Roots of Quadratic Equation

Q.

If one zero of $2x^2-3x+k$ is reciprocal to the other, then find the value of $k$.

Q.

Find the zeroes of the quadratic polynomial $2x^2-9-3x$ and verify the relationship between zeroes and the coefficients.

Q.

Find the zeroes of the quadratic polynomial $p\left(x\right)=ab{x}^2+(b^2-ac)x-bc$ and verify the relationship between the zeroes and coefficients.

Q. Solve the following quadratic equation for x:

$4x^2+4bx-(a^2-b^2)=0$

Q. Question 8
Show that $0.142857142857\dots \dots =\frac{1}{7}$

Q.

Find the zeroes of the polynomial $x^2+5x+6$ and verify the relationship between the zeroes and coefficients.

Q. The equation $(Cosp-1)x^2+\left(Cosp\right)x+Sinp=0,$ where x is a variable with real roots. then the interval of p may be any one of the following.

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

Q. If two zeros of the polynomial $f\left(x\right)=x^3-4x^2-3x+12$ are $\sqrt{3}$ and $-\sqrt{3}$, then find its third zero. [2 MARKS]

Q.

Find the zeroes of the quadratic polynomial $2x^2-3+5x$ and verify the relationship between zeroes and the coefficients.

Q. In the question, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice.

Assertion(A) : $x^3$ + $x$ has only one real zero.
Reason(R) : A polynomial of $n^{th}$ degree must have 𝒏 real zeros.

1. Both assertion (A) and reason (R) are true, but reason (R) is not the correct explanation of assertion (A).
   
2. Both assertion (A) and reason (R) are true, and reason (R) is the correct explanation of assertion (A).
3. Assertion (A) is false, but reason (R) is true.
   
4. Assertion (A) is true, but reason (R) is false.
   

Q.

A fast train takes $3$ hours less than a slow train for a journey of $600\mathrm{km}$. If the speed of the slow train is $10\mathrm{km}/\mathrm{hr}$ less than that of the first train, find the speeds of the two trains.

Q.

The number of real solutions of the equation ${\left|x\right|}^2-3\left|x\right|+2=0$ is

1. $4$

2. $1$

3. $3$

4. $2$

Q. Question 10
One of the zeroes of the polynomial is $2x^2+7x–4$ is

A) $2$
B) $\frac{1}{2}$
C) $-\frac{1}{2}$
D) $-2$

Q. If one root of the quadratic equation $2x^2+kx-6=0$ is 2, find the value of k. Also, find the other root.
[2 MARKS]

Q.

Find the value of k, if x - 1 is a factor p(x) in each of the following cases:
(i) $p\left(x\right)=x^2+x+k$
(ii) $p\left(x\right)=2x^2+kx+\sqrt{2}$
(iii) $p\left(x\right)=k{x}^2-\sqrt{2}x+1$
(iv) $p\left(x\right)=k{x}^2-3x+k$

Q.

Find the zeroes of the quadratic polynomial $4x^2-4x-3$ and verify the relation between the zeroes and their coefficients.

Q.

Find the complex cube root of $8$.

Q.

Use Venn diagrams to verify De Morgan’s law of complementation (A$\cup$B)’ = A’ $\cap$ B’

Q.

Factorise the following expression: $7x-42$

Q. 34. What are the conditions for common root or roots of 2 Quadratic equations?

Q. Question 4
Does there exist a quadratic equation whose coefficients are rational but both of its roots are irrational? Justify your answer.

Q.

Louis want to carpet the rectangular floor of his basement.

The basement has an area of $864$ square feet.

The width of the basement is $\frac{2}{3}$ its length.

What is the length of Louiss basement $?$

Q.

Like surds are those surds which have same radicand but different order.

1. zero is a natural number

2. -1 is a whole number

Q.

The number of non-zero integral solutions of the equation ${\left|1-i\right|}^x=2^x$ is

1. Infinite

2. $1$

3. $2$

4. None of these

Q. if the equations x^2+abx+c=0 and x^2+acx+b=0 have a common root the teir other roots satisfy the equati

Q.

Write the discriminant of $4x^2-20x+25=0$ and determine the nature of its root.

Q.

Let $a,b,c$ be real numbers, $a\ne 0$. If $\alpha$ is a root of $a^2{x}^2+bx+c=0$, $\beta$ is a root of $a^2{x}^2-bx-c=0$ and $0<\alpha <\beta$, then the equation $a^2{x}^2+2bx+2c=0$ has a root $\gamma$ that always satisfies.

1. $\gamma =\frac{[\alpha +\beta ]}{2}$

2. $\gamma =\left[\alpha \right]+\left[\frac{\beta }{2}\right]$

3. $\gamma =\left[\alpha \right]$

4. $\alpha <\gamma <\beta$

Q. In the question, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice.

Assertion(A) : $x^3$ + $x$ has only one real zero.
Reason(R) : A polynomial of $n^{th}$ degree must have 𝒏 real zeros.

Q.

Find a quadratic polynomial, the sum and product of whose zeroes are $\frac{5}{6}$ and $\frac{1}{3}$ respectively.

Q. Question 37

The area of a semi-circle of radius 4r is

(a) $8\pi r^2$
(b) $4\pi r^2$
(c) $12\pi r^2$
(d) $2\pi r^2$