Nature of Roots of a Cubic Polynomial Using Derivatives

Q. The equation $e^{x-1}$ + x - 2 = 0 has

1. one real root
2. two real roots
3. three real roots
4. four real roots.

Q. A function $f:R\to R$ is defined as $f\left(x\right)=-x^3+3x^2-2x+4.$ Which of the following is true about the function $f\left(x\right)$?

1. $f\left(x\right)$ is an onto function
2. $f\left(x\right)$ is an into function
3. $f\left(x\right)$ is a one-one function
4. $f\left(x\right)=4$ has only one solution

Q. Evaluate: ${\int }_0^1\frac{xdx}{\sqrt{1+x^2}}$

Q.

If the equation $x^2+\lambda x+\mu =0$ has equal roots and one root of the equation $x^2+\lambda x-12=0$ is 2, then $(\lambda ,\mu )=$

1. (4, -4)

2. (-4, -4)

3. (4, 4)

4. (-4, 4)

Q.

Let Z be a complex Numbers satisfying the equation $z^2$ - (11 + i) z+k+3i = 0 where k $ϵ$ R and i = $\sqrt{(-i)}$, suppose equation has one real and one non-real root. Find the non-real root.

1. 7 + i

2. 5 + i

3. 8 + i

4. 4 + i

Q. The least natural number a for which $x+a{x}^{-2}>2forallxϵR^{+}$

1. 5
2. 2
3. 9
4. 1

Q. The least natural number a for which $x+a{x}^{-2}>2forallxϵR^{+}$

Q.

For equation $x^4-2x^3-2x^2+18x-63=0$ if two of its roots are equal in magnitude but opposite in sign the other two roots are

1. Real and negative

2. Real and positive

3. Complex Number

4. None of these

Q.

If $a_1$ < $a_2$< $a_3$ < $a_4$ < $a_5$ < $a_6$, then the equation $(x-a_1)(x-a_3)(x-a_5)+2(x-a_2)(x-a_4)(x-a_6)$ = 0 has

1. Three real roots

2. One real root

3. None of these

4. One real root in each interval ( , ), ( , ) and ( , )

Q. If $\alpha ,\beta$ and $\gamma$ are the roots of the equation $x^3-3x^2+3x+7=0,$ and $\omega$ is the cube root of unity, then the value of $\frac{\alpha -1}{\beta -1}+\frac{\beta -1}{\gamma -1}+\frac{\gamma -1}{\alpha -1}$ is equal to

1. $3{\omega }^2$
2. $\frac{3}{\omega }$
3. None of these
4. $2{\omega }^2$

Q. The roots of the equation $x^3$ + 2$x^2$ + 5x + 2cos x = 0 in [0, 2$\pi$] is

1. only one root
2. only three roots
3. None of these
4. only two roots

Q. Which of the following could be the possible integral value(s) of a for which the equation $x^3-3x+a=0$ has three distinct roots?

1. 1
2. -1
3. \N
4. 3
5. -3

Q. The equation $e^{x-1}$ + x - 2 = 0 has

1. one real root
2. two real roots
3. three real roots
4. four real roots.

Q. If ${\int }_0^{3}f\left(x\right).dx={\int }_3^{0}f\left(x\right).dx$ then

1. f(x) is an exponential function
2. f(x) = 0
3. f(x) is a logarithmic function
4. f(x) is a constant function

Q.

$\alpha ,\beta$ are the roots of $a{x}^2+bx+c=0and\gamma ,\delta aretherootsofp{x}^2+qx+r=0and{D}_1,D_2$ be the respective discriminants of these equations. If $\alpha ,\beta ,\gamma ,and\delta$ are in A.P. then $D_1:D_2=(where\alpha ,\beta ,\gamma \delta ,ϵR\&a,b,c,p,q,rϵR)$

1. 

2. 

3. 

4. 

Q. Let α, β be the roots of the quadratic equation x² + px + p³ = 0. If (α, β) is a pt. on the parabola y² = x, then what are the roots of the qudratic equation?

Q.

Interval in which $‘a^{\prime }$ lies so that $x^3–3x+a=0$ has three real and distinct roots.

1. (-2, 2)

2. (-1 , 2)

3. (-2, 1)

4. (-1, 1)

Q. If $2a+3b+6c=0$, then at least one root of the equation $a{x}^2+bx+c=0$ lies in the interval

1. (2, 3)
2. (0, 1)
3. (1, 2)
4. (1, 3)

Q.

If the roots of the equation a$x^2$ + bx + c = 0 are real and of the form$\frac{(\propto -1)}{\propto }$ and $\frac{(\propto +1)}{\propto }$, then value of ${(a+b+c)}^2$ is

1. b² - 4ac

2. b² - 2ac

3. 2b² - ac

4. None

Q. If the roots of the equation $4x^2+5k=(5k+1)x$ differ by unity, then the negative value of $k$ is

1. $-3$
2. $-\frac{3}{5}$
3. $-\frac{1}{5}$
4. $Noneofthese$

Q.

If -1, $\alpha$, ${\alpha }^3$, ${\alpha }^5$, $\overline{\alpha }$, $\overline{{\alpha }^3}$, $\overline{{\alpha }^5}$ are the roots of the equation $z^7$ + 1 = 0. Find the value of $cos\frac{\pi }{7}$ $\times$ $cos\frac{3\pi }{7}$ $\times$ $cos\frac{5\pi }{7}$.

1. -

2. -

3. -

4. -

Q. The equation $2^{2x}+(a-1)2^{x+1}+a=0$ has roots of opposite signs, then exhaustive set of values of $a$ is

1. $a\in (-1,0)$
2. $a<0$
3. $a\in (-\infty ,\frac{1}{3})$
4. $a\in (0,\frac{1}{3})$

Q. Find the roots of equation:
$x^2-\frac{1}{12}x-\frac{1}{12}=0$

1. $\frac{-1}{4},\frac{1}{3}$
2. $4,3$
3. $\frac{-1}{4},\frac{-1}{3}$
4. $-4,-3$

Q. The roots of the equation $x^3$ + 2$x^2$ + 5x + 2cos x = 0 in [0, 2$\pi$] is

1. only one root
2. only two roots
3. only three roots
4. NO ROOTS EXIST

Q. The function $f\left(x\right)=3+2(a+1)x+(a^2+1)x^2-x^3$ has a local minimum at $x=x_1$ and local maximum at $x=x_2$ such that $x_1<2<x_2$, then $a$ belongs to interval(s).

1. $(-\infty ,-\frac{3}{2})$
2. $(-\frac{3}{2},1)$
3. $(0,\infty )$
4. $(1,\infty )$

Q. The least natural number a for which $x+a{x}^{-2}>2forallxϵR^{+}$

1. 1
2. 2
3. 5
4. 9

Q. If $\alpha ,\beta$ are the roots of $a{x}^2+bx+c=0$, form the equation whose roots are ${\alpha }^2+{\beta }^2$ and ${\alpha }^{-2}+{\beta }^{-2}$.

Q. Find the value of m for which both roots of equation $x^2-mx+1=0$ are less than unity.

Q. If the point $(3,4)$ lies on the line $ax+y=10$, then find the value of $a$.

1. $2$
2. $3$
3. $4$
4. $1$

Q.

Interval in which $‘a^{\prime }$ lies so that $x^3–3x+a=0$ has three real and distinct roots.

1. (-1, 1)

2. (-2, 2)

3. (-1 , 2)

4. (-2, 1)