AP EAPCET 2021 - 20th August Evening Shift

A spherical iron ball 10 cm in radius is coated
with a layer of ice of uniform thickness, which
melts at a rate of 50 cm\(^3\)
/min. When the
thickness of the ice is 15 cm, the rate at which
the thickness of ice decreases is ........ cm/min...

Find the minimum value of \(2x+3y\), when \(xy=6\).

The volume of a spherical balloon is increasing at the rate of \(30 \mathrm{~cm}^3\) per minute. Find the rate of change of surface area of the balloon, when its radius is \(6 \mathrm{~cm}\).

If \(g(x)=\frac{1}{6} f\left(3 x^2-1\right)+\frac{1}{2} f\left(1-x^2\right), \forall x \in R\), where \(f^{\prime \prime}(x) > 0, \forall x \in R\). Then, \(g(x)\) is increasing in the interval

If the function \(f(x)=2 x^3-9 a x^2+12 a^2 x+1\) attains its maximum and minimum at \(p\) and \(q\) respectively, such that \(p^2=q\), then \(a\) equals

The equation of the pair of straight lines parallel to \(x\)-axis and touching the circle \(x^2+y^2-6 x-4 y-12=0\) is

The points where the circle \(x^2+y^2-3 x -4 y+2=0\) cuts the \(X\)-axis are

The center and radius of the circle \(x^2+y^2+8 x+10 y-8=0\) respectively are and units

The poles of the tangents to the circle \(x^2+y^2=4\) with respect to the circle \((x+2)^2+y^2=8\), lie on

If the power of the point \((1,6)\) with respect to the circle \(x^2+y^2+4 x-6 y-a=0\) is \(-16\) then \(a\) equals

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