4. Quadratic Equations(11th)

SECTION A: Quadratic Equations

1. The sum of the squares of the roots of $|x-2|^2+|x-2|-2=0$ and the squares of the roots of $x^2-2|x-3|-5=0$ , is:

(A) 24

(B) 26

(D) 30

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2. The number of real roots of the equation $x|x-2|+3|x-3|+1=0$ is:

(A) 4

(B) 3

(D) 1

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3. Let the set of all values of $p \in \mathbb{R}$ , for which both the roots of the equation $x^2-(p+2) x+(2 p+9)=0$ are negative real numbers, be the interval $(\alpha, \beta]$ . Then $\beta-2 \alpha$ is equal to:

(A) 5

(B) 6

(D) 9

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4. Consider the equation $x^2+4 x-n=0$ , where $n \in [20,100]$ is a natural number. Then the number of all distinct values of $n$ , for which the given equation has integral roots, is equal to:

(A) 6

(B) 5

(D) 7

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5. Let the equation $x(x+2)(12-k)=2$ have equal roots. Then the distance of the point $\left(k, \frac{k}{2}\right)$ from the line $3 x+4 y+5=0$ is:

(A) 15

(B) 12

(D) $15 \sqrt{5}$

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6. Let $\alpha$ and $\beta$ be the roots of $x^2+\sqrt{3} x-16=0$ , and $\gamma$ and $\delta$ be the roots of $x^2+3 x-1=0$ . If $P_n=\alpha^n+\beta^n$ and $Q_n=\gamma^n+\delta^n$ , then $\frac{P_n+\sqrt{3} P_n}{2 P_n}+\frac{Q_n-Q_n}{4 P_n}$ is equal to:

(A) 4

(B) 3

(D) 7

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7. Let $P_n=\alpha^n+\beta^n$ , $n \in \mathbb{N}$ . If $P_{10}=123$ , $P_9=76$ , $P_8=47$ and $P_1=1$ , then the quadratic equation having roots $\frac{1}{\alpha}$ and $\frac{1}{\beta}$ is:

(A) $x^2+x-1=0$

(B) $x^2-x+1=0$

(D) $x^2-x-1=0$

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8. If the set of all $a \in \mathbb{R}$ , for which the equation $2 x^2+(a-5) x+15=3 a$ has no real root, is the interval $(\alpha, \beta)$ , and $X=\{x \in \mathbb{Z} : \alpha < x < \beta\}$ , then $\sum_{x \in X} x^2$ is equal to:

(A) 2139

(B) 2119

(D) 2129

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9. The number of solutions of the equation $\left( \frac{9}{x} - \frac{9}{\sqrt{x}} + 2 \right) \left( \frac{2}{x} - \frac{7}{\sqrt{x}} + 3 \right) = 0$ is:

(A) 3

(B) 2

(D) 4

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10. Let $f: \mathbb{R} \setminus \{0\} \to (-\infty, 1)$ be a polynomial of degree 2, satisfying $f(x) f\left(\frac{1}{x}\right)=f(x)+f\left(\frac{1}{x}\right)$ . If $f(K)=-2 K$ , then the sum of squares of all possible values of $K$ is:

(A) 9

(B) 1

(D) 7

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11. The sum of the squares of all the roots of the equation $x^2+|2 x-3|-4=0$ , is:

(A) $6(2-\sqrt{2})$

(B) $3(3-\sqrt{2})$

(D) $6(3-\sqrt{2})$

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12. The number of real solution(s) of the equation $x^2+3 x+2=\min \{|x-3|,|x+2|\}$ is:

(A) 2

(B) 3

(D) 0

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13. The product of all the rational roots of the equation $\left(x^2-9 x+11\right)^2-(x-4)(x-5)=3$ , is equal to:

(A) 7

(B) 21

(D) 14

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14. Let $\alpha_\theta$ and $\beta_\theta$ be the distinct roots of $2 x^2+(\cos \theta) x-1=0$ , $\theta \in (0,2 \pi)$ . If $m$ and $M$ are the minimum and the maximum values of $\alpha_\theta^4+\beta_\theta^4$ , then $16(M+m)$ equals:

(A) 27

(B) 17

(D) 24

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15. If the set of all $a \in \mathbb{R} \setminus \{1\}$ , for which the roots of the equation $(1-a) x^2+2(a-3) x+9=0$ are positive is $(-\infty,-\alpha] \cup [\beta, \gamma)$ , then $2 \alpha+\beta+\gamma$ is equal to:

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16. If the equation $a(b-c) x^2+b(c-a) x+c(a-b)=0$ has equal roots, where $a+c=15$ and $b=\frac{36}{5}$ , then $a^2+c^2$ is equal to:

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17. The number of distinct real roots of the equation $|x+1||x+3|-4|x+2|+5=0$ , is:

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18. Let $x_{1}, x_{2}, x_{3}, x_{4}$ be the solutions of the equation $4 x^{4}+8 x^{3}-17 x^{2}-12 x+9=0$ and $\left(4+x_{1}^{2}\right)\left(4+x_{2}^{2}\right)\left(4+x_{3}^{2}\right)\left(4+x_{4}^{2}\right)=\frac{125}{16} m$ . Then the value of $m$ is:

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19. The number of real solutions of the equation $x|x+5|+2|x+7|-2=0$ is:

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20. The number of distinct real roots of the equation $|x||x+2|-5|x+1|-1=0$ is:

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SECTION B: Quadratic Equations (Q21-60)

21. Let $a, b, c$ be the lengths of three sides of a triangle satisfying the condition $\left(a^{2}+b^{2}\right) x^{2}-2 b(a+c) x+\left(b^{2}+c^{2}\right)=0$ . If the set of all possible values of $x$ is the interval $(\alpha, \beta)$ , then $12\left(\alpha^{2}+\beta^{2}\right)$ is equal to:

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22. The number of real solutions of the equation $x\left(x^{2}+3|x|+5|x-1|+6|x-2|\right)=0$ is:

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23. Let $\alpha, \beta \in \mathbb{N}$ be roots of the equation $x^{2}-70 x+\lambda=0$ , where $\frac{\lambda}{2}, \frac{\lambda}{3} \notin \mathbb{N}$ . If $\lambda$ assumes the minimum possible value, then $\frac{(\sqrt{\alpha-1}+\sqrt{\beta-1})(\lambda+35)}{|\alpha-\beta|}$ is equal to:

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24. Let the set $C=\left\{(x, y) \mid x^{2}-2^{y}=2023, x, y \in \mathbb{N}\right\}$ . Then $\sum_{(x, y) \in C}(x+y)$ is equal to:

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25. Let $[\alpha]$ denote the greatest integer $\leq \alpha$ . Then $[\sqrt{1}]+[\sqrt{2}]+[\sqrt{3}]+\ldots+[\sqrt{120}]$ is equal to:

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26. The number of points where the curve $f(x)=\mathrm{e}^{8 x}-\mathrm{e}^{6 x}-3 \mathrm{e}^{4 x}-\mathrm{e}^{2 x}+1, x \in \mathbb{R}$ cuts the $x$ -axis, is equal to:

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27. If $a$ and $b$ are the roots of the equation $x^{2}-7 x-1=0$ , then the value of $\frac{a^{21}+b^{21}+a^{17}+b^{17}}{a^{19}+b^{19}}$ is equal to:

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28. Let $m$ and $n$ be the numbers of real roots of the quadratic equations $x^{2}-12 x+[x]+31=0$ and $x^{2}-5|x+2|-4=0$ respectively, where $[x]$ denotes the greatest integer $\leq x$ . Then $m^{2}+mn+n^{2}$ is equal to:

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29. If the value of real number $a>0$ for which $x^{2}-5 a x+1=0$ and $x^{2}-a x-5=0$ have a common real root is $\frac{3}{\sqrt{2 \beta}}$ , then $\beta$ is equal to:

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30. Let $\alpha_{1}, \alpha_{2}, \ldots, \alpha_{7}$ be the roots of the equation $x^{7}+3 x^{5}-13 x^{3}-15 x=0$ and $\left|\alpha_{1}\right| \geq\left|\alpha_{2}\right| \geq \ldots \geq\left|\alpha_{7}\right|$ . Then $\alpha_{1} \alpha_{2}-\alpha_{3} \alpha_{4}+\alpha_{5} \alpha_{6}$ is equal to:

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31. Let $\alpha \in \mathbb{R}$ and let $\alpha, \beta$ be the roots of the equation $x^{2}+60^{\frac{1}{4}} x+a=0$ . If $\alpha^{4}+\beta^{4}=-30$ , then the product of all possible values of $a$ is:

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32. Let $\lambda \in \mathbb{R}$ and let the equation $E$ be $|x|^{2}-2|x|+|\lambda-3|=0$ . Then the largest element in the set $S=\{x+\lambda: x \text{ is an integer solution of } E \}$ is:

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33. Let $\alpha, \beta (\alpha>\beta)$ be the roots of the quadratic equation $x^{2}-x-4=0$ . If $P_{n}=\alpha^{n}-\beta^{n}, n \in \mathbb{N}$ , then $\frac{P_{15} P_{16}-P_{14} P_{16}-P_{15}^{2}+P_{14} P_{15}}{P_{13} P_{14}}$ is equal to:

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34. The sum of all real values of $x$ for which $\frac{3 x^{2}-9 x+17}{x^{2}+3 x+10}=\frac{5 x^{2}-7 x+19}{3 x^{2}+5 x+12}$ is equal to:

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35. If for some $p, q, r \in \mathbb{R}$ , not all have same sign, one of the roots of the equation $\left(p^{2}+q^{2}\right) x^{2}-2 q(p+r) x+q^{2}+r^{2}=0$ is also a root of the equation $x^{2}+2 x-8=0$ , then $\frac{q^{2}+r^{2}}{p^{2}}$ is equal to:

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36. The number of distinct real roots of the equation $x^{5}\left(x^{3}-x^{2}-x+1\right)+x\left(3 x^{3}-4 x^{2}-2 x+4\right)-1=0$ is:

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37. The number of real solutions of the equation $e^{4 x}+4 e^{3 x}-58 e^{2 x}+4 e^{x}+1=0$ is:

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38. Let $\alpha, \beta$ be the roots of the equation $x^{2}-4 \lambda x+5=0$ and $\alpha, \gamma$ be the roots of the equation $x^{2}-(3 \sqrt{2}+2 \sqrt{3}) x+7+3 \lambda \sqrt{3}=0, \lambda>0$ . If $\beta+\gamma=3 \sqrt{2}$ , then $(\alpha+2 \beta+\gamma)^{2}$ is equal to:

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39. If the sum of all the roots of the equation $e^{2 x}-11 e^{x}-45 e^{-x}+\frac{81}{2}=0$ is $\log_{e} p$ , then $p$ is equal to:

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40. Let $p$ and $q$ be two real numbers such that $p+q=3$ and $p^{4}+q^{4}=369$ . Then $\left(\frac{1}{p}+\frac{1}{q}\right)^{-2}$ is equal to:

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41. The sum of the cubes of all the roots of the equation $x^{4}-3 x^{3}-2 x^{2}+3 x+1=0$ is:

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42. Let $f(x)$ be a polynomial of degree 3 such that $f(k)=-\frac{2}{k}$ for $k=2,3,4,5$ . Then the value of $52-10 f(10)$ is equal to:

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43. Let $\lambda \neq 0$ be in $\mathbb{R}$ . If $\alpha$ and $\beta$ are the roots of the equation $x^{2}-x+2 \lambda=0$ , and $\alpha$ and $\gamma$ are the roots of equation $3 x^{2}-10 x+27 \lambda=0$ , then $\frac{\beta \gamma}{\lambda}$ is equal to:

(A) 36

(B) 9

(D) 18

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44. The sum of all integral values of $k (k \neq 0)$ for which the equation $\frac{2}{x-1}-\frac{1}{x-2}=\frac{2}{k}$ in $x$ has no real roots, is:

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45. The number of real roots of the equation $e^{4 x}-e^{3 x}-4 e^{2 x}-e^{x}+1=0$ is equal to:

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46. If $a+b+c=1$ , $a b+b c+c a=2$ and $a b c=3$ , then the value of $a^{4}+b^{4}+c^{4}$ is equal to:

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47. If $\alpha, \beta$ are roots of the equation $x^{2}+5(\sqrt{2}) x+10=0$ , $\alpha>\beta$ and $P_{n}=\alpha^{n}-\beta^{n}$ for each positive integer $n$ , then the value of $\left(\frac{P_{17} P_{20}+5 \sqrt{2} P_{17} P_{19}}{P_{18} P_{19}+5 \sqrt{2} P_{18}^{2}}\right)$ is equal to:

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48. Let $\alpha$ and $\beta$ be two real numbers such that $\alpha+\beta=1$ and $\alpha \beta=-1$ . Let $p_{n}=(\alpha)^{n}+(\beta)^{n}$ , $p_{n-1}=11$ and $p_{n+1}=29$ for some integer $n \geq 1$ . Then, the value of $p_{n}^{2}$ is:

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49. The sum of the $162^{\text{th}}$ power of the roots of the equation $x^{3}-2 x^{2}+2 x-1=0$ is:

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50. The number of real roots of the equation $(x+1)^{2}+|x-5|=\frac{27}{4}$ is:

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51. The least positive value of $a$ for which the equation $2 x^{2}+(a-10) x+\frac{33}{2}=2 a$ has real roots is:

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52. Let $\alpha, \beta; \alpha>\beta$ , be the roots of the equation $x^{2}-\sqrt{2} x-\sqrt{3}=0$ . Let $P_{n}=\alpha^{n}-\beta^{n}, n \in \mathbb{N}$ . Then $(11 \sqrt{3}-10 \sqrt{2}) P_{10}+(11 \sqrt{2}+10) P_{11}-11 P_{12}$ is equal to:

(A) $10 \sqrt{3} P_{9}$

(B) $11 \sqrt{3} P_{9}$

(D) $10 \sqrt{2} P_{9}$

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53. Let $\alpha, \beta$ be the roots of the equation $x^{2}+2 \sqrt{2} x-1=0$ . The quadratic equation, whose roots are $\alpha^{4}+\beta^{4}$ and $\frac{1}{10}\left(\alpha^{6}+\beta^{6}\right)$ , is:

(A) $x^{2}-180 x+9506=0$

(B) $x^{2}-195 x+9506=0$

(D) $x^{2}-195 x+9466=0$

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54. The sum of all the solutions of the equation $(8)^{2 x}-16 \cdot (8)^{x}+48=0$ is:

(A) $1+\log_{8}(6)$

(B) $1+\log_{6}(8)$

(D) $\log_{8}(4)$

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55. Let $\alpha, \beta$ be the distinct roots of the equation $x^{2}-\left(t^{2}-5 t+6\right) x+1=0, t \in \mathbb{R}$ and $a_{n}=\alpha^{n}+\beta^{n}$ . Then the minimum value of $\frac{a_{2023}+a_{2025}}{a_{2024}}$ is:

(A) $-\frac{1}{2}$

(B) $-\frac{1}{4}$

(D) $\frac{1}{2}$

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56. The coefficients $a, b, c$ in the quadratic equation $a x^{2}+b x+c=0$ are from the set $\{1,2,3,4,5,6\}$ . If the probability of this equation having one real root bigger than the other is $p$ , then $216 p$ equals:

(A) 38

(B) 7

(D) 19

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57. If 2 and 6 are the roots of the equation $a x^{2}+b x+1=0$ , then the quadratic equation, whose roots are $\frac{1}{2 a+b}$ and $\frac{1}{6 a+b}$ , is:

(A) $x^{2}+8 x+12=0$

(B) $2 x^{2}+11 x+12=0$

(D) $x^{2}+10 x+16=0$

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58. Let $\alpha$ and $\beta$ be the roots of the equation $p x^{2}+q x-r=0$ , where $p \neq 0$ . If $p, q$ and $r$ be the consecutive terms of a non-constant G.P. and $\frac{1}{\alpha}+\frac{1}{\beta}=\frac{3}{4}$ , then the value of $(\alpha-\beta)^{2}$ is:

(A) 8

(B) 9

(D) $\frac{80}{9}$

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59. Let $S=\left\{x \in \mathbb{R} : (\sqrt{3}+\sqrt{2})^{x}+(\sqrt{3}-\sqrt{2})^{x}=10\right\}$ . Then the number of elements in $S$ is:

(A) 4

(B) 0

(D) 1

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60. Let $S$ be the set of positive integral values of $a$ for which $\frac{a x^{2}+2(a+1) x+9 a+4}{x^{2}-8 x+32}<0, \forall x \in \mathbb{R}$ . Then, the number of elements in $S$ is:

(A) 0

(B) $\infty$

(D) 1

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SECTION C: Quadratic Equations (Q61-80)

61. If $\alpha, \beta$ are the roots of the equation $x^{2}-x-1=0$ and $S_{n}=2023 \alpha^{n}+2024 \beta^{n}$ , then:

(A) $2 S_{12}=S_{11}+S_{10}$

(B) $S_{12}=S_{11}+S_{10}$

(D) $2 S_{11}=S_{12}+S_{10}$

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62. The number of real roots of the equation $x|x|-5|x+2|+6=0$ , is:

(A) 4

(B) 3

(D) 6

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63. Let $\alpha, \beta$ be the roots of the equation $x^{2}-\sqrt{2} x+2=0$ . Then $\alpha^{14}+\beta^{14}$ is equal to:

(A) -64

(B) $-64 \sqrt{2}$

(D) -128

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64. The set of all $a \in \mathbb{R}$ for which the equation $x|x-1|+|x+2|+a=0$ has exactly one real root, is:

(A) $(-\infty, \infty)$

(B) $(-6, \infty)$

(D) $(-6, -3)$

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65. Let $\alpha, \beta$ be the roots of the quadratic equation $x^{2}+\sqrt{6} x+3=0$ . Then $\frac{\alpha^{23}+\beta^{23}+\alpha^{14}+\beta^{14}}{\alpha^{15}+\beta^{15}+\alpha^{10}+\beta^{10}}$ is equal to:

(A) 72

(B) 9

(D) 81

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66. Let $\alpha, \beta, \gamma$ be the three roots of the equation $x^{3}+b x+c=0$ . If $\beta \gamma=1=-\alpha$ , then $b^{3}+2 c^{3}-3 \alpha^{3}-6 \beta^{3}-8 \gamma^{3}$ is equal to:

(A) 21

(B) 19

(D) $\frac{155}{8}$

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67. Let $A=\{x \in \mathbb{R} : [x+3]+[x+4] \leq 3\}$ , $B=\left\{x \in \mathbb{R} : 3^{x}\left(\sum_{r=1}^{\infty} \frac{3}{10^{r}}\right)^{x-3}<3^{-3 x}\right\}$ , where $[t]$ denotes greatest integer function. Then:

(A) $B \subset A, A \neq B$

(B) $A \subset B, A \neq B$

(D) $A \cap B=\emptyset$

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68. The sum of all the roots of the equation $\left|x^{2}-8 x+15\right|-2 x+7=0$ is:

(A) $11+\sqrt{3}$

(B) $9+\sqrt{3}$

(D) $11-\sqrt{3}$

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69. The number of integral values of $k$ , for which one root of the equation $2 x^{2}-8 x+k=0$ lies in the interval $(1,2)$ and its other root lies in the interval $(2,3)$, is:

(A) 2

(B) 0

(D) 3

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70. Let $S=\left\{x \in \mathbb{R} : (\sqrt{3}+\sqrt{2})^{x^{2}-4}+(\sqrt{3}-\sqrt{2})^{x^{2}-4}=10\right\}$ . Then $n(S)$ is equal to:

(A) 6

(B) 4

(D) 2

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71. The equation $\mathrm{e}^{4 x}+8 \mathrm{e}^{3 x}+13 \mathrm{e}^{2 x}-8 \mathrm{e}^{x}+1=0, x \in \mathbb{R}$ has:

(A) two solutions and both are negative

(B) two solutions and only one of them is negative

(D) no solution

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72. The number of real roots of the equation $\sqrt{x^{2}-4 x+3}+\sqrt{x^{2}-9}=\sqrt{4 x^{2}-14 x+6}$ , is:

(A) 0

(B) 1

(D) 2

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73. Let $\lambda \neq 0$ be a real number. Let $\alpha, \beta$ be the roots of the equation $14 x^{2}-31 x+3 \lambda=0$ and $\alpha, \gamma$ be the roots of the equation $35 x^{2}-53 x+4 \lambda=0$ . Then $\frac{3 \alpha}{\beta}$ and $\frac{4 \alpha}{\gamma}$ are the roots of the equation:

(A) $7 x^{2}-245 x+250=0$

(B) $49 x^{2}-245 x+250=0$

(D) $7 x^{2}+245 x-250=0$

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74. The number of real solutions of the equation $3\left(x^{2}+\frac{1}{x^{2}}\right)-2\left(x+\frac{1}{x}\right)+5=0$ , is:

(A) 3

(B) 4

(D) 2

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75. The equation $x^{2}-4 x+[x]+3=x[x]$ , where $[x]$ denotes the greatest integer function, has:

(A) exactly two solutions in $(-\infty, \infty)$

(B) no solution

(D) a unique solution in $(-\infty, 1)$

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76. If $\frac{1}{(20-a)(40-a)}+\frac{1}{(40-a)(60-a)}+\ldots+\frac{1}{(180-a)(200-a)}=\frac{1}{256}$ , then the maximum value of $a$ is:

(A) 198

(B) 202

(D) 218

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77. Let $S=\left\{x \in [-6,3] \setminus \{-2,2\} : \frac{|x+3|-1}{|x|-2} \geq 0\right\}$ and $T=\left\{x \in \mathbb{Z} : x^{2}-7|x|+9 \leq 0\right\}$ . Then the number of elements in $S \cap T$ is:

(A) 7

(B) 5

(D) 3

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78. Let $\alpha, \beta$ be the roots of the equation $x^{2}-\sqrt{2} x+\sqrt{6}=0$ and $\frac{1}{\alpha^{2}}+1, \frac{1}{\beta^{2}}+1$ be the roots of the equation $x^{2}+a x+b=0$ . Then the roots of the equation $x^{2}-(a+b-2) x+(a+b+2)=0$ are:

(A) non-real complex numbers

(B) real and both negative

(D) real and exactly one of them is positive

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79. If $\alpha, \beta$ are the roots of the equation $x^{2}-\left(5+3^{3^{\sqrt{\log_{3} 5}}}-5^{\sqrt{\log_{5} 3}}\right) x + 3\left(3^{\left(\log_{3} 5\right)^{\frac{1}{3}}}-5^{\left(\log_{5} 3\right)^{\frac{2}{3}}}-1\right)=0$ , then the equation, whose roots are $\alpha+\frac{1}{\beta}$ and $\beta+\frac{1}{\alpha}$ , is:

(A) $3 x^{2}-20 x-12=0$

(B) $3 x^{2}-10 x-4=0$

(D) $3 x^{2}-20 x+16=0$

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80. The minimum value of the sum of the squares of the roots of $x^{2}+(3-a) x+1=2 a$ is:

(A) 4

(B) 5

(D) 8

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