Given equation of the plane is \(5x + 2y - 3z = 17\). Then,
A. A point \((2, 2, -1)\) lies on the plane
B. The direction cosines of normal to plane are \((2, 2, -1)\)
C. Coordinates of foot of perpendicular from origin are \((5, 2, -3)\)
D. The plane does not pass through the origin
Choose the correct answer from the options given below:

Match List I with List II

LIST I	LIST II
A. \(\lim _{x \rightarrow 0} \frac{1-\cos 2 x}{3 x^2}\)	I. 2
B. \(\lim _{x \rightarrow 4} \frac{x^2-16}{x-4}\)	II. 8
C. \(\lim _{x \rightarrow 0} \frac{\sin a x+4 x}{a x+\sin 4 x}\)	III. 1
D. \(\lim _{z \rightarrow 1} \frac{z^{1 / 3}-1}{z^{1 / 6}-1}\)	IV. \(\frac{2}{3}\)

The function \(f(x)=\left|\begin{array}{cc}x^2 & x \\ 3 & 1\end{array}\right|, x \in R\) has a:

If \(f(x)=e^x g(x), g(0)=2, g^{\prime}(0)=1\) then \(f^{\prime}(0)\) is equal to:

Match List-I with List-II

List-I (Differential equation)	List-II (Order and Degree)
(A) \(\frac{d^3 y}{d x^3}+y^2+e^{d y / d x}=0\)	(I) order \(=3\), degree \(=1\)
(B) \(\left(\frac{d^2 y}{d x^2}\right)^3+\left(\frac{d y}{d x}\right)^2+\frac{d y}{d x}+1=0\)	(II) order \(=3\), degree not defined
(C) \(2 x^2 \frac{d^2 y}{d x^2}-3\left(\frac{d y}{d x}\right)^2+y=0\)	(III) order \(=2\), degree \(=3\)
(D) \(\frac{d^3 y}{d x^3}+2\left(\frac{d y}{d x}\right)^2+\frac{d y}{d x}=0\)	(IV) order \(=2\), degree \(=1\)

Choose the correct answer from the options given below :

Match List-I with List-II
Let \(\theta\) be the angle between the vector \(\vec{a}\) and \(\vec{b}\).

List-I	List-II
(A) \(\vec{a} \cdot \vec{b}\)	(I) \(\frac{1}{10} \log \left|\frac{5+x}{5-x}\right|+C\)
(B) \(\vec{a} \times \vec{b}\)	(II) \(\frac{\vec{a} \cdot \vec{b}}{|\vec{b}|^2} \vec{b}\)
(C) Projection vector of \(\vec{a}\) on \(\vec{b}(\neq 0)\)	(III) \(|\vec{a}||\vec{b}| \sin \theta \hat{n}\) where \(\hat{n}\) is a unit vector perpendicular to both \(\vec{a}\) and \(\vec{b}\)
(D) \(\vec{a}\) and \(\vec{b}\) are orthogonal vectors	(IV) \(|\vec{a}||\vec{b}| \cos \theta\)

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The equation of trend line for five years (2004-2008) for the profit of a company is \(y_t=150+8 x\)The predicted profit for the year 2011 is:

Arrange the following steps for replication of HIV in a correct sequence:
A. Production of new viral RNA by infected cell.
B. Infection of normal cell by virus
C. Production of viral DNA
D. Production of new viruses
Choose the correct answer from the options given below

Arrange the following transition metal ions based on decreasing number of unpaired electrons.
A. \(Fe ^{2+}\)
B. \(Mn ^{2+}\)
C. \(Co ^{2+}\)
D. \(Sc ^{3+}\)
E. \(Cu ^{2+}\)
Choose the correct answer from the options given below:

Extinction of Steller's sea Cow and passenger pigeon is due to

Arrange the amount of energy in a sequence from the highest trophic level to the lowest trophic level (Pyramid of Energy):
(A). 10,000 J
(B). 1000 J
(C). 10 J
(D). 100 J
Choose the correct answer from the options given below:

By using Einstein photoelectric equation, if we consider kinetic energy of photo electrons on y-axis and frequency of incident radiation on x-axis. The slope of the graph will represent (the symbols have their usual meanings)

Read the passage carefully and answer the questions
Sodium dichromate \(Na _2 Cr _2 O _7\) and Potassium dichromate \(K _2 Cr _2 O _7\) are strong oxidizing agents; in acidic solution, its oxidising action can be represented as follows :
\(Cr _2 O _7^{2-}+14 H ^{+}+6 e^{-} \rightarrow 2 Cr ^{3+}+7 H _2 O \left( E ^{\circ}=1.33 V\right)\)
The chromates and dichromates are interconvertible in aqueous solution depending upon pH of the solution.
The chromate ion is tetrahedral whereas the dichromate ion consists of two tetrahedra sharing one corner with \(Cr - O - Cr\) bond angle of \(126^{\circ}\)
Potassium permanganate is prepared by fusion of \(MnO _2\) with an alkali metal hydroxide and an oxidising agent like \(KNO _3\).
This produces the dark green \(K _2 MnO _4\) which disproportionates in a neutral or acidic solution to give permanganate. It is a strong oxidizing agent. Hydrogen ion concentration of the solution plays 
an important part in the reduction of permanganate to manganate, manganese dioxide and manganese(II) salt.
Potassium permanganate forms dark purple (almost black) crystals which are isostructural with those of \(KClO _4\).
The salt is not very soluble in water \((6.4 g / 100 g\) of water at 293 K ), but when heated it decomposes at 513 K.
The number of moles of KMnO₄ required to react with one mole of sulphide ion in acidic solution is