Quadratic Equation

Q. The set of all values of $k>-1,$ for which the equation $(3x^2+4x+3{)}^2-(k+1\left)\right(3x^2+4x+3\left)\right(3x^2+4x+2)+k(3x^2+4x+2{)}^2=0$ has real roots, is

1. $(\frac{1}{2},\frac{3}{1}]-\left\{1\right\}$
2. $[-\frac{1}{2},1)$
3. $[2,3)$
4. $(1,\frac{5}{2}]$

Q. Is x = 5 a solution of the equation $x^2-2x-15=0$?

Q. The sum of all the solutions of the equation $|505x-1010|+|1515x+505|=2020$, is

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

Q.

Let A = {1, 2, 3}, B = {1, 3, 5}. If relation R from A to B is given by

R = {(1, 3), (2, 5), (3, 3)}. Then, $R^{-1}$ is

1. {(1, 3), (2, 5), (3, 3), }

2. {(1, 3), (5, 2)}

3. None of these

4. {(3, 3), (3, 1), (5, 2), }

Q. Solve the equation $2x^2+x+1=0$

Q.

Using identities, evaluate $297x303$.

Q. If the roots of $(x-\alpha )(x-4+\beta )+(x-2+\alpha )(x+2-\beta )=0$ are $p$ and $q$, then the value of sum of the roots of $2(x-p)(x-q)-(x-\alpha )(x-4+\beta )=0$ and $2(x-p)(x-q)-(x-2+\alpha )(x+2-\beta )=0$ is

Q.

Let , $f\left(x\right)=a{x}^2+bx+c,g\left(x\right)=a{x}^2+px+qwherea,b,c,q,p,ϵRandb\ne p$. If their discriminants are equal and f(x) = g(x) has a root , $\alpha$ then

1. will be G.M of all the roots of f(x) = 0, g(x) = 0

2. will be A.M. of the roots of f(x) = 0, g(x) = 0

3. will be A.M of the roots of f(x) = 0 or g(x) = 0

4. will be G.M of the roots of f(x) = 0 or g(x) = 0

Q. Find which of the following equations are quadratic:

$\left(B\right)5x^2-8x=-3(7-2x)$

Q.

If $co{s}^2\frac{\pi }{8}$ is a root of equation $x^2$ + ax + b = 0 where a, b $ϵ$ Q then a + b = ___

1. $\frac{-7}{8}$

2. $\frac{8}{7}$

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

4. 2

Q. Sum of all real $x$ such that $\frac{4x^2+15x+17}{x^2+4x+12}=\frac{5x^2+16x+18}{2x^2+5x+13}$ is

1. $-\frac{17}{4}$
2. $-\frac{15}{3}$
3. $-\frac{13}{4}$
4. $-\frac{11}{3}$

Q.

If there is a term containing $x^{2r}in{(x+\frac{1}{x^2})}^{n-3}$, then :

1. n-2r is a positive integral multiple of 3

2. n-2r is even

3. n-2r is odd

4. n = 2r

Q. The value of $a$ for which the equation $(a^2-a-2)x^2+(a^2-4)x+(a^2-3a+2)=0$ have more than two roots

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

Q. Which of the following is/are quadratic equation?

1. $(x+1{)}^2=2(x-3)$
2. $(x-2)(x+1)=(x-1)(x+3)$
3. $x^2-2x=(-2)(3-x)$
4. $(x+2)(x-2)=-x^2$

Q. Which of the following option(s) is/are true for $f\left(x\right)=7$?

1. It is a constant polynomial
2. It is a polynomial of degree $0$
3. It is a monomial
4. The number of zeroes of the polynomial is $0$

Q. The number of distinct real roots of the equation $x^5(x^3-x^2-x+1)+x(3x^3-4x^2-2x+4)-1=0$ is

Q. Find which of the following equations are quadratic:

$\left(D\right)$ $x^2+5x-5=(x-3{)}^2$

Q. The value of ${\tan }^{-1}\left(\tan (-\frac{3\pi }{4})\right)+{\cot }^{-1}\left(\cot (-\frac{3\pi }{4})\right)$ is

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

Q.

Which of the following is an identity?

1. ${(x-2)}^2=x^2-4x+4$

2. $x^2-4x+4=0$

3. ${(x+3)}^2=x^2+5x+8$

4. $2x-1=0$

Q.

If $tan{\theta }_1,tan{\theta }_2,tan{\theta }_3$ are the real roots of $x^3-(a+1)x^2+(b-a)x-b=0where{\theta }_1,{\theta }_2,{\theta }_3$ are acute then ${\theta }_1+{\theta }_2+{\theta }_3=$

1. $\frac{5\pi }{6}$

2. $\frac{\pi }{2}$

3. $\frac{6\pi }{5}$

4. $\frac{\pi }{4}$

Q. Solve for $x$ and $y$:
$\frac{x}{a}+\frac{y}{b}=2$; $ax-by=a^2-b^2$

Q. Which of the following is/are quadratic equation?

1. $(x+1{)}^2=2(x-3)$
2. $(x-2)(x+1)=(x-1)(x+3)$
3. $x^2-2x=(-2)(3-x)$
4. $(x+2)(x-2)=-x^2$

Q. Out of the given equations, is not a quadratic equation.

1. $x^2-2=0$
2. ${(x+2)}^2=x^2$
3. $x^2-5x=-6$
4. $5t^2-\sqrt{3}t+7=\sqrt{5}$

Q. Find which of the following equations are quadratic:

$\left(E\right)$ $7x^3-2x^2+10=(2x-5{)}^2$

Q. The number of real roots of the equation, $e^{4x}+e^{3x}-4e^{2x}+e^x+1=0$ is :

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

Q. If $P\left(2\right)=0$ and $P^{\prime }\left(x\right)+20P\left(x\right)<0$ for all $x>0.$Then the number of solution(s) for $P\left(x\right)=1$ for $x>2,$ is

Q. Identify which of the following statement(s) is(are) correct for the equation ${\log }_{(x^2+2x-3)}(4-{\log }_2(|x-1|+|x-3|))=0$

1. The equation has exactly two solution
2. The equation has more than two solutions
3. The sum of all the solutions of the equation is $6$
4. The sum of all the solutions of the equation is $4$

Q. The general values of q for which $\theta =ta{n}^{-1}\left(2ta{n}^2\theta \right)-\frac{1}{2}si{n}^{-1}\left(\frac{3sin2\theta }{5+4cos2\theta }\right)$ holds true,

1. $\theta =n\pi$
2. $\theta =n\pi +\frac{\pi }{4}$
3. $\theta =n\pi +ta{n}^{-1}(-2)$
4. $\theta =n\pi +ta{n}^{-1}\left(2\right)$

Q.

If a and b are the roots of equation $x^2$ + 4x + p = 0 where P= ${\sum }_{r=0}^n$ $n_{c_r}$ $\frac{1+rx}{(1+nx{)}^r}$$(-1{)}^r$
then the value of $|a-b|$ is

1. 2

2. 6

3. None of these

4. 4

Q. The sum of all the solutions of the equation $|505x-1010|+|1515x+505|=2020$, is

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