Chapter 14

The set of values of \(x\) for which \(\log (1+x)0\) (C) \(0

Let \(f(x)=\cos 2 \pi x+x-[x]\), where \([-]\) denotes the greatest integer function. Then the number of points in \([0,10]\) at which \(f(x)\) assumes its local maximum value, is (A) 10 (B) 9 (C) 0 (D) infinite

The function \(f(x)=\frac{\sin x}{x}\) is decreasing in the interval (A) \(\left(-\frac{\pi}{2}, 0\right)\) (B) \(\left(0, \frac{\pi}{2}\right)\) (C) \((0, \pi)\) (D) None of these

If \(a x+\frac{b}{x} \geq c\) 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

If \(0<\alpha<\beta<\frac{\pi}{2}\) then (A) \(\frac{\tan \beta}{\tan \alpha}<\frac{\alpha}{\beta}\) (B) \(\frac{\tan \beta}{\tan \alpha}>\frac{\alpha}{\beta}\) (C) \(\frac{\tan \alpha}{\tan \beta}<\frac{\alpha}{\beta}\) (D) \(\frac{\tan \alpha}{\tan \beta}>\frac{\alpha}{\beta}\)

The range of values of \(a\) for which the function \(f(x)=x^{3}+(a+2) x^{2}+3 a x+5\) may be monotonic in \(R\), is (A) \(a<1\) (B) \(14\) (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) \(k<3\) (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

If \(f(x)=2 x^{3}+9 x^{2}+\lambda x+20\) is a decreasing function of \(x\) in the largest possible interval \((-2,-1)\) then \(\lambda\) is equal to (A) 12 (B) \(-12\) (C) 6 (D) None of these

Let \(f^{\prime}(x)>0\) and \(g^{\prime}(x)<0\) for all \(x \in R\). Then, (A) \(f[g(x)]>f[g(x-1)]\) (B) \(f[g(x)]>f[g(x+1)]\) (C) \(g[f(x)]>g[f(x-1)]\) (D) \(g[f(x)]

If the function \(f(x)=3 \cos |x|-6 a x+b\) increases for all \(x \in R\), then the range of values of \(a\) is given by (A) \(a>-\frac{1}{2}\) (B) \(a<-\frac{1}{2}\) (C) \(a \leq b\) (D) \(a \geq b\)

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) \(a<1\)

The two tangents to the curve \(a x^{2}+2 h x y+b y^{2}=1\), \(a>0\) at the points where it crosses \(x\)-axis, are (A) parallel (B) perpendicular (C) inclined at an angle \(\frac{\pi}{4}\) (D) None of these

The curve \(y-e^{x y}+x=0\) has a vertical tangent at the point (A) \((1,1)\) (B) at no point (C) \((0,1)\) (D) \((1,0)\)

The set of all values of \(a\) for which the function \(f(x)=\) \(\left(a^{2}-3 a+2\right)\left(\cos ^{2} x / 4-\sin ^{2} x / 4\right)+(a-1) x+\sin 1\) does not possess critical points is (A) \([1, \infty)\) (B) \((0,1) \cup(1,4)\) (C) \((-2,4)\) (D) \((1,3) \cup(3,5)\)

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

Tangent is drawn to the ellipse \(\frac{x^{2}}{27}+y^{2}=1\) at \((3 \sqrt{3} \cos \theta, \sin \theta)\), where \(\theta \in(0, \theta / 2)\). Then, the value of \(\theta\) such that sum of intercepts on axes made by this tangent is minimum, is (A) \(\frac{\pi}{3}\) (B) \(\frac{\pi}{6}\) (C) \(\frac{\pi}{8}\) (D) \(\frac{\pi}{4}\)

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) \(a=4 b\)

A function \(f\) is such that \(f^{\prime}(a)=f^{\prime \prime}(a)=f^{\prime \prime \prime}(a)=\ldots=\) \(f^{(2 n)}(a)=0\) and \(f\) has a local maximum value \(b\) at \(x=\) \(a\), if \(f(x)\) is (A) \((x-a)^{2 n+2}\) (B) \(b-1-(x+1-a)^{2 n-1}\) (C) \(b-(x-a)^{2 n+2}\) (D) \((x-a)^{2 n+2}-b\)

If the area of the triangle included between the axes and any tangent to the curve \(x^{n} y=a^{n}\) is constant, then \(n\) is equal to (A) 1 (B) 2 (C) \(\frac{3}{2}\) (D) \(\frac{1}{2}\)

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 atleast one \(x\) (C) \(f^{\prime}(x)=2 g^{\prime}(x)\) for atmost one \(x\) (D) None of these

For a differentiable curve \(y=f(x)\) having atleast two extremum in the interval \([a, b]\), (A) two of its maximum values occur successively (B) two of its minimum values occur successively (C) maximum and minimum values occur alternatively (D) None of the above

The points on the curve \(x y^{2}=1\) which are nearest to the origin are (A) \(\left[\left(\frac{1}{2}\right)^{1 / 3}, \pm\left(\frac{1}{2}\right)^{-1 / 6}\right]\) (B) \(\left[\left(\frac{1}{2}\right)^{1 / 3}, 2^{-1 / 6}\right]\) (C) \(\left(2^{1 / 3}, \pm\left(\frac{1}{2}\right)^{-1 / 6}\right)\) (D) None of these

N characters of information are held on magnetic tape, in batches of \(x\) characters each; the batch processing time is \(\alpha+\beta x^{2}\) seconds; \(\alpha, \beta\) are constants. The optimum value of \(x\) for fast processing is (A) \(\frac{\alpha}{\beta}\) (B) \(\frac{\beta}{\alpha}\) (C) \(\sqrt{\frac{\alpha}{\beta}}\) (D) \(\sqrt{\frac{\beta}{\alpha}}\)

\(A B\) is a diameter of a circle and \(C\) is any point on the circumference of the circle, then (A) area of \(\Delta A B C\) is maximum when it is an isosceles (B) area of \(\Delta A B C\) is minimum when it is an isosceles (C) the perimeter of \(\triangle A B C\) is minimum when it is isosceles (D) the perimeter of \(\Delta A B C\) is maximum when it is isosceles

Let \(f(x)=1+3 x^{2}+3^{2} x^{4}+\ldots+3^{30} \cdot x^{60} .\) Then \(f(x)\) has (A) atleast one maximum (B) exactly one maximum (C) atleast one minimum (D) exactly one minimum

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

The minimum value of the function \(f(x)=\frac{x^{p}}{p}+\frac{x^{-q}}{q}\), where \(\frac{1}{p}+\frac{1}{q}=1, p>1\) is (A) 1 (B) 0 (C) 2 (D) None of these

If \(f(x)=\frac{x^{2}-1}{x^{2}+1}\), for every real number \(x\), then the minimum value of \(f\) (A) does not exist because \(f\) is unboundecd (B) is not attained even though \(f\) is bounded (C) is equal to 1 (D) is equal to \(-1\)

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 \(b\) (D) all \(b\) if \(a=0\)

On the curve \(x^{3}=12 y\), the abscissa changes at a faster rate than the ordinate. Then, \(x\) belongs to the interval (A) \((-4,4)\) (B) \((-3,3)\) (C) \((-2,2)\) (D) None of these

The maximum value of radius vector where \(\frac{c^{4}}{r^{2}}=\frac{a^{2}}{\sin ^{2} t}+\frac{b^{2}}{\cos ^{2} t} ;(a, b>0)\) is (A) \((a+b)^{2}\) (B) \(\frac{c^{4}}{(a+b)^{2}}\) (C) \(\frac{c^{2}}{a+b}\) (D) \(c^{2}(a+b)\)

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 \(f^{\prime}(x)=\frac{1}{1+x^{2}}\) for all \(x\) and \(f(0)=0\), then (A) \(f(2)<0.4\) (B) \(f(2)>2\) (C) \(0.4

The interval in which \(\lambda\) should be if \(f(x)=\sin ^{3} x+\lambda\) \(\sin ^{2} x(-\pi / 2

Twenty metre of wire is available to fence off a flower bed in the form of a sector. If the flower bed has the maximum surface then radius is (A) 10 (B) \(5 / 2\) (C) 5 (D) \(15 / 2\)

The normal to the curve \(x=a(1+\cos \theta), y=a \sin \theta\) at \(\theta\) always passes through the fixed point (A) \((a, a)\) (B) \((a, 0)\) (C) \((0, a)\) (D) None of these

If the tangent to the curve \(2 y^{3}=a x^{2}+x^{3}\) at the point \((a, a)\) cuts off intercepts \(\alpha\) and \(\beta\) on the coordinate axes such that \(\alpha^{2}+\beta^{2}=61\), then \(a=\) (A) \(\pm 30\) (B) \(\pm 5\) (C) \(\pm 6\) (D) \(\pm 61\)

If \(x \in[0,2]\) and \(g(x)=f(x)+f(2-x)\). Also, \(f^{\prime \prime}(x)<0\) then \(g(x)\) (A) increases in \([0,2]\) (B) decreases in \([0,2]\) (C) decreases in \([0,1)\) and increases in \((1,2]\) (D) increases in \([0,1)\) and decreases in \((1,2]\)

A spherical balloon is filled with \(4500 \pi\) cubic meters of helium gas. If a leak in the balloon causes the gas to escape at the rate of \(72 \pi\) cubic meters per minute, then the rate (in meters per minute) at which the radius of the balloon decreases 40 minutes after the leakage began is (A) \(9 / 7\) (B) \(7 / 9\) (C) \(2 / 9\) (D) \(9 / 2\)

Let \(a, b \in R\) be such that the fucntion \(f\) given by \(f(x)=\) \(\ln |x|+b x^{2}+a x, x \neq 0\) has extreme values at \(x=-1\) and \(x=2\). Statement \(1: f\) has local maximum at \(x=-1\) and at \(x=2\). Statement 2: \(a=\frac{1}{2}\) and \(b=\frac{-1}{4}\). (A) Statement- 1 is false, Statement- 2 is true. (B) Statement \(-1\) is true, statement- 2 is true, statement-2 is a correct explanation for Statement-1. (C) Statement \(-1\) is true, statement \(-2\) is true; statement-2 is not a correct explanation for Statement-1. (D) Statement- 1 is true, statement- 2 is false.

Each side of a square is increasing at the uniform rate of \(1 \mathrm{~m} / \mathrm{sec}\). If after some time the area of the square is increasing at the rate of \(8 \mathrm{~m}^{2} / \mathrm{sec}\), then the area of square at that time in sq. meters is: (A) 4 (B) 9 (C) 16 (D) 25

Let \(a, b, c \in R, a>0\) and function \(f: R \rightarrow R\) be defined by \(f(x)=a x^{2}+b x+c\) Statement 1: \(b^{2}<4 a c \Rightarrow f(x)>0\), for every value of \(x\). Statement 2: \(f\) is strictly decreasing in the interval \(\left(-\infty, \frac{-b}{2 a}\right)\) and strictly increasing in the interval \(\left(\frac{-b}{2 a}, \infty\right)\) (A) Statement- 1 is true, Statement- 2 istrue, Statement-2 is a correct explanation for Statement- 1 . (B) Statement- 1 is true, Statement- 2 is true, Statement-2 is not a correct explanation for Statement-1. (C) Statement- 1 is true, Statement \(-2\) is false. (D) Statement- 1 is false, Statement- 2 is true.

How many real solutions does the equation \(x^{7}+14 x^{5}+\) \(16 x^{3}+30 x-560=0\) have? (A) 7 (B) 1 (C) 3 (D) 5

Given \(P(x)=x^{4}+a x^{3}+b x^{2}+c x+d\) such that \(x=0\) is the only real root of \(P^{\prime}(x)=0\). If \(P(-1)

Let \(f:[2,7] \rightarrow[0, \infty)\) be a continuous and differentiable function. Then, \((f(7)-f(2)) \frac{\left(f(7)^{2}+(f(2))^{2}+f(2) f(7)\right.}{3}\) is equal to (A) \(5 f^{2}(c) f^{\prime}(C)\) (B) \(5 f^{\prime}(c)\) (C) \(f(c) f^{\prime}(C)\) (D) None of these where \(c \in(2,7)\).

Let the function \(f(x)\) be defined as \(f(x)=\left\\{\begin{array}{lc}\tan ^{-1} \alpha-3 x^{2}, 0

Let \(f(x)=\left\\{\begin{array}{ll}|x|, & 0<|x| \leq 2 \\ 1, & x=0\end{array}\right.\). Then, at \(x=0, f\) has (A) a local maximum (B) no local maximum (C) a local minimum (D) no extremum