If \(\hat{a}\) is a unit vector perpendicular to both the vectors \(\vec{b}=\hat{j}+2 \hat{k}\) and \(\vec{c}=\hat{i}+2 \hat{j}\), then \(\hat{a}\) is equal to :

The function \(f(x)=\left\{\begin{array}{cl}\frac{x^2+2 x-3}{x-1} & , x \neq 1 \\ 2 & , x=1\end{array}\right.\) is :

Let \(f(x)=x^3-6 x^2+9 x-8\) be a function, then which of the following statements are TRUE?
(A) \(f^{\prime}(x)=3(x-1)(x-3)\)
(B) The critical points of the function are \(x=1\) and \(x=3\)
(C) \(x=1\) is the point of local minimum
(D) The local maximum value is -4
Choose the correct answer from the options below :

If A is a square matrix of order 3 and \(|A| = -2\) then, \(|-2A^{-1}|\) is:

If \(y = x^2 \log_e x\), then \(\frac{d^2y}{dx^2}\) is equal to:

If \(A\) is a square matrix of order 3 and \(|A|=5\), then the value of \(\left|-A A^T\right|\) is :

The corner points of the bounded feasible region associated with the LPP: Maximize \(Z=p x+q y, p, q>0\) are (0, 0), (3.5, 0), \(\left(\frac{112}{59}, \frac{135}{59}\right)\) and (0,3). If the optimum value of Z occurs at both \(\left(\frac{112}{59}, \frac{135}{59}\right)\) and (0,3), then

Arrange the following nucleophiles in the decreasing order of their attacking strength :
(A) \(C _6 H _5 COO ^{-}\)
(B) \(CH _3 CH _2 CH _2 COO ^{-}\)
(C) \(CH _3 CH _2 CH _2 CH _2 S^{-}\)
(D) \(CH _3 CH _2 CH _2 CH _2 O ^{-}\)
Choose the correct answer from the options given below :

The enzyme used to treat a plant cell for isolation of DNA is:

The electronic configuration of the last three orbitals in cerium (At. No. 58) in ground state is:

The length of the line segment joining the points with position vectors \(2 \hat{i}-2 \hat{j}+3 \hat{k}\) and \(5 \hat{i}+2 \hat{j}+3 \hat{k}\) is :

Answer the question on the basis of passage given below:
In the periodic table, the d-block contains the elements of group \(3\) to \(12\). The d-orbitals are progressively filled in each of the four long periods. The elements of d block referred as transition metals have partly filled d orbitals and exhibit certain characteristic properties such as variety of oxidation states, formation of coloured ions, act as catalyst and show paramagnetic behaviour.
The two inner transition metals series \(4f\) and \(5f\) are known as Lanthanoids and Actinoids respectively. The lanthanoids resemble one another more closely as compared to ordinary transition elements in any series.
In context of the Lanthanoids, which of the following statements is/are NOT correct?
A. There is a gradual decrease in the radii of the members with increasing atomic number in the series
B. Availability of \(4f\) electrons results in the formation of compounds in \(+4\) oxidation state for all the members of the series
C. Because of similar properties, the separation of lanthanoids is not easy
D. \(La(OH)_3\) is least basic among hydroxides of lanthanoids
E. Ce can act as an oxidising agent
Choose the correct answer from the options given below:

The kinetic energy of an electron in the ground state of hydrogen atom is \(1 3 . 6 ~ e V\). The potential energy (P.E) and total energy ( E ) of electron in this state is :