If \(\vec{a}=\hat{i}+\hat{j}+\hat{k}, \vec{b}=\hat{j}-\hat{k}\) and \(\vec{c}=\hat{i}+\hat{j}+\hat{k}\) such that \(\vec{r} \times \vec{b}=\vec{c} \times \vec{b}\) and \(\vec{r} \cdot \vec{a}=0\), then \(\vec{r}\) is :

If the objective function \(z = 4x + 3y\) has maximum value on a line joining points (3, a) and (b, 2) where \(a > 0, b > 0\) such that \(a - b = 2\) then the maximum value of z is:

The maximum value of \(f(x)=\frac{\log x}{x}\) for \(x\) in \([2, \infty]\) is

The point on the curve \(\frac{x^2}{4}+\frac{y^2}{9}=1\) at which the tangent to the curve is parallel to the \(x\)-aris is

The feasible region represented by the constraints \(x+y \leq 50,3 x+y \leq 90, x \geq 0, y \geq 0\) of an LPP is

Rahul can run 34.4 m in the given time as Amit runs 50 m. By how much distance Rahul is away from Amit at the winning point, in a two km race?

The value of \(\tan ^{-1} \frac{2}{11}+\tan ^{-1} \frac{7}{24}\) is :

A catalyst increases the rate of reaction by:

Which of the following statements are true?
(A) The vector equation of the line through the point (5, 2, -4) and parallel to the vector \(3 \hat{i}+2 \hat{j}-8 \hat{k}\) is \(\vec{r}=(5 \hat{i}+2 \hat{j}-4 \hat{k})+\lambda(3 \hat{i}+2 \hat{j}-8 \hat{k})\)
(B) Vector form of the equation of line \(\frac{x-5}{3}=\frac{y+4}{7}=\frac{z-6}{2}\) is \(\vec{r}=(5 \hat{i}-4 \hat{j}+6 \hat{k})+\lambda(3 \hat{i}+7 \hat{j}+2 \hat{k})\)
(C) The direction cosines of z-axis are (1, 1, 0).
(D) If a line has direction ratios 2, -1, -2, then its direction cosines are \(-2 / 3,-1 / 3,-2 / 3\).
Choose the correct answer from the options given below:

Mrs Rathna invested Rs. 2 lakh in an enterprise for 5 years. Her compound annual growth rate (CAGR) turned out to be 20.5%. The end balance would be (given \((1.205)^5=2.54\) ):

What is the color of Fe³⁺ (aq) ion?

Read the following passage carefully and answer the given questions.
Bioactive molecule, cyclosporin A, that is used as an immunosuppressive agent in organ-transplant patients, is produced by the fungus Trichoderma polysporum. Statins produced by the yeast Monascus purpureus have been commercialised as blood-cholesterol lowering agents. It acts by competitively inhibiting the enzyme responsible for synthesis of cholesterol. Baculoviruses are pathogens that attack insects and other arthropods. The majority of baculoviruses used as biological control agents are in the genus Nucleopolyhedrovirus. These viruses are excellent candidates for species-specific, narrow spectrum insecticidal applications. They have been shown to have no negative impacts on plants, mammals, birds, fish or even on non-target insects. BOD refers to the amount of the oxygen that would be consumed if all the organic matter in one liter of water were oxidised by bacteria. The sewage water is treated till the BOD is reduced. Cheese, is one of the oldest food items in which microbes were used. Different varieties of cheese are known by their characteristic texture, flavour and taste. For example, the large holes in 'Swiss cheese' are due to production of a large amount of CO₂ by a bacterium named Propionibacterium sharmanii.
Bioactive molecule, cyclosporin A, that is used as an immunosuppressive agent in organ-transplant patients, is produced by the fungus-

A 12 V battery is connected to a \( 6 \text{ } \Omega, 10 \text{ H} \) coil, through a closed switch. The switch is suddenly opened in 1 ms. The average induced emf across the coil will be