If $\mathrm{f}(\mathrm{x})$ is a twice differentiable function such that $\mathrm{f}(\mathrm{a})=0, \mathrm{f}(\mathrm{b})=2, \mathrm{f}(\mathrm{c})=-1, \mathrm{f}(\mathrm{d})=2, \mathrm{f}(\mathrm{e})=0$, where $\mathrm{a}<\mathrm{b}<\mathrm{c}<\mathrm{d}<\mathrm{e}$, then the minimum number of zeroes of $g(x)=(f(x))^2+f^{\prime \prime}(x) f(x)$ in the interval $[a, e]$ is

If $a_n=\frac{3}{4}-\left(\frac{3}{4}\right)^2+\left(\frac{3}{4}\right)^3+\cdots(-1)^{n-1}\left(\frac{3}{4}\right)^n$ and $b_n=1-a_n$, then find the minimum natural number $n_0$ such that $b_n>a_n \forall n>n_0$

The value of $5050 \frac{\int_0^1\left(1-x^{50}\right)^{100} d x}{\int_0^1\left(1-x^{50}\right)^{101} d x}$ is

If roots of the equation $x^2-10 c x-11 d=0$ are $a, b$ and those of $x^2-10 a x-11 b=0$ are...

The value of $\left[\begin{array}{lll}3 & 2 & 0\end{array}\right] \mathrm{U}\left[\begin{array}{l}3 \\ 2 \\ 0\end{array}\right]$ is

The sum of the elements of $\mathrm{U}^{-1}$ is

$A=\left[\begin{array}{lll}1 & 0 & 0 \\ 2 & 1 & 0 \\ 3 & 2 & 1\end{array}\right]$, if $U_1, U_2$...

How can the conversion of (i) to (ii) be brought about?

Find the number of times $\mathrm{y}_1+\mathrm{y}_2=0$ at $\mathrm{x}=0$ in 1 sec

Find wave velocity of louder sound

A line M through A is drawn parallel to BD . Point S moves such that its distances from the...

Two waves $\mathrm{y}_1=\mathrm{A} \cos (0.5 \pi \mathrm{x}-100 \pi \mathrm{t})$ and $y_2=A \cos (0.46 \pi x-92 \pi t)$ are travelling in...

$\mathrm{MgSO}_4$ on reaction with $\mathrm{NH}_4 \mathrm{OH}$ and $\mathrm{Na}_2 \mathrm{HPO}_4$ forms a white crystalline precipitate. What is its formula?

What are N and M ?

The smallest ketone and its next homologue are reacted with $\mathrm{NH}_2 \mathrm{OH}$ to form oxime.

The given graph represents the variation of Z (compressibility factor $=\frac{\mathrm{PV}}{\mathrm{nRT}}$ ) versus P , for three real gases $...

The major products P and Q are

A ball moves over a fixed track as shown in the figure. From $A$ to $B$ the ball rolls without...