2 R is reflexive.<br \/>\nLet (T<sub>1<\/sub>, T<sub>2<\/sub>) \u2208 R \u21d2 T<sub>1<\/sub> is 111 to T<sub>2<\/sub><br \/>\n\u21d2 T<sub>2<\/sub> 111 T<sub>1<\/sub><br \/>\n\u21d2 (T<sub>2<\/sub>, T<sub>1<\/sub>) \u2208 R<br \/>\n\u2234 R is symmetric<br \/>\nLet (T<sub>1<\/sub>, T<sub>2<\/sub>) \u2208 R; (T<sub>2<\/sub>, T<sub>3<\/sub>) \u2208 R<br \/>\n\u21d2 T<sub>1<\/sub> is 111 to T<sub>2<\/sub> and T<sub>2<\/sub> is 111 to T<sub>3<\/sub><br \/>\n\u21d2 T<sub>1<\/sub> is 111 to T<sub>3<\/sub><br \/>\n\u21d2 (T<sub>1<\/sub>, T<sub>2<\/sub>) \u2208 R<br \/>\n\u2234 R is transitive<br \/>\n\u2234 R is an equivalence relation.<\/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 2020\" width=\"172\" height=\"16\" \/><\/p>\n<p>Question 26.<br \/>\nProve that \\(2 \\tan ^{-1}\\left(\\frac{1}{2}\\right)+\\tan ^{-1}\\left(\\frac{1}{7}\\right)=\\tan ^{-1}\\left(\\frac{31}{17}\\right)\\)<br \/>\nSolution:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63154\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q26.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2020 Q26\" width=\"418\" height=\"237\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q26.png 418w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q26-300x170.png 300w\" sizes=\"(max-width: 418px) 100vw, 418px\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63131\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q26.1.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2020 Q26.1\" width=\"502\" height=\"102\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q26.1.png 502w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q26.1-300x61.png 300w\" sizes=\"(max-width: 502px) 100vw, 502px\" \/><\/p>\n<p>Question 27.<br \/>\nIf F(x) = \\(\\left[\\begin{array}{ccc}\\cos x &amp; -\\sin x &amp; 0 \\\\\\sin x &amp; \\cos x &amp; 0 \\\\0 &amp; 0 &amp; 1\\end{array}\\right]\\) then show that F(x) F(y) = F(x + y).<br \/>\nSolution:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63155\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q27.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2020 Q27\" width=\"535\" height=\"466\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q27.png 535w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q27-300x261.png 300w\" sizes=\"(max-width: 535px) 100vw, 535px\" \/><\/p>\n<p>Question 28.<br \/>\nIf x = 2at<sup>2<\/sup>, y = at<sup>4<\/sup> then find \\(\\frac{d y}{d x}\\)<br \/>\nSolution:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63156\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q28.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2020 Q28\" width=\"312\" height=\"50\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q28.png 312w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q28-300x48.png 300w\" sizes=\"(max-width: 312px) 100vw, 312px\" \/><\/p>\n<p>Question 29.<br \/>\nVerify Mean Value Theorem for the function f(x) = x<sup>2<\/sup> &#8211; 4x &#8211; 3, x \u2208 [1, 4].<br \/>\nSolution:<br \/>\nf(x) is a polynomial in x.<br \/>\nIt is continuous in {1, 4} and differentiable in {1, 4) and f'(x) = 2x &#8211; 4.<br \/>\nThere exists at least one value c \u2208 (1, 4) such that f'(c) = \\(\\frac{f(b)-f(a)}{b-a}\\)<br \/>\na = 1, f(a) = f(1) = 12 &#8211; 4(1) &#8211; 3 = 1 &#8211; 4 &#8211; 3 = 1 &#8211; 7 = -6<br \/>\nb = 4, f(b) = f(4) = 42 &#8211; 4(4) &#8211; 3 = 16 &#8211; 16 &#8211; 3 = -3<br \/>\nf'(c) = 2c &#8211; 4<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63157\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q29.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2020 Q29\" width=\"288\" height=\"209\" \/><br \/>\nMean Value theorem is verified.<\/p>\n<p>Question 30.<br \/>\nUse differential to approximate \u221a36.6.<br \/>\nSolution:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63158\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q30.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2020 Q30\" width=\"358\" height=\"182\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q30.png 358w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q30-300x153.png 300w\" sizes=\"(max-width: 358px) 100vw, 358px\" \/><\/p>\n<p>Question 31.<br \/>\nFind \\(\\int \\frac{(x-3)^{e^{x}}}{(x-1)^{3}} d x\\)<br \/>\nSolution:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63160\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q31.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2020 Q31\" width=\"458\" height=\"147\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q31.png 458w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q31-300x96.png 300w\" sizes=\"(max-width: 458px) 100vw, 458px\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63159\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q31.1.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2020 Q31.1\" width=\"507\" height=\"63\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q31.1.png 507w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q31.1-300x37.png 300w\" sizes=\"(max-width: 507px) 100vw, 507px\" \/><\/p>\n<p>Question 32.<br \/>\nEvaluate: \\(\\int_{0}^{\\pi \/ 2} \\cos ^{2} x d x\\)<br \/>\nSolution:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63161\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q32.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2020 Q32\" width=\"510\" height=\"65\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q32.png 510w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q32-300x38.png 300w\" sizes=\"(max-width: 510px) 100vw, 510px\" \/><\/p>\n<p>Question 33.<br \/>\nFind the area of the region bounded by x<sup>2<\/sup> = 4y, y = 2, y = 4 and the y-axes in the first quadrant.<br \/>\nSolution:<br \/>\nThe given curve x<sup>2<\/sup> = 4y is a parabola which is symmetrical about Y-axis (\u2235 it contains even power of x) only and passes through the origin.<br \/>\nThe area of the region bounded by the curve x<sup>2<\/sup> = 4y, y = 2 and y = 4 and the Y-axis are shown in the figure.<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63163\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q33.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2020 Q33\" width=\"238\" height=\"206\" \/><br \/>\nRequired area (shaded region) = \\(\\int_{y=a}^{y=b}|x| d y\\)<br \/>\n(Here, |x| = \u221a4y and a = 2, b = 4)<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63162\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q33.1.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2020 Q33.1\" width=\"675\" height=\"193\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q33.1.png 675w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q33.1-300x86.png 300w\" sizes=\"(max-width: 675px) 100vw, 675px\" \/><\/p>\n<p>Question 34.<br \/>\nFind the equation of a curve passing through the point (-2, 3), given that the slope of the tangent to the curve at any point (x, y) is \\(\\frac{2 x}{y^{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 2020\" width=\"172\" height=\"16\" \/><\/p>\n<p>Question 35.<br \/>\nFind a unit vector perpendicular to each of the vector \\((\\vec{a}+\\vec{b})\\) and \\((\\vec{a}+\\vec{b})\\) where \\(\\vec{a}=3 \\hat{i}+2 \\hat{j}+2 \\hat{k}\\) and \\(\\vec{b}=\\hat{i}+2 \\hat{j}-2 \\hat{k}\\).<br \/>\nSolution:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63165\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q35.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2020 Q35\" width=\"514\" height=\"374\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q35.png 514w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q35-300x218.png 300w\" sizes=\"(max-width: 514px) 100vw, 514px\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63164\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q35.1.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2020 Q35.1\" width=\"417\" height=\"184\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q35.1.png 417w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q35.1-300x132.png 300w\" sizes=\"(max-width: 417px) 100vw, 417px\" \/><\/p>\n<p>Question 36.<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-63166\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q36.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2020 Q36\" width=\"259\" height=\"235\" \/><\/p>\n<p>Question 37.<br \/>\nFind the equation of the plane through the intersection of the planes 3x &#8211; y + 2z &#8211; 4 = 0 and x + y + z &#8211; 2 = 0 and the point (2, 2, 1).<\/p>\n<p>Question 38.<br \/>\nA man is known to speak truth 3 out of 4 times. He throws a dice and reports that it is a six. Find the probability that it is actually a six.<br \/>\nSolution:<br \/>\nP(S<sub>1<\/sub>) = Probability that six occurs = \\(\\frac{1}{6}\\)<br \/>\nP(S<sub>2<\/sub>) = Probability that six does not occur = \\(\\frac{5}{6}\\)<br \/>\nP(E | S<sub>1<\/sub>) = Probability that man reports that six occurs when six has actually on the die = Probability that the truth occurs = \\(\\frac{3}{4}\\)<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63167\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q38.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2020 Q38\" width=\"346\" height=\"178\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q38.png 346w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q38-300x154.png 300w\" sizes=\"(max-width: 346px) 100vw, 346px\" \/><\/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 \/>\nShow that the function f : R \u2192 R given by f(x) = 4x + 3 is invertible. Find the inverse of f.<br \/>\nSolution:<br \/>\nLet f(x<sub>1<\/sub>) = f(x<sub>2<\/sub>) = 4x<sub>1<\/sub> + 3 = 4x<sub>2<\/sub> + 3<br \/>\n\u21d2 4x<sub>1<\/sub> = 4x<sub>2<\/sub><br \/>\n\u21d2 x<sub>1<\/sub> = x<sub>2<\/sub><br \/>\n\u2234 f is one-one.<br \/>\nLet y \u2208 Y \u2203 x \u2208 N such that y = 4x + 3<br \/>\n\u2234 f is onto.<br \/>\n\u2234 f is bijective.<br \/>\nConsider an arbitrary element y of Y. By the definition of Y, y = 4x + 3, for some x in the domain N.<br \/>\nx = \\(\\frac{y-3}{4}\\)<br \/>\nDefine a function g : Y \u2192 N by g(y) = \\(\\frac{y-3}{4}\\)<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63168\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q39.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2020 Q39\" width=\"456\" height=\"120\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q39.png 456w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q39-300x79.png 300w\" sizes=\"(max-width: 456px) 100vw, 456px\" \/><br \/>\n\u21d2 gof = I<sub>N<\/sub> and fog = I<sub>Y<\/sub><br \/>\n\u21d2 f is invertible and g is the inverse of f.<br \/>\n\u2234 f<sup>-1<\/sup> = g.<\/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 2020\" width=\"172\" height=\"16\" \/><\/p>\n<p>Question 40.<br \/>\nIf A = \\(\\left[\\begin{array}{ccc}1 &amp; 2 &amp; -3 \\\\5 &amp; 0 &amp; 2 \\\\1 &amp; -1 &amp; 1\\end{array}\\right]\\), B =\\(\\left[\\begin{array}{ccc}3 &amp; -1 &amp; 2 \\\\4 &amp; 2 &amp; 5 \\\\2 &amp; 0 &amp; 0\\end{array}\\right]\\) and C =\\(\\left[\\begin{array}{ccc}4 &amp; 1 &amp; 2 \\\\0 &amp; 3 &amp; 2 \\\\1 &amp; -2 &amp; 3\\end{array}\\right]\\) then compute (A + B) and (B &#8211; C). Also verify that A + (B &#8211; C) = (A + B) &#8211; C. Also verify (A + B) C = AC + BC.<br \/>\nSolution:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63170\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q40.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2020 Q40\" width=\"398\" height=\"107\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q40.png 398w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q40-300x81.png 300w\" sizes=\"(max-width: 398px) 100vw, 398px\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63169\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q40.1.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2020 Q40.1\" width=\"516\" height=\"213\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q40.1.png 516w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q40.1-300x124.png 300w\" sizes=\"(max-width: 516px) 100vw, 516px\" \/><\/p>\n<p>Question 41.<br \/>\nSolve the following system of linear equations by matrix method:<br \/>\n2x + 3y + 3z = 5<br \/>\nx &#8211; 2y + z = -4<br \/>\n3x &#8211; y &#8211; 2z = 3<br \/>\nSolution:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63172\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q41.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2020 Q41\" width=\"302\" height=\"297\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q41.png 302w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q41-300x295.png 300w\" sizes=\"(max-width: 302px) 100vw, 302px\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63171\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q41.1.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2020 Q41.1\" width=\"451\" height=\"185\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q41.1.png 451w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q41.1-300x123.png 300w\" sizes=\"(max-width: 451px) 100vw, 451px\" \/><br \/>\nx = 1, y = 2, z = -1<\/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>\u00a0+ 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-63173\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q42.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2020 Q42\" width=\"675\" height=\"310\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q42.png 675w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q42-300x138.png 300w\" sizes=\"(max-width: 675px) 100vw, 675px\" \/><\/p>\n<p>Question 43.<br \/>\nSand is pouring from a pipe at the rate of 12 cm<sup>3<\/sup>\/sec. The falling sand forms a cone on the ground in such a way that the height of the cone is always one-sixth of the radius of the base. How fast is the height of the sand cone increasing when the height is 4 cm?<br \/>\nSolution:<br \/>\nLet r be the radius, h be the height and V be the volume of sand c\u00f3ne at any time t.<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63174\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q43.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2020 Q43\" width=\"593\" height=\"349\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q43.png 593w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q43-300x177.png 300w\" sizes=\"(max-width: 593px) 100vw, 593px\" \/><br \/>\nHence, when the height of the sand cone is 4 cm, its height is increasing at the rate of \\(\\frac{1}{48 \\pi}\\) cm\/s<\/p>\n<p>Question 44.<br \/>\nFind the integral of \\(\\frac{1}{x^{2}+a^{2}}\\) w.r.t x and hence evaluate \\(\\int \\frac{1}{x^{2}+2 x+2} d x\\)<br \/>\nSolution:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63176\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q44.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2020 Q44\" width=\"495\" height=\"110\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q44.png 495w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q44-300x67.png 300w\" sizes=\"(max-width: 495px) 100vw, 495px\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63175\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q44.1.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2020 Q44.1\" width=\"535\" height=\"129\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q44.1.png 535w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q44.1-300x72.png 300w\" sizes=\"(max-width: 535px) 100vw, 535px\" \/><\/p>\n<p>Question 45.<br \/>\nUsing the method of integration, find the area of the smaller region bounded by the ellipse \\(\\frac{x^{2}}{9}+\\frac{y^{2}}{4}=1\\) and the line \\(\\frac{x}{3}+\\frac{y}{2}=1\\)<br \/>\nSolution:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63177\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q45.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2020 Q45\" width=\"364\" height=\"333\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q45.png 364w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q45-300x274.png 300w\" sizes=\"(max-width: 364px) 100vw, 364px\" \/><\/p>\n<p>Question 46.<br \/>\nFind the general solution of the differential equation x\\(\\frac{d y}{d x}\\) + 2y = x<sup>2<\/sup> log x.<br \/>\nSolution:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63179\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q46.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2020 Q46\" width=\"457\" height=\"306\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q46.png 457w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q46-300x201.png 300w\" sizes=\"(max-width: 457px) 100vw, 457px\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63178\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q46.1.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2020 Q46.1\" width=\"168\" height=\"191\" \/><\/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 2020\" width=\"172\" height=\"16\" \/><\/p>\n<p>Question 47.<br \/>\nDerive the equation of a line in space passing through a given point and parallel to a given Vector and Cartesian form.<br \/>\nSolution:<br \/>\nVector Form<br \/>\nLet \\(\\vec{a}\\) be the position vector of the given point A with respect to the origin O of the rectangular<br \/>\ncoordinate system. Let l be the line which passes through point A and is parallel to a given vector \\(\\vec{b}\\). Let \\(\\vec{r}\\) be the position vector of an arbitrary point P on the line Fig.<br \/>\nThen \\(\\vec{AP}\\) is parallel to the vector \\(\\vec{b}\\), i.e., \\(\\overrightarrow{\\mathrm{AP}}=\\lambda \\vec{b}\\), where \u03bb is some real number.<br \/>\nBut \\(\\overrightarrow{\\mathrm{AP}}=\\overrightarrow{\\mathrm{OP}}-\\overrightarrow{\\mathrm{AO}}\\)<br \/>\ni.e., \\(\\lambda \\vec{b}=\\vec{r}-\\vec{a}\\)<br \/>\nConversely, for each value of the parameter \u03bb, this equation gives the position vector of a point P on the line. Hence, the vector equation of the line is given by<br \/>\n\\(\\vec{r}=\\vec{a}+\\lambda \\vec{b}\\) &#8230;&#8230;. (1)<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63180\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q47.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2020 Q47\" width=\"331\" height=\"256\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q47.png 331w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q47-300x232.png 300w\" sizes=\"(max-width: 331px) 100vw, 331px\" \/><br \/>\nCartesian Form<br \/>\nLet the coordinates of the given point A be (x<sub>1<\/sub>, y<sub>1<\/sub>, z<sub>1<\/sub>) and the direction ratios of the line be a, b, c.<br \/>\nConsider the coordinates of any point P be (x, y, z). Then \\(\\vec{r}=x \\hat{i}+y \\hat{j}+z \\hat{k} ; \\vec{a}=x_{1} \\hat{i}+y_{1} \\hat{j}+z_{1} \\hat{k}\\) and \\(\\vec{b}=a \\hat{i}+b \\hat{j}+c \\hat{k}\\)<br \/>\nSubstituting these values in (1) and equating the coefficients of \\(\\hat{i}, \\hat{j} \\text { and } \\hat{k}\\), we get<br \/>\nx = x<sub>1<\/sub> + \u03bba ; y = y<sub>1<\/sub> + \u03bbb ; z = z<sub>1<\/sub> + \u03bbc &#8230;&#8230;.. (2)<br \/>\nThese are parametric equations of the line. Eliminating the parameter \u03bb from (2), we get<br \/>\n\\(\\frac{x-x_{1}}{a}=\\frac{y-y_{1}}{b}=\\frac{z-z_{1}}{c}\\)<br \/>\nThis is tle Cartesian equation of the line.<\/p>\n<p>Question 48.<br \/>\nA person buys a lottery ticket in 50 lotteries, in each of which his chance of winning a prize is \\(\\frac{1}{100}\\). What is the probability that he will win a prize (a) exactly once (b) at least once?<br \/>\nSolution:<br \/>\nLet X denotes the number of wins<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63183\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q48.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2020 Q48\" width=\"634\" height=\"252\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q48.png 634w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q48-300x119.png 300w\" sizes=\"(max-width: 634px) 100vw, 634px\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63181\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q48.1.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2020 Q48.1\" width=\"377\" height=\"245\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q48.1.png 377w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q48.1-300x195.png 300w\" sizes=\"(max-width: 377px) 100vw, 377px\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63182\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q48.2.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2020 Q48.2\" width=\"315\" height=\"68\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q48.2.png 315w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q48.2-300x65.png 300w\" sizes=\"(max-width: 315px) 100vw, 315px\" \/><\/p>\n<p style=\"text-align: center;\"><span style=\"color: #0000ff;\">Part &#8211; E<\/span><\/p>\n<p><span style=\"color: #0000ff;\">Answer any 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-63188\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q49a.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2020 Q49(a)\" width=\"554\" height=\"102\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q49a.png 554w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q49a-300x55.png 300w\" sizes=\"(max-width: 554px) 100vw, 554px\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63184\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q49a.1.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2020 Q49(a).1\" width=\"302\" height=\"64\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q49a.1.png 302w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q49a.1-300x64.png 300w\" sizes=\"(max-width: 302px) 100vw, 302px\" \/><br \/>\nSolution:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63185\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q49a.2.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2020 Q49(a).2\" width=\"408\" height=\"276\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q49a.2.png 408w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q49a.2-300x203.png 300w\" sizes=\"(max-width: 408px) 100vw, 408px\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63186\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q49a.3.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2020 Q49(a).3\" width=\"578\" height=\"260\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q49a.3.png 578w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q49a.3-300x135.png 300w\" sizes=\"(max-width: 578px) 100vw, 578px\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63187\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q49a.4.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2020 Q49(a).4\" width=\"569\" height=\"80\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q49a.4.png 569w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q49a.4-300x42.png 300w\" sizes=\"(max-width: 569px) 100vw, 569px\" \/><\/p>\n<p>Question 49(b).<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63191\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q49b.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2020 Q49(b)\" width=\"436\" height=\"86\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q49b.png 436w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q49b-300x59.png 300w\" sizes=\"(max-width: 436px) 100vw, 436px\" \/><br \/>\nSolution:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63189\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q49b.1.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2020 Q49(b).1\" width=\"524\" height=\"240\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q49b.1.png 524w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q49b.1-300x137.png 300w\" sizes=\"(max-width: 524px) 100vw, 524px\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63190\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q49b.2.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2020 Q49(b).2\" width=\"573\" height=\"167\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q49b.2.png 573w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q49b.2-300x87.png 300w\" sizes=\"(max-width: 573px) 100vw, 573px\" \/><\/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 2020\" width=\"172\" height=\"16\" \/><\/p>\n<p>Question 50(a).<br \/>\nMaximise z = 4x + y Subject to constraints: x + y \u2264 50, 3x = y \u2264 90, x \u2265 0, y \u2265 0 by graphical method.<br \/>\nSolution:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63193\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q50a.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2020 Q50(a)\" width=\"316\" height=\"274\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q50a.png 316w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q50a-300x260.png 300w\" sizes=\"(max-width: 316px) 100vw, 316px\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63192\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q50a.1.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2020 Q50(a).1\" width=\"452\" height=\"350\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q50a.1.png 452w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q50a.1-300x232.png 300w\" sizes=\"(max-width: 452px) 100vw, 452px\" \/><br \/>\nOptimal Solution:<br \/>\nZ is maximum at D(30, 0) &amp; Z<sub>max<\/sub> = 120<\/p>\n<p>Question 50(b).<br \/>\nFind the value of K,<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63195\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q50b.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2020 Q50(b)\" width=\"415\" height=\"65\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q50b.png 415w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q50b-300x47.png 300w\" sizes=\"(max-width: 415px) 100vw, 415px\" \/><br \/>\nSolution:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63194\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q50b.1.png\" alt=\"2nd PUC Maths Previous Year Question Paper March 2020 Q50(b).1\" width=\"650\" height=\"286\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q50b.1.png 650w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-March-2020-Q50b.1-300x132.png 300w\" sizes=\"(max-width: 650px) 100vw, 650px\" \/><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Students can Download 2nd PUC Maths Previous Year Question Paper March 2020, 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 2020 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\/27317"}],"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=27317"}],"version-history":[{"count":0,"href":"https:\/\/kseebsolutions.guru\/wp-json\/wp\/v2\/posts\/27317\/revisions"}],"wp:attachment":[{"href":"https:\/\/kseebsolutions.guru\/wp-json\/wp\/v2\/media?parent=27317"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/kseebsolutions.guru\/wp-json\/wp\/v2\/categories?post=27317"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/kseebsolutions.guru\/wp-json\/wp\/v2\/tags?post=27317"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}