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If f(9) = 9, f'(9) = 4, then $\underset{x\to 9}{lim}\frac{\sqrt{f\left(x\right)}-3}{\sqrt{x}-3}$ equals

The solution of $\frac{dy}{dx}=\frac{y}{x}+tan\frac{y}{x}$ is

The general solution of the inequality $-7\le \frac{3-2x}{5}\le 3$ is

The set of values of a for which the function $f\left(x\right)=(4a-3)(x+ln5)+2(a-7)cot\left(\frac{x}{2}\right)si{n}^2\left(\frac{x}{2}\right)$ does not possess critical point is

Find the sum to n terms

$\frac{1}{1\times 2}+\frac{1}{2\times 3}+\frac{1}{3\times 4}+\dots \dots$

Which of the following function is an onto function-

The values of x and y for which the numbers 3+i$x^2$y and $x^2$ +y+4i are conjugate complex are

If s be the sum of the coefficients in the expansion of $(px+qy+rz{)}^n,p,q,r>0$, then ${lim}_{n\to \infty }\frac{S}{(S^{\frac{1}{n}}+1{)}^n}$=

If $A(-2,1),B(2,3)$ and $C(-2,-4)$ are three points, then the acute angle between the lines $BA$ and $BC$ is

$f:R\to R$ is defined as $f\left(x\right)=x^4-6x^2+12$. The range of f(x) is

Let p,q, and r be any three logical statements. Which of the following is true?

For any two statements p and q, the statement $\sim (p\vee q)\vee (\sim p\wedge q)$ is equivalent to

If p is the length of perpendicular from the origin to the line whose intercepts on the axes are a and b then show that $\frac{1}{p^2}=\frac{1}{a^2}+\frac{1}{b^2}$

One root of the following given equation $2x^5-14x^4+31x^3-64x^2+19x+130=0$ is

Number of combination of different things taking r things at a time when,

The values of $\frac{1}{x}$ for x < -3 is

Equation of the hyperbola with length of the latusrectum 4 and e = 3 is

If $(1+x{)}^n=\sum _{r=0}^n{}^n{C}_r{x}^r$ and $\sum _{r=0}^n\frac{1}{{}^n{C}_r}=a,$ then the value of $\sum _{0\le i\le n}\sum _{0\le j\le n}(\frac{i}{{}^n{C}_i}+\frac{j}{{}^n{C}_j})$ is equal to

Given equation of the curve is $y=sinx$.

Using definite integral, find the area of the region between the given curve and the x-axis in the interval of $[0,\pi ]$.

If $f"\left(x\right)=kin[0,a],then{\int }_0^{a}f\left(x\right)dx-{\left\{\begin{array}{c}xf\left(x\right)-\frac{x^2}{2!}{f}^{\prime }\left(x\right)+\frac{x^3}{3!}f"\left(x\right)\end{array}\right\}}_0^a$ is

If $f\left(A\right)=\sum _{r=1}^8\tan \left(rA\right)\cdot \tan (r+1)A,$ then

If three complex numbers are in A.P., then they lie on

To find:

$1^2+2^2+3^2...................+n^2$

The relation between equilibrium constant $K_P$ and $K_C$ is

In a right-angled triangle, $a$ and $b$ are the lengths of the two sides and $c$ is the length of the hypotenuse. If $c+b$ and $c-b$ are numbers other than $1$, then ${\log }_{c+b}a+{\log }_{c-b}a=$

Fill in the blanks in following table:
$$\begin{array}{ccccc}& P\left(A\right)& P\left(B\right)& P(A\cap B)& P(A\cup B)\\ \left(i\right)& \frac{1}{3}& \frac{1}{5}& \frac{1}{15}& \cdots \\ \left(ii\right)& 0.35& \cdots & 0.25& 0.6\\ \left(iii\right)& 0.5& 0.35& \cdots & 0.7\end{array}$$

Find the coordinatesof the foce, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse.
$\frac{x^2}{4}+\frac{y^2}{25}=1.$

Let $z_1$ and $z_2$ be complex numbers such that $z_1\ne z_2$ and $|z_1|$ = $|z_2|$. If $z_1$ has positive real part and $z_2$ has negative imaginary part, then $\frac{z_1+z_2}{z_1-z_2}$ may be

The value of ${\sum }_{n=1}^{2015}{i}^n$ is

If the lengths of transverse and conjugate axis of the hypberbola are 4,2 then the distance Between the foci is

If A = {2, 3, 5, 6}, B = {4, 8, 15, 17} a R b $\Rightarrow$ a divides b. Find domain and range.

The statement $(p\to q)\to \left[\right(\sim p\to q)\to q]$ is

The value of $\sin {40}^{\circ }{35}^{\prime }\cos {19}^{\circ }{25}^{\prime }+\cos {40}^{\circ }{35}^{\prime }\sin {19}^{\circ }{25}^{\prime }$ is

Find the value of $\underset{n\to \infty }{lim}\frac{1+2+3+....n}{n^2}$

Let $S$ be the set of all column matrices $\left[\begin{array}{c}{b}_1\\ b_2\\ b_3\end{array}\right]$

such that $b_1,b_2,b_3\in R$ and the system of equations (in real variables)

$$\begin{array}{c}-x+2y+5z=b_1\\ 2x-4y+3z=b_2\\ x-2y+2z=b_3\end{array}$$

has at least one solution. Then, which of the following system(s) (in real variables) has (have) at least one solution for each $\left[\begin{array}{c}{b}_1\\ b_2\\ b_3\end{array}\right]\in S$?

What is the sentence type?

Dad fixed the car and drove himself to the supermarket.

Answer the following by appropriately matching the lists based on the information given in Column I and Column II

​​​​​​$\begin{array}{cc}Column1& Column2\\ \text{a.}f:R\to [\frac{3\pi }{4},\pi )\text{and}f\left(x\right)={\cot }^{-1}(2x-x^2-2),& \\ \text{then}f\text{is}& \text{p. one-one}\\ \text{b.}f:R\to R\text{and}f\left(x\right)=e^{px}\sin qx& \\ \text{where}p,q\in R^{+},\text{then}f\text{is}& \text{q. into}\\ \text{c.}f:R^{+}\to [4,\infty )\text{and}f\left(x\right)=4+3x^2,& \\ \text{then}f\text{is}& \text{r. many-one}\\ \text{d.}f:R\to R\text{and}f\left(f\right(x\left)\right)=x,\forall x\in R& \\ \text{then}f\text{is}& \text{s. onto}\end{array}$

The angle between the lines joining the origin to the points of intersection of the line y = 3x + 2with the curve $x^2$ + 2xy + ${3y}^2$ + 4x + 8y = 11, is

${lim}_{x\to 0}\frac{ta{n}^{-1}x-si{n}^{-1}x}{x^3}\text{is equal to}$

The sum of the series ${}^{20}{C}_0$ - ${}^{20}{C}_1$ + ${}^{20}{C}_2$ - ${}^{20}{C}_3$ ...............+ ${}^{20}{C}_{10}$ is

If |z|=2, then the points representing the complex numbers -1+5z will lie on a