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Recent questions tagged quadratic-equations

1 vote
1 answer
1
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, 2019 in Numerical Ability Arjun 129 views
1 vote
2 answers
2
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 opposite sign ... $b$ have the same sign while $c$ has the opposite sign. $a$ and $c$ have the same sign. $a$, $b$ and $c$ have the same sign.
asked Sep 23, 2019 in Numerical Ability Arjun 114 views
1 vote
1 answer
3
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, 2019 in Numerical Ability Arjun 92 views
0 votes
1 answer
4
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, 2019 in Calculus Arjun 45 views
0 votes
1 answer
5
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, 2019 in Numerical Ability Arjun 69 views
0 votes
1 answer
6
The number of real roots of the equation $2 \cos \left( \frac{x^2+x}{6} \right) = 2^x +2^{-x} \text{ is }$ $0$ $1$ $2$ infinitely many
asked Sep 23, 2019 in Numerical Ability Arjun 100 views
0 votes
1 answer
7
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, 2019 in Numerical Ability Arjun 111 views
1 vote
2 answers
8
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, 2019 in Numerical Ability Arjun 139 views
0 votes
1 answer
9
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, 2019 in Numerical Ability gatecse 66 views
1 vote
1 answer
10
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, 2019 in Numerical Ability gatecse 45 views
0 votes
1 answer
11
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, 2019 in Numerical Ability gatecse 36 views
0 votes
1 answer
12
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, 2019 in Numerical Ability gatecse 39 views
0 votes
1 answer
13
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, 2019 in Numerical Ability gatecse 43 views
0 votes
1 answer
14
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, 2019 in Numerical Ability gatecse 39 views
1 vote
1 answer
15
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, 2019 in Numerical Ability gatecse 43 views
1 vote
2 answers
16
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, 2019 in Numerical Ability gatecse 62 views
1 vote
1 answer
17
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, 2019 in Numerical Ability gatecse 55 views
1 vote
0 answers
18
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, 2019 in Numerical Ability gatecse 50 views
0 votes
1 answer
19
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, 2019 in Numerical Ability gatecse 44 views
0 votes
1 answer
20
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, 2019 in Numerical Ability gatecse 55 views
1 vote
1 answer
21
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 jothee 71 views
1 vote
1 answer
22
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 jothee 60 views
0 votes
0 answers
23
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 jothee 38 views
2 votes
2 answers
24
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 Lakshman Patel RJIT 424 views
0 votes
1 answer
25
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 Arjun 204 views
3 votes
1 answer
26
Find all real solutions of the equation $x^{2} - |x-1| - 3 = 0$
asked Mar 4, 2017 in Others Devasish Ghosh 269 views
5 votes
1 answer
27
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 makhdoom ghaya 583 views
1 vote
1 answer
29
The golden ratio $\varphi$ and its conjugate $\bar{\varphi}$ both satisfy the equation $x^3 &ndash;x-1=0$ $x^3 +x-1=0$ $x^2 &ndash;x-1=0$ $x^2 +x-1=0$
asked Jul 16, 2016 in Others jothee 537 views
1 vote
1 answer
30
If $\sqrt{3}$ + 1 is a root of equation 3 x$^{3}$ + ax$^{2}$ + bx + 12 = 0 where a and b are rational numbers, then b is equal to -6 2 6 10
asked Mar 23, 2016 in Numerical Ability vijaycs 185 views
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