AP EAPCET 2021 - 19th August Evening Shift

Given, \(f(x)=x^3-4x\), if x changes from 2 to 1.99, then the approximate change in the value of \(f(x)\) is

If the curves \(\frac{x^2}{a^2}+\frac{y^2}{4}=1\) and \(y^3=16 x\) intersect at right angles, then \(a^2\) is equal to

Let \(x\) and \(y\) be the sides of two squares such that, \(y=x-x^2\). The rate of change of area of the second square with respect to area of the first square is

If \(f^{\prime \prime}(x)\) is a positive function for all \(x \in R, f^{\prime}(3)=0\) and \(g(x)=f\left(\tan ^2(x)-2 \tan (x)+4\right)\) for \(0 < x <\frac{\pi}{2}\), then the interval in which \(g(x)\) is increasing is

Given, two fixed points \(A(-2,1)\) and \(B(3,0)\).
Find the locus of a point \(P\) which moves such that the angle \(\angle A P B\) is always a right angle.

The equations of the tangents to the circle \(x^2+y^2=4\) drawn from the point \((4,0)\) are

If \(P(-9,-1)\) is a point on the circle \(x^2+y^2+4 x+8 y-38=0\), then find equation of the tangent drawn at the other end of the diameter drawn through \(P\)

Find the equation of a circle whose radius is 5 units and passes through two points on the \(X\)-axis, which are at a distance of 4 units from the origin

If a foot of the normal from the point \((4,3)\) to a circle is \((2,1)\) and \(2 x-y-2=0\), is a diameter of the circle, then the equation of circle is

The length of the tangent from any point on the circle \((x-3)^2+(y+2)^2=5 r^2\) to the circle \((x-3)^2+(y+2)^2=r^2\) is 16 units, then the area between the two circles in square units is

The equation of the circle, which cuts orthogonally each of the three circles
$$\begin{aligned} & x^2+y^2-2 x+3 y-7=0, \\ & x^2+y^2+5 x-5 y+9=0 \text { and } \\ & x^2+y^2+7 x-9 y+29=0 \end{aligned}$$

If \(|z-2|=|z-1|\), where \(z\) is a complex number, then locus \(z\) is a straight line

If \({\left( {{{1 + i} \over {1 - i}}} \right)^m} = 1\), then m cannot be equal to

If \(\int_0^{\pi / 2} \tan ^n(x) d x=k \int_0^{\pi / 2} \cot ^n(x) d x\), then

\(\int_0^2 x e^x d x\) is equal to

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

If \(y=e^{x^2+e^{x^2+e^{x^2+\cdots \infty}}}\), then \(\frac{d y}{d x}\) is equal to

\(\frac{d}{d x}\left[\tan ^{-1}\left(\frac{\cos x}{1+\sin x}\right)\right]\) is equal to

If \(x^2+y^2=1\), then

In an ellipse, if the distance between the foci
is 6 units and the length of its minor axis is
8 units, then its eccentricity is

The real valued function \(f(x)=\frac{x}{e^x-1}+\frac{x}{2}+1\) defined on \(R /\{0\}\) is

The domain of the function \(f(x)=\frac{1}{[x]-1}\), where \([x]\) is greatest integer function of \(x\) is

Let \(f: R \rightarrow R\) be a function defined by \(f(x)=\frac{4^x}{4^x+2}\), what is the value of \(f\left(\frac{1}{4}\right)+2 f\left(\frac{1}{2}\right)+f\left(\frac{3}{4}\right)\) is equal to

Which of the following is partial fraction of \(\frac{-x^2+6 x+13}{(3 x+5)\left(x^2+4 x+4\right)}\) is equal to

\(\int \frac{1+x+\sqrt{x+x^2}}{\sqrt{x}+\sqrt{1+x}} d x\) is equal to

\(\int(\cos x) \log \cot \left(\frac{x}{2}\right) d x\) is equal to

\(\int \sqrt{e^{4 x}+e^{2 x}} d x\) is equal to

If \(\int \frac{1}{1+\sin x} d x=\tan (f(x))+c\), then \(f^{\prime}(0)\) is equal to

For how many distinct values of \(x\), the following \(\sin \left[2 \cos ^{-1} \cot \left(2 \tan ^{-1} x\right)\right]=0\) holds?

If $$f(x)=\left\{\begin{array}{cc}\frac{e^{\alpha x}-e^x-x}{x^2}, & x \neq 0 \\ \frac{3}{2}, & x=0\end{array}\right.$$
Find the value of \(\alpha\) for which the function \(f\) is continuous

The value of \(k(k > 0)\), for which the function \(f(x)=\frac{\left(e^x-1\right)^4}{\sin \left(\frac{x^2}{k^2}\right) \log \left(1+\frac{x^2}{2}\right)}\), where \(x \neq 0\) and \(f(0)=8\)

If \(f^{\prime \prime}(x)\) is continuous at \(x=0\) and \(f^{\prime \prime}(0)=4\), then find the following value.
\(\lim _\limits{x \rightarrow 0} \frac{2 f(x)-3 f(2 x)+f(4 x)}{x^2}\) is equal to

For what natural number \(n \in N\), the inequality \(2^n > n+1\) is valid?

The value of $$\left|\begin{array}{ccc}b+c & a & a \\ b & c+a & b \\ c & c & a+b\end{array}\right|$$ is

Let \(A, B, C, D\) be square real matrices such that \(C^T=D A B, D^{\mathrm{T}}=A B C\) and \(S=A B C D\), then \(S^2\) is equal to

$$A=\left[\begin{array}{ccc}a^2 & 15 & 31 \\ 12 & b^2 & 41 \\ 35 & 61 & c^2\end{array}\right]$$ and $$B=\left[\begin{array}{ccc}2 a & 3 & 5 \\ 2 & 2 b & 8 \\ 1 & 4 & 2 c-3\end{array}\right]$$ are two matrices such that the sum of the princi...

Let \(a, b\) and \(c\) be such that \(b+c \neq 0\) and
$$\begin{aligned}
& \left|\begin{array}{ccc}
a & a+1 & a-1 \\
-b & b+1 & b-1 \\
c & c-1 & c+1
\end{array}\right| \\
& +\left|\begin{array}{ccc}
a+1 & b+1 & c-1 \\
a-1 & b-1 & c+1 \\
(-1...

Find the equation of the parabola which
passes through (6, \(-\)2), has its vertex at the
origin and its axis along the Y-axis.

For \(1 \leq r \leq n, \frac{1}{r+1}\left\{{ }^n P_{r+1}-{ }^{(n-1)} P_{r+1}\right\}\) is equal to

In how many ways 4 balls can be picked from
6 black and 4 green coloured balls such that
at least one black ball is selected?

In how many ways can 9 examination papers
be arranged so, that the best and the worst
papers are never together?

P speaks truth in 70% of the cases and Q in
80% of the cases. In what percent of cases are
they likely to agree in stating the same fact

If \(A\) and \(B\) are two events with \(P(A \cap B)=\frac{1}{3}, P(A \cup B)=\frac{5}{6}\) and \(P\left(A^C\right)=\frac{1}{2}\), then the value of \(P\left(B^C\right)\) is

A coin is tossed 2020 times. The probability of getting head on 1947th toss is

A discrete random variable X takes values 10,
20, 30 and 40. with probability 0.3, 0.3, 0.2
and 0.2 respectively. Then the expected value
of X is

Let \(X\) be a random variable which takes values \(1,2,3,4\) such that \(P(X=r)=K r^3\) where \(r=1,2,3,4\) then

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 \(r_1=2, r_2=3, r_3=6\), what is the value of $$r_...

If \(\alpha\) and \(\beta\) are the roots of \(11 x^2+12 x-13=0\), then \(\frac{1}{\alpha^2}+\frac{1}{\beta^2}\) is equal to (approximately close to)

The value of \(a\) for which the equations \(x^3+a x+1=0\) and \(x^4+a x^2+1=0\) have a common root is

If \(a\) is a positive integer such that roots of the equation \(7 x^2-13 x+a=0\) are rational numbers, then the smallest possible value of \(a\) is