The area of the parallelogram of which \(\hat{i}\) and \(\hat{i}+\hat{j}\) are adjacent sides is :

A swimmer whose speed in swimming pool is 5 km/h, swims between two points in a river and returns back to starting point. He took 20 minutes more to cover the distance upstream than to cover downstream. If the speed of stream is 2 km/h, then the distance between two points is

Which of the following statements are correct about the Compound Annual Growth Rate (CAGR)?
(A) It can be used to compare historical returns on different investment portfolios.
(B) It helps smooth returns when growth rates are expected to be volatile and inconsistent.
(C) It is unable to track the performance of various business measures of one or multiple companies alongside one another.
(D) It can be used to calculate the average growth of a single investment.
Choose the correct answer from the options given below :

Consider the line \(\vec{r}=-2 \hat{i}+3 \hat{j}+\hat{k}+\lambda(5 \hat{i}-3 \hat{j}-\hat{k})\). Match List-I with List-II

List-I	List-II
(A) A point on the given line	(I) \(\left(\frac{5}{\sqrt{35}}, \frac{-3}{\sqrt{35}}, \frac{-1}{\sqrt{35}}\right)\)
(B) Direction ratios of the given line	(II) (2, 3, 1)
(C) Direction cosines of the given line	(III) (5, -3, -1)
(D) Direction ratios of a line perpendicular to given line	(IV) (-2, 3, 1)

Choose the correct answer from the options given below:

If the present value for a sequence of payments made at the end of each 6 months and continuing forever is ₹6000. If the money is worth 4% compounded semi-annually, then, the value of sequence of payment made is:

The integrating factor of the differential equation \(\left(x \log _e x\right) \frac{d y}{d x}+y=2 \log _e x\) is:

If \(x=t^3\) and \(y=t^2\), then \(\frac{d^2 y}{d x^2}\) is equal to :

Which of the following correctly expresses the equation for species- area relationships? Where( S=Species richness, A=Area, Z=slope of line (regression coefficient), C=Y- intercept

Match List-I with List-II

List-I (Products)	List-II (Microorganisms)
(A) Wine or beer	(I) Penicillium notatum
(B) Swiss Cheese	(II) Aspergillus niger
(C) Antibiotic	(III) Propionibacterium sharmanii
(D) Citric Acid	(IV) Sachharomyces cerevisiae

Choose the correct answer from the options given below:

The Galvanic cells will behave like an electrolytic cell when

The current sensitivity of a moving coil galvanometer can be increased by
A. Decreasing the magnetic field
B. Increasing the number of turns
C. Increasing the area of the coil
D. Inserting a soft iron core in the coil
E. Decreasing the area of the coil
Choose the correct answer from the options given below:

Read the passage carefully and answer the Questions
\(KMnO _4\) is prepared by the fusion of \(MnO _2\) with an alkali metal hydroxide and an oxidizing agent like \(KNO _3\) to give a dark-green manganate ion which disproportionate to give permanganate as follows.
\(\begin{array}{l}2 MnO _2+4 KOH + O _2 \rightarrow 2 K_2 MnO _4+2 H _2 O \\ 3 K_2 MnO _4+4 H ^{+} \rightarrow 2 KMnO _4+ MnO _2+2 H _2 O \end{array}\)
On heating \(K _2 MnO _4\) decomposes at 513 K to give \(K _2 MnO _4\). Permanganate ion is tetrahedral and diamagnetic
Acidified \(KMnO _4\) acts a strong oxidizing agent which oxidizes oxalic acid, ferrous ions, nitrite ion and iodides.
Reaction of iodide ion \(\left( I ^{-}\right)\)with \(KMnO _4\) will.....

The term "biodiversity" was popularised by: