12th Mathematics - Book Back Answers - Chapter 10 - English Medium Guides

  

 

Plus Two / 12th Mathematics - Book Back Answers - Chapter 10 - English Medium

Tamil Nadu Board 12th Standard Mathematics - Chapter 10: Book Back Answers and Solutions

    This post covers the book back answers and solutions for Chapter 10 from the Tamil Nadu State Board 12th Standard Mathematics textbook. These detailed answers have been carefully prepared by our expert teachers at KalviTips.com.

    We have explained each answer in a simple, easy-to-understand format, highlighting important points step by step under the relevant subtopics. Students are advised to read and memorize these subtopics thoroughly. Once you understand the main concepts, you’ll be able to connect other related points with real-life examples and confidently present them in your tests and exams.

    By going through this material, you’ll gain a strong understanding of Chapter 10 along with the corresponding book back questions and answers (PDF format).

Question Types Covered:

• 1 Mark Questions: Choose the correct answer, Fill in the blanks, Identify the correct statement, Match the following 
• 2 Mark Questions: Answer briefly 
• 3, 4, and 5 Mark Questions: Answer in detail

All answers are presented in a clear and student-friendly manner, focusing on key points to help you score full marks.

All the best, Class 12 students! Prepare well and aim for top scores. Thank you!

 

I. Multiple Choice Questions

1. The order and degree of the differential equation $\frac{{\mathbf{d}}^{\mathbf{2}}\mathbf{y}}{\mathbf{d}{\mathbf{x}}^{\mathbf{2}}}\mathbf{+}$$\left(\begin{array}{c}\mathrm{<math\; xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi\; mathvariant="bold">d</mi><mi\; mathvariant="bold">y</mi></mrow><mrow><mi\; mathvariant="bold">d</mi><mi\; mathvariant="bold">x</mi></mrow></mfrac></math>}\end{array}\right)$^1/2 +x^1/4 =0 are respectively
(1) 2, 3 
(2) 3, 3 
(3) 2, 6 
(4) 2, 4
Answer Key:
(1) 2, 3 

 
2. The differential equation representing the family of curves y = Acos(x + B), where A and B
are parameters, is
(1) $\frac{{\mathbf{d}}^{\mathbf{2}}\mathbf{y}}{\mathbf{d}{\mathbf{x}}^{\mathbf{2}}}$−y$\frac{{\mathbf{=0}}^{}}{}$
(2) $\frac{{\mathbf{d}}^{\mathbf{2}}\mathbf{y}}{\mathbf{d}{\mathbf{x}}^{\mathbf{2}}}$+y$\frac{{\mathbf{=0}}^{}}{}$
(3) $\frac{{\mathbf{d}}^{\mathbf{2}}\mathbf{y}}{\mathbf{d}{\mathbf{x}}^{\mathbf{2=0}}}$
(4) $\frac{{\mathbf{d}}^{\mathbf{2}}\mathbf{x}}{\mathbf{d}{\mathbf{y}}^{\mathbf{2}}}$$\frac{{\mathbf{=0}}^{}}{}$
Answer Key:
(2) $\frac{{\mathbf{d}}^{\mathbf{2}}\mathbf{y}}{\mathbf{d}{\mathbf{x}}^{\mathbf{2}}}$+y$\frac{{\mathbf{=0}}^{}}{}$

$\frac{{\mathbf{ }}^{}}{}$
3. The order and degree of the differential equation $\sqrt{\mathbf{sin}\mathbf{x}}$(dx + dy) = $\sqrt{\mathbf{cos}\mathbf{x}}$(dx − dy) is
(1) 1, 2 
(2) 2, 2 
(3) 1, 1 
(4) 2, 1
Answer Key:
(3) 1, 1 

 
4. The order of the differential equation of all circles with centre at (h, k ) and radius ‘a’ is
(1) 2 
(2) 3 
(3) 4 
(4) 1
Answer Key:
(2) 3 

 
5. The differential equation of the family of curves y = Ae^x + Be^-x , where A and B are arbitrary
constants is
(1) $\frac{{\mathbf{d}}^{\mathbf{2}}\mathbf{y}}{\mathbf{d}{\mathbf{x}}^{\mathbf{2+y=0}}}$
(2) $\frac{{\mathbf{d}}^{\mathbf{2}}\mathbf{y}}{\mathbf{d}{\mathbf{x}}^{\mathbf{2-y=0}}}$
(3) $\frac{\mathbf{d}\mathbf{y}}{\mathbf{d}\mathbf{x+y=0}}$
(4) $\frac{\mathbf{d}\mathbf{y}}{\mathbf{d}\mathbf{x-y=0}}$
Answer Key:
(2) $\frac{{\mathbf{d}}^{\mathbf{2}}\mathbf{y}}{\mathbf{d}{\mathbf{x}}^{\mathbf{2-y=0}}}$

$\frac{{\mathbf{ }}^{}}{}$
6. The general solution of the differential equation $\begin{array}{c}\mathrm{<math\; xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi\; mathvariant="bold">d</mi><mi\; mathvariant="bold">y</mi></mrow><mrow><mi\; mathvariant="bold">d</mi><mi\; mathvariant="bold">x</mi></mrow></mfrac></math>}\end{array}$$\frac{}{}$=$\frac{\mathbf{y}}{\mathbf{x}}$ is
(1) xy = k 
(2) y = k log x 
(3) y = kx 
(4) log y = kx
Answer Key:
(3) y = kx 

 
7. The solution of the differential equation 2x $\begin{array}{c}\mathrm{<math\; xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi\; mathvariant="bold">d</mi><mi\; mathvariant="bold">y</mi></mrow><mrow><mi\; mathvariant="bold">d</mi><mi\; mathvariant="bold">x</mi></mrow></mfrac></math>}\end{array}$$\frac{}{}$-y=3 represents
(1) straight lines 
(2) circles 
(3) parabola 
(4) ellipse
Answer Key:
(3) parabola

 
8. The solution of $\begin{array}{c}\mathrm{<math\; xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi\; mathvariant="bold">d</mi><mi\; mathvariant="bold">y</mi></mrow><mrow><mi\; mathvariant="bold">d</mi><mi\; mathvariant="bold">x</mi></mrow></mfrac></math>}\end{array}$$\frac{}{}$ + p(x) y = 0 is
(1) $\mathbf{y}\mathbf{=}\mathbf{c}{\mathbf{e}}^{\mathbf{\int }\mathbf{p}\mathbf{d}\mathbf{x}}$
(2) $\mathbf{y}\mathbf{=}\mathbf{c}{\mathbf{e}}^{\mathbf{-}\mathbf{\int }\mathbf{p}\mathbf{d}\mathbf{x}}$
(3) $\mathbf{y}\mathbf{=}\mathbf{c}{\mathbf{e}}^{\mathbf{-}\mathbf{\int }\mathbf{p}\mathbf{d}\mathbf{y}}$
(4) $\mathbf{y}\mathbf{=}\mathbf{c}{\mathbf{e}}^{\mathbf{\int }\mathbf{p}\mathbf{d}\mathbf{y}}$
Answer Key:
(2) $\mathbf{y}\mathbf{=}\mathbf{c}{\mathbf{e}}^{\mathbf{-}\mathbf{\int }\mathbf{p}\mathbf{d}\mathbf{x}}$

${\mathbf{ }}^{}$
9. The integrating factor of the differential equation $\begin{array}{c}\mathrm{<math\; xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi\; mathvariant="bold">d</mi><mi\; mathvariant="bold">y</mi></mrow><mrow><mi\; mathvariant="bold">d</mi><mi\; mathvariant="bold">x</mi></mrow></mfrac></math>}\end{array}$ + y = $\frac{\mathbf{1}\mathbf{+}\mathbf{y}}{\mathbf{\lambda }}$ is
(1) $\frac{\mathbf{x}}{{\mathbf{e}}^{\mathbf{\lambda }}}$
(2) $\frac{{\mathbf{e}}^{\mathbf{\lambda }}}{\mathrm{x}}$
(3) $\mathbf{\lambda }{\mathbf{e}}^{\mathbf{x}}$ 
(4) ${\mathbf{e}}^{\mathbf{x}}$
Answer Key:
(2) $\frac{{\mathbf{e}}^{\mathbf{\lambda }}}{\mathrm{x}}$

$\frac{\mathrm{ }}{}$
10. The integrating factor of the differential equation $\begin{array}{c}\mathrm{<math\; xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi\; mathvariant="bold">d</mi><mi\; mathvariant="bold">y</mi></mrow><mrow><mi\; mathvariant="bold">d</mi><mi\; mathvariant="bold">x</mi></mrow></mfrac></math>}\end{array}$ + P(x) y = Q(x) is x , then P(x)
(1) x 
(2) $\frac{{\mathbf{x}}^{\mathbf{2}}}{\mathbf{2}}$
(3) $\frac{\mathbf{1}}{\mathbf{x}}$
(4) $\frac{\mathbf{1}}{{\mathbf{x}}^{\mathbf{2}}}$
Answer Key:
(3) $\frac{\mathbf{1}}{\mathbf{x}}$

$\frac{\mathbf{ }}{}$
11. The degree of the differential equation y(x)=1+$\begin{array}{c}\mathrm{<math\; xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi\; mathvariant="bold">d</mi><mi\; mathvariant="bold">y</mi></mrow><mrow><mi\; mathvariant="bold">d</mi><mi\; mathvariant="bold">x</mi></mrow></mfrac></math>}\end{array}$$\frac{}{}$+$\frac{\mathbf{1}}{\mathbf{1.2}}$$\left(\begin{array}{c}\mathrm{<math\; xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi\; mathvariant="bold">d</mi><mi\; mathvariant="bold">y</mi></mrow><mrow><mi\; mathvariant="bold">d</mi><mi\; mathvariant="bold">x</mi></mrow></mfrac></math>}\end{array}\right)$² + $\frac{\mathbf{1}}{\mathbf{1.2}\mathbf{.}\mathbf{3}}$$\frac{}{}$$\frac{\mathbf{}}{}$$\left(\begin{array}{c}\mathrm{<math\; xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi\; mathvariant="bold">d</mi><mi\; mathvariant="bold">y</mi></mrow><mrow><mi\; mathvariant="bold">d</mi><mi\; mathvariant="bold">x</mi></mrow></mfrac></math>}\end{array}\right)$³ +... is

(1) 2 
(2) 3 
(3) 1 
(4) 4
Answer Key:
(3) 1 

 
12. If p and q are the order and degree of the differential equation y $\begin{array}{c}\mathrm{<math\; xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi\; mathvariant="bold">d</mi><mi\; mathvariant="bold">y</mi></mrow><mrow><mi\; mathvariant="bold">d</mi><mi\; mathvariant="bold">x</mi></mrow></mfrac></math>}\end{array}$+x³+xy=cosx,
when
(1) p < q 
(2) p = q 
(3) p > q 
(4) p exists and q does not exist
Answer Key:
(3) p > q

 
13. The solution of the differential equation$\begin{array}{c}\mathrm{<math\; xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi\; mathvariant="bold"> d</mi><mi\; mathvariant="bold">y</mi></mrow><mrow><mi\; mathvariant="bold">d</mi><mi\; mathvariant="bold">x </mi></mrow></mfrac></math>}\end{array}$+$\mathbf{ }\frac{\mathbf{1}}{\sqrt{\mathbf{1}\mathbf{-}{\mathbf{x}}^{\mathbf{2}}}}$=0 is 
(1) y + sin^-1 x=c
(2) x +  sin^-1 y=0
(3) y² + 2sin^-1 x=c
(4) x² + sin^-1y=0
Answer Key:
(1) y + sin^-1 x=c

 
14. The solution of the differential equation $\begin{array}{c}\mathrm{<math\; xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi\; mathvariant="bold">d</mi><mi\; mathvariant="bold">y</mi></mrow><mrow><mi\; mathvariant="bold">d</mi><mi\; mathvariant="bold">x</mi></mrow></mfrac></math>}\end{array}$= 2xy is
(1) y = Ce^x2 
(2) y = 2^x2 +C 
(3) y = Ce−^-x2 +C 
(4) y = ^x2 +C
Answer Key:
(1) y = Ce^x2

 
15. The general solution of the differential equation log$(\begin{array}{c}\mathrm{<math\; xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi\; mathvariant="bold">d</mi><mi\; mathvariant="bold">y</mi></mrow><mrow><mi\; mathvariant="bold">d</mi><mi\; mathvariant="bold">x</mi></mrow></mfrac></math>}\end{array})=x+y$ is
(1) e^x + e^y = C 
(2) e^x + e^-y = C 
(3) e^-x + e^y = C 
(4) e^-x + e^-y = C
Answer Key:
(2) e^x + e^-y = C

 
16. The solution of $\begin{array}{c}\mathrm{<math\; xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi\; mathvariant="bold">d</mi><mi\; mathvariant="bold">y</mi></mrow><mrow><mi\; mathvariant="bold">d</mi><mi\; mathvariant="bold">x </mi></mrow></mfrac></math>}\end{array}$= 2^y-x is
(1) ${\mathbf{2}}^{\mathbf{x}}\mathbf{+}{\mathbf{2}}^{\mathbf{y}}$ = C 
(2) ${\mathbf{2}}^{\mathbf{x}}-{\mathbf{2}}^{\mathbf{y}}$ = C 
(3) $\frac{\mathbf{1}}{{\mathbf{2}}^{\mathbf{x}}}\mathbf{-}\frac{\mathbf{1}}{{\mathbf{2}}^{\mathbf{y}}}$ = C 
(4) x+y = C
Answer Key:
(3) $\frac{\mathbf{1}}{{\mathbf{2}}^{\mathbf{x}}}\mathbf{-}\frac{\mathbf{1}}{{\mathbf{2}}^{\mathbf{y}}}$ = C 

 
17. The Solution of the differential equation 

(1)x$\begin{array}{c}\mathrm{<math\; xmlns="http://www.w3.org/1998/Math/MathML"><mi\; mathvariant="bold-italic">\varphi </mi></math>}\end{array}$$\left(\begin{array}{c}\mathrm{<math\; xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi\; mathvariant="bold"><math\; xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi\; mathvariant="bold-italic">y</mi><mi\; mathvariant="bold-italic">x</mi></mfrac></math></mi></mrow></mfrac></math>}\end{array}\right)$$=k$

(2)x$\begin{array}{c}\mathrm{<math\; xmlns="http://www.w3.org/1998/Math/MathML"><mi\; mathvariant="bold-italic">\varphi </mi></math>}\end{array}$$\left(\begin{array}{c}\mathrm{<math\; xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi\; mathvariant="bold"><math\; xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi\; mathvariant="bold-italic">y</mi><mi\; mathvariant="bold-italic">x</mi></mfrac></math></mi></mrow></mfrac></math>}\end{array}\right)$$=kx$ 
(3)y$\begin{array}{c}\mathrm{<math\; xmlns="http://www.w3.org/1998/Math/MathML"><mi\; mathvariant="bold-italic">\varphi </mi></math>}\end{array}$$\left(\begin{array}{c}\mathrm{<math\; xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi\; mathvariant="bold"><math\; xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi\; mathvariant="bold-italic">y</mi><mi\; mathvariant="bold-italic">x</mi></mfrac></math></mi></mrow></mfrac></math>}\end{array}\right)$$=k$ 
(4)$\begin{array}{c}\mathrm{<math\; xmlns="http://www.w3.org/1998/Math/MathML"><mi\; mathvariant="bold-italic">\varphi </mi></math>}\end{array}$$\left(\begin{array}{c}\mathrm{<math\; xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi\; mathvariant="bold"><math\; xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi\; mathvariant="bold-italic">y</mi><mi\; mathvariant="bold-italic">x</mi></mfrac></math></mi></mrow></mfrac></math>}\end{array}\right)$$=k$y 
Answer Key:
(2)x$\begin{array}{c}\mathrm{<math\; xmlns="http://www.w3.org/1998/Math/MathML"><mi\; mathvariant="bold-italic">\varphi </mi></math>}\end{array}$$\left(\begin{array}{c}\mathrm{<math\; xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi\; mathvariant="bold"><math\; xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi\; mathvariant="bold-italic">y</mi><mi\; mathvariant="bold-italic">x</mi></mfrac></math></mi></mrow></mfrac></math>}\end{array}\right)$$=kx$ 

 
18. If sin x is the integrating factor of the linear differential equation $\begin{array}{c}\mathrm{<math\; xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi\; mathvariant="bold">d</mi><mi\; mathvariant="bold">y</mi></mrow><mrow><mi\; mathvariant="bold">d</mi><mi\; mathvariant="bold">x</mi></mrow></mfrac></math>}\end{array}$ + Py = Q, then P is
(1) log sin x 
(2) cos x 
(3) tan x 
(4) cot x
Answer Key:
(4) cot x

 
19. The number of arbitrary constants in the general solutions of order n and n +1are respectively
(1) n −1, n 
(2) n, n +1 
(3) n +1, n + 2 
(4) n +1, n
Answer Key:
(2) n, n +1

 
20. The number of arbitrary constants in the particular solution of a differential equation of third
order is
(1) 3 
(2) 2 
(3) 1 
(4) 0
Answer Key:

(4) 0

 

21. Integrating factor of the differential equation $\begin{array}{c}\mathrm{<math\; xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi\; mathvariant="bold">d</mi><mi\; mathvariant="bold">y</mi></mrow><mrow><mi\; mathvariant="bold">d</mi><mi\; mathvariant="bold">x</mi></mrow></mfrac></math>}\end{array}$= $\frac{\mathit{x}\mathbf{+}\mathit{y}\mathbf{+}\mathbf{1}}{\mathit{x}\mathbf{+}\mathbf{1}}$ is
(1) $\frac{\mathbf{1}}{\mathit{x}\mathbf{+}\mathbf{1}}$
(2) x+1
(3) $\frac{\mathbf{1}}{\sqrt{\mathit{x}\mathbf{+}\mathbf{1}}}$
(4) $\sqrt{\mathit{x}\mathbf{+}\mathbf{1}}$
Answer Key:
(1) $\frac{\mathbf{1}}{\mathit{x}\mathbf{+}\mathbf{1}}$

$\frac{\mathbf{ }}{}$
22. The population P in any year t is such that the rate of increase in the population is proportional
to the population. Then
(1) P = $\mathit{C}{\mathit{e}}^{\mathit{k}\mathit{t}}$
(2) P = $\mathit{C}{\mathit{e}}^{\mathbf{-}\mathit{k}\mathit{t}}$
(3) P = Ckt 
(4) P = C
Answer Key:
(1) P = $\mathit{C}{\mathit{e}}^{\mathit{k}\mathit{t}}$

${\mathit{ }}^{}$
23. t is the amount of certain substance left in after time t. If the rate of evaporation of the
substance is proportional to the amount remaining, then
(1) P = $\mathit{C}{\mathit{e}}^{\mathit{k}\mathit{t}}$
(2) P = $\mathit{C}{\mathit{e}}^{\mathbf{-}\mathit{k}\mathit{t}}$ 
(3) P = Ckt 
(4) Pt = C
Answer Key:
(2) P = $\mathit{C}{\mathit{e}}^{\mathbf{-}\mathit{k}\mathit{t}}$ 

 
24. If the solution of the differential equation $\begin{array}{c}\mathrm{<math\; xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi\; mathvariant="bold">d</mi><mi\; mathvariant="bold">y</mi></mrow><mrow><mi\; mathvariant="bold">d</mi><mi\; mathvariant="bold">x</mi></mrow></mfrac></math>}\end{array}$=$\frac{\mathbf{a}\mathbf{x}\mathbf{+}\mathbf{3}}{\mathbf{2}\mathbf{y}\mathbf{+}\mathbf{f}}$ represents a circle, then the value of
a is
(1) 2 
(2) −2 
(3) 1 
(4) −1
Answer Key:
(2) −2 

 
25. The slope at any point of a curve y = f (x) is given by$\begin{array}{c}\mathrm{<math\; xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi\; mathvariant="bold"> d</mi><mi\; mathvariant="bold">y</mi></mrow><mrow><mi\; mathvariant="bold">d</mi><mi\; mathvariant="bold">x</mi></mrow></mfrac></math>}\end{array}$ = 3x² and it passes through (-1,1).
Then the equation of the curve is
(1) y = x³ + 2 
(2) y = 3x² + 4 
(3) y = 3x³ + 4 
(4) y = x³ + 5
Answer Key:
(1) y = x³ + 2

 

II.Short Answer Questions.

12th Mathematics

III. Long Answer Questions.

12th Mathematics

IV. Exercise.

12th Mathematics