AP EAPCET 2021 - 20th August Morning Shift

If \(y=4 x-6\) is a tangent to the curve \(y^2=a x^4+b\) at \((3,6)\), then the values of \(a\) and \(b\) are

Find the positive value of \(a\) for which the equality \(2 \alpha+\beta=8\) holds, where \(\alpha\) and \(\beta\) are the points of maximum and minimum, respectively, of the function \(f(x)=2 x^3-9 a x^2+12 a^2 x+1\).

If the radius of a sphere is measured as 9 cm
with an error of 0.03 cm, then find the
approximate error in calculating its surface
area.

The diameter and altitude of a right circular
cone, at a certain instant, were found to be
10 cm and 20 cm respectively. If its diameter
is increasing at a rate of 2 cm/s, then at what
rate must its altitude change, in order to keep
its vol...

The locus of a point, which is at a distance of 4 units from \((3,-2)\) in \(x y\)-plane is

Find the equation of the circle which passes through origin and cuts off the intercepts \(-\)2 and 3 over the \(X\) and \(Y\)-axes respectively.

The angle between the pair of tangents drawn from \((1,1)\) to the circle \(x^2+y^2+4 x+4 y-1=0\) is

If the circle \(x^2+y^2-4 x-8 y-5=0\) intersects the line \(3 x-4 y-m=0\) in two distinct points, then the number of integral values of '\(m\)' is

Let \(C\) be the circle center \((0,0)\) and radius 3 units. The equation of the locus of the mid-points of the chords of the circle \(c\) that subtends an angle of \(\frac{2 \pi}{3}\) at its centre is

The length of the common chord of the circles \(x^2+y^2+3x+5y+4=0\) and \(x^2+y^2+5x+3y+4=0\) is __________ units.

Find the equation of the circle which passes through the point \((1,2)\) and the points of intersection of the circles \(x^2+y^2-8 x-6 y+21=0\) and \(x^2+y^2-2 x-15=0\)

If \(z_1=2+3 i\) and \(z_2=3+2 i\), where \(i=\sqrt{-1}\), then $$\left[\begin{array}{cc}z_1 & z_2 \\ -\bar{z}_2 & \bar{z}_1\end{array}\right]\left[\begin{array}{cc}\bar{z}_1 & -z_2 \\ \bar{z}_2 & z_1\end{array}\right]$$ is equal to

The radius of the circle represented by \((1+i)(1+3i)(1+7i)=x+iy\) is \((i=\sqrt{-1})\).

If \(1, \alpha_1, \alpha_2, \alpha_3\) and \(\alpha_4\) are the roots of \(z^5-1=0\) and \(\omega\) is a cube root of units, then $$(\omega-1)\left(\omega-\alpha_1\right)\left(\omega-\alpha_2\right)\left(\omega-\alpha_3\right)\left(\omega-\...

If \(a > 0\) and \(z=x+i y\), then
\(\log _{\cos ^2 \theta}|z-a|>\log _{\cos ^2 \theta}|z-a i|,(\theta \in R)\)
implies

If one root of the equation \(i x^2-2(i+1) x+(2-i)=0\) is \((2-i)\), then the other root is

If \(\int_0^a {{{dx} \over {4 + {x^2}}} = {\pi \over 8}}\), then the value of a is equal to

\(\int_1^2 {{{{x^3} - 1} \over {{x^2}}}}\) is equal to

The solution of the differential equation \(2x\left(\frac{dy}{dx}\right)-y=4\) represents a family of

If \(f(x)=2x^2+3x-5\), then the value of \(f'(0)+3f'(-1)\) is equal to

If \(y=\left(1+\frac{1}{x}\right)\left(1+\frac{2}{x}\right)\left(1+\frac{3}{x}\right) \ldots\left(1+\frac{n}{x}\right)\) and \(x \neq 0\). When \(x=-1, \frac{d y}{d x}\) is equal to

If \(\tan \theta_1, \tan \theta_2=\frac{-a^2}{b^2}\), then the chord joining 2 points \(\theta_1\) and \(\theta_2\) one the ellipse \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\) will subtend a right angle at

\(f(x)=\sin x+\cos x \cdot g(x)=x^2-1\), then \(g(f(x))\) is invertible if

If \(f: z \rightarrow z\) is defined by \(f(x)=x^9-11 x^8-2 x^7+22 x^6+x^4 -12 x^3+11 x^2+x-3, \forall x \in z\), then \(f(11)\) is equal to

Let \(f(x)=x^3\) and \(g(x)=3^x\), then the quadratic equation whose roots are solutions of the equation \((f \circ g)(x)=(g \circ f)(x)\) (for \(x \neq 0\)) is

If one focus of a hyperbola is \((3,0)\), the equation of its directrix is \(4 x-3 y-3=0\) and its eccentricity \(e=5 / 4\), then the coordinates of its vertex is

Given, \(\frac{3 x-2}{(x+1)^2(x+3)}=\frac{A}{x+1} +\frac{B}{(x+1)^2}+\frac{C}{x+3}\), then \(4 A+2 B+4 C\) is equal to

\(\int \frac{\sin \alpha}{\sqrt{1+\cos \alpha}} d \alpha\) is equal to

If \(\int \frac{\cos 4 x+1}{\cot x-\tan x}=k \cos 4 x+C\), then \(k\) is equal to

If \(\int\left[\cos (x) \cdot \frac{d}{d x}(\operatorname{cosec}(x)] d x=f(x)+g(x)+c\right.\) then \(f(x) \cdot g(x)\) is equal to

If \(\int \frac{(2 x+1)^6}{(3 x+2)^8} d x=P\left(\frac{2 x+1}{3 x+2}\right)^Q+R\), then \(\frac{P}{Q}\) is equal to

If \(x=\sin \left(2 \tan ^{-1} 2\right), y=\cos \left(2 \tan ^{-1} 3\right)\) and \(z=\sec \left(3 \tan ^{-1} 4\right)\), then

\(\frac{d}{d x}\left\{\sin ^2\left(\cot ^{-1} \sqrt{\frac{1+x}{1-x}}\right)\right\}\) is equal to

If \(y=\tan ^{-1}\left\{\frac{a x-b}{b x+a}\right\}\), then \(y^{\prime}\) is equal to

\(\mathop {\lim }\limits_{n \to \infty } {{n{{(2n + 1)}^2}} \over {(n + 2)({n^2} + 3n - 1)}}\) is equal to

If the function \(f(x)\), defined below, is continuous on the interval \([0,8]\), then $$f(x)=\left\{\begin{array}{cc}x^2+a x+b & , \quad 0 \leq x < 2 \\ 3 x+2, & 2 \leq x \leq 4 \\ 2 a x+5 b & , 4 < x \leq 8\end{array}\right.$$

If \(f(x)\), defined below, is continuous at \(x=4\), then
$$f(x) = \left\{ {\matrix{
{{{x - 4} \over {|x - 4|}} + a} & , & {x < 4} \cr
{a + b} & , & {x = 4} \cr
{{{x - 4} \over {|x - 4|}} + b} & , & {x > 4} \cr

} } \right....

The trace of the matrix $$A=\left[\begin{array}{ccc}1 & -5 & 7 \\ 0 & 7 & 9 \\ 11 & 8 & 9\end{array}\right]$$ is

If \(A, B\) and \(C\) are the angles of a triangle, then the system of equations
\(-x+y \cos C+z \cos B=0, x \cos C-y+z \cos A=0\) and \(x \cos B+y \cos A-z=0\)

If $$\left[\begin{array}{cc}1 & -\tan \theta \\ \tan \theta & 1\end{array}\right]\left[\begin{array}{cc}1 & \tan \theta \\ -\tan \theta & 1\end{array}\right]^{-1} =\left[\begin{array}{cc}a & -b \\ b & a\end{array}\right]$$, then

What is the value of $$\left|\begin{array}{ccc}a & b & c \\ a-b & b-c & c-a \\ b+c & c+a & a+b\end{array}\right|$$ ?

The coordinates of the focus of the parabola described parametrically by \(x=5t^2+2\) and \(y=10t+4\) (where t is a parameter) are

The value of \({ }^6 P_4+4 \cdot{ }^6 P_3\) is

The number of ways in which 3 boys and
2 girls can sit on a bench so that no two boys
are adjacent is

In how many ways can 5 balls be placed in
4 tins if any number of balls can be placed in
any tin?

One card is selected at random from 27 cards
numbered form 1 to 27. What is the
probability that the number on the card is
even or divisible by 5.

Nine balls one drawn simultaneously from a bag containing 5 white and 7 black balls. The probability of drawing 3 white and 6 black balls is

The probabilities that \(A\) and \(B\) speak truth are \(\frac{4}{5}\) and \(\frac{3}{4}\) respectively. The probability that they contradict each other when asked to speak on a fact is

The mean and variance of a binomial variable
X are 2 and 1 respectively. The probability
that X takes values greater than 1 is

In \(\triangle A B C\), medians \(A D\) and \(B E\) are drawn. If \(A D=4, \angle D A B=\frac{\pi}{6}\) and \(\angle A B E=\frac{\pi}{3}\), then the area of \(\triangle A B C\) is