Question 1: Palette 1/84

AP-EAPCET 2025 MCQ

If $x^2-4 a x+5+a>0$ for all $x \in R$ whenever $a \in(\alpha, \beta)$, then $4 \beta+\alpha=$

A

0

B

4

C

5

D

8

Question 2: Palette 2/84

AP-EAPCET 2025 MCQ

If $\alpha, \beta, \gamma$ are the roots of the equation $x^3-12 x^2+k x-18=0$ and one of them is thrice the sum of the other two roots, then $\alpha^2+\beta^2+\gamma^2-k=$

A

115

B

41

C

56

D

57

Question 3: Palette 3/84

AP-EAPCET 2025 MCQ

The polynomial equation of degree 5 whose roots are the roots of the equation $x^5-3 x^4-x^3+11 x^2-12 x+4=0$ each increased by 2 , is

A

$x^5-13 x^4+63 x^3-135 x^2-108 x=0$

B

$x^5-13 x^4+63 x^3+135 x^2+108 x=0$

C

$x^5-13 x^4+63 x^3-135 x^2+108 x=0$

D

$x^5-13 x^4-63 x^3-135 x^2-108 x=0$

Question 4: Palette 4/84

AP-EAPCET 2025 MCQ

If the area of a square is 575 square units, then the approximate value of its side is

A

23.9792

B

23.7992

C

23.8687

D

23.7868

Question 5: Palette 5/84

AP-EAPCET 2025 MCQ

If $\alpha$ is the common root of the quadratic equations $x^2-5 x+4 a=0, x^2-2 a x-8=0$, where $a \in R$, then the value $\alpha^4-\alpha^3+68$ is

A

260

B

250

C

0

D

240

Question 6: Palette 6/84

AP-EAPCET 2025 MCQ

If $\alpha, \beta$ are the roots of $x^2-5 \gamma x-6 \delta=0$ and $\gamma, \delta$ are the roots of $x^2-5 \alpha x-6 \beta=0$, then $\alpha+\beta+\gamma+\delta=$

A

0

B

125

C

144

D

180

Question 7: Palette 7/84

AP-EAPCET 2025 MCQ

If $\alpha, \beta, \gamma$ are the roots of the equation $x^3+p x^2+q x+r=0$, then $(\alpha+\beta)(\beta+\gamma)(\gamma+\alpha)=$

A

$p-q r$

B

$q-r p$

C

$r-p q$

D

$r+p q$

Question 8: Palette 8/84

AP-EAPCET 2025 MCQ

If the difference of the roots of the equation $x^2-7 x+10=0$ is same as the difference of the roots of the equation $x^2-17 x+k=0$, then a divisor of $k$ is $x^2-7 x+10=0$

A

14

B

17

C

6

D

15

Question 9: Palette 9/84

AP-EAPCET 2025 MCQ

The product of all the real roots of the equation $|x|^2-5|x|+6=0$

A

25

B

36

C

4

D

16

Question 10: Palette 10/84

AP-EAPCET 2025 MCQ

If $\alpha, \beta$ and $\gamma$ are the roots of the equation $5 x^3-4 x^2+3 x-2=0$, then $\alpha^3+\beta^3+\gamma^3=$

A

$\frac{17}{25}$

B

$\frac{394}{125}$

C

$\frac{34}{125}$

D

$\frac{34}{25}$

Question 11: Palette 11/84

AP-EAPCET 2025 MCQ

After the roots of the equation $6 x^3+7 x^2-4 x-2=0$ are diminished by $h$, if the transformed equation does not contain $x$ term, then the product of all the possible value of $h$ is

A

$1 / 3$

B

$-2 / 3$

C

$-2 / 9$

D

$7 / 3$

Question 12: Palette 12/84

AP-EAPCET 2025 MCQ

The number of distinct quadratic equations $a x^2+b x+c=0$ with unequal real roots that can be formed by choosing the coefficients $a, b, c(a \neq b \neq c)$ from the set $\{0,1,2,4\}$ is

A

4

B

6

C

5

D

12

Question 13: Palette 13/84

AP-EAPCET 2025 MCQ

The number of solutions of the equation $\sqrt{3 x^2+x+5}=x-3$ is

A

2

B

1

C

0

D

4

Question 14: Palette 14/84

AP-EAPCET 2025 MCQ

The set of all real values of $x$ for which $\frac{x^2-1}{(x-4)(x-3)} \geq 1$ is

A

$[-1,1] \cup(3,4)$

B

$\left[\frac{13}{7}, 3\right) \cup(4, \infty)$

C

$\left(-\infty, \frac{13}{7}\right] \cup(3,4)$

D

$R-[3,4]$

Question 15: Palette 15/84

AP-EAPCET 2025 MCQ

If $\alpha, \beta$ and $\gamma$ are the roots of the equation $2 x^3+3 x^2-5 x-7=0$, then $\frac{1}{\alpha^2}+\frac{1}{\beta^2}+\frac{1}{\gamma^2}=$

A

$-\frac{17}{49}$

B

$-\frac{23}{49}$

C

$\frac{55}{49}$

D

$\frac{67}{49}$

Question 16: Palette 16/84

AP-EAPCET 2025 MCQ

Two roots of the equation, $a x^4+b x^3+c x^2+d x+e=0$ are positive and equal. If the product of the other two real roots is 1 , then

A

$b e^2=a^2 d$

B

$3 e+\frac{2 b \sqrt{e}}{\sqrt{a}}+c=a$

C

$e+2 b \sqrt{e}+3 c=a \sqrt{a}$

D

$b^2 e=a d^2$

Question 17: Palette 17/84

AP-EAPCET 2025 MCQ

Let $(a-3) x^2+12 x+(a+6)>0, \forall x \in R$ and $a \in(\ell, \infty)$. If $a$ is the least positive integral value of $a$, then the roots of $(\alpha-3) x^2+12 x+(\ell+2)=0$ are

A

1,2

B

2,3

C

$-1,-2$

D

$-2,-3$

Question 18: Palette 18/84

AP-EAPCET 2025 MCQ

If the roots of the equation $x^2+2 a x+b=0$ are real, distinct and differ atmost by 2 m , then $b$ lies in the interval

A

$\left(a^2, a^2+m^2\right)$

B

$\left(a^2+m^2, a^2\right)$

C

$\left[a^2, a^2+2 m^2\right]$

D

$\left[a^2-m^2, a^2\right)$

Question 19: Palette 19/84

AP-EAPCET 2025 MCQ

The cubic equation whose roots are the squares of the roots of the equation $x^3-2 x^2+3 x-4=0$ is

A

$x^3+2 x^2+7 x-16=0$

B

$x^3+2 x^2-7 x-16=0$

C

$x^3-2 x^2-7 x+16=0$

D

$x^3-2 x^2+7 x+16=0$

Question 20: Palette 20/84

AP-EAPCET 2025 MCQ

If $\alpha, \beta, \gamma$ are the roots of the equation $x^3+p x^2+q x+r=0$, then $\alpha^3+\beta^3+\gamma^3=$

A

$p^3-3 p q+r$

B

$p^2-2 p q+r$

C

$3 p q-3 r-p^3$

D

$3 p q+3 r+p^3$

Question 21: Palette 21/84

AP-EAPCET 2025 MCQ

If $\alpha, \beta$ are the roots of the equation $x^2+b x+c=0$ satisfying the conditions $\alpha+\beta=5$ and $\alpha^3+\beta^3=60$, then $3 c+2=$

A

$2 b$

B

$3 b$

C

$-3 b$

D

$-2 b$

Question 22: Palette 22/84

AP-EAPCET 2025 MCQ

If $\alpha, \beta, \gamma$ are the roots of the equation,

$ \begin{aligned} & x^3+a x^2+b x+c=0, \text { then }(\alpha+\beta-2 \gamma) \\ & (\beta+\gamma-2 \alpha)(\gamma+\alpha-2 \beta)= \end{aligned} $

A

$2 a^3+9 a b+27 c$

B

$2 a^3+9 a b-27 c$

C

$2 a^3-9 a b-27 c$

D

$2 a^3-9 a b+27 c$

Question 23: Palette 23/84

AP-EAPCET 2025 MCQ

If the sum of two roots of the equation $x^4+2 x^3-7 x^2-8 x+12=0$ is zero, then the sum of the squares of the other two roots is

A

5

B

10

C

13

D

25

Question 24: Palette 24/84

AP-EAPCET 2025 MCQ

$f(x)$ is a quadratic polynomial satisfying the condition $f(x)+f\left(\frac{1}{x}\right)=f(x) f\left(\frac{1}{x}\right)$. If $f(-1)=0$, then the range of $f$ is

A

$[1, \infty)$

B

$[-1,1]$

C

$(-\infty, 1]$

D

$R$

Question 25: Palette 25/84

AP-EAPCET 2025 MCQ

If $\alpha \neq 0$ and zero are the roots of the equation $x^2-5 k x+\left(6 k^2-2 k\right)=0$, then $\alpha=$

A

$\frac{1}{3}$

B

1

C

$\frac{5}{3}$

D

5

Question 26: Palette 26/84

AP-EAPCET 2025 MCQ

The set of all real values of $x$ satisfying the inequation $\frac{8 x^2-14 x-9}{3 x^2-7 x-6}>2$ is

A

$(-\infty, 1) \cup(3, \infty)$

B

$\left(-\infty,-\frac{2}{3}\right) \cup(2, \infty)$

C

$\left(-\frac{2}{3}, 2\right)$

D

$\left(-\infty,-\frac{2}{3}\right) \cup(3, \infty)$

Question 27: Palette 27/84

AP-EAPCET 2025 MCQ

When the roots of $x^3+\alpha x^2+\beta x+6=0$ are increased by 1 , if one of the resultant values is the least root of $x^4-6 x^3+11 x^2-6 x=0$, then

A

$\alpha-\beta+5=0$

B

$\alpha+\beta+7=0$

C

$2 \alpha+\beta+7=0$

D

$2 \alpha+3 \beta-1=0$

Question 28: Palette 28/84

AP-EAPCET 2025 MCQ

Let ' $a$ ' be a non-zero real number. If the equation whose roots are the squares of the roots of the cubic equation $x^3-a x^2+a x-1=0$ is identical with this cubic equation, then ' $a$ ' =

A

$\frac{1}{3}$

B

3

C

$\frac{1}{2}$

D

2

Question 29: Palette 29/84

AP-EAPCET 2025 MCQ

If $(2 k-1) x^2-2(3 k-2) x+4 k>0$ for every $x \in R$, then the sum of all possible integral values of $k$ is

A

21

B

27

C

36

D

28

Question 30: Palette 30/84

AP-EAPCET 2025 MCQ

If $\alpha$ is a repeated root of multiplicity 2 of the equation $18 x^3-33 x^2+20 x-4=0$, then

A

$3 \alpha^2-8 \alpha+4=0$

B

$3 \alpha^2+8 \alpha+4=0$

C

$3 \alpha^2-\alpha-4=0$

D

$3 \alpha^2+2 \alpha-4=0$

Question 31: Palette 31/84

AP-EAPCET 2025 MCQ

The equation $6 x^4-5 x^3+13 x^2-5 x+6=0$ will have

A

only real roots

B

only complex roots

C

two real and two complex roots

D

two real and two purely imaginary roots

Question 32: Palette 32/84

AP-EAPCET 2025 MCQ

The roots $\alpha, \beta$ of the equation $x^2-6(k-1) x+4(k-2)=0$ are equal in magnitude but opposite in sign, if $\alpha>\beta$, then the product of the roots of the equation $2 x^2-\alpha x+6 \beta(\alpha+1)=0$

A

12

B

-12

C

16

D

-18

Question 33: Palette 33/84

AP-EAPCET 2025 MCQ

If $a x^2+b x+c<0 \forall x \in R$ and the expressions $c x^2+a x+b$ and $a x^2+b x+c$ have their extreme values at the same point $x$, then for the expression $c x^2+a x+b$

A

Minimum value $=\frac{4 b}{3}$

B

Maximum value $=\frac{4 a}{3}$

C

Minimum value $=\frac{3 a}{4}$

D

Maximum value $=\frac{3 b}{4}$

Question 34: Palette 34/84

AP-EAPCET 2025 MCQ

If $x^2-5 x+6$ is a factor of $f(x)=x^4-17 x^3+k x^2-247 x+210$, then the other quadratic factor of $f(x)$ is

A

$x^2+12 x+35$

B

$x^2-12 x+35$

C

$x^2-6 x+35$

D

$x^2+6 x+35$

Question 35: Palette 35/84

AP-EAPCET 2025 MCQ

Given $f(x)=x^2-5 x+4$. Out of first 20 natural numbers, if a number $x$ is chosen at random, then the probability that the chosen $x$ satisfies the inequality $f(x)>10$ is

A

$\frac{1}{2}$

B

$\frac{3}{4}$

C

$\frac{7}{10}$

D

$\frac{13}{20}$

Question 36: Palette 36/84

AP-EAPCET 2025 MCQ

If the harmonic mean of the roots of the equation $\sqrt{2} x^2-b x+(8-2 \sqrt{5})=0$ is

A

3

B

2

C

$4-\sqrt{5}$

D

$4+\sqrt{5}$

Question 37: Palette 37/84

AP-EAPCET 2025 MCQ

All the values of $k$ such that the quadratic expression $2 k x^2-(4 k+1) x+2$ is negative for exactly three integrals values of $x$, lie in the interval

A

$\left[\frac{1}{12}, \frac{1}{10}\right)$

B

$\left(\frac{1}{6}, \frac{1}{5}\right)$

C

$[-1,2)$

D

$[2,6)$

Question 38: Palette 38/84

AP-EAPCET 2025 MCQ

If $\alpha, \beta$ and $\gamma$ are the roots of the equation $x^3-13 x^2+k x+189=0$ such that $\beta-\gamma=2$, then $\beta+\gamma: k+\alpha=$

A

$4: 3$

B

$2: 1$

C

$6: 5$

D

$3: 4$

Question 39: Palette 39/84

AP-EAPCET 2024 MCQ

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

A

$\left(-\infty,-\frac{2}{3}\right] \cup[3, \infty)$

B

$\left(-2,-\frac{2}{3}\right) \cup\left(\frac{3}{2}, 3\right)$

C

$(-\infty,-2) \cup\left(\frac{3}{2}, \infty\right)$

D

$\left[-\frac{2}{3}, \frac{3}{2}\right)$

Question 40: Palette 40/84

AP-EAPCET 2024 MCQ

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

A

$\left(-\frac{13}{7}, 1\right)$

B

$\left(-1, \frac{13}{7}\right)$

C

$\left(-\infty,-\frac{13}{7}\right) \cup(1, \infty)$

D

$(-\infty,-1) \cup\left(\frac{13}{7}, \infty\right)$

Question 41: Palette 41/84

AP-EAPCET 2024 MCQ

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

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

A

$x^3+376 x^2-121 x-9=0$

B

$144 x^3-400 x^2+121 x+98=0$

C

$144 x^3-376 x^2+121 x-9=0$

D

$x^3+400 x^2-121 x-98=0$

Question 42: Palette 42/84

AP-EAPCET 2024 MCQ

$\alpha, \beta$ 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

A

-44

B

-28

C

44

D

28

Question 43: Palette 43/84

AP-EAPCET 2024 MCQ

If ' $a$ ' is a rational number, then the roots of the equation $x^2-3 a x+a^2-2 a-4=0$ are

A

rational and equal numbers

B

different real numbers

C

different rational numbers only

D

not real numbers

Question 44: Palette 44/84

AP-EAPCET 2024 MCQ

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

A

$(-7,5)$

B

$(5, \infty)$

C

$(1,5)$

D

$(-\infty, 1)$

Question 45: Palette 45/84

AP-EAPCET 2024 MCQ

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

A

$3 x^2-7 x-21$

B

$3 x^3-7 x^2+21 x-45$

C

$3 x^4-7 x^3+21 x^2-45+114$

D

$114 x-143$

Question 46: Palette 46/84

AP-EAPCET 2024 MCQ

If both the roots of the equation $x^2-6 a x+2-2 a+9 a^2=0$ exceed 3 , then

A

$a<\frac{3}{2}$

B

$a>\frac{3}{2}$

C

$a<\frac{5}{2}$

D

$a>\frac{11}{9}$

Question 47: Palette 47/84

AP-EAPCET 2024 MCQ

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

A

$x^2-3 x+1=0$

B

$-x^4-5 x^2+5 x+1=0$

C

$-x^4+5 x^2-5 x+1=0$

D

$x^4-3 x^2+1=0$

Question 48: Palette 48/84

AP-EAPCET 2024 MCQ

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 $\alpha+\beta+\gamma+\lambda=$

A

0

B

-1

C

1

D

2

Question 49: Palette 49/84

AP-EAPCET 2024 MCQ

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 $\alpha^2+\beta^2=$

A

4

B

5

C

6

D

7

Question 50: Palette 50/84

AP-EAPCET 2024 MCQ

Roots of the equation $a(b-c) x^2+b(c-a) x+c(a-b)=0$ are

A

$\frac{a(b-c)}{c(a-b)}, 1$

B

$\frac{b(c-a)}{c(a-b)}, 1$

C

$\frac{c(a-b)}{a(b-c)}, 1$

D

$\frac{c(a-b)}{b(c-a)}, 1$

Question 51: Palette 51/84

AP-EAPCET 2024 MCQ

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

A

$x^4+x^3-3 x^2+6 x+4=0$

B

$x^4+9 x^3+27 x^2+38 x+28=0$

C

$x^4+5 x^3+6 x^2+7 x+9=0$

D

$x^4-5 x^3+6 x^2-7 x+9=0$

Question 52: Palette 52/84

AP-EAPCET 2024 MCQ

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

A

$L$

B

$M$

C

$L+M$

D

$M-L$

Question 53: Palette 53/84

AP-EAPCET 2024 MCQ

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

A

$\frac{3}{4}$

B

$\frac{3}{2}$

C

2

D

$-\frac{1}{2}$

Question 54: Palette 54/84

AP-EAPCET 2024 MCQ

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

A

$\frac{1}{2}$

B

4

C

$2 \pi$

D

$\sqrt{e+13}$

Question 55: Palette 55/84

AP-EAPCET 2024 MCQ

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

A

$\frac{69}{4}$

B

$-\frac{69}{4}$

C

$-\frac{35}{4}$

D

$\frac{35}{4}$

Question 56: Palette 56/84

AP-EAPCET 2024 MCQ

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

A

-35

B

7.5

C

-7.5

D

-1.5

Question 57: Palette 57/84

AP-EAPCET 2024 MCQ

If the sum of two roots of $x^3+p x^2+q x-5=0$ is equal to its third root, then $p\left(p^2-4 q\right)=$

A

-20

B

20

C

40

D

-40

Question 58: Palette 58/84

AP-EAPCET 2024 MCQ

$ 4+\frac{1}{4+\frac{1}{4+\frac{1}{4+\ldots \infty}}}= $

A

$(2+\sqrt{5}),(2-\sqrt{5})$

B

$2+\sqrt{5}$

C

$2-\sqrt{5}$

D

$2+\sqrt{3}$

Question 59: Palette 59/84

AP-EAPCET 2024 MCQ

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

A

3 (or) -3

B

2 (or) -2

C

2 (or) -3

D

-2 (or) 3

Question 60: Palette 60/84

AP-EAPCET 2024 MCQ

If $\alpha, \beta, \gamma$ are roots of equations $x^3+a x^2+b x+x=0$, then $\alpha^{-1}+\beta^{-1}+\gamma^{-1}=$

A

$\frac{a}{c}$

B

$-\frac{b}{c}$

C

$\frac{c}{a}$

D

$\frac{b}{a}$

Question 61: Palette 61/84

AP-EAPCET 2024 MCQ

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

A

17

B

6

C

7

D

8

Question 62: Palette 62/84

AP-EAPCET 2024 MCQ

If the roots of the quadratic equation $ x^2 - 35x + c = 0 $ are in the ratio 2 : 3 and $ c = 6K $, then $ K = $

A

49

B

14

C

21

D

7

Question 63: Palette 63/84

AP-EAPCET 2024 MCQ

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 $3 \gamma+2 \delta=$

A

0

B

1

C

3

D

5

Question 64: Palette 64/84

AP-EAPCET 2022 MCQ

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

A

2

B

3

C

4

D

infinite

Question 65: Palette 65/84

AP-EAPCET 2022 MCQ

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

A

$-$1

B

0

C

1

D

2

Question 66: Palette 66/84

AP-EAPCET 2022 MCQ

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

A

1

B

2

C

3

D

0

Question 67: Palette 67/84

AP-EAPCET 2022 MCQ

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

A

2

B

1

C

$-$1

D

$-$2

Question 68: Palette 68/84

AP-EAPCET 2022 MCQ

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

A

4

B

$-$4

C

2

D

$-$2

Question 69: Palette 69/84

AP-EAPCET 2022 MCQ

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

A

$a-b-4=0$

B

$a-b+4=0$

C

$a+b+4=0$

D

$a+b-4=0$

Question 70: Palette 70/84

AP-EAPCET 2022 MCQ

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

A

1990

B

1989

C

1991

D

1992

Question 71: Palette 71/84

AP-EAPCET 2022 MCQ

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

A

22/3

B

52

C

34

D

18/5

Question 72: Palette 72/84

AP-EAPCET 2022 MCQ

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, $\sum_\limits{n=1}^{10} f(n)=$

A

330

B

255

C

165

D

190

Question 73: Palette 73/84

AP-EAPCET 2022 MCQ

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

A

3

B

2

C

1

D

Infinitely many

Question 74: Palette 74/84

AP-EAPCET 2022 MCQ

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

A

1

B

2

C

3

D

4

Question 75: Palette 75/84

AP-EAPCET 2021 MCQ

For $a\ne b$, if the equation $x^2+ax+b=0$ and $x^2+bx+a=0$ have a common root, then the value of $a+b$ is equal to

A

$-$1

B

0

C

1

D

2

Question 76: Palette 76/84

AP-EAPCET 2021 MCQ

If the product of the roots of $9x^3+112x^2-120x+a=0$ is 12, then the value of $a$ is

A

$-$12

B

12

C

$-$108

D

108

Question 77: Palette 77/84

AP-EAPCET 2021 MCQ

$2+\sqrt{5}, 1$ are roots of the cubic equation given by

A

$x^3+3 x^2-3 x-1=0$

B

$x^3-3 x^2+3 x-1=0$

C

$x^3-5 x^2+3 x+1=0$

D

$x^3+5 x^2-3 x+1=0$

Question 78: Palette 78/84

AP-EAPCET 2021 MCQ

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

A

$x^2-x+1=0$

B

$x^2+x-1=0$

C

$x^2-x-1=0$

D

$x^2+x+1=0$

Question 79: Palette 79/84

AP-EAPCET 2021 MCQ

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

A

$\frac{7}{3}, \frac{4}{3}, \frac{4}{3}$

B

$\frac{5}{3}, \frac{2}{3}, \frac{2}{3}$

C

$\frac{-5}{3}, \frac{-2}{3}, \frac{-2}{3}$

D

$\frac{-7}{3}, \frac{-4}{3}, \frac{-4}{3}$

Question 80: Palette 80/84

AP-EAPCET 2021 MCQ

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

A

$-5,-30$

B

$-5,30$

C

5, 30

D

$5,-30$

Question 81: Palette 81/84

AP-EAPCET 2021 MCQ

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)

A

4.54

B

3.54

C

2.54

D

1.54

Question 82: Palette 82/84

AP-EAPCET 2021 MCQ

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

A

2

B

$-$2

C

0

D

1

Question 83: Palette 83/84

AP-EAPCET 2021 MCQ

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

A

5

B

6

C

7

D

8

Question 84: Palette 84/84

AP-EAPCET 2021 MCQ

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

A

$\log _e 10$

B

$2 \log _e 2$

C

$\log _2 29$

D

$2 \log _{10} 2$