State whether the following statement are TRUE or FALSE. Give reason in support of your answer.
a) Derivative of (1)/(x)\frac{1}{x} with respect to xx is 1 .
b) If AA is a matrix of order 2 by 3 and BB is a matrix of order 3 by 2 , then order of the matrix A+B\mathrm{A}+\mathrm{B} is 2 by 3 .
c) If vec(a)=3 hat(i)-5 hat(j)+4 hat(k)\vec{a}=3 \hat{i}-5 \hat{j}+4 \hat{k} and vec(b)=4 hat(i)+4 hat(j)+2 hat(k)\vec{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//21 / 2 and probability of the event EnnF\mathrm{E} \cap \mathrm{F} is 1//61 / 6, then probability of the event FF is 1//31 / 3, where events EE and FF are independent.
e) If the first term of an AP is 5 and 101 term of the APA P is 1005 then common difference of the AP will be 105.
Solve the following system of equations using Cramer’s rule.
a) Prove that 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] is an orthogonal matrix.
a) Evaluate int_(-5)^(5)(x^(3)+x)dx\int_{-5}^5\left(x^3+x\right) d x.
b) Evaluate int(ln x)/(x^(2))dx\int \frac{\ln x}{x^2} d x
a) Solve the differential equation sin x(dy)/(dx)+y cos x=1\sin x \frac{d y}{d x}+y \cos x=1.
b) If vec(a)=2 hat(i)+3 hat(j)+4 hat(k)\vec{a}=2 \hat{i}+3 \hat{j}+4 \hat{k} and vec(b)=3 hat(i)+5 hat(j)+2 hat(k)\vec{b}=3 \hat{i}+5 \hat{j}+2 \hat{k} then find vec(a)xx vec(b)\vec{a} \times \vec{b}.
c) In an iron determination (taking 1 g sample every time) the following four replicate results were obtained: 24.8,25.2,23.624.8,25.2,23.6 and 24.7 mg iron. Calculate the coefficient of variation and relative standard deviation in ppm of the given data.
a) In a factory there are three machines A, B, C which produce 10%10 \%, 40%40 \% and 50%50 \% items respectively. Past experience shows that percentage of defective items produced by machines A, B, C are 5%5 \%, 4%,2%4 \%, 2 \% 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?
b) Assume that in a population each person is equally likely to have a particular disease and disease status of each individual is independent of each other, then find the probability that out of the 5 randomly selected individuals who are tested for this particular disease exactly 3 have this disease.
c) A hospital specialising in heart surgery. In 2023 total of 1000 patients were admitted for treatment. The average payment made by a patient was Rs 1,00,0001,00,000 with a standard deviation of Rs 20000 . Under the assumption that payments follow a normal distribution, find the number of patients who paid between Rs 90,000 and Rs 1,10,000.
Expert Answer
MCH-014 Solved Assignment 2024
Question:-1
State whether the following statement are TRUE or FALSE. Give reason in support of your answer.
Question:-1(a)
Derivative of (1)/(x)\frac{1}{x} with respect to xx is 1.
Answer:
The statement "The derivative of (1)/(x)\frac{1}{x} with respect to xx is 1" is false.
To justify this, we can compute the derivative of (1)/(x)\frac{1}{x} using basic differentiation rules.
Given the function f(x)=(1)/(x)f(x) = \frac{1}{x}, we apply the power rule of differentiation. Rewriting f(x)f(x) in terms of exponents, we have:
f(x)=x^(-1)f(x) = x^{-1}
The power rule states that the derivative of x^(n)x^n is nx^(n-1)nx^{n-1}. Applying this rule:
Therefore, the derivative of (1)/(x)\frac{1}{x} with respect to xx is -(1)/(x^(2))-\frac{1}{x^2}, not 1.
Question:-1(b)
If AA is a matrix of order 2 by 3 and BB is a matrix of order 3 by 2, then order of the matrix A+B\mathrm{A}+\mathrm{B} is 2 by 3.
Answer:
The statement "If AA is a matrix of order 2 by 3 and BB is a matrix of order 3 by 2, then the order of the matrix A+BA + B is 2 by 3" is false.
To justify this, we need to understand the rules for matrix addition. Two matrices can only be added if they have the same dimensions.
Matrix AA has dimensions 2 by 3.
Matrix BB has dimensions 3 by 2.
Since AA and BB do not have the same dimensions, their addition A+BA + B is not defined. Therefore, it is impossible to form the matrix A+BA + B, and thus the statement about the order of A+BA + B is meaningless.
Question:-1(c)
If vec(a)=3 hat(i)-5 hat(j)+4 hat(k)\vec{a}=3 \hat{i}-5 \hat{j}+4 \hat{k} and vec(b)=4 hat(i)+4 hat(j)+2 hat(k)\vec{b}=4 \hat{i}+4 \hat{j}+2 \hat{k} they are perpendicular to each other.
Answer:
The statement "If vec(a)=3 hat(i)-5 hat(j)+4 hat(k)\vec{a}=3 \hat{i}-5 \hat{j}+4 \hat{k} and vec(b)=4 hat(i)+4 hat(j)+2 hat(k)\vec{b}=4 \hat{i}+4 \hat{j}+2 \hat{k}, they are perpendicular to each other" is false.
To justify this, we need to determine whether the vectors vec(a)\vec{a} and vec(b)\vec{b} are perpendicular by checking if their dot product is zero.
The dot product of vec(a)\vec{a} and vec(b)\vec{b} is calculated as follows:
Since the dot product vec(a)* vec(b)\vec{a} \cdot \vec{b} is indeed zero, the vectors vec(a)\vec{a} and vec(b)\vec{b} are perpendicular to each other. Therefore, the statement is true.
Question:-1(d)
If probability of an event E is 1//21 / 2 and probability of the event EnnF\mathrm{E} \cap \mathrm{F} is 1//61 / 6, then probability of the event FF is 1//31 / 3, where events EE and FF are independent.
Answer:
The statement "If the probability of an event EE is (1)/(2)\frac{1}{2} and the probability of the event E nn FE \cap F is (1)/(6)\frac{1}{6}, then the probability of the event FF is (1)/(3)\frac{1}{3}, where events EE and FF are independent" is true.
To justify this, we use the definition of independent events. For two independent events EE and FF, the probability of their intersection is given by the product of their individual probabilities:
P(E nn F)=P(E)*P(F)P(E \cap F) = P(E) \cdot P(F)
Given:
P(E)=(1)/(2)P(E) = \frac{1}{2}
P(E nn F)=(1)/(6)P(E \cap F) = \frac{1}{6}
We need to find P(F)P(F). Using the formula for independent events: