JEE Mains Previous Year Questions (PYQ)

JEE Main 2023 8th April Morning Shift

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

7.

The value of $ 3+\cfrac{1}{\,4+\cfrac{1}{\,3+\cfrac{1}{\,4+\cfrac{1}{\,3+\cdots\infty}}}} $ is equal to:

8.

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}-\left(3\sqrt{2}+2\sqrt{3}\right)x+7+3\lambda\sqrt{3}=0 $, $ \lambda>0 $. If $ \beta+\gamma=3\sqrt{2} $, then $ (\alpha+2\beta+\gamma)^{2} $ is equal to _______.

9.

Let $ f(x) $ be a quadratic polynomial with leading coefficient $ 1 $ such that $ f(0) = p, \; p \neq 0, $ and $ f(1) = \dfrac{1}{3} $. If the equations $ f(x) = 0 $ and $ fof of of(x) = 0 $ have a common real root, then $ f(-3) $ is equal to $ \_ \_ \_ \_ \_ \_ $.

10.

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

11.

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

12.

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

13.

$ \alpha=\sin 36^{\circ} $ is a root of which of the following equation?

14.

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

15.

Let $ f(x)=ax^{2}+bx+c $ be such that $ f(1)=3 $, $ f(-2)=\lambda $ and $ f(3)=4 $. If $ f(0)+f(1)+f(-2)+f(3)=14 $, then $ \lambda $ is equal to:

16.

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

17.

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

18.

If the sum of the squares of the reciprocals of the roots $ \alpha $ and $ \beta $ of the equation $ 3x^{2} + \lambda x - 1 = 0 $ is $ 15 $, then $ 6(\alpha^{3} + \beta^{3})^{2} $ is equal to:

19.

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

20.

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+3x-1=0 $. If $ P_n=\alpha^n+\beta^n $ and $ Q_n=\gamma^n+\delta^n $, then $ \dfrac{P_{25}+\sqrt{3}P_{24}}{2P_{23}}+\dfrac{Q_{25}-Q_{23}}{Q_{24}} $ is equal to:

21.

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

22.

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

23.

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

24.

If the set of all $ a \in \mathbb{R} - \{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 ________________.

25.

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

26.

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:

27.

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

28.

No. of integral value(s) of $ p $ for which the roots of the equation $ 3x^2 + 2x + p(p-4)(p-6)(p-8) = 0 $ are of opposite sign is/are ________.

29.

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

30.

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:

31.

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

32.

Let the sum of the maximum and the minimum values of the function $ f(x)=\dfrac{2x^{2}-3x+8}{2x^{2}+3x+8} $ be $ \dfrac{m}{n} $, where $ \gcd(m,n)=1 $. Then $ m+n $ is equal to:

33.

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:

34.

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 ________.

35.

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

36.

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 _______.

37.

Let $ \alpha, \beta $ be two roots of the equation $ x^{2} + (20)^{1/4}x + (5)^{1/2} = 0 $. Then $ \alpha^{8} + \beta^{8} $ is equal to:

38.

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

39.

The number of distinct real solutions of the equation $ x\lvert x + 4\rvert + 3\lvert x + 2\rvert + 10 = 0 $ is:

40.

If the quadratic equation $ (\lambda + 2)x^2 - 3\lambda x + 4\lambda = 0, \ \lambda \neq -2, $ has two positive roots, then the number of possible integral values of $ \lambda $ is:

41.

$ cosec 18^\circ $ is a root of the equation:

42.

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

43.

If the set of all values of $ a $, for which the equation $ 5x^{3} - 15x - a = 0 $ has three distinct real roots, is the interval $ (\alpha, \beta) $, then $ \beta - 2\alpha $ is equal to ________.

44.

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

45.

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

46.

Let $\displaystyle S=\left\{ x \in \left( -\frac{\pi}{2},\, \frac{\pi}{2} \right) : 9^{\,1-\tan^{2}x} + 9^{\,\tan^{2}x} = 10 \right\} $ and $ \beta =\displaystyle \sum\limits_{x\in S} \tan^{2}\!\left( \frac{x}{3} \right) $, then $ \dfrac{1}{6} (\beta - 14)^{2} $ is equal to:

47.

Let $ a \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 _____.

48.

Let the parabola $ y = x^2 + px + q $ passing through the point $ (1, -1) $ be such that the distance between its vertex and the $ x $-axis is minimum. Then the value of $ p^2 + q^2 $ is:

49.

Let $ a, b \in R $ be such that the equation $ a x^{2} - 2 b x + 15 = 0 $ has a repeated root $ \alpha $. If $ \alpha $ and $ \beta $ are the roots of the equation $ x^{2} - 2 b x + 21 = 0 $, then $ \alpha^{2} + \beta^{2} $ is equal to:

50.

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

51.

The sum of all the real roots of the equation $ (e^{2x} - 4)(6e^{2x} - 5e^{x} + 1) = 0 $ is:

52.

If $ a + b + c = 1 $, $ ab + bc + ca = 2 $ and $ abc = 3 $, then the value of $ a^4 + b^4 + c^4 $ is equal to ________.

53.

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

54.

Let $ S=\left\{\alpha:\log_{2}\!\left(9^{\,2\alpha-4}+13\right)-\log_{2}\!\left(\frac{5}{2}\cdot 3^{\,2\alpha-4}+1\right)=2\right\}. $ Then the maximum value of $ \beta $ for which the equation $ x^{2}-2\!\left(\ \displaystyle \sum_{a\in S} \alpha\right)^{2}x+\displaystyle \sum_{a\in S}(\alpha+1)^{2}\beta=0 $ has real roots, is _______.

55.

The domain of $ f(x) = \dfrac{\log_{(x+1)}(x-2)}{e^{2\log_e x} - (2x + 3)} , \ x \in \mathbb{R} $ is:

56.

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( \dfrac{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 __________ .

57.

The smallest positive integral value of $ a $, for which all the roots of $ x^4 - a x^2 + 9 = 0 $ are real and distinct, is equal to:

58.

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 ______.

59.

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

60.

The roots of the quadratic equation $ 3x^2 - px + q = 0 $ are $ 10^{\text{th}} $ and $ 11^{\text{th}} $ terms of an arithmetic progression with common difference $ \dfrac{3}{2} $. If the sum of the first $ 11 $ terms of this arithmetic progression is $ 88 $, then $ q - 2p $ is equal to _________.

61.

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 ______.

62.

Let $ a,b \in \mathbb{C} $. Let $ \alpha,\beta $ be the roots of the equation $ x^2 + \alpha x + b = 0 $. If $ \beta - \alpha = \sqrt{11} $ and $ \beta^2 - \alpha^2 = 3i\sqrt{11} $, then $ (\beta^3 - \alpha^3)^2 $ is equal to:

63.

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

64.

Let $ f:[-1,1]\rightarrow \mathbb{R} $ be defined as $ f(x)=ax^{2}+bx+c $ for all $ x\in [-1,1] $, where $ a,b,c\in \mathbb{R} $ such that $ f(-1)=2 $, $ f'(-1)=1 $ and for $ x\in [-1,1] $ the maximum value of $ f''(x) $ is $ \frac{1}{2} $. If $ f(x)\le \alpha,\, x\in[-1,1] $, then the least value of $ \alpha $ is equal to ________.

65.

If $ 2 $ and $ 6 $ are the roots of the equation $ ax^2+bx+1=0 $, then the quadratic equation, whose roots are $ \dfrac{1}{2a+b} $ and $ \dfrac{1}{6a+b} $, is:

66.

Let $ \alpha, \beta $ be the roots of the equation $ x^2 - 3x + r = 0 $, and $ \frac{\alpha}{2}, 2\beta $ be the roots of the equation $ x^2 + 3x + r = 0 $. If the roots of the equation $ x^2 + 6x = m $ are $ 2\alpha + \beta + 2r $ and $ \alpha - 2\beta - \frac{r}{2} $, then $ m $ is equal to:

67.

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

68.

The number of solutions, of the equation $ e^{\sin x}-2e^{-\sin x}=2 $, is:

69.

Let $ x_1, x_2, x_3, x_4 $ be the solution of the equation $ 4x^{4} + 8x^{3} - 17x^{2} - 12x + 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) = \dfrac{125}{16}\, m $. Then the value of $ m $ is:

70.

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:

71.

Let $ a, b, c $ be three distinct positive real numbers such that $ (2a)^{\log_e a} = (bc)^{\log_e b} $ and $ b^{\log_e 2} = a^{\log_e c} $. Then $ 6a + 5bc $ is equal to ________.

72.

If $ \alpha, \beta, \gamma $ are the roots of $ x^3 + 3x + 9 = 0 $, find the value of $ \left(\alpha^9 + \beta^9 + \gamma^9\right) $.

73.

If $ \alpha, \beta $ are roots of equation $ x^2 - 2x - 1 = 0 $, then value of $ 5\alpha^4 + 12\beta^3 $ is ________.

74.

The sum of the roots of the equation, $ x + 1 - 2\log_{2}(3 + 2^{x}) + 2\log_{4}(10 - 2^{-x}) = 0 $, is:

75.

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

76.

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

77.

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

78.

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

79.

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:

80.

If $ \alpha > \beta > 0 $ are the roots of the equation $ a x^{2}+bx+1=0 $, and $\displaystyle \lim_{x\to \dfrac{1}{\alpha}} \left(\frac{1-\cos(x^{2}+bx+a)}{2(1-\alpha x)^{2}}\right)^{\dfrac{1}{2}}=\frac{1}{k}\left(\frac{1}{\beta}-\frac{1}{\alpha}\right) $, then $ k $ is equal to:

81.

If $ \alpha $ and $ \beta $ are the distinct roots of the equation $ x^{2}+(3)^{1/4}x+3^{1/2}=0 $, then the value of $ \alpha^{96}\left(\alpha^{12}-1\right)^{9}+\beta^{96}\left(\beta^{12}-1\right)^{9} $ is equal to;

82.

The sum of all the roots of the equation $ |x^2 - 8x + 15| - 2x + 7 = 0 $ is:

83.

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

84.

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

85.

If $ \alpha, \beta $, where $ \alpha < \beta $, are the roots of the equation $ \lambda x^2 - (\lambda + 3)x + 3 = 0 $ such that $ \frac{1}{\alpha} - \frac{1}{\beta} = \frac{1}{3} $, then the sum of all possible values of $ \lambda $ is:

86.

Suppose $ a,b,c \in \mathbb{R} $ and $ b \neq c $. If $ \alpha, \beta $ are roots of $ x^2 + ax + b = 0 $ and $ \gamma, \delta $ are roots of $ x^2 + ax + c = 0 $, then value of $ \dfrac{(\alpha - \gamma)(\alpha - \delta)}{(\beta - \gamma)(\beta - \delta)} $ is independent of:

87.

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

88.

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

89.

The number of solutions of the equation $ \log_{(x+1)}\!\left(2x{^{2}}+7x+5\right)+\log_{(2x+5)}(x+1)^{2}-4=0, \; x>0, $ is ______.

90.

Let one root of the quadratic equation in $ x $:
$ (k^{2}-15k+27)x^{2}+9(k-1)x+18=0 $ be twice the other. Then the length of the latus rectum of the parabola $ y^{2}=6kx $ is equal to:

91.

The sum of all the real solutions of the equation $ \log_{(x+3)}(6x^2 + 28x + 30) = 5 - 2\log_{(6x+10)}(x^2 + 6x + 9) $ is equal to:

92.

The given equation is $ 3x^4 - 2x^3 + 4x^2 - 4x + 12 = 0 $. Now, which of the following options is correct?

93.

Let $ \alpha , \alpha+2 , \alpha \in \mathbb{Z} $, be the roots of the quadratic equation $ x(x+2) + (x+1)(x+3) + (x+2)(x+4) + \ldots + (x+n-1)(x+n+1) = 4n $ for some $ n \in \mathbb{N} $. Then $ n+\alpha $ is equal to:

94.

If $ x \in [5,7] $ the greatest value of expression $ -3x^2 + 54x - 200 $ is ________.

95.

The sum of all the roots of the equation $ (x - 1)^2 - 5|x - 1| + 6 = 0 $, is:

96.

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

97.

Let $ \displaystyle \sum_{n=0}^{\infty} \dfrac{n^{3}\,(2n)! + (2n-1)(n!)}{(n!)((2n)!)} = a e + \dfrac{b}{e} + c $, where $ a, b, c \in \mathbb{Z} $ and $ e = \displaystyle \sum_{n=0}^{\infty} \dfrac{1}{n!} $. Then $ a^{2} - b + c $ is equal to ________.

98.

Let $ x^2 - (m-3)x + m = 0 $ ($ m \in \mathbb{R} $) be a quadratic equation. Find the values of $ m $ for which both the roots are negative?

99.

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

100.

If $ \alpha $ and $ \beta (\alpha < \beta) $ are the roots of the equation $ (-2 + \sqrt{3})\lvert \sqrt{x} - 3 \rvert + (x - 6\sqrt{x}) + (9 - 2\sqrt{3}) = 0, \ \ x \ge 0 $, then $ \sqrt{\frac{\beta}{\alpha}} + \sqrt{\alpha \beta} $ is equal to:

101.

The value of $4 + \frac{1}{5 + \frac{1}{4 + \frac{1}{5 + \frac{1}{4 + \dots \infty}}}} \text{ is : }$

102.

103.

The parabolas: $ ax^{2}+2bx+cy=0 $ and $ dx^{2}+2ex+fy=0 $ intersect on the line $ y=1 $. If $ a,b,c,d,e,f $ are positive real numbers and $ a,b,c $ are in G.P., then:

104.

The set of all values of $ k > -1 $, for which the equation $ (3x^{2} + 4x + 3)^{2} - (k + 1)\,(3x^{2} + 4x + 3)\,(3x^{2} + 4x + 2) + k\,(3x^{2} + 4x + 2)^{2} = 0 $ has real roots, is:

105.

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

106.

The smallest positive integer $ p $ for which expression $ x^2 - 2px + 3p + 4 $ is negative for atleast one real $ x $ is:

107.

Consider the equation $ x^{2} + 4x - 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:

108.

The number of pairs $ (a,b) $ of real numbers, such that whenever $ \alpha $ is a root of the equation $ x^{2} + ax + b = 0 $, $ \alpha^{2} - 2 $ is also a root of this equation is:

109.

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

110.

Let $ f(x) $ and $ g(x) $ be two real polynomials of degree $ 2 $ and $ 1 $ respectively. If $ f(g(x)) = 8x^2 - 2x $, and $ g(f(x)) = 4x^2 + 6x + 1 $, then the value of $ f(2) + g(2) $ is ________.

111.

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 $ \left(11\sqrt{3}-10\sqrt{2}\right)P_{10}+\left(11\sqrt{2}+10\right)P_{11}-11P_{12} $ is equal to

112.

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

113.

Let $ \displaystyle \alpha = \max_{x \in R} \{8^{2 \sin 3x} \cdot 4^{4 \cos 3x}\} \text{ and } \beta = \min_{x \in R} \{8^{2 \sin 3x} \cdot 4^{4 \cos 3x}\} $. If $ 8x^{2} + bx + c = 0 $ is a quadratic equation whose roots are $ \alpha^{1/5} $ and $ \beta^{1/5} $, then the value $ c - b $ is equal to:

114.

Consider the quadratic equation $ (n^2-2n+2)x^2-3x+(n^2-2n+2)^2=0,\, n\in R $. Let $ \alpha $ be the minimum value of the product of its roots and $ \beta $ be the maximum value of the sum of its roots. Then the sum of the first six terms of the G.P., whose first term is $ \alpha $ and the common ratio is $ \dfrac{\alpha}{\beta} $, is:

115.

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 $ \dfrac{1}{10}(\alpha^6 + \beta^6) $, is:

116.

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 $ \dfrac{P_{15}P_{16}-P_{14}P_{16}-P_{15}^{2}+P_{14}P_{15}}{P_{13}P_{14}} $ is equal to __________.

117.

The maximum value of $ z $ in the following $ z = 6xy + y^2 $, where $ 3x + 4y \leq 100 $ and $ 4x + 3y \leq 75 $ for $ x \geq 0 $ and $ y \geq 0 $ is __________.

118.

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

119.

Let $ f(x) $ be a quadratic polynomial such that $ f(-2) + f(3) = 0 $. If one of the roots of $ f(x) = 0 $ is $ -1 $, then the sum of the roots of $ f(x) = 0 $ is equal to:

JEE MainsJEE AdvanceBITSATJEE Mains | PYQ

JEE Mains JEE Advance BITSAT
JEE Mains | PYQ