Quadratic Equations

Q.

The number of quadratic equations which remain unchanged by squaring their roots, is:

1. 2

2. 4

3. infinitely many

4. 0

Q. If p and q are the roots of $x^2+px+q=0$, then

1. p = 1 or 0
2. p = - 2
3. p =1
4. p = -2 or 0

Q.

$\frac{1}{\left(\sqrt{9}-\sqrt{8}\right)}-\frac{1}{\left(\sqrt{8}-\sqrt{7}\right)}+\frac{1}{\left(\sqrt{7}-\sqrt{6}\right)}-\frac{1}{\left(\sqrt{6}-\sqrt{5}\right)}+\frac{1}{\left(\sqrt{5}-\sqrt{4}\right)}$ is equal to:

1. 0

2. 1/3

3. 1

4. 5

5. 4

Q.

The geometric mean of two numbers is $6$ and their arithmetic mean is $6.5$ the numbers are

1. $(3,12)$

2. $(4,9)$

3. $(2,18)$

4. $(7,6)$

Q.

If the equation $x^4-4x^3+a{x}^2+bx+1=0$ has four positive roots, then the value of $|a-b|$is

1. $2$

2. $1$

3. $10$

4. $3$

Q.

If $x=\frac{\sqrt{3}+1}{\sqrt{3}-1}andy=\frac{\sqrt{3}-1}{\sqrt{3}+1},thenthevalueof\left(x^2+y^2\right)is:$

1. 15

2. 16

3. 13

4. 14

Q. If the roots of the equation $x^2-2ax+a^2+a-3=0$ are real and less than 3, then:

1. $a<2$
2. $2\le a\le 3$
3. $3<a\le 4$
4. $a>4$

Q.

Let$\alpha ,\beta$ be the roots of $a{x}^2+bx+c=0$ and ${\alpha }_1,-\beta$be the roots of $a_1{x}^2+b_{1}x+c_1=0$ then $\alpha ,{\alpha }_1$ are the roots of

1. $\frac{x^2}{\left[\left(\frac{b}{a}\right)+\left(\frac{b_1}{a_1}\right)\right]}+x+\frac{1}{\left[\left(\frac{b}{c}\right)+\left(\frac{b_1}{c_1}\right)\right]}=0$

2. $\frac{x^2}{\left[\left(\frac{b}{a}\right)+\left(\frac{b_1}{a_1}\right)\right]}+\frac{x}{\left[\left(\frac{b}{c}\right)+\left(\frac{b_1}{c_1}\right)\right]}+1=0$

3. $\frac{x^2}{\left[\left(\frac{b}{c}\right)+\left(\frac{b_1}{c_1}\right)\right]}+x+\frac{1}{\left[\left(\frac{b}{a}\right)+\left(\frac{b_1}{a_1}\right)\right]}=0$

4. $\frac{x^2}{\left[\left(\frac{b}{c}\right)+\left(\frac{b_1}{c_1}\right)\right]}+\frac{1}{\left[\left(\frac{b}{a}\right)+\left(\frac{b_1}{a_1}\right)\right]}+1=0$

Q. If $x=\sqrt{\frac{\sqrt{5}+1}{\sqrt{5}-1}}$, value of $x^2+x-1$ is

1. $\sqrt{5}$
2. $\sqrt{5}+1$
3. $-\sqrt{5}$
4. $2$

Q.

Which one is the correct square root of $56$.

1. $2\sqrt{14}$

2. $2\sqrt{7}$

3. none of these

4. $3\sqrt{14}$

Q. The least possible integral value of a for which the equation $x^2-2(a-1)x+(2a+1)=0$ has both the roots positive, is:

1. 6
2. 3
3. 4
4. 1

Q. If $x^2-ax-21=0$ and $x^2-3ax+35=0;a>0$ have a common root, then a is equal to:

1. 1
2. 2
3. 4
4. 5

Q.

The sum of $\frac{1}{\sqrt{2}+1}=\frac{1}{\sqrt{3}+\sqrt{2}}+\frac{1}{\sqrt{4}+\sqrt{3}}+.....+\frac{1}{\sqrt{100}+\sqrt{99}}$ is equal to

1. 11

2. 1

3. 10

4. 9

5. None of these

Q.

If two positive numbers differ by 5. If their product is 336, then the sum of the two numbers is:

1. 21

2. 37

3. 51

4. 19

5. 28

Q. The sum of all the real roots of the equation $|x-2{|}^2+|x-2|-2=0$ is:

1. 2
2. 3
3. 4
4. None of these

Q. $6(x^2+\frac{1}{x^2})-25(x-\frac{1}{x})+12=0$

1. , , 2, 3
2. , , -2, -3
3. , , -2, 3
4. , 2, -3
5. , , 2, 3

Q.

The minimum possible value of the sum of squares of the roots of the equation $x^2+(a+3)x-(a+5)=0$ is

1. 1

2. 2

3. 3

4. 4

Q. Equation $a{x}^2+2x+1$ has one double root if:

1. a = 1
2. a = 2
3. a = -1
4. a = 0

Q.

$Given,\sqrt{2}=\mathrm{1.414.}Then,thevalueof\sqrt{8}+2\sqrt{32}-3\sqrt{128}+4\sqrt{50}is$

1. 8.484

2. 8.426

3. 8.876

4. 8.7865

5. 8.526

Q.

If the arithmetic mean of the roots of a quadratic equation is $8$ and the geometric mean is $5$, then the equation is

1. $x^2-16x-25=0$

2. $x^2+16x-25=0$

3. $x^2-16x+25=0$

4. $x^2-8x-5=0$

Q. The equations ax + b = 0 and cx + d = 0 are consistent, if:

1. ad = bc
2. ad + bc = 0
3. ab - cd = 0
4. ab + cd = 0
5. None of these

Q. If the equation $(3x{)}^2+(27\times 3^{\frac{1}{k}}-15)x+4=0$ has equal roots, then k =

1. 0
2. $\frac{1}{2}$
3. $-2$
4. $-\frac{1}{2}$

Q. Study the following graph carefully and answer the questions that follow.


If the water level of river A in July is decreased by 20%, then what will be water level of river A in July?

1. None of these
2. 152 m
3. 156 m
4. 164 m
5. 162 m

Q. If $b^2>4ac,$ then $a(x^2+4x+4{)}^2+b(x^2+4x+4)+c=0$ has distinct real roots if ?

1. b < a < 0 < c
2. a < b < 0 < c
3. c < 0 < b < a
4. b < 0 < a < c

Q. The number of real solutions of the equation $2|x{|}^2-5|x|+2=0$ is:

1. 0
2. 4
3. 2
4. none of these

Q. The quadratic equation in x such that the arithmetic mean of the roots is p and the geometric mean is q is

1. none of these

2. 

3. 

4. 

Q.

The equation $\sqrt{\left({\left(x-2\right)}^2+y^2\right)}+\sqrt{\left({\left(x+2\right)}^2+y^2\right)}=4$ represents a/an

1. circle

2. straight lines

3. parabola

4. ellipse

Q.

If $\alpha$and $\beta$are the roots of the equation $a{x}^2+bx+c=0(c\ne 0)$, then the equation, whose roots are$1/[a\alpha +b]$and $1/[a\beta +b],$ is

1. $ac{x}^2–bx+1=0$

2. $x^2–acx+bc+1=0$

3. $ac{x}^2+bx–1=0$

4. $x^2+acx+11=0$

Q.

$Findthevalueofx:\sqrt{1+x\sqrt{x^2+24}}=x+1$

1. 1

2. 3

3. 4

4. 5

5. 2

Q. The set of values of p for which the roots of the equation $3x^2+2x+p(p-1)=0$ are of opposite sign is

1. $(-\infty ,0)$
2. $(0,1)$
3. $(1,\infty )$
4. $(0,\infty )$