AP PGECET Question Paper 2025 (Available) : Download Biotechnology Question Paper with Solution PDF

Which one of the following is most accurate for determining bacterial cell viability?

• (1) Gram staining
• (2) Colony-forming unit (CFU) count
• (3) Optical density
• (4) Total cell count using a hemocytometer

The most common mode of transmission for diphtheria, pneumonia, and tuberculosis is through:

• (1) Direct contact
• (2) Aerosols
• (3) Contaminated water
• (4) Insect vectors

Which enzyme allows retroviruses to transcribe RNA into DNA?

• (1) DNA polymerase
• (2) Reverse transcriptase
• (3) RNA polymerase
• (4) Ligase

Which growth phase is characterized by the most rapid increase in cell number and maximal metabolic activity?

• (1) Lag phase
• (2) Log (exponential) phase
• (3) Stationary phase
• (4) Death phase

The prominent group of microorganisms involved in marine biocorrosion is:

• (1) Sulphur oxidizing bacteria
• (2) Iron oxidizing bacteria
• (3) Sulphide oxidizing bacteria
• (4) Sulphate reducing bacteria

Which of the following is not a characteristic of anaerobic respiration in bacteria?

• (1) Use of oxygen as final electron acceptor
• (2) Generation of ATP via electron transport chain
• (3) Use of inorganic molecules like nitrate or sulfate
• (4) Less energy yield compared to aerobic respiration

Photosynthesis is a:

• (1) Reductive, endergonic, anabolic process
• (2) Reductive, exergonic, catabolic process
• (3) Reductive, exergonic, anabolic process
• (4) Reductive, endergonic, catabolic process

Which of the following is a selective medium for Gram-negative bacteria?

• (1) MacConkey agar
• (2) Chocolate agar
• (3) Blood agar
• (4) Nutrient agar

A transducing phage differs from a regular bacteriophage in that it:

• (1) Carries plasmid DNA
• (2) Forms lysogens in all hosts
• (3) Transfers bacterial DNA from one cell to another
• (4) Lyses host cells more rapidly

Which of the following statements about prions is true?

• (1) They are viruses with no envelope
• (2) They contain RNA
• (3) They replicate without nucleic acids
• (4) They are bacteria with unusual morphology

Agar-agar is a polymer of:

• (1) Glucose
• (2) Sulphated sugar
• (3) Pectin
• (4) Protein

Which of the following enzymes is responsible for converting glucose-6-phosphate to fructose-6-phosphate in glycolysis?

• (1) Hexokinase
• (2) Phosphoglucose isomerase
• (3) Aldolase
• (4) Phosphofructokinase-1

Porins:

• (1) Are cytoskeletal proteins
• (2) Form channels which allow passage of hydrophilic molecules
• (3) Are fatty acids
• (4) Are pores in the stem of a plant

Which vitamin is a precursor for the coenzyme NAD+?

• (1) Vitamin B1 (Thiamine)
• (2) Vitamin B2 (Riboflavin)
• (3) Vitamin B3 (Niacin)
• (4) Vitamin B6 (Pyridoxine)

A prominent prebiotic substance is:

• (1) Starch
• (2) Pectin
• (3) Fructo oligosaccharide
• (4) Cellulose

During the TCA cycle, which step results in substrate-level phosphorylation?

• (1) Isocitrate to α-ketoglutarate
• (2) α-ketoglutarate to Succinyl-CoA
• (3) Succinyl-CoA to Succinate
• (4) Malate to Oxaloacetate

Which of the following antibiotics is produced by chemical synthesis?

• (1) Penicillin
• (2) Streptomycin
• (3) Tetracycline
• (4) Chloramphenicol

In oxidative phosphorylation, inhibition of complex III would most directly lead to accumulation of which of the following?

• (1) NAD+
• (2) NADH
• (3) Reduced Ubiquinol (QH2)
• (4) Cytochrome c in oxidized form

The Bohr effect describes:

• (1) Increased O\(_2\) affinity of hemoglobin at low pH
• (2) Decreased O\(_2\) affinity of hemoglobin at low pH
• (3) Cooperative binding of O\(_2\) to hemoglobin
• (4) Allosteric inhibition of hemoglobin by CO\(_2\)

The pentose phosphate pathway produces:

• (1) NADH and ribose-5-phosphate
• (2) NADPH and ribose-5-phosphate
• (3) FADH\(_2\) and glyceraldehyde-3-phosphate
• (4) ATP and pyruvate

Which of the following would be most affected in a cell lacking functional phosphatidylinositol-4,5-bisphosphate (PIP2)?

• (1) Activation of protein kinase A
• (2) Activation of phospholipase C
• (3) DNA replication
• (4) Protein synthesis

The parts of proteins having the highest flexibility are:

• (1) α-helices
• (2) β-sheets
• (3) Peptide bonds
• (4) Surface side chains

The chemiosmotic hypothesis explains ATP synthesis by:

• (1) Substrate-level phosphorylation
• (2) Proton gradient-driven ATP synthase
• (3) Direct transfer of electrons to ADP
• (4) Hydrolysis of GTP

In bacterial operons, which mutation would prevent transcription of downstream structural genes but not affect the promoter region?

• (1) Mutation in the operator
• (2) Mutation in the promoter
• (3) Mutation in the repressor gene
• (4) Mutation in the Shine-Dalgarno sequence

In DNA replication, the leading strand is synthesized:

• (1) Discontinuously in Okazaki fragments
• (2) By reverse transcriptase
• (3) Only during mitosis
• (4) Continuously in the 5' → 3' direction

Major gluconeogenesis occurs in:

• (1) Liver and kidney
• (2) Liver and heart
• (3) Liver and skeletal muscle
• (4) Liver and adrenal gland

The central dogma of molecular biology describes:

• (1) DNA \(\rightarrow\) RNA \(\rightarrow\) Protein
• (2) RNA \(\rightarrow\) DNA \(\rightarrow\) Protein
• (3) Protein \(\rightarrow\) RNA \(\rightarrow\) DNA
• (4) DNA \(\rightarrow\) Protein \(\rightarrow\) RNA

A karyotype is used to visualize:

• (1) Protein structure
• (2) RNA sequences
• (3) Chromosome number and structure
• (4) Metabolic pathways

If side chains of amino acids interact with each other, which of the following would be termed as a salt bridge?

• (1) Tyr - Phe
• (2) Cys - Cys
• (3) Lys - Glu
• (4) Ala - Val

In eukaryotic transcription, the carboxy-terminal domain (CTD) of RNA polymerase II is essential for:

• (1) DNA binding specificity
• (2) Sigma factor recruitment
• (3) RNA splicing, capping, and polyadenylation coordination
• (4) Enhancer binding

Which of the following organisms typically get their carbon for biosynthesis from carbon dioxide?

• (1) Glucose fermenting bacteria
• (2) Anaerobic glucose respiring bacteria
• (3) Aerobic glucose respiring bacteria
• (4) Ammonia oxidizing bacteria

The Ti plasmid, used in plant genetic engineering, is naturally found in:

• (1) Agrobacterium tumefaciens
• (2) Escherichia coli
• (3) Bacillus thuringiensis
• (4) Saccharomyces cerevisiae

A child inherits two different mutant alleles for a recessive disease gene from each parent. This condition is known as:

• (1) Compound heterozygosity
• (2) Homozygosity
• (3) Dominant negative mutation
• (4) Heteroplasmy

Which of the following are components of a phospholipid?

• (1) Cholesterol, glycerol, fatty acids
• (2) Fatty acids, phosphate group, glycerol
• (3) Glycerol, amino acids, phosphate group
• (4) Phosphate group, cholesterol, monosaccharides

Which enzyme is crucial for the cleavage of DNA at specific sites in genetic engineering?

• (1) DNA ligase
• (2) Restriction endonuclease
• (3) DNA polymerase
• (4) Reverse transcriptase

In the production of citric acid by Aspergillus niger, the accumulation of citric acid is primarily due to:

• (1) Inhibition of the TCA cycle enzyme isocitrate dehydrogenase by Mn\(^{2+}\) deficiency
• (2) Increased expression of citrate synthase
• (3) Enhanced glycolytic flux
• (4) Addition of iron salts

The net charge of a protein may not be sufficient to determine whether a protein will bind to an ion exchanger. This is due to:

• (1) The presence of hydrophobic patches on the protein surface
• (2) Heterogeneous spatial distribution of charged amino acids
• (3) The presence of repeating motifs in some proteins
• (4) The strong hydration potential of protein

What is the primary role of calcium alginate in enzyme immobilization?

• (1) Provides covalent attachment to enzymes
• (2) Maintains high substrate concentration
• (3) Forms a gel matrix for entrapment
• (4) Increases enzyme turnover number

Which plant hormone is most commonly used to induce shoot formation in tissue culture?

• (1) Auxin
• (2) Gibberellin
• (3) Abscisic acid
• (4) Cytokinin

Which microorganism is most commonly used for large-scale industrial production of L-glutamic acid?

• (1) Bacillus subtilis
• (2) Escherichia coli
• (3) Corynebacterium glutamicum
• (4) Pseudomonas aeruginosa

In aerobic solid waste stabilization, oxygen transfer limitation primarily affects:

• (1) Nitrate reduction
• (2) Fungal colonization
• (3) Thermophilic bacterial activity
• (4) Anaerobic methanogenesis

Knockout mice are primarily used to study:

• (1) Gene function
• (2) Protein purification
• (3) Viral replication
• (4) Antibody production

Which one of the following tools of recombinant DNA technology is INCORRECTLY paired with its applications?

• (1) restriction endonuclease - production of DNA fragments for gene cloning.
• (2) DNA ligase - enzyme that cuts DNA, creating sticky ends
• (3) DNA polymerase - copies DNA sequences in the polymerase chain reaction
• (4) reverse transcriptase - production of cDNA from mRNA

The main purpose of using reporter genes in transgenic animals is to:

• (1) Track gene expression
• (2) Increase growth rate
• (3) Enhance disease resistance
• (4) Improve reproductive capacity

Which of the following is a common consequence of improper pH control in citric acid fermentation?

• (1) Inactivation of citrate synthase
• (2) Accumulation of oxalic acid
• (3) Enhanced ethanol production
• (4) Increased biomass yield

Which one of the following amino acids is optically inactive?

• (1) Glycine
• (2) Methionine
• (3) Phenylalanine
• (4) Glutamine

The chemical nature of covalent linkage in a disaccharide is known as:

• (1) Ester
• (2) Ether
• (3) Amide
• (4) Diester

Which technique is used to produce identical copies of a particular gene?

• (1) Gene therapy
• (2) Gene knockout
• (3) Gene silencing
• (4) Gene cloning

Under steady-state conditions, the average specific growth rate of the culture is:

• (1) Zero
• (2) Equal to maximum growth rate
• (3) Equal to dilution rate
• (4) Less than dilution rate

Which of the following receptors is not a signalling receptor?

• (1) Cytokine receptor
• (2) Chemokine receptor
• (3) T-cell receptor
• (4) Mannose receptor

Which one of the following phytohormones is produced under water-deficit and plays an important role in tolerance against drought?

• (1) Abscisic acid
• (2) Cytokinin
• (3) Ethylene
• (4) Gibberellin

In non-Newtonian fermentation broths, the flow behavior index (n) < 1 indicates:

• (1) Newtonian fluid
• (2) Pseudoplastic fluid
• (3) Dilatant fluid
• (4) Bingham plastic fluid

Which immunodeficiency is X-linked and affects BTK kinase?

• (1) SCID
• (2) CGD
• (3) Wiskott-Aldrich syndrome
• (4) Bruton's agammaglobulinemia

Exact mass and sequence of proteins and peptides can be measured by:

• (1) CD Spectroscopy
• (2) Proton NMR
• (3) X-Ray
• (4) Mass spectroscopy

A chemostat operating with two substrates exhibits diauxic growth if:

• (1) Both substrates are utilized simultaneously
• (2) One substrate represses the other’s uptake
• (3) They have the same saturation constants
• (4) Substrate concentrations are equal

A patient with hyper-IgM syndrome likely has a defect in:

• (1) RAG-1/2
• (2) CD40 ligand
• (3) IL-2 receptor
• (4) NADPH oxidase

Which of the following complementarity determining regions (CDRs) of antibodies is sequentially and conformationally the most variable?

• (1) CDR1 of Light chain
• (2) CDR3 of Light chain
• (3) CDR1 of Heavy chain
• (4) CDR3 of Heavy chain

Which of the following factors most strongly influences the expression of totipotency in plant tissue culture?

• (1) Light intensity
• (2) Carbon dioxide levels
• (3) Hormonal balance in media
• (4) Temperature variation

The plasmid pBR322 contains two antibiotic resistance genes:

• (1) ampR and tetR
• (2) kanR and catR
• (3) neoR and hygR
• (4) bla and specR

Which one of the following matrices can be used to identify distantly related homologs?

• (1) BLOSUM90
• (2) BLOSUM62
• (3) BLOSUM45
• (4) BLOSUM80

Vortexing in a stirred tank reactor can be prevented by using:

• (1) axial flow impeller
• (2) a turbine impeller
• (3) baffles in the reactor
• (4) multiple impellers

Which of the following is a plant growth inhibitor rather than a promoter?

• (1) Cytokinin
• (2) Gibberellin
• (3) Abscisic acid
• (4) Auxin

Polymerases are available with proof reading activity. Which of the following are the characteristics of these types of polymerases?

• (1) They add A residue at 3' end
• (2) They are obtained from Thermococcus litoralis
• (3) They can't be obtained from archaebacteria
• (4) The marine bacteria from which they are obtained grow at temperatures lower than that of Thermus aquatics

Which one of the following is a signaling receptor?

• (1) mannose receptor
• (2) toll-like receptor
• (3) scavenger receptor
• (4) LPS receptor

Optimization of which of the following nutrients is most critical for anthocyanin production in plant suspension cultures?

• (1) Nitrogen
• (2) Magnesium
• (3) Iron
• (4) Phosphate

Which one of the following is the most suitable type of impeller for mixing high viscosity (viscosity is greater than 10 power 5 cP) fluids?

• (1) Propeller
• (2) Flat blade turbine
• (3) Paddle
• (4) Helical ribbon

Cell suspension cultures are usually derived from:

• (1) Leaf discs
• (2) Root tips
• (3) Callus tissue
• (4) Zygotic embryos

What is the pH of the medium when sucrose is used as substrate for the production of citric acid?

• (1) 3
• (2) 4
• (3) 5
• (4) 6

Innate immunity is mediated by:

• (1) Toll like receptors
• (2) G protein coupled receptors
• (3) Integrins
• (4) FGF receptor

Which factor most critically determines the attachment efficiency of animal cells on microcarriers?

• (1) Oxygen concentration
• (2) Surface charge and coating of the microcarrier
• (3) Agitation speed
• (4) pH of the medium

What is the primary disadvantage of using spinner flasks for hybridoma culture?

• (1) Low oxygen transfer rate
• (2) Non-uniform pH
• (3) High shear stress
• (4) Inability to scale-up

An operon is:

• (1) Regulatory molecule that turns genes on and off
• (2) Cluster of regulatory sequences controlling transcription of protein coding genes.
• (3) Cluster of genes that are co-ordinately regulated
• (4) Promoter, an operator, and a group of linked structural genes

Which of the following precursors is added to the medium to get penicillin G?

• (1) Phenyl carbamic acid
• (2) Phenyl acetic acid
• (3) Ammonium sulphate
• (4) Ammonium chloride

The Cytokinin receptor is:

• (1) A G-protein coupled receptor
• (2) A tyrosine kinase
• (3) An acidic cytosolic protein
• (4) A two-component histidine kinase

Which strategy can be used to minimize the shear damage in bioreactors used for animal cell culture?

• (1) Using spargers
• (2) Increasing agitation rate
• (3) Adding Pluronic F-68
• (4) Increasing temperature

Which one of the following is not a deficiency disorder?

• (1) Beriberi
• (2) Night Blindness
• (3) Poliomyelitis
• (4) Pernicious Anemia

The overall stoichiometry for aerobic cell growth is:

3C\textsubscript{6}H\textsubscript{12}O\textsubscript{6} + 2.5NH\textsubscript{3} + O\textsubscript{2} \(\rightarrow\) 1.5C\textsubscript{6H\textsubscript{8O\textsubscript{4Na + 3CO\textsubscript{2 + 5H\textsubscript{2O


The elemental composition formula of the biomass is:

• (1) C\textsubscript{5}H\textsubscript{22}O\textsubscript{6}N\textsubscript{1.667}
• (2) C\textsubscript{10}H\textsubscript{12}O\textsubscript{6}N\textsubscript{2.667}
• (3) C\textsubscript{5}H\textsubscript{12}O\textsubscript{6}N\textsubscript{1.667}
• (4) C\textsubscript{10}H\textsubscript{22}O\textsubscript{6}N\textsubscript{1.667}

Inosine in the tRNA anticodon will base pair with all except which one of the following bases in the codon of mRNA?

• (1) Adenine
• (2) Uracil
• (3) Cytosine
• (4) Guanine

Which of the following best explains the absence of immune response to self-antigens under normal physiological conditions?

• (1) Clonal expansion
• (2) Clonal deletion
• (3) Somatic recombination
• (4) Affinity maturation

Which gene is typically inserted into host cells to facilitate monoclonal antibody production using hybridoma technology?

• (1) IgG heavy chain
• (2) Myc oncogene
• (3) HAT resistance gene
• (4) DHFR gene

The primary function of MHC class I molecules is to present:

• (1) Bacterial polysaccharides to B cells
• (2) Viral peptides to CD8+ T cells
• (3) Self-antigens to NK cells
• (4) Peptides to CD4+ T cells

Which algorithm is most commonly used for pairwise sequence alignment?

• (1) BLAST
• (2) Smith-Waterman
• (3) Needleman-Wunsch
• (4) Hidden Markov Model

Mixing time increases with the volume of the reactor because of increase in the:

• (1) Circulation time
• (2) Shear
• (3) Turbulence
• (4) Flow rate

In the thymus, positive selection of T cells ensures:

• (1) T cells do not recognize self-antigens
• (2) T cells recognize antigens presented by MHC molecules
• (3) Elimination of self-reactive clones
• (4) Clonal anergy

Which tool would you use to predict transmembrane helices in a protein?

• (1) TMHMM
• (2) PSIPRED
• (3) CLUSTAL OMEGA
• (4) MUSCLE

In Graft-Versus-Host-Disease (GVHD), the immunocompetent donor T cells:

• (1) Attack the host's transplanted organ
• (2) Are attacked by the host immune system
• (3) Attack the host's tissues
• (4) Tolerate the host's tissues

Genome-Wide Association Studies primarily identifies:

• (1) Protein-protein interactions
• (2) SNPs linked to phenotypes
• (3) Alternative splicing events
• (4) Horizontal gene transfer

Amphotericin B is clinically used against which one of the following pathogens?

• (1) Herpes simplex virus I
• (2) M. tuberculosis
• (3) Candida spp.
• (4) P. vivax

The ENCODE project aims to:

• (1) Annotate all functional elements in the human genome
• (2) Sequence all prokaryotic genomes
• (3) Develop new sequencing technologies
• (4) Catalog protein structures

Match the items in Group 1 with an appropriate description in Group 2:

• (1) P-4, Q-1, R-2, S-3
• (2) P-2, Q-4, R-1, S-3
• (3) P-2, Q-3, R-1, S-4
• (4) P-2, Q-1, R-4, S-3

Which of the following vectors can accommodate the largest foreign DNA insert?

• (1) Plasmid
• (2) Cosmid
• (3) Bacteriophage lambda
• (4) Yeast artificial chromosome (YAC)

Which one of the following hormones promotes production of seedless grapes?

• (1) IAA
• (2) IBA
• (3) BAP
• (4) GA3

The term "washout" in continuous culture refers to:

• (1) Cell death due to high pH
• (2) Removal of dead cells only
• (3) Cell loss exceeding growth rate
• (4) Incomplete mixing

Which transposable element uses a “copy-and-paste” mechanism to move within the genome?

• (1) DNA transposon
• (2) Retrotransposon
• (3) Ac-Ds element
• (4) Insertion sequence

Seedless fruits may arise as a result of:

• (1) Parthenocarpy
• (2) Sexual reproduction
• (3) Autogamy
• (4) Allogamy

In downstream processing, chromatography is primarily used for:

• (1) Cell harvesting
• (2) Cell lysis
• (3) Product concentration
• (4) Product purification

Which of the following techniques uses temperature cycling for DNA amplification?

• (1) DNA fingerprinting
• (2) PCR
• (3) RAPD
• (4) Northern blotting

High frequency heterokaryon formation is observed during protoplast fusion by the addition of:

• (1) Glycerol
• (2) PEG
• (3) NaNO₃
• (4) DMSO

The Monod equation relates microbial growth rate to:

• (1) Oxygen concentration
• (2) Enzyme activity
• (3) Substrate concentration
• (4) Biomass concentration

Site-directed mutagenesis is commonly used to:

• (1) Increase plasmid yield
• (2) Knock out specific genes in yeast
• (3) Introduce specific point mutations in DNA
• (4) Identify transposable elements

What type of sequence alignment does BLAST primarily perform?

• (1) Pairwise
• (2) Multiple sequence
• (3) Global
• (4) Local

The oxygen uptake rate (OUR) in a bioprocess is dependent on:

• (1) Substrate concentration only
• (2) Cell density and metabolic activity
• (3) Agitation rate only
• (4) Temperature alone

What region of an mRNA is most commonly associated with transcript destabilization?

• (1) The 5' untranslated region
• (2) The 3' untranslated region
• (3) The exonic coding regions
• (4) The intronic regions

Which scoring matrix is primarily used in amino acid sequence alignment?

• (1) PAM
• (2) BLAST
• (3) T-Coffee
• (4) GFF

High cell density fermentations often exhibit:

• (1) Newtonian behavior
• (2) Pseudoplastic (shear-thinning) behavior
• (3) Dilatant (shear-thickening) behavior
• (4) Rheopectic behavior

Which one of the following is the causative agent of Typhoid fever?

• (1) V. cholera
• (2) P. multocida
• (3) S. Typhi
• (4) E. coli

What is the main purpose of DNA microarrays in genomics?

• (1) To identify protein-protein interactions
• (2) To study gene expression levels across different conditions
• (3) To sequence the entire genome
• (4) To determine protein structures

Dissolved oxygen (DO) in a bioreactor is typically measured using a:

• (1) Thermocouple
• (2) pH electrode
• (3) Polarographic electrode
• (4) Conductivity meter

Which one of the following is the causative agent of fowl cholera?

• (1) V. cholera
• (2) P. multocida
• (3) E. coli
• (4) S. Pullorum

The main end products of anaerobic digestion of organic waste are:

• (1) CO\textsubscript{2} and H\textsubscript{2}O
• (2) Methane (CH\textsubscript{4}) and CO\textsubscript{2}
• (3) Nitrates and sulfates
• (4) Oxygen and biomass

If the determinant of the 3 x 3 matrix A = \(\begin{pmatrix} a & 1 & 2
b & 0 & -2
1 & -3 & 1 \end{pmatrix}\) is zero, then the values of a and b are:

• (1) a = 0, b = 0
• (2) a = \(\frac{3}{2}\), b = 1
• (3) a = \(\frac{1}{3}\), b = 0
• (4) a = \(-\frac{1}{6}\), b = \(-\frac{1}{7}\)

Let A = \(\begin{pmatrix} 0 & 1 & 1
1 & 0 & 1
1 & 1 & 0 \end{pmatrix}\) be a 3 x 3 matrix. If \(\alpha\) and \(\beta\) are the largest and smallest eigenvalues of A, respectively, then \(\alpha - \beta\) = ___

• (1) 0
• (2) 1
• (3) 2
• (4) 3

The value of the real variable x > 0 that minimizes the function \(f(x) = x^{-x}e^{x}\) is

• (1) e
• (2) \( \frac{1}{e} \)
• (3) \( \sqrt{e} \)
• (4) 1

If \(f(x) = |x - 1|\), then

• (1) \( f(x) \) is differentiable at \(x = 1\)
• (2) \( f(x) \) is not differentiable at \(x = 1\)
• (3) \( f(x) \) is not differentiable at \(x = 0\)
• (4) \( f(x) \) is not continuous at \(x = 0\)

The solution of the differential equation \(\frac{d^2y}{dx^2} - 3 \frac{dy}{dx} + 2y = 0\) satisfying \(y(0) = 0\), \(y'(0) = 1\), is

• (1) \( y(x) = e^x - e^{2x} \)
• (2) \( y(x) = e^x + e^{2x} \)
• (3) \( y(x) = -e^x - e^{2x} \)
• (4) \( y(x) = -e^x + e^{2x} \)

The inverse Laplace transformation of \( \frac{s+5}{s^2 + 4s + 4} \) for \( t \geq 0 \), is

• (1) \( 4e^{2t} \)
• (2) \( 4e^{-2t} \)
• (3) \( (1 + 3t)e^{2t} \)
• (4) \( (1 + 3t)e^{-2t} \)

The probability distribution of a random variable \( X \) is given as follows. Then, \( P(X = 50) - \frac{P(X \leq 30)}{P(X \geq 20)} \) =

• (1) \( \frac{2}{3} \)
• (2) \( \frac{5}{6} \)
• (3) \( \frac{1}{12} \)
• (4) 0

Simpson's \( \frac{1}{3} \) rule is applied when

• (1) the number of intervals is divisible by 3
• (2) the number of intervals is divisible by 2
• (3) the number of intervals is divisible by 5
• (4) the number of intervals is divisible by 7

Let a random variable \( X \) follow Poisson distribution such that \( P(X = 0) = 2P(X = 1) \). Then, \( P(X = 3) = ______

• (1) \( \frac{1}{6e} \)
• (2) \( \frac{1}{48\sqrt{e}} \)
• (3) \( \frac{4}{3e^2} \)
• (4) \( \frac{1}{2} \)

If \( A \) and \( B \) are two events having probabilities, \( P(A) = 0.6 \), \( P(B) = 0.3 \), and \( P(A \cap B) = 0.2 \), then the probability that neither \( A \) nor \( B \) occurs is ______

• (1) 0
• (2) 0.3
• (3) 0.7
• (4) 0.8