(i) Draw a schematic diagram of the cloning vector pBR322 and label:
(1) BamHI site
(2) Gene for ampicillin resistance
(3) Ori
(4) Rop gene
The pBR322 vector is a widely used plasmid in genetic engineering. Key features include:
- BamHI site: A restriction site for inserting foreign DNA.
- Ampicillin resistance gene: Enables selection of transformed cells that can grow in the presence of ampicillin.
- Ori (Origin of replication): Ensures plasmid replication within the host cells.
- Rop gene: Regulates plasmid copy number for stable cloning. These components make pBR322 an efficient and versatile vector for molecular cloning, simplifying the process of inserting and propagating foreign DNA in host cells.
(b) Why do the symptoms of malaria not appear in a person immediately after being bitten by an infected female Anopheles? Give one reason. Explain when and how the symptoms of the disease would appear.
Read the following passage and answer the questions that follow:
“Mosquitoes are drastically affecting human health in almost all developing tropical countries. Different species of mosquitoes cause fatal diseases, leading to loss of life or reduced productivity. Consequently, the health index of these countries suffers.”
(a) Name the form in which Plasmodium gains entry into (1) human body (2) the female {Anopheles} body.
Self-study helps students to build confidence in learning. It boosts the self-esteem of the learners. Recent surveys suggested that close to 50% of learners were self-taught using internet resources and upskilled themselves.
A student may spend 1 hour to 6 hours in a day in upskilling self. The probability distribution of the number of hours spent by a student is given below:
\[ P(X = x) = \begin{cases} kx^2, & \text{for } x = 1, 2, 3, \\ 2kx, & \text{for } x = 4, 5, 6, \\ 0, & \text{otherwise.} \end{cases} \]
where \( x \) denotes the number of hours. Based on the above information, answer the following questions:
(i) Express the probability distribution given above in the form of a probability distribution table.
(ii) Find the value of \( k \).
(iii)(a) Find the mean number of hours spent by the student.
(iii)(b) Find \( P(1 < X < 6) \).
A bacteria sample of certain number of bacteria is observed to grow exponentially in a given amount of time. Using the exponential growth model, the rate of growth of this sample of bacteria is calculated.
The differential equation representing the growth of bacteria is given as:
\[ \frac{dP}{dt} = kP, \] where \( P \) is the population of bacteria at any time \( t \).
Based on the above information, answer the following questions:
(i) Obtain the general solution of the given differential equation and express it as an exponential function of \( t \).
(ii) If the population of bacteria is 1000 at \( t = 0 \), and 2000 at \( t = 1 \), find the value of \( k \).
A scholarship is a sum of money provided to a student to help him or her pay for education. Some students are granted scholarships based on their academic achievements, while others are rewarded based on their financial needs.
Every year, a school offers scholarships to girl children and meritorious achievers based on certain criteria. In the session 2022–23, the school offered a monthly scholarship of ₹3,000 each to some girl students and ₹4,000 each to meritorious achievers in academics as well as sports.
In all, 50 students were given the scholarships, and the monthly expenditure incurred by the school on scholarships was ₹1,80,000.
Based on the above information, answer the following questions:
(i) Express the given information algebraically using matrices.
(ii) Check whether the system of matrix equations so obtained is consistent or not.
(iii)(a) Find the number of scholarships of each kind given by the school using matrices.
(iii)(b) Had the amount of scholarship given to each girl child and meritorious student been interchanged, what would be the monthly expenditure incurred by the school?
Show that \( R \) is an equivalence relation. Also, write the equivalence class \([2]\).