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If the sum of the coefficients in the expansion of $(x+y{)}^n$ is 8192, then the greatest coefficient in the expansion is

$\underset{x\to 0}{lim}\left[\frac{Incosx}{\sqrt[4]{41+x^2}-1}\right]\text{is equal to}$

Let $z_1$ and $z_2$ be two complex numbers such that $\overline{z_1}+i\overline{z_2}=0,\arg \left(z_1{z}_2\right)=\pi$. Then which of the following is/are ture?

Plot the graph of $f(x-1)$ if the graph of $f\left(x\right)$ looks like

Solve $\sqrt{2}secx+tanx=1$

Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

A = {1, 2, 3, 4}

Then, $A^c$ is

Supposse $a,b,c$ are in A.P. and $a^2,b^2,c^2$ are in G.P. If $a<b<c$ and $a+b+c=\frac{3}{2}$, then the value of $a$ is

(i) How many words can be formed with the letters of the word, 'HARYANA'? How many of these

(ii) have H and N together ?

(iii) begin with H and end with N ?

(iv) have 3 vowels together?

The number of $5$ digit numbers (without repetition) using digits $0,1,2,3,4,5$ such that they are even

The equation of the tangent to the parabola $y^2=16x$ inclined at an angle of ${60}^{\circ }$ to the positive $x-$axis is

If $E_1,E_2$ are two events with $E_1\cap E_2=\varphi ,$ then $P({\overline{E}}_1\cap {\overline{E}}_2)=$

Solve for $x$:

$lo{g}_3\left(\frac{x}{3}\right)+lo{g}_{1/9}\left(x\right)<1$

The value of $\underset{x\to \infty }{lim}{\left(\frac{p^{1/x}+q^{1/x}+r^{1/x}+s^{1/x}}{4}\right)}^{3x},$ where $p,q,r,s>0$ is equal to

Equation of the locus of all points such that the difference of its distances from $(-3,-7)$ and $(-3,3)$ is $8$ units is

The extremities of the latus rectum of an ellipse $\frac{x^2}{16}+\frac{y^2}{9}=1$ is

The set of real values of $x$ satisfying $||x-1|-1|\le 2,$ is

A line cuts the x - axis at A(7, 0) and the y - axis at B(0, -5). A variable line PQ is drawn perpendicular to AB. Cutting the x - axis at P and the y - axis at Q.

If AQ and BP intersect at R, the locus of R is

Find the product of the following

(1)

(2)

(3)

(4)

Let $\theta \in (0,\frac{\pi }{2})andx=Xcos\theta +ysin\theta ,y=Xsin\theta -Ycos\theta$ such that $x^2+2xy+y^2=a{X}^2+b{Y}^2$ Where a and b are constants then

The number of points with non-negative integral coordinates that lie in the interior of the region common to the circle $x^2+y^2=16$ and the parabola $y^2=4x,$ is

Find the maximum value of $5+(\sin x-4{)}^2$

If $z=(-\sqrt{3}+\sqrt{-2})(2\sqrt{3}-i)$, then

To receive grade A in a course one must obtain an average of 90 marks or more in five papers, each of 100 marks. If Tanvy scored 89, 93, 95 and 91 marks in first four papers, find the minimum marks that she must score in the last paper to get grade A in the course.

$ta{n}^{-1}\frac{1}{\sqrt{x^2-1}}=$

If $f\left(x\right)=x+\sin x;g\left(x\right)=e^{-x};u=\sqrt{c+1}-\sqrt{c};v=\sqrt{c}-\sqrt{c-1};(c>1),$ then

The value of $C_0+2C_1+3C_2+4C_3+\dots +(n+1)C_n$ is

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( where $C_r={}^n{C}_r$)

Graph of f(x) is given. Draw the graph of $f^{-1}\left(x\right)$

If a, b, c are distinct positive real numbers such that b(a + c) = 2ac then the roots of $a{x}^2+bx+c=0$ are

Show that the statement

p : "If x is a real number such that $x^3+4x=0$ then x is 0" is true by (i) direct method (ii) Method of contrapositive.

Let $S=\left\{\right(\lambda ,\mu )\in R\times R:f(t)=(|\lambda |e^{|t|}-\mu )\cdot \sin (2|t|),t\in R,\text{is a differentiable function}\}.$

Then $S$ is a subset of :

Volume of parallelopiped formed by vectors $\overrightarrow{a}\times \overrightarrow{b},\overrightarrow{b}\times \overrightarrow{c}$ and $\overrightarrow{c}\times \overrightarrow{a}$ is $36$ cubic units. Based on the given information above match the following by appropriately matching the lists given in Column I and Column II.



$\begin{array}{cc}Column1& Column2\\ \text{a. Volume of parallelopiped formed by vectors}& \text{p.}0\text{cubic units}\\ \overrightarrow{a},\overrightarrow{b}\text{and}\overrightarrow{c}\text{is}& \\ \text{b. Volume of tetrahedron formed by vectors}& \text{q.}12\text{cubic units}\\ \overrightarrow{a},\overrightarrow{b}\text{and}\overrightarrow{c}\text{is}& \\ \text{c. Volume of parallelopiped formed by vectors}& \text{r.}6\text{cubic units}\\ \overrightarrow{a}+\overrightarrow{b},\overrightarrow{b}+\overrightarrow{c}\text{and}\overrightarrow{c}+\overrightarrow{a}\text{is}& \\ \text{d. Volume of parallelopiped formed by vectors}& \text{s.}1\text{cubic unit}\\ \overrightarrow{a}-\overrightarrow{b},\overrightarrow{b}-\overrightarrow{c}\text{and}\overrightarrow{c}-\overrightarrow{a}\text{is}& \end{array}$

Express the following in standard form: (2 + 3i)(2 - 3i)(1 - i)²

The circle passing through three distinct points $(1,t),(t,1)$ and $(t,t)$ for all values of $t$, also passes through the point:

The coefficient of $x^{256}$ in the expansion of $(1-x{)}^{101}(x^2+x+1{)}^{100}$ is

A line makes angles $\alpha ,\beta ,\gamma$ with the coordinate axes. If $\alpha +\beta =\frac{\pi }{2},$ then $(\cos \alpha +\cos \beta +\cos \gamma {)}^2$ is equal to

A point $P$ inside the $△ABC$, splits the area of $△ABC$ into three equal parts, where $A\equiv (3,4)$. If line joining the vertex $A$ to $P$ cuts the side $BC$ at the point $(-1,0)$, then coordinates of $P$ is

The graph of $y=|x^3+1|$ is

A normal inclined at an angle of $\frac{\pi }{4}$ to the x-axis of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ is drawn. It meets the major and minor axes in P and Q respectively. If C is the centre of the ellipse then the area of the triangle CPQ is

If $A(x_1,y_1)$, $B(x_2,y_2)$ and $C(x_3,y_3)$ are the vertices of a triangle, then the excentre with respect to B is

The critical constants for water are $374{}^{\circ }C,217$ atm and 0.05 L mol${}^{-1}$. calculate $a$ and $b$.

If $co{s}^{-1}x>si{n}^{-1}x$ then

If f(x) and g(x) have equal derivatives, then f(x) - g(x) is a ___________ function

If sinA=sinB and cosA=cosB, then

Equation $6x^2-5xy+y^2=0$ represents____________________