{"id":19290,"date":"2021-02-26T11:10:37","date_gmt":"2021-02-26T05:40:37","guid":{"rendered":"https:\/\/ncertsolutions.guru\/?p=19290"},"modified":"2021-02-26T11:10:37","modified_gmt":"2021-02-26T05:40:37","slug":"mcq-questions-for-class-12-maths-chapter-9","status":"publish","type":"post","link":"https:\/\/ncertsolutions.guru\/mcq-questions-for-class-12-maths-chapter-9\/","title":{"rendered":"MCQ Questions for Class 12 Maths Chapter 9 Differential Equations with Answers"},"content":{"rendered":"

Students can access the NCERT MCQ Questions for Class 12 Maths Chapter 9 Differential Equations with Answers Pdf free download aids in your exam preparation and you can get a good hold of the chapter. Use MCQ Questions for Class 12 Maths with Answers<\/a> during preparation and score maximum marks in the exam. Students can download the Differential Equations Class 12 MCQs Questions with Answers from here and test their problem-solving skills. Clear all the fundamentals and prepare thoroughly for the exam taking help from Class 12 Maths Chapter 9 Differential Equations Objective Questions.<\/p>\n

Differential Equations Class 12 MCQs Questions with Answers<\/h2>\n

Students are advised to solve the Differential Equations Multiple Choice Questions of Class 12 Maths to know different concepts. Practicing the MCQ Questions on Differential Equations Class 12 with answers will boost your confidence thereby helping you score well in the exam.<\/p>\n

Explore numerous MCQ Questions of Differential Equations Class 12 with answers provided with detailed solutions by looking below.<\/p>\n

Question 1.
\nThe degree of the differential equation:
\n(\\(\\frac { d^2y }{dx^2}\\))\u00b3 + (\\(\\frac { dy }{dx}\\))\u00b2 + sin (\\(\\frac { dy }{dx}\\)) + 1 = 0 is
\n(a) 3
\n(b) 2
\n(c) 1
\n(d) not defined.<\/p>\n

\nAnswer<\/span><\/summary>\n

Answer: (a) 3<\/p>\n<\/details>\n

\n

Question 2.
\nThe order of the differential equation:
\n2x\u00b2 \\(\\frac { d^2y }{dx^2}\\) – 3 \\(\\frac { dy }{dx}\\) + y = 0 is
\n(a) 2
\n(b) 1
\n(c) 0
\n(d) not defined.<\/p>\n

\nAnswer<\/span><\/summary>\n

Answer: (a) 2<\/p>\n<\/details>\n

\n

Question 3.
\nThe number of arbitrary constants in the general solution of a differential equation of fourth order is:
\n(a) 0
\n(b) 2
\n(c) 3
\n(d) 4.<\/p>\n

\nAnswer<\/span><\/summary>\n

Answer: (d) 4.<\/p>\n<\/details>\n

\n

Question 4.
\nThe number of arbitrary constants in the particular solution of a differential equation of third order is:
\n(a) 3
\n(b) 2
\n(c) 1
\n(d) 0.<\/p>\n

\nAnswer<\/span><\/summary>\n

Answer: (d) 0.<\/p>\n<\/details>\n

\n

Question 5.
\nWhich of the following differential equations has y = c1<\/sub> ex<\/sup>+ c2<\/sub> e-x<\/sup> as the general solution?
\n(a) \\(\\frac { d^2y }{dx^2}\\) + y = 0
\n(b) \\(\\frac { d^2y }{dx^2}\\) – y = 0
\n(c) \\(\\frac { d^2y }{dx^2}\\) + 1 = 0
\n(d) \\(\\frac { d^2y }{dx^2}\\) – 1 = 0<\/p>\n

\nAnswer<\/span><\/summary>\n

Answer: (b) \\(\\frac { d^2y }{dx^2}\\) – y = 0<\/p>\n<\/details>\n

\n

Question 6.
\nWhich of the following differential equations has y = x as one of its particular solutions?
\n(a) \\(\\frac { d^2y }{dx^2}\\) – x\u00b2 \\(\\frac { dy }{dx}\\) + xy = x
\n(b) \\(\\frac { d^2y }{dx^2}\\) + x \\(\\frac { dy }{dx}\\) + xy = x
\n(c) \\(\\frac { d^2y }{dx^2}\\) – x\u00b2 \\(\\frac { dy }{dx}\\) + xy = 0
\n(d) \\(\\frac { d^2y }{dx^2}\\) + x \\(\\frac { dy }{dx}\\) + xy = 0<\/p>\n

\nAnswer<\/span><\/summary>\n

Answer: (c) \\(\\frac { d^2y }{dx^2}\\) – x\u00b2 \\(\\frac { dy }{dx}\\) + xy = 0<\/p>\n<\/details>\n

\n

Question 7.
\nThe general solution of the differential equation \\(\\frac { dy }{dx}\\) = ex+y<\/sup> is
\n(a) ex<\/sup> + e-y<\/sup> = c
\n(b) ex<\/sup> + ey<\/sup> = c
\n(c) e-x<\/sup> + ey<\/sup> = c
\n(d) e-x<\/sup> + e-y<\/sup> = c.<\/p>\n

\nAnswer<\/span><\/summary>\n

Answer: (a) ex<\/sup> + e-y<\/sup> = c<\/p>\n<\/details>\n

\n

Question 8.
\nWhich of the following differential equations cannot be solved, using variable separable method?
\n(a) \\(\\frac { dy }{dx}\\) + ex+y<\/sup> + e-x+y<\/sup>
\n(b) (y\u00b2 – 2xy) dx = (x\u00b2 – 2xy) dy
\n(c) xy \\(\\frac { dy }{dx}\\) = 1 + x + y + xy
\n(d) \\(\\frac { dy }{dx}\\) + y = 2.<\/p>\n

\nAnswer<\/span><\/summary>\n

Answer: (b) (y\u00b2 – 2xy) dx = (x\u00b2 – 2xy) dy<\/p>\n<\/details>\n

\n

Question 9.
\nA homogeneous differential equation of the form \\(\\frac { dy }{dx}\\) = h(\\(\\frac { x }{y}\\)) can be solved by making the substitution.
\n(a) y = vx
\n(b) v = yx
\n(c) x = vy
\n(d) x = v<\/p>\n

\nAnswer<\/span><\/summary>\n

Answer: (c) x = vy<\/p>\n<\/details>\n

\n

Question 10.
\nWhich of the following is a homogeneous differential equation?
\n(a) (4x + 6y + 5)dy – (3y + 2x + 4)dx = 0
\n(b) xy dx – (x\u00b3 + y\u00b2)dy = Q
\n(c) (x\u00b3 + 2y\u00b2) dx + 2xy dy = 0
\n(d) y\u00b2 dx + (x\u00b2 – xy – y\u00b2)dy = 0.<\/p>\n

\nAnswer<\/span><\/summary>\n

Answer: (d) y\u00b2 dx + (x\u00b2 – xy – y\u00b2)dy = 0.<\/p>\n<\/details>\n

\n

Question 11.
\nThe integrating factor of the differential equation x\\(\\frac { dy }{dx}\\) – y = 2x\u00b2 is
\n(a) e-x<\/sup>
\n(b) e-y<\/sup>
\n(c) \\(\\frac { 1 }{x}\\)
\n(d) x<\/p>\n

\nAnswer<\/span><\/summary>\n

Answer: (c) \\(\\frac { 1 }{x}\\)<\/p>\n<\/details>\n

\n

Question 12.
\nThe integrating factor of the differential equation
\n(1 – y\u00b2) \\(\\frac { dy }{dx}\\) + yx = ay(-1 < y < 1) is
\n(a) \\(\\frac { 1 }{y^2-1}\\)
\n(b) \\(\\frac { 1 }{\\sqrt{y^2-1}}\\)
\n(c) \\(\\frac { 1 }{1-y^2}\\)
\n(d) \\(\\frac { 1 }{\\sqrt{1-y^2}}\\)<\/p>\n

\nAnswer<\/span><\/summary>\n

Answer: (d) \\(\\frac { 1 }{\\sqrt{1-y^2}}\\)<\/p>\n<\/details>\n

\n

Question 13.
\nThe general solution of the differential equation \\(\\frac { y dx – x dy }{y}\\) = 0 is
\n(a) xy = c
\n(b) x = cy\u00b2
\n(c) y = cx
\n(d) y = cx\u00b2.<\/p>\n

\nAnswer<\/span><\/summary>\n

Answer: (c) y = cx<\/p>\n<\/details>\n

\n

Question 14.
\nThe general solution of a differential equation of the type \\(\\frac { dy }{dx}\\) + P1<\/sub> x = Q1<\/sub> is:
\n(a) y e\u222bp1<\/sub> dy<\/sup> = \u222b(Q1<\/sub> e\u222bp1<\/sub> dy<\/sup>) dy + c
\n(b) y e\u222bp1<\/sub> dx<\/sup> = \u222b(Q1<\/sub> e\u222bp1<\/sub> dx<\/sup>) dx + c
\n(c) x e\u222bp1<\/sub> dy<\/sup> = \u222b(Q1<\/sub> e\u222bp1<\/sub> dy<\/sup>) dy + c
\n(d) x e\u222bp1<\/sub> dx<\/sup> = \u222b(Q1<\/sub> e\u222bp1<\/sub> dx<\/sup>) dx + c<\/p>\n

\nAnswer<\/span><\/summary>\n

Answer: (c) x e\u222bp1<\/sub> dy<\/sup> = \u222b(Q1<\/sub> e\u222bp1<\/sub> dy<\/sup>) dy + c<\/p>\n<\/details>\n

\n

Question 15.
\nThe general solution of the differential equation
\nex<\/sup> dy + (y ex<\/sup> + 2x) dx = 0 is
\n(a) x ex<\/sup> + x\u00b2 = c
\n(b) x ey<\/sup> + y\u00b2 = c
\n(c) y ex<\/sup> + x\u00b2 = c
\n(d) y ex<\/sup> + x\u00b2 = c.<\/p>\n

\nAnswer<\/span><\/summary>\n

Answer: (c) y ex<\/sup> + x\u00b2 = c<\/p>\n<\/details>\n

\n

Question 16.
\nThe degree of the differential equation representing the family of curves (x – a)\u00b2 + y\u00b2 = 16 is
\n(a) 0
\n(b) 2
\n(c) 3
\n(d) 1.<\/p>\n

\nAnswer<\/span><\/summary>\n

Answer: (d) 1.<\/p>\n<\/details>\n

\n

Question 17.
\nThe degree of the differential equation
\n\\(\\frac { d^2y }{dx^2}\\) + 3(\\(\\frac { dy }{dx}\\))\u00b2 = x\u00b2 log (\\(\\frac { d^2y }{dx^2}\\)) is
\n(a) 1
\n(b) 2
\n(c) 3
\n(d) not defined<\/p>\n

\nAnswer<\/span><\/summary>\n

Answer: (d) not defined<\/p>\n<\/details>\n

\n

Question 18.
\nThe order and degree of the differential equation
\n[1 + (\\(\\frac { dy }{dx}\\))\u00b2]\u00b2 = \\(\\frac { d^2y }{dx^2}\\)
\n(a) 1, 2
\n(b) 2, 2
\n(c) 2, 1
\n(d) 4, 2.<\/p>\n

\nAnswer<\/span><\/summary>\n

Answer: (c) 2, 1<\/p>\n<\/details>\n

\n

Question 19.
\nThe solution of the differential equation:
\n2x \\(\\frac { dy }{dx}\\) – y = 3 represents a family of:
\n(a) straight lines
\n(b) circles
\n(c) parabolas
\n(d) ellipses.<\/p>\n

\nAnswer<\/span><\/summary>\n

Answer: (c) parabolas<\/p>\n<\/details>\n

\n

Question 20.
\nA solution of the differential equation:
\n(\\(\\frac { dy }{dx}\\))\u00b2 – x \\(\\frac { dy }{dx}\\) + y = 0 is
\n(a) y = 2
\n(b) y = 2x
\n(c) y = 2x – 4
\n(d) y = 2x\u00b2 – 4.<\/p>\n

\nAnswer<\/span><\/summary>\n

Answer: (c) y = 2x – 4<\/p>\n<\/details>\n

\n

Question 21.
\nThe solution of the differential equation:
\nx\\(\\frac { dy }{dx}\\) + 2y = x\u00b2 is
\n(a) y = \\(\\frac { x^2+c }{4x^2}\\)
\n(b) y = \\(\\frac { x^2 }{4}\\) + c
\n(c) y = y = \\(\\frac { x^4+c }{x^2}\\)
\n(d) y = y = \\(\\frac { x^4+c }{4x^2}\\)<\/p>\n

\nAnswer<\/span><\/summary>\n

Answer: (d) y = y = \\(\\frac { x^4+c }{4x^2}\\)<\/p>\n<\/details>\n

\n

Question 22.
\nThe differential equation of the family of circles with fixed radius 5 units and centre on the line y = 2 is
\n(a) (x – 2)\u00b2 y’\u00b2 = 25 – (y – 2)\u00b2
\n(b) (x – 2) y’\u00b2 = 25 – (y – 2)\u00b2
\n(c) (y – 2) y’\u00b2 =25 – (y – 2)\u00b2.
\n(d) (y – 2)\u00b2 y’\u00b2 = 25 – (y – 2)\u00b2.<\/p>\n

\nAnswer<\/span><\/summary>\n

Answer: (d) (y – 2)\u00b2 y’\u00b2 = 25 – (y – 2)\u00b2.
\nHint:
\nThe equation of the circle is
\n(x – c)\u00b2 + (y – 2)\u00b2 = 25 …………… (1)
\nDiff. w.r.t. x,
\n2(x – c) + 2(y – 2)y’ = 0
\n\u21d2 (x – c) = -(y – 2)y’
\nPutting in (1),
\n(y – 2)\u00b2 y’\u00b2 + (y – 2)\u00b2 = 25
\n(y – 2)\u00b2 y’\u00b2 = 25 – (y – 2)\u00b2.<\/p>\n<\/details>\n

\n

Question 23.
\nThe differential equation which represents the family of curves y = ec2<\/sub>x<\/sup>, where c1<\/sub> and c2<\/sub> are arbitrary constants, is:
\n(a) y” = y’y
\n(b) yy” = y’
\n(c) yy” = (y’)\u00b2
\n(d) y’ = y\u00b2<\/p>\n

\nAnswer<\/span><\/summary>\n

Answer: (d) y’ = y\u00b2
\nHint:
\nWe have y = c1<\/sub> ec2<\/sub>x<\/sup> …………… (1)
\nDiff. w.r.t. x, y’ = c1<\/sub>c2<\/sub> ec2<\/sub>x<\/sup>
\n\u21d2 y\u2019 = c2<\/sub>y ………… (2) [Using(1)]
\nAgain diff. w.r.t. x,
\ny\u201d = c2<\/sub>y’ …………… (3)
\nFrom (2) and (3),
\n\\(\\frac { y” }{y’}\\) = \\(\\frac { y’ }{y}\\)
\n\u21d2 yy” = (y’)\u00b2<\/p>\n<\/details>\n

\n

Question 24.
\nSolution of the differential equation:
\ncos x dy = y (sin x – y) dx, 0 < x < \\(\\frac { \u03c0 }{2}\\) is
\n(a) sec x = (tan x + c)y
\n(b) y sec x = tan x + c
\n(c) y tan x = sec x + c
\n(d) tan x = (sec x + c)y.<\/p>\n

\nAnswer<\/span><\/summary>\n

Answer: (a) sec x = (tan x + c)y
\nHint:
\nHere cos x dy =y (sin x – y)dx
\n\u21d2 cos x dy – y sin x dx = – y\u00b2 dx
\n\u21d2 d(y cos x) = – y\u00b2 dx.
\nIntegrating, \u222b\\(\\frac { d(y cos x) }{y^2 cos^2 x}\\) = -\u222b\\(\\frac { 1 }{cos^2 x}\\) dx
\n\u21d2 –\\(\\frac { 1 }{y cos x}\\) = -tan x – c
\n\u21d2 sec x = (tan x + c)y.<\/p>\n<\/details>\n

\n

Question 25.
\nAt present, a firm is manufacturing 2000 items. It is estimated that the rate of change of production P w.r.t. additional number of worker x is given by:
\n\\(\\frac { dP }{dx}\\) = 100 – 12\u221ax
\nIf the firm employs 25 more workers, then the new level of production of items is:
\n(a) 3000
\n(b) 3500
\n(c) 4500
\n(d) 2500.<\/p>\n

\nAnswer<\/span><\/summary>\n

Answer: (b) 3500
\nHint:
\nWe have: \\(\\frac { dP }{dx}\\) = 100 – 12\u221ax
\nIntegrating,
\n\\(\\int_{2000}^{P}\\) dP = \\(\\int_{0}^{25}\\) (100 – 12\u221ax)dx
\n\u21d2 [P]\\(_{2000}^{P}\\) = [100x – 12\\(\\frac { x^{3\/2} }{3\/2}\\)]\\(_{0}^{25}\\)
\n\u21d2 P – 2000 = 100(25)-8(25)3\/2<\/sup>
\n\u21d2 P – 2000 = 2500 – 1000
\n\u21d2 P = 3500.<\/p>\n<\/details>\n

\n

Fill in the Blanks<\/span><\/p>\n

Question 1.
\nThe degree of the differential equation:
\nx\u00b2(\\(\\frac { d^2y }{dx^2}\\))\u00b3 + y(\\(\\frac { dy }{dx}\\))4<\/sup> + x\u00b3 = 0 is …………….<\/p>\n

\nAnswer<\/span><\/summary>\n

Answer: 3.<\/p>\n<\/details>\n

\n

Question 2.
\nThe degree and order of the differential equation:
\n(\\(\\frac { ds }{dt}\\))4<\/sup> + 3s\\(\\frac { d^2s }{dt^2}\\) is ……………. and ………………..<\/p>\n

\nAnswer<\/span><\/summary>\n

Answer: 2, 1<\/p>\n<\/details>\n

\n

Question 3.
\nDifferential equation of the family of lines passing through the origin is …………………<\/p>\n

\nAnswer<\/span><\/summary>\n

Answer: \\(\\frac { dy }{dx}\\) = \\(\\frac { y }{x}\\).<\/p>\n<\/details>\n

\n

Question 4.
\nThe differential equation of which y = 2 (x\u00b2 – 1) + ce-x<\/sup> is a solution is ……………….<\/p>\n

\nAnswer<\/span><\/summary>\n

Answer: \\(\\frac { dy }{dx}\\) + 2xy = 4x\u00b3<\/p>\n<\/details>\n

\n

Question 5.
\nGeneral solution of (x\u00b2 + 1)\\(\\frac { dy }{dx}\\) = 2 is ………………….<\/p>\n

\nAnswer<\/span><\/summary>\n

Answer: y = 2 tan-1<\/sup> x + c.<\/p>\n<\/details>\n

\n

Question 6.
\nSolution of \\(\\frac { dy }{dx}\\) = \\(\\sqrt { 4 – y^2}\\) (- 2 < y < 2) is …………….<\/p>\n

\nAnswer<\/span><\/summary>\n

Answer: sin-1<\/sup> \\(\\frac { y }{2}\\) = x + c.<\/p>\n<\/details>\n

\n

Question 7.
\nSolution of \\(\\frac { dy }{dx}\\) = \\(\\frac { y }{x}\\) is ……………….<\/p>\n

\nAnswer<\/span><\/summary>\n

Answer: y = cx.<\/p>\n<\/details>\n

\n

Question 8.
\nThe differential equation \\(\\frac { dy }{dx}\\) = \\(\\frac { x-y }{x+y}\\) is ………………. equation.<\/p>\n

\nAnswer<\/span><\/summary>\n

Answer: homogeneous.<\/p>\n<\/details>\n

\n

Question 9.
\nThe integrating factor of x log x \\(\\frac { dy }{dx}\\) + y = 2 log x is …………………<\/p>\n

\nAnswer<\/span><\/summary>\n

Answer: log x.<\/p>\n<\/details>\n

\n

Question 10.
\nThe integrating factor of (\\(\\frac { e^{-2\u221ax} }{\u221ax}\\) – \\(\\frac { y }{\u221ax}\\))\\(\\frac { dy }{dx}\\) = 1 is …………………<\/p>\n

\nAnswer<\/span><\/summary>\n

Answer: e2\u221ax<\/sup><\/p>\n<\/details>\n

\n

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Students can access the NCERT MCQ Questions for Class 12 Maths Chapter 9 Differential Equations with Answers Pdf free download aids in your exam preparation and you can get a good hold of the chapter. Use MCQ Questions for Class 12 Maths with Answers during preparation and score maximum marks in the exam. Students can …<\/p>\n

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