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Let ${\lambda }_1$ be the area of the region on the plane bounded by $max(\left|x\right|,\left|y\right|)\le 1$ and $xy\le \frac{1}{2},$ and ${\lambda }_2$ be the length of $y-$intercept of the plane which is passing through the intersection of the planes $x+2y+3z+5=0$ and $2x-3y+7z+1=0$ and is parallel to the line $\overrightarrow{r}=\overrightarrow{i}+2\hat{j}+t(8\hat{i}-7\hat{j}-4\hat{k}),$ where $t\in R.$ Then $\left[{\lambda }_1\right]+\left[{\lambda }_2\right]$ equals

($[.]$ denotes the greatest integer function)

If ${\log }_{10}7=0.8451$, then the position of the first significant figure of $7^{-20}$, is

$If\frac{2sin\alpha }{1+cos\alpha +sin\alpha }=y,then\frac{1-cos\alpha +sin\alpha }{1+sin\alpha }isequalto$

Find the sum to n terms of the series whose nth term is given by

$n^2+2^n$

The number of subsets of the set $A=\{1,2,3,\dots ,9\}$ containing at least one odd number is

The solution of the inequality $\frac{4x+4}{2}>-x$ is

For any $3\times 3$ matrix $M$, let $|M|$ denote the determinant of $M$. Let $I$ be the $3\times 3$ identify matrix. Let $E$ and $F$ be two $3\times 3$ matrices such that $(I-EF)$ is invertible. If $G=(I-EF{)}^{-1}$, then which of the following statements is(are) TRUE?

Find the mean deviation taking deviation from the mean for the given individual observation 10, 20, 30, 40, 50, 60, 70

In a $\Delta ABC,$ if cos A = $\frac{sinA}{2sinC},$ show that the triangle is isosceles.

The sum of the first three terms of a G.P. is $6$ and the sum of its first three odd terms is $10.5$. Then the sum of its first term and the common ratio can be

If the area of the triangle whose one vertex is at the vertex of the parabola, $y^2+4(x-a^2)=0$ and the other two vertices are the points of intersection of the parabola and $y$-axis, is $250$ sq. units, then a value of '$a$' is :

Let an ellipse and a hyperbola have same foci. If the length of conjugate axis of the hyperbola is equal to the length of minor axis of the ellipse, then the value of $\frac{1}{e_1^2}+\frac{1}{e_2^2}$ is $(e_1$ and $e_2$ denote the eccentricities of the two conics$)$

If $\alpha$ and $\beta$ are the roots of a quadratic equation satisfying the conditons $\alpha \beta =4$ and $\frac{\alpha }{\alpha -1}+\frac{\beta }{\beta -1}=\frac{a^2-7}{a^2-4},\alpha ,\beta ,a\in R$. For what values of ${}^{\prime }{a}^{\prime }$ will the quadratic equation have equal roots?

Let $S=1+\frac{4}{5}+\frac{7}{5^2}+\frac{10}{5^3}+\cdots \infty .$ Then the value of $S$ is

Find the interval of $x$ such that $\left|\frac{x^2-3x-1}{x^2+x+1}\right|<3$ is satisfied is

Let $p={lim}_{x\to 0^{+}}(1+ta{n}^2\sqrt{x}{)}^{\frac{1}{2x}}$ then log p is equal to:

The minimum value of cos(cos x) is

Write the component statements of the compound statement and check whether the compound statement is true or false.

'A line is straight and extends indefinitely in both directions.'

If $a,b$ and $c$ are the greatest values of ${}^{19}{C}_p,{}^{20}{C}_q,{}^{21}{C}_r$ respectively, then:

If $f\left(x\right)=x^2$ and $g\left(x\right)=2^x$, then the solution set of the equation $f\left(2^x\right)=g\left(x^2\right)$ is

Team of four students is to be selected from a total of 12 students for a quiz competition. In how many ways it can be selected if two particular students should be included and two particular students don't want to be in the team.

$\begin{array}{cc}\text{Trigonometric Equations}& \text{General Solutions}\\ 1.\text{tan x = 2}& \text{P.}2n\pi \pm \frac{2\pi }{3},n\in I\\ 2.\text{sin x =}\frac{\sqrt{3}}{2}& \text{Q.}n\pi -\frac{\pi }{4},n\in I\\ 3.\text{cos x =}\frac{-1}{2}& \text{R.}n\pi +(-1{)}^n\frac{\pi }{3},n\in I\\ 4.\text{cot x = -1}& S.n\pi +ta{n}^{-1}\left(2\right),n\in I\end{array}$

What will be the molarity of $C{l}^{-}$ ions in an $\frac{M}{30}$ solution of $FeC{l}_3$?

If the coefficients of $x^7$ and $x^8$ in the expansion of ${(2+\frac{x}{3})}^n$ are equal, then the value of $n$ is

If the vertices of a triangle be (a, b - c), (b, c - a) and (c, a - b), then the centroid of the triangle lies

If $\arg \left(z\right)<0,$ then $\arg (-z)-\arg \left(z\right)=$

In the following figure , if AB = 2BC and $\alpha =\frac{\pi }{2}$, then find the co-ordinates of A($Z_2$)

If $x\in [-5,3],$ then the correct option(s) is (are)

Find the equation of the hyperbola satisfying the given conditions,

Vertices $(\pm 2,0),foci(\pm 3,0)$

Number of planes possible satisfying the condition that it should be parallel to 2 given vectors.

The straight line $y=2x+\lambda$ does not meet the parabola $y^2=2x$, if

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

If $\frac{|x-3|}{x-3}>0$, then

If the line $ky-2x-k^2+2h=0\&$ parabola $x^2=4y$ touches each other, then

Prove that cos A 2A cos 4A cos 8A = $\frac{sin16A}{16sinA}$.

If $a_1,a_2,a_3.....a_n$ are in A.P. Where $a_i>0$ for all i, then the value of
$\frac{1}{\sqrt{a_1}+\sqrt{a_2}}+\frac{1}{\sqrt{a_2}+\sqrt{a_3}}+.........+\frac{1}{\sqrt{a_{n-1}}+\sqrt{a_n}}=$

The equation of the curve passing through the origin and satisfying the equation $x\frac{dy}{dx}+\sin 2y=x^4{\cos }^{2}y$ is

The Boolean expression $(p\wedge \sim q)\vee q\vee (\sim p\wedge q)$ is equivalent to

Consider the parabola whose focus at $(0,0)$ and tangent at vertex is $x-y+1=0$.

Tangents drawn to the parabola at the extremities of the chord $3x+2y=0$ intersects at an angle

Which among the following relations defined on $R$ is/are functions

A cylinder is inscribed in a sphere of radius $3$ units. Find the curved surface area of the cylinder which has the maximum volume.

The least value of the expression $2{\log }_{10}x-{\log }_x\left(0.01\right)$, for $x>1$, is