Sample Solution

MCH-014 Solved Assignment 2024

1. State whether the following statement are TRUE or FALSE. Give reason in support of your answer.
   a) Derivative of $\frac{1}{x}$$\frac{1}{x}$(1)/(x)\frac{1}{x}$\frac{1}{x}$ with respect to $x$$x$xx$x$ is 1 .
   b) If $A$$A$AA$A$ is a matrix of order 2 by 3 and $B$$B$BB$B$ is a matrix of order 3 by 2 , then order of the matrix $\mathrm{A}+\mathrm{B}$$\mathrm{A}+\mathrm{B}$A+B\mathrm{A}+\mathrm{B}$\mathrm{A}+\mathrm{B}$ is 2 by 3 .
   c) If $\overrightarrow{a}=3\hat{i}-5\hat{j}+4\hat{k}$$\overrightarrow{a}=3\widehat{i}-5\widehat{j}+4\widehat{k}$vec(a)=3 hat(i)-5 hat(j)+4 hat(k)\vec{a}=3 \hat{i}-5 \hat{j}+4 \hat{k}$\overrightarrow{a}=3\hat{i}-5\hat{j}+4\hat{k}$ and $\overrightarrow{b}=4\hat{i}+4\hat{j}+2\hat{k}$$\overrightarrow{b}=4\widehat{i}+4\widehat{j}+2\widehat{k}$vec(b)=4 hat(i)+4 hat(j)+2 hat(k)\vec{b}=4 \hat{i}+4 \hat{j}+2 \hat{k}$\overrightarrow{b}=4\hat{i}+4\hat{j}+2\hat{k}$ they are perpendicular to each other.
   d) If probability of an event E is $1/2$$1/2$1//21 / 2$1/2$ and probability of the event $\mathrm{E}\cap \mathrm{F}$$\mathrm{E}\cap \mathrm{F}$EnnF\mathrm{E} \cap \mathrm{F}$\mathrm{E}\cap \mathrm{F}$ is $1/6$$1/6$1//61 / 6$1/6$, then probability of the event $F$$F$FF$F$ is $1/3$$1/3$1//31 / 3$1/3$, where events $E$$E$EE$E$ and $F$$F$FF$F$ are independent.
   e) If the first term of an AP is 5 and 101 term of the $AP$$AP$APA P$AP$ is 1005 then common difference of the AP will be 105.
2. Solve the following system of equations using Cramer’s rule.

3. a) Prove that $A=\left[\begin{array}{cc}\cos 2\alpha & -\sin 2\alpha \\ \sin 2\alpha & \cos 2\alpha \end{array}\right]$$A=\left[\begin{array}{cc}\cos 2\alpha & -\sin 2\alpha \\ \sin 2\alpha & \cos 2\alpha \end{array}\right]$A=[[cos 2alpha,-sin 2alpha],[sin 2alpha,cos 2alpha]]A=\left[\begin{array}{cc}\cos 2 \alpha & -\sin 2 \alpha \\ \sin 2 \alpha & \cos 2 \alpha\end{array}\right]$A=\left[\begin{array}{cc}\cos 2\alpha & -\sin 2\alpha \\ \sin 2\alpha & \cos 2\alpha \end{array}\right]$ is an orthogonal matrix.
4. a) Evaluate ${\int }_{-5}^5(x^3+x)dx$${\int }_{-5}^5 \left(x^3+x\right)dx$int_(-5)^(5)(x^(3)+x)dx\int_{-5}^5\left(x^3+x\right) d x${\int }_{-5}^5(x^3+x)dx$.

5. a) Solve the differential equation $\sin x\frac{dy}{dx}+y\cos x=1$$\sin x\frac{dy}{dx}+y\cos x=1$sin x(dy)/(dx)+y cos x=1\sin x \frac{d y}{d x}+y \cos x=1$\sin x\frac{dy}{dx}+y\cos x=1$.
   b) If $\overrightarrow{a}=2\hat{i}+3\hat{j}+4\hat{k}$$\overrightarrow{a}=2\widehat{i}+3\widehat{j}+4\widehat{k}$vec(a)=2 hat(i)+3 hat(j)+4 hat(k)\vec{a}=2 \hat{i}+3 \hat{j}+4 \hat{k}$\overrightarrow{a}=2\hat{i}+3\hat{j}+4\hat{k}$ and $\overrightarrow{b}=3\hat{i}+5\hat{j}+2\hat{k}$$\overrightarrow{b}=3\widehat{i}+5\widehat{j}+2\widehat{k}$vec(b)=3 hat(i)+5 hat(j)+2 hat(k)\vec{b}=3 \hat{i}+5 \hat{j}+2 \hat{k}$\overrightarrow{b}=3\hat{i}+5\hat{j}+2\hat{k}$ then find $\overrightarrow{a}\times \overrightarrow{b}$$\overrightarrow{a}\times \overrightarrow{b}$vec(a)xx vec(b)\vec{a} \times \vec{b}$\overrightarrow{a}\times \overrightarrow{b}$.
   c) In an iron determination (taking 1 g sample every time) the following four replicate results were obtained: $24.8,25.2,23.6$$24.8,25.2,23.6$24.8,25.2,23.624.8,25.2,23.6$24.8,25.2,23.6$ and 24.7 mg iron. Calculate the coefficient of variation and relative standard deviation in ppm of the given data.
6. a) In a factory there are three machines A, B, C which produce $10\mathrm{\%}$$10\mathrm{\%}$10%10 \%$10\mathrm{\%}$, $40\mathrm{\%}$$40\mathrm{\%}$40%40 \%$40\mathrm{\%}$ and $50\mathrm{\%}$$50\mathrm{\%}$50%50 \%$50\mathrm{\%}$ items respectively. Past experience shows that percentage of defective items produced by machines A, B, C are $5\mathrm{\%}$$5\mathrm{\%}$5%5 \%$5\mathrm{\%}$, $4\mathrm{\%},2\mathrm{\%}$$4\mathrm{\%},2\mathrm{\%}$4%,2%4 \%, 2 \%$4\mathrm{\%},2\mathrm{\%}$ respectively. An item from the production of these machines is selected at random and it is found defective. What is the probability that it is produced by machine A?

Expert Answer

MCH-014 Solved Assignment 2024

State whether the following statement are TRUE or FALSE. Give reason in support of your answer.

Derivative of $\frac{1}{x}$$\frac{1}{x}$(1)/(x)\frac{1}{x}$\frac{1}{x}$ with respect to $x$$x$xx$x$ is 1.

Answer:

If $A$$A$AA$A$ is a matrix of order 2 by 3 and $B$$B$BB$B$ is a matrix of order 3 by 2, then order of the matrix $\mathrm{A}+\mathrm{B}$$\mathrm{A}+\mathrm{B}$A+B\mathrm{A}+\mathrm{B}$\mathrm{A}+\mathrm{B}$ is 2 by 3.

Answer:

• Matrix $A$$A$AA$A$ has dimensions 2 by 3.
• Matrix $B$$B$BB$B$ has dimensions 3 by 2.

If $\overrightarrow{a}=3\hat{i}-5\hat{j}+4\hat{k}$$\overrightarrow{a}=3\widehat{i}-5\widehat{j}+4\widehat{k}$vec(a)=3 hat(i)-5 hat(j)+4 hat(k)\vec{a}=3 \hat{i}-5 \hat{j}+4 \hat{k}$\overrightarrow{a}=3\hat{i}-5\hat{j}+4\hat{k}$ and $\overrightarrow{b}=4\hat{i}+4\hat{j}+2\hat{k}$$\overrightarrow{b}=4\widehat{i}+4\widehat{j}+2\widehat{k}$vec(b)=4 hat(i)+4 hat(j)+2 hat(k)\vec{b}=4 \hat{i}+4 \hat{j}+2 \hat{k}$\overrightarrow{b}=4\hat{i}+4\hat{j}+2\hat{k}$ they are perpendicular to each other.

Answer:

If probability of an event E is $1/2$$1/2$1//21 / 2$1/2$ and probability of the event $\mathrm{E}\cap \mathrm{F}$$\mathrm{E}\cap \mathrm{F}$EnnF\mathrm{E} \cap \mathrm{F}$\mathrm{E}\cap \mathrm{F}$ is $1/6$$1/6$1//61 / 6$1/6$, then probability of the event $F$$F$FF$F$ is $1/3$$1/3$1//31 / 3$1/3$, where events $E$$E$EE$E$ and $F$$F$FF$F$ are independent.

Answer:

If the first term of an AP is 5 and 101 term of the $AP$$AP$APA P$AP$ is 1005 then common difference of the AP will be 105.

Answer:

• The first term $a=5$$a=5$a=5a = 5$a=5$
• The 101st term $a_{101}=1005$$a_{101}=1005$a_(101)=1005a_{101} = 1005$a_{101}=1005$