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Beyond Advanced · Limits, Continuity and Differentiability

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1
A function f:[0,3]→[0,2]f:[0,3]\to[0,2] is graphed in the figure. Its rule is: f(x)=1+xf(x)=1+x for 0≤x<10\le x<1; f(1)=0f(1)=0; f(x)=3−xf(x)=3-x for 1<x<21<x<2; f(2)=2f(2)=2; f(x)=x−2f(x)=x-2 for 2<x≤32<x\le3. Open dots mark excluded endpoints and filled dots mark the actual values. Let h(x)=f(f(x))h(x)=f(f(x)) wherever defined. Which of the following statements are correct?
Not attempted
xyO12312
A
lim⁡x→1h(x)\lim_{x\to1}h(x) exists and equals 11
B
lim⁡x→2h(x)\lim_{x\to2}h(x) exists
C
lim⁡x→0+h(x)=2\lim_{x\to0^{+}}h(x)=2
D
lim⁡x→2+h(x)=h(2)\lim_{x\to2^{+}}h(x)=h(2)
2
Let D(x)=1D(x)=1 if xx is rational and D(x)=0D(x)=0 if xx is irrational, and let f(x)=x D(x)f(x)=x\,D(x). Which of the following statements are correct?
Not attempted
A
lim⁡x→0f(x)x\lim_{x\to0}\dfrac{f(x)}{x} does not exist
B
lim⁡x→0f(x)∣x∣=0\lim_{x\to0}\dfrac{f(x)}{\sqrt{|x|}}=0
C
lim⁡x→1f(x)\lim_{x\to1}f(x) exists
D
lim⁡x→0(f(x))2x=0\lim_{x\to0}\dfrac{\big(f(x)\big)^{2}}{x}=0
3
Evaluate lim⁡x→−∞[x3+3x2+73+x2−4x+9+xx2+1x2−3]\displaystyle\lim_{x\to-\infty}\left[\sqrt[3]{x^{3}+3x^{2}+7}+\sqrt{x^{2}-4x+9}+\frac{x\sqrt{x^{2}+1}}{x^{2}-3}\right].
Not attempted
Your answer: —
Correct answer: 2
4
Let [⋅][\cdot] denote the greatest integer function. Evaluate lim⁡x→0([sin⁡xx]+[tan⁡xx]+[xsin⁡x]+[sin⁡−1xx]+[xtan⁡−1x]+[tan⁡−1xx])\displaystyle\lim_{x\to0}\left(\left[\frac{\sin x}{x}\right]+\left[\frac{\tan x}{x}\right]+\left[\frac{x}{\sin x}\right]+\left[\frac{\sin^{-1}x}{x}\right]+\left[\frac{x}{\tan^{-1}x}\right]+\left[\frac{\tan^{-1}x}{x}\right]\right).
Not attempted
Your answer: —
Correct answer: 4
5
Let f(x)=[x]2−[x2]f(x)=[x]^{2}-[x^{2}], where [⋅][\cdot] is the greatest integer function. Find the number of points in the open interval (−3,3)(-3,3) at which lim⁡x→af(x)\lim_{x\to a}f(x) does NOT exist.
Not attempted
Your answer: —
Correct answer: 16
6
Evaluate lim⁡x→0sin⁡(πcos⁡2x)ln⁡(cos⁡x)\displaystyle\lim_{x\to0}\frac{\sin\left(\pi\cos^{2}x\right)}{\ln(\cos x)}.
Not attempted
A
2π2\pi
B
−2π-2\pi
C
−π-\pi
D
00
7
Let f,gf,g be real functions defined near aa (not necessarily at aa). Which of the following implications are TRUE?
Not attempted
A
If lim⁡x→a(f(x)+g(x))\lim_{x\to a}\big(f(x)+g(x)\big) and lim⁡x→a(f(x)−g(x))\lim_{x\to a}\big(f(x)-g(x)\big) both exist (finite), then lim⁡x→af(x)\lim_{x\to a}f(x) exists
B
If lim⁡x→af(x)g(x)\lim_{x\to a}f(x)g(x) and lim⁡x→af(x)g(x)\lim_{x\to a}\dfrac{f(x)}{g(x)} both exist (finite), then lim⁡x→af(x)\lim_{x\to a}f(x) exists
C
If lim⁡x→a∣f(x)∣=0\lim_{x\to a}|f(x)|=0, then lim⁡x→af(x)=0\lim_{x\to a}f(x)=0
D
If lim⁡x→a(f(x)+g(x))\lim_{x\to a}\big(f(x)+g(x)\big) and lim⁡x→af(x)g(x)\lim_{x\to a}f(x)g(x) both exist (finite), then lim⁡x→af(x)\lim_{x\to a}f(x) exists
8
Evaluate lim⁡x→1(x3−1)(x4−1)(x6−1)(x−1)3\displaystyle\lim_{x\to1}\frac{\left(\sqrt[3]{x}-1\right)\left(\sqrt[4]{x}-1\right)\left(\sqrt[6]{x}-1\right)}{\left(\sqrt{x}-1\right)^{3}}. Give the answer correct to two decimal places.
Not attempted
Your answer: —
Correct answer: 0.11
9
Here x∘x^{\circ} denotes xx degrees, and sin⁡\sin, tan⁡\tan of a pure number are in radians. Evaluate lim⁡x→0sin⁡(x∘) tan⁡(3x∘)x sin⁡(5x)\displaystyle\lim_{x\to0}\frac{\sin(x^{\circ})\,\tan(3x^{\circ})}{x\,\sin(5x)}.
Not attempted
A
35\dfrac{3}{5}
B
π254000\dfrac{\pi^{2}}{54000}
C
π300\dfrac{\pi}{300}
D
π210800\dfrac{\pi^{2}}{10800}
10
Evaluate lim⁡x→01−cos⁡x cos⁡2x cos⁡3x⋯cos⁡10xx2\displaystyle\lim_{x\to0}\frac{1-\cos x\,\cos2x\,\cos3x\cdots\cos10x}{x^{2}}.
Not attempted
Your answer: —
Correct answer: 192.5
11
Evaluate lim⁡x→0(2x+4x+8x3)2sin⁡x\displaystyle\lim_{x\to0}\left(\frac{2^{x}+4^{x}+8^{x}}{3}\right)^{\frac{2}{\sin x}}.
Not attempted
Your answer: —
Correct answer: 16
12
Evaluate lim⁡x→0+(cot⁡x)1/ln⁡x\displaystyle\lim_{x\to0^{+}}(\cot x)^{1/\ln x}.
Not attempted
A
ee
B
11
C
1e\dfrac{1}{e}
D
00
13
Which of the following limits are correctly evaluated?
Not attempted
A
lim⁡x→0+xsin⁡x=1\displaystyle\lim_{x\to0^{+}}x^{\sin x}=1
B
lim⁡x→∞(1+1x)x2ex=e−1/2\displaystyle\lim_{x\to\infty}\dfrac{\left(1+\frac1x\right)^{x^{2}}}{e^{x}}=e^{-1/2}
C
lim⁡x→∞(ln⁡x)10x0.001=∞\displaystyle\lim_{x\to\infty}\dfrac{(\ln x)^{10}}{x^{0.001}}=\infty
D
lim⁡x→∞(x2+1x2−1)x=e2\displaystyle\lim_{x\to\infty}\left(\dfrac{x^{2}+1}{x^{2}-1}\right)^{x}=e^{2}
14
Evaluate lim⁡x→0e−(1+x)1/xx\displaystyle\lim_{x\to0}\frac{e-(1+x)^{1/x}}{x}, correct to two decimal places.
Not attempted
Your answer: —
Correct answer: 1.36
15
Evaluate lim⁡x→∞(ln⁡(x+1)ln⁡x)xln⁡x\displaystyle\lim_{x\to\infty}\left(\frac{\ln(x+1)}{\ln x}\right)^{x\ln x}.
Not attempted
A
ee
B
11
C
e2e^{2}
D
∞\infty
16
Match each limit in List-I with its value in List-II. List-IList-IIP)lim⁡x→0(1+3x)2/x1)e6Q)lim⁡x→∞(x+2x−3)x2)e5R)lim⁡x→0+x(xx)3)0S)lim⁡x→0(cos⁡x)cot⁡2x4)15)e−1/2\begin{array}{ll} \textbf{List-I} & \textbf{List-II} \\ \text{P)} \displaystyle\lim_{x\to0}(1+3x)^{2/x} & \text{1)} e^{6} \\ \text{Q)} \displaystyle\lim_{x\to\infty}\left(\frac{x+2}{x-3}\right)^{x} & \text{2)} e^{5} \\ \text{R)} \displaystyle\lim_{x\to0^{+}}x^{\left(x^{x}\right)} & \text{3)} 0 \\ \text{S)} \displaystyle\lim_{x\to0}(\cos x)^{\cot^{2}x} & \text{4)} 1 \\ & \text{5)} e^{-1/2} \end{array}
Not attempted
A
P-2, Q-1, R-3, S-5
B
P-1, Q-2, R-4, S-3
C
P-5, Q-2, R-3, S-1
D
P-1, Q-2, R-3, S-5
17
Let L=lim⁡x→0sin⁡(tan⁡x)−tan⁡(sin⁡x)x7L=\displaystyle\lim_{x\to0}\frac{\sin(\tan x)-\tan(\sin x)}{x^{7}}. Find ∣1L∣\left|\dfrac1L\right|.
Not attempted
Your answer: —
Correct answer: 30
18
Real constants a,b,ca,b,c satisfy lim⁡x→0asin⁡x+bsin⁡2x+csin⁡3x−xx5=120\displaystyle\lim_{x\to0}\frac{a\sin x+b\sin2x+c\sin3x-x}{x^{5}}=\frac1{20}. Find a+b+ca+b+c.
Not attempted
A
43\dfrac43
B
11
C
53\dfrac53
D
74\dfrac74
19
Which of the following statements are correct?
Not attempted
A
lim⁡x→∞x+sin⁡xx−sin⁡x=1\displaystyle\lim_{x\to\infty}\frac{x+\sin x}{x-\sin x}=1, although the ratio of derivatives 1+cos⁡x1−cos⁡x\dfrac{1+\cos x}{1-\cos x} has no limit as x→∞x\to\infty
B
lim⁡x→0x2sin⁡(1/x)sin⁡x=0\displaystyle\lim_{x\to0}\frac{x^{2}\sin(1/x)}{\sin x}=0, although the ratio of derivatives of numerator and denominator has no limit at 00
C
lim⁡x→0e−1/x2xn=0\displaystyle\lim_{x\to0}\frac{e^{-1/x^{2}}}{x^{n}}=0 for every positive integer nn
D
lim⁡x→0sin⁡x−xcos⁡xx3=12\displaystyle\lim_{x\to0}\frac{\sin x-x\cos x}{x^{3}}=\frac12
20
Evaluate lim⁡x→0−  x([12x]+[22x]+[32x]+⋯+[122x])\displaystyle\lim_{x\to0^{-}}\;x\left(\left[\frac{1^{2}}{x}\right]+\left[\frac{2^{2}}{x}\right]+\left[\frac{3^{2}}{x}\right]+\cdots+\left[\frac{12^{2}}{x}\right]\right), where [⋅][\cdot] is the greatest integer function.
Not attempted
Your answer: —
Correct answer: 650
21
Evaluate lim⁡x→0cos⁡x−e−x2/2x4\displaystyle\lim_{x\to0}\frac{\cos x-e^{-x^{2}/2}}{x^{4}}.
Not attempted
A
112\dfrac1{12}
B
−18-\dfrac18
C
−112-\dfrac1{12}
D
124\dfrac1{24}
22
Match each limit in List-I with its value in List-II. List-IList-IIP)lim⁡x→0tan⁡x−sin⁡xx31)16Q)lim⁡x→0x−sin⁡xx32)12R)lim⁡x→0tan⁡x−xx−sin⁡x3)1S)lim⁡x→0sin⁡−1x−tan⁡−1xtan⁡x−sin⁡x4)25)13\begin{array}{ll} \textbf{List-I} & \textbf{List-II} \\ \text{P)} \displaystyle\lim_{x\to0}\frac{\tan x-\sin x}{x^{3}} & \text{1)} \frac16 \\ \text{Q)} \displaystyle\lim_{x\to0}\frac{x-\sin x}{x^{3}} & \text{2)} \frac12 \\ \text{R)} \displaystyle\lim_{x\to0}\frac{\tan x-x}{x-\sin x} & \text{3)} 1 \\ \text{S)} \displaystyle\lim_{x\to0}\frac{\sin^{-1}x-\tan^{-1}x}{\tan x-\sin x} & \text{4)} 2 \\ & \text{5)} \frac13 \end{array}
Not attempted
A
P-1, Q-2, R-4, S-3
B
P-2, Q-1, R-3, S-4
C
P-5, Q-1, R-4, S-2
D
P-2, Q-1, R-4, S-3
23
Let f(x)=[x2]−[x]2f(x)=[x^{2}]-[x]^{2} for 0≤x<20\le x<2, where [⋅][\cdot] is the greatest integer function. Each option shows a graph on [0,2)[0,2) (filled dot: value attained; open dot: value not attained). Which graph is that of ff?
Not attempted
A
Graph P
xyO12-2-1123Graph P
B
Graph Q
xyO12-2-1123Graph Q
C
Graph R
xyO12-2-1123Graph R
D
Graph S
xyO12-2-1123Graph S
24
Let f(x)=[2sin⁡x]f(x)=[2\sin x] for 0≤x≤2π0\le x\le2\pi, where [⋅][\cdot] is the greatest integer function. Find the number of points of [0,2π][0,2\pi] at which ff is discontinuous (continuity at an endpoint means continuity from inside the interval).
Not attempted
Your answer: —
Correct answer: 7
25
For which values of aa is the following true? \"For EVERY continuous function f:[0,1]→Rf:[0,1]\to\mathbb R with f(0)=f(1)f(0)=f(1) there exist xx and x+ax+a in [0,1][0,1] with f(x)=f(x+a)f(x)=f(x+a).\"
Not attempted
A
a=12a=\dfrac12
B
a=23a=\dfrac23
C
a=14a=\dfrac14
D
a=35a=\dfrac35
26
For real constants a,ba,b define f(x)=lim⁡n→∞(ax+b)+(x2+3x) x2n1+x2n\displaystyle f(x)=\lim_{n\to\infty}\frac{(ax+b)+(x^{2}+3x)\,x^{2n}}{1+x^{2n}} for every real xx at which the limit exists. If ff is continuous at both x=1x=1 and x=−1x=-1, find a2+b2a^{2}+b^{2}.
Not attempted
A
55
B
1313
C
1010
D
88
27
Which of the following statements are TRUE?
Not attempted
A
There is a continuous f:R→Rf:\mathbb R\to\mathbb R with f(f(x))=−xf(f(x))=-x for all xx
B
There is a continuous f:R→Rf:\mathbb R\to\mathbb R with f(f(x))=x3f(f(x))=x^{3} for all xx
C
There is a continuous bijection f:[0,1]→[0,1]f:[0,1]\to[0,1] that is neither increasing nor decreasing
D
There is a continuous f:R→Rf:\mathbb R\to\mathbb R with f(f(x))=2xf(f(x))=2x for all xx
28
Match each function in List-I (where [⋅][\cdot] is the greatest integer function) with the behaviour at x=0x=0 in List-II. List-IList-IIP)f(x)=[cos⁡x] with f(0)=[cos⁡0]1)Removable discontinuity at 0Q)f(x)=sin⁡x∣x∣ for x≠0, f(0)=02)Jump discontinuity at 0R)f(x)=1xsin⁡1x for x≠0, f(0)=03)Infinite discontinuity at 0S)f(x)=x2sin⁡1x for x≠0, f(0)=04)Oscillatory discontinuity at 05)Continuous at 0\begin{array}{ll} \textbf{List-I} & \textbf{List-II} \\ \text{P)} f(x)=[\cos x]\ \text{with}\ f(0)=[\cos0] & \text{1)} \text{Removable discontinuity at }0 \\ \text{Q)} f(x)=\dfrac{\sin x}{|x|}\ \text{for}\ x\neq0,\ f(0)=0 & \text{2)} \text{Jump discontinuity at }0 \\ \text{R)} f(x)=\dfrac1x\sin\dfrac1x\ \text{for}\ x\neq0,\ f(0)=0 & \text{3)} \text{Infinite discontinuity at }0 \\ \text{S)} f(x)=x^{2}\sin\dfrac1x\ \text{for}\ x\neq0,\ f(0)=0 & \text{4)} \text{Oscillatory discontinuity at }0 \\ & \text{5)} \text{Continuous at }0 \end{array}
Not attempted
A
P-2, Q-1, R-4, S-5
B
P-1, Q-2, R-4, S-5
C
P-1, Q-2, R-5, S-4
D
P-5, Q-2, R-4, S-1
29
Let D(x)=1D(x)=1 for rational xx and D(x)=0D(x)=0 for irrational xx, and define f(x)=x2D(x)+(2x−1)(1−D(x))f(x)=x^{2}D(x)+(2x-1)\big(1-D(x)\big). Which statement about differentiability of ff is correct?
Not attempted
A
ff is differentiable at exactly one point, x=1x=1, and f′(1)=0f'(1)=0
B
ff is continuous only at x=1x=1, and not differentiable there
C
ff is differentiable nowhere
D
ff is differentiable at exactly one point, x=1x=1, and f′(1)=2f'(1)=2
30
Let f(x)=∣∣x2−4x∣−4∣f(x)=\big||x^{2}-4x|-4\big|. At how many points of R\mathbb R is ff NOT differentiable?
Not attempted
A
55
B
33
C
66
D
44
31
Let f(x)=x3sin⁡1xf(x)=x^{3}\sin\dfrac1x for x≠0x\ne0 and f(0)=0f(0)=0. Which of the following statements are correct?
Not attempted
xyO
A
f′(0)f'(0) exists and equals 00
B
f′f' is continuous at 00
C
f′′(0)f''(0) exists
D
f′′(x)f''(x) is unbounded in every neighbourhood of 00 (for x≠0x\ne0)
32
Let f(x)=sin⁡xf(x)=\sin x for x≤π6x\le\dfrac\pi6 and f(x)=ax2+bx+cf(x)=ax^{2}+bx+c for x>π6x>\dfrac\pi6, where a,b,ca,b,c are constants chosen so that ff is twice differentiable at x=π6x=\dfrac\pi6. Find f ⁣(π2)f\!\left(\dfrac\pi2\right) correct to two decimal places. (Use π=3.1416\pi=3.1416, 3=1.7321\sqrt3=1.7321.)
Not attempted
Your answer: —
Correct answer: 1.13
33
A function f:R→Rf:\mathbb R\to\mathbb R satisfies f(x+y)=f(x)f(y)f(x+y)=f(x)f(y) for all real x,yx,y, and f(x)=1+x g(x)f(x)=1+x\,g(x) where lim⁡x→0g(x)=ln⁡3\displaystyle\lim_{x\to0}g(x)=\ln3. Find f′(2)f'(2), correct to two decimal places.
Not attempted
Your answer: —
Correct answer: 9.89
34
Here [⋅][\cdot] denotes the greatest integer function. Which of the following are correct?
Not attempted
A
(x−1) ∣x−1∣(x-1)\,|x-1| is twice differentiable at x=1x=1
B
[x]sin⁡(πx)[x]\sin(\pi x) is differentiable at every integer
C
[x]sin⁡2(πx)[x]\sin^{2}(\pi x) is differentiable at every integer
D
sin⁡∣x∣\sin|x| is differentiable at x=0x=0
35
Match each function (all at x=0x=0) in List-I with the correct description in List-II. List-IList-IIP)f(x)=x2sin⁡1x, f(0)=01)differentiable at 0, and f′ is continuous at 0Q)f(x)=∣x∣3/22)differentiable at 0, but f′ is not continuous at 0R)f(x)=xsin⁡1x, f(0)=03)corner at 0S)f(x)=∣x∣4)one-sided derivatives at 0 are infinite5)continuous at 0, not differentiable, and the difference quotient oscillates\begin{array}{ll} \textbf{List-I} & \textbf{List-II} \\ \text{P)} f(x)=x^{2}\sin\dfrac1x,\ f(0)=0 & \text{1)} \text{differentiable at }0\text{, and }f'\text{ is continuous at }0 \\ \text{Q)} f(x)=|x|^{3/2} & \text{2)} \text{differentiable at }0\text{, but }f'\text{ is not continuous at }0 \\ \text{R)} f(x)=x\sin\dfrac1x,\ f(0)=0 & \text{3)} \text{corner at }0 \\ \text{S)} f(x)=\sqrt{|x|} & \text{4)} \text{one-sided derivatives at }0\text{ are infinite} \\ & \text{5)} \text{continuous at }0\text{, not differentiable, and the difference quotient oscillates} \end{array}
Not attempted
A
P-2, Q-1, R-5, S-4
B
P-1, Q-2, R-5, S-4
C
P-2, Q-1, R-4, S-5
D
P-5, Q-1, R-2, S-4
36
Let u(x)=sin⁡−12x1+x2u(x)=\sin^{-1}\dfrac{2x}{1+x^{2}} and v(x)=tan⁡−12x1−x2v(x)=\tan^{-1}\dfrac{2x}{1-x^{2}} (principal values). Find the derivative of uu with respect to vv at x=3x=\sqrt3.
Not attempted
A
−1-1
B
11
C
22
D
−2-2
37
Let f(x)=x3+3xf(x)=x^{3}+3x, which is strictly increasing, and let g=f−1g=f^{-1}. Find g′′(4)g''(4).
Not attempted
A
136\dfrac1{36}
B
−16-\dfrac16
C
−136-\dfrac1{36}
D
−1216-\dfrac1{216}
38
Which of the following statements are correct (in each case on the stated domain, with principal values)?
Not attempted
A
ddxsec⁡−1x=1∣x∣x2−1\dfrac{d}{dx}\sec^{-1}x=\dfrac{1}{|x|\sqrt{x^{2}-1}} for all ∣x∣>1|x|>1
B
ddxcos⁡−1 ⁣(4x3−3x)=−31−x2\dfrac{d}{dx}\cos^{-1}\!\left(4x^{3}-3x\right)=-\dfrac{3}{\sqrt{1-x^{2}}} for every x∈(−1,1)x\in(-1,1)
C
ddx[tan⁡−1x+tan⁡−11x]=0\dfrac{d}{dx}\left[\tan^{-1}x+\tan^{-1}\dfrac1x\right]=0 for every x≠0x\neq0
D
ddxln⁡∣x2−1∣=2xx2−1\dfrac{d}{dx}\ln\left|x^{2}-1\right|=\dfrac{2x}{x^{2}-1} for every x≠±1x\neq\pm1
39
Let f(x)=x(xx)f(x)=x^{\left(x^{x}\right)} and g(x)=(xx)xg(x)=\left(x^{x}\right)^{x} for x>0x>0. Find f′(2)−g′(2)f'(2)-g'(2) correct to two decimal places. (Use ln⁡2=0.6931\ln2=0.6931.)
Not attempted
Your answer: —
Correct answer: 30.75
40
Let φ(x)=sin⁡−1(sin⁡x)+cos⁡−1(cos⁡x)+tan⁡−1(tan⁡x)\varphi(x)=\sin^{-1}(\sin x)+\cos^{-1}(\cos x)+\tan^{-1}(\tan x) (principal values, xx in radians, where defined). Find φ′(2)+φ′(4)+φ′(5)+φ′(7)\varphi'(2)+\varphi'(4)+\varphi'(5)+\varphi'(7). (Use π=3.14\pi=3.14.)
Not attempted
Your answer: —
Correct answer: 4
41
The curve x3+y3=3xyx^{3}+y^{3}=3xy passes through the point (32,32)\left(\tfrac32,\tfrac32\right). Find d2ydx2\dfrac{d^{2}y}{dx^{2}} at this point.
Not attempted
xyO(3/2, 3/2)
A
323\dfrac{32}{3}
B
−323-\dfrac{32}{3}
C
−163-\dfrac{16}{3}
D
−83-\dfrac{8}{3}
42
A curve is given by x=etcos⁡tx=e^{t}\cos t, y=etsin⁡ty=e^{t}\sin t. Find the value of eπ/6(d2ydx2)2e^{\pi/6}\left(\dfrac{d^{2}y}{dx^{2}}\right)^{2} at t=π12t=\dfrac{\pi}{12}.
Not attempted
Your answer: —
Correct answer: 32
43
Which of the following statements are correct?
Not attempted
A
If y=(sin⁡x)cos⁡xy=(\sin x)^{\cos x}, then y′ ⁣(π2)=1y'\!\left(\dfrac\pi2\right)=1
B
If y=xx2y=x^{x^{2}} (x>0x>0), then y′(1)=1y'(1)=1
C
If y=(ln⁡x)xy=(\ln x)^{x} (x>1x>1), then y′(e)=1y'(e)=1
D
The tenth derivative of x2exx^{2}e^{x} at x=0x=0 equals 4545
44
Let y=(sin⁡−1x)2y=\left(\sin^{-1}x\right)^{2} and let yny_{n} denote dnydxn\dfrac{d^{n}y}{dx^{n}}. Find y8(0)y_{8}(0).
Not attempted
Your answer: —
Correct answer: 4608
45
For 1<x≤21<x\le\sqrt2 the infinite power tower y=xxx⋅⋅⋅y=x^{x^{x^{\cdot^{\cdot^{\cdot}}}}} converges. Find dydx\dfrac{dy}{dx} at x=2x=\sqrt2.
Not attempted
A
221+ln⁡2\dfrac{2\sqrt2}{1+\ln2}
B
21−ln⁡2\dfrac{\sqrt2}{1-\ln2}
C
42 (1−2ln⁡2)\dfrac{4}{\sqrt2\,(1-2\ln2)}
D
221−ln⁡2\dfrac{2\sqrt2}{1-\ln2}
46
The relation x=y+eyx=y+e^{y} defines yy implicitly as a differentiable function of xx. Find d2ydx2\dfrac{d^{2}y}{dx^{2}} at x=1x=1.
Not attempted
A
18\dfrac18
B
−14-\dfrac14
C
−12-\dfrac12
D
−18-\dfrac18
47
For an integer kk, let N(k)N(k) be the number of distinct tangent lines to the curve y=x3−3xy=x^{3}-3x that pass through the point (2,k)(2,k). How many integers kk satisfy N(k)=3N(k)=3?
Not attempted
xyOx = 2−11
Your answer: —
Correct answer: 7
48
Which of the following statements are correct?
Not attempted
xyOO(4, 4)y² = 4xx² = 4y
A
The curves x2−y2=5x^{2}-y^{2}=5 and x218+y28=1\dfrac{x^{2}}{18}+\dfrac{y^{2}}{8}=1 intersect at right angles at each point of intersection
B
The curves y2=4xy^{2}=4x and x2=4yx^{2}=4y intersect at (4,4)(4,4) at an angle tan⁡−134\tan^{-1}\dfrac34
C
The curves y=sin⁡xy=\sin x and y=cos⁡xy=\cos x intersect at x=π4x=\dfrac\pi4 at an angle tan⁡−1(22)\tan^{-1}\left(2\sqrt2\right)
D
The curves y=x2y=x^{2} and y=x3y=x^{3} cut each other orthogonally at (1,1)(1,1)
49
For a>1a>1 the curves y=axy=a^{x} and y=log⁡axy=\log_{a}x touch each other at exactly one point. Find aa.
Not attempted
xyOy = aˣy = log_a xy = x
A
eee^{e}
B
e1/ee^{1/e}
C
e2/ee^{2/e}
D
e\sqrt{e}
50
A ladder of length 1010 m leans against a vertical wall with its foot on level ground. The foot is pulled away from the wall at a constant speed of 22 m/s. Find the speed (in m/s) of the midpoint of the ladder at the instant when the foot is 66 m from the wall.
Not attempted
Your answer: —
Correct answer: 1.25
51
Paragraph I A right circular cone has base radius r=3.00 cmr=3.00\ \text{cm} and height h=4.00 cmh=4.00\ \text{cm}, so its slant height is l=r2+h2=5.00 cml=\sqrt{r^{2}+h^{2}}=5.00\ \text{cm}. The radius and the height are each measured with maximum absolute errors ∣Δr∣=0.03 cm|\Delta r|=0.03\ \text{cm} and ∣Δh∣=0.02 cm|\Delta h|=0.02\ \text{cm}. Treat all errors as small and use differentials (first-order approximations). The errors in rr and hh are independent and may have either sign, and the slant height is not measured separately but computed from rr and hh. Question 1 of 2: Find the maximum percentage error in the volume V=13πr2hV=\frac13\pi r^{2}h.
Not attempted
hrl
Your answer: —
Correct answer: 2.5
52
Paragraph I A right circular cone has base radius r=3.00 cmr=3.00\ \text{cm} and height h=4.00 cmh=4.00\ \text{cm}, so its slant height is l=r2+h2=5.00 cml=\sqrt{r^{2}+h^{2}}=5.00\ \text{cm}. The radius and the height are each measured with maximum absolute errors ∣Δr∣=0.03 cm|\Delta r|=0.03\ \text{cm} and ∣Δh∣=0.02 cm|\Delta h|=0.02\ \text{cm}. Treat all errors as small and use differentials (first-order approximations). The errors in rr and hh are independent and may have either sign, and the slant height is not measured separately but computed from rr and hh. Question 2 of 2: Find the maximum percentage error in the curved surface area S=πrlS=\pi r l of the cone, where l=r2+h2l=\sqrt{r^{2}+h^{2}}.
Not attempted
hrl
Your answer: —
Correct answer: 1.68
53
Let f(x)=(x−1)(x−2)2(x−3)3⋯(x−10)10f(x)=(x-1)(x-2)^{2}(x-3)^{3}\cdots(x-10)^{10}. Find the number of distinct real zeros of f′(x)f'(x).
Not attempted
Your answer: —
Correct answer: 18
54
Let f:[a,b]→Rf:[a,b]\to\mathbb R be continuous, differentiable on (a,b)(a,b), with f(a)=f(b)=0f(a)=f(b)=0. Which of the following statements must be true for EVERY such ff?
Not attempted
A
There is c∈(a,b)c\in(a,b) with f′(c)=f(c)f'(c)=f(c)
B
There is c∈(a,b)c\in(a,b) with f′(c)+f(c)=0f'(c)+f(c)=0
C
There is c∈(a,b)c\in(a,b) with f′(c)=c f(c)f'(c)=c\,f(c)
D
There is c∈(a,b)c\in(a,b) with f′(c)=c2f(c)f'(c)=c^{2}f(c)
55
The graph of the derivative f′(x)=(x+2)(x−1)2(x−3)10f'(x)=\dfrac{(x+2)(x-1)^{2}(x-3)}{10} of a differentiable function f:R→Rf:\mathbb R\to\mathbb R is shown in the figure (it cuts the xx-axis at −2-2 and 33 and only touches it at 11), and f(0)=0f(0)=0. Which option shows the graph of ff for −3≤x≤5-3\le x\le5? (All options use the same scales.)
Not attempted
xy = f ′(x)O-213
A
Graph Q
xyO-213Graph Q
B
Graph R
xyO-213Graph R
C
Graph P
xyO-213Graph P
D
Graph S
xyO-213Graph S
56
Find the number of integers kk for which the equation x4−6x2+k=0x^{4}-6x^{2}+k=0 has four distinct real roots.
Not attempted
xy = x⁴ − 6x²O-2-112
Your answer: —
Correct answer: 8
57
For which of the following does the conclusion of the Lagrange mean value theorem FAIL, i.e. there is NO cc in the open interval with f′(c)=f(b)−f(a)b−af'(c)=\dfrac{f(b)-f(a)}{b-a}?
Not attempted
A
f(x)=x1/3f(x)=x^{1/3} on [−1,1][-1,1]
B
f(x)=∣x∣f(x)=|x| on [−1,2][-1,2]
C
f(x)=x2/3f(x)=x^{2/3} on [−1,8][-1,8]
D
f(x)=x∣x∣f(x)=x|x| on [−1,2][-1,2]
58
Match each statement in List-I with the largest open interval in List-II (inside the domain of the function) on which it is true. List-IList-IIP)x e−x is strictly decreasing on1)(1,e)Q)x2e−x is strictly increasing on2)(0,2)R)ln⁡xx is strictly increasing on3)(0,e)S)xln⁡x (x>1) is strictly decreasing on4)(e,∞)5)(1,∞)\begin{array}{ll} \textbf{List-I} & \textbf{List-II} \\ \text{P)} x\,e^{-x}\ \text{is strictly decreasing on} & \text{1)} (1,e) \\ \text{Q)} x^{2}e^{-x}\ \text{is strictly increasing on} & \text{2)} (0,2) \\ \text{R)} \dfrac{\ln x}{x}\ \text{is strictly increasing on} & \text{3)} (0,e) \\ \text{S)} \dfrac{x}{\ln x}\ (x>1)\ \text{is strictly decreasing on} & \text{4)} (e,\infty) \\ & \text{5)} (1,\infty) \end{array}
Not attempted
A
P-1, Q-2, R-3, S-5
B
P-5, Q-2, R-3, S-1
C
P-5, Q-3, R-2, S-1
D
P-2, Q-5, R-3, S-4
59
A right circular cone is inscribed in a sphere of radius 33 (its vertex and the whole circumference of its base lie on the sphere). Find the maximum possible curved (lateral) surface area of such a cone, correct to two decimal places. (Use π=3.1416\pi=3.1416 and 3=1.7321\sqrt3=1.7321.)
Not attempted
O3vertexbase circle
Your answer: —
Correct answer: 43.53
60
Which of the following statements are correct?
Not attempted
A
For f(x)=x2sin⁡1xf(x)=x^{2}\sin\dfrac1x (x≠0x\ne0), f(0)=0f(0)=0: f′(0)=0f'(0)=0 but ff has neither a local maximum nor a local minimum at 00
B
For f(x)=e−1/x2f(x)=e^{-1/x^{2}} (x≠0x\ne0), f(0)=0f(0)=0: every derivative of ff vanishes at 00, yet ff has a (global) minimum at 00
C
For f(x)=xsin⁡1xf(x)=x\sin\dfrac1x (x≠0x\ne0), f(0)=0f(0)=0: ff has a local minimum at 00 because f(0)=0f(0)=0 is the smallest value of ∣f∣|f|
D
There is a function with f′(0)=f′′(0)=0f'(0)=f''(0)=0 that has a local extremum at 00
61
Find the minimum value of f(x)=x2−2x+5+x2−8x+25f(x)=\sqrt{x^{2}-2x+5}+\sqrt{x^{2}-8x+25} for real xx.
Not attempted
xyA (1, 2)B (4, 3)P (x, 0)
A
34\sqrt{34}
B
10\sqrt{10}
C
55
D
77
62
Find the shortest distance from the point (0,3)(0,3) to the parabola y=x2y=x^{2}.
Not attempted
xyO(0, 3)y = x²
A
112\dfrac{\sqrt{11}}{2}
B
33
C
22
D
114\dfrac{\sqrt{11}}{4}
63
Paragraph II Two corridors of constant widths 2727 m (horizontal corridor) and 6464 m (vertical corridor) meet at a right angle, as in the figure. A thin straight rod is carried horizontally around the corner, always touching the inner corner of the junction, with its ends on the outer walls. When the rod makes an angle θ\theta (0<θ<π20<\theta<\frac\pi2) with the horizontal, its length from outer wall to outer wall is L(θ)=27sin⁡θ+64cos⁡θL(\theta)=\dfrac{27}{\sin\theta}+\dfrac{64}{\cos\theta}. A rod can be carried round the corner only if it is no longer than the minimum of L(θ)L(\theta). Question 1 of 2: Find the length (in metres) of the longest rod that can be carried around the corner.
Not attempted
width 27width 64θcorner
Your answer: —
Correct answer: 125
64
Paragraph II Two corridors of constant widths 2727 m (horizontal corridor) and 6464 m (vertical corridor) meet at a right angle, as in the figure. A thin straight rod is carried horizontally around the corner, always touching the inner corner of the junction, with its ends on the outer walls. When the rod makes an angle θ\theta (0<θ<π20<\theta<\frac\pi2) with the horizontal, its length from outer wall to outer wall is L(θ)=27sin⁡θ+64cos⁡θL(\theta)=\dfrac{27}{\sin\theta}+\dfrac{64}{\cos\theta}. A rod can be carried round the corner only if it is no longer than the minimum of L(θ)L(\theta). Question 2 of 2: Find the angle θ\theta, in degrees, at which the rod of maximum carriable length is oriented when it is passing the corner in its tightest position. (Use tan⁡−1(0.75)\tan^{-1}(0.75) in degrees =36.87∘=36.87^\circ if needed.)
Not attempted
width 27width 64θcorner
Your answer: —
Correct answer: 36.87
65
Which of the following is the graph of f(x)=x e−xf(x)=x\,e^{-x} for −2≤x≤4-2\le x\le4? The scale is the same in every option.
Not attempted
A
Graph P
xyO-1123-2-11Graph P
B
Graph Q
xyO-1123-2-11Graph Q
C
Graph R
xyO-1123-2-11Graph R
D
Graph S
xyO-1123-2-11Graph S
66
For each integer kk from −3-3 to 55 (inclusive) let N(k)N(k) be the number of real solutions of ex=kxe^{x}=kx. Find ∑k=−35N(k)\displaystyle\sum_{k=-3}^{5}N(k).
Not attempted
Your answer: —
Correct answer: 9
67
Which of the following statements are correct?
Not attempted
A
The function f(x)=x4−4x3f(x)=x^{4}-4x^{3} has exactly one point of inflection
B
The function f(x)=x5/3f(x)=x^{5/3} has a point of inflection at x=0x=0 although f′′(0)f''(0) does not exist
C
The function f(x)=x6f(x)=x^{6} has a point of inflection at x=0x=0 because f′′(0)=f′′′(0)=0f''(0)=f'''(0)=0
D
The function f(x)=e−x2f(x)=e^{-x^{2}} has exactly two points of inflection
68
Which of the following is the complete set of asymptotes of the curve y=x e1/xy=x\,e^{1/x}?
Not attempted
A
y=x+1y=x+1 only
B
x=0x=0 and y=xy=x
C
x=0x=0 and y=x+1y=x+1
D
x=0x=0, y=x+1y=x+1 and y=x−1y=x-1
69
Paragraph III Let g(x)=x e−x2/2g(x)=x\,e^{-x^{2}/2} for real xx; its graph is shown in the figure. It is an odd function, g′(x)=(1−x2)e−x2/2g'(x)=(1-x^{2})e^{-x^{2}/2} and g′′(x)=x(x2−3)e−x2/2g''(x)=x(x^{2}-3)e^{-x^{2}/2}. For 0<a<10<a<1 the line y=axy=ax meets the graph at the origin and at two further points. Question 1 of 2: Let MM be the maximum value and mm the minimum value of the slope g′(x)g'(x) over all real xx. Find ∣Mm∣\left|\dfrac{M}{m}\right|, correct to two decimal places. (Use e1.5=4.4817e^{1.5}=4.4817.)
Not attempted
xyO-3-2-11230.5−0.5y = g(x)
Your answer: —
Correct answer: 2.24
70
Paragraph III Let g(x)=x e−x2/2g(x)=x\,e^{-x^{2}/2} for real xx; its graph is shown in the figure. It is an odd function, g′(x)=(1−x2)e−x2/2g'(x)=(1-x^{2})e^{-x^{2}/2} and g′′(x)=x(x2−3)e−x2/2g''(x)=x(x^{2}-3)e^{-x^{2}/2}. For 0<a<10<a<1 the line y=axy=ax meets the graph at the origin and at two further points. Question 2 of 2: For 0<a<10<a<1, the line y=axy=ax meets the graph of gg at three points; the two outer points are at a distance 222\sqrt2 from each other. Find aa, correct to two decimal places. (Use e−1=0.3679e^{-1}=0.3679.)
Not attempted
xyO-3-2-11230.5−0.5y = g(x)
Your answer: —
Correct answer: 0.37
Solve g(x)=axg(x)=ax: one root is x=0x=0; for the others divide by xx.
xe−x2/2=axxe^{-x^{2}/2}=ax gives x=0x=0 or e−x2/2=ae^{-x^{2}/2}=a, i.e. x2=2ln⁡1ax^{2}=2\ln\frac1a, so x=±2ln⁡(1/a)x=\pm\sqrt{2\ln(1/a)} (real since 0<a<10<a<1). The distance between the outer points is 22ln⁡(1/a)=222\sqrt{2\ln(1/a)}=2\sqrt2, so ln⁡1a=1\ln\frac1a=1 and a=e−1≈0.37a=e^{-1}\approx0.37. The graph (odd, with maximum slope 11 at the origin) lies above y=axy=ax for small x>0x>0 only when a<1a<1, which is why three intersections occur exactly for 0<a<10<a<1.