{"id":27431,"date":"2020-06-30T17:20:06","date_gmt":"2020-06-30T11:50:06","guid":{"rendered":"https:\/\/kseebsolutions.guru\/?p=27431"},"modified":"2021-07-02T15:07:09","modified_gmt":"2021-07-02T09:37:09","slug":"2nd-puc-maths-previous-year-question-paper-june-2017","status":"publish","type":"post","link":"https:\/\/kseebsolutions.guru\/2nd-puc-maths-previous-year-question-paper-june-2017\/","title":{"rendered":"2nd PUC Maths Previous Year Question Paper June 2017"},"content":{"rendered":"<p>Students can Download 2nd PUC Maths Previous Year Question Paper June 2017, Karnataka <a href=\"https:\/\/kseebsolutions.guru\/2nd-puc-maths-model-question-papers-with-answers\/\">2nd PUC Maths Model Question Papers with Answers<\/a> helps you to revise the complete Karnataka State Board Syllabus and score more marks in your examinations.<\/p>\n<h2>Karnataka 2nd PUC Maths Previous Year Question Paper June 2017<\/h2>\n<p>Time: 3 Hrs 15 Min<br \/>\nMax. Marks: 100<\/p>\n<p>Instructions<\/p>\n<ul>\n<li>The question paper has five parts namely A, B, C, D, and E. Answer all the parts.<\/li>\n<li>Use the graph sheet for the question on Linear programming in Part &#8211; E<\/li>\n<\/ul>\n<p style=\"text-align: center;\"><span style=\"color: #0000ff;\">Part &#8211; A<\/span><\/p>\n<p><span style=\"color: #0000ff;\">Answer ALL the following questions: (10 \u00d7 1 = 10)<\/span><\/p>\n<p>Question 1.<br \/>\nFind the identity element for the binary operation *, defined on the set Q of rational numbers by a * b = \\(\\frac{a b}{4}\\)<br \/>\nSolution:<br \/>\nLet e be the identity element.<br \/>\na * e = a<br \/>\n\u21d2 \\(\\frac{a e}{4}\\) = a<br \/>\n\u21d2 e = 4<\/p>\n<p>Question 2.<br \/>\nWrite the values of x for which \\(\\tan ^{-1} \\frac{1}{x}=\\cot ^{-1} x\\), holds.<br \/>\nSolution:<br \/>\nx &gt; 0<\/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 2017\" width=\"172\" height=\"16\" \/><\/p>\n<p>Question 3.<br \/>\nConstruct a 2 \u00d7 2 matrix, A = [a<sub>ij<\/sub>], whose elements are given by, a<sub>ij<\/sub> = \\(\\frac{i}{j}\\)<br \/>\nSolution:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63579\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q3.png\" alt=\"2nd PUC Maths Previous Year Question Paper June 2017 Q3\" width=\"245\" height=\"131\" \/><\/p>\n<p>Question 4.<br \/>\nFind the values of x for which \\(\\left|\\begin{array}{ll}3 &amp; x \\\\x &amp; 1\\end{array}\\right|=\\left|\\begin{array}{ll}3 &amp; 2 \\\\4 &amp; 1\\end{array}\\right|\\)<br \/>\nSolution:<br \/>\n3 &#8211; x<sup>2<\/sup> = 3 &#8211; 8<br \/>\n\u21d2 x<sup>2<\/sup> = 8<br \/>\n\u21d2 x = \u00b1\u221a8 = \u00b12\u221a2<\/p>\n<p>Question 5.<br \/>\nFind \\(\\frac{d y}{d x}\\), if y = sin (x<sup>2<\/sup>)<br \/>\nSolution:<br \/>\n\\(\\frac{d y}{d x}\\) = cos(x<sup>2<\/sup>) . 2x<\/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 2017\" width=\"172\" height=\"16\" \/><\/p>\n<p>Question 6.<br \/>\nFind \u222bcos 3x dx<br \/>\nSolution:<br \/>\n\\(\\frac{\\sin 3 x}{3}+c\\)<\/p>\n<p>Question 7.<br \/>\nFind unit vector in the direction of vector \\(\\hat{i}+\\hat{j}+2 \\hat{k}\\)<br \/>\nSolution:<br \/>\n\\(\\hat{a}=\\frac{\\vec{a}}{|\\vec{a}|}=\\frac{\\hat{i}+\\hat{j}+2 \\hat{k}}{\\sqrt{6}}\\)<\/p>\n<p>Question 8.<br \/>\nWrite the direction cosines of the y-axis.<br \/>\nSolution:<br \/>\n0, 1, 0<\/p>\n<p>Question 9.<br \/>\nDefine the 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 2017\" width=\"172\" height=\"16\" \/><\/p>\n<p>Question 10.<br \/>\nIf P(A) = 0.8 and P(B|A) = 0.4 then find P(A\u2229B).<br \/>\nSolution:<br \/>\nP(A\u2229B) = P(A) \u00d7 P(B\/A)= 0.8 \u00d7 0.4 = 0.32<\/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 \/>\nShow that if f : A \u2192 B and g : B \u2192 C are onto, then gof : A \u2192 C is also onto.<br \/>\nSolution:<br \/>\nGiven an arbitrary element z &#8211; C, there exists a pre-image y of z under g such that g(y) = z i.e., g is onto. Further, for y \u2208 B, there exists an element x in A with f(x) = y, since f is onto. Therefore, gof(x) = g(fx)) = g(y) = z, showing that gof is onto.<\/p>\n<p>Question 12.<br \/>\nShow that \\(2 \\tan ^{-1} x=\\cos ^{-1} \\frac{1-x^{2}}{1+x^{2}}, x \\geq 0\\)<br \/>\nSolution:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63580\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q12.png\" alt=\"2nd PUC Maths Previous Year Question Paper June 2017 Q12\" width=\"487\" height=\"132\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q12.png 487w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q12-300x81.png 300w\" sizes=\"(max-width: 487px) 100vw, 487px\" \/><\/p>\n<p>Question 13.<br \/>\nFind the value of \\(\\sin ^{-1}\\left(\\sin \\frac{3 \\pi}{5}\\right)\\)<br \/>\nSolution:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63581\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q13.png\" alt=\"2nd PUC Maths Previous Year Question Paper June 2017 Q13\" width=\"524\" height=\"79\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q13.png 524w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q13-300x45.png 300w\" sizes=\"(max-width: 524px) 100vw, 524px\" \/><\/p>\n<p>Question 14.<br \/>\nUsing the determinant method, find the area of the triangle whose vertices are (1, 0), (6, 0) and (4, 3).<br \/>\nSolution:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63582\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q14.png\" alt=\"2nd PUC Maths Previous Year Question Paper June 2017 Q14\" width=\"606\" height=\"208\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q14.png 606w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q14-300x103.png 300w\" sizes=\"(max-width: 606px) 100vw, 606px\" \/><\/p>\n<p>Question 15.<br \/>\nDifferentiate (sin x)<sup>x<\/sup> with respect to x.<br \/>\nSolution:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63583\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q15.png\" alt=\"2nd PUC Maths Previous Year Question Paper June 2017 Q15\" width=\"262\" height=\"260\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q15.png 262w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q15-150x150.png 150w\" sizes=\"(max-width: 262px) 100vw, 262px\" \/><\/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 2017\" width=\"172\" height=\"16\" \/><\/p>\n<p>Question 16.<br \/>\nFind \\(\\frac{d y}{d x}\\), if 2x + 3y = sin y.<br \/>\nSolution:<br \/>\nDifferentiate both sides w.r.t x<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63584\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q16.png\" alt=\"2nd PUC Maths Previous Year Question Paper June 2017 Q16\" width=\"611\" height=\"62\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q16.png 611w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q16-300x30.png 300w\" sizes=\"(max-width: 611px) 100vw, 611px\" \/><\/p>\n<p>Question 17.<br \/>\nFind the point on the curve \\(\\frac{x^{2}}{4}+\\frac{y^{2}}{25}=1\\) at which the tangents are parallel to x-axis.<br \/>\nSolution:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63585\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q17.png\" alt=\"2nd PUC Maths Previous Year Question Paper June 2017 Q17\" width=\"482\" height=\"266\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q17.png 482w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q17-300x166.png 300w\" sizes=\"(max-width: 482px) 100vw, 482px\" \/><\/p>\n<p>Question 18.<br \/>\nEvaluate: \\(\\int \\frac{\\sqrt{\\tan x}}{\\sin x \\cos x} d x\\)<br \/>\nSolution:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63587\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q18.png\" alt=\"2nd PUC Maths Previous Year Question Paper June 2017 Q18\" width=\"495\" height=\"202\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q18.png 495w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q18-300x122.png 300w\" sizes=\"(max-width: 495px) 100vw, 495px\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63586\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q18.1.png\" alt=\"2nd PUC Maths Previous Year Question Paper June 2017 Q18.1\" width=\"228\" height=\"129\" \/><\/p>\n<p>Question 19.<br \/>\nEvaluate: \\(\\int \\frac{x-3}{(x-1)^{3}} e^{x} d x\\)<br \/>\nSolution:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63588\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q19.png\" alt=\"2nd PUC Maths Previous Year Question Paper June 2017 Q19\" width=\"386\" height=\"263\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q19.png 386w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q19-300x204.png 300w\" sizes=\"(max-width: 386px) 100vw, 386px\" \/><\/p>\n<p>Question 20.<br \/>\nFind the order and degree, if defined of the differential equation \\(\\frac{d^{4} y}{d x^{4}}+\\sin \\left(\\frac{d^{3} y}{d x^{3}}\\right)=0\\)<br \/>\nSolution:<br \/>\nOrder = 4<br \/>\nDegree = not defined.<\/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 2017\" width=\"172\" height=\"16\" \/><\/p>\n<p>Question 21.<br \/>\nIf \\(\\vec{a}\\) is a unit vector and \\((\\vec{x}-\\vec{a}) \\cdot(\\vec{x}+\\vec{a})=8\\), then find \\(\\vec{x}\\)|.<br \/>\nSolution:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63591\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q21.png\" alt=\"2nd PUC Maths Previous Year Question Paper June 2017 Q21\" width=\"110\" height=\"145\" \/><\/p>\n<p>Question 22.<br \/>\nFind the area of the parallelogram whose adjacent sides are given by the vector \\(\\vec{a}=3 \\hat{i}+\\hat{j}+4 \\hat{k}\\) and \\(\\vec{b}=\\hat{i}-\\hat{j}+\\hat{k}\\)<\/p>\n<p>Question 23.<br \/>\nFind the angle between the pair of lines given by \\(\\vec{r}=2 \\hat{i}-5 \\hat{j}+\\hat{k}+\\lambda(3 \\hat{i}+2 \\hat{j}+6 \\hat{k})\\) and \\(\\vec{r}=7 \\hat{i}-6 \\hat{k}+\\mu(\\hat{i}+2 \\hat{j}+2 \\hat{k})\\)<\/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 2017\" width=\"172\" height=\"16\" \/><\/p>\n<p>Question 24.<br \/>\nIf A and B are two independent events, then prove that the probability of occurrence of at least one of A and B is given by 1 &#8211; P(A&#8217;) P(B&#8217;).<br \/>\nSolution:<br \/>\nP(Atleast one of A or B) = P(A \u222a B) [\u2235 A and B are independent events]<br \/>\n= P(A) + P(B) &#8211; P(A \u2229 B)<br \/>\n= P(A) + P(B) &#8211; P(A) . P(B)<br \/>\n= P(A) + P(B) &#8211; (1 &#8211; P(A))<br \/>\n= 1 &#8211; P(A&#8217;) + P(B) P(A&#8217;)<br \/>\n= 1 &#8211; [P(A&#8217;) + P(B)P(A&#8217;)]<br \/>\n= 1 &#8211; P(A&#8217;) [1 &#8211; P(B)]<br \/>\n= 1 &#8211; P(A&#8217;) P(B&#8217;)<br \/>\n= RHS.<\/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 \/>\nCheck whether the relation R defined in the set {1, 2, 3, 4, 5, 6} as R = {(a, b) : b = a + 1} is reflexive or symmetric.<br \/>\nSolution:<br \/>\nR = {(1, 2), (2, 3), (3, 4), (4, 5), (5, 6)}<br \/>\n2 \u2208 A, But (2, 2) \u2209 R<br \/>\n\u2234 R is not reflexive<br \/>\n(2, 3) \u2208 R, But (3, 2) \u2209 R<br \/>\n\u2234 R is not Symmetric<\/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 2017\" width=\"172\" height=\"16\" \/><\/p>\n<p>Question 26.<br \/>\nSolve : tan<sup>-1<\/sup> 2x + tan<sup>-1<\/sup> 3x = \\(\\frac{\\pi}{4}\\). Find the value of &#8216;x&#8217;.<br \/>\nSolution:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63592\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q26.png\" alt=\"2nd PUC Maths Previous Year Question Paper June 2017 Q26\" width=\"498\" height=\"184\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q26.png 498w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q26-300x111.png 300w\" sizes=\"(max-width: 498px) 100vw, 498px\" \/><\/p>\n<p>Since 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 transformations, find the inverse of A = \\(\\left[\\begin{array}{rr}1 &amp; 2 \\\\2 &amp; -1\\end{array}\\right]\\)<br \/>\nSolution:<br \/>\nA = I A<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63594\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q27.png\" alt=\"2nd PUC Maths Previous Year Question Paper June 2017 Q27\" width=\"168\" height=\"64\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63593\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q27.1.png\" alt=\"2nd PUC Maths Previous Year Question Paper June 2017 Q27.1\" width=\"300\" height=\"298\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q27.1.png 300w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q27.1-150x150.png 150w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/p>\n<p>Question 28.<br \/>\nFind \\(\\frac{d y}{d x}\\), if x = a (cos t + log tan\\(\\frac{t}{2}\\)], y = a sin t.<br \/>\nSolution:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63596\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q28.png\" alt=\"2nd PUC Maths Previous Year Question Paper June 2017 Q28\" width=\"352\" height=\"444\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q28.png 352w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q28-238x300.png 238w\" sizes=\"(max-width: 352px) 100vw, 352px\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63595\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q28.1.png\" alt=\"2nd PUC Maths Previous Year Question Paper June 2017 Q28.1\" width=\"313\" height=\"169\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q28.1.png 313w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q28.1-300x162.png 300w\" sizes=\"(max-width: 313px) 100vw, 313px\" \/><\/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 2017\" width=\"172\" height=\"16\" \/><\/p>\n<p>Question 29.<br \/>\nVerify Mean Value Theorem for the function f(x) = x<sup>2<\/sup> in the interval [2, 4].<br \/>\nSolution:<br \/>\nThe function f(x) = x<sup>2<\/sup> is continuous in {2, 4} and differentiable in {2, 4} as its derivative f(x) = 2x is defined in (2, 4).<br \/>\nNow f(2) = 4 and f(4) = 16. Hence<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63597\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q29.png\" alt=\"2nd PUC Maths Previous Year Question Paper June 2017 Q29\" width=\"188\" height=\"58\" \/><br \/>\nMVT states that there is a point c \u2208 (2, 4) such that f'(c) = 6. But f'(x) = 2x which implies c = 3.<br \/>\nThus at c = 3 \u2208 (2, 4), we have f'(c) = 6.<\/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 \\frac{x}{(x+1)(x+2)} d x\\)<br \/>\nSolution:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63598\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q31.png\" alt=\"2nd PUC Maths Previous Year Question Paper June 2017 Q31\" width=\"598\" height=\"363\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q31.png 598w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q31-300x182.png 300w\" sizes=\"(max-width: 598px) 100vw, 598px\" \/><\/p>\n<p>Question 32.<br \/>\nEvaluate: \\(\\int \\frac{x \\cos ^{-1} x}{\\sqrt{1-x^{2}}} d x\\)<br \/>\nSolution:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63600\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q32.png\" alt=\"2nd PUC Maths Previous Year Question Paper June 2017 Q32\" width=\"314\" height=\"129\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q32.png 314w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q32-300x123.png 300w\" sizes=\"(max-width: 314px) 100vw, 314px\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63599\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q32.1.png\" alt=\"2nd PUC Maths Previous Year Question Paper June 2017 Q32.1\" width=\"585\" height=\"46\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q32.1.png 585w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q32.1-300x24.png 300w\" sizes=\"(max-width: 585px) 100vw, 585px\" \/><\/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 2017\" width=\"172\" height=\"16\" \/><\/p>\n<p>Question 33.<br \/>\nFind the area bounded by the curve y = cos x between x = 0 and x = 2\u03c0.<br \/>\nSolution:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63601\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q33.png\" alt=\"2nd PUC Maths Previous Year Question Paper June 2017 Q33\" width=\"588\" height=\"221\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q33.png 588w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q33-300x113.png 300w\" sizes=\"(max-width: 588px) 100vw, 588px\" \/><\/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}}\\)<br \/>\nSolution:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63602\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q34.png\" alt=\"2nd PUC Maths Previous Year Question Paper June 2017 Q34\" width=\"348\" height=\"161\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q34.png 348w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q34-300x139.png 300w\" sizes=\"(max-width: 348px) 100vw, 348px\" \/><\/p>\n<p>Question 35.<br \/>\nShow that the position vector fo thepoint P, which divides the line joining the points A and B having position vectors \\(\\vec{a}\\) and \\(\\vec{b}\\) internally in the ratio m : n is \\(\\frac{m \\vec{b}+n \\vec{a}}{m+n}\\).<\/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 2017\" width=\"172\" height=\"16\" \/><\/p>\n<p>Question 36.<br \/>\nFind x such that the four point 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-63604\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q36.png\" alt=\"2nd PUC Maths Previous Year Question Paper June 2017 Q36\" width=\"154\" height=\"72\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63603\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q36.1.png\" alt=\"2nd PUC Maths Previous Year Question Paper June 2017 Q36.1\" width=\"622\" height=\"270\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q36.1.png 622w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q36.1-300x130.png 300w\" sizes=\"(max-width: 622px) 100vw, 622px\" \/><\/p>\n<p>Question 37.<br \/>\nFind the vector and cartesian equations of the plane which passes through the points (5, 2, -4) and perpendicular to the line with direction ratios 2, 3, -1.<br \/>\nSolution:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63605\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q37.png\" alt=\"2nd PUC Maths Previous Year Question Paper June 2017 Q37\" width=\"678\" height=\"185\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q37.png 678w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q37-300x82.png 300w\" sizes=\"(max-width: 678px) 100vw, 678px\" \/><\/p>\n<p>Question 38.<br \/>\nA man is known to speak truth 3 out of 4 times. He throws a die 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-63606\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q38.png\" alt=\"2nd PUC Maths Previous Year Question Paper June 2017 Q38\" width=\"350\" height=\"178\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q38.png 350w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q38-300x153.png 300w\" sizes=\"(max-width: 350px) 100vw, 350px\" \/><\/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 \/>\nProve that the function f : R \u2192 R defined by f(x) = 4x + 3 is invertible and find the inverse of f.<\/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 2017\" width=\"172\" height=\"16\" \/><\/p>\n<p>Question 40.<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63610\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q40.png\" alt=\"2nd PUC Maths Previous Year Question Paper June 2017 Q40\" width=\"368\" height=\"93\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q40.png 368w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q40-300x76.png 300w\" sizes=\"(max-width: 368px) 100vw, 368px\" \/><br \/>\nCalculate 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-63607\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q40.1.png\" alt=\"2nd PUC Maths Previous Year Question Paper June 2017 Q40.1\" width=\"386\" height=\"207\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q40.1.png 386w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q40.1-300x161.png 300w\" sizes=\"(max-width: 386px) 100vw, 386px\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63608\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q40.2.png\" alt=\"2nd PUC Maths Previous Year Question Paper June 2017 Q40.2\" width=\"416\" height=\"299\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q40.2.png 416w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q40.2-300x216.png 300w\" sizes=\"(max-width: 416px) 100vw, 416px\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63609\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q40.3.png\" alt=\"2nd PUC Maths Previous Year Question Paper June 2017 Q40.3\" width=\"640\" height=\"263\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q40.3.png 640w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q40.3-300x123.png 300w\" sizes=\"(max-width: 640px) 100vw, 640px\" \/><\/p>\n<p>Question 41.<br \/>\nSolve the following system of equations by matrix method.<br \/>\nx &#8211; y + z = 6<br \/>\ny + 3z = 11<br \/>\nx &#8211; 2y + z = 0<br \/>\nSolution:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63612\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q41.png\" alt=\"2nd PUC Maths Previous Year Question Paper June 2017 Q41\" width=\"396\" height=\"473\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q41.png 396w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q41-251x300.png 251w\" sizes=\"(max-width: 396px) 100vw, 396px\" \/><\/p>\n<p>Question 42.<br \/>\nIf y = 3 cos(log x) + 4 sin(log x), show that x<sup>2<\/sup> y<sub>2<\/sub> + x y<sub>1<\/sub> + y = 0.<br \/>\nSolution:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63614\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q42.png\" alt=\"2nd PUC Maths Previous Year Question Paper June 2017 Q42\" width=\"685\" height=\"286\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q42.png 685w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q42-300x125.png 300w\" sizes=\"(max-width: 685px) 100vw, 685px\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63613\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q42.1.png\" alt=\"2nd PUC Maths Previous Year Question Paper June 2017 Q42.1\" width=\"682\" height=\"220\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q42.1.png 682w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q42.1-300x97.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 2017\" width=\"172\" height=\"16\" \/><\/p>\n<p>Question 43.<br \/>\nSand is pouring from a pipe at the rate of 12 cm<sup>3<\/sup>\/s. 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 the sand cone at any time t.<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63615\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q43.png\" alt=\"2nd PUC Maths Previous Year Question Paper June 2017 Q43\" width=\"594\" height=\"346\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q43.png 594w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q43-300x175.png 300w\" sizes=\"(max-width: 594px) 100vw, 594px\" \/><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 \\(\\sqrt{a^{2}+x^{2}}\\) with respect to x and hence evaluate \\(\\int \\sqrt{1+x^{2}} d x\\)<br \/>\nSolution:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63616\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q44.png\" alt=\"2nd PUC Maths Previous Year Question Paper June 2017 Q44\" width=\"601\" height=\"422\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q44.png 601w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q44-300x211.png 300w\" sizes=\"(max-width: 601px) 100vw, 601px\" \/><\/p>\n<p>Question 45.<br \/>\nFind the smaller area enclosed by the circle x<sup>2<\/sup> + y<sup>2<\/sup> = 4 and the line x + y = 2.<br \/>\nSolution:<br \/>\nThe smaller area enclosed by the circle, x<sup>2<\/sup> + y<sup>2<\/sup> = 4 and the line x + y = 2 is represented by the shaded area ACBA.<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63618\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q45.png\" alt=\"2nd PUC Maths Previous Year Question Paper June 2017 Q45\" width=\"265\" height=\"236\" \/><br \/>\nThe intersection points of circle and line are A(2, 0) and B(0, 2).<br \/>\nRequired area (shown \u00a1n shaded region)<br \/>\n= Area OACBO &#8211; Area (\u2206OAB)<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63617\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q45.1.png\" alt=\"2nd PUC Maths Previous Year Question Paper June 2017 Q45.1\" width=\"444\" height=\"390\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q45.1.png 444w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q45.1-300x264.png 300w\" sizes=\"(max-width: 444px) 100vw, 444px\" \/><\/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 2017\" width=\"172\" height=\"16\" \/><\/p>\n<p>Question 46.<br \/>\nFind the general solution of the differential equation y dx &#8211; (x + 2y<sup>2<\/sup>) dy = 0.<br \/>\nSolution:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63620\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q46.png\" alt=\"2nd PUC Maths Previous Year Question Paper June 2017 Q46\" width=\"178\" height=\"148\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63619\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q46.1.png\" alt=\"2nd PUC Maths Previous Year Question Paper June 2017 Q46.1\" width=\"576\" height=\"314\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q46.1.png 576w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q46.1-300x164.png 300w\" sizes=\"(max-width: 576px) 100vw, 576px\" \/><\/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 \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63622\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q47.png\" alt=\"2nd PUC Maths Previous Year Question Paper June 2017 Q47\" width=\"691\" height=\"441\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q47.png 691w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q47-300x191.png 300w\" sizes=\"(max-width: 691px) 100vw, 691px\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63621\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q47.1.png\" alt=\"2nd PUC Maths Previous Year Question Paper June 2017 Q47.1\" width=\"445\" height=\"169\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q47.1.png 445w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q47.1-300x114.png 300w\" sizes=\"(max-width: 445px) 100vw, 445px\" \/><\/p>\n<p>Question 48.<br \/>\nIf a fair coin is tossed 10 times, find the probability of<br \/>\n(i) exactly six heads and<br \/>\n(ii) at least six heads.<br \/>\nSolution:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63623\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q48.png\" alt=\"2nd PUC Maths Previous Year Question Paper June 2017 Q48\" width=\"596\" height=\"190\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q48.png 596w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q48-300x96.png 300w\" sizes=\"(max-width: 596px) 100vw, 596px\" \/><\/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 \/>\nMinimize z = -3x + 4y subject to the constraints<br \/>\nx + 2y \u2264 8<br \/>\n3x + 2y \u2264 12<br \/>\nx \u2265 0, y \u2265 0<br \/>\nby graphical method.<br \/>\nSolution:<br \/>\nDraw the graph of the line, x + 2y = 8<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63627\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q49a.png\" alt=\"2nd PUC Maths Previous Year Question Paper June 2017 Q49(a)\" width=\"237\" height=\"93\" \/><br \/>\nPutting (0, 0) in the inequality x + 2y \u2264 8, we have<br \/>\n0 + 0 \u2264 8<br \/>\n0 \u2264 8 (which is true)<br \/>\nSo, the half-plane is towards the origin,<br \/>\nSince, x, y \u2265 0<br \/>\nSo, the feasible region lies in the first quadrant.<br \/>\nDraw the graph of the line, 3x + 2y = 12<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63624\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q49a.1.png\" alt=\"2nd PUC Maths Previous Year Question Paper June 2017 Q49(a).1\" width=\"241\" height=\"91\" \/><br \/>\nPutting (0, 0) in the inequality 3x + 2y \u2264 12, we have<br \/>\n3 x 0 + 2 x 0 \u2264 12 \u21d2 0 \u2264 12 (which is true)<br \/>\nSo, the half-plane is towards the origin,<br \/>\nThe feasible region is OABCO.<br \/>\nOn solving equations x + 2y = 8 and x = 2 and y = 3<br \/>\nIntersection point B is (2, 3)<br \/>\nThe corner points of the feasible region are O(0, 0), A (4, 0) and B(2, 3) and C(0, 4).<br \/>\nThe values of Z at these points are as follows:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63625\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q49a.2.png\" alt=\"2nd PUC Maths Previous Year Question Paper June 2017 Q49(a).2\" width=\"409\" height=\"161\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q49a.2.png 409w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q49a.2-300x118.png 300w\" sizes=\"(max-width: 409px) 100vw, 409px\" \/><br \/>\nTherefore, the maximum value of Z is -12 at point A(4, 0).<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63626\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q49a.3.png\" alt=\"2nd PUC Maths Previous Year Question Paper June 2017 Q49(a).3\" width=\"415\" height=\"350\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q49a.3.png 415w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q49a.3-300x253.png 300w\" sizes=\"(max-width: 415px) 100vw, 415px\" \/><\/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 2017\" width=\"172\" height=\"16\" \/><\/p>\n<p>Question 49(b).<br \/>\nProve that: \\(\\left|\\begin{array}{ccc}1 &amp; a &amp; a^{2} \\\\1 &amp; b &amp; b^{2} \\\\1 &amp; c &amp; c^{2}\\end{array}\\right|=(a-b)(b-c)(c-a)\\)<br \/>\nSolution:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63629\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q49b.png\" alt=\"2nd PUC Maths Previous Year Question Paper June 2017 Q49(b)\" width=\"250\" height=\"384\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q49b.png 250w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q49b-195x300.png 195w\" sizes=\"(max-width: 250px) 100vw, 250px\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63628\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q49b.1.png\" alt=\"2nd PUC Maths Previous Year Question Paper June 2017 Q49(b).1\" width=\"497\" height=\"113\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q49b.1.png 497w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q49b.1-300x68.png 300w\" sizes=\"(max-width: 497px) 100vw, 497px\" \/><\/p>\n<p>Question 50(a).<br \/>\nProve that:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63631\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q50a.png\" alt=\"2nd PUC Maths Previous Year Question Paper June 2017 Q50(a)\" width=\"415\" height=\"160\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q50a.png 415w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q50a-300x116.png 300w\" sizes=\"(max-width: 415px) 100vw, 415px\" \/><br \/>\nSolution:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63630\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q50a.1.png\" alt=\"2nd PUC Maths Previous Year Question Paper June 2017 Q50(a).1\" width=\"545\" height=\"483\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q50a.1.png 545w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q50a.1-300x266.png 300w\" sizes=\"(max-width: 545px) 100vw, 545px\" \/><br \/>\nf(x) = x<sup>3<\/sup> + x cos x<br \/>\nf(-x) = (-x)<sup>3<\/sup> &#8211; x cos (-x) = -x<sup>3<\/sup> &#8211; x cos x = -[x<sup>3<\/sup> + x cos x] = -f(x)<br \/>\n\u2234 f(x) is odd<br \/>\n\u2234 \\(\\int_{-\\pi \/ 2}^{\\pi \/ 2}\\left(x^{3}+x \\cos x\\right) d x=0\\)<\/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 2017\" width=\"172\" height=\"16\" \/><\/p>\n<p>Question 50(b).<br \/>\nFor what value of \u03bb is the function defined by<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63633\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q50b.png\" alt=\"2nd PUC Maths Previous Year Question Paper June 2017 Q50(b)\" width=\"419\" height=\"64\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q50b.png 419w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q50b-300x46.png 300w\" sizes=\"(max-width: 419px) 100vw, 419px\" \/><br \/>\nSolution:<br \/>\nWhen x &lt; 0, f(x) = \u03bb(x<sup>2<\/sup> &#8211; 2x) being a polynomial is continous. When x &gt; 0, f(x) = 4x + 1 being a polynomial is continous.<br \/>\nAt x = 0<br \/>\nf(0) = \u03bb(0<sup>2<\/sup> &#8211; 2 \u00d7 0) = 0<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-63632\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q50b.1.png\" alt=\"2nd PUC Maths Previous Year Question Paper June 2017 Q50(b).1\" width=\"378\" height=\"81\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q50b.1.png 378w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2020\/06\/2nd-PUC-Maths-Previous-Year-Question-Paper-June-2017-Q50b.1-300x64.png 300w\" sizes=\"(max-width: 378px) 100vw, 378px\" \/><br \/>\n\u2234 LHL \u2260 RHL<br \/>\n\u2234 lim f(x) does not exists for x = 0<br \/>\n\u2234 f is discontinous at x = 0 for any value of \u03bb.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Students can Download 2nd PUC Maths Previous Year Question Paper June 2017, Karnataka 2nd PUC Maths Model Question Papers with Answers helps you to revise the complete Karnataka State Board Syllabus and score more marks in your examinations. Karnataka 2nd PUC Maths Previous Year Question Paper June 2017 Time: 3 Hrs 15 Min Max. Marks: &hellip;<\/p>\n","protected":false},"author":5,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[82],"tags":[],"jetpack_sharing_enabled":true,"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/kseebsolutions.guru\/wp-json\/wp\/v2\/posts\/27431"}],"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=27431"}],"version-history":[{"count":0,"href":"https:\/\/kseebsolutions.guru\/wp-json\/wp\/v2\/posts\/27431\/revisions"}],"wp:attachment":[{"href":"https:\/\/kseebsolutions.guru\/wp-json\/wp\/v2\/media?parent=27431"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/kseebsolutions.guru\/wp-json\/wp\/v2\/categories?post=27431"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/kseebsolutions.guru\/wp-json\/wp\/v2\/tags?post=27431"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}