Nature of Roots

Q.

The integer $k$, for which the inequality $x^2-2(3k-1)x+8k^2-7>0$ is valid for every $x$ in $R$ is :

1. $3$

2. $2$

3. $4$

4. $0$

Q.

The equation $x^2+ax-a^2$ - 1 = 0 will have roots of opposite signs if :

1. a [-1 , 1 ]

2. a

3. a

4. None of these

Q.

If the roots of the equation q$x^2$ + p$x$ + q = 0 where p, q are real, are complex, then the roots of the equation $x^2$ - 4q$x$ + $p^2$ = 0 are

1. Real and unequal

2. Real and equal

3. Imaginary

4. nothing can be said in particular

Q. Suppose $a,b,c$ are three non-zero real numbers. Then the equation $x^2+(a+b+c)x+a^2+b^2+c^2=0$ has

1. rational and unequal roots.
2. irrational roots.
3. rational and equal roots.
4. no real roots.

Q.

If $\left|z+8\right|+\left|z-8\right|=16$, where $z$ is a complex number, then the point $z$ will lie on

1. a circle

2. an ellipse

3. a straight line

4. None of the above

Q. Consider the equation $x^2-4|x|+3-|k-1|=0$, where $k$ is a real parameter.
$$\begin{array}{cccc}& \text{Column I}& & \text{Column II}\\ \text{(A)}& \text{No real roots}& \text{(P)}& k\in (-2,4)\\ \text{(B)}& \text{Two distinct real roots}& \text{(Q)}& k\in \{-2,4\}\\ \text{(C)}& \text{Three distinct real roots}& \text{(R)}& k\in (-\infty ,-2)\cup (4,\infty )\\ \text{(D)}& \text{Four distinct real roots}& \text{(S)}& k\in \varphi \\ & & \text{(T)}& k\in [-2,4]\\ & & \text{(U)}& k\in (-\infty ,-2]\cup [4,\infty )\end{array}$$
Which of the following is the only CORRECT combination?

1. $\left(C\right)\to \left(Q\right)$
2. $\left(D\right)\to \left(S\right)$
3. $\left(B\right)\to \left(T\right)$
4. $\left(A\right)\to \left(U\right)$

Q.

Select all the functions whose graphs include the point $(16,4)$.

1. $\mathrm{y}=2\mathrm{x}$

2. $\mathrm{y}={\mathrm{x}}^2$

3. $\mathrm{y}=\mathrm{x}+12$

4. $\mathrm{y}=\mathrm{x}-12$

5. $\mathrm{y}=14\mathrm{x}$

Q. If the roots of the quadratic equation a(b- c)x2 + b(c - a)x + c(ab)0 are equal anda, b, c>0, then prove that 2/b=1/a+1/bi.e., a, b, c are in H.P.bac

Q. If the roots of the equation $x^2-8x+a^2-6a=0$ are real and distinct, then the number of integral value(s) of $a$ is

Q. If the roots of the equation $(a-1)(x^2+x+1{)}^2=(a+1\left)\right(x^4+x^2+1)$ are real and distinct, then $a$ belongs to

1. $(-\infty ,3]$
2. $(-\infty ,-2)\cup (2,\infty )$
3. $[-2,2]$
4. $\varphi$

Q. If ax^2+bx+c does not have any real root and 4a+c<2b then the number of real roots of the equation x^4+3x^2-a=

Q.

Let f(x) = 2x + 1. Then the number of real values of x for which the three numbers f(x), f(2x), f(4x) are in G.P. is

__

Q. If the roots of the equation $6x^2-7x+k=0,k>-3$ are rational, then possible integral value(s) of $k$ is/are

1. $-1$
2. $-1,-2$
3. $-2$
4. $0,1,2$

Q.

In a quadratic equation with leading coefficient 1, a student reads the coefficient 16 of x wrongly as 19 and obtain the roots as -15 and -4, the correct roots are

1. 6, 10

2. -7, -9

3. 15, 4

4. -6, -10

Q. If both the roots of the quadratic equation $x^2-(2n+18)x-n-11=0,n\in Z$ are rational, then the value(s) of $n$ is/are

1. $-8$
2. $-10$
3. $-11$
4. $-12$

Q. If $a,b,c\in R^{+}$, then both the roots of the equation $(x-b)(x-c)+(x-a)(x-c)+(x-a)(x-b)=0$ are always

1. real and negative
2. imaginary
3. real and positive
4. real and equal

Q. if for x>0 f(x)=(2-x^n)^{1/n} and g(x)=x^{2 }+rx+s; r, s∈ R it is given g(x)-x=0 has imaginary roots then the number of real roots of the equation g(g(x))-f(f(x))=0 i

Q. If the equation $x^2-4x+m=0$ has real roots, then the maximum value of $m$ is

Q. If $(a{x}^2+bx+c)y+a^{\prime }{x}^2+b^{\prime }x+c^{\prime }=0$, then the condition that x may be rational function of $y$ is

1. $(a{b}^{\prime }-a^{\prime }b{)}^2=(a{c}^{\prime }-a^{\prime }c\left)\right(b{c}^{\prime }-b^{\prime }c)$
2. $(b{c}^{\prime }-b^{\prime }c{)}^2=(a{b}^{\prime }-a^{\prime }b\left)\right(a{c}^{\prime }-a^{\prime }c)$
3. $(a{c}^{\prime }-a^{\prime }c{)}^2=(a{b}^{\prime }-a^{\prime }b\left)\right(b{c}^{\prime }-b^{\prime }c)$
4. None of the above

Q. If roots of the quadratic equation a(b-c)x^2+b(c-a)x+c(a-b)=0 are equal and a, b, c is greater than 2÷b=1÷a+1÷c ie a, b, c are in h.p

Q. $\begin{array}{cc}\text{(I) If}{x}^2+x-a=0\text{has integral roots}& \left(P\right)2\\ \text{and}a\in N,\text{then}a\text{can be equal to}& \\ \text{(II) If the equation}a{x}^2+2bx+4c=16& \left(Q\right)12\\ \text{has no real roots and}a+c>b+4,& \\ \text{then the integral value of}c\text{can be}& \\ \text{(III) If equation}{x}^2+2bx+9b-14=0& \left(R\right)1\\ \text{has only negative roots, then the integral}& \\ \text{values of}b\text{can be}& \\ \text{(IV) If}N\text{be the number of solutions of}& \left(S\right)30\\ \text{the equation}|x-|4-x||-2x=4,\text{then}& \\ \text{the value of}N\text{is}& \end{array}$

Which of the following is the only CORRECT combination?

1. $\left(II\right)\to \left(P\right),\left(Q\right)$
2. $\left(III\right)\to \left(R\right)$
3. $\left(I\right)\to \left(P\right),\left(Q\right),\left(R\right)$
4. $\left(I\right)\to \left(P\right),\left(Q\right),\left(S\right)$

Q. If one root of the quadratic equation $a{x}^2-bx-c=0;a,b,c\in R$ is reciprocal of the other, then which one of the following is correct?

1. $a+b=0$
2. $b=c$
3. $a=c$
4. $a+c=0$

Q. The domain of $f\left(x\right)={\left(\frac{x^4-2x^3+3x^2-2x+2}{2x^2-2x+1}\right)}^{1/2022}$ is

1. $R$
2. $[0,\infty )$
3. $R-\{\frac{1}{2},1\}$
4. $R-\left\{\frac{1}{2}\right\}$

Q. The nature of roots of the quadratic equation $(x-\sin \alpha )(x-\cos \alpha )+\sqrt{3}=0$ are

1. Rational and equal.
2. Real and equal.
3. Real and unequal.
4. Imaginary.

Q. If both the roots of the quadratic equation $a{x}^2+bx+c=0$ are negative, then

1. $a,b,c$ can be any positive rational numbers
2. $a,b,c$ can be any integers
3. $a,b,c$ can be any negative integers
4. $a,b,c$ can be any real numbers

Q. If both the roots of the quadratic equation $a{x}^2+bx+c=0$ are zero, then

1. $b=c=0,a\ne 0$
2. $c=0,a\ne 0$
3. $a=b=c=0$
4. $b=0,a\ne 0$

Q. If $a,c\in N$ and both the roots of the equation $a{x}^2-5x+c=0$ are real, then both roots must be

1. Positive
2. Data in sufficient
3. Opposite in sign
4. Negative

Q. if f(x)=ax^2+bx+c, g(x)= -ax^2 +bx+ c where ac not equal to zero then prove that f(x).g(x)=0 has at least two real root.

Q.

The complete set of values of $k$, for which the quadratic equation $x^2-kx+k+2=0$ has equal roots, consists of

1. $2+\sqrt{2}$

2. $2\overline{+}\sqrt{1}2$

3. $2-\sqrt{1}2$

4. $-2-\sqrt{1}2$

Q. If roots of the equation $a{x}^2+2(a+b)x+(a+2b+c)=0$ are imaginary, then find nature of roots of the equation $a{x}^2+2bx+c=0$.