UFM Pure

grandes-ecoles 2025 Q5 Reciprocal and antireciprocal polynomial properties View

Let $R$ be a non-constant polynomial of $\mathbf{C}[X]$ having the following property: Every root $a$ of $R$ is nonzero and $\frac{1}{a}$ is a root of $R$ with the same multiplicity as $a$. Deduce that $R$ is reciprocal or antireciprocal.

grandes-ecoles 2025 Q8 Eigenvalue-root connection for matrices or linear operators View

8. Calculate the characteristic polynomial of the matrix $A$ (you may reason by induction on $d$).

iran-konkur 2013 Q105 Vieta's formulas: compute symmetric functions of roots View

105- If $\alpha, \beta$ are the roots of the equation $2x^2 - 3x - 4 = 0$, the equation whose roots are $\left\{\dfrac{1}{\alpha}+1,\ \dfrac{1}{\beta}+1\right\}$ is:

[(1)] $4x^2 - \Delta x + 1 = 0$
[(2)] $4x^2 - 3x + 1 = 0$
[(3)] $4x^2 - \Delta x - 1 = 0$
[(4)] $4x^2 - 3x - 1 = 0$

iran-konkur 2021 Q101 Vieta's formulas: compute symmetric functions of roots View

101- If the sum and product of the real roots of the equation $x^4 - 7x^2 - 5 = 0$ are $S$ and $P$ respectively, what is the value of $2P^2 - 3SP + 2S$?
(1) $59 - 7\sqrt{69}$ (2) $7 + \sqrt{69}$ (3) $50$ (4) $59 + 7\sqrt{69}$

isi-entrance 2012 Q30 Vieta's formulas: compute symmetric functions of roots View

Let $s, sr, sr^2, sr^3$ be the roots of $x^4 + ax^3 + bx^2 + cx + d = 0$ (roots in geometric progression). Show that $c^2 = a^2 d$.

isi-entrance 2020 Q17 Number of Solutions / Roots via Curve Analysis View

The number of real roots of the polynomial $$p ( x ) = \left( x ^ { 2020 } + 2020 x ^ { 2 } + 2020 \right) \left( x ^ { 3 } - 2020 \right) \left( x ^ { 2 } - 2020 \right)$$ is
(A) 2
(B) 3
(C) 2023
(D) 2025 .

jee-advanced 2007 Q50 Vieta's formulas: compute symmetric functions of roots View

Let $\alpha, \beta$ be the roots of the equation $x^2 - px + r = 0$ and $\frac{\alpha}{2}, 2\beta$ be the roots of the equation $x^2 - qx + r = 0$. Then the value of $r$ is
(A) $\frac{2}{9}(p-q)(2q-p)$
(B) $\frac{2}{9}(q-p)(2p-q)$
(C) $\frac{2}{9}(q-2p)(2q-p)$
(D) $\frac{2}{9}(2p-q)(2q-p)$

jee-advanced 2024 Q9 4 marks Vieta's formulas: compute symmetric functions of roots View

Let $f ( x ) = x ^ { 4 } + a x ^ { 3 } + b x ^ { 2 } + c$ be a polynomial with real coefficients such that $f ( 1 ) = - 9$. Suppose that $i \sqrt { 3 }$ is a root of the equation $4 x ^ { 3 } + 3 a x ^ { 2 } + 2 b x = 0$, where $i = \sqrt { - 1 }$. If $\alpha _ { 1 } , \alpha _ { 2 } , \alpha _ { 3 }$, and $\alpha _ { 4 }$ are all the roots of the equation $f ( x ) = 0$, then $\left| \alpha _ { 1 } \right| ^ { 2 } + \left| \alpha _ { 2 } \right| ^ { 2 } + \left| \alpha _ { 3 } \right| ^ { 2 } + \left| \alpha _ { 4 } \right| ^ { 2 }$ is equal to $\_\_\_\_$ .

jee-main 2014 Q61 Finding roots or coefficients of a quadratic using Vieta's relations View

If $\frac { 1 } { \sqrt { \alpha } } , \frac { 1 } { \sqrt { \beta } }$ are the roots of the equation $a x ^ { 2 } + b x + 1 = 0 , ( a \neq 0 , a , b \in R )$, then the equation $x \left( x + b ^ { 3 } \right) + \left( a ^ { 3 } - 3 a b x \right) = 0$ has roots:
(1) $\sqrt { \alpha \beta }$ and $\alpha \beta$
(2) $\alpha ^ { - \frac { 3 } { 2 } }$ and $\beta ^ { - \frac { 3 } { 2 } }$
(3) $\alpha \beta ^ { \frac { 1 } { 2 } }$ and $\alpha ^ { \frac { 1 } { 2 } } \beta$
(4) $\alpha ^ { \frac { 3 } { 2 } }$ and $\beta ^ { \frac { 3 } { 2 } }$

jee-main 2014 Q62 Finding roots or coefficients of a quadratic using Vieta's relations View

If equations $a x ^ { 2 } + b x + c = 0 , ( a , b , c \in R , a \neq 0 )$ and $2 x ^ { 2 } + 3 x + 4 = 0$ have a common root, then $a : b : c$ equals :
(1) $2 : 3 : 4$
(2) $4 : 3 : 2$
(3) $1 : 2 : 3$
(4) $3 : 2 : 1$

jee-main 2018 Q61 Vieta's formulas: compute symmetric functions of roots View

If $\lambda \in \mathrm { R }$ is such that the sum of the cubes of the roots of the equation, $x ^ { 2 } + ( 2 - \lambda ) x + ( 10 - \lambda ) = 0$ is minimum, then the magnitude of the difference of the roots of this equation is
(1) 20
(2) $2 \sqrt { 5 }$
(3) $2 \sqrt { 7 }$
(4) $4 \sqrt { 2 }$

jee-main 2020 Q51 Finding roots or coefficients of a quadratic using Vieta's relations View

Let $\lambda \neq 0$ be in $R$. If $\alpha$ and $\beta$ are the roots of the equation, $x ^ { 2 } - x + 2 \lambda = 0$ and $\alpha$ and $\gamma$ are the roots of the equation, $3 x ^ { 2 } - 10 x + 27 \lambda = 0$, then $\frac { \beta \gamma } { \lambda }$ is equal to:
(1) 27
(2) 18
(3) 9
(4) 36

jee-main 2020 Q51 Evaluating an algebraic expression given a constraint View

If $\alpha$ and $\beta$ be two roots of the equation $x ^ { 2 } - 64 x + 256 = 0$. Then the value of $\left( \frac { \alpha ^ { 3 } } { \beta ^ { 5 } } \right) ^ { \frac { 1 } { 8 } } + \left( \frac { \beta ^ { 3 } } { \alpha ^ { 5 } } \right) ^ { \frac { 1 } { 8 } }$ is :
(1) 2
(2) 3
(3) 1
(4) 4

jee-main 2020 Q51 Finding roots or coefficients of a quadratic using Vieta's relations View

If $\alpha$ and $\beta$ are the roots of the equation $2\mathrm{x}(2\mathrm{x}+1)=1$, then $\beta$ is equal to:
(1) $2\alpha(\alpha+1)$
(2) $-2\alpha(\alpha+1)$
(3) $2\alpha(\alpha-1)$
(4) $2\alpha^{2}$

jee-main 2021 Q61 Powers of i or Complex Number Integer Powers View

If $\alpha$ and $\beta$ are the distinct roots of the equation $x ^ { 2 } + ( 3 ) ^ { \frac { 1 } { 4 } } x + 3 ^ { \frac { 1 } { 2 } } = 0$, then the value of $\alpha ^ { 96 } \left( \alpha ^ { 12 } - 1 \right) + \beta ^ { 96 } \left( \beta ^ { 12 } - 1 \right)$ is equal to:
(1) $56 \times 3 ^ { 25 }$
(2) $56 \times 3 ^ { 24 }$
(3) $52 \times 3 ^ { 24 }$
(4) $28 \times 3 ^ { 25 }$

jee-main 2022 Q61 Vieta's formulas: compute symmetric functions of roots View

If the sum of the squares of the reciprocals of the roots $\alpha$ and $\beta$ of the equation $3 x ^ { 2 } + \lambda x - 1 = 0$ is 15 , then $6 \left( \alpha ^ { 3 } + \beta ^ { 3 } \right) ^ { 2 }$ is equal to
(1) 46
(2) 36
(3) 24
(4) 18

jee-main 2022 Q61 Vieta's formulas: compute symmetric functions of roots View

If $\alpha , \beta , \gamma , \delta$ are the roots of the equation $x ^ { 4 } + x ^ { 3 } + x ^ { 2 } + x + 1 = 0$, then $\alpha ^ { 2021 } + \beta ^ { 2021 } + \gamma ^ { 2021 } + \delta ^ { 2021 }$ is equal to
(1) 4
(2) 1
(3) - 4
(4) - 1

jee-main 2022 Q81 Vieta's formulas: compute symmetric functions of roots View

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 $\_\_\_\_$.

jee-main 2023 Q61 Vieta's formulas: compute symmetric functions of roots View

Let $\alpha , \beta$ be the roots of the equation $x ^ { 2 } - \sqrt { 2 } x + 2 = 0$. Then $\alpha ^ { 14 } + \beta ^ { 14 }$ is equal to
(1) $- 64$
(2) $- 64 \sqrt { 2 }$
(3) $- 128$
(4) $- 128 \sqrt { 2 }$

jee-main 2023 Q61 Determine coefficients or parameters from root conditions View

Let $a \in R$ 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 $\_\_\_\_$.

jee-main 2023 Q61 Determine coefficients or parameters from root conditions View

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 :
(1) $7 x ^ { 2 } + 245 x - 250 = 0$
(2) $7 x ^ { 2 } - 245 x + 250 = 0$
(3) $49 x ^ { 2 } - 245 x + 250 = 0$
(4) $49 x ^ { 2 } + 245 x + 250 = 0$

jee-main 2023 Q61 Vieta's formulas: compute symmetric functions of roots View

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 $\_\_\_\_$

jee-main 2023 Q61 Vieta's formulas: compute symmetric functions of roots View

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
(1) $\frac { 155 } { 8 }$
(2) 21
(3) $\frac { 169 } { 8 }$
(4) 19

jee-main 2023 Q61 Vieta's formulas: compute symmetric functions of roots View

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
(1) 81
(2) 9
(3) 72
(4) 729

jee-main 2025 Q6 Polynomial identity or factoring to simplify a given expression View

The product of all the rational roots of the equation $\left(x^2 - 9x + 11\right)^2 - (x-4)(x-5) = 3$, is equal to
(1) 14
(2) 21
(3) 28
(4) 7

UFM Pure

Sequences and series, recurrence and convergence 563

Roots of polynomials 109

Polar coordinates 22

Conic sections 223

Taylor series 190

Hyperbolic functions 5

Integration using inverse trig and hyperbolic functions 17

Integration with Partial Fractions 28

Vectors: Lines & Planes 205

Vectors: Cross Product & Distances 40

First order differential equations (integrating factor) 217

Complex numbers 2 47

Second order differential equations 155

Systems of differential equations 8