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What is the equation of a curve given by the parametric form $x=9+6sec\theta ;y=-2-4tan\theta$.

An aeroplane moving horizontally with a speed of 720 km/h drops a food pocket, while flying at a height of 396.9 m.the time taken by a food pocket to reach the ground and its horizontal range is (Take g = 9.8 m/$se{c}^2$)

cosec A - 2cot 2A cos A =

$\text{If}\underset{x\to \infty }{lim}(\sqrt{x^2-x+1}-ax-b)=0\text{then the values of a and b are given by}$

Simplify the expression
${(1+x)}^{1000}+x(1+x{)}^{999}+x^2(1+x{)}^{998}+\cdots +x^{1000}$ and find the coefficient of $x^{50}$

Find the product of the identity function by the modulus function.

If sinx + cosx = $\frac{1}{5}$ then cot x is equal to ________.

"$$\begin{array}{cc}TrigonometricEquations& GeneralSolutions\\ 1.7co{s}^{2}x+si{n}^{2}x=4& P.n\pi \pm \frac{\pi }{4},wheren\in I\\ 2.si{n}^{2}x=\frac{1}{2}& Q.n\pi \pm \frac{\pi }{3},wheren\in I\\ 3.ta{n}^{2}x+3=0& R.n\pi \pm \frac{\pi }{6},wheren\in I\\ 4.3ta{n}^{2}x-1=0& S.Nosolution\end{array}$$ "

Solve: ( $5^x$ - 1)( $2^x$ - 8) < 0

The numbers of diagonals in a do-decagon will be__.

If f is a function satisfying $f(x+y)=f\left(x\right)f\left(y\right)$ for all, $x,yϵN$ such that f(1) = 3 and ${\sum }_{x=1}^{n}f\left(x\right)=120$, find the value of n.

Find the value of tan 3A in terms of tanA & tan2A

If $lo{g}_{10}x=a$, find the value of ${10}^{a-1}$ in terms of x.

If the ${10}^{th}$ term of an A.P. is $\frac{1}{20}$ and its ${20}^{th}$ term is $\frac{1}{10}$, then the sum of its first $200$ terms is

${10}^{(2n-1)}+1$ is divisible by

If $x=a+\frac{a}{r}+\frac{a}{r^2}+\cdots \infty ,$ $y=b-\frac{b}{r}+\frac{b}{r^2}-\cdots \infty$ and $z=c+\frac{c}{r^2}+\frac{c}{r^4}+\cdots \infty$ for $|r|>1,$ then the value of $\frac{xy}{z}$ is

Prove that the coefficient of $x^n$ in the binomial expansion of $(1+x{)}^{2n}$ is twice the coefficient of $x^n$ in the binomial expansion of $(1+x{)}^{2n-1}$.

$cos\frac{2\pi }{7}+cos\frac{4\pi }{7}+cos\frac{6\pi }{7}$ =

If y = 2x is a chord of the circle $x^2+y^2-10x=0$, find the equation of a circle with this chord as diameter.

Or

Find the equation of ellipse whose foci are (2, 3) and (- 2, 3) and whose semi-minor axis is $\sqrt{5}$

A straight line meet the axes in A and B such that the centroid of triangle OAB is (a,a). Then the equation of the line AB is

For logarithm ${\log }_{0.2}$ $\frac{x+2}{x}$ to be defined. Which of the following is/are true?

A natural number x is chosen at random from the first 100 natural numbers. Then the probability that, $x+\frac{100}{x}>50$ is:

Find the derivative of the following functions from first principle.

(i) $x^3$ - 27
(ii) (x-1) (x-2)

(iii) $\frac{1}{x^2}$
(iv) $\frac{x+1}{x-1}$

The equation of directrix of a conic is
$L:x+y-1=0$ and the focus is the point $(0,0)$. Find the equation of the conic if its eccentricity is
$\frac{1}{\sqrt{2}}$

The number of solutions of the pair of equations 2 $si{n}^2$θ - $co{s}^2$θ = 0 and 2 $si{n}^2$θ - 3 sin θ = 0, in the interval [0, 2π] is

Three coins are tossed. Describe
(i) Two events whicha are mutually exclusive.
(ii) Three events which are mutually exclusive and exhaustive.
(iii) Two events, which are not mutually exclusive.
(iv) Two events which are mutuallyexclusive but not exhaustive.
(v) Three events which are mutually exclusive but not exhaustive.

If z is a complex number $\overline{z^{-1}}\left(\overline{z}\right)$ = , then

If |z - i Re(z)| = |z - Im(z)|, then :

Using the principle of mathematical induction, prove that

$\mathrm{1.2.3}+\mathrm{2.3.4}+\mathrm{3.4.5}+...+n(n+1)(n+2)=\frac{n(n+1)(n+2)(n+3)}{4},$ for all $n\in N.$

A large tank filled with water to a height 'h' is to be emptied through a small hole at the bottom. The ratio of time taken for the level of water to fall from h to h/2 and from h/2 to zero is

If $A=\{x:x=2n+1,nϵZ\}$ and $B=\{x:x=2n,nϵZ\}$ then find $A\cup B$.

For any complex numbers $z,z_1$ and $z_2$ prove that:
(i) $arg\left(\overline{z}\right)=-arg\left(z\right)$
(ii) $arg\left(z_1{z}_2\right)=arg\left(z_1\right)+arg\left(z_2\right)$
(iii) $arg\left(z_1\overline{z_2}\right)=arg\left(z_1\right)-arg\left(z_2\right)$
(iv) $arg\left(\frac{z_1}{z_2}\right)=arg\left(z_1\right)-arg\left(z_2\right)$

If a, b, c are three distinvt positive real numbers which are in H.P., then $\frac{3a+2b}{2a-b}+\frac{3c+2b}{2c-b}$ is

Draw the graph of the identity function $f:R\to R:f\left(x\right)=x$ for all $xϵR$

A liquid cools down from ${70}^{\circ }C$ to ${60}^{\circ }C$ in 5 minutes.

The time taken to cool it from ${60}^{\circ }C$ to ${50}^{\circ }C$ will be

If the coefficient of $(2r+4{)}^{th}$ and $(r-2{)}^{th}$ terms in the

expansion of ($1+x{)}^{18}$ are equal, then r =

The sides of a right angled triangle are in arithmetic progression. If the triangle has an area of 24 sq. units, then what is the length of its smallest side?___

$Ifaco{s}^3\alpha +3acos\alpha si{n}^2\alpha =mandasi{n}^3\alpha +3aco{s}^2\alpha sin\alpha =n,then(m+n{)}^{\frac{2}{3}}+(m-n{)}^{\frac{2}{3}}isequalto$

What can be the shape of graph of $y=2x^2+bx+cb,cϵR$

If $|x|=1,|y|=2,$ then the least value of $|x-y|$ is

Derivative of $f\left(x\right)=x^2\sin x$ is

Mean and variance of $20$ observations are $10$ and $4$, respectively. It was found, that in place of $11,9$ was taken by mistake, then correct variance is

Let $f\left(x\right)=\sqrt{x}$ and $g\left(x\right)=x$ be two functions defined over the set of non-negative real numbers. Find:

(i) $(f+g)\left(x\right)$

(ii) $(f-g)\left(x\right)$

(iii) $\left(fg\right)\left(x\right)$

(iv) $\frac{f}{g}\left(x\right)$