12th Business Mathematics and Statistics - Book Back Answers - Chapter 4

Plus Two / 12th Business Mathematics and Statistics - Book Back Answers - Chapter 4 - English Medium

Tamil Nadu Board 12th Standard Business Mathematics and Statistics - Chapter 4: Book Back Answers and Solutions

This post covers the book back answers and solutions for Chapter 4 – 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 4 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 4:

I. Multiple Choice Questions

1. The degree of the differential equation $\frac{d^{4} y}{d x^{4}} -$ $(\begin{matrix} \frac{d^{2} y}{d x^{2}} \end{matrix})$ ⁴ $+ \frac{d y}{d x} = 3$
(a) 1
(b) 2
(c) 3
(d) 4
Answer Key:
(a) 1

2. The order and degree of the differential equation $\sqrt{\frac{d^{2} y}{d x^{2}}} = \sqrt{\frac{d y}{d x} + 5}$ are respectively
(a) 2 and 3
(b) 3 and 2
(c) 2 and 1
(d) 2 and 2
Answer Key:
(c) 2 and 1

3. The order and degree of the differential equation $(\begin{matrix} \frac{d^{2} y}{d x^{2}} \end{matrix})$ ^{$\frac{3}{2}$} - $\sqrt$ $(\begin{matrix} \frac{d y}{d x} \end{matrix})$ -4=0 are respectively.
(a) 2 and 6
(b) 3 and 6
(c) 1 and 4
(d) 2 and 4
Answer Key:
(a) 2 and 6

4. The differential equation $(\begin{matrix} \frac{d y}{d x} \end{matrix})$ ³ + $2 y^{\frac{1}{2}}$ =x is
(a) of order 2 and degree 1
(b) of order 1 and degree 3
(c) of order 1 and degree 6
(d) of order 1 and degree 2
Answer Key:
(b) of order 1 and degree 3

5. The differential equation formed by eliminating a and b from y = ae^x + be^-x is
(a)

\begin{matrix} \frac{d^{2} y}{d x^{2 -y}} \end{matrix}

$\begin{matrix} = \end{matrix}$

\begin{matrix} 0 \end{matrix}

(b)

\begin{matrix} \frac{d^{2} y}{d x^{2-}} \end{matrix}

\begin{matrix} \frac{d y}{d x=0} \end{matrix}

(c)

\begin{matrix} \frac{d^{2} y}{d x^{2=0}} \end{matrix}

(d)

\begin{matrix} \frac{d^{2} y}{d x^{2-x=0}} \end{matrix}

Answer Key:
(a)

\begin{matrix} \frac{d^{2} y}{d x^{2 -y}} \end{matrix}

$\begin{matrix} = \end{matrix}$

\begin{matrix} 0 \end{matrix}

\begin{matrix}  \end{matrix}

6. If y = cx + c − c³ then its differential equation is
(a) y = x

\begin{matrix} \frac{d y}{d x+} \end{matrix}

\begin{matrix} \frac{d y}{d x-} \end{matrix}

(\begin{matrix} \frac{d y}{d x} \end{matrix})

³
(b) y+

(\begin{matrix} \frac{d y}{d x} \end{matrix})

³= x

\begin{matrix} \frac{d y}{d x-} \end{matrix}

\begin{matrix} \frac{d y}{d x} \end{matrix}

(c)

\begin{matrix} \frac{d y}{d x+y=} \end{matrix}

(\begin{matrix} \frac{d y}{d x} \end{matrix})

\begin{matrix} \frac{-x d y}{d x} \end{matrix}

(d)

\frac{d^{3} y}{d x^{3=0}}

Answer Key:
(a) y = x

\begin{matrix} \frac{d y}{d x+} \end{matrix}

\begin{matrix} \frac{d y}{d x-} \end{matrix}

(\begin{matrix} \frac{d y}{d x} \end{matrix})

7. The integrating factor of the differential equation $\begin{matrix} \frac{d y}{d x} \end{matrix}$ + Px = Q is
(a)

e^{\int p d x}

(b)

^{\int p d x}

(c)

\int p dy

(d)

e^{\int p dy}

Answer Key:
(d)

e^{\int p dy}

8. The complementary function of (D² + 4)y = e^2x is
(a) (Ax + B)e^2x
(b) (Ax + B)e^-2x
(c) Acos2x + Bsin2x
(d) Ae^-2x + Be^2x
Answer Key:
(c) Acos2x + Bsin2x

9. The differential equation of y = mx + c is (m and c are arbitrary constants)
(a)

\begin{matrix} \frac{d^{2} y}{d x^{2 = 0}} \end{matrix}

(b) y =

\begin{matrix} \frac{d y}{d x+c} \end{matrix}

\begin{matrix} \frac{d^{2} y}{d x^{2 = 0}} \end{matrix}

\begin{matrix}  \end{matrix}

10. The particular integral of the differential equation is $\begin{matrix} \frac{d^{2} y}{d x^{2 - 8}} \end{matrix}$ $\begin{matrix} \frac{d y}{d x + 16y = 2e} \end{matrix}$ ^4x
(a)

\frac{x^{2} e^{4 x}}{2!}

(b)

\frac{e^{4 x}}{2!}

(c)

x^{2} e^{4 x}

(d)

x e^{4 x}

Answer Key:
(c)

x^{2} e^{4 x}

11. Solution of

\begin{matrix} d yd x + \end{matrix}

Px = 0
(a) x = ce^py
(b) x = ce^-py
(c) x = py + c
(d) x = cy
Answer Key:
(b) x = ce^-py

12. If sec² x is an integrating factor of the differential equation

\begin{matrix} d yd x \end{matrix}

+ Py = Q then P =
(a) 2tan x
(b) sec x
(c) cos² x
(d) tan² x
Answer Key:
(a) 2tan x

13. The integrating factor of x $\begin{matrix} \frac{d y}{d x} \end{matrix}$ − y = x2 is
(a)

\frac{- 1}{x}

(b)

\frac{1}{x}

\frac{1}{x}

14. The solution of the differential equation $\begin{matrix} \frac{d y}{d x} \end{matrix}$ + Py = Q where P and Q are the
function of x is
(a) y =

\int Q e^{\int p d x} d x + c

(b) y =

\int Q e^{-\int p d x} d x + c

^{\int p d x}

=

\int Q e^{\int p d x} d x + c

(d) ye

^{\int p d x}

= \int Q e^{-\int p d x} d x + c

Answer Key:
(c) ye

^{\int p d x}

=

\int Q e^{\int p d x} d x + c

15. The differential equation formed by eliminating A and B from y = e^-2x (Acos x + Bsin x)is
(a) y₂ − 4y₁ + 5 = 0
(b) y₂+ 4y – 5 = 0
(c) y₂ − 4y₁ − 5 = 0
(d) y₂+ 4y₁ + 5 = 0
Answer Key:
(d) y₂+ 4y₁ + 5 = 0

16. The particular integral of the differential equation f (D) y = e^ax where f(D) = (D − a)²
(a)

\frac{x^{2}}{2}

e^ax

(b) xe^ax
(c)

\frac{x}{2}

e^ax

(d) x²eax
Answer Key:
(a)

\frac{x^{2}}{2}

e^ax

17. The differential equation of x² + y² = a²
(a) xdy+ydx=0
(b) ydx–xdy=0
(c) xdx–ydx=0
(d) xdx+ydy=0
Answer Key:
(d) xdx+ydy=0

18. The complementary function of

\begin{matrix} d2y \end{matrix}

dx2 = 0 is
(a) A + Be^x
(b) (A + B)e^x
(c) (Ax + B)e^x
(d) Ae^x + B
Answer Key:
(a) A + Be^x

19. The P.I of (3D² + D − 14)y =13e^2x is
(a)

\frac{x}{2}

e^2x
(b) xe^2x
(c)

\frac{x^{2}}{2}

e^2x
(d) 13xe^2x
Answer Key:
(b) xe^2x

20. The general solution of the differential equation $\begin{matrix} \frac{d y}{d x} \end{matrix}$ = cos x is
(a) y = sin x +1 (b) y = sin x - 2
(c) y = cos x + c, c is an arbitrary constant
(d) y = sin x + c, c is an arbitrary constant
Answer Key:
(d) y = sin x + c, c is an arbitrary constant

21. A homogeneous differential equation of the form $\begin{matrix} \frac{d y}{d x} \end{matrix}$ $\begin{matrix} = f \end{matrix}$ $(\begin{matrix} \frac{y}{x} \end{matrix})$ can be solved by making substitution,
(a) y = v x

(b) v = y x
(c) x = v y

(d) x = v

Answer Key:

(a) y = v x

22. A homogeneous differential equation of the form $\begin{matrix} \frac{d y}{d x} \end{matrix}$ $\begin{matrix} = f \end{matrix}$ $(\begin{matrix} \frac{x}{y} \end{matrix})$ can be solved by making substitution,
(a) x = v y

(b) y = v x
(c) y = v

(d) x = v

Answer Key:

(a) x = v y

23. The variable separable form of $\begin{matrix} \frac{d y}{d x} \end{matrix}$ $\begin{matrix} =\frac{y (x - y)}{x (x + y)} \end{matrix}$ by taking y = vx and $\begin{matrix} \frac{d y}{d x} \end{matrix}$ $\begin{matrix} = V+x \end{matrix}$ $\begin{matrix} \frac{d v}{d x} \end{matrix}$ is
(a)

\frac{2 v^{2}}{1 + v} d v = \frac{d x}{x}

(b)

\frac{2 v^{2}}{1 + v} d v = - \frac{d x}{x}

(c)

\frac{2 v^{2}}{1 - v} d v = - \frac{d x}{x}

(d)

\frac{1 + v}{2 v^{2}} d v = - \frac{d x}{x}

Answer Key:

(d)

\frac{1 + v}{2 v^{2}} d v = - \frac{d x}{x}

24. Which of the following is the homogeneous differential equation?
(a) (3x − 5) dx = (4y −1) dy
(b) xy dx − (x³ + y³ ) dy = 0
(c) y² dx + (x² − xy − y² ) dy = 0
(d) (x² + y) dx =(y² + x) dy
Answer Key:

25. The solution of the differential equation $\begin{matrix} \frac{d y}{d x =} \end{matrix}$ $\frac{y}{x} +$

(

$\frac{y}{x}$

)

(

$\frac{y}{x}$

)

is
(a) f

(\begin{matrix} \frac{y}{x} \end{matrix})

=kx

(b) x f

(\begin{matrix} \frac{y}{x} \end{matrix})=k

(\begin{matrix} \frac{y}{x} \end{matrix})=ky

(d) yf

(\begin{matrix} \frac{y}{x} \end{matrix})=k

Answer Key:

(a) f

(\begin{matrix} \frac{y}{x} \end{matrix})

=kx

II. Short answer questions

12th BM

III. Long answer questions

12th BM