Nature of Roots

Q.

Rohan's mother is 26 years older than him. The product of their ages (in years) 3 years from now will be 360. Find Rohan's present age.

1. 7 years

2. 9 years.

3. 4 years

4. 1 year

Q.

If one root of the polynomial p(y)=5y² +13y + m is reciprocal, find the value of m.

Q.

If the roots of the equation $(a^2+b^2)x^2-2(ac+bd)x+(c^2+d^2)=0$ are equal, prove that $\frac{a}{b}=\frac{c}{d}$

Q.

If the zeroes of the quadratic polynomial ${\mathrm{x}}^2+(\mathrm{a}+1)\mathrm{x}+\mathrm{b}$ are $2$ and $-3$, then

1. $\mathrm{a}=-7$, $\mathrm{b}=-1$

2. $\mathrm{a}=5$, $\mathrm{b}=-1$

3. $\mathrm{a}=2$, $\mathrm{b}=-6$

4. $\mathrm{a}=0$, $\mathrm{b}=-6$

Q.

If-5 is a root of the quadratic equation $2x^2+px-15=0$ and the quadratic equation $p(x^2+x)+k=0$ has equal roots, find the value of k.

Q. The roots of the equation ax2 + bx + c =0 will be reciprocal of each other if
(a) a=b
(b) c=a
(c) b=c
(d) b=0

Q. Which of the following is not the graph of a quadratic polynomial?

1. 
2. 
   
3. 
4. 

Q.

The positive value of K for which the equation x² + kx + 64 = 0 has equal roots is :

1. 16

2. -16

3. -8

4. 8

Q.

find the value of k for which the following equation has equal roots (k-12 )x^2+2(k-12)x+2=0

Q.

For what value of $k,(4-k)x^2+(2k+4)x+(8k+1)=0$, is a perfect square.

Q.

If-5 is a root of the quadratic equation 2x2+px−15=0 and the quadratic equation p(x2+x)+k=0 has equal roots, find the value of k

Q.

The number of real roots of the equation, $e^{4x}+e^{3x}-4e^{2x}+e^x+1=0$ is

1. $3$

2. $4$

3. $1$

4. $2$

Q.

Find the values of p for which the quadratic equation $(2p+1)x^2-(7p+2)x+(7p-3)=0$ has real and equal roots.

Q. Find a quadratic equation whose one of the root is 2 and whose sum of the roots of zero

Q. if the roots of the equation a(b-c)x^2+b(c-a)x+c(a-b)=0 are equal and a, b, c>0, then prove that 2/b=1/a+1/c, i.e., a, b, c are in H.P.

Q.

If x= -2 is a root of equation 3x ² + 7x + p = 0, then find the value of k so that the roots of the equation x² + k(4x + k - 1) + p=0 are equal.

Q.

The quadratic equation whose roots are real and equal is:

1. 3x² - 5x + 2 = 0

2. 2x² - 4x + 3 = 0

3. x² - 4x + 4 = 0

4. x² - 2√2 - 6 = 0

Q. If the equation $5x^2–4x+p=0$ has no real roots, then values of $p$ cannot be equal to

1. $\frac{1}{2}$
   
2. $\frac{5}{6}$
   
3. $1$
4. $\frac{6}{5}$
   

Q.

Find the value of k for which each of the following quadratic equation has equal roots.

(k-4)x^{​​​​​​2} +2(k-4)x +4=0.

Q.

The equation $x^2+x-5=0$ has two distinct real roots.

1. True

2. False

Q. The quadratic equation $n^2–an–5=0$ has (where a is a real number)

1. No real root
2. Two distinct real roots
3. Two equal roots
4. Three real roots

Q.

It the roots of the quadratic equation x²+2px+mn=o are the real and equal

Q. If one root the equation 2x² + kx + 4 = 0 is 2, then the other root is

(a) 6
(b) −6
(c) −1
(d) 1

Q.

If p, q are real and $p\ne q,$ then show that the roots of the equation $(p-q)x^2+5(p+q)x-2(p-q)=0$ are real and unequal.

Q. If the equation ax² + 2x + a = 0 has two distinct roots, if

(a) a = ±1
(b) a = 0
(c) a = 0, 1
(d) a = −1, 0

Q.

Find the roots of the Equation $x^2-5x+6=0$ using the quadratic formula.

Q.

Find the value of $k$ if the equation $k^2{x}^2-2\left(2k-1\right)x+4=0$ has equal roots.

Q.

Solve by factorising $\frac{1}{(x+4)}-\frac{1}{(x-7)}=\frac{11}{30}$ where $x$ is not equal to $-4,7$.

Q. 8. Ifx^2+k(4x+k-1)+2=0has equal roots , then k=

Q. Find the value of p for which the roots of the equation px (x − 2) + 6 = 0 are equal.