12th Business Mathematics and Statistics - Book Back Answers - Chapter 2 - English Medium Guides

 

 

Plus Two / 12th Business Mathematics and Statistics - Book Back Answers - Chapter 2 - English Medium

Tamil Nadu Board 12th Standard Business Mathematics and Statistics - Chapter 2: Book Back Answers and Solutions

    This post covers the book back answers and solutions for Chapter 2 – Business Mathematics and Statistics from the Tamil Nadu State Board 12th Standard Business Mathematics and Statistics 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 Business Mathematics and Statistics  Chapter 2 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!

Chapter 2:

I. Multiple Choice Questions

1. $\mathbf{\int }\frac{\mathbf{1}}{{\mathbf{x}}^{\mathbf{3}}}\mathbf{d}\mathbf{x}$ is

(a)$\frac{\mathbf{-}\mathbf{3}}{{\mathbf{x}}^{\mathbf{2}}}\mathbf{+}\mathbf{c}$

(b)$\frac{\mathbf{-}\mathbf{1}}{\mathbf{2}{\mathbf{x}}^{\mathbf{2}}}\mathbf{+}\mathbf{c}$

(c)$\frac{\mathbf{-}\mathbf{1}}{\mathbf{3}{\mathbf{x}}^{\mathbf{2}}}\mathbf{+}\mathbf{c}$

(d)$\frac{\mathbf{-}\mathbf{2}}{{\mathbf{x}}^{\mathbf{2}}}\mathbf{+}\mathbf{c}$

Answer Key:

(b)$\frac{\mathbf{-}\mathbf{1}}{\mathbf{2}{\mathbf{x}}^{\mathbf{2}}}\mathbf{+}\mathbf{c}$

$\mathbf{ }$ 

2. $\mathbf{\int }{\mathbf{2}}^{\mathbf{x}}\mathbf{d}\mathbf{x}$ is

(a)${\mathbf{2}}^{\mathbf{xlog2+c}}$

(b)${\mathbf{2}}^{\mathbf{x+c}}$

(c)$\frac{{\mathbf{2}}^{\mathbf{x}}}{\mathbf{log}\mathbf{2}}\mathbf{+}\mathbf{c}$

(d)$\frac{\mathbf{log}\mathbf{2}}{{\mathbf{2}}^{\mathbf{x}}}\mathbf{+}\mathbf{c}$

Answer Key:

(c)$\frac{{\mathbf{2}}^{\mathbf{x}}}{\mathbf{log}\mathbf{2}}\mathbf{+}\mathbf{c}$

$\mathbf{ }$ 

3. $\mathbf{\int }\frac{\mathbf{sin} 2\mathrm{x}}{\mathbf{2}\mathbf{s}\mathbf{i}\mathbf{n}\mathbf{x}}\mathbf{d}\mathbf{x}$ is

(a)sinx+c

(b)$\frac{\mathbf{1}}{\mathbf{2}}$ sinx+c

(c)cosx+c

(d)$\frac{\mathbf{1}}{\mathbf{2}}$ cosx+c

Answer Key: 

(a)sinx+c

  

4. $\mathbf{\int }\frac{\mathbf{sin}\mathbf{5}\mathbf{x}\mathbf{-}\mathbf{s}\mathbf{i}\mathbf{n}\mathbf{x}}{\mathbf{cos}\mathbf{3}\mathbf{x}}\mathbf{d}\mathbf{x}$ is

(a) -cos2x+c

(b) -cos2x+c

(c) -$\frac{\mathbf{1}}{\mathbf{4}}$cos2x+c

(d) -4cos2x+c

Answer Key:

(a) -cos2x+c

  

5.  $\mathbf{\int }\frac{\mathbf{log}\mathbf{x}}{\mathbf{x}}\mathbf{d}\mathbf{x,}$ x>0 is

(a)$\frac{\mathbf{1}}{\mathbf{2}}\mathbf{(}\mathbf{l}\mathbf{o}\mathbf{g}\mathbf{x}{\mathbf{)}}^{\mathbf{2}}\mathbf{+}\mathbf{c}$

(b)$-\frac{\mathbf{1}}{\mathbf{2}}\mathbf{(}\mathbf{l}\mathbf{o}\mathbf{g}\mathbf{x}{\mathbf{)}}^{\mathbf{2}}\mathbf{+}\mathbf{c}$

(c)$\frac{\mathbf{2}}{{\mathbf{x}}^{\mathbf{2}}}\mathbf{+}\mathbf{c}$

(d)$\frac{\mathbf{2}}{{\mathbf{x}}^{\mathbf{2}}}\mathbf{+}\mathbf{c}$

Answer Key: 

(a)$\frac{\mathbf{1}}{\mathbf{2}}\mathbf{(}\mathbf{l}\mathbf{o}\mathbf{g}\mathbf{x}{\mathbf{)}}^{\mathbf{2}}\mathbf{+}\mathbf{c}$

$\mathbf{ }$ 

6. $\mathbf{\int }\frac{{\mathbf{e}}^{\mathbf{x}}}{\sqrt{\mathbf{1}\mathbf{+}{\mathbf{e}}^{\mathbf{x}}}}\mathbf{d}\mathbf{x}$ is 

(a)$\frac{{\mathbf{e}}^{\mathbf{x}}}{\sqrt{\mathbf{1}\mathbf{+}{\mathbf{e}}^{\mathbf{x}}}}\mathbf{+}\mathbf{c}$

(b)$\mathbf{2}\sqrt{\mathbf{1}\mathbf{+}{\mathit{e}}^{\mathit{x}}}\mathbf{+}\mathbf{c}$

(c)$\sqrt{\mathbf{1}\mathbf{+}{\mathit{e}}^{\mathit{x}}}\mathbf{+}\mathbf{c}$

(d)${\mathit{e}}^{\mathit{x}}\sqrt{\mathbf{1}\mathbf{+}{\mathbf{e}}^{\mathbf{x}}}\mathbf{+}\mathbf{c}$

Answer Key:  

(b)$\mathbf{2}\sqrt{\mathbf{1}\mathbf{+}{\mathit{e}}^{\mathit{x}}}\mathbf{+}\mathbf{c}$

$\mathbf{ }$

7. $\mathit{\int }\sqrt{{\mathit{e}}^{\mathit{x}}}\mathit{d}\mathit{x}$ is

(a)$\sqrt{{\mathit{e}}^{\mathit{x}}}\mathbf{+}\mathit{c}$

(b)$\mathbf{2}\sqrt{{\mathit{e}}^{\mathit{x}}}\mathbf{+}\mathit{c}$

(c)$\frac{\mathbf{1}}{\mathbf{2}}\sqrt{{\mathit{e}}^{\mathit{x}}}\mathbf{+}\mathit{c}$

(d)$\frac{\mathbf{1}}{\mathbf{2}\sqrt{{\mathit{e}}^{\mathit{x}}}}\mathbf{+}\mathbf{c}$

Answer Key: 

(b)$\mathbf{2}\sqrt{{\mathit{e}}^{\mathit{x}}}\mathbf{+}\mathit{c}$

$\mathit{ }$

8. $$∫e2x[2x2+2x]dx

(a)${\mathbf{e}}^{\mathbf{2}\mathbf{x}}{\mathbf{x}}^{\mathbf{2}}\mathbf{+}\mathit{c}$

(b)$\mathbf{x}{\mathbf{e}}^{\mathbf{2}\mathbf{x}}\mathbf{+}\mathit{c}$

(c)$\mathbf{2}{\mathbf{x}}^{\mathbf{2}}{\mathit{e}}^{\mathbf{2}}\mathbf{+}\mathit{c}$

(d)$\frac{{\mathbf{x}}^{\mathbf{2}}{\mathbf{e}}^{\mathbf{2}\mathbf{x}}}{\mathbf{2}}\mathbf{+}\mathbf{c}$

Answer Key: 

(a)${\mathbf{e}}^{\mathbf{2}\mathbf{x}}{\mathbf{x}}^{\mathbf{2}}\mathbf{+}\mathit{c}$

$\mathit{ }$

9. $\mathbf{\int }\frac{{\mathbf{e}}^{\mathbf{x}}}{{\mathbf{e}}^{\mathbf{x}}+1}\mathbf{d}\mathbf{x}$ is

(a)log|$\frac{{\mathbf{e}}^{\mathbf{x}}}{{\mathbf{e}}^{\mathbf{x}}+1}$|+c

(b)log|$\frac{{\mathbf{e}}^{\mathbf{x}}+1}{{\mathbf{e}}^{\mathbf{x}}}$|+c

(c)log|${\mathbf{e}}^{\mathbf{x}}$|${\mathbf{}}^{}$+c

(d)log|${\mathbf{e}}^{\mathbf{x}}+1$|+c

Answer Key: 

(d)log|${\mathbf{e}}^{\mathbf{x}}+1$|+c

 

 

10.dx is 

(a)log |x-3| - log |x+1| +c

(b)log |x-3|+ log |x+1| +c

(c)9log |x-3|- log |x+1| +c

(d)9log |x-3-|+ log |x+1| +c

Answer Key: 

(c)9log |x-3|- log |x+1| +c

 

11. $\int \frac{2{\mathrm{x}}^3}{4+{\mathrm{x}}^4}$ is

(a)$\mathbf{log|}$$\mathbf{4}\mathbf{+}{\mathbf{x}}^{\mathbf{4}}\mathbf{|}\mathbf{+}\mathbf{c}$

(b)$\frac{1}{2}\log \mathbf{|}\mathbf{4}\mathbf{+}{\mathbf{x}}^{\mathbf{4}}\mathbf{|}\mathbf{+}\mathbf{c}$

(c)$\frac{1}{4}\log \mathbf{|}\mathbf{4}\mathbf{+}{\mathbf{x}}^{\mathbf{4}}\mathbf{|}\mathbf{+}\mathbf{c}$

(d)$\log$$\mathbf{|}$$\frac{2{\mathrm{x}}^3}{\mathbf{4}\mathbf{+}{\mathbf{x}}^{\mathbf{4}}}$$\mathbf{|}$$\frac{{\mathbf{}}^{}}{}+\mathrm{c}$

Answer Key: 

(b)$\frac{1}{2}\log \mathbf{|}\mathbf{4}\mathbf{+}{\mathbf{x}}^{\mathbf{4}}\mathbf{|}\mathbf{+}\mathbf{c}$

$\mathbf{ }$

12. $\int \frac{\mathrm{dx}}{\sqrt{{\mathrm{x}}^2-36}}$ is

(a)${\mathbf{x}}^{\mathbf{2}}\mathbf{-}\mathbf{36}\mathbf{+}\mathbf{c}$

(b)$\mathbf{log}$$\mathbf{|}$$\mathbf{}\mathbf{x}\mathbf{+}\sqrt{{\mathbf{x}}^{\mathbf{2}}\mathbf{-}\mathbf{36}}$$\mathbf{|}$$\mathbf{+}\mathbf{c}$

(c)$\mathbf{log}$$\mathbf{|}$$\mathbf{x}\mathbf{-}\sqrt{{\mathbf{x}}^{\mathbf{2}}\mathbf{-}\mathbf{36}}$$\mathbf{|}$$\sqrt{\mathbf{}}+c$

(d) $\mathbf{log}$$\mathbf{|}$$\mathbf{}{\mathbf{x}}^{\mathbf{2}}\mathbf{+}\sqrt{{\mathbf{x}}^{\mathbf{2}}\mathbf{-}\mathbf{36}}$$\mathbf{|}$$\sqrt{\mathbf{}}+c$

Answer Key: 

(b)$\mathbf{log}$$\mathbf{|}$$\mathbf{}\mathbf{x}\mathbf{+}\sqrt{{\mathbf{x}}^{\mathbf{2}}\mathbf{-}\mathbf{36}}$$\mathbf{|}$$\mathbf{+}\mathbf{c}$

$\mathbf{ }$

13. $\mathbf{\int }\frac{\mathbf{2}\mathbf{x}\mathbf{+}\mathbf{3}}{\sqrt{{\mathbf{x}}^{\mathbf{2}}\mathbf{+}\mathbf{3}\mathbf{x}\mathbf{+}\mathbf{2}}}$ is 

(a)$\mathbf{ }\sqrt{{\mathbf{x}}^{\mathbf{2}}\mathbf{+}\mathbf{3}\mathbf{x}\mathbf{+}\mathbf{2}}+c$

(b)$2\sqrt{{\mathbf{x}}^{\mathbf{2}}\mathbf{+}\mathbf{3}\mathbf{x}\mathbf{+}\mathbf{2}}+c$

(c)$\mathbf{log}\mathbf{(}{\mathbf{x}}^{\mathbf{2}}\mathbf{+}\mathbf{3}\mathbf{x}\mathbf{+}\mathbf{2}\mathbf{)}\mathbf{+}\mathbf{c}$

(d)$\frac{\mathbf{2}}{\mathbf{3}}\mathbf{\left(}{\mathbf{x}}^{\mathbf{2}}\mathbf{+}\mathbf{3}\mathbf{x}\mathbf{+}\mathbf{2}\mathbf{\right)}\mathbf{+}\mathbf{c}$

Answer Key: 

(b)$2\sqrt{{\mathbf{x}}^{\mathbf{2}}\mathbf{+}\mathbf{3}\mathbf{x}\mathbf{+}\mathbf{2}}+c$

$$

14. $\underset{\mathbf{0}}{\overset{\mathbf{1}}{\mathbf{\int }}}\mathbf{(}\mathbf{2}\mathbf{x}\mathbf{+}\mathbf{1}\mathbf{)} \mathbf{d}\mathbf{x}$ is

(a) 1

(b) 2

(c) 3

(d) 4

Answer Key: 

(b) 2

 

15. $\underset{\mathbf{2}}{\overset{\mathbf{4}}{\mathbf{\int }}}\frac{\mathbf{d}\mathbf{x}}{\mathbf{x}}$ is

(a) log 4

(b) 0

(c) log 2

(d) log 8

Answer Key: 

(c) log 2

 

16. $\underset{\mathbf{0}}{\overset{\mathbf{\infty }}{\mathbf{\int }}}{\mathbf{e}}^{\mathbf{-}\mathbf{2}\mathbf{x}}$ dx is 

(a) 0

(b) 1

(c) 2 

(d) $\frac{\mathbf{1}}{\mathbf{2}}$

Answer Key:

(d) $\frac{\mathbf{1}}{\mathbf{2}}$

$\frac{\mathbf{ }}{}$ 

17. $\underset{\mathbf{-}\mathbf{1}}{\overset{\mathbf{1}}{\mathbf{\int }}}{\mathbf{x}}^{\mathbf{3}} {\mathbf{e}}^{{\mathbf{x}}^{\mathbf{4}}} \mathrm{dx}$ is 

(a) 1

(b) $\mathbf{2} \underset{0}{\overset{1}{\int }}{\mathbf{x}}^{\mathbf{3}} {\mathbf{e}}^{{\mathbf{x}}^{\mathbf{4}}} \mathrm{dx}$

(c) 0

(d) ${\mathbf{e}}^{{\mathbf{x}}^{\mathbf{4}}}$ 

Answer Key: 

(c) 0

  

18. If f(x0 is A continuous function and a<c<b, then $\underset{\mathbf{a}}{\overset{\mathbf{c}}{\mathbf{\int }}}\mathbf{f}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{ }\mathbf{d}\mathbf{x}\mathbf{+}\underset{\mathbf{c}}{\overset{\mathbf{b}}{\mathbf{\int }}}\mathbf{f}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{ }\mathbf{d}\mathbf{x}$ is

(a) $\underset{\mathbf{a}}{\overset{\mathbf{b}}{\mathbf{\int }}}\mathbf{f}\left(\mathbf{x}\right)\mathbf{ }\mathbf{d}\mathbf{x}\mathbf{-}\underset{\mathbf{a}}{\overset{\mathbf{c}}{\mathbf{\int }}}\mathbf{f}\left(\mathbf{x}\right)\mathbf{ }\mathbf{d}\mathbf{x}$

(b) $\underset{\mathbf{a}}{\overset{\mathbf{c}}{\mathbf{\int }}}\mathbf{f}\left(\mathbf{x}\right)\mathbf{ }\mathbf{d}\mathbf{x}\mathbf{-}\underset{\mathbf{a}}{\overset{\mathbf{b}}{\mathbf{\int }}}\mathbf{f}\left(\mathbf{x}\right)\mathbf{ }\mathbf{d}\mathbf{x}$

(c) $\underset{\mathbf{a}}{\overset{\mathbf{b}}{\mathbf{\int }}}\mathbf{f}\left(\mathbf{x}\right)\mathbf{ }\mathbf{d}\mathbf{x}$

(d) 0

Answer Key:

(c) $\underset{\mathbf{a}}{\overset{\mathbf{b}}{\mathbf{\int }}}\mathbf{f}\left(\mathbf{x}\right)\mathbf{ }\mathbf{d}\mathbf{x}$

$\mathbf{ }$ 

19. The Value of $\underset{\mathbf{-}\frac{\mathbf{\pi }}{\mathbf{2}}}{\overset{\frac{\mathbf{\pi }}{\mathbf{2}}}{\mathbf{\int }}}$ cosx dx is 

(a) 0

(b) 2

(c) 1

(d) 4

Answer Key: 

(b) 2

  

20. $\underset{\mathbf{0}}{\overset{\mathbf{1}}{\mathbf{\int }}}\sqrt{{\mathbf{x}}^{\mathbf{4}}\mathbf{(}\mathbf{1}\mathbf{-}\mathbf{x}{\mathbf{)}}^{\mathbf{2}}}\mathbf{d}\mathbf{x}$ is

(a)$\frac{\mathbf{1}}{\mathbf{12}}$

(b)$\frac{\mathbf{-}\mathbf{7}}{\mathbf{12}}$

(c)$\frac{\mathbf{7}}{\mathbf{12}}$

(d)$\frac{\mathbf{-}\mathbf{1}}{\mathbf{12}}$

Answer Key:

(a)$\frac{\mathbf{1}}{\mathbf{12}}$

$\frac{\mathbf{ }}{}$  

21. If $\underset{\mathbf{0}}{\overset{\mathbf{1\; f(x)\; dx=1, }}{\mathbf{\int }}}$$\underset{\mathbf{0}}{\overset{\mathbf{1\; x }}{\mathbf{\int }}}$$\underset{}{\overset{}{\mathbf{f(x)\; dx=a\; and }}}$$\underset{}{\overset{}{\mathbf{}}}$

$\underset{}{\overset{}{\mathbf{(a)\; 4}}}$

$\underset{}{\overset{}{\mathbf{(b)\; 0}}}$

$\underset{}{\overset{}{\mathbf{(c)\; 2}}}$

$\underset{}{\overset{}{\mathbf{(d)\; 1}}}$

Answer Key:

$\underset{}{\overset{}{\mathbf{(b)\; 0}}}$

$\underset{}{\overset{}{\mathbf{ }}}$ $\underset{}{\overset{}{\mathbf{ }}}$

22. $\underset{\mathbf{2}}{\overset{\mathbf{3}}{\mathbf{\int }}}\mathbf{f}\mathbf{(}\mathbf{5}\mathbf{-}\mathbf{x}\mathbf{)}\mathbf{d}\mathbf{x}\mathbf{ }\mathbf{-}\underset{\mathbf{2}}{\overset{\mathbf{3}}{\mathbf{\int }}}\mathbf{f}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{d}\mathbf{x}$ is 

(a) 1

(b) 0

(c) -1

(d) 5

Answer Key: 

(b) 0

  

23. $\begin{array}{c}\mathrm{<math\; xmlns="http://www.w3.org/1998/Math/MathML"><mi\; mathvariant="bold">d</mi><mi\; mathvariant="bold">x </mi></math>}\end{array}$is 

(a) $\frac{20}{3}$

(b) $\frac{21}{3}$

(c) $\frac{28}{3}$

(d) $\frac{1}{3}$

Answer Key: 

(c) $\frac{28}{3}$

$\frac{\mathrm{ }}{}$ 

24. $\underset{\mathbf{0}}{\overset{\frac{\mathbf{\pi }}{\mathbf{3}}}{\mathbf{\int }}}$ tanx dx is 

(a) log 2

(b) 0

(c) log $\sqrt{\mathbf{2}}$ 

(d) 2 log 2

Answer Key: 

(a) log 2

  

25. Using the factorial representation of  the gamma function, which of the following is the solution for the gamma function Γ(n) when n=8

(a) 5040

(b) 5400

(c) 4500

(d) 5540

Answer Key: 

(a) 5040

  

26.  Γ(n) is

(a) (n-1)!

(b) n!

(c) nΓ(n)

(d) (n-1)Γ(n)

Answer Key: 

(a) (n-1)!

  

27.  Γ(1) is

(a) 0

(b) 1

(c) n

(d) n!

Answer Key: 

(b) 1

  

28. If n>0, then Γ(n) is 

(a) $\underset{\mathbf{0}}{\overset{\mathbf{1}}{\mathbf{\int }}}{\mathbf{e}}^{\mathbf{-}\mathbf{x}}\mathbf{ }{\mathbf{x}}^{\mathbf{n}\mathbf{-}\mathbf{1}}\mathbf{d}\mathbf{x}$

(b) $\underset{\mathbf{0}}{\overset{\mathbf{1}}{\mathbf{\int }}}{\mathbf{e}}^{\mathbf{-}\mathbf{x}}\mathbf{ }{\mathbf{x}}^{\mathbf{n}}\mathbf{ }\mathbf{d}\mathbf{x}$

(c) $\underset{\mathbf{0}}{\overset{\mathbf{\infty }}{\mathbf{\int }}}{\mathbf{e}}^{\mathbf{x}} {\mathrm{x}}^{-\mathrm{n}} \mathrm{dx}$

(d) $\underset{\mathbf{0}}{\overset{\mathbf{\infty }}{\mathbf{\int }}}{\mathbf{e}}^{\mathbf{-}\mathbf{x}} {\mathbf{x}}^{\mathbf{n}\mathbf{-}\mathbf{1}} \mathbf{d}\mathbf{x}$

Answer Key: 

(d) $\underset{\mathbf{0}}{\overset{\mathbf{\infty }}{\mathbf{\int }}}{\mathbf{e}}^{\mathbf{-}\mathbf{x}} {\mathbf{x}}^{\mathbf{n}\mathbf{-}\mathbf{1}} \mathbf{d}\mathbf{x}$

$\mathbf{ }$ 

29.  Γ$($$\frac{\mathbf{3}}{\mathbf{2}}$$)$

$(a)\; <math\; xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mi\; mathvariant="bold">\pi </mi></msqrt></math>$

$(b)\; <math\; xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msqrt><mi\; mathvariant="bold">\pi </mi></msqrt><mn\; mathvariant="bold">2</mn></mfrac></math>$

$(c)\; <math\; xmlns="http://www.w3.org/1998/Math/MathML"><mn\; mathvariant="bold">2</mn><msqrt><mi\; mathvariant="bold">\pi </mi></msqrt><mspace\; linebreak="newline"></mspace></math>$

$(d)\; <math\; xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn\; mathvariant="bold">3</mn><mn\; mathvariant="bold">2</mn></mfrac></math>$

Answer Key:$$

$(b)\; <math\; xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msqrt><mi\; mathvariant="bold">\pi </mi></msqrt><mn\; mathvariant="bold">2</mn></mfrac></math>$

$<math\; xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn\; mathvariant="bold"> </mn></mfrac></math>$ 

30. $\underset{\mathbf{0}}{\overset{\mathbf{\infty }}{\mathbf{\int }}}{\mathbf{x}}^{\mathbf{4}}\mathbf{ }{\mathbf{e}}^{\mathbf{-}\mathbf{x}}\mathbf{ }\mathbf{d}\mathbf{x}$ is

(a) 12

(b) 4

(c) 4!

(d) 64

Answer Key:$$ 

(c) 4! 

II. Short answer questions

12th BM

III. Long answer questions

12th BM