{"id":19860,"date":"2020-01-16T12:50:02","date_gmt":"2020-01-16T07:20:02","guid":{"rendered":"https:\/\/kseebsolutions.guru\/?p=19860"},"modified":"2021-07-02T15:28:55","modified_gmt":"2021-07-02T09:58:55","slug":"2nd-puc-maths-question-bank-chapter-6-miscellaneous-ex","status":"publish","type":"post","link":"https:\/\/kseebsolutions.guru\/2nd-puc-maths-question-bank-chapter-6-miscellaneous-ex\/","title":{"rendered":"2nd PUC Maths Question Bank Chapter 6 Application of Derivatives Miscellaneous Exercise"},"content":{"rendered":"

Students can Download Maths Chapter 6 Application of Derivatives Miscellaneous Exercise Questions and Answers, Notes Pdf, 2nd PUC Maths Question Bank with Answers<\/a>\u00a0helps you to revise the complete Karnataka State Board Syllabus and score more marks in your examinations.<\/p>\n

Karnataka 2nd PUC Maths Question Bank Chapter 6 Application of Derivatives Miscellaneous Exercise<\/h2>\n

Question 1.
\nUsing differentials, find the approximate value of each of the following<\/p>\n

(a) \\(\\left(\\frac{17}{81}\\right)^{1 \/ 4}\\)
\nAnswer:
\n\"2nd<\/p>\n

\"KSEEB<\/p>\n

(b) (33)-1\/5<\/sup>
\nAnswer:
\n\"2nd
\n\"2nd<\/p>\n

Question 2.
\nShow that the function given by log has maximum at \\(f(x)=\\frac{\\log x}{x}\\) has maximum at x = e
\nAnswer:
\n\"2nd<\/p>\n

Question 3.
\nThe two equal sides of an isosceles triangle with fixed base b are decreasing at the rate of 3 cm per second. How fast is the area decreasing when the two equal sides are equal to the base ?
\nAnswer:
\n\"2nd
\n\"2nd<\/p>\n

\"2nd<\/p>\n

Question 4.
\nFind the equation of the normal to curve y2<\/sup> = 4x which passes through the point (1, 2).
\nAnswer:
\n\"2nd<\/p>\n

Question 5.
\nShow that the normal at any point 6, to the curve x = a cos \u03b8 +a \u03b8 sin \u03b8 ,y = a sin \u03b8 –
\na \u03b8 cos \u03b8 is at a constant distance from the origin.
\nAnswer:
\n\"2nd
\n\u2234 equation of normal is
\ny – (a sin \u03b8 – a \u03b8 cos \u03b8)
\n= – cot \u03b8 (x – a cos \u03b8 – a \u03b8 sin \u03b8)
\ny + x cot \u03b8= a cot \u03b8 cos \u03b8+a \u03b8 cot \u03b8 sin \u03b8 – a \u03b8 cos \u03b8
\n\\(y+\\frac{x \\cos \\theta}{\\sin \\theta}=\\frac{a \\cos ^{2} \\theta}{\\sin \\theta}+\\frac{a \\sin ^{2} \\theta}{\\sin \\theta}\\)
\nx cos \u03b8 a cos2<\/sup>\u03b8, a sin2<\/sup>\u03b8
\nnormal form is x cos \u03b8+ y sin \u03b8 = p when p is the normal from the normal to the given curve is at a consistent distance \u2018a\u2019 from the origin.<\/p>\n

\"KSEEB<\/p>\n

Question 6.
\nFind the intervals in which the function f given by
\n\\(f(x)=\\frac{4 \\sin x-2 x-x \\cos x}{2+\\cos x}\\)
\n(i) increasing
\n(ii) decreasing
\nAnswer:
\n\"2nd<\/p>\n

Question 7.
\nFind the intervals in which the function f given by
\nx3<\/sup> + -1\/x3<\/sup>, x \u2260 0
\n(i) increasing
\n(ii) decreasing
\nAnswer:
\n\"2nd<\/p>\n

\"2nd<\/p>\n

\"KSEEB<\/p>\n

Question 8.
\nFind the maximum area of an isosceles triangle inscribed in the ellipse
\n\\(\\frac{x^{2}}{a^{2}}+\\frac{y^{2}}{b^{2}}=1\\) with its vertex at one end of the major axis.
\nAnswer:
\n\"2nd
\n\"2nd<\/p>\n

\"2nd<\/p>\n

Question 9.
\n‘A tank with rectangular base and rectangular sides, open at the top is to be constructed so that its depth is 2 m and volume is 8 m3<\/sup>. If building of tank costs \u20b9 70 per sq metres for the base and \u20b9 45 per square metre for sides. What is the cost of least expensive tank?
\nAnswer:
\n\"2nd<\/p>\n

\"KSEEB<\/p>\n

Question 10.
\nThe sum of the perimeter of a circle and square is k, where k is some constant. Prove that the sum of their areas is least when the side of square is double the radius of the circle.
\nAnswer:
\n\"2nd
\n\"2nd<\/p>\n

Question 11.
\nA window is in the form of a rectangle surmounted by a semicircular opening. The total perimeter of the window is 10 m. Find the dimensions of the window to admit maximum light through the whole opening.
\nAnswer:
\n\"2nd
\n\"2nd<\/p>\n

\"KSEEB<\/p>\n

Question 12.
\nA point on the hypotenuse of a triangle is at distance a and b from the sides of the triangle. Show that the maximum length of the hypotenuse is
\n\\(\\left(a^{\\frac{3}{2}}+b^{\\frac{2}{3}}\\right)^{\\frac{3}{2}}\\)
\nAnswer:
\n\"2nd
\n\"2nd
\n\"2nd<\/p>\n

Question 13.
\nFind the points at which the function f given by f (x) = (x – 2)4<\/sup> (x + 1)3<\/sup> has
\n(i) local maxima
\n(ii) local minima
\n(iii) point of inflexion
\nAnswer:
\nf (x) = (x – 2)4<\/sup> (x + 1)3
\n<\/sup>f’ (x) = (x – 2)4<\/sup> 3 (x + 1)2<\/sup> + (x + 1)3<\/sup> 4(x – 2)3
\n<\/sup>= (x – 2)3<\/sup> (x + 1) [3 (x -2) + 4 (x + 1)]
\n= (x – 2)3<\/sup> (x + 1)2<\/sup> [3x – 6 + 4x + 4]
\n= (x – 2)3<\/sup> (x + 1)2<\/sup> [7x – 2]
\nf’ (x) = 0 \u21d2 x = 2, x = -1, x = 2\/7
\nf’ (x)=(x – 2)3<\/sup> (x+1)2<\/sup> x 7 + (x – 2)3<\/sup> (7x – 2) 2 (x +1) + (x + 1)2<\/sup> (7x – 2)3 (x – 2)2
\n<\/sup>here second demivalive test fails as f” (x) = 0 then we apply first demivalive test
\nAt x = 2 f'(x) changes from negative to positive.
\n\u2234 function has minimum value and the local minimum value is f (2) = 0
\nAt x = -1
\nf’ (x) does not change (positive to negative)
\n\u2234 fx<\/sup> has neither maximum value nor minimum value.
\n\u2234 x = -1 is the point of inflection at x = 2\/7
\nf’ (x) change from positive to negative
\nhence function has been maximum at x = 2\/7 and local maximum value is
\n\"2nd<\/p>\n

\"KSEEB<\/p>\n

Question 14.
\nFind the absolute maximum and minimum values of the function f given by
\nf (x) = cos2<\/sup> x + sin x, x \u2208 [o,\u03c0]
\nAnswer:
\n\"2nd<\/p>\n

Question 15.
\nShow that the altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius r is \\(\\frac{4 r}{3}\\)
\nAnswer:
\nLet the radius of sphere be ‘r’
\n\"2nd
\n\"2nd<\/p>\n

Question 16.
\nLet f be a function defined on [a, b] such that f’ (x) > 0, for all xe (a, b). Then prove that f is an increasing function on (a, b).
\nAnswer:
\nLet x1<\/sub>, x2<\/sub> be any two real number [a, b] such that x1 <\/sub><\u00a0 x2<\/sub>, then f (x) satisfies the conditions of L.M.V theorem. \u220b C \u2208\u00a0 (a, b)
\n\"2nd<\/p>\n

Question 17.
\nShow that the height of the cylinder of maximum volume that can be inscribed in sphere of radius R is \\(\\frac{2 \\mathbf{R}}{\\sqrt{3}}\\). Also find the maximum volume.
\nAnswer:
\nLet r be the radius of the cylinder and height 2x
\n\"2nd<\/p>\n

\"KSEEB<\/p>\n

Question 18.
\nShow that height of the cylinder of greatest volume which can be inscribed in a right circular cone of height h and semi vertical angle a is one-third that of the cone and \u03b1 the greatest volume of cylinder is \\(\\frac{4}{27} \\pi \\mathrm{h}^{3}\\)
\nAnswer:
\nHeight of cone – h
\nRadius of cone – r
\nHeight of cylinder – y
\nRadius of cylinder – x
\n\"2nd
\n\"2nd
\n\"2nd
\nChoose the correct answer in the Exercises from 19 to 24.<\/p>\n

Question 19.
\nA cylindrical tank of radius 10 m is being filled with wheat at the rate of 314 cubic metre per hour. Then the depth of the wheat is increasing at the rate of
\n(A) 1 m3<\/sup>\/h
\n(B) 0.1 m3<\/sup>h
\n(C) 1.1 m3<\/sup>\/h
\n(D) 0.5 m3<\/sup>\/h
\nAnswer:
\n\"2nd<\/p>\n

Question 20.
\nThe slope of the tangent to the curve x = t2<\/sup> + 3t – 8, y = 2t2<\/sup> – 2t – 5 at the point (2,-1) is
\n(A) \\(\\frac{22}{7}\\)
\n(B) \\(\\frac{6}{7}\\)
\n(C) \\(\\frac{7}{6}\\)
\n(D)\\(\\frac{-6}{7}\\)
\nAnswer:
\n\"2nd<\/p>\n

Question 21.
\nThe line y = mx + 1 is a tangent to the curve y2<\/sup> = 4x if the value of m is\u00a0 ………..
\n(A) 1
\n(B) 2
\n(C) 3
\n(D) \\(\\frac{1}{2}\\)
\nAnswer:
\n\"2nd
\n\"2nd<\/p>\n

\"KSEEB<\/p>\n

Question 22.
\nThe normal at the point (1,1) on the curve 2y + x2<\/sup> = 3 is ……………….
\n(A) x + y = 0
\n(B) x – y = 0
\n(C) x + y = 1
\n(D) x – y = 1
\nAnswer:
\n\"2nd<\/p>\n

Question 23.
\nThe normal to the curve x2<\/sup> = 4y passing (1,2) is
\n(A) x + y = 3
\n(B) x – y = 3
\n(C) x + y = 1
\n(D) x – y = 1
\nAnswer:
\n\"2nd
\n\"2nd<\/p>\n

Question 24.
\nThe points on the curve 9y2<\/sup> = x3<\/sup>, where the normal to the curve makes equal intercepts with the axes are ………………
\n(A) \\(\\left(4, \\pm \\frac{8}{3}\\right)\\)
\n(B) \\(\\left(4, \\frac{-8}{3}\\right)\\)
\n(C) \\(\\left(4,+\\frac{3}{8}\\right)\\)
\n(D) \\(\\left(\\pm 4, \\frac{8}{3}\\right)\\)
\nAnswer:
\n\"2nd<\/p>\n

2nd PUC Maths Application of Derivatives Miscellaneous Exercise Additional Questions and Answers<\/h3>\n

Question 1.
\nIf the length of the three sides of a trapezium other than the base is 10cm, find the area of the trapezium when it is maximum (CBSE 2010)
\nAnswer:
\n\"2nd
\n\"2nd
\n\"2nd<\/p>\n

Question 2.
\nFind the intervals in which the function sinx + cos x x \u2208 [0,\u03c0] is (1) strictly increasing (2) strictly decreasing. (CBSE 2010)
\nAnswer:
\n\"2nd<\/p>\n

\"KSEEB<\/p>\n

Question 3.
\nIf the radius of a sphere is measured as 9 cm with an error of 0.03 cm, find the apps error in. Calculating its surface area, (CBSE 2011)
\nAnswer:
\n\"2nd<\/p>\n

Question 4.
\nFind the relationship between a and b so that the function of defined by
\n\\(f(x)=\\left\\{\\begin{array}{l}{a x+1, \\text { If } x \\leq 3 \\text { is continous at } x=3} \\\\{b x+3 \\text { if } x>3} \\end{array}\\right.\\)
\nAnswer:
\n\"PUC<\/p>\n

Question 5.
\nAn open box with a square box is to be made out of a given quantity of sheet of area a2<\/sup>. Show that the maximum volume of the box is \\(a^{3} \/ 6 \\sqrt{3}\\) (PUC 2011)
\nAnswer:
\n\"2nd
\n\"2nd<\/p>\n

Question 6.
\nShow that the rectangle of maximum perimeter which can be insenibed in a circle of radius a is a square of side \\(\\sqrt{2}\\) a. (CBSE 2008)
\nAnswer:
\n\"2nd
\n\"2nd
\n\"2nd<\/p>\n

\"KSEEB<\/p>\n

Question 7.
\nDiscuss the continuity of
\n\\(f(x)=\\left\\{\\begin{array}{cl}{\\frac{1-\\cos x}{x^{2}}} & {x \\neq 0} \\\\{1} & {x=0} \\end{array}\\right.\\) (KPUC 2008)
\nAnswer:
\n\"2nd<\/p>\n","protected":false},"excerpt":{"rendered":"

Students can Download Maths Chapter 6 Application of Derivatives Miscellaneous Exercise Questions and Answers, Notes Pdf, 2nd PUC Maths Question Bank with Answers\u00a0helps you to revise the complete Karnataka State Board Syllabus and score more marks in your examinations. Karnataka 2nd PUC Maths Question Bank Chapter 6 Application of Derivatives Miscellaneous Exercise Question 1. Using …<\/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\/19860"}],"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=19860"}],"version-history":[{"count":0,"href":"https:\/\/kseebsolutions.guru\/wp-json\/wp\/v2\/posts\/19860\/revisions"}],"wp:attachment":[{"href":"https:\/\/kseebsolutions.guru\/wp-json\/wp\/v2\/media?parent=19860"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/kseebsolutions.guru\/wp-json\/wp\/v2\/categories?post=19860"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/kseebsolutions.guru\/wp-json\/wp\/v2\/tags?post=19860"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}