Standard Form of a Quadratic Equation

Q. If $a{x}^2+bx+c=0$ is a quadratic equation, then which of the following conditions is always correct?

1. $a,b,c$ are integers
2. $a=0$
3. $a\ne 0$ and $a,b,c$ are real numbers
4. $b,c\ne 0$

Q. A quadratic polynomial may not have zeroes.

1. True
2. False

Q. If discriminant of a quadratic equation is less than zero, then the equation has

1. real and distinct roots
2. real and equal roots
3. no real roots
4. None of the above

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

Assertion: The value of 𝒙 are $\frac{-a}{2}$, 𝒂 for a quadratic equation $2x^2+ax-a^2=0$.

Reason: For quadratic equation $a{x}^2+bx+c=0$, $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

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

Q.

Check Whether The Following Are Quadratic Equations$(2\mathrm{X}-1)(\mathrm{X}-3)=(\mathrm{X}+5)(\mathrm{X}-1)$.

Q. What is the degree of a quadratic polynomial?

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

Q.

What is the degree of a quadratic equation?

Q.

What are quadratic equation? Give example

Q.

Kumar and Kavya together have 45 marbles. Both of them lost 5 marbles each, and the product of the number of marbles they now have is 124. Representing the problem mathematically gives the quadratic equation _____.

1. x² – 5x + 32 = 0

2. x² – 40x + 124 = 0

3. x² – 45x + 324 = 0

4. x² – 4x+ 32 = 0

Q.

A rectangular field has an area of $3sq.units$. The length is one unit more than twice the breadth. Frame an equation to represent this. [Assume breadth is $x$].

1. $2x^2+x-3=0$

2. $2x^2+x-6=0$

3. $x^2–2x+3=0$

4. $x^2–2x+6=0$

Q. A real number $\alpha$ is said to be the root of the quadratic equation $a{x}^2+bx+c=0$, if $a{\alpha }^2+b\alpha +c=0$

1. True
2. False

Q. Match the value of the expression $x^{99}+x^{100}+x$ for the given values of $x$.

1. 3
2. -1
3. \N

Q. Question 2
Which of the following is not a quadratic equation?
(a) $2(x-1{)}^2=4x^2-2x+1$
(b) $2x-x^2=x^2+5$
(c) ${(\sqrt{2}x+\sqrt{3})}^3=3x^2-5x$​​​​​​​
(d) $(x^2+2x{)}^2=x^4+3+4x^2$​​​​​​​

Q.

Which of the following is the standard form of a quadratic equation?

1. $a{x}^2+c=0$; $a\ne 0$; a $\&$ c are real

2. $a{x}^2+bx+c=0;a\ge 0$; a, b, c are real

3. $a{x}^2+bx+c=0$; a $\ne$ 0; a, b, c are real

4. $a{x}^2+bx+c=0;a\le 0$; a, b, c are real.

Q. Question 1(iii)
Check whether the following are quadratic equations:
$\left(iii\right)(x-2)(x+1)=(x-1)(x+3)$

Q.

Which of the following is not a quadratic equation?

1. $0x^2$ – $2x=x^2+3x+5$

2. $2(x+1)+8=(x+2)(x-2)$

3. $x^2$ + $4x+4=x^2–4$

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

Q. The coefficient of $x$ in the quadratic equation $a{x}^2+bx+c=0$ is called the linear coefficient.

1. True
2. False

Q. If $a{x}^2+bx+c=0$ is a quadratic equation, then which of the following conditions is always correct?

1. $a,b,c$ are integers
2. $a=0$
3. $a\ne 0$ and $a,b,c$ are real numbers
4. $b,c\ne 0$

Q.

The cost of production of each toy (in rupees) was found to be 55 minus the number of toys produced in a day. Let’s say $x$ toys were produced on a day, what is the total cost of production in terms of $x$?

1. (x-55)(x+55)

2. x² - 55x

3. x² + 55x

4. 55x – x²

Q. If discriminant of a quadratic equation is equal to zero, then the equation has

1. real and equal roots
2. no real roots
3. real and distinct roots
4. None of the above

Q. What are the condition(s) for $(a^2-4)x^{d-2}+bx+c=0$
to be a quadratic equation in $x$?

1. a can be ANY real number other than 2, -2
2. a, b, c are ALL real numbers where a cannot be 4 or -4
3. d = 4
4. a, b, c are real

Q.

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

1. 0

2. -1

3. $\pi$

4. 1

Q.

The expression $x^4+8x^2+16$ can be factorized as:

1. $(x^2+x+16{)}^2$

2. $(x^2+x+1{)}^2$

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

4. $(x^2+x-4)(x^2-x+4)$

Q.

A cottage industry produces a certain number of toys in a day. The cost of production of each toy (in rupees) was found to be 55 minus the number of toys produced in a day. On a particular day, the total cost of production was Rs 750. The mathematical representation to find out the number of toys produced on that day is:

1. $x^2$ – $50x+320=0$

2. $x^2$ – $40x+124=0$

3. $x^2$ – $45x+324=0$

4. $x^2$– $55x+750=0$

Q. The coefficient of $x$ in the quadratic equation $a{x}^2+bx+c=0$ is called the linear coefficient.

1. True
2. False

Q.

Which of the following is the reduced form of the equation $(x-2{)}^3$ = $(2x-1{)}^2+x^3$ ?

1. 

2. 

3. 

4. 

Q.

Check whether the following is a quadratic equation: $(x+2{)}^3=2x(x^2-1)$

Q. If $a{x}^n+b{x}^{n-1}+c=0$ is a quadratic equation, then which of the given options is not correct?

1. a, b and c are real numbers
2. n = 2
3. Coefficient 'a' must always be 1
4. a$\ne$0

Q. If $a{x}^2+bx+c=0$ is a quadratic equation, then which of the following conditions is always correct?

1. $a,b,c$ are integers
2. $a=0$
3. $a\ne 0$ and $a,b,c$ are real numbers
4. $b,c\ne 0$

Q.

Add

$8x^2+5xy-3y^2,-5x^2+3xy+5xyand-3x^2+2xy+4y^2$