AP EAPCET 2021 - 20th August Evening Shift

The equation of radical axis of the circles \(x^2+y^2+4 x+6 y+7=0\) and \(4 x^2+4 y^2+8 x+12 y-9=0\) is

The radical axis of the circles \(S_1: x^2+y^2-4 x+6 y-10=0\) and \(S_2 : x^2+y^2+2 x-6 y+2=0\), cut the circle \(S_1\) in

Let \(Z_1, Z_2\) and \(Z_3\) be three non zero complex numbers such that \(a=\left|Z_1\right|, b=\left|Z_2\right|\) and \(c=\left|Z_3\right|\), if the determinant $$\left|\begin{array}{lll}a & b & c \\ b & c & a \\ c & a & b\end{array}\righ...

If \(\left|z_1+z_2\right|^2=\left|z_1\right|^2+\left|z_2\right|^2\), where \(z_1\) and \(z_2\) are two complex numbers, then

A real value of \(x\) will satisfy the equation, \(\left(\frac{3-4 i x}{3+4 i x}\right)=\alpha-i \beta,(\alpha, \beta\) are real \()\), if

What is the value of \((1-i \sqrt{3})^9\) is equal to

\(\left(\frac{\sqrt{6}-\sqrt{2}}{4}+\frac{\sqrt{6}+\sqrt{2}}{4} i\right)^{2020}\) is equal to

If \(\int_\limits0^\pi \log (\sin x) d x=8 k\), then \(\int_\limits0^{\frac{\pi}{4}} \log (1+\tan x) d x\) is equal to

If \(\int_\limits0^1 x^m(1-x)^n d x=k \int_\limits0^1 x^n(1-x)^m d x\), then the value of $k$ equals

If \(y=\sin (\sin x)\) and \(y^{\prime \prime}+f(x) \cdot y^{\prime}+g(x) \cdot y=0\), then \(f(x) \cdot g(x)\) is equal to

The equation of the curve passing through the point \(\left(0, \frac{\pi}{4}\right)\) and satisfying the differential equation \(\left(e^x \tan y\right) d x\left.+\left(1+e^x\right) \sec ^2 y\right) d y=0\) is given by

If \(x=\sec \theta-\cos \theta\) and \(y=\sec ^n \theta-\cos ^n \theta\), then \(\left(x^2+4\right)\left(\frac{d y}{d x}\right)^2\) is equal to

If \(y=\log _{\cot x} \tan x-\log _{\tan x} \cot x +\tan ^{-1}\left(\frac{4 x}{4-x^2}\right)\), then \(\frac{d y}{d x}\) is equal to

A point moves so that the sum of its distances from \((a e, 0)\) and \((-a e, 0)\) is \(2 a\), then the equation to its locus, where \(b^2=a^2\left(1-e^2\right)\) is

Let \(f(x)=(x+2)^2-2, x \geq-2\). Then, \(f^{-1}(x)\) is equal to

If \(f\) is the greatest integers function defined on \(R\) as \(f(x)=[x]\) and \(g\) is the modulus function defined on $R$ as \(g(x)=|x|\), then the value of \((g \circ f)\left(\frac{-5}{3}\right)\) is

If \(f: R \rightarrow R\) and \(g: R \rightarrow R\) are two functions defined by \(f(x)=a x+b(a \neq 0), \forall x \in R\) and \(g(x)=c x^3+d(c \neq 0), \forall x \in R\), then \((f \circ g)^{-1}(x)\) is equal to

If \(f(10-x)=3 x^2+4 x-5\) and \(f(x)=p x^2+q x+r\), then \(p+q+r\) is equal to

If the focal chord of the hyperbola subtends a right angle at the center, then its eccentricity is

If
$$\begin{aligned} \frac{2 x^4-x^3+3 x^2-x+4}{x^2-3 x+2} =f(x)+\frac{A}{x-1}+\frac{B}{x-2}\end{aligned}$$, then

If \(f^{\prime}(x)=x+\frac{1}{x}\), then \(f(x)\) is equal to

If \(f(x)=\frac{1}{\left(\cos ^2 x\right) \sqrt{1+\tan x}}\), then its antiderivative \(F(x)=\) ........, given, \(F(0)=4\)

If the primitive of \(\cos (\log x)\) is \(f(x)\{\cos (g(x))+\sin (h(x))\}\), then which among the following is true?

\(\int \frac{\sec x}{\sqrt{\sin (2 x+\theta)+\sin \theta}} d x\) is equal to

\(\tan ^{-1}(-2)-\tan ^{-1}(3)\) is equal to

If \(\lim _\limits{x \rightarrow 0}\left(\frac{11 x^3-3 x+4}{13 x^3-5 x^2-7}\right)=\frac{a}{b}\), then the value of \(a+b\) equals

\(\lim _\limits{x \rightarrow 1} \frac{(1-x)\left(1-x^2\right) \ldots\left(1-x^{2 n}\right)}{\left\{(1-x)\left(1-x^2\right) \ldots \ldots\left(1-x^n\right)\right\}^2}=\) _____________, \(\forall n \in N\)

If \(f(x)=\frac{\log _e\left(1+x^2(\tan x)\right)}{\sin x^3}, x \neq 0\) is to be continuous at \(x=0\), then \(f(0)\) must be equal to

If \(k \in R\) and $$\operatorname{det} A=\left|\begin{array}{lll}a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3\end{array}\right|=k$$, then $$\operatorname{det} B=\left|\begin{array}{ccc}a_1 & b_1 & c_1 \\ a_2+2 a_1 & b_2+2 b_1 & c_...

If $$A=\left[\begin{array}{llll}\sqrt{2020} & \sqrt{2021} & \sqrt{2021} & \sqrt{2023} \\ \sqrt{4040} & \sqrt{4042} & \sqrt{4044} & \sqrt{4046} \\ \sqrt{6060} & \sqrt{6063} & \sqrt{6066} & \sqrt{6069} \\ \sqrt{8080} & \sqrt{8084} & \sqrt{808...

If $$\left|\begin{array}{lll}x & x^2 & 1+x^3 \\ y & y^2 & 1+y^3 \\ z & z^2 & 1+z^3\end{array}\right|=0$$ and \(x, y\) and \(z\) are all distinct, then \(x y z\) is equal to

Let A be a \(n\times n\) matrix such that A is upper-triangular. Then, \(adj (A)\) is equal to

If $$f(x)=\left|\begin{array}{ccc}x & x^2 & x^3 \\ 1 & 2 x & 3 x^2 \\ 0 & 2 & 6 x\end{array}\right|$$, then the ratio \(f^{\prime \prime}(x): f^{\prime}(x)\) is equal to

The point of intersection of the latus rectum and axis of the parabola \(y^2+4 x+2 y-8=0\) is

A set contains 11 elements. The number of subsets of the set which contain at most 5 elements is

A coin is tossed until a head appears or it has
been tossed thrice. Given that head doesn’t
appear on the first toss, the probability that
coin tossed thrice is

Box-I contains 3 cards bearing numbers 1, 2, 3 , Box II contains 5 cards bearing numbers 1 , 2, 3, 4, 5 and Box III contains 7 cards bearing numbers 1, 2, 3, 4, 5, 6, 7. One card is drawn at random from each of the boxes. If \(x_i\) be the ...

The range of a random variable \(X\) is \(\{1,2,3, \ldots\}\) and \(P(X=x)=\frac{C^x}{x !}\). for \(x=1,2,3\), ... Then, the value of \(C\) is

Tom and Jerry play a game of alternately
throwing an unfair coin. First one to get head
wins. If Tom starts the game, he has 62.5%
chance of winning the game. Suppose this
coin is tossed 5 times, then the probability of
getting exactly 3 he...

What is the value of \((a-b)^2 \cos ^2 \frac{c}{2}+(a+b)^2 \sin ^2 \frac{c}{2}\) is equal to

In \(\triangle A B C\), suppose the radius of the circle opposite to an angle \(A\) is denoted by \(r_1\), similarly \(r_2 \leftrightarrow\) angle \(B, r_3 \leftrightarrow\) angle \(C\). If \(r_1=2, r_2=3\) and \(r_3=6\), then what is $$(a,...

If in \(\triangle A B C, a \tan A+b \tan B=(a+b). \tan \left(\frac{A+B}{2}\right)\), then which of the following holds?

For \(a\ne b\), if the equation \(x^2+ax+b=0\) and \(x^2+bx+a=0\) have a common root, then the value of \(a+b\) is equal to

If the product of the roots of \(9x^3+112x^2-120x+a=0\) is 12, then the value of \(a\) is

\(2+\sqrt{5}, 1\) are roots of the cubic equation given by

Using mathematical induction, the numbers \(a_n^{\prime}\) s are defined by \(a_0=1, a_{n+1}=3 n^2+n+a_n (n \geq 0)\), then \(a_n\) is equal to

If the mean of a data x is 10 and if all the
observations are multiplied by 2, then the
mean of new data is

The point to which the origin should be shifted in order to eliminate the \(x\) and \(y\) terms from the equation \(9 x^2+4 y^2+10 x+12 y+1=0\) is

If \(A(1,3)\) and \(C(7,5)\) are two opposite vertices of a square, then find the equation of a side passing through \(A\).

\(C\) is the centroid of the triangle with vertices \((3,-1),(1,3)\) and \((2,4)\). Let \(P\) be the point of intersection of the lines \(x+3 y-1=0\) and \(3 x-y+1=0\). Then a line which passes through both points \(C\) and \(P\) would also...