\(A\) and \(B\) are two independent events. The probability that both events \(A\) and \(B\) occur is \(\frac{1}{6}\) and the probability that neither of them occur is \(\frac{1}{3}\). If \(P(A)=x, P(B)=y\) then the value of \(x+y\) is.

Out of the given statement, choose the correct statement.
(A) The direction ratios of the vector \(\vec{a}=3 \hat{i}-\hat{j}+4 \hat{k}\) is \(3,-1,4\).
(B) If \(\theta\) is the angle between two vectors \(\vec{a}\) and \(\vec{b}\), then their cross product is given as \(\vec{a} \times \vec{b}=|\vec{a}||\vec{b}| \cos \theta\).
(C) The unit vector in the direction of vector \(\vec{a}=\hat{i}+2 \hat{j}-2 \hat{k}\) is \(\hat{a}=\frac{1}{3}(\hat{i}+2 \hat{j}-2 \hat{k})\).
(D) If \(\vec{a}=3 \hat{i}\) and \(\vec{b}=4 \hat{j}\) then \(\vec{a} \cdot \vec{b}=12\).
(E) If \(\vec{a}\) and \(\vec{b}\) represent the adjacent sides of a triangle then its area is given of \(\frac{1}{2}|\vec{a} \times \vec{b}|\).
Choose the correct answer from the options given below :

Match List-I with List-II (where \(c\) is an arbitrary constant)

List-I (Definite integral)	List-II (Value)
(A) \(\int_1^e \frac{\log x}{x} d x\)	(I) 4
(B) \(\int_{-2}^2 x^3\left(1-x^2\right) d x\)	(II) \(\frac{1}{2}\)
(C) \(\int_1^2 x d x\)	(III) \(0\)
(D) \(\int_{-2}^2|x| d x\)	(IV) \(\frac{3}{2}\)

Choose the correct answer from the options given below :

For \(x>e\),\(\int \frac{d x}{x-\sqrt{x}}\) is equal to

If \(y=\log x^5\), then \(\frac{d^2 y}{d x^2}\) is given by :

The corner points of the feasible region determined by a system of linear constraints are (0,0), (0, 40), (20, 40), (60, 20), (60, 0). If the objective function is \(Z = 4x + 3y\) then which one of the following is true?

The value of \(|\text{adj}(2A)|\), if \(A = [a_{ij}]_{3 \times 3}\) and \(|A| = 2\) is:

Three resistors, each of resistance R, are arranged in the following different combinations with a source of emf E. Arrange them in the decreasing order of power dissipation:
(A) Three resistors in series
(B) Three resistors in parallel
(C) One resistor in series with two resistors in parallel
(D) One resistor in parallel with two resistors in series
Choose the correct answer from the options given below:

If \(f(x)=\cos 3 x, 0 \leq x \leq \frac{\pi}{2}\), then :
(A) \(f\) is strictly increasing on \(\left[0, \frac{\pi}{3}\right]\)
(B) \(f\) is strictly decreasing on \(\left(\frac{\pi}{3}, \frac{\pi}{2}\right]\)
(C)\(f\) is strictly increasing on \(\left(\frac{\pi}{3}, \frac{\pi}{2}\right]\)
(D) \(f\) is strictly decreasing on \(\left[0, \frac{\pi}{3}\right]\)
(E) \(f\) is strictly increasing on \(\left[0, \frac{\pi}{2}\right]\)
Choose the correct answerfrom the ptions given below:

The ratio of radii of the nuclei with the mass numbers given as 2, 16, 54 and 128, respectively, will be

Match List I with List II

List I (Technique/term)	List II (Features)
(A) Amniocentesis	(I) Ligation of fallopian tube
(B) Maternal mortality rate	(II) Ligation of vas deferens
(C) Vasectomy	(III) Death rate
(D) Tubectomy	(IV) Testing of sex

Choose the correct options:

After the completion of replication, each DNA molecule will have one parental strand and one newly synthesized strand. This scheme is known as:

Match List - I with List - II.

LIST I (Causal Agent)	LIST II (Disease)
(A) Salmonella typhi	(I) Typhoid
(B) Streptococcus pneumoniae	(II) Pneumonia
(C) Rhino Viruses	(III) Common cold
(D) Plasmodium	(IV) Malaria

Choose the correct answer from the options given below: