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Recent questions tagged roots

1 votes
1 answer
2
ISI2014-DCG-54
The number of real roots of the equation $1+\cos ^2x+\cos ^3 x – \cos^4x=5$ is equal to $0$ $1$ $3$ $4$
The number of real roots of the equation $1+\cos ^2x+\cos ^3 x – \cos^4x=5$ is equal to$0$$1$$3$$4$
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1 votes
1 answer
3
ISI2015-MMA-12
Consider the polynomial $x^5+ax^4+bx^3+cx^2+dx+4$ where $a,b,c,d$ are real numbers. If $(1+2i)$ and $(3-2i)$ are two two roots of this polynomial then the value of $a$ is $-524/65$ $524/65$ $-1/65$ $1/65$
Consider the polynomial $x^5+ax^4+bx^3+cx^2+dx+4$ where $a,b,c,d$ are real numbers. If $(1+2i)$ and $(3-2i)$ are two two roots of this polynomial then the value of $a$ i...
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0 votes
1 answer
4
ISI2015-DCG-7
Let $x^2-2(4k-1)x+15k^2-2k-7>0$ for any real value of $x$. Then the integer value of $k$ is $2$ $4$ $3$ $1$
Let $x^2-2(4k-1)x+15k^2-2k-7>0$ for any real value of $x$. Then the integer value of $k$ is$2$$4$$3$$1$
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1 votes
1 answer
5
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
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$$x^2-63x+2=...
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0 votes
1 answer
6
ISI2015-DCG-26
If $r$ be the ratio of the roots of the equation $ax^{2}+bx+c=0,$ then $\frac{r}{b}=\frac{r+1}{ac}$ $\frac{r+1}{b}=\frac{r}{ac}$ $\frac{(r+1)^{2}}{r}=\frac{b^{2}}{ac}$ $\left(\frac{r}{b}\right)^{2}=\frac{r+1}{ac}$
If $r$ be the ratio of the roots of the equation $ax^{2}+bx+c=0,$ then $\frac{r}{b}=\frac{r+1}{ac}$$\frac{r+1}{b}=\frac{r}{ac}$$\frac{(r+1)^{2}}{r}=\frac{b^{2}}{ac}$$\lef...
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0 votes
1 answer
9
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 $2$ $4$ $3$ $1$
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$$1$
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1 votes
1 answer
10
ISI2016-DCG-25
If $\alpha$ and $\beta$ be the roots of the equation $x^{2}+3x+4=0,$ then the equation with roots $(\alpha+\beta)^{2}$ and $(\alpha-\beta)^{2}$ is $x^{2}+2x+63=0$ $x^{2}-63x+2=0$ $x^{2}-2x-63=0$ None of these
If $\alpha$ and $\beta$ be the roots of the equation $x^{2}+3x+4=0,$ then the equation with roots $(\alpha+\beta)^{2}$ and $(\alpha-\beta)^{2}$ is$x^{2}+2x+63=0$$x^{2}-63...
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1 votes
2 answers
11
ISI2016-DCG-26
If $r$ be the ratio of the roots of the equation $ax^{2}+bx+c=0,$ then $\frac{r}{b}=\frac{r+1}{ac}$ $\frac{r+1}{b}=\frac{r}{ac}$ $\frac{(r+1)^{2}}{r}=\frac{b^{2}}{ac}$ $\left(\frac{r}{b}\right)^{2}=\frac{r+1}{ac}$
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}$$\lef...
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1 votes
0 answers
13
ISI2016-DCG-29
The condition that ensures that the roots of the equation $x^{3}-px^{2}+qx-r=0$ are in H.P. is $r^{2}-9pqr+q^{3}=0$ $27r^{2}-9pqr+3q^{3}=0$ $3r^{3}-27pqr-9q^{3}=0$ $27r^{2}-9pqr+2q^{3}=0$
The condition that ensures that the roots of the equation $x^{3}-px^{2}+qx-r=0$ are in H.P. is$r^{2}-9pqr+q^{3}=0$$27r^{2}-9pqr+3q^{3}=0$$3r^{3}-27pqr-9q^{3}=0$$27r^{2}-...
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0 votes
1 answer
15
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 $0$ $1$ $2$ $5$
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$$5$
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1 votes
0 answers
16
ISI2018-DCG-11
The sum of $99^{th}$ power of all the roots of $x^7-1=0$ is equal to $1$ $2$ $-1$ $0$
The sum of $99^{th}$ power of all the roots of $x^7-1=0$ is equal to$1$$2$$-1$$0$
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0 votes
1 answer
17
ISI2017-PCB-A-1
Suppose all the roots of the equation $x^3 +bx-2017=0$ (where $b$ is a real number) are real. Prove that exactly one root is positive.
Suppose all the roots of the equation $x^3 +bx-2017=0$ (where $b$ is a real number) are real. Prove that exactly one root is positive.
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1 votes
1 answer
18
ISI2016-PCB-A-1
If $\alpha, \beta, \gamma$ are the roots of the equation $x^3+6x+1=0$, then prove that $\frac{\alpha}{\beta} + \frac{\beta}{\alpha} + \frac{\beta}{\gamma}+ \frac{\gamma}{\beta} + \frac{\gamma}{\alpha}+ \frac{\alpha}{\gamma}=-3.$
If $\alpha, \beta, \gamma$ are the roots of the equation $x^3+6x+1=0$, then prove that $$\frac{\alpha}{\beta} + \frac{\beta}{\alpha} + \frac{\beta}{\gamma}+ \frac{\gamma}...
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0 votes
0 answers
19
ISI2017-MMA-3
If $\alpha, \beta$ and $\gamma$ are the roots of the equation $x^3+3x^2-8x+1=0$, then an equation whose roots are $\alpha+1, \beta+1$ and $\gamma+1$ is given by $y^3-11y+11=0$ $y^3-11y-11=0$ $y^3+13y+13=0$ $y^3+6y^2+y-3=0$
If $\alpha, \beta$ and $\gamma$ are the roots of the equation $x^3+3x^2-8x+1=0$, then an equation whose roots are $\alpha+1, \beta+1$ and $\gamma+1$ is given by $y^3-11y+...
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1 votes
2 answers
20
ISI2016-MMA-3
The number of real roots of the equation $2 \cos \big(\frac{x^2+x}{6}\big)=2^x+2^{-x}$ is $0$ $1$ $2$ $\infty$
The number of real roots of the equation $2 \cos \big(\frac{x^2+x}{6}\big)=2^x+2^{-x}$ is$0$$1$$2$$\infty$
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0 votes
0 answers
21
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 $a<-\frac{2}{3}$ $a=0$ $0<a<\frac{3}{4}$ $a \geq \frac{3}{4}$
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}$
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