Ajesh has set up a sinking fund in order to have ₹ 10,00,000 after 10 years for his son's education. The amount should be set aside at the end of every 6 months into an account paying \(5\%\) per annum compounded half yearly is : [given \((1.025)^{20}=1.6386\) ]

If \(y=\tan ^{-1} x\) then \(\frac{d y}{d x}\) at \(y=\frac{\pi}{3}\) is equal to :

If \(x=a \cos ^3 \theta, y=a \sin ^3 \theta\). The value of \(\frac{d^2 y}{d x^2}\) at \(\theta=\frac{\pi}{6}\) is :

Match List-I with List-II

List - I	List - II
(A) Angle between \(\hat{i}-\hat{j}\) and \(\hat{i}+\hat{j}\)	(I) \(\pi\)
(B) Angle between \(\hat{i}-\hat{j}+\hat{k}\) and \(-\hat{i}+\hat{j}-\hat{k}\)	(II) \(\frac{3 \pi}{4}\)
(C) Angle between \(\hat{i}+\hat{j}\) and \(-\hat{i}\)	(III) \(\frac{\pi}{4}\)
(D) Angle between \(\hat{i}+\hat{k}\) and \(\hat{k}\)	(IV) \(\frac{\pi}{2}\)

Choose the correct answer from the options given below :

If \(P\left[\begin{array}{lll}1 & 2 & 3 \\ 4 & 5 & 6\end{array}\right]=\left[\begin{array}{ccc}-7 & -8 & -9 \\ 2 & 4 & 6\end{array}\right]\), then matrix \(P\) is equal to :

The maximum value of Z = 3x + y subject to the constraints
x + y \(\le\) 7,
x \(\le\) 5,
y \(\le\) 4,
x \(\ge\) 0, y \(\ge\) 0 is:

The maximum value of the objective function \(Z=8 x+2 y\) of an L.P.P. subject to the constraints: \(2 x+y \leq 3,2 x+3 y \leq 6, x \geq 0, y \geq 0 \) is:

The interval in which the function \(g(x)=x^2 e^{-x}\) is increasing is :

Arrange the following plants as per the correct order of their evolution.
(A). Psilophyton
(B). Tracheophyte ancestors
(C). Rhynia-type plants
(D). Chlorophyte ancestors
Choose the correct answer from the options given below:

Read the following passage given below and answer the question.
Aldehydes and ketones undergo nucleophilic addition reactions. A nucleophile attacks the electrophilic carbon atom of polar carbonyl group from perpendicular to plane of \(sp^2\) hybridised orbitals of carbonyl carbon. The hybridisation of carbon changes from \(sp^2\) to \(sp^3\) and a tetrahedral alkoxide intermediate is formed. This captures \(H^+\) to give neutral product. In carboxylic acids, carbonyl carbon is less electrophilic than carbonyl carbon in aldehydes and ketones.
An organic compound \(C_3H_6O\) does not give a precipitate with 2,4-dinitrophenylhydrazine and also does not react with metallic sodium. It could be :

The tropical Amazonian rain forest is located in:

Expand "GEAC":

Which of the following statements are correct ?
(A) Electrostatic field lines form closed loops.
(B) In an electric dipole, the direction from charge to q is the direction of the dipole.
(C) Water molecule is an example of polar molecule.
(D) The electric field due to a uniformly charged thin shell is zero at some points inside the shell.
(E) Electrostatic force is conservative in nature.
Choose the correct answer from the options given below :