Quadratic Equations

Q.

For each of the following, find a quadratic polynomial whose sum and product respectively of the zeroes are as given. Also find the zeroes of these polynomials by factorization

$\left(\frac{21}{8}\right),\left(\frac{5}{16}\right)$

Q. If $a{x}^2+bx+c=0$ is a quadratic equation, then “a” cannot be:

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

Q. Solve the following quadratic equations by factorization:

$\frac{16}{x}-1=\frac{15}{x+1};x\ne 0,-1$

Q.

Find a quadratic polynomial whose zeroes are $2$ and$-6$.

Q.

Find a quadratic polynomial whose zeroes are $-3$ and $4$.

Q. Write the number of real roots of the equation x² + 3 |x| + 2 = 0.

Q.

Find a quadratic polynomial, the sum of whose zeroes is $4$ and one zero is $5$.

Q. A car covers a distance of 400 km at a certain speed. Had the speed been 12 km/h more, the time taken for the journey would have been 1 hour 40 minutes less. Find the original speed of the car.
[4 MARKS]

Q.

Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively $4,1$.

Q. If the quadratic equation $p{x}^2-2\sqrt{5}px+15=0$, has two roots then find the value of p.

Q. Question 6
A cottage industry produces a certain number of pottery articles in a day. It was observed on a particular day that the cost of production of each article (in rupees) was 3 more than twice the number of articles produced on that day. If the total cost of production on that day was Rs 90, find the number of articles produced and the cost of each article.

Q. Solve the following quadratic equations by factorization:

$\frac{x+1}{x-1}+\frac{x-2}{x+2}=4-\frac{2x+3}{x-2};x\ne 1,-2,2$

Q.

State whether ${(x-2)}^2+1=2x-3$ is a quadratic equation.

Q.

Which of the following are quadratic equations?
$\left(i\right)x^2+6x-4=0$
($ii)\sqrt{3x^2}-2x+\frac{1}{2}=0$
$\left(iii\right)x^2+\frac{1}{x^2}=5$
$\left(iv\right)x-\frac{3}{x}=x^2$
$\left(v\right)2x^2-\sqrt{3x}+9=0$
$\left(vi\right)x^2-2x-\sqrt{x}-5=0$
$\left(vii\right)3x^2-5x+9=x^2-7x+3$
$\left(viii\right)x+\frac{1}{x}=1$
$\left(ix\right)x^2-3x=0$
$\left(x\right){(x+\frac{1}{x})}^2=3(x+\frac{1}{x})+4$
$\left(xi\right)(2x+1)(3x+2)=6(x-1)(x-2)$
$\left(xii\right)x+\frac{1}{x}=x^2,x\ne 0$
$\left(xiii\right)16x^2-3=(2x+5)(5x-3)$
$\left(xiv\right)(x+2{)}^3=x^3-4$
$\left(xv\right)x(x+1)+8=(x÷2)(x-2)$

Q.

If one zero of the quadratic polynomial $p\left(x\right)=4x^2-8xk-9$ is negative of the other, find the value of $k$.

Q. Question 5
If the temperature of a liquid can be measured in Kelvin units as $x^{\circ }$ K or in Fahrenheit units as $y^{\circ }$ F, the relation between the two systems of measurement of temperature is given by the linear equation.
$y=\frac{9}{5}(x-273)+32$

(i) Find the temperature of the liquid in Fahrenheit, if the temperature of the liquid is 313 k.

(ii) If the temperature is ${158}^{\circ }$ F, then find the temperature in Kelvin.

Q.

If the difference of two number is $8$ and the difference of their square is $160$. What are the numbers?

Q. Find the quadratic polynomial, the sum of whose root is $\sqrt{2}$ and their product is $\frac{1}{3}$

Q.

What should be subtracted from $-\frac{5}{3}$ to get $\frac{5}{6}$?

Q. Solve the following inequality:
$\frac{x-1}{x}-\frac{x+1}{x-1}<2$

1. (0, 0) ∪ (½, 1)
2. (-1, 0) ∪ (½, 2)
3. (-2, 0) ∪ (½, 1)
4. (-1, 0) ∪ (½, 1)

Q. John and Jivanti together have 45 marbles. Both of them lost 5 marbles each, and the product of the number of marbles they now have is 128. Form the quadratic equation to find how many marbles they had to start with, if John and x marbles.

Q. Two pipes can together fill a tank in 40/13 minutes. If one pipe take 3 minutes more than the other to fill it, find the time in which each pipe can fill the tank.

Q.

By what number should $\frac{-44}{9}$ be divided to get $\frac{-11}{3}$

Q. Solve the following inequality :
$\frac{x-1}{x^2-4x+3}<1$

1. (1, 3) ∪ (5, ∞)
2. (1, 3) ∪ (6, ∞)
3. (1, 3) ∪ (4, ∞)
4. (1, 2) ∪ (4, ∞)

Q.

State whether $x(x+1)+8=(x+2)(x-2)$ is a quadratic equation.

Q. What is the minimum number of terms and maximum number of terms can a quadratic equation have?
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Q.

If $(2,0)$ is a solution of linear equation $2\mathrm{x}+3\mathrm{y}=\mathrm{k}$ then the value of $\mathrm{k}$ is

1. $4$

2. $5$

3. $2$

4. $1$

Q. Question 1(vi)
Check whether the following are quadratic equations:
(vi) $x^2+3x+1=(x-2{)}^2$

Q.

Two pipes running together can fill a tank in 100\9 minutes, if one pipe takes 5 minutes more than other to fill the tank find the time in which each pipe would fill the tank.

Q.

A shopkeeper buys some books to Rs.80. If he had bought 4 more books for the same amount each book would have cost Re.1 less. Find the number of books he bought