If \(A=\left[\begin{array}{ll}1 & -1 \\ 2 & -1\end{array}\right], B=\left[\begin{array}{cc}a & 1 \\ b & -1\end{array}\right]\) and \((A+B)^2=A^2+B^2\) then

Two pipes can fill a cistern in 8 and 12 hours respectively. The pipes are opened simultaneously, and it takes 12 minutes more to fill the cistern due to leakage. If the cistern is full, what will be the time taken by the leakage to empty it?

The area (in sq. units) of the region enclosed by the ellipse \(16 x^2+25 y^2=400\) is:

Area of the triangle ABC with vertices \(\left(a, a^2\right),\left(b, b^2\right)\) and \(\left(c, c^2\right)\) is equal to :

The value of the determinant \(\left|\begin{array}{ccc}\cos ^2 \theta & \cos \theta \sin \theta & 0 \\ -\sin \theta & \cos \theta & 0 \\ 0 & 0 & 1\end{array}\right|\) is equal to

The real valued function \(f(x)=x^{15}+5 x^9+10\) is increasing for __________ .

A fisherman is rowing a boat. He takes 6 hours to row 48 km upstream whereas he takes 3 hours to go to the same distance downstream.
Based on the above information, which of the following are correct?
(A) His speed of rowing in still water is 10 km/hour.
(B) The speed of the stream is 3 km/hour.
(C) The average speed of the fisherman is 32/3 km/hour.
(D) The speed of the stream is 4 km/hour.
Choose the correct answer from the options given below :

Ethanol is treated with concentrated sulphuric acid. The product formed is :

The reason for the survival of the melanised moth, during the post-industrialization period in England was:

Consider the LPP, Min \(Z=3 x-2 y\) subject to the conditions
\(
\begin{array}{l}
4 x+3 y \geq 12 \\
4 x+y \geq 4 \\
x \geq 0, y \geq 0
\end{array}
\)
then which of the following is correct?

Identify the role of deoxyribonucleoside triphosphates in DNA replication.
A. Initiate replication.
B. Act as substrate.
C. Provide energy for polymerisation.
D. Activate ligases.
Choose the correct answer from the options given below:

Benzene ring in phenol is expected to give the characteristic substitution or replacement reactions i.e. halogenation, nitration, sulphonation, alkylation etc. In these reactions, the attacking reagent is electrophile. The phenolic group activates the ortho and para positions in the ring due to conjugation. Disubstitution reactions are quite fast as the electron density in the ring has increased considerably. Phenols do not respond to nucleophilic substitution reactions as it is influenced by phenolic group.
Victor Meyer test gives following observations. Match List I with List II:

List-I	List-II
A. Blood red colouration	I. Tertiary alcohol
B. Blue colouration	II. Phenols
C. Colourless solution	III. Primary alcohol
D. Violet colouration with \(\text{FeCl}_3\)	IV. Secondary alcohol

Choose the correct answer from the options given below:

If \(x=\frac{1-t}{1+t}\) and \(y=\frac{2 t}{1+t}\), then \(\frac{d^2 y}{d x^2}\) is equal to :