Let box I contains 3 black and 4 white balls, box II contains 2 black and 2 white balls, box III contains 4 black and 3 white balls. A. box is selected at random and then a ball is randomly drawn from the selected box. If the color of the ball is black, then the probability that the ball is drawn from box III, is:

Which of the following statements are correct?
(A) If elements of a row (or column) of a determinant are multiplied with cofactors of any other row (or column), then their sum is zero.
(B) The value of determinant is equal to the sum of the product of elements of any row (or column) with their corresponding cofactors.
(C) If any two rows (or columns) of a determinant are interchanged, their sign of determinant does not change.
(D) The value of determinant remains unchanged if its rows and columns are interchanged.
(E) A square matrix A has inverse if and only if A is singular.
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The equation of normal to the curve \(2 y+x^2=3\) at the point (1, 1) is:

In the context of differential equation
Match List I with List II

LIST I	LIST II
A. \(\frac{d y}{d x}=\frac{1+y^2}{1+x^2}\)	I. Not a differential equation
B. \(x^2 \frac{d y}{d x}=x^2-2 y^2+x y\)	II. Linear first order
C. \(\sin x+y=\cos (x+y)\)	III. Variable separable
D. \((x+y) \frac{d y}{d x}=1\)	IV. Homogenous

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The shortest distance between the lines \(\vec{r}_1=\hat{i}+\hat{j}+\lambda(2 \hat{i}-\hat{j}+\hat{k})\) and \(\vec{r}_2=2 \hat{i}+\hat{j}-\hat{k}+\mu(4 \hat{i}-2 \hat{j}+2 \hat{k})\) is :

The distance between the y-axis and the point (1, 2, 3) is:

The plane \(2 x-(1+\lambda) y+3 \lambda z=0, \lambda \in R\) passes through the intersection of planes :

Bacillus thuringiensis produces proteins (Bt toxin) that kill certain insects.
Identify the correct statements.
(A) Bt toxin protein exists as an inactive form (protoxin) in Bacillus
thuringiensis.
(B) Bt toxin protein exists as an active form toxin in the insect gut.
(C) The alkaline pH of the gut of an insect solubilize the toxin and converts
in an inactive form.
(D) The toxin is coded by a gene crylac named cry.
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Two nuclei have mass numbers in the ratio 1:8. The ratio of their nuclear radii would be :

During the process of spermatogenesis, the secondary spermatocytes
undergo ___________ division to produce ___________ spermatids

The focal length of the objective of a microscope is \( f_o \) and that of the eye piece is \( f_e \). The aperture of objective is \( d_o \) and that of eye piece is \( d_e \). Which of the following relation is correct ?

Match List - I with List - II.

List - I	List - II
(A) Sexually transmitted disease	(I) Assisted reproductive technology
(B) Intra-uterine devices	(II) Detection of genetic disorder
(C) In-vitro fertilization	(III) Contraception method
(D) Amniocentesis	(IV) AIDS

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Which of the following is true for species area relationship equation log S = log C + Z log A