Class 12th

If $\mathrm{c}$ $$ is a point at which Rolle's theorem holds for the function, ${}_{}\left(\frac{{}^{}}{}\right)$ $f\left(x\right)={\mathrm{l}\mathrm{o}\mathrm{g}}_e?\left(\frac{x^2+\alpha }{7x}\right)$ in the interval $\left[\mathrm{3,4}\right]$ $$ , where $$ , $\alpha \in R$ then ${}^{}$ $f^{\mathrm{\text{'}}\mathrm{\text{'}}}\left(c\right)$ is equal to:

If ${\theta }_1$ and ${\theta }_2$ be respectively the smallest and the largest values of $\theta$ in $\left(\mathrm{0,2}\pi \right)-\left\{\pi \right\}$ which satisfy the equation, $2{\mathrm{c}\mathrm{o}\mathrm{t}}^2\theta -\frac{5}{\mathrm{s}\mathrm{i}\mathrm{n}\theta }+4=0$ , then ${\int }_{{\theta }_1}^{{\theta }_2}?{\mathrm{c}\mathrm{o}\mathrm{s}}^2?3\theta d\theta$ is equal to:

Let the vectors a,b,c be such that |a|=2, |b|=4 and |c|=4. If the projection of b on a is equal to the projection of c on a and b is perpendicular to c, then the value of |a+b-c| is

Does the rate of a reaction also depend on the phase of reactants?

If the lines x+y=a and x-y=b touch the curve y=x²-3x+2 at the points where the curve intersects the x-axis, then a/b is equal to

How come catalysts are able to speed up the reaction without themselves getting consumed?

In a workshop, there are five machines and the probability of any one of them to be out of service on a day is $\frac{1}{4}$ . If the probability that at most two machines will be out of service on the same day is ${\left(\frac{3}{4}\right)}^{3}k$ , then $k$  is equal to:

If x=1 is a critical point of the function f(x)=(3x²+ax-2-a)eˣ, then

Let $A$ $$ and $$ $B$ be two independent events such that $P\left(A\right)=\frac{1}{3}$ $\frac{}{}$ and $P\left(B\right)=\frac{1}{6}$ $\frac{}{}$ . Then, which of the following is TRUE?

The inverse function of $f\left(x\right)=\frac{8^{2x}-8^{-2x}}{8^{2x}+8^{-2x}},x\in (-\mathrm{1,1})$ $\frac{{}^{}{}^{}}{{}^{}{}^{}}$ , is