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Using elementary transformations, find the inverse of matrix $\left[\begin{array}{cc}3& 10\\ 2& 7\end{array}\right]$, if it exists.

In the Taylor series expansion of $exp\left(x\right)+sin\left(x\right)$ about the point $x=\pi$, the coefficient of $(x-\pi {)}^2$ is

If $7^x=3^{{\log }_{9}7}\cdot 5^{{\log }_{25}49}$, then the value of $x$ is

Point on the hyperbola $\frac{x^2}{24}-\frac{y^2}{18}=1$ which is nearest to the line $3x+2y+1=0$ is

Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse

The values of $\lambda ,\mu$ for which the system of equations $x+y+z=6,x+2y+3z=10,x+2y+\lambda z=\mu$ has an infinite number of solutions, is

Following data is available about 3 nuclei P, Q and R. Arrange them in decreasing order of stability.

$\begin{array}{cccc}& P& Q& R\\ No.ofprotons& 5& 2& 3\\ No.ofneutrons& 5& 3& 3\\ Bindingenergy\left(MeV\right)& 100& 60& 66\end{array}$

A tangent PT is drawn to the circle $x^2+y^2=4$ at the point $P(\sqrt{3},1)$. A straight line L, perpendicular to PT is a tangent to the circle $(x-3{)}^2+y^2=1$

A possible equation of L is

A ball is thrown at a speed of 40 m/s at an angle of ${60}^0$ with the horizontal. Find

(a) The maximum height reached and

(b) The range of the ball. Take g = 10 $m/s^2.$

A straight line L through the point (3, -2) is inclined at an angle ${60}^{\circ }$ to the line $\sqrt{3}x+y=1$. If L also intersects the X-axis, then the equation of L is

If $\cos x-y^2-\sqrt{y-x^2-1}\ge 0$, then

The ellipse $E_1:\frac{x^2}{9}+\frac{y^2}{4}=1$ is inscribed in a rectangle R whose sides are parallel to the coordinate axes. Another ellipse $E_2$ passing through the point (0,4) circumscribes the rectangle R. The Eccentricity of the ellipse $E_2$ is

Two tangents are drawn from a point $P$ to the circle $x^2+y^2-2x-4y+4=0,$ such that the angle between these tangents is ${\tan }^{-1}\left(\frac{12}{5}\right),$ where ${\tan }^{-1}\left(\frac{12}{5}\right)\in (0,\pi ).$ If the centre of the circle is denoted by $C$ and these tangents touch the circle at points $A$ and $B$, then the ratio of the areas of $\Delta PAB$ and $\Delta CAB$ is:

The number of solutions of the equation :

3cos² x sin²x - sin⁴x - cos²x = 0 in the interval [0, 2] is:

Find the value of expression

$\sin (-\theta )+\cos (-\theta )+\sec (-\theta )$

$+\sin (\pi -\theta )+\cos (\pi -\theta )+\sec (\pi -\theta )$

If $z$ is a non-zero complex number, then the area of the quadrilateral formed by the points $z,\overline{z},-z$ and $-\overline{z}$ is

If the line segment joining the points $A(a,b)$ and $B(c,d)$ subtends an angle $\theta$ at the origin, then $\cos \theta$ is equal to

Prove that the lines $2x+3y=19$ and $2x+3y+7=0$ are equidistant from the line $2x+3y=6$

If the eccentricities of the hyperbolas $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1and\frac{y^2}{b^2}-\frac{x^2}{a^2}=1$ be e and $e_1$, then $\frac{1}{e^2}+\frac{1}{e_1^2}=$

Domain of $f\left(x\right)=\frac{\sqrt{9-x^2}}{\sqrt{\left[x\right]+3}}$ is

The trace of a square matrix is defined as the sum of the principal diagonal elements. For real numbers $a$ and $b,$ if the trace of matrices $A=\left[\begin{array}{cc}2a^2& 5\\ 3& 9-6b\end{array}\right]$ and $B=\left[\begin{array}{cc}-b^2& 2\\ 3& 8a-8\end{array}\right]$ are equal, then $2a-b$ is equal to

The sum of the series ${\cot }^{-1}(2^2+\frac{1}{2})+{\cot }^{-1}(2^3+\frac{1}{2^2})+{\cot }^{-1}(2^4+\frac{1}{2^3})+......\infty$ is equal to

The rate of growth of bacteria in a culture is proportional to the number of bacteria present and the bacteria count is $1000$ at initial time $t=0.$ The number of bacteria is increased by $20$% in $2$ hours. If the population of bacteria is $2000$ after $\frac{k}{{\log }_e\left(\frac{6}{5}\right)}$ hours, then ${\left(\frac{k}{{\log }_{e}2}\right)}^2$ is equal to :

If sin A =, calculate cos A and tan A.

If $A=\left[\begin{array}{cc}1& 2\\ 4& 9\end{array}\right],$ then $A^{-1}$ is



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The sides of a triangle are three consecutive natural numbers and its largest angle is twice the smallest one, then the sides are

The plane through the intersection of the planes $x+y+z=1$ and $2x+3y-z+4=0$ and parallel to $y$-axis also passes through the point:

Differentiate the
function with respect to x.

At an election, a voter may vote for any number of candidates, not greater than the number to be elected. There are $10$ candidates and $4$ are to be elected. If a voter votes for atleast one candidate, then the number of ways in which he can vote, is

The area, enclosed by the curves $y=\sin x+\cos x$ and $y=|\cos x–\sin x|$ and the lines $x=0,$ $x=\frac{\pi }{2}$, is:

${lim}_{x\to 1}\{\frac{x-2}{x^2-x}-\frac{1}{x^3-3x^2+2x}\}$