Chapter 4

If the roots of the equation \(x^{2}-2 a x+a^{2}+a-3=0\) are real and less than 3 , then (A) \(a<2\) (B) \(2 \leq a \leq 3\) (B) \(3 \leq a \leq 4\) (D) \(a>4\)

For all real \(x\), the minimum value of \(\frac{1-x+x^{2}}{1+x+x^{2}}\) is (A) 0 (B) \(\frac{1}{3}\) (C) 1 (D) 3

Given that, for all real \(x\), the expression \(\frac{x^{2}-2 x+4}{x^{2}+2 x+4}\) lies between \(\frac{1}{3}\) and 3 . The values between which the expression \(\frac{9 \cdot 3^{2 x}+6 \cdot 3^{x}+4}{9 \cdot 3^{2 x}-6 \cdot 3^{x}+4}\) lies are (A) 0 and 2 (B) \(-1\) and 1 (C) \(-2\) and 0 (D) \(\frac{1}{3}\) and 3 .

The value of \(k\) for which the number 3 lies between the roots of the equation \(x^{2}+(1-2 k) x+\left(k^{2}-\right.\) \(k-2)=0\) is given by (A) \(25\)

The number of negative integral solutions of \(x^{2} \cdot 2^{x+1}\) \(+2^{|x-3|+2}=x^{2} \cdot 2^{(x-3 \mid+4)}+2^{x-1}\) is (A) 4 (B) 2 (C) 1 (D) 0

If \(\alpha\) and \(\beta(\alpha<\beta)\), are the roots of the equation \(x^{2}+\) \(b x+c=0\), where \(c<0

If the ratio of the roots of \(x^{2}+b x+c=0\) and \(x^{2}+q x+\) \(r=0\) be the same, then (A) \(r^{2} c=b^{2} q\) (B) \(r^{2} b=c^{2} q\) (C) \(r b^{2}=c q^{2}\) (D) \(r^{2}=b q^{2}\)

The number of solutions of \(|[x]-2 x|=4\), where \([x]\) is the greatest integer \(\leq x\), is (A) 2 (B) 4 (C) 1 (D) infinite

If \(\alpha, \beta\) are the roots of the equation \(x^{2}+p x+q=0\) then \(\frac{\alpha}{\beta}\) is a root of the equation (A) \(p x^{2}+\left(2 q-p^{2}\right) x+p=0\) (B) \(q x^{2}+\left(p^{2}-2 q\right) x+q=0\) (C) \(q x^{2}+\left(2 q-p^{2}\right) x+q=0\) (D) None of these

If \(a x^{2}+b x+c=0, a \neq 0, a, b, c \in R\) has distinct real roots in \((1,2)\) then \(a\) and \(5 a+2 b+c\) have (A) same sign (B) opposite sign (C) not determined (D) None of these

If \(a<0\), the positive root of the equation \(x^{2}-2 a\) \(|x-a|-3 a^{2}=0\) is (A) \(a(-1-\sqrt{6})\) (B) \(a(-1+\sqrt{6})\) (C) \(a(1-\sqrt{2})\) (D) None of these

If \(p x^{2}+q x+r=0\) has no real roots and \(p, q, r\) are real such that \(p+r>0\), then (A) \(p-q+r \leq 0\) (B) \(p+r \geq q\) (C) \(p+r=q\) (D) None of these

Given \(l x^{2}-m x+5=0\) does not have two distinct real roots, the minimum value of \(5 l+m\) is (A) 5 (B) \(-5\) (C) 1 (D) \(-1\)

If 1 lies between the roots of \(3 x^{2}-3 \sin \theta-2 \cos ^{2} \theta=0\) then (A) \(\frac{-1}{2}<\sin \theta<\frac{1}{2}\) (B) \(\frac{-1}{2}<\sin \theta<0\) (C) \(\frac{1}{2}<\sin \theta<1\) (D) None of these

If \(\alpha, \beta\) are the roots of the equation \(375 x^{2}-25 x-2=0\) and \(S_{n}=\alpha^{n}+\beta^{n}\), then \(\underset{n \rightarrow \infty}{\mathrm{Lt}} \sum_{r=1}^{n} S_{r}\) is (A) \(\frac{7}{12}\) (B) \(\frac{1}{12}\) (C) \(\frac{35}{12}\) (D) None of these

If \(a x^{2}+b x+6=0\) does not have two distinct real roots \(a \in R, b \in R\), then the least value of \(3 a+b\) is (A) 4 (B) \(-1\) (C) 1 (D) \(-2\)

If the ratio of the roots of \(\lambda x^{2}+\mu x+v=0\) is equal to the ratio of the roots of \(x^{2}+x+1=0\), then \(\lambda, \mu, v\) are in (A) A.P. (B) G.P. (C) H.P. (D) None of these

If the roots of \(x^{2}+a x+b=0\) are \(c\) and \(d\) then roots of \(x^{2}+(2 c+a) x+c^{2}+a c+b=0\) are (A) \(1, d-c\) (B) \(0, d-c\) (C) \(1, c-d\) (D) None of these

The solution set of \((x)^{2}+(x+1)^{2}=25\), where \((x)\) is the least integer greater than or equal to \(x\), is (A) \((2,4)\) (B) \((-5,4] \cup(2,3]\) (C) \([-4,-3) \cup[3,4)\) (D) None of these

Solution of \(2^{x}+2^{|x|} \geq 2 \sqrt{2}\) is (A) \(\left(-\infty, \log _{2}(\sqrt{2}+1)\right.\) (B) \((0,8)\) (C) \(\left(\frac{1}{2}, \log _{2}(\sqrt{2}-1)\right)\) (D) \(\left(-\infty, \log _{2}(\sqrt{2}-1)\right] \cup\left[\frac{1}{2}, \infty\right)\)

The solution set of \(\left|\frac{x+1}{x}\right|+|x+1|=\frac{(x+1)^{2}}{|x|}\) is (A) \(\\{x \mid x \geq 0\\}\) (B) \(\\{x \mid x>0\\} \cup\\{-1\\}\) (C) \(\\{-1,1\\}\) (D) \(\\{x \mid x \geq 1\) or \(x \leq-1\\}\)

If \(\alpha, \beta\) are the roots of the equation \(a x^{2}+b x+c=0\), \((a \neq 0)\) and \(\alpha+\delta, \beta+\delta\) are the roots of \(A x^{2}+B x+\) \(C=0,(A \neq 0)\) for some constant \(\delta\), then (A) \(\frac{b^{2}-4 a c}{a^{2}}=\frac{B^{2}-4 A C}{A^{2}}\) (B) \(\frac{b^{2}-2 a c}{a^{2}}=\frac{B^{2}-2 A C}{A^{2}}\) (C) \(\frac{b^{2}-8 a c}{a^{2}}=\frac{B^{2}-8 A C}{A^{2}}\) (D) None of these

Let \(a, b, c\) be real, if \(a x^{2}+b x+c=0\) has two real roots \(\alpha\) and \(\beta\), where \(\alpha<-1\) and \(\beta>1\) then \(1+\frac{c}{a}+\left|\frac{b}{a}\right|\) is (A) \(<0\) (B) \(\geq 0\) (C) \(\leq 0\) (D) None of these.

If \(a, b, c\) are in G.P., then the equations \(a x^{2}+2 b x+c=\) 0 and \(d x^{2}+2 e x+f=0\) have a common root if \(\frac{d}{a}, \frac{e}{b}\), \(\frac{f}{c}\) are in (A) H.P. (B) G.P. (C) A.P. (D) None of these

If the equations \(x^{2}+a b x+c=0\) and \(x^{2}+a c x+b=0\) have a common root, then their other roots satisfy the equation (A) \(x^{2}+a(b+c) x+a^{2} b c=0\) (B) \(x^{2}-a(b+c) x+a^{2} b c=0\) (C) \(x^{2}-a(b+c) x-a^{2} b c=0\) (D) None of these

If \(\left(a x^{2}+b x+c\right) y+a^{\prime} x^{2}+b^{\prime} x+c^{\prime}=0\), then the condi- tion that \(x\) may be a rational function of \(y\) is (A) \(\left(a c^{\prime}-a^{\prime} c\right)^{2}=\left(a b^{\prime}-a^{\prime} b\right)\left(b c^{\prime}-b^{\prime} c\right)\) (B) \(\left(a b^{\prime}-a^{\prime} b\right)^{2}=\left(a c^{\prime}-a^{\prime} c\right)\left(b c^{\prime}-b^{\prime} c\right)\) (C) \(\left(b c^{\prime}-b^{\prime} c\right)^{2}=\left(a b^{\prime}-a^{\prime} b\right)\left(a c^{\prime}-a^{\prime} c\right)\) (D) None of these

If \(n\) and \(r\) are positive integers such that \(0

If the equations \(x^{2}-p x+q=0\) and \(x^{2}-a x+b=0\) have a common root and the other root of the second equation is the reciprocal of the other root of the first, then \((q-b)^{2}=\) (A) \(a q(p-b)^{2}\) (B) \(b q(p-a)^{2}\) (C) \(b q(p-b)^{2}\) (D) None of these

If the two equations \(a x^{2}+b x+c=0\) and \(2 x^{2}-3 x+\) \(4=0\) have a common root, then (A) \(6 a=4 b=-3 c\) (B) \(3 a=-4 b=3 c\) (C) \(6 a=-4 b=3 c\) (D) None of these

If \(a, b, c\) are rational and \(a x^{2}+b x+c=0\) and \(3 x^{2}+\) \(x-5=0\) have a common root, then \(3 a+b+2 c=\) (A) 0 (B) 1 (C) 2 (D) None of these

If \(a x^{2}+2 b x+c=0\) and \(a_{1} x^{2}+2 b_{1} x+c_{1}=0\) have a common root and \(\frac{a}{a_{1}}, \frac{b}{b_{1}}, \frac{c}{c_{1}}\) are in A.P., then \(a_{1}, b_{1}\), \(c_{1}\) are in (A) A.P. (B) G.P. (C) H.P. (D) None of these

If \(x\) is real, then the minimum value of \(\frac{(a+x)(b+x)}{(c+x)}(x>-c)\), for \(a>c, b>c\) is (A) \((\sqrt{a-b}+\sqrt{c-b})^{2}\) (B) \((\sqrt{a-c}+\sqrt{b-c})^{2}\) (C) \((\sqrt{a-c}-\sqrt{b-c})^{2}\) (D) None of these

If the ratio of the roots of \(a_{1} x^{2}+b_{1} x+c_{1}=0\) be equal to the ratio of the roots of \(a_{2} x^{2}+b_{2} x+c_{2}=0\), then \(\frac{a_{1}}{a_{2}}\), \(\frac{b_{1}}{b_{2}}, \frac{c_{1}}{c_{2}}\) are in (A) A.P. (B) G.P. (C) H.P. (D) None of these

If \(\alpha, \beta\) be the roots of the equation \(x^{2}-p x+q=0\) and \(\alpha>0, \beta>0\), then the value of \(\alpha^{1 / 4}+\beta^{1 / 4}\) is \(\left(p+6 \sqrt{q}+4 q^{1 / 4} \sqrt{p+2 \sqrt{q}}\right)^{k}\), where \(k\) is equal to (A) 1 (B) \(\frac{1}{2}\) (C) \(\frac{1}{3}\) (D) \(\frac{1}{4}\)

If \(a, b\) are the roots of the equation \(x^{2}+p x+1=0\) and \(c, d\) are the roots of the equation \(x^{2}+q x+1=0\), then \((a-c)(b-c)(a+d)(b+d)=\) (A) \(p^{2}-q^{2}\) (B) \(q^{2}-p^{2}\) (C) \(p^{2}+q^{2}\) (D) \(2\left(p^{2}-q^{2}\right)\)

If \(q \neq 0\) and the equation \(x^{3}+p x^{2}+q=0\) has a root of multiplicity 2 , then \(p\) and \(q\) are connected by (A) \(p^{2}+2 q=0\) (B) \(p^{2}-2 q=0\) (C) \(4 p^{3}+27 q+1=0\) (D) \(4 p^{3}+27 q=0\)

If the roots of the equation \(a x^{2}+b x+c=0\), are of the form \(\frac{\alpha}{\alpha-1}\) and \(\frac{\alpha+1}{\alpha}\), then the value of \((a+b+c)^{2}\) is (A) \(b^{2}-2 a c\) (B) \(2 b^{2}-a c\) (C) \(b^{2}-4 a c\) (D) \(4 b^{2}-2 a c\)

If the sum of the roots of the quadratic equation \(a x^{2}+\) \(b x+c=0\) is equal to the sum of the squares of their reciprocals, then \(\frac{a}{c}, \frac{b}{a}\) and \(\frac{c}{b}\) are in (A) arithmetic progression (B) geometric progression (C) harmonic progression (D) arithmetico-geometric progression

If both the roots of the quadratic equation \(x^{2}-2 k x+\) \(k^{2}+k-5=0\) are less than 5, then \(k\) lies in the interval (A) \((-\infty, 4)\) (B) \([4,5]\) (C) \((5,6]\) (D) \((6, \infty)\)

If for real number \(a\), the equation \((a-2)(x-[x])^{2}+\) \(2(x-[x])+a^{2}=0\) (where \([x]\) denotes the greatest integer \(\leq x\) ) has no integral solution and has exactly one solution in \((2,3)\), then \(a\) lies in the interval (A) \((-1,2)\) (B) \((0,1)\) (C) \((-1,0)\) (D) \((2,3)\)

Let \(a, b, c\) be distinct positive numbers such that each of the quadratics \(a x^{2}+b x+c, b x^{2}+c x+a\) and \(c x^{2}+a x+b\) is non-negative for all \(x \in R\). If \(R=\frac{a^{2}+b^{2}+c^{2}}{a b+b c+c a}\), then (A) \(1 \leq R<4\) (B) \(1

The set of values of \(a\) for which the equation \(\left(x^{2}+x\right.\) \(+2)^{2}-(a-3)\left(x^{2}+x+2\right)\left(x^{2}+x+1\right)+(a-4)\left(x^{2}+\right.\) \(x+1)^{2}=0\) has at least one real root is (A) \(\left(5, \frac{19}{3}\right)\) (B) \(\left[5, \frac{19}{3}\right]\) (C) \(\left[5, \frac{19}{3}\right)\) (D) \(\left(5, \frac{19}{3}\right]\)

If all real values of \(x\) obtained from the equation \(4^{x}-(a-3) 2^{x}+a-4=0\) are non-positive, then \(a\) belongs to (A) \([4,5]\) (B) \((4,5]\) (C) \([4,5)\) (D) \((4,5)\)

Let \(f(x)=x^{2}+a x+b\) be a quadratic polynomial, where \(a\) and \(b\) are integers. If for a given integer \(n\), \(f(n) f(n+1)=f(m)\) for some integer \(m\), then the value of \(m\) is (A) \(n(n+1)+a n+b\) (B) \(n(n+1)+a+b n\) (C) \(n(n+1)+a+b\) (D) None of these

If for any real \(x\), we have \(-1 \leq \frac{x^{2}+n x-2}{x^{2}-3 x+4} \leq 2\), then \(n\) belongs to (A) \([-\sqrt{40}+6,-1]\) (B) \([-\sqrt{40}+6, \sqrt{40}-6]\) (C) \([-1, \sqrt{40}-6]\) (D) None of these

If \(b>a\), then the equation \((x-a)(x-b)-1=0\) has (A) both roots in \((-\infty, a)\) (B) one root in \((-\infty, a)\) and other in \((b, \infty)\) (C) both roots in \((b, \infty)\) (D) both roots in \([a, b]\)

The quadratic equation \(\frac{(x+b)(x+c)}{(b-a)(c-a)}+\frac{(x+c)(x+a)}{(c-b)(a-b)}+\frac{(x+a)(x+b)}{(a-c)(b-c)}=1\) has (A) two real and distinct roots (B) imaginary roots (C) equal roots (D) infinite roots

The equation \(a x^{4}-2 x^{2}-(a-1)=0\) will have real and unequal roots if (A) \(a<0, a \neq 1\) (B) \(a>0, a \neq 1\) (C) \(0

If the equation \(x^{2}+\left[a^{2}-5 a+b+4\right] x+b=0\) has roots \(-5\) and 1 , where \([a]\) denotes the greatest integer less than or equal to \(a\), then the set of values of \(a\) is (A) \(\left(\frac{5-3 \sqrt{5}}{2}, \frac{5+3 \sqrt{5}}{2}\right)\) (B) \(\left(0, \frac{5+3 \sqrt{5}}{2}\right)\) (C) \(\left(-1, \frac{5-3 \sqrt{5}}{2}\right] \cup\left[\frac{5+3 \sqrt{5}}{2}, 6\right)\) (D) None of these