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Solve the following system of inequalities graphically: 2x – y > 1, x – 2y < –1

If $x=acos\theta ,y=bsin\theta$, then $\frac{d^{3}y}{d{x}^3}$ is equal to

If $xϵ[-1,1],thenrangeofta{n}^{-1}(-x)$ is

The equation of common tangent to the curve $y^2=8x$ and $xy=-1$ is

If $\underset{0}{\overset{\pi }{\int }}\sqrt{1+4{\sin }^2\frac{x}{2}-4\sin \frac{x}{2}}dx=4\sqrt{a}-2b+\frac{\pi }{a},$ then which of the following statements is/are true :

(where $a,b$ are prime numbers)

Match the following range of $y=\sin x$ for the given intervals.

Let a circle is touching exactly two sides of a square $ABCD$ and passes through exactly one of its vertices. If area of the square $ABCD$ is $1$ sq. units, then the radius of the circle (in units) is

In $△ABC$, the length of side $AB$ is $2$ units and $\angle ABC=\frac{\pi }{3}.$ If $B$ and the mid-point of $BC$ have the coordinates $(0,0)$ and $(2,0)$ respectively, then the orthocentre of $△ABC$ is

A matrix ‘B’ is singular if

The value of $\underset{x\to \frac{\pi }{2}}{lim}{(2^{\frac{1}{{\cos }^{2}x}}+3^{\frac{1}{{\cos }^{2}x}}+4^{\frac{1}{{\cos }^{2}x}}+5^{\frac{1}{{\cos }^{2}x}}+6^{\frac{1}{{\cos }^{2}x}})}^{2{\cos }^{2}x}$ is

Consider a pyramid OPQRS located in the first octant $(x\ge 0,y\ge 0,z\ge 0)$ with O as origin, and OP and OR along the X-axis and the Y-axis, respectively. The base OPQR of the pyramid is a square with OP = 3. The point S is directly above the mid-point T of diagonal OQ such that TS = 3. Then.

Let $A$ be a nilpotent matrix of index $m.$ If $(I-A{)}^n=I+A+A^2+\cdots +A^{m-1},$ where $I$ is the identity matrix of order same as matrix $A,$ then the value of $n$ is

Find the sum of the following series up to n terms:

The sum of the series $\frac{1}{x+1}+\frac{2}{x^2+1}+\frac{2^2}{x^4+1}+\cdots +\frac{2^{100}}{x^{2^{100}}+1}$ when $x=2$ is :

If $f\left(x\right)=\frac{x^2}{|x|}$, write $\frac{d}{dx}\left(f\right(x\left)\right).$

Which of the following intervals belong(s) to the domain of the function $f\left(x\right)=\frac{\sqrt{-{\log }_{0.4}(x^4-1)}}{x^2+2x+8}$?

If $\frac{1}{(x^2+\frac{1}{x}{)}^{\frac{4}{3}}}$ can be expanded by binomial theorem if

If $\underset{x\to 0}{lim}(1+ax{)}^{\frac{b}{x}}=e^4$ where $a$ and $b$ are natural numbers, then

If function $f\left(x\right)=\lambda \sin x+\cos x$ has two extremum points in $[0,2\pi ],$ then the value of $\lambda$ can be

If $f\left(x\right)=min\{|x-1|,|x-2|,|x-3|\},$ then the value of $I=\underset{0}{\overset{3}{\int }}f\left(x\right)dx$ equals

The integral $\int {\left(\frac{x}{x\sin x+\cos x}\right)}^{2}dx$ is equal to

(where $C$ is a constant of integration)

If $f\left(x\right)=\frac{si{n}^{4}x+co{s}^{2}x}{si{n}^{2}x+co{s}^{4}x}$ for $xϵR,$ then $f\left(2002\right)$.