Sum of the minor and the cofactor of the element 6 in the determinant \(\left|\begin{array}{lll}2 & 3 & 1 \\ 4 & 5 & 6 \\ 7 & 8 & 9\end{array}\right|\) is :

Match (List I) with (List II)

(List I)	(List II)
A. For any vector \(\vec{a}, \vec{a} \times \vec{a}\)	\(I\). \(|\vec{a}|^2\)
B. For any vector \(\vec{a}, \vec{a} \cdot \vec{a}\)	\(II.\) \(0\)
C. If either \(\vec{a}=\overrightarrow{0}\) or \(\vec{b}=\overrightarrow{0}\), then \(\vec{a} \cdot \vec{b}\) is	\(III.\) \(|\vec{a}||\vec{b}|\)
D. If the angle between \(\vec{a}\) and \(\vec{b}\) is \(\theta=0\), then \(\vec{a} \cdot \vec{b}\) is	\(IV.\) \(\overrightarrow{0}\)

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For the LPP :
Maximise \(z=4 x+y\)
subject to constraints
\(x+y \leq 50, \quad 3 x+y \leq 90, \quad x \geq 0, y \geq 0\) the maximum value of \(z\) is :

If an asset costs Rs. 50,000 with an estimated useful life of 6 years and a scrap value of Rs. 5000. Then by using a linear depreciation method, the annual depreciation of the asset will be:

If \(\int \frac{2 x}{x^2+3 x+2} d x=m \log |x+2|+n \log |x+1|+C\), then the values of m and n are:

For the function \(f(x)=x^{1 / x}, x>0\), which of the following are correct?
(A) \(x=0\) is the only point where extremum may occur.
(B) The given function is maximum at \(x=e\).
(C) The function has no extreme value for \(x>0\).
(D) The maximum value of the function \(f(x)\) is \(e^{1 / e}\)
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If \(A=\left[\begin{array}{ll}x & 3 \\ 2 & y\end{array}\right], \quad B=\left[\begin{array}{cc}2 y & 1 \\ 3 & x\end{array}\right]\) and \(A+2 B=\left[\begin{array}{cc}6 & 5 \\ 8 & -2\end{array}\right]\), then the value of \((x, y)\) is:

In a reaction \(2A \to B\), the concentration of A decreases from 0.64 M to 0.22 M in 5 minutes. What is the rate of reaction (in \(M s^{-1}\)) during this interval ?

Two point charges of \( 1 \times 10^{-6} \, C \) and \( 4 \times 10^{-6} \, C \) are placed at a certain distance away from each other. The resultant electric field intensity is zero at a distance of 15 cm from the charge \( 1 \times 10^{-6} \, C \). The distance between the two point charges will be :

Answer the questions on the basis of passage given below :
As aldehydes ketones possess the carbonyl functional group, they show similar reactions. They exhibit nucleophilic addition reactions, reduction, oxidations, reactions due to alpha-Hydrogen atom, and many other reactions, like Cannizzaro reaction to electrophilic substitution reactions (in case of aromatic aldehydes ketones).
These reactions help in the preparation of a number of important compounds, like acetic acid, vinyl acetate, polymers and in perfume and dye industries.
Nucleophilic addition reactions form cyanohydrins, acetals, alcohols and N -substituted derivatives like imine, oxime, semicarbazone, phenylhydrazone Schiff's base. Reactions due to alpha-hydrogen atoms help in formation of aldols and alpha, beta-unsaturated aldehydes ketones.
Which of the following combinations of Grignard's reagent and carbonyl compound can give \(C _6 H _5- C ( OH )\left( CH _3\right)- CH \left( CH _3\right)_2\)
A. \(C _6 H _5 CH _2 MgBr +\left( CH _3\right)_2 CH - CHO\)
B. \(C _6 H _5 MgBr + CH _3- CO - CH \left( CH _3\right)_2\)
C. \(CH _3 MgBr + C _6 H _5- CO - CH \left( CH _3\right)_2\)
D. \(C _6 H _5- CO - CH _3+\left( CH _3\right)_2 CHMgBr\)
E. \(C _6 H _5 CH ( MgBr ) CH _3+\left( CH _3\right)_2 CO\)
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Arrange the given organisms involved in an ecosystem in correct sequence from lower to higher order.
(A) Fish
(B) Phytoplankton
(C) Man
(D) Zooplankton
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The integral value of \(k\) for which the system of linear equations
\(\begin{array}{l}k x+y+2 z=0 \\k y=x-3 z \\2 x+y+k z=0\end{array}\)
has a non-zero solution is :

With reference to sampling, which of the following are correct?
(A) Simple random sampling is probability sampling
(B) Snow-ball sampling is non-probability sampling
(C) Stratified sampling is probability sampling
(D) Cluster sampling is non-probability sampling
Choose the correct answer from the options given below: