Chapter 4

The roots of the equation \(2^{x+2} \cdot 3^{\frac{3 x}{x-1}}=9\) are given by (A) \(\log _{2},\left(\frac{2}{3}\right)-2\) (B) \(3,-3\) (C) \(-2,1-\frac{\log 3}{\log 2}\) (D) \(1-\log _{2} 3,2\)

If \(a, b, c\) are positive real numbers, then the number of real roots of the equation \(a x^{2}+b|x|+c=0\) is (A) 0 (B) 2 (C) 4 (D) None of these

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

If \(\alpha\) and \(\beta\) are the roots of \(x^{2}+p x+q=0\) and \(\alpha^{4}\) and \(\beta^{4}\) are the roots of \(x^{2}-r x+s=0\), then the equation \(x^{2}-4 q x+2 q^{2}-r=0\) has always (A) two real roots (B) two positive roots (C) two negative roots (D) one positive and one negative root

If \(a, b, c, d\) and \(p\) are distinct real numbers such that \(\left(a^{2}+b^{2}+c^{2}\right) p^{2}-2(a b+b c+c d) p+\left(b^{2}+c^{2}+d^{2}\right) \leq 0\) then \(a, b, c\) and \(d\) (A) are in A.P. (B) are in G.P. (C) are in H.P. (D) satisfy \(a b=c d\)

Let \(S\) denotes the set of all values of \(x\) for which the equation \(2 x^{2}-2(2 a+1) x+a(a+1)=0\) has one root less than \(a\) and other root greater than \(a\), then \(S\) equals (A) \((0,1)\) (B) \((-1,0)\) (C) \((0,1 / 2)\) (D) None of these

Let \(a, b, c\) be positive real numbers, such that \(b x^{2}+\) \(\left(\sqrt{(a+c)^{2}+4 b^{2}}\right) x+(a+c) \geq 0, \forall x \in R\), then \(a, b, c\) are in: (A) G.P. (B) A.P. (C) H.P. (D) None of these

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) \(r c^{2}=b q^{2}\)

If \(0 \leq x<\frac{\pi}{2}\), then the solution of the equation \(16^{\sin ^{2} x}+16^{\cos ^{2} x}=10\) is given by \(x\) equal to (A) \(\frac{\pi}{6}, \frac{\pi}{3}\) (B) \(\frac{\pi}{3}, \frac{\pi}{2}\) (C) \(\frac{\pi}{6}, \frac{\pi}{2}\) (D) None of these

If one of the roots of the equation \(x^{2}-(p+1) x+p^{2}+\) \(p-8=0\) is greater than 2 and the other root is smaller than 2 , then \(p\) is such that (A) \(-\frac{11}{3}

The common roots of the equations \(x^{3}+2 x^{2}+2 x+\) \(1=0\) and \(1+x^{130}+x^{1988}=0\) are (where \(\omega\) is a none real cube root of unity) (A) \(\omega\) (B) \(\omega^{2}\) (C) \(-1\) (D) None of these

If \(^{4} x\) ' satisfies \(\left|x^{2}-3 x+2\right|+|x-1|=x-3\), then (A) \(x \in \phi\) (B) \(x \in[1,2]\) (C) \(x \in[3, \infty)\) (D) \(x \in(-\infty, \infty)\)

The number of solutions (s) of the equation \(\sqrt{3 x^{2}+6 x+7}+\sqrt{5 x^{2}+10 x+14} \leq 4-2 x-x^{2}\) is (A) one (B) two (C) four (D) infinite

If \(\left(a^{2}-1\right) x^{2}+(a-1) x+a^{2}-4 a+3=0\) is an identity in \(x\), then the value of \(a\) is (A) 1 (B) 3 (C) \(-1\) (D) \(-3\)

Both the roots of the equation \((x-b)(x-c)+(x-a)\) \((x-c)+(x-a)(x-b)=0\) are always (A) positive (B) negative (C) real (D) None of these

If \(a, b, c \in R\) and quadratic equation \(x^{2}+(a+b) x+\) \(c=0\) has no real roots then (A) \(c(a+b+c)>0\) (B) \(c+c(a+b+c)>0\) (C) \(c+c(a+b-c)>0\) (D) \(c(a+b-c)>0\)

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) \(-1\)

The set of possible values of \(\lambda\) for which \(\lambda^{2}-\left(\lambda^{2}-\right.\) \(5 \lambda+5) x+\left(2 \lambda^{2}-3 \lambda-4\right)=0\) has roots whose sum and product are both less than 1 is (A) \(\left(-1, \frac{5}{2}\right)\) (B) \((1,4)\) (C) \(\left[1, \frac{5}{2}\right]\) (D) \(\left(1, \frac{5}{2}\right)\)

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 \(\operatorname{Lt}_{n \rightarrow \infty} \sum_{r=1}^{n} S_{r}\) is (A) \(\frac{7}{12}\) (B) \(\frac{1}{12}\) (C) \(\frac{35}{12}\) (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

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 \(\alpha, \beta\) be the roots of \(x^{2}+p x-q=0\) and \(\gamma, \delta\) be the roots of \(x^{2}+p x+r=0, q+r \neq 0\), then \(\frac{(\alpha-\gamma)(\alpha-\delta)}{(\beta-\gamma)(\beta-\delta)}=\) (A) 1 (B) \(q\) (C) \(r\) (D) \(q+r\)

Number of integral solutions of \(\frac{x+2}{x^{2}+1}>\frac{1}{2}\) is (A) 0 (B) 1 (C) 2 (D) 3

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 \(c

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) \(b^{2}-4 a c\) (C) \(4 b^{2}-a c\) (D) \(2 b^{2}-a c\)

If \(\alpha, \beta\) be roots of \(x^{2}+p x+1=0\) and \(\gamma \delta \delta\) be the roots of \(x^{2}+q x+1=0\), then \((\alpha-\gamma)(\beta-\gamma)(\alpha+\delta)(\beta+\delta)=\) (A) \(p^{2}+q^{2}\) (B) \(p^{2}-q^{2}\) (C) \(q^{2}-p^{2}\) (D) None of these

If \(a\) and \(b\) are odd integers then \([x]^{2}+a[x]+b=0\) (where [. ] denotes greatest integer function) has (A) finite number of roots (B) infinite number of roots (C) no roots (D) None of these

If \(\log _{9}\left(x^{2}-5 x+6\right)>\log _{3}(x-4), x\) belongs to (A) \((-\infty, 4)\) (B) \((4, \infty)\) (C) \((-\infty,-4) \cup(4, \infty)\) (D) no real value of \(x\)

Let \(a, b, c\) be real numbers, \(a \neq 0 .\) If \(\alpha\) is a root of \(a^{2} x^{2}+b x+c=0, \beta\) is a root of \(a^{2} x^{2}-b x-c=0\) and \(0<\alpha<\beta\), then the equation \(a^{2} x^{2}+2 b x+2 c=0\) has a root \(\gamma\) that always satisfies (A) \(\gamma=\frac{\alpha+\beta}{2}\) (B) \(\gamma=\alpha+\frac{\beta}{2}\) (C) \(\gamma=\alpha\) (D) \(\alpha<\gamma<\beta\)

Number of solutions of the equation \(x^{2}-2-2[x]=0\) ([-] denotes greatest integer function) is (A) 1 (B) 2 (C) 3 (D) None of these

The number of real roots of the equation \(2^{\sin ^{4} x}-2^{\cos ^{2} x}=\) 1 is (A) 2 (B) 1 (C) infinite (D) None of these

If the absolute value of the difference of roots of the equation \(x^{2}+p x+1=0\) exceeds, \(\sqrt{3 p}\), then (A) \(p<-1\) or \(p>4\) (B) \(p>4\) (C) \(-1

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

If the equation \(x^{2}+2(k+1) x+9 k-5=0\) has only negative roots, then (A) \(k \leq 0\) (B) \(k \geq 0\) (C) \(k \geq 6\) (D) \(k \leq 6\)

If the product of the roots of the equation \(x^{2}-3 k x+\) \(2 e^{2 \ln k}-1=0\) is 7 , then for real roots the value of \(k\) is equal to (A) 1 (B) 2 (C) 3 (D) 4

The solution set of $$ \left(\frac{3}{5}\right)^{x}=x-x^{2}-9 \text { is } $$ (A) \(\\{0\\}\) (B) \(\\{1\\}\) (C) \(\phi\) (D) None of these

The equation \(e^{\sin x}-e^{-\sin x}-4=0\) has (A) infinite number of real roots (B) no real roots (C) exactly one real root (D) exactly four real roots

Suppose the cube \(x^{3}-p x+q\) has three distinct real roots where \(p>0\) and \(q>0\). Then which one of the following holds? (A) The cubic has minima at \(\sqrt{\frac{p}{3}}\) and maxima at \(-\sqrt{\frac{p}{3}}\) (B) The cubic has minima at \(-\sqrt{\frac{p}{3}}\) and maxima at \(\sqrt{\frac{p}{3}}\) (C) The cubic has minima at both \(\sqrt{\frac{p}{3}}\) and \(-\sqrt{\frac{p}{3}}\) (D) The cubic has maxima at both \(\sqrt{\frac{p}{3}}\) and \(-\sqrt{\frac{p}{3}}\)

The quadratic equations \(x^{2}-6 x+a=0\) and \(x^{2}-c x\) \(+6=0\) have one root in common. The other roots of the first and second equations are integers in the ratio \(4: 3\). Then the common root is (A) 1 (B) 4 (C) 3 (D) 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 \(-4 a b\)

The equation \(\sqrt{x+3-4 \sqrt{x-1}}+\sqrt{x+8-6 \sqrt{x-1}}=1\) has (A) no solution (B) one solution (C) two solutions (D) more than two solutions

If \(x, y \in[0,10]\), then the number of solutions \((x, y)\) of the inequation \(3^{\sec ^{2} x-1} \sqrt{9 y^{2}-6 y+2} \leq 1\) is (A) 2 (B) 4 (C) 6 (D) infinite

If \(f(x)=x-[x], x(\neq 0) \in R\), where \([x]\) is the greatest integer less than or equal to \(x\), then the number of solutions of \(f(x)+f\left(\frac{1}{x}\right)=1\) are (A) 0 (B) 1 (C) infinite (D) 2

If \(x^{2}-(a+b+c) x+(a b+b c+c a)=0\) has imaginary roots, where \(a, b, c \in R^{+}\), then \(\sqrt{a}, \sqrt{b}, \sqrt{c}\) (A) can be the sides of a triangle (B) cannot be the sides of a triangle (C) nothing can be said (D) None of these

If \(x_{1}, x_{2}, x_{3}, \ldots, x_{n}\) are the roots of the equation \(x^{n}+a x+\) \(b=0\), then the value of \(\left(x_{1}-x_{2}\right)\left(x_{1}-x_{3}\right)\left(x_{1}-x_{4}\right) \ldots\) \(\left(x_{1}-x_{n}\right)\) is equal to (A) \(n x_{1}^{n-1}+a\) (B) \(n\left(x_{1}\right)^{n-1}\) (C) \(n x_{1}+b\) (D) \(n x_{1}^{n-1}+b\)