{"id":27355,"date":"2020-06-29T16:13:36","date_gmt":"2020-06-29T10:43:36","guid":{"rendered":"https:\/\/kseebsolutions.guru\/?p=27355"},"modified":"2021-07-02T15:07:10","modified_gmt":"2021-07-02T09:37:10","slug":"2nd-puc-maths-previous-year-question-paper-june-2018","status":"publish","type":"post","link":"https:\/\/kseebsolutions.guru\/2nd-puc-maths-previous-year-question-paper-june-2018\/","title":{"rendered":"2nd PUC Maths Previous Year Question Paper June 2018"},"content":{"rendered":"<p>Students can Download 2nd PUC Maths Previous Year Question Paper June 2018, 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 June 2018<\/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 \/>\nThe relation R on set A = {1, 2, 3} is defined as R {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3)} is not transitive. Why?<br \/>\nSolution:<br \/>\n(1, 2) \u2208 R, (2, 3) \u2208 R. But (1, 3) \u2209 R<\/p>\n<p>Question 2.<br \/>\nWrite the range of y = cos<sup>-1<\/sup> x<br \/>\nSolution:<br \/>\n[0, \u03c0]<\/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, 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 June 2018\" 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> = 36 \u21d2 x = \u00b16<\/p>\n<p>Question 5.<br \/>\nFind \\(\\frac{d y}{d x}\\), if y = sin(ax + b)<br \/>\nSolution:<br \/>\n\\(\\frac{d y}{d x}\\) = a cos(ax + b)<\/p>\n<p>Question 6.<br \/>\nEvaluate: \u222bsec x (sec x + tan x) dx<br \/>\nSolution:<br \/>\nI = \u222bsec<sup>2<\/sup> x dx + \u222b sec x tan x dx = tan x + sec x + c<\/p>\n<p>Question 7.<br \/>\nDefine the negative of a vector.<br \/>\nSolution:<br \/>\nNegative of the vector is obtained by changing the direction of the given vector into the opposite direction or by multiplying the given vector by -1.<br \/>\nSay \\(\\vec{a}\\) be the given vector so negative of \\(\\vec{a}\\) is &#8211;\\(\\vec{a}\\)<br \/>\n\\(\\vec{a}\\) and &#8211;\\(\\vec{a}\\) will have the same magnitude but opposite direction.<\/p>\n<p>Question 8.<br \/>\nThe Cartesian equation of a line is \\(\\frac{x-5}{3}=\\frac{y-4}{7}=\\frac{z-6}{2}\\). Write its vector form.<br \/>\nSolution:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63462\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q8.png\" alt=\"2nd PUC Maths Previous Year Question Paper June 2018 Q8\" width=\"303\" height=\"91\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q8.png 303w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q8-300x90.png 300w\" sizes=\"(max-width: 303px) 100vw, 303px\" \/><\/p>\n<p>Question 9.<br \/>\nDefine optimal solution in a 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><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 June 2018\" width=\"172\" height=\"16\" \/><\/p>\n<p>Question 10.<br \/>\nFind P(A\/B), if P(B) = 0.5 and P(A\u2229B) = 0.32.<br \/>\nSolution:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63463\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q10.png\" alt=\"2nd PUC Maths Previous Year Question Paper June 2018 Q10\" width=\"259\" height=\"58\" \/><\/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 \/>\nDefine binary operation on a set. Verify whether the operation * is defined on Q set of rational numbers by a * b = ab + 1, \u2200 a, b \u2208 Q is binary or not.<br \/>\nSolution:<br \/>\nA binary operation on a set A is a function<br \/>\n* : A \u00d7 A \u2192 A<br \/>\na * b = ab + 1 \u2208 Q<br \/>\n\u2234 * is a Binary Operation on Q.<\/p>\n<p>Question 12.<br \/>\nWrite \\(\\tan ^{-1}(\\sqrt{\\frac{1-\\cos x}{1+\\cos x}})\\), 0 &lt; x &lt; \u03c0 in the simplest form.<br \/>\nSolution:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63464\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q12.png\" alt=\"2nd PUC Maths Previous Year Question Paper June 2018 Q12\" width=\"299\" height=\"198\" \/><\/p>\n<p>Question 13.<br \/>\nFind the value of \\(\\cos ^{-1}\\left(\\cos \\frac{13 \\pi}{6}\\right)\\)<br \/>\nSolution:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63465\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q13.png\" alt=\"2nd PUC Maths Previous Year Question Paper June 2018 Q13\" width=\"315\" height=\"59\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q13.png 315w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q13-300x56.png 300w\" sizes=\"(max-width: 315px) 100vw, 315px\" \/><\/p>\n<p>Question 14.<br \/>\nIf the area of the triangle with vertices (2, -6), (5, 4) and (K, 4) is 35 sq. units, then find the values of K, using determinants.<br \/>\nSolution:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63466\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q14.png\" alt=\"2nd PUC Maths Previous Year Question Paper June 2018 Q14\" width=\"464\" height=\"260\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q14.png 464w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q14-300x168.png 300w\" sizes=\"(max-width: 464px) 100vw, 464px\" \/><\/p>\n<p>Question 15.<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63468\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q15.png\" alt=\"2nd PUC Maths Previous Year Question Paper June 2018 Q15\" width=\"346\" height=\"60\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q15.png 346w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q15-300x52.png 300w\" sizes=\"(max-width: 346px) 100vw, 346px\" \/><br \/>\nSolution:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63467\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q15.1.png\" alt=\"2nd PUC Maths Previous Year Question Paper June 2018 Q15.1\" width=\"676\" height=\"263\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q15.1.png 676w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q15.1-300x117.png 300w\" sizes=\"(max-width: 676px) 100vw, 676px\" \/><\/p>\n<p>Question 16.<br \/>\nDifferentiate (sin x)<sup>cos x<\/sup> with respect to x.<br \/>\nSolution:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63469\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q16.png\" alt=\"2nd PUC Maths Previous Year Question Paper June 2018 Q16\" width=\"333\" height=\"253\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q16.png 333w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q16-300x228.png 300w\" sizes=\"(max-width: 333px) 100vw, 333px\" \/><\/p>\n<p>Question 17.<br \/>\nIf the radius of a sphere is measured as 7 m with an error of 0.02 m, then find the approximate error in calculating its volume.<br \/>\nSolution:<br \/>\nLet r be the radius of the sphere and \u2206r be the error in measuring radius.<br \/>\nThen, r = 7 m and \u2206r = 0.02 m<br \/>\nNow, volume of a sphere is given by V = \\(\\frac{4}{3} \\pi r^{3}\\)<br \/>\nOn differentiate w.r.t r, we get \\(\\frac{d V}{d r}=\\left(\\frac{4}{3} \\pi\\right)\\left(3 r^{2}\\right)=4 \\pi r^{2}\\)<br \/>\n\u0394V = (\\(\\frac{d V}{d r}\\)) \u0394r<br \/>\n= (4\u03c0r<sup>2<\/sup>) \u0394r<br \/>\n= 4\u03c0 \u00d7 7<sup>2<\/sup> \u00d7 0.002<br \/>\n= 3.92\u03c0 m<sup>3<\/sup><br \/>\nHence, the approximate error in calculating the volume is 3.92\u03c0 m<sup>3<\/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 June 2018\" width=\"172\" height=\"16\" \/><\/p>\n<p>Question 18.<br \/>\nEvaluate: \\(\\int \\cos 6 x \\sqrt{1+\\sin 6 x} d x\\)<br \/>\nSolution:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63470\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q18.png\" alt=\"2nd PUC Maths Previous Year Question Paper June 2018 Q18\" width=\"323\" height=\"153\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q18.png 323w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q18-300x142.png 300w\" sizes=\"(max-width: 323px) 100vw, 323px\" \/><\/p>\n<p>Question 19.<br \/>\nIntegrate \\(\\frac{x e^{x}}{(1+x)^{2}}\\) with respect to x.<br \/>\nSolution:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63472\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q19.png\" alt=\"2nd PUC Maths Previous Year Question Paper June 2018 Q19\" width=\"211\" height=\"152\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63471\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q19.1.png\" alt=\"2nd PUC Maths Previous Year Question Paper June 2018 Q19.1\" width=\"187\" height=\"121\" \/><\/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-63473\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q20.png\" alt=\"2nd PUC Maths Previous Year Question Paper June 2018 Q20\" width=\"203\" height=\"59\" \/><br \/>\nSolution:<br \/>\nOrder = 3, Degree = 1<\/p>\n<p>Question 21.<br \/>\nFind the projection of the vector \\(\\vec{a}=\\hat{i}-\\hat{j}+3 \\hat{k}\\) on the vector \\(\\vec{b}=2 \\hat{i}+3 \\hat{j}+2 \\hat{k}\\).<br \/>\nSolution:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63474\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q21.png\" alt=\"2nd PUC Maths Previous Year Question Paper June 2018 Q21\" width=\"369\" height=\"71\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q21.png 369w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q21-300x58.png 300w\" sizes=\"(max-width: 369px) 100vw, 369px\" \/><\/p>\n<p>Question 22.<br \/>\nFind the area of the parallelogram whose adjacent sides are given 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 \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63475\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q22.png\" alt=\"2nd PUC Maths Previous Year Question Paper June 2018 Q22\" width=\"528\" height=\"150\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q22.png 528w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q22-300x85.png 300w\" sizes=\"(max-width: 528px) 100vw, 528px\" \/><\/p>\n<p>Question 23.<br \/>\nFind the angle between the line \\(\\frac{x+1}{2}=\\frac{y}{3}=\\frac{z-3}{6}\\) and the plane 10x + 2y &#8211; 11z = 3<br \/>\nSolution:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63476\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q23.png\" alt=\"2nd PUC Maths Previous Year Question Paper June 2018 Q23\" width=\"556\" height=\"358\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q23.png 556w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q23-300x193.png 300w\" sizes=\"(max-width: 556px) 100vw, 556px\" \/><\/p>\n<p>Question 24.<br \/>\nThe random variable X has a probability distribution P(X) of the following form, where K is some number.<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63477\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q24.png\" alt=\"2nd PUC Maths Previous Year Question Paper June 2018 Q24\" width=\"220\" height=\"117\" \/><br \/>\n(a) Determine the value of K.<br \/>\n(b) Find P(X &lt; 2).<br \/>\nSolution:<br \/>\n(a) \u03a3P(X) = 1<br \/>\n\u21d2 6K = 1<br \/>\n\u21d2 K = \\(\\frac{1}{6}\\)<br \/>\n(b) P(X &lt; 2) = P(X = 0) + P(X = 1)<br \/>\n= K + 2K<br \/>\n= 3K<br \/>\n= 3(\\(\\frac{1}{6}\\))<br \/>\n= \\(\\frac{1}{2}\\)<\/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 \/>\nIf f : R \u2192 R and g : R \u2192 R are given by f(x) = cos x and g(x) = 3x<sup>2<\/sup>, then show that gof \u2260 fog.<\/p>\n<p>Question 26.<br \/>\nSolve : tan<sup>-1<\/sup> 2x + tan<sup>-1<\/sup> 3x = \\(\\frac{\\pi}{4}\\)<br \/>\nSolution:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63478\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q26.png\" alt=\"2nd PUC Maths Previous Year Question Paper June 2018 Q26\" width=\"501\" height=\"183\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q26.png 501w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q26-300x110.png 300w\" sizes=\"(max-width: 501px) 100vw, 501px\" \/><br \/>\nSince x = -1 does not satisfy the equation, x = \\(\\frac{1}{6}\\) is the only solution of the given equation.<\/p>\n<p>Question 27.<br \/>\nBy using elementary operations, find the inverse of the matrix A = \\(\\left[\\begin{array}{cc}3 &amp; -1 \\\\-4 &amp; 2\\end{array}\\right]\\)<br \/>\nSolution:<br \/>\nA = I A<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63479\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q27.png\" alt=\"2nd PUC Maths Previous Year Question Paper June 2018 Q27\" width=\"318\" height=\"482\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q27.png 318w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q27-198x300.png 198w\" sizes=\"(max-width: 318px) 100vw, 318px\" \/><\/p>\n<p>Question 28.<br \/>\nIf x = a(\u03b8 &#8211; sin \u03b8) and y = a(1 + cos \u03b8), then prove that \\(\\frac{d y}{d x}=-\\cot (\\theta \/ 2)\\)<br \/>\nSolution:<br \/>\nGiven x = a(\u03b8 &#8211; sin \u03b8) and y = a(1 + cos \u03b8)<br \/>\nDifferentiating w.r.t \u03b8, we get<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63480\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q28.png\" alt=\"2nd PUC Maths Previous Year Question Paper June 2018 Q28\" width=\"552\" height=\"163\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q28.png 552w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q28-300x89.png 300w\" sizes=\"(max-width: 552px) 100vw, 552px\" \/><\/p>\n<p>Question 29.<br \/>\nVerify Mean Value Theorem if f(x) = x<sup>2<\/sup> &#8211; 4x + 3 in the interval x \u2208 [a, b], where a = 1 and b = 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-63481\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q29.png\" alt=\"2nd PUC Maths Previous Year Question Paper June 2018 Q29\" width=\"287\" height=\"213\" \/><br \/>\n\u2234 Mean Value 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 June 2018\" width=\"172\" height=\"16\" \/><\/p>\n<p>Question 30.<br \/>\nFind two positive numbers whose sum is 15 and the sum of whose squares is minimum.<\/p>\n<p>Question 31.<br \/>\nEvaluate: \\(\\int_{0}^{1} \\frac{\\tan ^{-1} x}{1+x^{2}} d x\\)<br \/>\nSolution:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63482\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q31.png\" alt=\"2nd PUC Maths Previous Year Question Paper June 2018 Q31\" width=\"247\" height=\"257\" \/><\/p>\n<p>Question 32.<br \/>\nIntegrate \\(\\frac{d x}{x\\left(x^{2}+1\\right)}\\) with respect to x.<br \/>\nSolution:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63484\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q32.png\" alt=\"2nd PUC Maths Previous Year Question Paper June 2018 Q32\" width=\"332\" height=\"250\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q32.png 332w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q32-300x226.png 300w\" sizes=\"(max-width: 332px) 100vw, 332px\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63483\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q32.1.png\" alt=\"2nd PUC Maths Previous Year Question Paper June 2018 Q32.1\" width=\"484\" height=\"96\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q32.1.png 484w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q32.1-300x60.png 300w\" sizes=\"(max-width: 484px) 100vw, 484px\" \/><\/p>\n<p>Question 33.<br \/>\nFind the area of the parabola y<sup>2<\/sup> = 4ax bounded by its latus rectum.<br \/>\nSolution:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63486\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q33.png\" alt=\"2nd PUC Maths Previous Year Question Paper June 2018 Q33\" width=\"167\" height=\"136\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63485\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q33.1.png\" alt=\"2nd PUC Maths Previous Year Question Paper June 2018 Q33.1\" width=\"592\" height=\"67\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q33.1.png 592w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q33.1-300x34.png 300w\" sizes=\"(max-width: 592px) 100vw, 592px\" \/><\/p>\n<p>Question 34.<br \/>\nFind the differential equation representing the family of curves y = a sin(x + b), where a, b are arbitrary constants.<\/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 June 2018\" width=\"172\" height=\"16\" \/><\/p>\n<p>Question 35.<br \/>\nFind a unit vector perpendicular to each of the vectors \\((\\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-63488\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q35.png\" alt=\"2nd PUC Maths Previous Year Question Paper June 2018 Q35\" width=\"511\" height=\"375\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q35.png 511w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q35-300x220.png 300w\" sizes=\"(max-width: 511px) 100vw, 511px\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63487\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q35.1.png\" alt=\"2nd PUC Maths Previous Year Question Paper June 2018 Q35.1\" width=\"421\" height=\"187\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q35.1.png 421w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q35.1-300x133.png 300w\" sizes=\"(max-width: 421px) 100vw, 421px\" \/><\/p>\n<p>Question 36.<br \/>\nProve that \\([\\vec{a}+\\vec{b}, \\vec{b}+\\vec{c}, \\vec{c}+\\vec{a}]=2[\\vec{a}, \\vec{b}, \\vec{c}]\\)<br \/>\nSolution:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63489\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q36.png\" alt=\"2nd PUC Maths Previous Year Question Paper June 2018 Q36\" width=\"640\" height=\"210\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q36.png 640w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q36-300x98.png 300w\" sizes=\"(max-width: 640px) 100vw, 640px\" \/><\/p>\n<p>Question 37.<br \/>\nFind the equation of the plane through the intersection of the planes 3x &#8211; y + 2z = 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 4 out of 5 times. He tossed a coin and reports that it is head. Find the probability that it is actually head.<br \/>\nSolution:<br \/>\nE<sub>1<\/sub> : coin shows a head<br \/>\nE<sub>2<\/sub> : coins shows a tail<br \/>\nP(E<sub>1<\/sub>) = P(E<sub>2<\/sub>) = \\(\\frac{1}{2}\\)<br \/>\nS = {H, T}<br \/>\nE<sub>1<\/sub> = {H}, E<sub>2<\/sub> = {T}<br \/>\nLet E : A reports that a head appears<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63490\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q38.png\" alt=\"2nd PUC Maths Previous Year Question Paper June 2018 Q38\" width=\"335\" height=\"234\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q38.png 335w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q38-300x210.png 300w\" sizes=\"(max-width: 335px) 100vw, 335px\" \/><\/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 R<sub>+<\/sub> be the set of all non-negative real numbers. Show that the function f : R<sub>+<\/sub> \u2192 [4, \u221e] given by f(x) = x<sup>2<\/sup> + 4 is invertible and write the inverse of f.<br \/>\nSolution:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63493\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q39.png\" alt=\"2nd PUC Maths Previous Year Question Paper June 2018 Q39\" width=\"681\" height=\"338\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q39.png 681w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q39-300x149.png 300w\" sizes=\"(max-width: 681px) 100vw, 681px\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63491\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q39.1.png\" alt=\"2nd PUC Maths Previous Year Question Paper June 2018 Q39.1\" width=\"561\" height=\"254\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q39.1.png 561w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q39.1-300x136.png 300w\" sizes=\"(max-width: 561px) 100vw, 561px\" \/><\/p>\n<p>Question 40.<br \/>\nIf A = \\(\\left[\\begin{array}{c}1 \\\\-4 \\\\3\\end{array}\\right]\\), B = \\(\\left[\\begin{array}{lll}-1 &amp; 2 &amp; 1\\end{array}\\right]\\), verify that (AB)&#8217; = B&#8217;A&#8217;. Calculate AC, BC and (A + B)C. Also, verify that (A + B)C = AC + BC.<br \/>\nSolution:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63495\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q40.png\" alt=\"2nd PUC Maths Previous Year Question Paper June 2018 Q40\" width=\"290\" height=\"285\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63494\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q40.1.png\" alt=\"2nd PUC Maths Previous Year Question Paper June 2018 Q40.1\" width=\"139\" height=\"94\" \/><\/p>\n<p>Question 41.<br \/>\nSolve the following system of equations by matrix method.<br \/>\n4x + 3y + 2z = 60<br \/>\n2x + 4y + 6z = 90<br \/>\n6x + 2y + 3z = 70<br \/>\nSolution:<br \/>\nThis system of equations can be written as AX = B, where<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63498\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q41.png\" alt=\"2nd PUC Maths Previous Year Question Paper June 2018 Q41\" width=\"433\" height=\"224\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q41.png 433w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q41-300x155.png 300w\" sizes=\"(max-width: 433px) 100vw, 433px\" \/><br \/>\nThus, A is non-singular, Therefore, its inverse exists.<br \/>\nTherefore, the given system is consistent and has a unique solution given by X = A<sup>-1<\/sup> B.<br \/>\nCofactors of A are<br \/>\nA<sub>11<\/sub> = 12 &#8211; 12 = 0<br \/>\nA<sub>12<\/sub> = -(6 &#8211; 36) = 30<br \/>\nA<sub>13<\/sub> = 4 &#8211; 24 = -20<br \/>\nA<sub>21<\/sub> = -(9 &#8211; 4) = -5<br \/>\nA<sub>22<\/sub> = 12 &#8211; 12 = 0<br \/>\nA<sub>23<\/sub> = -(8 &#8211; 18) = 10<br \/>\nA<sub>31<\/sub> = (18 &#8211; 8) = 10<br \/>\nA<sub>32<\/sub>\u00a0= -(24 &#8211; 4) = -20<br \/>\nA<sub>33<\/sub> = 16 &#8211; 6 = 10<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63496\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q41.1.png\" alt=\"2nd PUC Maths Previous Year Question Paper June 2018 Q41.1\" width=\"368\" height=\"93\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q41.1.png 368w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q41.1-300x76.png 300w\" sizes=\"(max-width: 368px) 100vw, 368px\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63497\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q41.2.png\" alt=\"2nd PUC Maths Previous Year Question Paper June 2018 Q41.2\" width=\"432\" height=\"457\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q41.2.png 432w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q41.2-284x300.png 284w\" sizes=\"(max-width: 432px) 100vw, 432px\" \/><\/p>\n<p>Question 42.<br \/>\nIf Y = A e<sup>mx<\/sup> + B e<sup>nx<\/sup> show that \\(\\frac{d^{2} y}{d x^{2}}-(m+n) \\frac{d y}{d x}+m n y=0\\)<br \/>\nSolution:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63499\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q42.png\" alt=\"2nd PUC Maths Previous Year Question Paper June 2018 Q42\" width=\"682\" height=\"360\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q42.png 682w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q42-300x158.png 300w\" sizes=\"(max-width: 682px) 100vw, 682px\" \/><\/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 June 2018\" width=\"172\" height=\"16\" \/><\/p>\n<p>Question 43.<br \/>\nA particle moves along the curve 6y = x<sup>3<\/sup> + 2. Find the points on the curve at which y-coordinate is changing 8 times as fast as the x-coordinate.<br \/>\nSolution:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63501\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q43.png\" alt=\"2nd PUC Maths Previous Year Question Paper June 2018 Q43\" width=\"346\" height=\"179\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q43.png 346w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q43-300x155.png 300w\" sizes=\"(max-width: 346px) 100vw, 346px\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63500\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q43.1.png\" alt=\"2nd PUC Maths Previous Year Question Paper June 2018 Q43.1\" width=\"549\" height=\"190\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q43.1.png 549w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q43.1-300x104.png 300w\" sizes=\"(max-width: 549px) 100vw, 549px\" \/><\/p>\n<p>Question 44.<br \/>\nFind the integral of \\(\\frac{1}{\\sqrt{a^{2}-x^{2}}}\\) with respect to x and hence find \\(\\int \\frac{1}{\\sqrt{7-6 x-x^{2}}} d x\\)<br \/>\nSolution:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63502\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q44.png\" alt=\"2nd PUC Maths Previous Year Question Paper June 2018 Q44\" width=\"547\" height=\"432\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q44.png 547w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q44-300x237.png 300w\" sizes=\"(max-width: 547px) 100vw, 547px\" \/><\/p>\n<p>Question 45.<br \/>\nFind the area eiiclosed by the ellipse \\(\\frac{x^{2}}{a^{2}}+\\frac{y^{2}}{b^{2}}=1\\) by the method of integration.<br \/>\nSolution:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63504\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q45.png\" alt=\"2nd PUC Maths Previous Year Question Paper June 2018 Q45\" width=\"243\" height=\"245\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q45.png 243w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q45-150x150.png 150w\" sizes=\"(max-width: 243px) 100vw, 243px\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63503\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q45.1.png\" alt=\"2nd PUC Maths Previous Year Question Paper June 2018 Q45.1\" width=\"241\" height=\"129\" \/><\/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>, (x \u2260 0)<br \/>\nSolution:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63505\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q46.png\" alt=\"2nd PUC Maths Previous Year Question Paper June 2018 Q46\" width=\"302\" height=\"366\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q46.png 302w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q46-248x300.png 248w\" sizes=\"(max-width: 302px) 100vw, 302px\" \/><\/p>\n<p>Question 47.<br \/>\nDerive the equation of a line in space passing through two given points both in vector and Cartesian form.<br \/>\nSolution:<br \/>\nVector Form<br \/>\nLet \\(\\overrightarrow{\\mathrm{a}}\\) and \\(\\overrightarrow{\\mathrm{b}}\\) be the position vectors of two points A(x<sub>1<\/sub>, y<sub>1<\/sub>, z<sub>1<\/sub>) and B(x<sub>2<\/sub>, y<sub>2<\/sub>, z<sub>2<\/sub>) respectively that are lying on a line.<br \/>\nLet \\(\\overrightarrow{\\mathrm{r}}\\) be the position vector of an arbitrary point P (x, y, z), then P is a point on the line if and only if \\(\\overrightarrow{\\mathrm{AP}}=\\overrightarrow{\\mathrm{r}}-\\overrightarrow{\\mathrm{a}}\\) and \\(\\overrightarrow{\\mathrm{AB}}=\\overrightarrow{\\mathrm{b}}-\\overrightarrow{\\mathrm{a}}\\) are collinear vectors. Therefore, P is on the line if an only<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63508\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q47.png\" alt=\"2nd PUC Maths Previous Year Question Paper June 2018 Q47\" width=\"360\" height=\"197\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q47.png 360w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q47-300x164.png 300w\" sizes=\"(max-width: 360px) 100vw, 360px\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63506\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q47.1.png\" alt=\"2nd PUC Maths Previous Year Question Paper June 2018 Q47.1\" width=\"506\" height=\"217\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q47.1.png 506w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q47.1-300x129.png 300w\" sizes=\"(max-width: 506px) 100vw, 506px\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63507\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q47.2.png\" alt=\"2nd PUC Maths Previous Year Question Paper June 2018 Q47.2\" width=\"319\" height=\"221\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q47.2.png 319w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q47.2-300x208.png 300w\" sizes=\"(max-width: 319px) 100vw, 319px\" \/><\/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 June 2018\" width=\"172\" height=\"16\" \/><\/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<br \/>\n(a) at least once<br \/>\n(b) exactly once<br \/>\nSolution:<br \/>\nLet X denote the number of wins<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63509\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q48.png\" alt=\"2nd PUC Maths Previous Year Question Paper June 2018 Q48\" width=\"638\" height=\"258\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q48.png 638w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q48-300x121.png 300w\" sizes=\"(max-width: 638px) 100vw, 638px\" \/><\/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-63513\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q49a.png\" alt=\"2nd PUC Maths Previous Year Question Paper June 2018 Q49(a)\" width=\"561\" height=\"137\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q49a.png 561w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q49a-300x73.png 300w\" sizes=\"(max-width: 561px) 100vw, 561px\" \/><br \/>\nSolution:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63510\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q49a.1.png\" alt=\"2nd PUC Maths Previous Year Question Paper June 2018 Q49(a).1\" width=\"408\" height=\"280\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q49a.1.png 408w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q49a.1-300x206.png 300w\" sizes=\"(max-width: 408px) 100vw, 408px\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63511\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q49a.2.png\" alt=\"2nd PUC Maths Previous Year Question Paper June 2018 Q49(a).2\" width=\"581\" height=\"257\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q49a.2.png 581w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q49a.2-300x133.png 300w\" sizes=\"(max-width: 581px) 100vw, 581px\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63512\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q49a.3.png\" alt=\"2nd PUC Maths Previous Year Question Paper June 2018 Q49(a).3\" width=\"569\" height=\"78\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q49a.3.png 569w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q49a.3-300x41.png 300w\" sizes=\"(max-width: 569px) 100vw, 569px\" \/><\/p>\n<p>Question 49(b).<br \/>\nFind the value of K, if <img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63515\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q49b.png\" alt=\"2nd PUC Maths Previous Year Question Paper June 2018 Q49(b)\" width=\"182\" height=\"63\" \/><br \/>\nSolution:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63514\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q49b.1.png\" alt=\"2nd PUC Maths Previous Year Question Paper June 2018 Q49(b).1\" width=\"470\" height=\"198\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q49b.1.png 470w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q49b.1-300x126.png 300w\" sizes=\"(max-width: 470px) 100vw, 470px\" \/><\/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 June 2018\" width=\"172\" height=\"16\" \/><\/p>\n<p>Question 50(a).<br \/>\nSolve the following problem graphically:<br \/>\nMaximize and minimize<br \/>\nZ = 10500x + 9000y<br \/>\nSubject to the constraints<br \/>\nx + y \u2264 50<br \/>\n2x + y \u2264 80<br \/>\nx \u2265 0, y \u2265 0<br \/>\nSolution:<br \/>\nx + y = 50<br \/>\ny = 0 \u21d2 x = 50<br \/>\n\u2234 A = (50, 0), B = (0, 50)<br \/>\n(0, 0) lies on x + y \u2264 50<br \/>\n2x + y = 80<br \/>\ny = 0, x = 40 \u21d2 C(40, 0)<br \/>\nx = 0, y = 80 \u21d2 D(0, 80)<br \/>\nOrigin lies on 2x + y \u2264 80<br \/>\nSolving x + y = 50<br \/>\n2x + y = 80 we get x = 30, y = 20<br \/>\nE = (30, 20)<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63517\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q50a.png\" alt=\"2nd PUC Maths Previous Year Question Paper June 2018 Q50(a)\" width=\"292\" height=\"259\" \/><br \/>\nOBEC is the feasible region.<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63516\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q50a.1.png\" alt=\"2nd PUC Maths Previous Year Question Paper June 2018 Q50(a).1\" width=\"360\" height=\"149\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q50a.1.png 360w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q50a.1-300x124.png 300w\" sizes=\"(max-width: 360px) 100vw, 360px\" \/><\/p>\n<p>Question 50(b).<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63519\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q50b.png\" alt=\"2nd PUC Maths Previous Year Question Paper June 2018 Q50(b)\" width=\"486\" height=\"87\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q50b.png 486w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q50b-300x54.png 300w\" sizes=\"(max-width: 486px) 100vw, 486px\" \/><br \/>\nSolution:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63518\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q50b.1.png\" alt=\"2nd PUC Maths Previous Year Question Paper June 2018 Q50(b).1\" width=\"556\" height=\"371\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q50b.1.png 556w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2018-Q50b.1-300x200.png 300w\" sizes=\"(max-width: 556px) 100vw, 556px\" \/><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Students can Download 2nd PUC Maths Previous Year Question Paper June 2018, 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 June 2018 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\/27355"}],"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=27355"}],"version-history":[{"count":0,"href":"https:\/\/kseebsolutions.guru\/wp-json\/wp\/v2\/posts\/27355\/revisions"}],"wp:attachment":[{"href":"https:\/\/kseebsolutions.guru\/wp-json\/wp\/v2\/media?parent=27355"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/kseebsolutions.guru\/wp-json\/wp\/v2\/categories?post=27355"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/kseebsolutions.guru\/wp-json\/wp\/v2\/tags?post=27355"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}