Discriminant

Q. A quadratic equation with rational coefficients if one of its roots is ${\cot }^2{18}^{\circ }$ is

1. $x^2+10x-5=0$
2. $x^2+10x+5=0$
3. $x^2-10x-5=0$
4. $x^2-10x+5=0$

Q. The least integral value of k such that $(k-2)x^2+8x+k+4$ is positive for all real values of x is

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

Q. If $a,b,c,d\in R,$ then the equation $(x^2+ax-3b)(x^2-cx+b)(x^2-dx+2b)=0$ has

1. no real roots
2. at least four real roots
3. at least two real roots
4. exactly six real roots

Q. Let p, q $\in$ {1, 2, 3, 4}. The number of equation of the form $p{x}^2+qx+1=0$ having real roots is

1. 15
2. 8
3. 9
4. 7

Q.

The value of $k$ for which the quadratic equation $k{x}^2+1=kx+3x-11x^2$ has real and equal roots are

1. $\{-11,-3\}$

2. $\{5,7\}$

3. $\{5,-7\}$

4. $\{-5,-7\}$

Q.

The domain of $f\left(x\right)=co{t}^{-1}\left[\frac{x}{\sqrt{x^2-\left[x^2\right]}}\right]$, $x\in R$ is

1. $R$

2. $R-\left[0\right]$

3. $R-\left[\pm \sqrt{n},n\ge 0,\ne Z\right]$

4. None of these

Q. Total number of values of $a$ so that $x^2-x-a=0$ has integral roots, where $a\in N$ and $6\le a\le 100$, is equal to

Q.

Write the discriminant of the equation $2x^2-5x-4=0$ and determine the nature of the roots.

Q. If the roots of the quadratic equation $a{x}^2+bx+c=0$ are opposite in sign and positive root is greater in magnitude, then

1. $a>0,b>0\text{and}c<0$
2. $a<0,b<0\text{and}c>0$
3. $b>0,c>0\text{and}a<0$
4. $b<0,c<0\text{and}a>0$

Q. If the roots of the equation $\frac{x^2+4x}{8x+6}=\frac{k-1}{k+1}$ are equal in magnitude and opposite in sign, then the value of $k$ is

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

Q.

If $\sin 2x=4\cos x$, then $x$ is equal to

1. $\left(\frac{n\mathrm{\pi }}{2}\right)\pm \frac{\mathrm{\pi }}{4};n\in Z$

2. no value

3. $n\mathrm{\pi }+{\left(-1\right)}^{\mathrm{n}}\frac{\mathrm{\pi }}{2};\mathrm{n}\in \mathrm{Z}$

4. $2n\mathrm{\pi }+\frac{\mathrm{\pi }}{2};\mathrm{n}\in \mathrm{Z}$

Q. If $x^2+b_{1}x+c_1=0$ and $x^2+b_{2}x+c_2=0$ are the two quadratic equations such that $b_1{b}_2=2(c_1+c_2)$ and $b_1,b_2,c_1,c_2\in R,$ then which of the following is always true ?

1. at least one equation has imaginary roots
2. at least one equation has real roots
3. exactly one equation has real roots
4. both the equations have real roots

Q. If $\alpha ,\beta$ are the roots of the equation $2x^2-35x+2=0$, then the value of $(2\alpha -35{)}^2(2\beta -35{)}^2$ is

1. $16$
2. $8$
3. $64$
4. None of these

Q.

If x is real, then the maximum and minimum values of expression $\frac{x^2+14x+9}{x^2+2x+3}$will be

1. 4, - 5

2. 5, - 4

3. - 4, 5

4. - 4, - 5

Q.

Let $x_1,x_2$ be the roots of $a{x}^2+bx+c=0$ and $x_1.x_2<0$, $x_1+x_2$ is non - zero. Roots of $x_1(x-x_2{)}^2+x_2(x-x_1{)}^2=0$ are

1. Negative

2. Real and of opposite signs

3. positive

4. non real

Q.

If a, b, c are distinct positive real numbers such that b(a + c) = 2ac then the roots of $a{x}^2+bx+c=0$ are

1. real and equal

2. real and distinct

3. Imaginary

4. data insufficient

Q. The equation $(cosp-1)x^2+cospx+sinp=0$ in x has real roots. Then the set of values of p is

1. $[0,2\pi ]$
2. $[-\pi ,0]$
3. $[-\frac{\pi }{2},\frac{\pi }{2}]$
4. $[0,\pi ]$

Q.

If both the roots of the quadratic equation $x^2$-2kx+$k^2$+k-5=0 are less than 5, then k lies in the interval.

1. (-∞, 4)

2. [4, 5]

3. [5, 6]

4. (6, ∞)

Q. If x is real, the expression $\frac{x+2}{2x^2+3x+6}$ takes all value in the interval

1. $(-\frac{1}{3},\frac{1}{‘3})$
2. $(\frac{1}{13},\frac{1}{3})$
3. $[-\frac{1}{13},\frac{1}{3}]$
4. None of these

Q. The discriminant of the quadratic equation $3x^2-4\sqrt{3}x+4=0$ is

1. $D=4$
2. $D=-2$
3. $D=0$
4. $D=96$

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

Q. If $a:b=1:5,$ then the roots of the equation $a{x}^2-bx+4a=0$ is/are

1. $1$
2. $4$
3. $5$
4. $-1$

Q. If both the roots of the quadratic equation $x^2-2kx+k^2+k-5=0$ are less than 5, then k lies in the interval

1. $(6,\infty )$
2. (5, 6]
3. $(-\infty ,4)$
4. $[4,5]$

Q. If $\alpha ,\beta$ are the roots of the equation $2x^2-35x+2=0$, then the value of $(2\alpha -35{)}^2(2\beta -35{)}^2$ is

1. $16$
2. $8$
3. $64$
4. None of these

Q. Let $a\in R$. For possible real values of $x$, $\frac{x^2+ax+3}{x^2+x+a}$ takes all real values, then

1. $4a^3+39<0$
2. $4a^3+39\ge 0$
3. $a\ge \frac{1}{4}$
4. $a<\frac{1}{4}$

Q. If the roots of the equation $x^2-2ax+a^2+a-3=0$ are real and less than 3, then

1. $a<2$
2. $a\le a\le 3$
3. $3<a\le 4$
4. $a>2$

Q. If the roots of the equation $x^2-2ax+a^2+a-3=0$ are real and less than 3, then

1. $a<2$
2. $a\le a\le 3$
3. $3<a\le 4$
4. $a>2$

Q.

The correct statement about the roots of the equation $x^2-4\sqrt{2}+8=0$

1. Real and unequal

2. Unequal and rational

3. Real and equal

4. Imaginary

Q. Which of the following is/are true for $f\left(x\right)=\frac{1-x+x^2}{1+x+x^2}(x\in R)$

1. Minimum value of $f\left(x\right)$ is $1$
2. Maximum value of $f\left(x\right)$ is $3$
3. $x=1$ is point of minima
4. $x=-1$ is point of maxima

Q. Let p, q $\in$ {1, 2, 3, 4}. The number of equation of the form $p{x}^2+qx+1=0$ having real roots is

1. 15
2. 9
3. 7
4. 8