Quadratic Formula for Finding Roots

Q.

Find the value of$k$in $2x^2+kx+3=0$so that they have two equal roots.

Q. Let $\alpha$ and $\beta$ be the roots of the quadratic equation $x^2\sin \theta -x(\sin \theta \cos \theta +1)+\cos \theta =0(0<\theta <{45}^{\circ })$, and $\alpha <\beta$. Then
$\sum _{\infty }^{n=0}({\alpha }^n+\frac{(-1{)}^n}{{\beta }^n})$ is equal to :

1. $\frac{1}{1+\cos \theta }-\frac{1}{1-\sin \theta }$
2. $\frac{1}{1-\cos \theta }-\frac{1}{1+\sin \theta }$
3. $\frac{1}{1+\cos \theta }+\frac{1}{1-\sin \theta }$
4. $\frac{1}{1-\cos \theta }+\frac{1}{1+\sin \theta }$

Q.

Find the values of $k$ for the following quadratic equation, so that it has two equal roots.

$2x^2+kx+3=0$

Q.

Solve the equation $x^2-\sqrt{3}x+1=0$:

1. $x=\frac{(-\sqrt{3}\pm 2i)}{2}$

2. $x=\frac{\left(-\sqrt{3}\pm i\right)}{2}$

3. $x=\left(\sqrt{3}\pm i\right)$

4. $x=\frac{\left(\sqrt{3}\pm i\right)}{2}$

Q. The equation $e^{sinx}-e^{-sinx}-4=0$ has:

1. exactly one real root
2. exactly four real roots
3. infinite number of real roots
4. no real roots

Q.

Using the quadratic formula, solve $9x^2-3(a^2+b^2)x+a^2{b}^2=0$ for $x$.

Q. Sum of the non-real roots of $(x^2+x-2)(x^2+x-3)=12$ is

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

Q.

Solve the following quadratics $21x^2-28x+10=0$

Q. The set of values of $a$ for which the function $f\left(x\right)=\frac{a{x}^3}{3}+(a+2)x^2+(a-1)x+2$ possesses a negative point of inflection is

1. $(-\infty ,-2)\cup (0,\infty )$
2. $\{-4,5\}$
3. $(-2,0)$
4. $\varphi$

Q.

$x^2+x+\frac{1}{\sqrt{2}}=0$

Q. The sum of the real roots of the equation $(7+4\sqrt{3}{)}^{x^2-8}+(7-4\sqrt{3}{)}^{x^2-8}=14$ is

Q. Let $\alpha$ and $\beta$ be the roots of the equation $x^2+x+1=0$. The equation whose roots are ${\alpha }^{19},{\beta }^7$ is

1. x² - x - 1 = 0
2. x² + x - 1 = 0
3. x² + x + 1 = 0
4. x² - x + 1 = 0

Q. The equation $x^2$ – 2 = [sin x], where [·] denotes the greatest integer function, has

1. exactly two roots

2. infinitely many roots
3. only one root which is an integer
4. only one root which is irrational

Q.

If $\alpha ,\beta$ are the roots of the equation a$x^2$ +2bx +c = 0 and $\alpha +\delta$, $\beta +\delta$ are the roots of A$x^2$ + 2Bx + C= 0 , then $\frac{b^2–ac}{B^2–AC}$ =

1. a / A

2. A / a

3. (a / A)²

4. (A / a)²

Q. The number of quadratic equation(s), with real roots which remain unchanged even after squaring its roots, is

1. $1$
2. $0$
3. $2$
4. $3$

Q.

Integrate the following functions.
$\int \frac{5x+3}{x^2+4x+10}dx.$

Q.

Integrate the following functions.
$\int \frac{6x+7}{\sqrt{(x-5)(x-4)}}dx$

Q.

Solve the following quadratics $21x^2+9x+1=0$

Q. Solve the following quadratic equation:
(ii) $x^2-(5-i)x+(18+i)=0$

Q. Solve the quadratic: $\sqrt{5}{x}^2+x+\sqrt{5}=0$

Q. A strictly binary tree with n leaves always contains ................nodes.

1. 2n - 1
2. n*n
3. $n^2$ - 1
4. 2n

Q. The degree of the differential equation ${\left(\frac{d^{2}y}{d{x}^2}\right)}^3+{\left(\frac{dy}{dx}\right)}^2+\sin \left(\frac{dy}{dx}\right)+1=0$
A. $3$
B. $2$
C. $1$
D. Not defined

Q. In $\Delta ABC$, if $a,b$ and $A$ are given and there are two possibilities for the third side $c_1$ and $c_2,$ then the value of $(c_1-c_2{)}^2+(c_1+c_2{)}^2{\tan }^{2}A$ is equal to

1. $4a^2$
2. $4b^2$
3. $a^2$
4. $b^2$

Q.

Find the roots of the equation $\frac{x^2-1}{x}=3$$,x\ne 0$.

Q.

If the function $f:(1,\infty )\to (1,\infty )$ is defined by $f\left(x\right)=2^{x(x-1)}$ , then $f^{-1}\left(x\right)$ is

1. 

2. 

3. 

4. Not defined

Q. Solve the following quadratic equation:
(i) $x^2-(3\sqrt{2}+2i)x+6\sqrt{2i}=0$

Q. Let $f:R\to R$ be the function defined by $f\left(x\right)=\sin (3x+2)\forall xϵR$. Then $f$ is invertible.

Q.

If $\alpha and\beta$ are the roots of the equation $x^2-x+1=0then{\alpha }^{2009}+{\beta }^{2009}=$

1. 2

2. -1

3. -2

4. 1

Q. Solve the following quadratic equation:
(iv) $x^2-(2+i)x-(1-7i)=0$

Q. The value of $2+\frac{1}{2+\frac{1}{2+.....\infty }}$ is

1. 
2. 
3. None of these
4.