Chapter 14

If the equation \(a_{n} x^{n}+a_{n-1} x^{n-1}+\ldots+a_{1} x=0\) has a positive root \(x=\alpha\), then the equation \(n a_{n} x^{n-1}+(n-1) a_{n-1} x^{n-2}+\ldots+a_{1}=0\) has a positive root, which is (A) smaller than \(\alpha\) (B) greater than \(\alpha\) (C) equal to \(\alpha\) (D) greater than or equal to \(\alpha\)

Let \(f\) be a function which is continuous and differentiable for all real \(x\). If \(f(2)=-4\) and \(f^{\prime}(x) \geq 6\) for all \(x \in[2,4]\), then (A) \(f(4)<8\) (B) \(f(4) \geq 8\) (C) \(f(4) \geq 12\) (D) None of these

If \(a x+\frac{D}{x} \geq c\) for all positive \(x\), where \(a, b>0\), then (A) \(a b<\frac{c^{2}}{4}\) (B) \(a b \geq \frac{c^{2}}{4}\) (C) \(a b \geq \frac{c}{4}\) (D) None of these

Let \(f\) be \(a\) continuous, diferentiable and bijective function. If the tangent to \(y=f(x)\) at \(x=a\) is also the normal to \(y=f(x)\) at \(x=b\), then there exists at least one \(c \in(a, b)\) such that (A) \(f^{\prime}(c)=0\) (B) \(f^{\prime}(c)>0\) (C) \(f^{\prime}(c)<0\) (D) None of these

The values of \(k\) for which the function \(f(x)=k x^{3}-9 x^{2}+9 x+3\) may be increasing on \(R\) are (A) \(k>3\) (B) \(k \leq 3\) (C) \(k \leq 3\) (D) None of these

The least possible value of \(k\) for which the function \(f(x)=x^{2}+k x+1\) may be increasing on \([1,2]\) is (A) 2 (B) \(-2\) (C) 0 (D) None of these

Let \(a+b=4, a<2\) and \(g(x)\) be a monotonically increasing function of \(x\). Then, \(f(a)=\int_{0}^{a} g(x) d x+\int_{0}^{b} g(x) d x\) (A) increases with increase in \((b-a)\) (B) decreases with increase in \((b-a)\) (C) increases with decrease in \((b-a)\) (D) None of these

The equation \(x+e^{x}=0\) has (A) only one real root (B) only two real roots (C) no real root (D) None of these

The value of \(a\) in order that \(f(x)=\sin x-\cos x-a x+b\) decreases for all real values is given by (A) \(a \geq \sqrt{2}\) (B) \(a<\sqrt{2}\) (C) \(a \geq 1\) (D) \(a<1\)

If \(f^{\prime \prime}(x)<0 \forall x \in(a, b)\), then \(f^{\prime}(x)=0\) (A) exactly once in \((a, b)\) (B) at most once in \((a, b)\) (C) at least once in \((a, b)\) (D) None of these

The minimum value of \(a \tan ^{2} x+b \cot ^{2} x\) equals the maximum value of \(a \sin ^{2} \theta+b \cos ^{2} \theta\) where \(a>b>0\), when (A) \(a=b\) (B) \(a=2 b\) (C) \(a=3 b\) (D) \(a=4 b\)

If \(f(x)=\frac{x^{2}-1}{x^{2}+1}\), for every real number, then minimum value of \(\vec{f}\) (A) Does not exist (B) Is note attained even through \(f\) is bounded (C) Is equal to 1 (D) Is equal to \(-1\)

If \(y=a \log |x|+b x^{2}+x\) has its extremum values at \(x=-1\) and \(x=2\), then (A) \(a=2, b=-1\) (B) \(a=2, b=-1 / 2\) (C) \(a=-2, b=1 / 2\) (D) None of these

If \(f(x)\) and \(g(x)\) are differentiable functions for \(0 \leq x \leq 1\) such that \(f(0)=2, g(0)=0, f(1)=6, g(1)=2\), then in the interval \((0,1)\), (A) \(f^{\prime}(x)=0\) for all \(x\) (B) \(f^{\prime}(x)=2 g^{\prime}(x)\) for at least one \(x\) (C) \(f^{\prime}(x)=2 g^{\prime}(x)\) for at most one \(x\) (D) None of these

The difference between the greatest and least values of the function \(f(x)=\cos x+\frac{1}{2} \cos 2 x-\frac{1}{3} \cos 3 x\) is (A) \(2 / 3\) (B) \(8 / 7\) (C) \(9 / 4\) (D) \(3 / 8\)

A function \(f\) is such that \(f^{\prime}(4)=f^{\prime \prime}(4)=0\) and \(f\) has minimum value 10 at \(x=4\). Then \(f(x)=\) (A) \(4+(x-4)^{4}\) (B) \(10+(x-4)^{4}\) (C) \((x-4)^{4}\) (D) None of these

The range of values of \(k\) for which the function \(f(x)=\left(k^{2}-7 k+12\right) \cos x+2(k-4) x+\log 2\) does not possess critical points, is (A) \((1,5)\) (B) \((1,5)-\\{4\\}\) (C) \((1,4)\) (D) None of these

If a differentiable function \(f(x)\) has a relative minimum at \(x=0\), then the function \(y=f(x)+a x+b\) has a relative minimum at \(x=0\) for (A) all \(a>0\) (B) all \(b>0\) (C) all \(a\) and \(\bar{b}\) (D) all \(b\) if \(a=0\)

Let \(f(x)\) and \(g(x)\) be defined and differentiable for \(x \geq x_{0}\) and \(f\left(x_{0}\right)=g\left(x_{0}\right), f^{\prime}(x)>g^{\prime}(x)\) for \(x>x_{0}\), then (A) \(f(x)x_{0}\) (b) \(f(x)=g(x), x>x_{0}\) (C) \(f(x)>g(x), x>x_{0}\) (d) None of these

If \(\alpha\) and \(\beta(\alpha<\beta)\) be two different real roots of the equation \(a x^{2}+b x+c=0\), then (A) \(\alpha>-\frac{b}{2 a}\) (B) \(\beta<-\frac{b}{2 a}\) (C) \(\alpha<-\frac{b}{2 a}<\beta\) (D) \(\beta<-\frac{b}{2 a}<\alpha\)

If \(p(x)=a_{0}+a_{1} x+a_{2} x^{2}+\ldots+a_{n} x^{n}\) and \(|p(x)| \leq / e^{x-1}-1 \mid\) for all \(x \geq 0\), then \(\left|a_{1}+2 a_{2}+3 a_{3}+\ldots+n a_{n}\right|\) (A) \(\leq 1\) (B) \(\geq 1\) (C) \(\geq 0\) (D) \(\leq 0\)

Let \(f(x)=\left\\{\begin{array}{cc}-x^{3}+\log _{2} b & 0

The second drivative \(f^{\prime \prime}(x)\) of the function \(f(x)\) exists for all \(x\) in \([0,1]\) and satisfies \(\left|f^{\prime \prime}(x)\right| \leq 1\). If \(f(0)=f(1)\), then for all \(x\) in \([0,1]\) (A) \(\left|f^{\prime}(x)\right|<1\) (B) \(\left|f^{\prime}(x)\right|>1\) (C) \(\left|f^{\prime}(x)\right|=1\) (D) \(f(x)\) is constant

Let the function \(f\) be defined as \(f(x)=\left\\{\begin{aligned} \frac{P(x)}{x-2}, & x \neq 2 \\ 7, & x=2 \end{aligned}\right.\) where \(P(x)\) is a polynomial such that \(P^{\prime \prime \prime}(x)\) is identically equal to 0 and \(P(3)=9\). If \(f(x)\) is continuous at \(x=2\), then (A) \(P(x)=2 x^{2}-x-6\) (B) \(P(x)=2 x^{2}+x-6\) (C) \(P(x)=2 x^{2}-x+6\) (D) None of these

The equation \(x^{5}-3 x-1=0\) has, in the interval \([1,2]\) (A) at least one root (B) at most one root (C) no root (D) a unique root

If the equation \(x-\sin x=k\) has a unique root in \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\), then the range of values of \(k\) are (A) \(\left(1-\frac{\pi}{2}, \frac{\pi}{2}-1\right)\) (B) \(\left[1-\frac{\pi}{2}, \frac{\pi}{2}-1\right]\) (C) \(\left[0, \frac{\pi}{2}+1\right]\) (D) None of these

The largest term in the sequence \(a_{n}=\frac{n}{n^{2}+10}, n \in N\) is (A) \(\frac{4}{26}\) (B) \(\frac{3}{19}\) (C) \(\frac{7}{18}\) (D) None of these

The range of values of \(a\) for which all roots of the equation \(3 x^{4}+4 x^{3}-12 x^{2}+a=0\) are real and distinct is (A) \((0,5)\) (B) \((1,4)\) (C) \((-1,5)\) (D) None of these

If \(\phi(x)=f(x)+f(1-x)\) and \(f^{\prime \prime}(x)<0\) in \((-1,1)\), then \(\phi(x)\) strictly increases in the interval (A) \(\left(0, \frac{1}{2}\right)\) (B) \(\left(\frac{1}{2}, 1\right)\) (C) \((-1,0)\) (D) \((0,1)\)

\(f(x)\) is a cubic function with \(f(1)=-6, f(-1)=10\) and has maxima at \(x=-1\). If \(f^{\prime}(x)\) has minima at \(x=1\), then (A) \(f(x)=x^{3}+3 x^{2}-9 x+5\) (B) \(f(x)=x^{3}-3 x^{2}-9 x+5\) (C) \(f(x)=x^{3}-3 x^{2}+9 x+5\) (D) \(f(x)=x^{3}-3 x^{2}-9 x+5\)

If the function \(f(x)= \begin{cases}-x^{3}+\frac{b^{3}-b^{2}+b-1}{b^{2}+3 b+2}, & 0 \leq x<1 \\ 2 x-3, & 1 \leq x \leq 3\end{cases}\) has the least value at \(x=1\), then all possible real values of \(b\) are (A) \((-1,1)\) (B) \((-2,-1) \cup[1, \infty)\) (C) \((-2,1)\) (D) None of these

The function \(f(x)=\frac{|x+1|}{x^{2}}\) is strictly decreasing in the interval (A) \((-\infty,-2) \cup(0,1)\) (B) \((-2,0) \cup(1, \infty)\) (C) \((-2,-1) \cup(0, \infty)\) (D) None of these

The range of values of \(a\) so that the equation \(x^{3}-3 x+\) \(a=0\) has three real and distinct roots is (A) \((-\infty,-2) \cup(2, \infty)\) (B) \((-2,0)\) (C) \((-2,0)\) (D) \((-2,2)\)

The curves \(\frac{x^{2}}{a}+\frac{y^{2}}{b}=1\) and \(\frac{x^{2}}{a_{1}}+\frac{y^{2}}{b_{1}}=1\) will cut orthogonally if (A) \(a+b=a_{1}+b_{1}\) (B) \(a-b=a_{1}-b_{1}\) (C) \(\frac{1}{a}-\frac{1}{b}=\frac{1}{a_{1}}-\frac{1}{b_{1}}\) (D) None of these

The point on the curve \(3 x^{2}-4 y^{2}=72\) which is nearest to the line \(3 x+2 y+1=0\) is (A) \((6,-3)\) (B) \((6,3)\) (C) \((-6,3)\) (D) \((-6,-3)\)

If the function \(f(x)=\left(a^{2}-3 a+2\right) \cos \frac{x}{2}+(a-1) x\) possesses critical points, then \(a\) belongs to the interval (A) \((-\infty, 0) \cup(4, \infty)\) (B) \((-\infty, 0] \cup[4, \infty)\) (C) \((-\infty, 0] \cup\\{1\\} \cup[4, \infty)\) (D) None of these

If the function \(f(x)=\left(1-\frac{\sqrt{21-4 b-b^{2}}}{b+1}\right) x^{3}\) \(+5 x+\sqrt{16}\) increases for all \(x\), then (A) \(b \in(-1,2)\) (B) \(b \in(-7,3)-\\{-1\\}\) (C) \(b \in(-7,-1) \cup(2,3)\) (D) None of these

Let \(f(x)=(x-3)(x-4)(x-4)(x-5)(x-6)\), then (A) \(f^{\prime}(x)\) has four roots (B) three roots of \(f^{\prime \prime}(x)=0\) lie in \((3,4) \cup(4,5) \cup(5,6)\) (C) the equation \(f^{\prime}(x)=0\) has only one root (D) three roots of \(f^{\prime}(x)=0\) lie in \((2,3) \cup(3,4) \cup(4,5)\)

For \(a \in[\pi, 2 \pi]\) and \(n \in Z\), the critical points of \(f(x)=\) \(\frac{1}{3} \sin a \tan ^{3} x+(\sin a-1) \tan x+\sqrt{\frac{a-2}{8-a}}\) are (A) \(x=n \pi\) (B) \(x=2 n \pi\) (C) \(x=(2 n+1) \pi\) (D) None of these

Let \(f^{\prime \prime}(x)>0 \forall x \in R\) and \(g(x)=f(2-x)+f(4+x)\). Then, \(g(x)\) is increasing in (A) \((-\infty,-1)\) (B) \((-\infty, 0)\) (C) \((-1, \infty)\) (D) None of these

The curves \(x^{2}-4 y^{2}+c=0\) and \(y^{2}=4 x\) will cut orthogonally for (A) \(c \in(0,16)\) (B) \(c \in(-3,4)\) (C) \(c \in(3,4)\) (D) None of these

Which of the following is not true? The function \(f(x)=x^{2}+\frac{\lambda}{x}\) has a (A) minimum at \(x=2\) if \(\lambda=16\) (B) maximum at \(x=2\) if \(\lambda=16\) (C) maximum for no real value of \(\lambda\) (D) point of inflexion at \(x=1\) if \(\lambda=-1\)

If the parabola \(y=f(x)\), having axis parallel to the \(y\)-axis, touches the line \(y=x\) at \((1,1)\), then (A) \(2 f^{\prime}(0)+f(0)=1\) (B) \(2 f(0)+f^{\prime}(0)=1\) (C) \(2 f(0)-f^{\prime}(0)=1\) (D) \(2 f^{\prime}(0)-f(0)=1\)

The angle between the tangents at any point \(P\) and the line joining \(P\) to the origin \(O\), where \(P\) is a point on the curve \(\ln \left(x^{2}+y^{2}\right)=c \tan ^{-1} \frac{y}{x}, c\) is a constant (A) varies as \(\tan ^{-1} x\) (B) varies as \(\tan ^{-1} y\) (C) is a constant (D) None of these

If the equation \(a x^{2}+b x+c=0\) has two distinct positive roots, then the equation \(a x^{2}+(b+6 a) x+\) \((c+3 b)=0\) has (A) two positive roots (B) exactly one positive root (C) at least one positive root (D) no positive root

If \(f(x)\) is continuous in \([a, b]\) and differentiable in \((a, b)\) then there exists at least one \(c \in(a, b)\) such that \(\frac{f(b)-f(a)}{b^{3}-a^{3}}\) equals (A) \(3 c^{2} f^{\prime}(\mathrm{C})\) (B) \(\frac{f^{\prime}(c)}{3 c^{2}}\) (C) \(f(c) f^{\prime}(C)\) (D) None of these

Let \(f(x)=\ln x\) and \(g(x)=x^{2}\). If \(c \in(4,5)\), then \(c \ln \left(\frac{4^{25}}{5^{16}}\right)\) equals (A) \(c \ln 5-8\) (B) \(2\left(c^{2} \ln 4-8\right)\) (C) \(2\left(c^{2} \ln 5-8\right)\) (D) \(c \ln 4-8\)

If \(0\frac{\sin x}{x}\) (B) \(\frac{2}{\pi}<\frac{\sin x}{x}\) (C) \(\frac{\sin x}{x}<1\) (D) \(\frac{\sin x}{x}>1\)

If \(0\cos x\) (B) \(\cos (\sin x)<\cos x\) (C) \(\cos (\sin x)>\sin (\cos x)\) (D) \(\cos (\sin x)<\sin (\cos x)\)

\((1+x)^{p} \leq 1+x^{p}\), where (A) \(p>1\) (B) \(0 \leq p \leq 1\) (C) \(x>0\) (D) \(x<0\)