Recent questions tagged quadratic-equations

+1 vote

1 answer

1

ISI2014-DCG-11
Let $x_1$ and $x_2$ be the roots of the quadratic equation $x^2-3x+a=0$, and $x_3$ and $x_4$ be the roots of the quadratic equation $x^2-12x+b=0$. If $x_1, x_2, x_3$ and $x_4 \: (0 < x_1 < x_2 < x_3 < x_4)$ are in $G.P.,$ then $ab$ equals $64$ $5184$ $-64$ $-5184$

asked Sep 23 in Numerical Ability by Arjun Veteran (424k points) | 24 views

+1 vote

2 answers

2

ISI2014-DCG-22
The conditions on $a$, $b$ and $c$ under which the roots of the quadratic equation $ax^2+bx+c=0 \: ,a \neq 0, \: b \neq 0 $ and $c \neq 0$, are unequal magnitude but of the opposite signs, are the following: $a$ and $c$ have the same sign while $b$ has the ... $c$ has the opposite sign. $a$ and $c$ have the same sign. $a$, $b$ and $c$ have the same sign.

asked Sep 23 in Numerical Ability by Arjun Veteran (424k points) | 27 views

+1 vote

1 answer

3

ISI2014-DCG-30
Consider the equation $P(x) =x^3+px^2+qx+r=0$ where $p,q$ and $r$ are all real and positive. State which of the following statements is always correct. All roots of $P(x) = 0$ are real The equation $P(x)=0$ has at least one real root The equation $P(x)=0$ has no negative real root The equation $P(x)=0$ must have one positive and one negative real root

asked Sep 23 in Numerical Ability by Arjun Veteran (424k points) | 20 views

0 votes

0 answers

4

ISI2014-DCG-48
If $x$ is real, the set of real values of $a$ for which the function $y=x^2-ax+1-2a^2$ is always greater than zero is $- \frac{2}{3} < a \leq \frac{2}{3}$ $- \frac{2}{3} \leq a < \frac{2}{3}$ $- \frac{2}{3} < a < \frac{2}{3}$ None of these

asked Sep 23 in Calculus by Arjun Veteran (424k points) | 7 views

0 votes

1 answer

5

ISI2014-DCG-55
If $a,b,c$ are sides of a triangle $ABC$ such that $x^2-2(a+b+c)x+3 \lambda (ab+bc+ca)=0$ has real roots then $\lambda < \frac{4}{3}$ $\lambda > \frac{5}{3}$ $\lambda \in \big( \frac{4}{3}, \frac{5}{3}\big)$ $\lambda \in \big( \frac{1}{3}, \frac{5}{3}\big)$

asked Sep 23 in Numerical Ability by Arjun Veteran (424k points) | 18 views

0 votes

1 answer

6

ISI2015-MMA-13
The number of real roots of the equation $2 \cos \bigg( \frac{x^2+x}{6} \bigg) = 2^x +2^{-x} \text{ is }$ $0$ $1$ $2$ infinitely many

asked Sep 23 in Numerical Ability by Arjun Veteran (424k points) | 12 views

0 votes

1 answer

7

ISI2015-MMA-15
The number of real solutions of the equations $(9/10)^x = -3+x-x^2$ is $2$ $0$ $1$ none of the above

asked Sep 23 in Numerical Ability by Arjun Veteran (424k points) | 15 views

+1 vote

2 answers

8

ISI2015-MMA-16
If two real polynomials $f(x)$ and $g(x)$ of degrees $m\: (\geq 2)$ and $n\: (\geq 1)$ respectively, satisfy $f(x^2+1)=f(x)g(x),$ for every $x \in \mathbb{R}$, then $f$ has exactly one real root $x_0$ such that $f’(x_0) \neq 0$ $f$ has exactly one real root $x_0$ such that $f’(x_0) = 0$ $f$ has $m$ distinct real roots $f$ has no real root

asked Sep 23 in Numerical Ability by Arjun Veteran (424k points) | 42 views

0 votes

1 answer

9

ISI2015-DCG-7
Let $x^2-2(4k-1)x+15k^2-2k-7>0$ for any real value of $x$. Then the integer value of $k$ is $2$ $4$ $3$ $1$

asked Sep 18 in Numerical Ability by gatecse Boss (16.8k points) | 33 views

+1 vote

1 answer

10

ISI2015-DCG-25
If $\alpha$ and $\beta$ be the roots of the equation $x^2+3x+4=0$, then the equation with roots $(\alpha + \beta)^2$ and $(\alpha – \beta)^2$ is $x^2+2x+63=0$ $x^2-63x+2=0$ $x^2-2x-63=0$ None of the above

asked Sep 18 in Numerical Ability by gatecse Boss (16.8k points) | 14 views

0 votes

1 answer

11

ISI2015-DCG-26
If $r$ be the ratio of the roots of the equation $ax^{2}+bx+c=0,$ then $\frac{r}{b}=\frac{r+1}{ac}$ $\frac{r+1}{b}=\frac{r}{ac}$ $\frac{(r+1)^{2}}{r}=\frac{b^{2}}{ac}$ $\left(\frac{r}{b}\right)^{2}=\frac{r+1}{ac}$

asked Sep 18 in Numerical Ability by gatecse Boss (16.8k points) | 10 views

0 votes

1 answer

12

ISI2015-DCG-28
If one root of a quadratic equation $ax^2+bx+c=0$ be equal to the $n^{th}$ power of the other, then $(ac)^{\frac{n}{n+1}} +b=0$ $(ac)^{\frac{n+1}{n}} +b=0$ $(ac^{n})^{\frac{1}{n+1}} +(a^nc)^{\frac{1}{n+1}}+b=0$ $(ac^{\frac{1}{n+1}})^n +(a^{\frac{1}{n+1}}c)^{n+1}+b=0$

asked Sep 18 in Numerical Ability by gatecse Boss (16.8k points) | 13 views

+2 votes

1 answer

13

ISI2015-DCG-29
The condition that ensures that the roots of the equation $x^3-px^2+qx-r=0$ are in $H.P.$ is $r^2-9pqr+q^3=0$ $27r^2-9pqr+3q^3=0$ $3r^3-27pqr-9q^3=0$ $27r^2-9pqr+2q^3=0$

asked Sep 18 in Numerical Ability by gatecse Boss (16.8k points) | 24 views

0 votes

1 answer

14

ISI2015-DCG-30
Let $p,q,r,s$ be real numbers such that $pr=2(q+s)$. Consider the equations $x^2+px+q=0$ and $x^2+rx+s=0$. Then at least one of the equations has real roots both these equations have real roots neither of these equations have real roots given data is not sufficient to arrive at any conclusion

asked Sep 18 in Numerical Ability by gatecse Boss (16.8k points) | 15 views

0 votes

1 answer

15

ISI2016-DCG-7
Let $x^{2}-2(4k-1)x+15k^{2}-2k-7>0$ for any real value of $x$. Then the integer value of $k$ is $2$ $4$ $3$ $1$

asked Sep 18 in Numerical Ability by gatecse Boss (16.8k points) | 13 views

+1 vote

1 answer

16

ISI2016-DCG-25
If $\alpha$ and $\beta$ be the roots of the equation $x^{2}+3x+4=0,$ then the equation with roots $(\alpha+\beta)^{2}$ and $(\alpha-\beta)^{2}$ is $x^{2}+2x+63=0$ $x^{2}-63x+2=0$ $x^{2}-2x-63=0$ None of these

asked Sep 18 in Numerical Ability by gatecse Boss (16.8k points) | 15 views

+1 vote

2 answers

17

ISI2016-DCG-26
If $r$ be the ratio of the roots of the equation $ax^{2}+bx+c=0,$ then $\frac{r}{b}=\frac{r+1}{ac}$ $\frac{r+1}{b}=\frac{r}{ac}$ $\frac{(r+1)^{2}}{r}=\frac{b^{2}}{ac}$ $\left(\frac{r}{b}\right)^{2}=\frac{r+1}{ac}$

asked Sep 18 in Numerical Ability by gatecse Boss (16.8k points) | 30 views

+1 vote

1 answer

18

ISI2016-DCG-28
If one root of a quadratic equation $ax^{2}+bx+c=0$ be equal to the n th power of the other, then $(ac)^{\frac{n}{n+1}}+b=0$ $(ac)^{\frac{n+1}{n}}+b=0$ $(ac^{n})^{\frac{1}{n+1}}+(a^{n}c)^{\frac{1}{n+1}}+b=0$ $(ac^\frac{1}{n+1})^{n}+(a^\frac{1}{n+1}c)^{n+1}+b=0$

asked Sep 18 in Numerical Ability by gatecse Boss (16.8k points) | 14 views

+1 vote

0 answers

19

ISI2016-DCG-29
The condition that ensures that the roots of the equation $x^{3}-px^{2}+qx-r=0$ are in H.P. is $r^{2}-9pqr+q^{3}=0$ $27r^{2}-9pqr+3q^{3}=0$ $3r^{3}-27pqr-9q^{3}=0$ $27r^{2}-9pqr+2q^{3}=0$

asked Sep 18 in Numerical Ability by gatecse Boss (16.8k points) | 9 views

0 votes

1 answer

20

ISI2016-DCG-30
Let $p,q,r,s$ be real numbers such that $pr=2(q+s).$ Consider the equations $x^{2}+px+q=0$ and $x^{2}+rx+s=0.$ Then at least one of the equations has real roots. both these equations have real roots. neither of these equations have real roots. given data is not sufficient to arrive at any conclusion.

asked Sep 18 in Numerical Ability by gatecse Boss (16.8k points) | 17 views

0 votes

1 answer

21

ISI2017-DCG-5
The sum of the squares of the roots of $x^2-(a-2)x-a-1=0$ becomes minimum when $a$ is $0$ $1$ $2$ $5$

asked Sep 18 in Numerical Ability by gatecse Boss (16.8k points) | 19 views

+1 vote

1 answer

22

ISI2016-PCB-A-1
If $\alpha, \beta, \gamma$ are the roots of the equation $x^3+6x+1=0$, then prove that $\frac{\alpha}{\beta} + \frac{\beta}{\alpha} + \frac{\beta}{\gamma}+ \frac{\gamma}{\beta} + \frac{\gamma}{\alpha}+ \frac{\alpha}{\gamma}=-3.$

asked Sep 18, 2018 in Numerical Ability by jothee Veteran (105k points) | 34 views

+1 vote

1 answer

23

ISI2016-MMA-3
The number of real roots of the equation $2 \cos \big(\frac{x^2+x}{6}\big)=2^x+2^{-x}$ is $0$ $1$ $2$ $\infty$

asked Sep 13, 2018 in Numerical Ability by jothee Veteran (105k points) | 22 views

0 votes

0 answers

24

ISI2016-MMA-29
Suppose $a$ is a real number for which all the roots of the equation $x^4 -2ax^2+x+a^2-a=0$ are real. Then $a<-\frac{2}{3}$ $a=0$ $0<a<\frac{3}{4}$ $a \geq \frac{3}{4}$

asked Sep 13, 2018 in Numerical Ability by jothee Veteran (105k points) | 10 views

+2 votes

2 answers

25

GATE2018 EE: GA-4
For what values of $k$ given below is $\dfrac{(k + 2)^2}{(k - 3)}$ an integer? $4, 8, 18$ $4, 10, 16$ $4, 8, 28$ $8, 26, 28$

asked Feb 21, 2018 in Numerical Ability by Lakshman Patel RJIT Veteran (54.8k points) | 312 views

0 votes

1 answer

26

GATE2018 ME-1: GA-7
Given that $a$ and $b$ are integers and $a+a^2 b^3$ is odd, which of the following statements is correct? $a$ and $b$ are both odd $a$ and $b$ are both even $a$ is even and $b$ is odd $a$ is odd and $b$ is even

asked Feb 17, 2018 in Numerical Ability by Arjun Veteran (424k points) | 62 views

+3 votes

1 answer

27

ISI 2015 PCB A2
Find all real solutions of the equation $x^{2} - |x-1| - 3 = 0$

asked Mar 4, 2017 in Others by Devasish Ghosh Junior (815 points) | 217 views

+2 votes

1 answer

28

GATE2016 EC-2: GA-4
Given $(9 \text{ inches}) ^{\frac{1}{2}} = (0.25\text{ yards}) ^{\frac{1}{2}},$ which one of the following statements is TRUE? $3$ inches = $0.5$ yards $9$ inches = $1.5$ yards $9$ inches = $0.25$ yards $81$ inches = $0.0625$ yards

asked Jan 21, 2017 in Numerical Ability by makhdoom ghaya Boss (30.1k points) | 299 views

+8 votes

1 answer

29

Radix+Quadratic equation

asked Oct 12, 2016 in Digital Logic by Rahul Jain25 Boss (11.1k points) | 731 views

+1 vote

1 answer

30

UGCNET-June2013-III-11
The golden ratio $\varphi$ and its conjugate $\bar{\varphi}$ both satisfy the equation $x^3 –x-1=0$ $x^3 +x-1=0$ $x^2 –x-1=0$ $x^2 +x-1=0$

asked Jul 16, 2016 in Others by jothee Veteran (105k points) | 372 views

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