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Practice Questions
MCQ (Single Correct Answer)
1
The set of all real values of $x$ satisfying the inequality $\frac{7 x^2-5 x-18}{2 x^2+x-6}<2$ is
AP EAPCET 2024 - 23th May Morning Shift
2
The set of all values of $k$ for which the inequality $x^2-(3 k+1) x+4 k^2+3 k-3>0$ is true for all real values of $x$, is
AP EAPCET 2024 - 23th May Morning Shift
3

The cubic equation whose roots are the square of the roots of the equation is

$$ 12 x^3-20 x^2+x+3=0 $$

AP EAPCET 2024 - 23th May Morning Shift
4
$\alpha, \beta$ and $\gamma$ are the roots of the equation $x^3+3 x^2-10 x-24=0$ If $\alpha(\beta+\gamma), \beta(\gamma+\alpha)$ and $\gamma(\alpha+\beta)$ are the roots of the equation $x^3+p x^2+q x+r=0$, then $q$ is equal to
AP EAPCET 2024 - 23th May Morning Shift
5
If ' $a$ ' is a rational number, then the roots of the equation $x^2-3 a x+a^2-2 a-4=0$ are
AP EAPCET 2024 - 22th May Evening Shift
6

The set of all real values ' $a$ ' for which $-1<\frac{2 x^2+a x+2}{x^2+x+1}<3$ holds for all real values of $x$ is

AP EAPCET 2024 - 22th May Evening Shift
7

The quotient, when $3 x^5-4 x^4+5 x^3-3 x^2+6 x-8$ is divided by $x^2+x-3$ is

AP EAPCET 2024 - 22th May Evening Shift
8
If both the roots of the equation $x^2-6 a x+2-2 a+9 a^2=0$ exceed 3 , then
AP EAPCET 2024 - 22th May Morning Shift
9
If $\alpha$ and $\beta$ are two distinct negative roots of $x^5-5 x^3+5 x^2-1=0$, then the equation of least degree with integer coefficients having $\sqrt{-\alpha}$ and $\sqrt{-\beta}$ as its roots, is
AP EAPCET 2024 - 22th May Morning Shift
10
If $\alpha$ is a common root of $x^2-5 x+\lambda=0$ and $x^2-8 x-2 \lambda=0(\lambda \neq 0)$ and $\beta, \gamma$ are the other roots of them, then $\alpha+\beta+\gamma+\lambda=$
AP EAPCET 2024 - 21th May Evening Shift
11
The equation $x^4-x^3-6 x^2+4 x+8=0$ has two equal roots. If $\alpha, \beta$ are the other two roots of this equation, then $\alpha^2+\beta^2=$
AP EAPCET 2024 - 21th May Evening Shift
12
Roots of the equation $a(b-c) x^2+b(c-a) x+c(a-b)=0$ are
AP EAPCET 2024 - 21th May Morning Shift
13
The algebraic equation of degree 4 whose roots are translate of the roots of the equation. $x^4+5 x^3+6 x^2+7 x+9=0$ by -1 is
AP EAPCET 2024 - 21th May Morning Shift
14
Let $[r]$ denote the largest integer not exceeditio $r$ and the roots of the equation $3 x^2+6 x+5+\alpha\left(x^2+2 x+2\right)=0$ are complex number when ever $\alpha>L$ and $\alpha
AP EAPCET 2024 - 20th May Evening Shift
15
For any real value of $x$. If $\frac{11 x^2+12 x+6}{x^2+4 x+2} \notin(a, b)$, then the value $x$ for which $\frac{11 x^2+12 x+6}{x^2+4 x+2}=b-a+3$ is
AP EAPCET 2024 - 20th May Evening Shift
16
If the roots of $\sqrt{\frac{1-y}{y}}+\sqrt{\frac{y}{1-y}}=\frac{5}{2}$ are $\alpha$ and $\beta(\beta>\alpha)$ and the equation $(\alpha+\beta) x^4-25 \alpha \beta x^2+(\gamma+\beta-\alpha)=0$ has real roots, then a possible value of $\gamma$ is
AP EAPCET 2024 - 20th May Evening Shift
17
If $\alpha$ and $\beta$ are two double roots of $x^2+3(a+3) x-9 a=0$ for different values of $a(\alpha>\beta)$, then the minimum value of $x^2+\alpha x-\beta=0$ is
AP EAPCET 2024 - 20th May Morning Shift
18
If $2 x^2+3 x-2=0$ and $3 x^2+\alpha x-2=0$ have one common root, then the sum of all possible values of $\alpha$ is
AP EAPCET 2024 - 20th May Morning Shift
19
If the sum of two roots of $x^3+p x^2+q x-5=0$ is equal to its third root, then $p\left(p^2-4 q\right)=$
AP EAPCET 2024 - 20th May Morning Shift
20
$$ 4+\frac{1}{4+\frac{1}{4+\frac{1}{4+\ldots \infty}}}= $$
AP EAPCET 2024 - 19th May Evening Shift
21
If $x^2+5 a x+6=0$ and $x^2+3 a x+2=0$ have a common root, then that common root is
AP EAPCET 2024 - 19th May Evening Shift
22
If $\alpha, \beta, \gamma$ are roots of equations $x^3+a x^2+b x+x=0$, then $\alpha^{-1}+\beta^{-1}+\gamma^{-1}=$
AP EAPCET 2024 - 19th May Evening Shift
23
For all positive integers $ n $ if $ 3^{2n+1} + 2^{n+1} $ is divisible by $ k $, then the number of prime numbers less than or equal to $ k $ is
AP EAPCET 2024 - 18th May Morning Shift
24
If the roots of the quadratic equation $ x^2 - 35x + c = 0 $ are in the ratio 2 : 3 and $ c = 6K $, then $ K = $
AP EAPCET 2024 - 18th May Morning Shift
25
If the sum of two roots $\alpha, \beta$ of the equation $x^4-x^3-8 x^2+2 x+12=0$ is zero and $\gamma, \delta(\gamma>\delta)$ are its other roots, then $3 \gamma+2 \delta=$
AP EAPCET 2024 - 18th May Morning Shift
26

If $$S=\left\{m \in R: x^2-2(1+3 m) x+7(3+2 m)=0\right.$$ has distinct roots}, then the number of elements in $$S$$ is

AP EAPCET 2022 - 5th July Morning Shift
27

The sum of the real roots of the equation $$x^4-2 x^3+x-380=0$$ is

AP EAPCET 2022 - 5th July Morning Shift
28

If one root of the cubic equation $$x^3+36=7 x^2$$ is double of another, then the number of negative roots are

AP EAPCET 2022 - 5th July Morning Shift
29

If $$f(f(0))=0$$, where $$f(x)=x^2+a x+b, b \neq 0$$, then $$a+b=$$

AP EAPCET 2022 - 4th July Evening Shift
30

The sum of the real roots of the equation $$|x-2|^2+|x-2|-2=0$$ is

AP EAPCET 2022 - 4th July Evening Shift
31

If the difference between the roots of $$x^2+a x+b=0$$ and that of the roots of $$x^2+b x+a=0$$ is same and $$a \neq b$$, then

AP EAPCET 2022 - 4th July Evening Shift
32

For what values of $$a \in Z$$, the quadratic expression $$(x+a)(x+1991)+1$$ can be factorised as $$(x+b)(x+c)$$, where $$b, c \in Z$$ ?

AP EAPCET 2022 - 4th July Evening Shift
33

If $$\frac{13 x+43}{2 x^2+17 x+30}=\frac{A}{2 x+5}+\frac{B}{x+6}$$, then $$A^2+B^2=$$

AP EAPCET 2022 - 4th July Evening Shift
34

If $$f(x)=a x^2+b x+c$$ for some $$a, b, c \in R$$ with $$a+b+c=3$$ and $$f(x+y)=f(x)+f(y)+x y, \forall x, y \in R$$. Then, $$\sum_\limits{n=1}^{10} f(n)=$$

AP EAPCET 2022 - 4th July Morning Shift
35

The number of positive real roots of the equation $$3^{x+1}+3^{-x+1}=10$$ is

AP EAPCET 2022 - 4th July Morning Shift
36

The number of real roots of the equation $$\sqrt{\frac{x}{1-x}}+\sqrt{\frac{1-x}{x}}=\frac{13}{6}$$ is

AP EAPCET 2022 - 4th July Morning Shift
37

If $$\alpha$$ and $$\beta$$ are the roots of the quadratic equation $$x^2+x+1=0$$, then the equation whose roots are $$\alpha^{2021}, \beta^{2021}$$ is given by

AP EAPCET 2021 - 20th August Morning Shift
38

If $$2, 1$$ and $$1$$ are roots of the equation $$x^3-4 x^2+5 x-2=0$$, then the roots of $$\left(x+\frac{1}{3}\right)^3-4\left(x+\frac{1}{3}\right)^2+5\left(x+\frac{1}{3}\right)-2=0$$

AP EAPCET 2021 - 20th August Morning Shift
39

If $$f(x)=2x^3+mx^2-13x+n$$ and 2, 3 are the roots of the equation $$f(x)=0$$, then the values of m and n are

AP EAPCET 2021 - 20th August Morning Shift
40

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)

AP EAPCET 2021 - 19th August Evening Shift
41

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

AP EAPCET 2021 - 19th August Evening Shift
42

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

AP EAPCET 2021 - 19th August Evening Shift
43

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

AP EAPCET 2021 - 19th August Morning Shift