AP EAPCET 2021 - 19th August Morning Shift

\((\sin \theta-i \cos \theta)^3\) is equal to

Real part of \((\cos 4+i \sin 4+1)^{2020}\) is

\(\int_2^4\{|x-2|+|x-3|\} d x\) is equal to

\(\int\limits_{-1 / 2}^{1 / 2}\left\{[x]+\log \left(\frac{1+x}{1-x}\right)\right\} d x\) is equal to

The solution of the differential equation \(\frac{d^2 y}{d x^2}+y=0\) is

If \(y=x+\frac{1}{x}\), then which among the following holds?

If \(3 \sin x y+4 \cos x y=5\), then \(\frac{d y}{d x}\) is equal to

\(f(x)=\sqrt{x^2+1}: g(x)=\frac{x+1}{x^2+1}: h(x)=2 x-3\), then the value of \(f^{\prime}\left[h^{\prime}\left(g^{\prime}(x)\right)\right]\) is equal to

For which value(s) of \(a\) \(f(x)=-x^3+4 a x^2+2 x-5\) is decreasing for every \(x\) ?

If a point \(P(x, y)\) moves along the ellipse \(\frac{x^2}{25}+\frac{y^2}{16}=1\) and if \(C\) is the center of the ellipse, then the sum of maximum and minimum values of \(C P\) is

Let \(f: R \rightarrow R\) and \(g: R \rightarrow R\) be defined by \(f(x)=2 x+1\) and \(g(x)=x^2-2\) determine \((g \circ f)(x)\) is equal to

Given, the function \(f(x)=\frac{a^x+a^{-x}}{2},(a>2)\), then \(f(x+y)+f(x-y)\) is equal to

If \(f\) is a function defined on \((0,1)\) by \(f(x)=\min \{x-[x],-x-[x]\}\), then \((f \circ f o f o f)(x)\) is equal to \(\rightarrow([\cdot]\) greatest integer function)

If \({({x^2} + 5x + 5)^{x + 5}} = 1\), then the number of integers satisfying this equation is

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

Which statement among the following is true?
(i) the function \(f(x)=x|x|\) is strictly increasing on \(R-\{0\}\).
(ii) the function \(f(x)=\log _{(1 / 4)} x\) is strictly increasing on \((0, \infty)\).
(iii) a one-one function is always an...

The asymptotes of the hyperbola \(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\), with any tangent to the hyperbola form a triangle whose area is \(a^2 \tan (\alpha)\). Then, its eccentricity equals

\(\int \frac{e^x(x+3)}{(x+5)^3} d x\) is equal to

If \(\int \frac{(x-1)^2}{\left(x^2+1\right)^2} d x=\tan ^{-1}(x)+g(x)+k\), then \(g(x)\) is equal to

If \(\int \frac{1-(\cot x)^{2021}}{\tan x+(\cot x)^{2022}} d x=\frac{1}{A} \log\left|(\sin x)^{2023}+(\cos x)^{2023}\right|+c\), then \(A\) is equal to

If \(\tan ^{-1}\left[\frac{1}{1+1 \cdot 2}\right]+\tan ^{-1}\left[\frac{1}{1+2 \cdot 3}\right]+\ldots+\tan ^{-1} \left[\frac{1}{1+n(1+1)}\right]=\tan ^{-1}[x]\), then \(x\) is equal to

If \(y=\tan ^{-1}\left(\frac{\sqrt{1+x^2}+\sqrt{1-x^2}}{\sqrt{1+x^2}-\sqrt{1-x^2}}\right)\), where \(x^2 \leq 1\). Then, find \(\frac{d y}{d x}\) is equal to

If \(\int \frac{d x}{x\left(\sqrt{\left.x^4-1\right)}\right.}=\frac{1}{k} \sec ^{-1}\left(x^k\right)\), then the value of \(k\) is equal to

\(\lim _\limits{z \rightarrow 1} \frac{z^{(1 / 3)}-1}{z^{(1 / 6)}-1}\) is equal to

$$f(x)=\left\{\begin{array}{cc} \frac{72^x-9^x-8^x+1}{\sqrt{2}-\sqrt{1+\cos x}}, & x \neq 0 \\ K \log 2 \log 3, & x=0 \end{array}\right.$$
Find the value of \(k\) for which the function \(f\) is continuous.

If the function \(f(x)\), defined below is continuous in the interval \([0, \pi]\), then $$f(x)=\left\{\begin{array}{cc}x+a \sqrt{2}(\sin x) & , \quad 0 \leq x < \frac{\pi}{4} \\ 2 x(\cot x)+b, & \frac{\pi}{4} \leq x \leq \frac{\pi}{2} \\ a...

\(n \in N\) then, the statement \(8 n+16 \leq 2^n\) is true for

The equation whose roots are the values of the equation \(\left| {\matrix{ 1 & { - 3} & 1 \cr 1 & 6 & 4 \cr 1 & {3x} & {{x^2}} \cr } } \right| = 0\) is

Let a and b be non-zero real numbers such that \(ab=5/2\) and given \(A = \left[ {\matrix{ a & { - b} \cr b & a \cr } } \right]\) and \(A{A^T} = 20I\) (\(l\) is unit matrix), then the equation whose roots are a and b is

If $$A=\left[\begin{array}{ccc}1 & -1 & 1 \\ 2 & 1 & -3 \\ 1 & 1 & 1\end{array}\right], 10 B=\left[\begin{array}{ccc}4 & 2 & 2 \\ -5 & 0 & \alpha \\ 1 & -2 & 3\end{array}\right]$$ and \(B=A^{-1}\), then the value of \(\alpha\) is

The rank of the matrix $$\left[\begin{array}{ccc}4 & 2 & (1-x) \\ 5 & k & 1 \\ 6 & 3 & (1+x)\end{array}\right]$$ is 1 , then,

If \(a_1, a_2, \ldots . a_9\) are in GP, then $$\left|\begin{array}{lll}\log a_1 & \log a_2 & \log a_3 \\ \log a_4 & \log a_5 & \log a_6 \\ \log a_7 & \log a_8 & \log a_9\end{array}\right|$$ is equal to

If \(\mathbf{a}=2 \hat{\mathbf{i}}+\hat{\mathbf{j}}+3 \hat{\mathbf{k}}, \mathbf{b}=\hat{\mathbf{i}}+3 \hat{\mathbf{j}}-\hat{\mathbf{k}}\) and \(\mathbf{c}=3 \hat{\mathbf{i}}-\hat{\mathbf{j}}-2 \hat{\mathbf{k}}\), then the value of $$\left|\...

If one end of focal chord of the parabola \(y^2=8x\) is \(\left(\frac{1}{2},2\right)\), then the length of the focal chord is ................ units.

If a person has 3 coins of different denominations, the number of different sums can be formed is

There are 7 identical white balls and 3 identical black balls. The number of distinguishable arrangements in a row of all the balls, so that no two black balls are adjacent is

The number of ways of distributing eight identical rings to three different girls so that every girl gets at least one ring is

If the letters of the word REGULATIONS be
arranged in such a way that relative positions
of the letters of the word GULATIONS
remain the same, then the probability that
there are exactly 4 letters between R and E is

12 balls are distributed among 3 boxes, then the probability that the first box will contain 3 balls is

A random variable X has the probability distribution

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A die is tossed thrice. If event of getting an even number is a success, then the probability of getting at least 2 successes is

In a \(\Delta ABC\), if a = 3, b = 4 and \(\sin A=\frac{3}{4}\), then \(\angle CBA\) is equal to

In \(\Delta ABC,A=75\Upsilon\) and \(B=45\Upsilon\), then the value of \(b+c\sqrt2\) 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 \(r_2 \leftrightarrow \angle B\) and \(r_3 \leftrightarrow \angle C\). If \(r\) is the radius of inscribed circle, then, w...

If D, E and F are respectively mid-points of AB, AC and BC in \(\Delta ABC\), then BE + AF is equal to

The sum of the roots of the equation \(e^{4 t}-10 e^{3 t}+29 e^{2 t}-22 e^t+4=0\) is

Let \(f(x)=x^3+a x^2+b x+c\) be polynomial with integer coefficients. If the roots of \(f(x)\) are integer and are in Arithmetic Progression, then \(a\) cannot take the value

The variance of the variates 112, 116, 120, 125 and 132 about their AM is

Which of the following set of data has least standard deviation?

If the axes are rotated through an angle \(45 \Upsilon\), the coordinates of the point \((2 \sqrt{2},-3 \sqrt{2})\) in the new system are