The general solution of the differential equation \(\frac{d y}{d x}+4 x y^2=0\) is :

\(\int \frac{1}{x^2\left(x^4+1\right)^{3 / 4}} d x\) is equal to :

If \(a, b\) and \(c\) are positive real numbers, then
Match List-I with List-II

List-I (Expression)	List-II (The Least value of the expression)
(A) \((a+b)(b+c)(c+a)\)	(I) \(8 a b c\)
(B) \((a+b+c)(a b+b c+c a)\)	(II) \(9 a^2 b^2 c^2\)
(C) \(\left(a^2+b^2+c^2\right)\left(a b^2+b c^2+c a^2\right)\)	(III) \(9 a b c\)
(D) \((a+b)^2(b+c)^2(c+a)^2\)	(IV) \(64 a^2 b^2 c^2\)

Choose the correct answer from the options given below:

Choose the correct answer from the options given below:
A. Equation of the line passing through the point (1, 2, 3) and parallel to the vector \(3 \hat{i}+2 \hat{j}-2 \hat{k}\) is \(\frac{x-1}{3}=\frac{y-2}{2}=\frac{y-3}{-2}\)
B. Equation of line passing through (1, 2, 3) and parallel to the line given by \(\frac{x+3}{3}=\frac{4-y}{5}=\frac{z+8}{6}\) is \(\frac{x-1}{3}=\frac{y-2}{5}=\frac{z+3}{6}\).
C. Equation of line passing through the origin and (5, -2,3) is \(\frac{x}{5}=\frac{y}{-2}=\frac{z}{3}\).
D. Equation of plane passing through the point (1, 2, 3) and perpendicular to the line with direction ratio's 2, 3, -1 is 2(x - 1) + 3(y - 2) - (z - 3) = 0.
E. Equation of plane with intercepts 2, 3 and 4 on x, y and z-axis respectively is 2x + 3y + 4z = 1

If \(A\) is a square matrix and \(|A|=4\), then the value of \(\left|A A^{\prime}\right|\) where \(A^{\prime}\) is transpose of \(A\) is:

\(\int \frac{1}{x \sqrt{x^2-9}} d x\) is equal to (given that C is constant of integration)

If a line makes angles \(\alpha, \beta\) and \(\gamma\) with the positive directions of coordinate axes respectively, then \(\cos 2 \alpha+\cos 2 \beta+\cos 2 \gamma\) is equal to

The value of \(\int_{-1}^1\left|x^3-x\right| d x\) is

A novel strategy to prevent infestation of roots of tobacco plant with the nematode Meloidegyne incognitia is :

A closely wound solenoid of 1000 turns and area of cross section \( 1.2 \text{ cm}^2 \) carries a current of 5 A. It is suspended horizontally in uniform magnetic field of \( 5 \times 10^{-2} \text{ T} \) making an angle of \( 30^\circ \) with the axis of solenoid. The torque acting on the solenoid is:

The minimum value of \(a x+b y\), where \(x y=c^2\) and \(a, b, c\) are positive, is:

Consider the non-empty set consisting of children in a family and a relation R is defined as aRb if a is a brother of b. Then R is:

The shortest distance between the following lines :
\(\vec{r}=(\hat{i}+\hat{j}-\hat{k})+s(2 \hat{i}+\hat{j}+\hat{k})\)
\(\vec{r}=(\hat{i}+\hat{j}+2 \hat{k})+t(4 \hat{i}+2 \hat{j}+2 \hat{k})\), where \(s\) and \(t\) are scalars, is :