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Let $f\left(x\right)=(1-x{)}^{2}si{n}^{2}x+x^2$ for all $x\in R$. Consider the statements:
P: There exists some $x\in R$ such that $f\left(x\right)+2x=2(1+x^2).$
Q:There exists some $x\in R$ such that $2f\left(x\right)+1=2x(1+x).$
Then

If x satisfies $|x-1|+|x-2|+|x-3|\ge 6$, then

Pick the graph(s) which corresponds to a function.

If $a{\cos }^3\alpha +3a\cos \alpha {\sin }^2\alpha =m$ and

$a{\sin }^3\alpha +3a{\cos }^2\alpha \sin \alpha =n$, Then $(m+n{)}^{\frac{2}{3}}+(m-n{)}^{\frac{2}{3}}$

is equal to

The solution set of the inequality ${\log }_{10}(x^2-16)\le {\log }_{10}(4x-11)$ is

If the parabola $y^2=4ax$ passes through (-3,2), then length of its latus rectum is

In an equilateral triangle ABC , match the following ratios to the values

Write the first five terms of each of the sequences in obtain the corresponding series :

$a_1=a_2=2,a_n=a_{n-1}-1,n>2$

If (a-b) sin($\theta +\varphi$) = (a+b) sin($\theta -\varphi$),find tan$\theta$

Find the equation of a line that cuts off equal intercepts on the cordinate axis and passes through the point (2, 3).

Find the domain of the function:

$f\left(x\right)=\frac{x^2+2x+1}{x^2-8x+12}$

If $a(\frac{1}{b}+\frac{1}{c}),b(\frac{1}{c}+\frac{1}{a})c(\frac{1}{a}+\frac{1}{b})$ are in A.P., prove that a, b, c are in A.P.

Find the indicated terms in each of the sequences in where $n^{th}$ terms are :

$a_n=\frac{n^2}{2^n};a_7$

In the quadratic equation $a{x}^2+bx+c=0,\Delta =b^2-4acand\alpha +\beta ,{\alpha }^2+{\beta }^2,{\alpha }^3+{\beta }^3,$ are in G.P. where $\alpha ,\beta$ are the root of $a{x}^2+bx+c=0,$ then

If$\frac{sin(x+y)}{sin(x-y)}=\frac{a+b}{a-b}\text{then}\frac{tanx}{tany}=$

In the sum of first n terms of an A.P. is $c{n}^2$. then the sum of squares of these n terms is

Find the general solution for the following equation:
sin 2x + cos x = 0

Match the following:

$\begin{array}{cc}\text{Column A}& \text{Column B}\\ 1.\{1,2,4,8\}& A.\{x:x\text{is a natural even number less than 10}}\\ 2.\left\{2\right\}& B.\{x:x\text{is a prime number and a divisor of 8}}\\ 3.\{2,4,6,8\}& C.\{x:x\text{is a divisor of 8}}\\ 4.\{4,6,8,9\}& D.\{x:x\text{is a composite number less than 10}\}\end{array}$

Principal value of argument of-2+2i is:

Find the value of $xϵ(0,\pi )$ which satisfies the equation sin x +$\sqrt{3}$ cos x = $\sqrt{2}$

(i) Find the derivative of $x^{2}cosx+5$, using first principle.

(ii) For the function $f\left(x\right)=\{\begin{array}{cc}a+bx,& x<2\\ 4,& x=2\\ b-ax,& x>2\end{array}$

${lim}_{x\to 2}f\left(x\right)=f\left(2\right)$. Find the values of a and b.

If x = $e^{y+e^{y+e^{y+e^{y+...\infty }}}}$, then $\frac{dy}{dx}$ is

In how many ways can a party of 4 men and 4 women be seated at a circular table so that no two women are adjacent?

Find the coordinates of the points which divide the line segment joining A(-2, 2) and B(2, 8) into four equal parts.

Let a${}_1$, a${}_2$, a${}_3$, a${}_4$ and a${}_5$ be such that a${}_1$, a${}_2$ and a${}_3$ are in A.P., a${}_2$, a${}_3$ and a${}_4$ are in G.P., and a${}_3$, a${}_4$ and a${}_5$ are in H.P. Then log${}_e$ a${}_1$, log${}_e$$a_3$ and log${}_e$ a${}_5$ are in

Differentiate $\left(\frac{x^2+5x-6}{4x^2-x+3}\right)$

The term that is independent of x, in the expansion of ${(\frac{3}{2}{x}^2-\frac{1}{3x})}^9$ is

2(2x+3)-10 < 6 (x-2)

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 $\frac{1}{e^2}+\frac{1}{e_1^2}=$

A bag contains 6 white, 7 red and 5 blue balls. Three balls are drawn at random. Find the probability of the event 'balls drawn are one of each color'.

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

The sum of coefficients of the last six terms in the expansion of $(1+x{)}^{11}$ is

Express the complex numbers in the form of a + ib:

$i^{-39}$

If $z=x+iy$ and $w=\frac{1-zi}{z-i},$ show that $|W|=1\Rightarrow z$ is purely real.

${\sin }^2{5}^{\circ }+{\sin }^2{10}^{\circ }+{\sin }^2{15}^{\circ }+...+{\sin }^2{85}^{\circ }+{\sin }^2{90}^{\circ }=$

If y=lnx, find $\frac{dy}{dx}$

The half-life of ${}_6{C}^{14}$ if its K or \lambda is $2.31\times {10}^{-4}$ is

Let $A=\{a,b,c,d,e\}$ and $B=\{1,2,3,4,5\}$. The number of relations which are not functions from $A$ to $B$ is

The graph of expression f(x) = -2 $x^2$ + 5x + 7 looks like

In how many ways 6 letters can be put in 6 addressed envelopes so that only 5 of them will go in the correct envelopes.

The largest term common to the sequences 1, 11, 21, 31, .... To 100 terms and 31, 36, 41, 46,.... to 100 terms is

If f(x) be a continuous function defined for $1\le x\le 3.f\left(x\right)ϵQ\forall xϵ[1,3]andf\left(2\right)=10$ (Where Q is a set of all rational numbers). Then, f(1.8) is

If z lies on the circle $|$z$|$ = 1, then $\frac{2}{z}$ lies on

If $\alpha ,\beta and\gamma$ are the roots of the equation $x^3+3x+2=0$ , Find the equation whose roots are $\alpha -\beta \left)\right(\alpha -\beta ),(\beta -\gamma \left)\right(\beta -\alpha ),(\gamma -\alpha \left)\right(\gamma -\beta$.

If $5^{40}$ is divided by 11 then the remainder is