Chapter 4

If \(t_{n}\) denotes the \(n\)th term of an A.P. and \(t_{p}=\frac{1}{q}\) and \(t_{q}\) \(=\frac{1}{p}\), then which of the following is necessarily a root of the equation \((p+2 q-3 r) x^{2}+(q+2 r-3 p) x+(r\) \(+2 p-3 q)=0\) (A) \(t_{p}\) (B) \(t_{q}\) (C) \(t_{p q}\) (D) \(t_{p+q}\)

If the roots of the equation \(4 x^{2}+4 a x+b=0\) are real and differ at most by \(a\), then \(b\) lies in (A) \(\left(0, \frac{a^{2}}{2}\right)\) (B) \(\left(\frac{a^{2}}{2}, a^{2}\right)\) (C) \(\left[0, a^{2}\right]\) (D) \(\left(0, a^{2}\right)\)

The roots of the equation \(a x^{2}+b x+c=0\), where \(a \in R^{+}\), are two consecutive odd positive integers, then (A) \(|b| \leq 4 a\) (B) \(|b| \geq 4 a\) (C) \(|b|=2 a\) (D) None of these

If \(a, b, c, d\) are real numbers, then the number of real roots of the equation \(\left(x^{2}+a x-3 b\right)\left(x^{2}-c x+b\right)\left(x^{2}-\right.\) \(d x+2 b)=0\) are (A) 3 (B) 4 (C) 6 (D) at least 2

If \(a \leq 0\), then the root of the equation \(x^{2}-2 a|x-a|-\) \(3 a^{2}=0\) is (A) \((1-\sqrt{2}) a\) (B) \((-1+\sqrt{6}) a\) (C) \((1+\sqrt{2}) a\) (D) \(-(1+\sqrt{6}) a\)

If \(c \neq 0\) and the equation \(\frac{p}{2 x}=\frac{a}{x+c}+\frac{b}{x-c}\) has two equal roots, then \(p\) can be (A) \((\sqrt{a}-\sqrt{b})^{2}\) (B) \((\sqrt{a}+\sqrt{b})^{2}\) (C) \(a+b\) (D) \(a-b\)

For \(a>0\), the roots of the equation \(\log _{a x} a+\log _{x} a^{2}+\) \(\log _{a^{2} x} a^{3}=0\), are given by (A) \(a^{1 / 2}\) (B) \(a^{-1 / 2}\) (C) \(a^{4 / 3}\) (D) \(a^{-4 / 3}\)

Solution of \(\left|x^{2}+4 x+3\right|+2 x+5=0\) is (A) 4 (B) \(-4\) (C) \(-1-\sqrt{3}\) (D) \(1+\sqrt{3}\)

If the equation \(x^{2}+9 y^{2}-4 x+3=0\) is satisfied for real values of \(x\) and \(y\), then (A) \(1 \leq x \leq 3\) (B) \(2 \leq x \leq 3\) (C) \(-\frac{1}{3} \leq y \leq \frac{1}{3}\) (D) \(\frac{1}{3} \leq x \leq 1\).

If \(\alpha, \beta\) are the roots of \(a x^{2}+b x+c=0\) and \(\alpha^{4}, \beta^{4}\) are the roots of \(l x^{2}+m x+n=0\), then the roots of the equation \(a^{2} l x^{2}-4 a c l x+2 c^{2} l+a^{2} m=0\) are (A) real (B) imaginary (C) opposite in sign (D) equal

If \(a, b, c\) are positive rational numbers such that \(a>b>c\) and the quadratic equation \((a+b-2 c) x^{2}+\) \((b+c-2 a) x+(c+a-2 b)=0\) has a root in the interval \((-1,0,\), then (A) \(c+a<2 b\) (B) both roots of the given equation are rational (C) the equation \(a x^{2}+2 b x+c=0\) has both negative real roots (D) the equation \(c x^{2}+2 a x+b=0\) has both negative real roots

If the equation \(x^{2}+a^{2} x+b^{2}=0\) has two roots each of which exceeds a number \(c\), then (A) \(a^{4}>4 b^{2}\) (B) \(c^{2}+a^{2} c+b^{2}>0\) (C) \(-\frac{a^{2}}{2}>c\) (D) \(-\frac{a^{2}}{2}

If the equation \(x^{2}+(a-b) x-a-b+1=0\), where \(a\), \(b \in \mathrm{R}\), has unequal real roots for all \(b \in R\), then (A) \(a<0\) (B) \(a>0\) (C) \(a>1\) (D) \(a<1\)

Let \(k\) be any point such that \(k \in R\) and \(\alpha, \beta\) are the roots of the quadratic equation \(f(x)=a x^{2}+b x+c=0\). If \(k\) lies outside and is less than both the roots then the equation must have real and distinct roots and the sign of \(f(k)\) is same as the sign of ' \(a\) '. Also, \(k\) is less than the \(x\)-coordinate of the vertex of the parabola \(y=a x^{2}+b x+c\). If \(k\) lies between both the roots, then the sign of \(f(k)\) is opposite to the sign of ' \(a\) '. If \(k\) lies outside and is greater than both the roots, then the sign of \(f(k)\) is same as the sign of ' \(a\) '. Also, \(k\) is greater than the \(x\)-coordinate of the vertex of the parabola \(y=a x^{2}+b x+c\). If both the roots of the equation lie between two real numbers \(k_{1}\) and \(k_{2}\), then equation must have real and distinct roots and the sign of \(f\left(k_{1}\right)\) and \(f\left(k_{2}\right)\) is same as the sign of \(a\). Also, the \(x\)-coordinate of the vertex of the parabola \(y=a x^{2}+b x+c\) lies between \(k_{1}\) and \(k_{2}\). The values of ' \(a\) ' for which the roots of the equation \((a+1) x^{2}-3 a x+4 a=0(a \neq-1)\) to be greater than unity are (A) \(\frac{-16}{7} \leq a<-1\) (B) \(-2

Let \(k\) be any point such that \(k \in R\) and \(\alpha, \beta\) are the roots of the quadratic equation \(f(x)=a x^{2}+b x+c=0\). If \(k\) lies outside and is less than both the roots then the equation must have real and distinct roots and the sign of \(f(k)\) is same as the sign of ' \(a\) '. Also, \(k\) is less than the \(x\)-coordinate of the vertex of the parabola \(y=a x^{2}+b x+c\). If \(k\) lies between both the roots, then the sign of \(f(k)\) is opposite to the sign of ' \(a\) '. If \(k\) lies outside and is greater than both the roots, then the sign of \(f(k)\) is same as the sign of ' \(a\) '. Also, \(k\) is greater than the \(x\)-coordinate of the vertex of the parabola \(y=a x^{2}+b x+c\). If both the roots of the equation lie between two real numbers \(k_{1}\) and \(k_{2}\), then equation must have real and distinct roots and the sign of \(f\left(k_{1}\right)\) and \(f\left(k_{2}\right)\) is same as the sign of \(a\). Also, the \(x\)-coordinate of the vertex of the parabola \(y=a x^{2}+b x+c\) lies between \(k_{1}\) and \(k_{2}\). The values of ' \(a\) ' so that 6 lies between the roots of the equation \(x^{2}+2(a-3) x+9=0\), are (A) \(a>-\frac{3}{4}\) (B) \(a<-\frac{3}{4}\) (C) \(a>\frac{3}{4}\) (D) \(a<\frac{3}{4}\)

Let \(k\) be any point such that \(k \in R\) and \(\alpha, \beta\) are the roots of the quadratic equation \(f(x)=a x^{2}+b x+c=0\). If \(k\) lies outside and is less than both the roots then the equation must have real and distinct roots and the sign of \(f(k)\) is same as the sign of ' \(a\) '. Also, \(k\) is less than the \(x\)-coordinate of the vertex of the parabola \(y=a x^{2}+b x+c\). If \(k\) lies between both the roots, then the sign of \(f(k)\) is opposite to the sign of ' \(a\) '. If \(k\) lies outside and is greater than both the roots, then the sign of \(f(k)\) is same as the sign of ' \(a\) '. Also, \(k\) is greater than the \(x\)-coordinate of the vertex of the parabola \(y=a x^{2}+b x+c\). If both the roots of the equation lie between two real numbers \(k_{1}\) and \(k_{2}\), then equation must have real and distinct roots and the sign of \(f\left(k_{1}\right)\) and \(f\left(k_{2}\right)\) is same as the sign of \(a\). Also, the \(x\)-coordinate of the vertex of the parabola \(y=a x^{2}+b x+c\) lies between \(k_{1}\) and \(k_{2}\). The value of \(a\) for which the equation \(\left(1-a^{2}\right) x^{2}+\) \(2 a x-1=0\) has roots belonging to \((0,1)\) is (A) \(a>\frac{1+\sqrt{5}}{2}\) (B) \(a>2\) (C) \(\frac{1+\sqrt{5}}{2}\sqrt{2}\)

Let \(k\) be any point such that \(k \in R\) and \(\alpha, \beta\) are the roots of the quadratic equation \(f(x)=a x^{2}+b x+c=0\). If \(k\) lies outside and is less than both the roots then the equation must have real and distinct roots and the sign of \(f(k)\) is same as the sign of ' \(a\) '. Also, \(k\) is less than the \(x\)-coordinate of the vertex of the parabola \(y=a x^{2}+b x+c\). If \(k\) lies between both the roots, then the sign of \(f(k)\) is opposite to the sign of ' \(a\) '. If \(k\) lies outside and is greater than both the roots, then the sign of \(f(k)\) is same as the sign of ' \(a\) '. Also, \(k\) is greater than the \(x\)-coordinate of the vertex of the parabola \(y=a x^{2}+b x+c\). If both the roots of the equation lie between two real numbers \(k_{1}\) and \(k_{2}\), then equation must have real and distinct roots and the sign of \(f\left(k_{1}\right)\) and \(f\left(k_{2}\right)\) is same as the sign of \(a\). Also, the \(x\)-coordinate of the vertex of the parabola \(y=a x^{2}+b x+c\) lies between \(k_{1}\) and \(k_{2}\). The values of \(a\) for which each one of the roots of \(x^{2}-4 a x+2 a^{2}-3 a+5=0\) is greater than 2, are (A) \(a \in(1, \infty)\) (B) \(a=1\) (C) \(a \in(-\infty, 1)\) (D) \(a \in(9 / 2, \infty)\)

The maximum number of positive real roots of a polynomial equation \(f(x)=0\) is the number of changes of signs from positive to negative and negative to positive in \(f(x)\). For example, consider the equation \(f(x)=x^{3}+6 x^{2}+11 x-\) \(6=0\). The signs of the various terms are: $$ +++- $$ Clearly, there is only one change of sign in the given expression. So, the given equation has at most one positive Ireal root. The maximum number of negative real roots of a polynomial equation \(f(x)=0\) is the number of changes of signs from positive to negative and negative to positive in \(f(-x)\). For example, for the equation \(f(x)=x^{4}+x^{3}+x^{2}-x-1=\) 0 , there are three changes of signs in \(f(-x)\). So, the given equation has atmost three negative real roots. If \(f(x)\) and \(f(-x)\) do not have any changes of signs, the equation \(f(x)=0\) has no real roots. Now, consider the polynomial $$ P_{n}(x)=1+2 x+3 x^{2}+\ldots+(n+1) x^{n} . $$ If \(n\) is even, the number of real roots of \(P_{n}(x)\) is (A) 0 (B) \(n\) (C) 1 (D) None of these

In the following questions an Assertion (A) is given followed by a Reason \((R) .\) Mark your responses from the following options: (A) Assertion(A) is True and Reason(R) is True; Reason(R) is a correct explanation for Assertion(A) (B) Assertion(A) is True, Reason(R) is True; Reason(R) is not a correct explanation for Assertion(A) (C) Assertion(A) is True, Reason( \(\mathrm{R}\) ) is False (D) Assertion(A) is False, Reason(R) is True Assertion: If the roots of the equations \(x^{2}-b x+c=0\) and \(x^{2}-c x+b=0\) differ by the same quantity, then \(b+c\) is equal to \(-4\). Reason: If \(\alpha, \beta\) are the roots of the equation \(A x^{2}+\) \(B x+C=0\), then \(\alpha-\beta=\frac{\sqrt{B^{2}-4 A C}}{A}\)

In the following questions an Assertion (A) is given followed by a Reason \((R) .\) Mark your responses from the following options: (A) Assertion(A) is True and Reason(R) is True; Reason(R) is a correct explanation for Assertion(A) (B) Assertion(A) is True, Reason(R) is True; Reason(R) is not a correct explanation for Assertion(A) (C) Assertion(A) is True, Reason( \(\mathrm{R}\) ) is False (D) Assertion(A) is False, Reason(R) is True Assertion: If the equation \(x^{2}+2(k+1) x+9 k-5=0\) has only negative roots, then \(k \leq 6\) Reason: The equation \(f(x)=0\) will have both roots negative if and only if (i) Discriminant \(\geq 0\), (ii) Sum of roots \(<0\), (iii) Product of roots \(>0\)

In the following questions an Assertion (A) is given followed by a Reason \((R) .\) Mark your responses from the following options: (A) Assertion(A) is True and Reason(R) is True; Reason(R) is a correct explanation for Assertion(A) (B) Assertion(A) is True, Reason(R) is True; Reason(R) is not a correct explanation for Assertion(A) (C) Assertion(A) is True, Reason( \(\mathrm{R}\) ) is False (D) Assertion(A) is False, Reason(R) is True Assertion: If the equations \(x^{2}+b x+c a=0\) and \(x^{2}+\) \(c x+a b=0\) have a common root, then their other roots will satisfy the equation \(x^{2}+a x+b c=0\) Reason: If the equations \(x^{2}+b x+c a=0\) and \(x^{2}+\) \(c x+a b=0\) have a common root, then \(a+b+c=0\)

If \(\alpha \neq \beta\) with \(a^{2}=5 \alpha-3\) and \(\beta^{2}=5 \beta-3\), then \(t h\) equation having \(\alpha^{\prime} \beta\) and \(\beta^{\prime} \alpha\) as its roots, is \(\mathbf{\\{} \mathbf{2 0 0 2}\) (A) \(3 x^{2}+19 x+3=0\) (B) \(3 x^{2}-19 x+3=0\) (C) \(3 x^{2}-19 x-3=0\) (D) \(x^{2}-16 x+1=0\)

The number of real solutions of the equation \(x^{2}-3\) \(|x|+2=0\) is (A) 2 (B) 4 (C) 1 (D) 3

The value of ' \(a\) ' for which one root of the quadratic equation \(\left(a^{2}-5 a+3\right) x^{2}+(3 a-1) x+2=0\) is twice as large as the other, is (A) \(\frac{2}{3}\) (B) \(-\frac{2}{3}\) (C) \(\frac{1}{3}\) (D) \(-\frac{1}{3}\)

If \((1-p)\) is a root of quadratic equation \(x^{2}+p x+\) \((1-p)=0\), then its roots are (A) 0,1 (B) \(-1,2\) (C) \(0,-1\) (D) \(-1,1\)

If one root of the equation \(x^{2}+p x+12=0\) is 4, while the equation \(x^{2}+p x+q=0\) has equal roots, then the value of ' \(q\) ' is (A) \(\frac{49}{4}\) (B) 4 (C) 3 (D) 12

If \(2 a+3 b+6 c=0\), then at least one root of the equation \(a x^{2}+b x+c=0\) lies in the interval \(\quad\) [2004] (A) \((0,1)\) (B) \((1,2)\) (C) \((2,3)\) (D) \((1,3)\)

The values of \(\alpha\) for which the sum of the squares of the roots of the equation \(x^{2}-(a-2) x-a-1=0\) assume the least value is (A) 1 (B) 0 (C) 3 (D) 2

If roots of the equation \(x^{2}-b x+c=0\) be two consectutive integers, then \(b^{2}-4 c\) equals \(\quad\) [2005] (A) \(-2\) (B) 3 (C) 2 (D) 1

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 [2005] (A) \((5,6]\) (B) \((6, \infty)\) (C) \((-\infty, 4)\) (D) \([4,5]\)

All the values of \(m\) for which both roots of the equations \(x^{2}-2 m x+m^{2}-1=0\) are greater than \(-2\) but less than 4, lie in the interval (A) \(-23\) (C) \(-1

If \(x\) is real, the maximum value of \(\frac{3 x^{2}+9 x+17}{3 x^{2}+9 x+7}\) is [2006] (A) \(1 / 4\) (B) 41 (C) 1 (D) \(17 / 7\)

If the difference between the roots of the equation \(x^{2}+a x+1=0\) is less than \(\sqrt{5}\), then the set of possible values of a is (A) \((-3,3)\) (B) \((-3, \infty)\) (C) \((3, \infty)\) (D) \((-\infty,-3)\)

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 \(-4 a b\)

If \(a\) and \(\beta\) are the roots of the equation \(x^{2}-x+1=0\) then the value of \(\alpha^{2009}+\beta^{2009}=\) [2010] (A) \(-1\) (B) 1 (C) 2 (D) \(-2\)

The equation \(e^{\sin x}-e^{-\sin x}-4=0\), for \(x\) real, has \(\mathbf{[ 2 0 1 2}]\) (A) infinite number of roots (B) no roots (C) exactly one root (D) exactly four roots

The real number \(k\) for which the equation, \(2 x^{3}+3 x+k=0\) has two distinct real roots in \([0,1]\) \([\mathbf{2 0 1 3}]\) (A) lies between 2 and 3 (B) lies between \(-1\) and 0 (C) does not exist (D) lies between 1 and 2

If the equations \(x^{2}+2 x+3=0\) and \(a x^{2}+b x+c=0, a, b, c \in R\) have a common root, then \(a: b: c\) is (A) \(3: 2: 1\) (B) \(1: 3: 2\) (C) \(3: \overline{1: 2}\) (D) \(1: 2: 3\)

If \(a \in R\) and the equation \(-3(x-[x])^{2}+2(x-[x])\) \(+a^{2}=0\) (where \([x]\) denotes the greatest integer \(\leq x\) ) has no integral solution, then all possible values of a lie in the interval (A) \((-1,0) \cup(0,1)\) (B) \((1,2)\) (C) \((-2,-1)\) (D) \((-\infty,-2) \cup(2, \infty)\)

Let \(\alpha\) and \(\beta\) be the roots of equation \(x^{2}-6 x-2=0\). If \(a_{n}=\alpha^{n}-\beta^{n}\), for \(n \geq 1\),then the value of \(\frac{a_{10}-2 a_{8}}{2 a_{9}}\) is equal to (A) \(-6\) (B) 3 (C) \(-3\) (D) 6