JEE Advanced Maths PYQ: Complex Numbers & Quadratic Equations (Chapterwise Questions)

Q 1 :

Suppose $a, b$ denote the distinct real roots of the quadratic polynomial $x^{2} + 20 x - 2020$ and suppose $c, d$ denote the distinct complex roots of the quadratic polynomial $x^{2} - 20 x + 2020$ . Then the value of $a c (a - c) + a d (a - d) + b c (b - c) + b d (b - d)$ is [2020]

0
8000
8080
16000

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(4)

Consider the quadratic polynomials in the form of equation

$x^{2} + 20 x - 2020 = 0 \dots (i)$

$x^{2} - 20 x + 2020 = 0 \dots (i i)$

$Since, a and b are roots of the equation (i), then$

$a + b = - 20, a b = - 2020$

$∵$ $c and d are the roots of the equation (i i), then$

$c + d = 20, c d = 2020$

$Now,$

$a c (a - c) + a d (a - d) + b c (b - c) + b d (b - d)$

$= a^{2} c - a c^{2} + a^{2} d - a d^{2} + b^{2} c - b c^{2} + b^{2} d - b d^{2}$

$= a^{2} (c + d) + b^{2} (c + d) - c^{2} (a + b) - d^{2} (a + b)$

$= (c + d) (a^{2} + b^{2}) - (a + b) (c^{2} + d^{2})$

$= (c + d) ({(a + b)}^{2} - 2 a b) - (a + b) ({(c + d)}^{2} - 2 c d)$

$= 20 [{(20)}^{2} + 4040] + 20 [{(20)}^{2} - 4040]$

$= 20 \times 800 = 16000$

Q 2 :

Let $- \frac{π}{6} < θ < - \frac{π}{12}$ . Suppose $α_{1}$ and $β_{1}$ are the roots of the equation $x^{2} - 2 x \sec θ + 1 = 0$ and $α_{2}$ and $β_{2}$ are the roots of the equation $x^{2} + 2 x \tan θ - 1 = 0 .$ If $α_{1} > β_{1}$ and $α_{2} > β_{2}$ , then $α_{1} + β_{2}$ equals [2016]

$2 (\sec θ - \tan θ)$
$2 \sec θ$
$- 2 \tan θ$
$0$

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(3)

$x^{2} - 2 x \sec θ + 1 = 0 \Rightarrow x = \sec θ \pm \tan θ$

$and x^{2} + 2 x \tan θ - 1 = 0 \Rightarrow x = - \tan θ \pm \sec θ$

$∵$ $- \frac{π}{6} < θ < - \frac{π}{12} \Rightarrow \sec \frac{π}{6} > \sec θ > \sec \frac{π}{12}$

$and - \tan \frac{π}{6} < \tan θ < - \frac{\tan π}{12}$

$Also \tan \frac{π}{12} < - \tan θ < \tan \frac{π}{6}$

$Since, α_{1}, β_{1} are roots of x^{2} - 2 x \sec θ + 1 = 0 and α_{1} > β_{1}$

$∴$ $α_{1} = \sec θ - \tan θ and β_{1} = \sec θ + \tan θ$

$Since, α_{2}, β_{2} are roots of x^{2} + 2 x \tan θ - 1 = 0 and α_{2} > β_{2}$

$∴$ $α_{2} = - \tan θ + \sec θ, β_{2} = - \tan θ - \sec θ$

$∴$ $α_{1} + β_{2} = \sec θ - \tan θ - \tan θ - \sec θ = - 2 \tan θ$

Q 3 :

The quadratic equation $p (x) = 0$ with real coefficients has purely imaginary roots. Then the equation $p (p (x)) = 0$ has [2014]

one purely imaginary root
all real roots
two real and two purely imaginary roots
neither real nor purely imaginary roots

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(4)

Quadratic equation with real coefficients and purely imaginary roots can be considered as

$p (x) = x^{2} + a = 0 where a > 0 and a \in ℝ$

$Then p [p (x)] = 0 \Rightarrow {(x^{2} + a)}^{2} + a = 0$

$\Rightarrow x^{4} + 2 a x^{2} + (a^{2} + a) = 0$

$\Rightarrow x^{2} = \frac{- 2 a \pm \sqrt{4 a^{2} - 4 (a^{2} + a)}}{2}$ $= \frac{- 2 a \pm \sqrt{4 a^{2} - 4 a^{2} - 4 a}}{2}$

$= \frac{- 2 a \pm \sqrt{- 4 a}}{2}$ $\Rightarrow x^{2} = - a \pm \sqrt{a} i$

$\Rightarrow x = \sqrt{- a \pm \sqrt{a} i} = α \pm i β, where α, β \neq 0$

$∴$ $p [p (x)] = 0 has complex roots which are neither purely real nor purely imaginary.$

Q 4 :

If $a \in R$ and the equation $- 3 {(x - [x])}^{2} + 2 (x - [x]) + a^{2} = 0$ (where $[x]$ denotes the greatest integer $\leq x$ ) has no integral solution, then all possible values of $a$ lie in the interval: [2014]

$(- 2, - 1)$
$(- \infty, - 2) \cup (2, \infty)$
$(- 1, 0) \cup (0, 1)$
$(1, 2)$

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(3)

$Consider - 3 {(x - [x])}^{2} + 2 [x - [x]] + a^{2} = 0$

$\Rightarrow 3 {x}^{2} - 2 {x} - a^{2} = 0 (∵ x - [x] = {x})$

$\Rightarrow 3 ({x}^{2} - \frac{2}{3} {x}) = a^{2}, a \neq 0$

$\Rightarrow a^{2} = 3 {({x} - \frac{1}{3})}^{2} - \frac{1}{3} (∵ 0 \leq {x} < 1)$

$- \frac{1}{3} \leq {x} - \frac{1}{3} < \frac{2}{3} \Rightarrow 0 \leq 3 ({x} \frac{- 1}{3}) < \frac{4}{3}$

$- \frac{1}{3} \leq 3 ({x} \frac{- 1}{3}) - \frac{1}{3} < 1$

$For non-integral solution 0 < a^{2} < 1$

$\Rightarrow a \in (- 1, 0) \cup (0, 1)$

Q 5 :

Let $α$ and $β$ be the roots of $x^{2} - 6 x - 2 = 0$ , with $α > β$ . If $a_{n} = α^{n} - β^{n} for n \geq 1,$ then the value of $\frac{a_{10} - 2 a_{8}}{2 a_{9}}$ is [2011]

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(3)

$∵$ $α, β are the roots of x^{2} - 6 x - 2 = 0$

$∴$ $α^{2} - 6 α - 2 = 0$

$\Rightarrow α^{10} - 6 α^{9} - 2 α^{8} = 0$

$\Rightarrow α^{10} - 2 α^{8} = 6 α^{9} \dots (i)$

$Similarly β^{10} - 2 β^{8} = 6 β^{9} \dots (i i)$

$On subtracting (ii) from (i),$

$α^{10} - β^{10} - 2 (α^{8} - β^{8}) = 6 (α^{9} - β^{9})$

$\Rightarrow a_{10} - 2 a_{8} = 6 a_{9}$

$\Rightarrow \frac{a_{10} - 2 a_{8}}{2 a_{9}} = 3$

Q 6 :

Let $(x_{0}, y_{0})$ be the solution of the following equations ${(2 x)}^{\ln 2} = {(3 y)}^{\ln 3}; 3^{\ln x} = 2^{\ln y} .$ Then $x_{0}$ is [2011]

$\frac{1}{6}$
$\frac{1}{3}$
$\frac{1}{2}$
$6$

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(3)

$Given : {(2 x)}^{\ln 2} = {(3 y)}^{\ln 3}$

$\Rightarrow \ln 2 \cdot \ln (2 x) = \ln 3 \cdot \ln (3 y)$

$\Rightarrow \ln 2 \cdot \ln (2 x) = \ln 3 \cdot (\ln 3 + \ln y) \dots (i)$

$Also given : 3^{\ln x} = 2^{\ln y}$

$\Rightarrow \ln x \cdot \ln 3 = \ln y \cdot \ln 2 \Rightarrow \ln y = \frac{\ln x \cdot \ln 3}{\ln 2} \dots (i i)$

$From equation (i) and (ii), we get$

$\ln 2 \cdot \ln (2 x) = \ln 3 [\ln 3 + \frac{\ln x \cdot \ln 3}{\ln 2}]$

$\Rightarrow {(\ln 2)}^{2} \ln (2 x) = {(\ln 3)}^{2} \ln 2 + {(\ln 3)}^{2} \ln x$

$\Rightarrow {(\ln 2)}^{2} \ln (2 x) = {(\ln 3)}^{2} (\ln 2 + \ln x)$

$\Rightarrow {(\ln 2)}^{2} \ln (2 x) - {(\ln 3)}^{2} \ln (2 x) = 0$

$\Rightarrow [{(\ln 2)}^{2} - {(\ln 3)}^{2}] \ln (2 x) = 0$ $\Rightarrow \ln (2 x) = 0$

$\Rightarrow 2 x = 1 \Rightarrow x = \frac{1}{2}$

Q 7 :

Let $p$ and $q$ be real numbers such that $p \neq 0$ , $p^{3} \neq q$ and $p^{3} \neq - q$ . If $α$ and $β$ are nonzero complex numbers satisfying $α + β = - p$ and $α^{3} + β^{3} = q$ , then a quadratic equation having $\frac{α}{β}$ and $\frac{β}{α}$ as its roots is [2010]

$(p^{3} + q) x^{2} - (p^{3} + 2 q) x + (p^{3} + q) = 0$
$(p^{3} + q) x^{2} - (p^{3} - 2 q) x + (p^{3} + q) = 0$
$(p^{3} - q) x^{2} - (5 p^{3} - 2 q) x + (p^{3} - q) = 0$
$(p^{3} - q) x^{2} - (5 p^{3} + 2 q) x + (p^{3} - q) = 0$

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(2)

$Given : α + β = - p and α^{3} + β^{3} = q$

$\Rightarrow {(α + β)}^{3} - 3 α β (α + β) = q$

$\Rightarrow - p^{3} - 3 α β (- p) = q \Rightarrow α β = \frac{p^{3} + q}{3 p}$

$Now for required quadratic equation,$

$Sum of roots = \frac{α}{β} + \frac{β}{α} = \frac{α^{2} + β^{2}}{α β}$

$= \frac{{(α + β)}^{2} - 2 α β}{α β}$ $= \frac{p^{2} - 2 (\frac{p^{3} + q}{3 p})}{\frac{p^{3} + q}{3 p}} = \frac{p^{3} - 2 q}{p^{3} + q}$

$and Product of roots = \frac{α}{β} \cdot \frac{β}{α} = 1$

$∴$ $Required equation is$ $x^{2} - (\frac{p^{3} - 2 q}{p^{3} + q}) x + 1 = 0$

$\Rightarrow (p^{3} + q) x^{2} - (p^{3} - 2 q) x + (p^{3} + q) = 0$

Q 8 :

Let $α, β$ be the roots of the equation $x^{2} - p x + r = 0$ and $\frac{α}{2}, 2 β$ be the roots of the equation $x^{2} - q x + r = 0$ . Then the value of $r$ is [2007]

$\frac{2}{9} (p - q) (2 q - p)$
$\frac{2}{9} (q - p) (2 p - q)$
$\frac{2}{9} (q - 2 p) (2 q - p)$
$\frac{2}{9} (2 p - q) (2 q - p)$

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(4)

$Since α and β are the roots of x^{2} - p x + r = 0$

$∴$ $α + β = p \dots (i)$

$and α β = r \dots (i i)$

$Also \frac{α}{2} and 2 β are the roots of x^{2} - q x + r = 0$

$∴$ $\frac{α}{2} + 2 β = q \Rightarrow α + 4 β = 2 q \dots (i i i)$

$Solving (i) and (iii) for α and β, we get$

$β = \frac{1}{3} (2 q - p) and α = \frac{2}{3} (2 p - q)$

$On substituting the values of α and β in equation (ii),$

$we get \frac{2}{9} (2 p - q) (2 q - p) = r .$

Q 9 :

Let $a, b, c$ be the sides of a triangle where $a \neq b \neq c$ and $λ \in R$ . If the roots of the equation $x^{2} + 2 (a + b + c) x + 3 λ (a b + b c + c a) = 0$ are real, then [2006]

$λ < \frac{4}{3}$
$λ > \frac{5}{3}$
$λ \in (\frac{1}{3}, \frac{5}{3})$
$λ \in (\frac{4}{3}, \frac{5}{3})$

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(1)

$∵$ $a, b, c are sides of a triangle and a \neq b \neq c$

$∴$ $| a - b |$ $<$ $| c |$ $\Rightarrow a^{2} + b^{2} - 2 a b < c^{2} \dots (i)$

$Similarly,$

$b^{2} + c^{2} - 2 b c < a^{2} \dots (i i), c^{2} + a^{2} - 2 c a < b^{2} \dots (i i i)$

$On adding (i), (ii) and (iii) we get$

$a^{2} + b^{2} + c^{2} < 2 (a b + b c + c a)$

$\Rightarrow \frac{a^{2} + b^{2} + c^{2}}{a b + b c + c a} < 2 \dots (i v)$

$∵$ $Roots of the given equation are real$

$∴$ ${(a + b + c)}^{2} - 3 λ (a b + b c + c a) \geq 0$

$\Rightarrow \frac{a^{2} + b^{2} + c^{2}}{a b + b c + c a} \geq 3 λ - 2 \dots (v)$

$From (iv) and (v), 3 λ - 2 < 2 \Rightarrow λ < \frac{4}{3}$

Q 10 :

If one root is square of the other root of the equation $x^{2} + p x + q = 0,$ then the relation between $p$ and $q$ is [2004]

$p^{3} - q (3 p - 1) + q^{2} = 0$
$p^{3} - q (3 p + 1) + q^{2} = 0$
$p^{3} + q (3 p - 1) + q^{2} = 0$
$p^{3} + q (3 p + 1) + q^{2} = 0$

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(1)

$x^{2} + p x + q = 0$

$Let roots be α and α^{2}, then$

$α + α^{2} = - p, α \cdot α^{2} = q \Rightarrow α = q^{1 / 3}$

$∴$ ${(q)}^{1 / 3} + {(q^{1 / 3})}^{2} = - p$

$On taking cube on both sides, we get$

$q + q^{2} + 3 q (q^{1 / 3} + q^{2 / 3}) = - p^{3}$

$\Rightarrow q + q^{2} - 3 p q = - p^{3}$

$\Rightarrow p^{3} + q^{2} - q (3 p - 1) = 0$

Q 11 :

For the equation $3 x^{2} + p x + 3 = 0$ , $p > 0$ , if one of the root is square of the other, then $p$ is equal to [2000]

1/3
1
3
2/3

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(3)

$Let α, α^{2} be the roots of 3 x^{2} + p x + 3 = 0$

$∴$ $α + α^{2} = - \frac{p}{3} and α^{3} = 1$

$\Rightarrow (α - 1) (α^{2} + α + 1) = 0$ $\Rightarrow α = 1 or α^{2} + α = - 1$

$If α = 1, then p = - 6, which is not possible as p > 0$

$If α^{2} + α = - 1 \Rightarrow - \frac{p}{3} = - 1 \Rightarrow p = 3$

Q 12 :

If $b > a$ , then the equation $(x - a) (x - b) - 1 = 0$ has [2000]

both roots in $(a, b)$
both roots in $(- \infty, a)$
both roots in $(b, + \infty)$
one root in $(- \infty, a)$ and the other in $(b, + \infty)$

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(4)

$Given : (x - a) (x - b) - 1 = 0, b > a$

$or x^{2} - (a + b) x + (a b - 1) = 0$

$Let f (x) = x^{2} - (a + b) x + (a b - 1)$

$D = {(a + b)}^{2} - 4 (a b - 1)$

$= {(a - b)}^{2} + 1 > 0$

$Since coefficient of x^{2} i.e. 1 > 0,$ $∴$ $f (x) represents upward parabola,$

$intersecting x-axis at two points corresponding to two real roots, D being positive.$

$Also f (a) = f (b) = - 1$

$\Rightarrow curve is below x-axis at a and b$

$∴$ $a and b both lie between the roots.$

$Therefore, the graph of given equation is as shown.$

$It is clear from graph, that one root of the equation lies in (- \infty, a) and other in (b, \infty) .$

Q 13 :

If $α$ and $β$ $(α < β)$ are the roots of the equation $x^{2} + b x + c = 0,$ where $c < 0 < b$ , then [2000]

$0 < α < β$
$α < 0 < β < | α |$
$α < β < 0$
$α < 0 < | α | < β$

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(2)

$Given : c < 0 < b and α + β = - b \dots (i)$

$α β = c \dots (i i)$

$From (ii), c < 0 \Rightarrow α β < 0 \Rightarrow Either α is negative or β is negative and second quantity is positive.$

$From (i), b > 0 \Rightarrow - b < 0 \Rightarrow α + β < 0$

$\Rightarrow the sum is negative$

$\Rightarrow (Modulus of negative quantity) > (Modulus of positive quantity)$

$But given α < β . Therefore, it is clear that α is negative and β is positive$

$and modulus of α is greater than modulus of β$

$\Rightarrow α < 0 < β < | α |$

Q 14 :

The product of all positive real values of $x$ satisfying the equation $x^{(16 {(\log_{5} x)}^{3} - 68 \log_{5} x)} = 5^{- 16}$ is ________. [2022]

(1)

Taking log with base 5 on both sides

$(16 {(\log_{5} x)}^{3} - 68 (\log_{5} x)) (\log_{5} x) = - 16$

$Let \log_{5} x = t$

$\Rightarrow 16 t^{4} - 68 t^{2} + 16 = 0$

$\Rightarrow 4 t^{4} - 17 t^{2} + 4 = 0$

$\Rightarrow (4 t^{2} - 1) (t^{2} - 4) = 0$

$\Rightarrow t = \pm \frac{1}{2}, \pm 2$

$So \log_{5} x = \pm \frac{1}{2}, \pm 2$ $\Rightarrow x = 5^{1 / 2}, 5^{- 1 / 2}, 5^{2}, 5^{- 2}$

$Product = 5^{1 / 2} \cdot 5^{- 1 / 2} \cdot 5^{2} \cdot 5^{- 2} = 1$

Q 15 :

For $x \in ℝ$ , then the number of real roots of the equation $3 x^{2} - 4 | x^{2} - 1 | + x - 1 = 0$ is ________. [2021]

(4)

$3 x^{2} + x - 1 = 4 | x^{2} - 1 |$

$Case 1: If x \in [- 1, 1],$

$3 x^{2} + x - 1 = - 4 x^{2} + 4$

$\Rightarrow 7 x^{2} + x - 5 = 0$ $∵ D = 141 > 0 ∴ Equation has two roots$

$Case 2: If x \in (- \infty, - 1] \cup [1, \infty)$

$3 x^{2} + x - 1 = 4 x^{2} - 4$

$\Rightarrow x^{2} - x - 3 = 0 ∵ D = 13 > 0$

$∴$ $Equation has two roots$

$So, total 4 roots.$

Q 16 :

The smallest value of $k$ , for which both the roots of the equation $x^{2} - 8 k x + 16 (k^{2} - k + 1) = 0$ are real, distinct and have values at least 4, is [2009]

(2)

The given equation is $x^{2} - 8 k x + 16 (k^{2} - k + 1) = 0$

$∵$ $Both the roots are real and distinct.$

$∴$ $D > 0 \Rightarrow {(8 k)}^{2} - 4 \times 16 (k^{2} - k + 1) > 0$

$\Rightarrow k > 1 ...(i)$

$∵$ $Both the roots are greater than or equal to 4$

$∴$ $α + β > 8 and f (4) \geq 0$

$\Rightarrow k > 1 ...(ii)$

$and 16 - 32 k + 16 (k^{2} - k + 1) \geq 0$

$\Rightarrow k^{2} - 3 k + 2 \geq 0 \Rightarrow (k - 1) (k - 2) \geq 0$

$\Rightarrow k \in (- \infty, 1] \cup [2, \infty) ...(iii)$

$Combining (i), (ii) and (iii), we get k \geq 2$

$∴$ $Smallest value of k = 2$

Q 17 :

Let $R^{2}$ denote $R \times R$ . Let $S = {(a, b, c) : a, b, c \in R$ and $a x^{2} + 2 b x y + c y^{2} > 0$ for all $(x, y) \in R^{2} - {(0, 0)} .$ Then which of the following statements is (are) TRUE? [2024]

$(2, \frac{7}{2}, 6) \in S$
If $(3, b, \frac{1}{12}) \in S$ , then $| 2 b | < 1$ .
For any given $(a, b, c) \in S$ , the system of linear equations $a x + b y = 1, b y + c y = - 1$ has a unique solution.
For any given $(a, b, c) \in S$ , the system of linear equations $(a + 1) x + b y = 0, b x + (c + 1) y = 0$ has a unique solution.

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Select one or more options

(2, 3, 4)

Given that $a x^{2} + 2 b x y + c y^{2} > 0$

$and x, y \in ℝ - {0}$

$\Rightarrow c {(\frac{y}{x})}^{2} + 2 b (\frac{y}{x}) + a > 0 \Rightarrow c > 0, D < 0$

$4 b^{2} - 4 a c < 0 \Rightarrow b^{2} < a c$

$(1) (2, \frac{7}{2}, 6)$
${(\frac{7}{2})}^{2} > 2 \times 6$
$∴$ $Option (1) is incorrect$

$(2) If (3, b, \frac{1}{12}) \in S$
$\Rightarrow b^{2} < 3 \cdot \frac{1}{12} \Rightarrow b^{2} < \frac{1}{4} \Rightarrow 4 b^{2} < 1$
$\Rightarrow | 2 b | < 1 Option (2) is correct$

$(3) a x + b y = 1$

$b x + c y = - 1$

$D = | \begin{matrix} a & b \\ b & c \end{matrix} | = a c - b^{2} \neq 0$

$∴$ $unique solution, option (3) is correct$

$(4) (a + 1) x + b y = 0$

$b x + (c + 1) y = 0$

$D = | \begin{matrix} a + 1 & b \\ b & c + 1 \end{matrix} |$

$= (a + 1) (c + 1) - b^{2} = a c - b^{2} + a + c + 1$

$Since a c - b^{2} > 0 \Rightarrow b^{2} < a c \Rightarrow a c is positive$

$\Rightarrow a and c are positive, then (a c - b^{2}) + a + c + 1 > 0$

$∴ unique solution \Rightarrow option (4) is correct$

Q 18 :

If $3^{x} = 4^{x - 1},$ then $x =$ [2013]

$\frac{2 \log_{3} 2}{2 \log_{3} 2 - 1}$
$\frac{2}{2 - \log_{2} 3}$
$\frac{1}{1 - \log_{4} 3}$
$\frac{2 \log_{2} 3}{2 \log_{2} 3 - 1}$

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Select one or more options

(1, 2, 3)

$3^{x} = 4^{x - 1} \Rightarrow x \log 3 = 2 (x - 1) \log 2$

$\Rightarrow x = \frac{2 \log 2}{2 \log 2 - \log 3}$ $\Rightarrow x = \frac{2 \log_{3} 2}{2 \log_{3} 2 - 1} = \frac{2}{2 - \log_{2} 3}$

$Also x = \frac{1}{1 - \frac{1}{2} \log_{2} 3} = \frac{1}{1 - \log_{4} 3}$

Q 19 :

Let $p, q$ be integers and let $α, β$ be the roots of the equation $x^{2} - x - 1 = 0,$ where $α \neq β$ . For $n = 0, 1, 2, \dots$ , let $a_{n} = p α^{n} + q β^{n} .$

FACT: If $a$ and $b$ are rational numbers and $a + b \sqrt{5} = 0,$ then $a = 0 = b .$ [2017]

Q. $a_{12} =$

$a_{11} - a_{10}$
$a_{11} + a_{10}$
$2 a_{11} + a_{10}$
$a_{11} + 2 a_{10}$

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(2)

$α^{2} = α + 1$

$β^{2} = β + 1$

$a_{n} = p α^{n} + q β^{n}$

$= p (α^{n - 1} + α^{n - 2}) + q (β^{n - 1} + β^{n - 2})$

$= a_{n - 1} + a_{n - 2}$

$∴$ $a_{12} = a_{11} + a_{10}$

Q 20 :

Let $p, q$ be integers and let $α, β$ be the roots of the equation $x^{2} - x - 1 = 0,$ where $α \neq β .$ For $n = 0, 1, 2, \dots,$ let $a_{n} = p α^{n} + q β^{n} .$

FACT: If $a$ and $b$ are rational numbers and $a + b \sqrt{5} = 0,$ then $a = 0 = b .$ [2017]

Q. $If a_{4} = 28, then p + 2 q =$

21
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7
12

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(4)

$α = \frac{1 + \sqrt{5}}{2}, β = \frac{1 - \sqrt{5}}{2}$

$a_{4} = a_{3} + a_{2}$

$= 2 a_{2} + a_{1}$

$= 3 a_{1} + 2 a_{0}$

$28 = p (3 α + 2) + q (3 β + 2)$

$28 = (p + q) (\frac{3}{2} + 2) + (p - q) (\frac{3 \sqrt{5}}{2})$

$∴$ $p - q = 0 and (p + q) \cdot \frac{7}{2} = 28$

$\Rightarrow p + q = 8 \Rightarrow p = q = 4$

$∴$ $p + 2 q = 12$

Q 21 :

Let $a, b, c, p, q$ be real numbers. Suppose $α, β$ are the roots of the equation $x^{2} + 2 p x + q = 0$ and $α, \frac{1}{β}$ are the roots of the equation $a x^{2} + 2 b x + c = 0,$ where $β^{2} \notin {- 1, 0, 1} .$

$STATEMENT-1: (p^{2} - q) (b^{2} - a c) \geq 0$

$and$

$STATEMENT-2: b \neq p a or c \neq q a$ [2008]

STATEMENT - 1 is True, STATEMENT - 2 is True; STATEMENT - 2 is a correct explanation for STATEMENT - 1
STATEMENT - 1 is True, STATEMENT - 2 is True; STATEMENT - 2 is NOT a correct explanation for STATEMENT - 1
STATEMENT - 1 is True, STATEMENT - 2 is False
STATEMENT - 1 is False, STATEMENT - 2 is True

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(2)

As $a, b, c, p, q \in R$ and the two given equations have exactly one common root

$\Rightarrow$ Either both equations have real roots or both equations have imaginary roots

$\Rightarrow Either D_{1} \geq 0 and D_{2} \geq 0 or D_{1} \leq 0 and D_{2} \leq 0$

$\Rightarrow p^{2} - q \geq 0 and b^{2} - a c \geq 0$ $or p^{2} - q \leq 0 and b^{2} - a c \leq 0$

$\Rightarrow (p^{2} - q) (b^{2} - a c) \geq 0$

$∴$ $Statement 1 is true.$

$Also we have α β = q and \frac{α}{β} = \frac{c}{a}$

$∴$ $\frac{α β}{α / β} = \frac{q}{c} \times a \Rightarrow β^{2} = \frac{q a}{c}$

$As β \neq 1 or - 1 \Rightarrow β^{2} \neq 1 \Rightarrow \frac{q a}{c} \neq 1 or c \neq q a$

$Again, as exactly one root α is common, and β \neq 1$

$∴$ $α + β \neq α + \frac{1}{β} \Rightarrow \frac{- 2 b}{a} \neq - 2 p \Rightarrow b \neq a p$

$∴$ $Statement 2 is correct.$

$But Statement 2 is not a correct explanation of Statement 1.$

Topic Question Set

Q 1 :

Suppose $a, b$ denote the distinct real roots of the quadratic polynomial $x^{2} + 20 x - 2020$ and suppose $c, d$ denote the distinct complex roots of the quadratic polynomial $x^{2} - 20 x + 2020$ . Then the value of $a c (a - c) + a d (a - d) + b c (b - c) + b d (b - d)$ is [2020]

Q 3 :

The quadratic equation $p (x) = 0$ with real coefficients has purely imaginary roots. Then the equation $p (p (x)) = 0$ has [2014]

Q 4 :

If $a \in R$ and the equation $- 3 {(x - [x])}^{2} + 2 (x - [x]) + a^{2} = 0$ (where $[x]$ denotes the greatest integer $\leq x$ ) has no integral solution, then all possible values of $a$ lie in the interval: [2014]

Q 5 :

Let $α$ and $β$ be the roots of $x^{2} - 6 x - 2 = 0$ , with $α > β$ . If $a_{n} = α^{n} - β^{n} for n \geq 1,$ then the value of $\frac{a_{10} - 2 a_{8}}{2 a_{9}}$ is [2011]

Q 6 :

Let $(x_{0}, y_{0})$ be the solution of the following equations ${(2 x)}^{\ln 2} = {(3 y)}^{\ln 3}; 3^{\ln x} = 2^{\ln y} .$ Then $x_{0}$ is [2011]

Q 7 :

Let $p$ and $q$ be real numbers such that $p \neq 0$ , $p^{3} \neq q$ and $p^{3} \neq - q$ . If $α$ and $β$ are nonzero complex numbers satisfying $α + β = - p$ and $α^{3} + β^{3} = q$ , then a quadratic equation having $\frac{α}{β}$ and $\frac{β}{α}$ as its roots is [2010]

Q 8 :

Let $α, β$ be the roots of the equation $x^{2} - p x + r = 0$ and $\frac{α}{2}, 2 β$ be the roots of the equation $x^{2} - q x + r = 0$ . Then the value of $r$ is [2007]

Q 9 :

Let $a, b, c$ be the sides of a triangle where $a \neq b \neq c$ and $λ \in R$ . If the roots of the equation $x^{2} + 2 (a + b + c) x + 3 λ (a b + b c + c a) = 0$ are real, then [2006]

Q 10 :

If one root is square of the other root of the equation $x^{2} + p x + q = 0,$ then the relation between $p$ and $q$ is [2004]

Q 11 :

For the equation $3 x^{2} + p x + 3 = 0$ , $p > 0$ , if one of the root is square of the other, then $p$ is equal to [2000]

Q 12 :

If $b > a$ , then the equation $(x - a) (x - b) - 1 = 0$ has [2000]

Q 13 :

If $α$ and $β$ $(α < β)$ are the roots of the equation $x^{2} + b x + c = 0,$ where $c < 0 < b$ , then [2000]

Q 14 :

The product of all positive real values of $x$ satisfying the equation $x^{(16 {(\log_{5} x)}^{3} - 68 \log_{5} x)} = 5^{- 16}$ is ________. [2022]

Q 15 :

For $x \in ℝ$ , then the number of real roots of the equation $3 x^{2} - 4 | x^{2} - 1 | + x - 1 = 0$ is ________. [2021]

Q 16 :

The smallest value of $k$ , for which both the roots of the equation $x^{2} - 8 k x + 16 (k^{2} - k + 1) = 0$ are real, distinct and have values at least 4, is [2009]

Q 17 :

Let $R^{2}$ denote $R \times R$ . Let $S = {(a, b, c) : a, b, c \in R$ and $a x^{2} + 2 b x y + c y^{2} > 0$ for all $(x, y) \in R^{2} - {(0, 0)} .$ Then which of the following statements is (are) TRUE? [2024]

Q 18 :

If $3^{x} = 4^{x - 1},$ then $x =$ [2013]

Q 19 :

Let $p, q$ be integers and let $α, β$ be the roots of the equation $x^{2} - x - 1 = 0,$ where $α \neq β$ . For $n = 0, 1, 2, \dots$ , let $a_{n} = p α^{n} + q β^{n} .$

FACT: If $a$ and $b$ are rational numbers and $a + b \sqrt{5} = 0,$ then $a = 0 = b .$ [2017]

Q. $a_{12} =$

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Topic Question Set

Q 1 : Suppose a,b denote the distinct real roots of the quadratic polynomial x2+20x-2020 and suppose c,d denote the distinct complex roots of the quadratic polynomial x2-20x+2020. Then the value of ac(a-c)+ad(a-d)+bc(b-c)+bd(b-d) is [2020]

Q 2 : Let -π6<θ<-π12. Suppose α1 and β1 are the roots of the equation x2-2xsecθ+1=0 and α2 and β2 are the roots of the equation x2+2xtanθ-1=0. If α1>β1 and α2>β2, then α1+β2 equals [2016]

Q 3 : The quadratic equation p(x)=0 with real coefficients has purely imaginary roots. Then the equation p(p(x))=0 has [2014]

Q 4 : If a∈R and the equation -3(x-[x])2+2(x-[x])+a2=0 (where [x] denotes the greatest integer ≤x) has no integral solution, then all possible values of a lie in the interval: [2014]

Q 5 : Let α and β be the roots of x2-6x-2=0, with α>β. If an=αn-βn for n≥1, then the value of a10-2a82a9 is [2011]

Q 6 : Let (x0,y0) be the solution of the following equations (2x)ln2=(3y)ln3; 3lnx=2lny. Then x0 is [2011]

Q 7 : Let p and q be real numbers such that p≠0, p3≠q and p3≠-q. If α and β are nonzero complex numbers satisfying α+β=-p and α3+β3=q, then a quadratic equation having αβ and βα as its roots is [2010]

Q 8 : Let α,β be the roots of the equation x2-px+r=0 and α2,2β be the roots of the equation x2-qx+r=0. Then the value of r is [2007]

Q 9 : Let a,b,c be the sides of a triangle where a≠b≠c and λ∈R. If the roots of the equation x2+2(a+b+c)x+3λ(ab+bc+ca)=0 are real, then [2006]

Q 10 : If one root is square of the other root of the equation x2+px+q=0, then the relation between p and q is [2004]

Q 11 : For the equation 3x2+px+3=0, p>0, if one of the root is square of the other, then p is equal to [2000]

Q 12 : If b>a, then the equation (x-a)(x-b)-1=0 has [2000]

Q 13 : If α and β(α<β) are the roots of the equation x2+bx+c=0, where c<0<b, then [2000]

Q 14 : The product of all positive real values of x satisfying the equation x(16(log5x)3-68log5x)=5-16 is ________. [2022]

Q 15 : For x∈ℝ, then the number of real roots of the equation 3x2-4|x2-1|+x-1=0 is ________. [2021]

Q 16 : The smallest value of k, for which both the roots of the equation x2-8kx+16(k2-k+1)=0 are real, distinct and have values at least 4, is [2009]

Q 17 : Let R2 denote R×R. Let S={(a,b,c):a,b,c∈R and ax2+2bxy+cy2>0 for all (x,y)∈R2-{(0,0)}. Then which of the following statements is (are) TRUE? [2024]

Q 18 : If 3x=4x-1, then x= [2013]

Q 19 : Let p, q be integers and let α,β be the roots of the equation x2-x-1=0, where α≠β. For n=0,1,2,…, let an=pαn+qβn. FACT: If a and b are rational numbers and a+b5=0, then a=0=b. [2017] Q. a12=

Q 20 : Let p,q be integers and let α,β be the roots of the equation x2-x-1=0, where α≠β. For n=0,1,2,…, let an=pαn+qβn. FACT: If a and b are rational numbers and a+b5=0, then a=0=b. [2017] Q. If a4=28, then p+2q=

Q 21 : Let a,b,c,p,q be real numbers. Suppose α,β are the roots of the equation x2+2px+q=0 and α,1β are the roots of the equation ax2+2bx+c=0, where β2∉{-1,0,1}. STATEMENT-1: (p2-q)(b2-ac)≥0 and STATEMENT-2: b≠pa or c≠qa [2008]

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Q 1 :

Suppose $a, b$ denote the distinct real roots of the quadratic polynomial $x^{2} + 20 x - 2020$ and suppose $c, d$ denote the distinct complex roots of the quadratic polynomial $x^{2} - 20 x + 2020$ . Then the value of $a c (a - c) + a d (a - d) + b c (b - c) + b d (b - d)$ is [2020]

Q 3 :

The quadratic equation $p (x) = 0$ with real coefficients has purely imaginary roots. Then the equation $p (p (x)) = 0$ has [2014]

Q 4 :

If $a \in R$ and the equation $- 3 {(x - [x])}^{2} + 2 (x - [x]) + a^{2} = 0$ (where $[x]$ denotes the greatest integer $\leq x$ ) has no integral solution, then all possible values of $a$ lie in the interval: [2014]

Q 5 :

Let $α$ and $β$ be the roots of $x^{2} - 6 x - 2 = 0$ , with $α > β$ . If $a_{n} = α^{n} - β^{n} for n \geq 1,$ then the value of $\frac{a_{10} - 2 a_{8}}{2 a_{9}}$ is [2011]

Q 6 :

Let $(x_{0}, y_{0})$ be the solution of the following equations ${(2 x)}^{\ln 2} = {(3 y)}^{\ln 3}; 3^{\ln x} = 2^{\ln y} .$ Then $x_{0}$ is [2011]

Q 7 :

Let $p$ and $q$ be real numbers such that $p \neq 0$ , $p^{3} \neq q$ and $p^{3} \neq - q$ . If $α$ and $β$ are nonzero complex numbers satisfying $α + β = - p$ and $α^{3} + β^{3} = q$ , then a quadratic equation having $\frac{α}{β}$ and $\frac{β}{α}$ as its roots is [2010]

Q 8 :

Let $α, β$ be the roots of the equation $x^{2} - p x + r = 0$ and $\frac{α}{2}, 2 β$ be the roots of the equation $x^{2} - q x + r = 0$ . Then the value of $r$ is [2007]

Q 9 :

Let $a, b, c$ be the sides of a triangle where $a \neq b \neq c$ and $λ \in R$ . If the roots of the equation $x^{2} + 2 (a + b + c) x + 3 λ (a b + b c + c a) = 0$ are real, then [2006]

Q 10 :

If one root is square of the other root of the equation $x^{2} + p x + q = 0,$ then the relation between $p$ and $q$ is [2004]

Q 11 :

For the equation $3 x^{2} + p x + 3 = 0$ , $p > 0$ , if one of the root is square of the other, then $p$ is equal to [2000]

Q 12 :

If $b > a$ , then the equation $(x - a) (x - b) - 1 = 0$ has [2000]

Q 13 :

If $α$ and $β$ $(α < β)$ are the roots of the equation $x^{2} + b x + c = 0,$ where $c < 0 < b$ , then [2000]

Q 14 :

The product of all positive real values of $x$ satisfying the equation $x^{(16 {(\log_{5} x)}^{3} - 68 \log_{5} x)} = 5^{- 16}$ is ________. [2022]

Q 15 :

For $x \in ℝ$ , then the number of real roots of the equation $3 x^{2} - 4 | x^{2} - 1 | + x - 1 = 0$ is ________. [2021]

Q 16 :

The smallest value of $k$ , for which both the roots of the equation $x^{2} - 8 k x + 16 (k^{2} - k + 1) = 0$ are real, distinct and have values at least 4, is [2009]

Q 17 :

Let $R^{2}$ denote $R \times R$ . Let $S = {(a, b, c) : a, b, c \in R$ and $a x^{2} + 2 b x y + c y^{2} > 0$ for all $(x, y) \in R^{2} - {(0, 0)} .$ Then which of the following statements is (are) TRUE? [2024]

Q 18 :

If $3^{x} = 4^{x - 1},$ then $x =$ [2013]

Q 19 :

Let $p, q$ be integers and let $α, β$ be the roots of the equation $x^{2} - x - 1 = 0,$ where $α \neq β$ . For $n = 0, 1, 2, \dots$ , let $a_{n} = p α^{n} + q β^{n} .$

FACT: If $a$ and $b$ are rational numbers and $a + b \sqrt{5} = 0,$ then $a = 0 = b .$ [2017]

Q. $a_{12} =$