Framing Second Degree Equations

Q. Which of the following equations are quadratic equations?

1. $(x^2+1{)}^2=x^4$
2. $(x+1{)}^2=x^2$
3. $(5x^2+2x)-(7+5x^2)=x^2+2$
4. $(x+2{)}^3=4-x^3$

Q. Question 2(ii)
Represent the following situations in the form of quadratic equations.
(ii) The product of two consecutive positive integers is $306$. We need to find the integers.

Q. A mathematician trying to cross a street happened to witness a bank robbery. When the police questioned him, he stated that the number plate of the van in which the thieves escaped had its last four digits as follows:
The first digit is 5. The last digit is the square of the second digit. The third digit is twice the second digit. Also he noticed the sum of digits to be “9”. What is the quadratic equation that the police need to frame to find the $2^{nd}$ digit, given that ‘b’ is the second digit?

1. $b^2+2b-3=0$
2. $4b+5=9$
3. $b^3-9=0$
4. $b^2+3b-4=0$

Q.

What is the coefficient of $z$ in $(z-5{)}^3$?

1. 125

2. -25

3. 5

4. 75

Q. Which of the given equations are quadratic?

1. $-x+x^3+4x^2=0$
2. $2x^2=-1$
3. $-x^2+x+4x^2=-3$
4. $-x+\frac{1}{2}=0$

Q. The cost of $2kg$ of apples and $1kg$ of grapes on a day was found to be $Rs160$. After a month, the cost of $4kg$ of apples and $2kg$ of grapes is $Rs300$. Represent the situation algebraically and geometrically.

Q.

State whether $(2x+1)(3x+2)=6(x-1)(x-2)$ is a quadratic equation.

Q. This section contains 1 Assertion-Reason type question, which has 4 choices (a), (b), (c) and (d) out of which ONLY ONE is correct.

इस खण्ड में 1 कथन-कारण प्रकार का प्रश्न है, जिसमें 4 विकल्प (a), (b), (c) तथा (d) दिये गये हैं, जिनमें से केवल एक सही है।

A : Roots of the equation x² + 4x + 7 = 0 are imaginary.

A : समीकरण x² + 4x + 7 = 0 के मूल काल्पनिक हैं।

R : If the discriminant of a quadratic equation is negative and the coefficients are real, then its roots are imaginary.

R : यदि द्विघात समीकरण का विविक्तकर ऋणात्मक है तथा गुणांक वास्तविक हैं, तब इसके मूल काल्पनिक हैं।

1. Both (A) and (R) are true and (R) is the correct explanation of (A)
   
   (A) तथा (R) दोनों सही हैं तथा (R), (A) का सही स्पष्टीकरण है
2. Both (A) and (R) are true but (R) is not the correct explanation of (A)
   
   (A) तथा (R) दोनों सही हैं लेकिन (R), (A) का सही स्पष्टीकरण नहीं है
3. (A) is true but (R) is false
   
   (A) सही है लेकिन (R) गलत है
4. (A) is false but (R) is true
   
   (A) गलत है लेकिन (R) सही है

Q. Which of the following equations are quadratic equations?

1. $(x^2+1{)}^2=x^4$
2. $(x+1{)}^2=x^2$
3. $(5x^2+2x)-(7+5x^2)=x^2+2$
4. $(x+2{)}^3=4-x^3$

Q. The coefficient of the quadratic term of $(x+2{)}^2=10x^2$ is ___.

1. 8
2. 10
3. 1
4. 9

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

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

Q.

Check whether given equation is a quadratic equation or not : $\frac{x+5}{1}=\frac{2x+10}{x-6}$

1. False

2. True

Q.

The product of two consecutive natural numbers is 72. Frame the quadratic equation.

1. (x + 1)(x+2)

2. x² + x

3. x² -x

4. 2x² +1

Q. Find the value of $a$ for following quadratic equation by comparing with standard form $(a{x}^2+bx+c)$

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

Q.

$x^2+2x+1=(3+x{)}^2+3$ represents a quadratic equation.

1. True

2. False.

Q. The zeros of the equation $x^2+8x+9=0$, can be

1. $x=4\pm \sqrt{7}$
2. $x=-2\pm \sqrt{7}$
3. $x=-4\pm \sqrt{7}$
4. $x=-1\pm \sqrt{7}$

Q. Find the value of $a$ for given quadratic equation by comparing with standard form $(a{x}^2+bx+c)$.

Q. Find the value of $a$ for given quadratic equation by comparing with standard form$(a{x}^2+bx+c)$.

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

Q. If $a{x}^n+b{x}^{n-1}+c=0$ is a quadratic equation, then $n=$.

1. \N
2. 1
3. 2

Q. $4x^2-9x+1=0$

1. $1$
2. $0$
3. $-9$
4. $4$

Q. The tower of a bridge, hung in the form of a parabola, have their tops $30$ metres above the road-way are $200$ metres apart. If the cable is $5$ metres above the road way at the centre of the bridge, then the length of the vertical supporting cable $30$ metres from the centre is

1. $100m$
2. $\frac{29}{4}m$
3. $\frac{9}{4}m$
4. $30m$

Q.

Following equation is a linear equation
$2(x-1{)}^2=4x^2-2x+1$

1. True

2. False

Q. If each root of the equation $x^2+11x+13=0$ is diminished by $4$, then the resulting equation is

1. $x^2+3x+73=0$
2. $x^2-3x-4=0$
3. $x^2+19x+73=0$
4. $x^2+3x-15=0$

Q. $x^2-7x+4=0$

1. $4$
2. $1$
3. $0$
4. $-7$

Q. $x^2+5x-4=0$

Q. The reduced form of $(2x-1{)}^2=x^2+2$ is not a quadratic equation.

1. True
2. False

Q.

Which of the following is not a quadratic equation?

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

2. $x^2+5x=10$

3. $(\sqrt{2}x+\sqrt{3}{)}^2=3x^2-5x$

4. $2x-x^2=x^2+5$

Q.

Quadratic equation is a polynomial of degree:

1. 2

2. 1

3. 3

4. 4

Q. If the solutions of the equation $x^2+3x-18=0$ are -6, 3, then the roots of the equation $2(x^2+3x-18)=0$ are _____.

1. -6, 6
2. -12, 6
3. -6, 3
4. 3, 3