If \(f(x)=a \log _e|x|+b x^2+x\) has critical points at \(x=-2\) and \(x=1\), then

A and B invested ₹ 10,000 and ₹ ₹ 12,000 in a business two years ago. After 1 year their net loss was ₹ 880 . And during the 2 nd year their net profit was ₹ 2,530. So the presen

Which of the following statement(s) is/are TRUE?
(A) Skew symmetric matrix of even order is always symmetric
(B) Skew symmetric matrix of odd order is non-singular
(C) Skew symmetric matrix of odd order is singular
(D) Skew symmetric matrix is always square matrix
Choose the correct answer from the options given below:

\(\int \frac{\sqrt{16+(\log x)^2}}{x} d x\) is equal to (where \(C\) is an arbitrary constant)

The order of \(\sqrt{1+\left(\frac{d y}{d x}\right)^2}=\left[a \frac{d^2 y}{d x^2}\right]^{\frac{1}{3}}\) is

Let \(A=\left[\begin{array}{cc}0 & 2 \alpha+1 \\ 1 & \beta\end{array}\right]\) and \(B=\left[b_{i j}\right]\) be a skew symmetric matrix of order 2 such that \(b_{12}=1\).
If \(A B=I_2\), where \(I_2\) is the identity matrix of order 2 , then :

Evaluate \(\int_0^{\frac{\pi}{4}} \frac{\sin 2 x}{\cos ^4 x+\sin ^4 x} d x=\)

A photoelectric surface is illuminated successively by monochrounstic light of wavelength \(\lambda\) and \(0.5 \lambda\). What will be the work function of the photosenaltive material of the marimum linetic energy of the photo electrons in the second case is 5 times that of the first case?

First meiotic division of primary follicles completes in the

\(\int \frac{\cos x-\sin x}{1+\sin 2 x} d x\) is equal to

Identify the incorrect statement(s) among the following:
when a dielectric medium is introduced between the plates of a charged capacitor:
(A) If a plate of dielectric constant k, thickness t is placed between the plates of a charged capacitor of area A and distance d, then the new capacitance will be \( \frac{A\varepsilon_0}{d - t(1 - 1/k)} \)
(B) The electric field between plates of the charged capacitor when the dielectric is introduced will be \( E = (E_0 - E_p) \)
(C) Dielectric constant of the dielectric placed between plates of charged capacitor will be \( k = \frac{E_0}{E_0 - E_p} \)
(D) Dielectric strength of the dielectric is the maximum value of the electric field which may be applied before its breakdown.
Choose the correct answer from the options given below :

Arrange the following elements in decreasing order of their radii:
(A) La
(B) Pr
(C) Gd
(D) Yb
Choose the correct answer from the options given below:

The equation of normal at the point (1, 1) on the curve \(2y + x^3 = 3\) is: