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The equation of a standing wave is $y=4\sin \left(\frac{\pi x}{5}\right)\cos \left(100\pi t\right)$, where $y$ and $x$ are in $\text{cm}$ and $t$ in $\text{s}$. The wave is formed using a string of length $20\text{cm}$. The second and $4^{th}$ antinodes are located at positions (in $\text{cm}$)

Differentiate the given functions w.r.t. x.

$(logx{)}^x+x^{logx}$

Out of the given equations, is not a quadratic equation.

$ta{n}^{-1}x+ta{n}^{-1}y+ta{n}^{-1}z=\frac{\pi }{2}\Rightarrow 1-xy-yz-zx=$

In a triangle ABC, points X and Y are on AB and AC, respectively, such that XY is parallel to BC. Which of the two following always hold(s) good? (Here [PQR] denotes the area of triangle PQR.)

(I) [BCX] = [BCY]

(II) [ACX] .[ABY] = [AXY] .[ABC]

Select the correct graph of $|4\cos x-3|.$

$\sqrt{\frac{y}{x}}+\sqrt{\frac{x}{y}}=2\Rightarrow \frac{dy}{dx}=$

Prove that the number of subsets of a set containing n distinct elements is $2^n$ for all $nϵN$.

If the function $f$ defined as $f\left(x\right)=\frac{1}{x}-\frac{k-1}{e^{2x}-1},x\ne 0$ is continuous at $x=0$, then

$li{m}_{x\to \infty }$$\frac{\left(\sqrt{n}\right)}{(\sqrt{n}+(\sqrt{n+1}\left)\right)}$ =

If $S_n=\sum _{r=0}^n\frac{1}{{}^n{C}_r}$ and $P_n=\sum _{r=0}^n\frac{r}{{}^n{C}_r}$ then $\frac{S_n}{P_n}$ is

The graph of a quadratic polynomial $f\left(x\right)$ is shown below:





Which of the following options is/are correct?

Given that
is
the mean and σ²
is the variance of n observations x₁, x₂
… x_n. Prove that the mean and
variance of the observations ax₁, ax₂,
ax₃ …ax_n are

and a² σ²,
respectively (a ≠ 0).

A plane meets the co-ordinates axes in A, B, C and $(\alpha ,\beta ,\gamma )$ is the centroid of the $\Delta ABC$. Then the equation of the plane is :

If $\alpha ,\beta$ are the roots of the equation $m{x}^2+6x+(2m-1)=0\forall m\in R-\{0,\frac{1}{2}\},$ then the quadratic equation with roots as $\frac{1}{\alpha },\frac{1}{\beta }$ is:

If $\cos x=\frac{3}{5}$ where $x\in (\frac{3\pi }{2},2\pi )$, then the value of $\sin 4x$ is

If $f\left(x\right)=\int \frac{5x^8+7x^6}{{(x^2+1+2x^7)}^2}dx,(x\ge 0)$, and $f\left(0\right)=0$, then the value of $f\left(1\right)$ is :

Let P and Q be distinct points on the parabola y²=2x such that a circle with PQ as diameter passes through the vertex O of the parabola. If P lies in the first quadrant and the area of the triangle OPQ is 3√2, then which of the following is the coordinates of P?

If $\Delta =\left|\begin{array}{ccc}5& 3& 8\\ 2& 8& 1\\ 1& 2& 3\end{array}\right|,$ then minor of the element in $3^{\text{rd}}$ row and $2^{\text{nd}}$ column is



[1 mark]

If f: N $\to$ N be a function defined by f (x) =$\{\begin{array}{cc}\frac{n+1}{2},& ifnisodd\\ \frac{n}{2},& ifniseven\end{array}$ , the f is

Let P(x) and Q(x) be arbitrary predicates. Which of the following statements is always TRUE?

The distance between the parallel lines $8x+6y+5=0$ and $4x+3y-25=0$ is

Calculate the enthalpy change when infinitely diluted solution of $CaC{l}_2$ and $N{a}_{2}C{O}_3$ are mixed. $\Delta H_f^0$ For $C{a}^{2+}\left(aq\right)$, $C{O}_3^{2-}\left(aq\right)$ and $CaC{O}_3\left(s\right)$ are $–129.80,-161.7,-288.5Kcalmo{l}^{-1}$ respectively.___

Solution of the equation ${\cos }^{2}x\frac{dy}{dx}-\left(\tan 2x\right)y={\cos }^{4}x,|x|<\frac{\pi }{4},$ when $y\left(\frac{\pi }{6}\right)=\frac{3\sqrt{3}}{8}$ is