Chapter 4

If equations \(a x^{2}+b x+c=0\) and \(c x^{2}+b x\) \(+a=0\) have one root common, show that either \(a+b+c=0\) or \(a-b+c=0\).

The value of a for which the sum of the squares of the roots of the equation \(x^{2}-\) \((a-2) x-(a+1)=0\) has the least value, is (a) 0 (b) 1 (c) 2 (d) 3

The maximum value of \(5+20 x-4 x^{2}, x \in\) \(R\) is (a) 25 (b) 30 (c) 5 (d) 1

If \(\alpha, \beta\) are roots of the quadratic equation \(a x^{2}+2 b x+c=0\), then prove that \(\sqrt{\alpha / \beta}+\sqrt{\beta / \alpha}=\frac{-2 b}{\sqrt{a c}} .\)

If \(x\) be real, then least value of \(3 x^{2}+7 x+10\) is (a) 10 (b) \(10 / 3\) (c) \(7 / 3\) (d) \(71 / 12\)

If ratio of the roots of \(x^{2}+p x+q=0\) be same as ratio of the roots of \(x 2+p^{\prime} x+q^{\prime}=\) 0 , then prove that \(p^{2} q^{\prime}=p^{\prime 2} q\).

If \(x\) is real, then the maximum and minimum values of the expression \(\frac{x^{2}-3 x+4}{x^{2}+3 x+4}\) will be (a) 2,1 (b) \(5,1 / 5\) (c) \(7,1 / 7\) (d) none of these

If \(\alpha, \beta\) are roots of the quadratic equation \(x^{2}\) \(+p x+p^{2}+q=0\), then prove that \(\alpha^{2}+\alpha \beta+\) \(\beta^{2}+q=0\)

The quadratic in \(t\), such that \(\mathrm{A} \cdot \mathrm{M}\). of its roots in \(A\) and G.M. is \(G\), is (a) \(t^{2}-2 A t+G^{2}=0\) (b) \(t^{2}-2 A t-G^{2}=0\) (c) \(t^{2}+2 A t+G^{2}=0\) (d) none of these

If both roots of equations \(K\left(6 x^{2}+3\right)+r x+\) \(2 x^{2}-1=0\) and \(6 K\left(2 x^{2}+1\right)+p x+4 x^{2}-2=\) 0 are common, then prove that \(2 r-p=0\).

Let \(f(x)=x^{2}+4 x+1\), then (a) \(f(x)>0\) for all \(x\) (b) \(f(x)>1\) when \(x \geq 0\) (c) \(f(x) \geq 1\) when \(x \leq-4\) (d) \(f(x)=f(-x)\) for all \(x\)

If the roots of the equation \(x^{2}-8 x+\left(a^{2}-6 a\right)\) \(=0\) are real, then (a) \(-2

The number of roots of the equation \(|x|^{2}-7\) \(|x|+12=0\) is (a) 1 (b) 2 (c) 3 (d) 4

Product of real roots of the equation \(t^{2} x^{2}+\) \(|x|+9=0\), (a) is always positive (b) is always negative (c) does not exist (d) none of these

The number of roots of the equation \(|x|=x^{2}\) \(+x-4\) is (a) 4 (b) 3 (c) 1 (d) 2

$$ \begin{aligned} &x^{2}-3 x+2 \text { be a factor of } x^{4}-p x^{2}+q \text {, then }\\\ &(p, q)= \end{aligned} $$ (a) \((3,4)\) (b) \((4,5)\) (c) \((4,3)\) (d) \((5,4)\)

If \((x+a)\) is a factor of both the quadratic polynomials \(x^{2}+p x+q\) and \(x^{2}+l x+m\), where \(p, q, l\) and \(m\) are constants, then which one of the following is correct? (a) \(a=(m-q) /(l-p)(l \neq p)\) (b) \(a=(m+q) /(l+p)(l \neq-p)\) (c) \(l=(m-q) /(a-p)(a \neq p)\) (d) \(p=(m-q) /(a-l)(a \neq l)\)

What is the value of \(x\) satisfying the equation \(16\left(\frac{a-x}{a+x}\right)^{3}=\frac{a+x}{a-x} ?\) (a) \(a / 2\) (b) \(a / 3\) (c) \(a / 4\) (d) 0

If \(\alpha, \beta\) be the roots of \(x^{2}+p x+q=0\) and \(\alpha+h, \beta+h\) are the roots of \(x^{2}+r x+s=0\) then (a) \(\frac{p}{r}=\frac{q}{s}\) (b) \(2 h=\left[\frac{p}{q}+\frac{r}{s}\right]\) (c) \(p^{2}-4 q=r^{2}-4 s\) (d) \(p r^{2}=q s^{2}\)

\(x^{2}-11 x+a\) and \(x^{2}-14 x+2 a\) will have a common factor, if \(a=\) (a) 24 (b) 0,24 (c) 3,24 (d) 0,3

The real root of the equation \(x^{3}-6 x+9=0\) is (a) 6 (b) \(-3\) (c) \(-6\) (d) \(-9\)

If \(\alpha, \beta\) are the roots of the quadratic equation \(x^{2}+b x-c=0\), then the equation whose roots are \(b\) and \(c\) is (a) \(x^{2}+\alpha x-\beta=0\) (b) \(x^{2}-[(\alpha+\beta)+\alpha \beta] x-\alpha \beta(\alpha+\beta)=0\) (c) \(x^{2}-[(\alpha+\beta)+\alpha \beta] x+\alpha \beta(\alpha+\beta)=0\) (d) \(x^{2}+[\alpha \beta+(\alpha+\beta)] x-\alpha \beta(\alpha+\beta)=0\)

Let \(\alpha, \beta\) be the roots of the equation \(a x^{2}\) \(+2 b x+c=0\) and \(\gamma, \delta\) be the roots of the equation \(p x^{2}+2 q x+r=0 .\) If \(\alpha, \beta, \gamma, \delta\) are in G.P., then (a) \(q^{2} a c=b^{2} p r\) (b) \(q a c=b p r\) (c) \(c^{2} p q=r^{2} a b\) (d) \(p^{2} a b=a^{2} q r\)

The value of \(k\) for which the equation \((k-2) x^{2}+8 x+k+4=0\) has both roots real, distinct and negative is (a) 0 (b) 2 (c) 3 (d) \(-4\)

The set of values of \(\lambda\) for which the equation \(3 x^{2}+2 x+\lambda(\lambda-1)=0\) are of opposite signs is (a) \((0,1)\) (b) \([0,1]\) (c) \([0,1)\) (d) \((0,1]\)

The values of \(a\) for which one root of the equation \(x^{2}-(a+1) x+a^{2}+a-8=0\) exceeds 2 and the other is lesser than 2 , are given by (a) \(a>3\) (b) \(9

The value of \(p\) for which both the roots of the equation \(4 x^{2}-20 p x+\left(25 p^{2}+15 p-66\right)\) \(=0\) are less than 2 , lies in the interval (a) \((-1,-4 / 5)\) (b) \((-\infty,-1)\) (c) \((2, \infty)\) (d) none of these

If both the roots of \(a x^{2}+b x+c=0\) are positive, then (a) \(-\frac{b}{a}>0\) (b) \(\frac{c}{a}>0\) (c) \(b^{2} \geq 4 a c\) (d) \(a c>0\)

The value of \(a\) for which the quadratic equation \(3 x^{2}+2\left(a^{2}+1\right) x+\left(a^{2}-3 a+2\right)=0\) passesses roots with opposite sign, lies in (a) \((-\infty, 1)\) (b) \((-\infty, 0)\) (c) \((1,2)\) (d) \((3 / 2,2)\)

If the roots of the equation \(b x^{2}+c x+a=0\) be imaginary, then for all real values of \(x\), the expression \(3 b^{2} x^{2}+6 b c x+2 c^{2}\) is (a) greater than \(4 a b\) (b) less than \(4 a b\) (c) greater than \(-4 a b\) (d) less than \(-4 a b\)