Nature of Roots

Q. The roots of the equation px^2-2(p+2)x+3p=0 are alpha and beta .If alpha- beta=2, calculate the value of Alpha, beta and p

Q. If the roots of the equation (a²+b²)x² -2b(a+c)x + (b²+c²) = 0 are equal, then A. 2b = a+c B. b² = ac C. b = 2ac/ a+c D. b = ac

Q. 64.Product of two roots x4 - 11x3 + kx2 + 269x - 2001 is -69 , then find the value of k

Q. the value of a for which the equation (1-a^2)x^2 + 2ax -1 =0 has roots belonging to (0, 1) is

Q. If roots of quadratic equation x^2+ax+b+1=0 are positive integers then a^2+ b^2 can be equal to A. 170 B. 37 C. 61 D. 19

Q.

When the square of a number is decreased by 15, it is equal to twice the original number. Find the possible values of the number.

1. 8, 7

2. 5, -3

3. 9, 6

4. 2, -7

Q.

Question 2 (iv)
Write whether the given statement is true or false. Justify your answer.
Every quadratic equation has at most two roots.

Q. 48. If (a, 0) is a point on a diameter of the circle x²+y²=4 then x²-4x-a²=0 must have : A. Exactly one real root in [-9/10, 1/10] B. Exactly one real root in [4, 49/10] C. Exactly one real root in [0, 2] D. Two distinct real roots in [-1, 5]

Q. The quadratic equation 11/4x^2 - 11(p+q)x+(10p^2 +24pq+10q^2) =0, where pis not equal to +or-q has what type of roots Real and equal Real and distinct No real roots

Q. Find the value of k so that the quadratic equation has equal roots: (k+3)x² + 2(k+3)x + 4 = 0

Q. If x^2-2px+q=0 has two equal roots then the equation (1+y)x^2 -2(p+y)x +(q+y)=0 will have real and distinct roots when

Q. The roots of following equation \sqrt{x+3-4\sqrt{x-1}}+\sqrt{x+8-6\sqrt{x-1}}=1 ar

Q. (p+4)x^2+(p+1)x+1=0 find the value of p if the equation has equal roots

Q. If alpha, beta are roots of ax2 + bx + c = 0, then the equation ax2 - bx (x - 1) + c(x - 1)2 = 0 has roots 1) alpha/(1-alpha), beta/(1-beta) 2) (1- alpha)/alpha, (1-beta)/beta 3) alpha/(1+alpha), beta/(1+beta) 4) (1+alpha)/alpha, (1+beta)/beta

Q. If the roots of the equation (a²+b²)x² -2(ac+bd)x + (c²+d²) = 0 are equal then prove that a/b = c/d.

Q.

Find the value of a for which the equation has coincident roots

a^2x^2+2(a+1)x+4=0

Q. {m^2+n^2}x^2 - 2{mp+nq}x + p^2 + q^2 This quadratic equation has equal roots. Prove that mn = √pq

Q. Solve the following quadratic equation by factorization method
$x^2-5x-36=0$

Q. Show that the equation [a-2] x^2 + [2-b]x + [b-a] = 0 has equal roots , if 2a = b+2

Q. . If the roots of the equation (b-c) x2 + (c-a) x + (a-b) = 0 are equal, then prove that 2b = a+c.

Q. the values of k for which the eqn |x|^2(|x|^2-2k+1)=1-k^2 has 1.no real root k belongs to 2.exactly 2 roots 3.repeated roots when k belongs to

Q. if alpha and beta are the roots of the quadratic equation 3 x square - 4 x + 1 is equal to zero then find the quadratic equation whose roots are 1) alpha/beta and beta/alpha 2) (alpha)^2/ beta and (beta)^2/ alpha

Q. If the equation x^2(a^2+b^2)-2b(a-c)x+b^2+c^2=0 has equal roots , then A.2b=ac B.b^2=ac C.b=2ac/(a+c) D.b=ac

Q. Find $p$ and $q$, If the equation $2x^2+4xy-p{y}^2+4x+qy+1=0$. represents a pair of perpendicular line.

Q. If α, β and γ are the real roots of a³ − 6a²+ 3a + 1 = 0, determine the sum of possible values of α²β + β²γ + γ²α.

Q. If (1 + k)x2 – 4x – 1 + k = 0 has real roots tanα and tanβ, then

Q. Assertion :If one root is $\sqrt{3}-\sqrt{2}$, then the equation of lowest degree with rational coefficients $x^4-10x^2+1=0$ Reason: For polynomial equation with rational coefficients irrational roots occurs in pairs.

1. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
2. Both Assertion and Reason are correct , but Reason is not the correct explanation for Assertion
3. Assertion is correct but Reason is incorrect
4. Assertion is incorrect but Reason is correct

Q. 34. If the equation x⁴-4x³+ax²+bx+1=0 has 4 positive roots, then the value of (|a|+|b|)/(a+b) is. 5/3/-5/-3

Q. The equation of locus of $P$, if $A=(4,0),B(-4,0)$ and $|PA+PB|=4$ is

1. $3x^2-y^2=12$
2. $x^2+y^2=4$
3. $x^2+3y^2=20$`
4. $3x^2+y^2=9$

Q. If $P\left(x\right)$ is a polynomial of degree less than or equal to $2$ and $S$ is the set of all such polynomials so that $P\left(1\right)=1,P\left(0\right)=0$ and $P^{\prime }\left(x\right)>0$ $\forall$ $x\in [0,1]$, then

1. $S=\varphi$
2. $S=(1-a)x^2+ax$ for $0<a<2$
3. $(1-a)$ $x^2+ax$ for $a\in (O,\infty )$
4. $S=\left\{\right(1-a)x^2+ax$ for $0<a<1$