TS EAMCET 2022 (Online) 19th July Morning Shift

If $\cos \alpha$ is the common value of $(-1)^{\frac{1}{4}}$ and $(-i)^{\frac{1}{2}}$ then $\tan \alpha=$

The equation of lowest degree with rational coefficients having roots $\sqrt{3}+\sqrt{2} i$ and $\sqrt{3}-\sqrt{2}$ is

It is given that $\frac{d}{d t}(t \log t-t)=\log t$, then $\exp \left(\int_0^1 2 x \log \left(1+x^2\right) d x\right)=$

\(\int_0^{2 a} f(x) d x=\)

The equation of any member of the family of all the ellipses whose axes are along the coordinate axes satisfies the differential equation

The degree of the differential equation $\left(\frac{d^2 y}{d x^2}\right)^{\frac{4}{3}}+x\left(\frac{d y}{d x}\right)^2-y \cos \left(\frac{d y}{d x}\right)=0$ is

The general solution of the differential equation $\frac{d y}{d x}=\frac{2 x-3 y+5}{6 x-9 y+7}$ is

If $\frac{3 x+5}{(x+1)\left(2 x^2+3\right)}=\frac{A}{x+1}+\frac{B x+C}{2 x^2+3}$ and $f(x)=A x^3+B x^2+7 x+C$, then $5 C-f^{\prime}(-2)=$

Let $f(x)=\sin x, g(x)=\cos x, h(x)=x^2$, then $\lim _{x \rightarrow 1} \frac{f(g(h(x)))-f(g(h(1)))}{x-1}=$

If $x \cos (k+y)=\cos y$, then $\frac{d y}{d x}$ at $y=\frac{\pi}{2}$ is

If $x=a(\cos \theta+\theta \sin \theta), y=f(\theta), f(2 \pi)=0$, $\frac{d y}{d x}=\frac{\tan \theta}{\theta}, \theta \neq 0$ and $\theta \neq(2 n+1) \frac{\pi}{2}$, then $f\left(\frac{\pi}{3}\right)=$

If $a x^2+b y^2=15$ is the equation of the ellipse for which distance between its foci is 2 and distance between its directrices is 5 , then $a+b=$

Assertion (A) The image of $\frac{x^2}{25}+\frac{y^2}{16}=1$ in the line $x+y=10$ is $\frac{(x-10)^2}{16}+\frac{(y-10)^2}{25}=1$
Reason ( $\mathbf{R}$ ) The image of a curve ' $C$ ' in a line $L$ is the locus of the image of every point of ...

The equation of the normal to the curve $4 x^2+9 y^2=36$ at the point $P\left(\frac{7 \pi}{4}\right)$ is

The set of all real values of $x$ for which $f(x)=\log _2\left(2^x-2\right)+\sqrt{1-x}$ is also real is

Let $f(x)=1-x, g(x)=\frac{1}{1-x}, h(x)=\frac{1}{x}$ be three functions, for $x \neq(0,1)$. If a function $F(x)$ satisfies $f(F(h(x)))=g(x)$, then

If the minimum value of $\cos (\sinh (\log x)+\cosh (\log x))$ is $k$, then $\cosh (k+1)=$

Let $P\left(\frac{\pi}{4}\right), Q\left(\frac{5 \pi}{4}\right), R\left(\frac{3 \pi}{4}\right), T\left(\frac{7 \pi}{4}\right)$ be the points on the hyperbola $x^2-4 y^2-4=0$ in the parametric form. Then the area of the quadrilateral $P Q R ...

If the latusrectum of a hyperbola subtends an angle of $120^{\circ}$ at its centre, then its eccentricity is

If $\frac{42-13 x}{x^2+x-6}=\frac{A}{l x+m}+\frac{B}{p x+q}$, where $l m>0$ and $p q<0$, then $\frac{A l p}{B m q}=$

\(\int \frac{\sin x \cdot \sec ^2 x-\tan x \cdot \sin x+\cos x}{(1-\cos 2 x)} d x=\)

If $f(x)=\int \frac{16 x^7+5 x^{10}}{\left(x^3+2+3 x^8\right)^2} d x(x \geq 0)$ and $f(0)=1$, then the value of $f(-1)$ is

Given that $\frac{d}{d x}\left(\tan ^{-1} x\right)=\frac{1}{1+x^2}$ and $\frac{d}{d x}\left(\sin h^{-1} x\right)=\frac{1}{\sqrt{1+x^2}}$. Then, $\int \frac{3 x^6-2 x^4+x^2-2}{x^2+1} d x=$

\(\mathop {\lim }\limits_{x \to 0} \frac{\tan ^2\left(\pi \sec ^4 x\right)}{\pi^2 x^4}=\)

\(\mathop {\lim }\limits_{x \to 0}\left(\frac{4!}{x^8}\left(1-\cos \frac{x^2}{3}-\cos \frac{x^2}{4}+\cos \frac{x^2}{3} \cos \frac{x^2}{4}\right)\right)=\)

If $\left[\begin{array}{ccc}0 & 2 & a \\ b & 0 & 4 \\ -3 & c & 0\end{array}\right]$ is a skew-symmetric matrix, then $\left[\begin{array}{ll}a & b \\ b & a\end{array}\right]\left[\begin{array}{ll}b & c \\ c & b\end{array}\right]=$

If $\left[\begin{array}{ccc}-1 & 2 & b \\ a & 5 & 6 \\ 3 & c & 7\end{array}\right]$ is a symmetric matrix, then $\left|\begin{array}{lll}a & b & c \\ b & c & a \\ c & a & b\end{array}\right|=$

If the matrix $A=\left[\begin{array}{lll}1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1\end{array}\right]$ satisfies the matrix equation $A^2-4 A-5 I=0$, then $A^{-1}=$

Consider the simultaneous linear equations $A X=B$ and $A Y=Q$. If $A$ is an invertible matrix and $B$ is the unique solution of $A Y=Q$, then the solution of $A X=B$ is

If $f(x)=\left|\begin{array}{ccc}-\sin x & 2 \sin 2 x & 4 \cos ^2 x \\ \cos x & 4 \sin ^2 x & 2 \sin 2 x \\ 0 & -\cos x & \sin x\end{array}\right|$, then $f\left(\frac{5 \pi}{4}\right)+f^{\prime}\left(\frac{5 \pi}{4}\right)=$

If the distance from a variable point $P$ to a fixed point $A(a, 0)$ is equal to the perpendicular distance from $P$ to the line $x+y=0$, then the equation of the locus of $P$ is

The point to which the origin is to be shifted by translation of axes so that the transformed equation of $y^2+4 y+8 x-2=0$ will not contain $y$ term and constant term is

Statement $14 x^2+y^2-4 x y-30 x-50 y+40=0$ is the equation of parabola having $(2,3)$ as its focus and $x+2 y+5=0$ as its directrix.
Statement II The equation of the directrix of the parabola $x^2-4 x+16 y+52=0$ is $y+1=0$
Which of the abo...

The cartesian eql tion of the parabola $x=-2+2 t^2, y=2+4 t$ is

Let $N$ be the set of positive integers. The number of distinct triplets $(x, y, z)$ satisfying $x, y, z \in N, x

A question paper has 3 parts and each part contains 4 questions. The number of different ways in which a candidate can answer 8 questions choosing at least two from each part is

When two dice are thrown, the probability of getting an ordered pair $(x, y)$ such that $x^2+y^2 \leq 25$ where $x$ and $y$ are numbers that show up on the two dice, is

If two cards are drawn simultaneously from a well shuffled pack of 52 cards, then the probability of getting a card having a prime number and a card having a number which is a multiple of 5 is

If $A$ and $B$ are two events of a random experiment such that $P(\bar{A})=\frac{2}{3}, P(B)=\frac{4}{15}$ and $P(A \cap \bar{B})=\frac{1}{5}$, then $\sqrt{195[P(B \mid(A \cup \bar{B}))+P(A \cup B)]}=$

A random variable $X$ has the range $\{0,1,2, \ldots$.$\} . If P(X=r)=k(1+r) 3^{-r}$, for $r=0,1,2, \ldots$ where $k>0$ is a real number, then $P(X=0)+P(X=1)+P(X=2)=$

In an experiment a person gets success $\alpha$ times out of $\beta$ trails. If the experiment consists of $n$ trials, then the probability that he fails at least $(n-1)$ times is

In a $\triangle A B C$, if $a=7, c=11, \cos A=\frac{17}{22}$, $\cos C=\frac{1}{14}$, then $b \tan \frac{B}{2} \tan \frac{C-A}{2}=$

In any $\triangle A B C, r^2 \cot \frac{A}{2} \cot \frac{B}{2} \cot \frac{C}{2}=$

If the sides of a $\triangle A B C$ whose perimeter is 42 are in arithmetic progression, its circumradius is $\frac{65}{8}$ and $B

If $\alpha$ and $\beta$ are the roots of the equation $x^2-2 \sqrt{3} x+4=0$, then $\alpha^6+\beta^6=$

When $b=17$, it is found that the roots of the equation $x^2+b x+c=0$ are -2 and -15 . If $\alpha$ and $\beta$ are the roots of the same equation when $b=13$, then $|\alpha-\beta|=$

Let $x$ be a real number. Malch the following:


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If $\alpha, \beta$ and $\gamma$ are the roots of the equation $5 x^3-2 x-4=0$, then $\alpha^3+\beta^3+\gamma^3=$

If the roots of $x^5-a x^4+b x^3-c x^2+d x-1=0$ are all positive such that their arithmetic mean and geometric mean are equal, then $a+b+c+d=$

The number of non-real roots of the equation $x^{10}-3 x^8+5 x^6-5 x^4+3 x^2-1=0$ is