If \(y=\sin ^{-1}\left(\frac{1-x^2}{1+x^2}\right)\) then \(\frac{d y}{d x}=\)

Let \(y(x)=a(x+1) \log (x+1)+b x+5\) be the solution of the differential equation \(e^{\frac{d y}{d x}}=x+1, y(0)=5\), then the value of \((a+b)\) is :

A boat can row at the speed of 16 km/hr in still water. If the river is flowing at 8 km/hr, and it takes 8 hours for a round trip, then the distance between the two places is :

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 area of the triangle with vertices \(A(2, k), B(1,1), C(10,8)\) is \(\frac{47}{2}\) sq. units, then one of the values of \(k\) is:

\(\int_0^{\frac{\pi}{3}} \frac{\sin x}{(2-\cos x)^2} d x=\)

If \(3\left[\begin{array}{cc}x & y \\ z & w\end{array}\right]=\left[\begin{array}{cc}x & 6 \\ -1 & 2 w\end{array}\right]+\left[\begin{array}{cc}4 & x+y \\ z+w & 3\end{array}\right]\), then the values of \(x, y, z\) and \(w\) are

For the linear programming problem (LPP),
Maximize \(Z=4 x+y\)
Subject to:
\(x+y \leq 5\)
\(3 x+y \leq 9\)
\(x, y \geq 0\)
Which of the following are NOT true?
(A) The given LPP has unbounded feasible region.
(B) The corner points of the feasible region are \((0,0),(0,5),(3,2)\) and \((3,0)\).
(C) The optimal value of the objective function is 12.
(D) The given LPP has a unique optimal solution.
Choose the correct answer from the options given below:

Down's syndrome is caused by:

The major product formed when acetanilide undergoes bromination in presence of acetic acid is:

Which of the following reactions has fractional order with respect to one of the reactants?

Decide which of the following atomic numbers are the atomic numbers of the inner transition elements
(A) 59
(B) 95
(C) 102
(D) 74
Choose the correct answer from the options given below:

The hormone which control the processes of burning of fats, proteins and carbohydrates, with liberation of energy in the body is :