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JEE Advanced · Mathematics · Limits, Continuity and Differentiability

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1
For α∈R\alpha\in\mathbb R, let [α][\alpha] denote the greatest integer ≤α\le\alpha. Consider f,g:(−3,3)→(−∞,∞)f,g:(-3,3)\to(-\infty,\infty) defined by f(x)=[x3]log⁡e(1+sin⁡2(π(x−[x])))f(x)=[x^3]\log_e(1+\sin^2(\pi(x-[x]))) and g(x)=x3sin⁡2(πlog⁡e(1+x−[x]))g(x)=x^3\sin^2(\pi\log_e(1+x-[x])). Let A={x∈(−3,3):f discontinuous at x}A=\{x\in(-3,3): f \text{ discontinuous at } x\} and B={x∈(−3,3):g discontinuous at x}B=\{x\in(-3,3): g \text{ discontinuous at }x\}. Find ∣A∣+2∣B∣−∣A∩B∣|A|+2|B|-|A\cap B|.
Not attempted
Your answer: —
Correct answer: 56
2
Consider the function f:(0,∞)→(−∞,∞)f: (0, \infty) \to (-\infty, \infty) given by f(x)=x log⁡e(x)−x+1f(x) = \sqrt{x}\,\log_e(x) - x + 1. Then which one of the following statements is TRUE?
Not attempted
A
The derivative of the function ff is decreasing in the interval (0,1)(0,1)
B
The function ff has a local maximum at some point a∈(0,∞)a \in (0,\infty)
C
The function ff has a local minimum at some point b∈(0,∞)b \in (0,\infty)
D
The function ff has NEITHER a point of local maximum NOR a point of local minimum in (0,∞)(0,\infty)
3
Let f:R→Rf:\mathbb R\to\mathbb R be an arbitrary function and let g:R→Rg:\mathbb R\to\mathbb R be defined by g(x)=xf(x)g(x)=xf(x) for all x∈Rx\in\mathbb R. Then which of the following statements is (are) TRUE?
Not attempted
A
The function gg is always continuous at x=0x=0
B
If ff is continuous at x=0x=0, then gg is differentiable at x=0x=0
C
If gg is differentiable at x=0x=0, then ff is continuous at x=0x=0
D
If gg is differentiable at x=0x=0, then lim⁡x→0f(x)\lim_{x\to0}f(x) exists
4
Consider f:(−π2,π2)→(−∞,∞)f:\left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right)\to(-\infty,\infty) defined by f(x)=(∣x∣+∣x−1∣)sin⁡x+[xsin⁡x]f(x)=(|x|+|x-1|)\sin x + [x\sin x], where [⋅][\cdot] is the greatest integer function. Let α\alpha be the number of points in (−π2,π2)\left(-\dfrac\pi2,\dfrac\pi2\right) where ff is NOT continuous, and β\beta the number of points where ff is NOT differentiable. Find α+β\alpha+\beta.
Not attempted
Your answer: —
Correct answer: 5
5
Let x0x_0 be the real number such that ex0+x0=0e^{x_0} + x_0 = 0. For a given real number α\alpha, define g(x)=3xex+3x−αex−αx3(ex+1)g(x) = \dfrac{3xe^{x} + 3x - \alpha e^{x} - \alpha x}{3(e^{x}+1)} for all real numbers xx. Then which one of the following statements is TRUE?
Not attempted
A
For α=2\alpha=2, lim⁡x→x0∣g(x)+ex0x−x0∣=0\displaystyle\lim_{x\to x_0}\left|\dfrac{g(x)+e^{x_0}}{x-x_0}\right| = 0
B
For α=2\alpha=2, lim⁡x→x0∣g(x)+ex0x−x0∣=1\displaystyle\lim_{x\to x_0}\left|\dfrac{g(x)+e^{x_0}}{x-x_0}\right| = 1
C
For α=3\alpha=3, lim⁡x→x0∣g(x)+ex0x−x0∣=0\displaystyle\lim_{x\to x_0}\left|\dfrac{g(x)+e^{x_0}}{x-x_0}\right| = 0
D
For α=3\alpha=3, lim⁡x→x0∣g(x)+ex0x−x0∣=23\displaystyle\lim_{x\to x_0}\left|\dfrac{g(x)+e^{x_0}}{x-x_0}\right| = \dfrac{2}{3}
6
Let R\mathbb{R} denote the set of all real numbers. Let f:R→Rf:\mathbb{R}\to\mathbb{R} be defined by f(x)={6x+sin⁡x2x+sin⁡xif x≠0,73if x=0.f(x)=\begin{cases}\dfrac{6x+\sin x}{2x+\sin x} & \text{if } x\ne0,\\[4pt] \dfrac73 & \text{if } x=0.\end{cases} Then which of the following statements is (are) TRUE?
Not attempted
A
The point x=0x=0 is a point of local maxima of ff
B
The point x=0x=0 is a point of local minima of ff
C
Number of points of local maxima of ff in the interval [π,6π][\pi,6\pi] is 33
D
Number of points of local minima of ff in the interval [2π,4π][2\pi,4\pi] is 11
7
Let R\mathbb{R} denote the set of all real numbers. Let f:R→Rf:\mathbb{R}\to\mathbb{R} and g:R→(0,4)g:\mathbb{R}\to(0,4) be functions defined by f(x)=log⁡e(x2+2x+4),g(x)=41+e−2x.f(x)=\log_e(x^2+2x+4),\qquad g(x)=\dfrac{4}{1+e^{-2x}}. Define the composite function f∘g−1f\circ g^{-1} by (f∘g−1)(x)=f(g−1(x))(f\circ g^{-1})(x)=f(g^{-1}(x)), where g−1g^{-1} is the inverse of the function gg. Then the value of the derivative of the composite function f∘g−1f\circ g^{-1} at x=2x=2 is ______________.
Not attempted
Your answer: —
Correct answer: 0.25
8
Let R\mathbb{R} denote the set of all real numbers. Define the function f:R→Rf:\mathbb{R}\to\mathbb{R} by f(x)={2−2x2−x2sin⁡1xif x≠0,2if x=0.f(x) = \begin{cases} 2 - 2x^2 - x^2\sin\dfrac{1}{x} & \text{if } x\neq 0, \\ 2 & \text{if } x = 0. \end{cases} Then which one of the following statements is TRUE?
Not attempted
A
The function ff is NOT differentiable at x=0x=0
B
There is a positive real number δ\delta, such that ff is a decreasing function on the interval (0,δ)(0,\delta)
C
For any positive real number δ\delta, the function ff is NOT an increasing function on the interval (−δ,0)(-\delta,0)
D
x=0x=0 is a point of local minima of ff
9
Let α\alpha and β\beta be real numbers such that lim⁡x→01x3(α2∫0x11−t2 dt  +  βxcos⁡x)=2\displaystyle\lim_{x\to 0}\frac{1}{x^3}\left(\frac{\alpha}{2}\int_0^x \frac{1}{1-t^2}\,dt \;+\; \beta x\cos x\right) = 2. Then the value of α+β\alpha+\beta is __________.
Not attempted
Your answer: —
Correct answer: 2.4
10
Let R\mathbb{R} denote the set of all real numbers. For a real number xx, let [x][x] denote the greatest integer less than or equal to xx. Let nn denote a natural number. Match each entry in List-I to the correct entry in List-II and choose the correct option. List-I (P) The minimum value of nn for which the function f(x)=[10x3−45x2+60x+35n]f(x) = \left[\dfrac{10x^3-45x^2+60x+35}{n}\right] is continuous on the interval [1,2][1,2], is (Q) The minimum value of nn for which g(x)=(2n2−13n−15)(x3+3x)g(x)=(2n^2-13n-15)(x^3+3x), x∈Rx\in\mathbb{R}, is an increasing function on R\mathbb{R}, is (R) The smallest natural number nn which is greater than 55, such that x=3x=3 is a point of local minima of h(x)=(x2−9)n(x2+2x+3)h(x)=(x^2-9)^n(x^2+2x+3), is (S) Number of x0∈Rx_0\in\mathbb{R} such that l(x)=∑k=04(sin⁡∣x−k∣+cos⁡∣x−k+12∣)l(x) = \displaystyle\sum_{k=0}^{4}\left(\sin|x-k|+\cos\left|x-k+\dfrac12\right|\right), x∈Rx\in\mathbb{R}, is NOT differentiable at x0x_0, is List-II (1) 88 (2) 99 (3) 55 (4) 66 (5) 1010
Not attempted
A
(P) →\to (1) (Q) →\to (3) (R) →\to (2) (S) →\to (5)
B
(P) →\to (2) (Q) →\to (1) (R) →\to (4) (S) →\to (3)
C
(P) →\to (5) (Q) →\to (1) (R) →\to (4) (S) →\to (3)
D
(P) →\to (2) (Q) →\to (3) (R) →\to (1) (S) →\to (5)
11
Let k∈Rk\in\mathbb{R}. If lim⁡x→0+(sin⁡(sin⁡kx)+cos⁡x+x)2x=e6\displaystyle\lim_{x\to0^{+}}\left(\sin(\sin kx)+\cos x+x\right)^{\frac{2}{x}}=e^{6}, then the value of kk is
Not attempted
A
11
B
22
C
33
D
44
12
Let f:R→Rf:\mathbb{R}\to\mathbb{R} be a function defined by f(x)={x2sin⁡(πx2),if x≠0,0,if x=0.f(x)=\begin{cases} x^2\sin\left(\dfrac{\pi}{x^2}\right), & \text{if } x\ne0, \\ 0, & \text{if } x=0.\end{cases} Then which of the following statements is TRUE?
Not attempted
A
f(x)=0f(x)=0 has infinitely many solutions in the interval [11010,∞)\left[\dfrac{1}{10^{10}},\infty\right).
B
f(x)=0f(x)=0 has no solutions in the interval [1π,∞)\left[\dfrac{1}{\pi},\infty\right).
C
The set of solutions of f(x)=0f(x)=0 in the interval (0,11010)\left(0,\dfrac{1}{10^{10}}\right) is finite.
D
f(x)=0f(x)=0 has more than 2525 solutions in the interval (1π2,1π)\left(\dfrac{1}{\pi^2},\dfrac{1}{\pi}\right).
13
Let SS be the set of all (α,β)∈R×R(\alpha,\beta)\in\mathbb{R}\times\mathbb{R} such that lim⁡x→∞sin⁡(x2)(log⁡ex)αsin⁡(1x2)xαβ(log⁡e(1+x))β=0\displaystyle\lim_{x\to\infty}\frac{\sin(x^2)(\log_e x)^{\alpha}\sin\left(\dfrac{1}{x^2}\right)}{x^{\alpha\beta}(\log_e(1+x))^{\beta}}=0. Then which of the following is (are) correct?
Not attempted
A
(−1,3)∈S(-1,3)\in S
B
(−1,1)∈S(-1,1)\in S
C
(1,−1)∈S(1,-1)\in S
D
(1,−2)∈S(1,-2)\in S
14
Let the function f:R→Rf:\mathbb{R}\to\mathbb{R} be defined by f(x)=sin⁡xeπx⋅(x2023+2024x+2025)(x2−x+3)+2eπx⋅(x2023+2024x+2025)(x2−x+3)f(x)=\dfrac{\sin x}{e^{\pi x}}\cdot\dfrac{\left(x^{2023}+2024x+2025\right)}{\left(x^2-x+3\right)}+\dfrac{2}{e^{\pi x}}\cdot\dfrac{\left(x^{2023}+2024x+2025\right)}{\left(x^2-x+3\right)}. Then the number of solutions of f(x)=0f(x)=0 in R\mathbb{R} is ______.
Not attempted
Your answer: —
Correct answer: 1
15
Let f:(0,1)→Rf:(0,1)\to\mathbb{R} be the function defined as f(x)=[4x](x−14)2(x−12)f(x)=[4x]\left(x-\dfrac14\right)^2\left(x-\dfrac12\right), where [x][x] denotes the greatest integer less than or equal to xx. Then which of the following statements is(are) true?
Not attempted
A
The function ff is discontinuous exactly at one point in (0,1)(0,1)
B
There is exactly one point in (0,1)(0,1) at which the function ff is continuous but NOT differentiable
C
The function ff is NOT differentiable at more than three points in (0,1)(0,1)
D
The minimum value of the function ff is −1512-\dfrac{1}{512}
16
Let SS be the set of all twice differentiable functions ff from R\mathbb{R} to R\mathbb{R} such that d2fdx2(x)>0\dfrac{d^2f}{dx^2}(x)>0 for all x∈(−1,1)x\in(-1,1). For f∈Sf\in S, let XfX_f be the number of points x∈(−1,1)x\in(-1,1) for which f(x)=xf(x)=x. Then which of the following statements is(are) true?
Not attempted
A
There exists a function f∈Sf\in S such that Xf=0X_f=0
B
For every function f∈Sf\in S, we have Xf≤2X_f\le2
C
There exists a function f∈Sf\in S such that Xf=2X_f=2
D
There does NOT exist any function ff in SS such that Xf=1X_f=1
17
Let f:(0,1)→Rf:(0,1)\to\mathbb{R} be the function defined as f(x)=nf(x)=\sqrt{n} if x∈[1n+1,1n)x\in\left[\dfrac{1}{n+1},\dfrac{1}{n}\right), where n∈Nn\in\mathbb{N}. Let g:(0,1)→Rg:(0,1)\to\mathbb{R} be a function such that ∫x2x1−tt dt<g(x)<2x\displaystyle\int_{x^2}^{x}\sqrt{\dfrac{1-t}{t}}\,dt<g(x)<2\sqrt{x} for all x∈(0,1)x\in(0,1). Then lim⁡x→0f(x)g(x)\displaystyle\lim_{x\to0}f(x)g(x)
Not attempted
A
does NOT exist
B
is equal to 11
C
is equal to 22
D
is equal to 33
18
For positive integer nn, define f(n)=n+16+5n−3n24n+3n2+32+n−3n28n+3n2+48−3n−3n212n+3n2+⋯+25n−7n27n2f(n)=n+\dfrac{16+5n-3n^2}{4n+3n^2}+\dfrac{32+n-3n^2}{8n+3n^2}+\dfrac{48-3n-3n^2}{12n+3n^2}+\cdots+\dfrac{25n-7n^2}{7n^2}. Then, the value of lim⁡n→∞f(n)\displaystyle\lim_{n\to\infty}f(n) is equal to
Not attempted
A
3+43log⁡e73+\dfrac43\log_e7
B
4−34log⁡e(73)4-\dfrac34\log_e\left(\dfrac73\right)
C
4−43log⁡e(73)4-\dfrac43\log_e\left(\dfrac73\right)
D
3+34log⁡e73+\dfrac34\log_e7
19
If β=lim⁡x→0ex3−(1−x3)1/3+((1−x2)1/2−1)sin⁡xxsin⁡2x\beta = \displaystyle\lim_{x\to0} \dfrac{e^{x^3}-(1-x^3)^{1/3}+\left((1-x^2)^{1/2}-1\right)\sin x}{x\sin^2 x}, then the value of 6β6\beta is _______.
Not attempted
Your answer: —
Correct answer: 5
20
Let α\alpha be a positive real number. Let f:R→Rf:\mathbb{R}\to\mathbb{R} and g:(α,∞)→Rg:(\alpha,\infty)\to\mathbb{R} be the functions defined by f(x)=sin⁡(πx12)f(x)=\sin\left(\dfrac{\pi x}{12}\right) and g(x)=2log⁡e(x−α)log⁡e(ex−eα)g(x)=\dfrac{2\log_e(\sqrt{x}-\sqrt{\alpha})}{\log_e(e^{\sqrt{x}}-e^{\sqrt{\alpha}})}. Then the value of lim⁡x→α+f(g(x))\displaystyle\lim_{x\to\alpha^+} f(g(x)) is __________.
Not attempted
Your answer: —
Correct answer: 0.5
21
Let f1:(0,∞)→Rf_1:(0,\infty)\to\mathbb{R} and f2:(0,∞)→Rf_2:(0,\infty)\to\mathbb{R} be defined by f1(x)=∫0x∏j=121(t−j)j dtf_1(x) = \int_0^x \prod_{j=1}^{21}(t-j)^j\,dt, x>0x>0 and f2(x)=98(x−1)50−600(x−1)49+2450f_2(x) = 98(x-1)^{50}-600(x-1)^{49}+2450, x>0x>0, where, for any positive integer nn and real numbers a1,a2,…,ana_1,a_2,\ldots,a_n, ∏i=1nai\prod_{i=1}^n a_i denotes the product of a1,a2,…,ana_1,a_2,\ldots,a_n. Let mim_i and nin_i, respectively, denote the number of points of local minima and the number of points of local maxima of fif_i, i=1,2i=1,2, in the interval (0,∞)(0,\infty). The value of 2m1+3n1+m1n12m_1+3n_1+m_1n_1 is ___.
Not attempted
Your answer: —
Correct answer: 57
22
Let f1:(0,∞)→Rf_1:(0,\infty)\to\mathbb{R} and f2:(0,∞)→Rf_2:(0,\infty)\to\mathbb{R} be defined by f1(x)=∫0x∏j=121(t−j)j dtf_1(x) = \int_0^x \prod_{j=1}^{21}(t-j)^j\,dt, x>0x>0 and f2(x)=98(x−1)50−600(x−1)49+2450f_2(x) = 98(x-1)^{50}-600(x-1)^{49}+2450, x>0x>0, where, for any positive integer nn and real numbers a1,a2,…,ana_1,a_2,\ldots,a_n, ∏i=1nai\prod_{i=1}^n a_i denotes the product of a1,a2,…,ana_1,a_2,\ldots,a_n. Let mim_i and nin_i, respectively, denote the number of points of local minima and the number of points of local maxima of fif_i, i=1,2i=1,2, in the interval (0,∞)(0,\infty). The value of 6m2+4n2+8m2n26m_2+4n_2+8m_2n_2 is ___.
Not attempted
Your answer: —
Correct answer: 6
23
Let f:R→Rf:\mathbb{R}\to\mathbb{R} be defined by f(x)=x2−3x−6x2+2x+4f(x)=\frac{x^2-3x-6}{x^2+2x+4} Then which of the following statements is (are) TRUE?
Not attempted
A
ff is decreasing in the interval (−2,−1)(-2,-1)
B
ff is increasing in the interval (1,2)(1,2)
C
ff is onto
D
Range of ff is [−32,2]\left[-\frac{3}{2},2\right]
24
Let the function f:(0,π)→Rf:(0,\pi) \to \mathbb{R} be defined by f(θ)=(sin⁡θ+cos⁡θ)2+(sin⁡θ−cos⁡θ)4f(\theta) = (\sin\theta+\cos\theta)^2 + (\sin\theta-\cos\theta)^4. Suppose the function ff has a local minimum at θ\theta precisely when θ∈{λ1π,…,λrπ}\theta \in \{\lambda_1\pi,\ldots,\lambda_r\pi\}, where 0<λ1<⋯<λr<10<\lambda_1<\cdots<\lambda_r<1. Then the value of λ1+⋯+λr\lambda_1+\cdots+\lambda_r is ____
Not attempted
Your answer: —
Correct answer: 0.5
25
Let f:R→Rf:\mathbb{R}\to\mathbb{R} and g:R→Rg:\mathbb{R}\to\mathbb{R} be functions satisfying f(x+y)=f(x)+f(y)+f(x)f(y)f(x+y)=f(x)+f(y)+f(x)f(y) and f(x)=xg(x)f(x)=xg(x) for all x,y∈Rx,y \in \mathbb{R}. If lim⁡x→0g(x)=1\displaystyle\lim_{x\to0} g(x)=1, then which of the following statements is/are TRUE?
Not attempted
A
ff is differentiable at every x∈Rx \in \mathbb{R}
B
If g(0)=1g(0)=1, then gg is differentiable at every x∈Rx \in \mathbb{R}
C
The derivative f′(1)f'(1) is equal to 11
D
The derivative f′(0)f'(0) is equal to 11
26
Let the functions f:(−1,1)→Rf:(-1,1) \to \mathbb{R} and g:(−1,1)→(−1,1)g:(-1,1) \to (-1,1) be defined by f(x)=∣2x−1∣+∣2x+1∣f(x)=|2x-1|+|2x+1| and g(x)=x−[x]g(x)=x-[x], where [x][x] denotes the greatest integer less than or equal to xx. Let f∘g:(−1,1)→Rf \circ g : (-1,1) \to \mathbb{R} be the composite function defined by (f∘g)(x)=f(g(x))(f\circ g)(x) = f(g(x)). Suppose cc is the number of points in the interval (−1,1)(-1,1) at which f∘gf\circ g is NOT continuous, and suppose dd is the number of points in the interval (−1,1)(-1,1) at which f∘gf\circ g is NOT differentiable. Then the value of c+dc+d is ____
Not attempted
Your answer: —
Correct answer: 4
27
The value of the limit lim⁡x→π242(sin⁡3x+sin⁡x)(2sin⁡2xsin⁡3x2+cos⁡5x2)−(2+2cos⁡2x+cos⁡3x2)\displaystyle\lim_{x \to \frac{\pi}{2}} \dfrac{4\sqrt2(\sin3x+\sin x)}{\left(2\sin2x\sin\frac{3x}{2}+\cos\frac{5x}{2}\right) - \left(\sqrt2+\sqrt2\cos2x+\cos\frac{3x}{2}\right)} is ____
Not attempted
Your answer: —
Correct answer: 8
28
For a polynomial g(x)g(x) with real coefficients, let mgm_g denote the number of distinct real roots of g(x)g(x). Suppose SS is the set of polynomials with real coefficients defined by S={(x2−1)2(a0+a1x+a2x2+a3x3):a0,a1,a2,a3∈R}S = \{(x^2-1)^2(a_0+a_1x+a_2x^2+a_3x^3) : a_0,a_1,a_2,a_3 \in \mathbb{R}\}. For a polynomial ff, let f′f' and f′′f'' denote its first and second order derivatives, respectively. Then the minimum possible value of (mf′+mf′′)(m_{f'} + m_{f''}), where f∈Sf \in S, is
Not attempted
Your answer: —
Correct answer: 5
29
Let ee denote the base of the natural logarithm. The value of the real number aa for which the right hand limit lim⁡x→0+(1−x)1/x−e−1xa\displaystyle\lim_{x\to0^+} \frac{(1-x)^{1/x} - e^{-1}}{x^a} is equal to a nonzero real number, is
Not attempted
Your answer: —
Correct answer: 1
30
Consider all rectangles lying in the region {(x,y)∈R×R:0≤x≤π2 and 0≤y≤2sin⁡(2x)}\{(x, y) \in \mathbb{R} \times \mathbb{R} : 0 \leq x \leq \frac{\pi}{2} \text{ and } 0 \leq y \leq 2\sin(2x)\} and having one side on the xx-axis. The area of the rectangle which has the maximum perimeter among all such rectangles, is
Not attempted
A
3π2\frac{3\pi}{2}
B
π\pi
C
π23\frac{\pi}{2\sqrt{3}}
D
π32\frac{\pi\sqrt{3}}{2}
31
Let the function f:R→Rf: \mathbb{R} \to \mathbb{R} be defined by f(x)=x3−x2+(x−1)sin⁡xf(x) = x^3 - x^2 + (x-1)\sin x and let g:R→Rg: \mathbb{R} \to \mathbb{R} be an arbitrary function. Let fg:R→Rfg: \mathbb{R} \to \mathbb{R} be the product function defined by (fg)(x)=f(x)g(x)(fg)(x) = f(x)g(x). Then which of the following statements is/are TRUE?
Not attempted
A
If gg is continuous at x=1x=1, then fgfg is differentiable at x=1x=1
B
If fgfg is differentiable at x=1x=1, then gg is continuous at x=1x=1
C
If gg is differentiable at x=1x=1, then fgfg is differentiable at x=1x=1
D
If fgfg is differentiable at x=1x=1, then gg is differentiable at x=1x=1
32
Let f(x)=sin⁡πxx2f(x)=\dfrac{\sin\pi x}{x^2}, x>0x>0. Let x1<x2<x3<⋯<xn<⋯x_1<x_2<x_3<\cdots<x_n<\cdots be all the points of local maximum of ff, and y1<y2<y3<⋯<yn<⋯y_1<y_2<y_3<\cdots<y_n<\cdots be all the points of local minimum of ff. Then which of the following options is/are correct?
Not attempted
A
∣xn−yn∣>1|x_n-y_n|>1 for every nn
B
x1<y1x_1<y_1
C
xn∈(2n, 2n+12)x_n\in\left(2n,\,2n+\tfrac12\right) for every nn
D
xn+1−xn>2x_{n+1}-x_n>2 for every nn
33
Let f:R→Rf:\mathbb{R}\to\mathbb{R} be a function. We say that ff has PROPERTY 1 if lim⁡h→0f(h)−f(0)∣h∣\lim_{h\to0}\dfrac{f(h)-f(0)}{\sqrt{|h|}} exists and is finite, and PROPERTY 2 if lim⁡h→0f(h)−f(0)h2\lim_{h\to0}\dfrac{f(h)-f(0)}{h^2} exists and is finite. Then which of the following options is/are correct?
Not attempted
A
f(x)=x∣x∣f(x)=x|x| has PROPERTY 2
B
f(x)=x2/3f(x)=x^{2/3} has PROPERTY 1
C
f(x)=sin⁡xf(x)=\sin x has PROPERTY 2
D
f(x)=∣x∣f(x)=|x| has PROPERTY 1
34
Let f:R→Rf:\mathbb{R}\to\mathbb{R} be given by f(x)={x5+5x4+10x3+10x2+3x+1,x<0x2−x+1,0≤x<123x3−4x2+7x−83,1≤x<3(x−2)log⁡e(x−2)−x+103,x≥3f(x)=\begin{cases}x^5+5x^4+10x^3+10x^2+3x+1, & x<0\\x^2-x+1, & 0\le x<1\\\dfrac23x^3-4x^2+7x-\dfrac83, & 1\le x<3\\(x-2)\log_e(x-2)-x+\dfrac{10}{3}, & x\ge3\end{cases}. Then which of the following options is/are correct?
Not attempted
A
f′f' has a local maximum at x=1x=1
B
ff is onto
C
ff is increasing on (−∞,0)(-\infty,0)
D
f′f' is NOT differentiable at x=1x=1
35
Let f1:R→Rf_1:\mathbb{R}\to\mathbb{R}, f2:(−π2,π2)→Rf_2:\left(-\frac{\pi}{2},\frac{\pi}{2}\right)\to\mathbb{R}, f3:(−1,eπ/2−2)→Rf_3:\left(-1,e^{\pi/2}-2\right)\to\mathbb{R} and f4:R→Rf_4:\mathbb{R}\to\mathbb{R} be functions defined by: (i) f1(x)=sin⁡(1−e−x2)f_1(x)=\sin\left(\sqrt{1-e^{-x^2}}\right), (ii) f2(x)={∣sin⁡x∣tan⁡−1xif x≠01if x=0f_2(x)=\begin{cases}\frac{|\sin x|}{\tan^{-1}x} & \text{if } x\neq0 \\ 1 & \text{if } x=0\end{cases}, where the inverse trigonometric function tan⁡−1x\tan^{-1}x assumes values in (−π2,π2)\left(-\frac{\pi}{2},\frac{\pi}{2}\right), (iii) f3(x)=[sin⁡(log⁡e(x+2))]f_3(x)=\left[\sin(\log_e(x+2))\right], where, for t∈Rt\in\mathbb{R}, [t][t] denotes the greatest integer less than or equal to tt, (iv) f4(x)={x2sin⁡(1x)if x≠00if x=0f_4(x)=\begin{cases}x^2\sin\left(\frac1x\right) & \text{if } x\neq0 \\ 0 & \text{if } x=0\end{cases}. Match each entry in LIST-I to the correct entry in LIST-II. LIST-I\textbf{LIST-I} P. The function f1f_1 is Q. The function f2f_2 is R. The function f3f_3 is S. The function f4f_4 is LIST-II\textbf{LIST-II} 1. NOT continuous at x=0x=0 2. continuous at x=0x=0 and NOT differentiable at x=0x=0 3. differentiable at x=0x=0 and its derivative is NOT continuous at x=0x=0 4. differentiable at x=0x=0 and its derivative is continuous at x=0x=0
Not attempted
A
P→\to2; Q→\to3; R→\to1; S→\to4
B
P→\to4; Q→\to1; R→\to2; S→\to3
C
P→\to4; Q→\to2; R→\to1; S→\to3
D
P→\to2; Q→\to1; R→\to4; S→\to3
36
Let f:R→Rf:\mathbb{R}\to\mathbb{R} be a differentiable function with f(0)=1f(0)=1 and satisfying the equation f(x+y)=f(x)f′(y)+f′(x)f(y)f(x+y)=f(x)f'(y)+f'(x)f(y) for all x,y∈Rx,y\in\mathbb{R}. Then, the value of log⁡e(f(4))\log_e(f(4)) is _____.
Not attempted
Your answer: —
Correct answer: 2
37
Let f:(0,π)→Rf:(0,\pi)\to\mathbb{R} be a twice differentiable function such that lim⁡t→xf(x)sin⁡t−f(t)sin⁡xt−x=sin⁡2x\lim_{t\to x}\frac{f(x)\sin t - f(t)\sin x}{t-x}=\sin^2 x for all x∈(0,π)x\in(0,\pi). If f(π6)=−π12f\left(\frac{\pi}{6}\right)=-\frac{\pi}{12}, then which of the following statement(s) is (are) TRUE?
Not attempted
A
f(π4)=π42f\left(\frac{\pi}{4}\right) = \frac{\pi}{4\sqrt2}
B
f(x)<x46−x2f(x) < \frac{x^4}{6} - x^2 for all x∈(0,π)x\in(0,\pi)
C
There exists α∈(0,π)\alpha\in(0,\pi) such that f′(α)=0f'(\alpha)=0
D
f′′(π2)+f(π2)=0f''\left(\frac{\pi}{2}\right) + f\left(\frac{\pi}{2}\right) = 0
38
For each positive integer nn, let yn=1n((n+1)(n+2)⋯(n+n))1/ny_n = \dfrac{1}{n}\Big((n+1)(n+2)\cdots(n+n)\Big)^{1/n}. For x∈Rx\in\mathbb{R}, let [x][x] be the greatest integer less than or equal to xx. If lim⁡n→∞yn=L\displaystyle\lim_{n\to\infty} y_n = L, then the value of [L][L] is ______.
Not attempted
Your answer: —
Correct answer: 1
39
For every twice differentiable function f:R→[−2,2]f:\mathbb{R}\to[-2,2] with (f(0))2+(f′(0))2=85(f(0))^2+(f'(0))^2=85, which of the following statement(s) is (are) TRUE?
Not attempted
A
There exist r,s∈Rr,s\in\mathbb{R}, where r<sr<s, such that ff is one-one on the open interval (r,s)(r,s)
B
There exists x0∈(−4,0)x_0\in(-4,0) such that ∣f′(x0)∣≤1|f'(x_0)|\le 1
C
lim⁡x→∞f(x)=1\displaystyle\lim_{x\to\infty} f(x)=1
D
There exists α∈(−4,4)\alpha\in(-4,4) such that f(α)+f′′(α)=0f(\alpha)+f''(\alpha)=0 and f′(α)≠0f'(\alpha)\ne0
40
If f:R→Rf: \mathbb{R} \to \mathbb{R} is a twice differentiable function such that f′′(x)>0f''(x) > 0 for all x∈Rx \in \mathbb{R}, and f(12)=12f\left(\frac{1}{2}\right) = \frac{1}{2}, f(1)=1f(1) = 1, then
Not attempted
A
f′(1)≤0f'(1) \le 0
B
0<f′(1)≤120 < f'(1) \le \frac{1}{2}
C
12<f′(1)≤1\frac{1}{2} < f'(1) \le 1
D
f′(1)>1f'(1) > 1
41
Let f(x)=1−x(1+∣1−x∣)∣1−x∣cos⁡(11−x)f(x) = \dfrac{1-x(1+|1-x|)}{|1-x|}\cos\left(\dfrac{1}{1-x}\right) for x≠1x \ne 1. Then
Not attempted
A
lim⁡x→1−f(x)=0\lim_{x\to 1^-} f(x) = 0
B
lim⁡x→1−f(x)\lim_{x\to 1^-} f(x) does not exist
C
lim⁡x→1+f(x)=0\lim_{x\to 1^+} f(x) = 0
D
lim⁡x→1+f(x)\lim_{x\to 1^+} f(x) does not exist
42
Let f(x)=x+ln⁡x−xln⁡x, x∈(0,∞)f(x)=x+\ln x-x\ln x,\ x\in(0,\infty). Column 1 concerns zeros of f,f′,f′′f,f',f''; Column 2 concerns the limiting behaviour of f,f′,f′′f,f',f'' at infinity; Column 3 concerns the increasing/decreasing nature of ff and f′f'. (I) f(x)=0 for some x∈(1,e2)(i) lim⁡x→∞f(x)=0(P) f is increasing in (0,1)(II) f′(x)=0 for some x∈(1,e)(ii) lim⁡x→∞f(x)=−∞(Q) f is decreasing in (e,e2)(III) f′(x)=0 for some x∈(0,1)(iii) lim⁡x→∞f′(x)=−∞(R) f′ is increasing in (0,1)(IV) f′′(x)=0 for some x∈(1,e)(iv) lim⁡x→∞f′′(x)=0(S) f′ is decreasing in (e,e2)\begin{array}{lll} \text{(I) } f(x)=0 \text{ for some } x\in(1,e^2) & \text{(i) } \lim\limits_{x\to\infty}f(x)=0 & \text{(P) } f \text{ is increasing in } (0,1)\\ \text{(II) } f'(x)=0 \text{ for some } x\in(1,e) & \text{(ii) } \lim\limits_{x\to\infty}f(x)=-\infty & \text{(Q) } f \text{ is decreasing in } (e,e^2)\\ \text{(III) } f'(x)=0 \text{ for some } x\in(0,1) & \text{(iii) } \lim\limits_{x\to\infty}f'(x)=-\infty & \text{(R) } f' \text{ is increasing in } (0,1)\\ \text{(IV) } f''(x)=0 \text{ for some } x\in(1,e) & \text{(iv) } \lim\limits_{x\to\infty}f''(x)=0 & \text{(S) } f' \text{ is decreasing in } (e,e^2) \end{array} Which of the following options is the only CORRECT combination?
Not attempted
A
(I) (i) (P)
B
(II) (ii) (Q)
C
(III) (iii) (R)
D
(IV) (iv) (S)
43
Let f(x)=x+ln⁡x−xln⁡x, x∈(0,∞)f(x)=x+\ln x-x\ln x,\ x\in(0,\infty). Column 1 concerns zeros of f,f′,f′′f,f',f''; Column 2 concerns the limiting behaviour of f,f′,f′′f,f',f'' at infinity; Column 3 concerns the increasing/decreasing nature of ff and f′f'. (I) f(x)=0 for some x∈(1,e2)(i) lim⁡x→∞f(x)=0(P) f is increasing in (0,1)(II) f′(x)=0 for some x∈(1,e)(ii) lim⁡x→∞f(x)=−∞(Q) f is decreasing in (e,e2)(III) f′(x)=0 for some x∈(0,1)(iii) lim⁡x→∞f′(x)=−∞(R) f′ is increasing in (0,1)(IV) f′′(x)=0 for some x∈(1,e)(iv) lim⁡x→∞f′′(x)=0(S) f′ is decreasing in (e,e2)\begin{array}{lll} \text{(I) } f(x)=0 \text{ for some } x\in(1,e^2) & \text{(i) } \lim\limits_{x\to\infty}f(x)=0 & \text{(P) } f \text{ is increasing in } (0,1)\\ \text{(II) } f'(x)=0 \text{ for some } x\in(1,e) & \text{(ii) } \lim\limits_{x\to\infty}f(x)=-\infty & \text{(Q) } f \text{ is decreasing in } (e,e^2)\\ \text{(III) } f'(x)=0 \text{ for some } x\in(0,1) & \text{(iii) } \lim\limits_{x\to\infty}f'(x)=-\infty & \text{(R) } f' \text{ is increasing in } (0,1)\\ \text{(IV) } f''(x)=0 \text{ for some } x\in(1,e) & \text{(iv) } \lim\limits_{x\to\infty}f''(x)=0 & \text{(S) } f' \text{ is decreasing in } (e,e^2) \end{array} Which of the following options is the only CORRECT combination?
Not attempted
A
(I) (ii) (R)
B
(II) (iii) (S)
C
(III) (iv) (P)
D
(IV) (i) (S)
44
Let f(x)=x+ln⁡x−xln⁡x, x∈(0,∞)f(x)=x+\ln x-x\ln x,\ x\in(0,\infty). Column 1 concerns zeros of f,f′,f′′f,f',f''; Column 2 concerns the limiting behaviour of f,f′,f′′f,f',f'' at infinity; Column 3 concerns the increasing/decreasing nature of ff and f′f'. (I) f(x)=0 for some x∈(1,e2)(i) lim⁡x→∞f(x)=0(P) f is increasing in (0,1)(II) f′(x)=0 for some x∈(1,e)(ii) lim⁡x→∞f(x)=−∞(Q) f is decreasing in (e,e2)(III) f′(x)=0 for some x∈(0,1)(iii) lim⁡x→∞f′(x)=−∞(R) f′ is increasing in (0,1)(IV) f′′(x)=0 for some x∈(1,e)(iv) lim⁡x→∞f′′(x)=0(S) f′ is decreasing in (e,e2)\begin{array}{lll} \text{(I) } f(x)=0 \text{ for some } x\in(1,e^2) & \text{(i) } \lim\limits_{x\to\infty}f(x)=0 & \text{(P) } f \text{ is increasing in } (0,1)\\ \text{(II) } f'(x)=0 \text{ for some } x\in(1,e) & \text{(ii) } \lim\limits_{x\to\infty}f(x)=-\infty & \text{(Q) } f \text{ is decreasing in } (e,e^2)\\ \text{(III) } f'(x)=0 \text{ for some } x\in(0,1) & \text{(iii) } \lim\limits_{x\to\infty}f'(x)=-\infty & \text{(R) } f' \text{ is increasing in } (0,1)\\ \text{(IV) } f''(x)=0 \text{ for some } x\in(1,e) & \text{(iv) } \lim\limits_{x\to\infty}f''(x)=0 & \text{(S) } f' \text{ is decreasing in } (e,e^2) \end{array} Which of the following options is the only INCORRECT combination?
Not attempted
A
(I) (iii) (P)
B
(II) (iv) (Q)
C
(III) (i) (R)
D
(II) (iii) (P)
45
Let [x][x] be the greatest integer less than or equal to xx. Then, at which of the following point(s) is the function f(x)=xcos⁡(π(x+[x]))f(x) = x\cos(\pi(x+[x])) discontinuous?
Not attempted
A
x=−1x=-1
B
x=0x=0
C
x=1x=1
D
x=2x=2
46
Let f:R→(0,1)f:\mathbb{R}\to(0,1) be a continuous function. Then, which of the following function(s) has(have) the value zero at some point in the interval (0,1)(0,1)?
Not attempted
A
x9−f(x)x^9 - f(x)
B
x−∫0π2−xf(t)cos⁡t dtx - \displaystyle\int_0^{\frac{\pi}{2}-x} f(t)\cos t\, dt
C
ex−∫0xf(t)sin⁡t dte^x - \displaystyle\int_0^{x} f(t)\sin t\, dt
D
f(x)+∫0π2f(t)sin⁡t dtf(x) + \displaystyle\int_0^{\frac{\pi}{2}} f(t)\sin t\, dt
47
Let f(x)=lim⁡n→∞(nn(x+n)(x+n2)⋯(x+nn)n!(x2+n2)(x2+n24)⋯(x2+n2n2))x/nf(x)=\displaystyle\lim_{n\to\infty}\left(\dfrac{n^n(x+n)\left(x+\dfrac{n}{2}\right)\cdots\left(x+\dfrac{n}{n}\right)}{n!\left(x^2+n^2\right)\left(x^2+\dfrac{n^2}{4}\right)\cdots\left(x^2+\dfrac{n^2}{n^2}\right)}\right)^{x/n}, for all x>0x>0. Then
Not attempted
A
f ⁣(12)≥f(1)f\!\left(\dfrac12\right)\ge f(1)
B
f ⁣(13)≤f ⁣(23)f\!\left(\dfrac13\right)\le f\!\left(\dfrac23\right)
C
f′(2)≤0f'(2)\le0
D
f′(3)f(3)≥f′(2)f(2)\dfrac{f'(3)}{f(3)}\ge\dfrac{f'(2)}{f(2)}
48
Let a,b∈Ra,b\in\mathbb{R} and f:R→Rf:\mathbb{R}\to\mathbb{R} be defined by f(x)=acos⁡(∣x3−x∣)+b∣x∣sin⁡(∣x3+x∣)f(x)=a\cos(|x^3-x|)+b|x|\sin(|x^3+x|). Then ff is
Not attempted
A
differentiable at x=0x=0 if a=0a=0 and b=1b=1
B
differentiable at x=1x=1 if a=1a=1 and b=0b=0
C
NOT differentiable at x=0x=0 if a=1a=1 and b=0b=0
D
NOT differentiable at x=1x=1 if a=1a=1 and b=1b=1
49
Let f:R→(0,∞)f:\mathbb{R}\to(0,\infty) and g:R→Rg:\mathbb{R}\to\mathbb{R} be twice differentiable functions such that f′′f'' and g′′g'' are continuous functions on R\mathbb{R}. Suppose f′(2)=g(2)=0f'(2)=g(2)=0, f′′(2)≠0f''(2)\ne0 and g′(2)≠0g'(2)\ne0. If lim⁡x→2f(x)g(x)f′(x)g′(x)=1\displaystyle\lim_{x\to2}\dfrac{f(x)g(x)}{f'(x)g'(x)}=1, then
Not attempted
A
ff has a local minimum at x=2x=2
B
ff has a local maximum at x=2x=2
C
f′′(2)>f(2)f''(2)>f(2)
D
f(x)−f′′(x)=0f(x)-f''(x)=0 for at least one x∈Rx\in\mathbb{R}
50
Let f:[−12,2]→Rf:\left[-\dfrac12,2\right]\to\mathbb{R} and g:[−12,2]→Rg:\left[-\dfrac12,2\right]\to\mathbb{R} be functions defined by f(x)=[x2−3]f(x)=[x^2-3] and g(x)=∣x∣f(x)+∣4x−7∣f(x)g(x)=|x|f(x)+|4x-7|f(x), where [y][y] denotes the greatest integer less than or equal to yy for y∈Ry\in\mathbb{R}. Then
Not attempted
A
ff is discontinuous exactly at three points in [−12,2]\left[-\dfrac12,2\right]
B
ff is discontinuous exactly at four points in [−12,2]\left[-\dfrac12,2\right]
C
gg is NOT differentiable exactly at four points in (−12,2)\left(-\dfrac12,2\right)
D
gg is NOT differentiable exactly at five points in (−12,2)\left(-\dfrac12,2\right)
51
Let α,β∈R\alpha,\beta\in\mathbb{R} be such that lim⁡x→0x2sin⁡(βx)αx−sin⁡x=1\displaystyle\lim_{x\to0}\dfrac{x^2\sin(\beta x)}{\alpha x-\sin x}=1. Then 6(α+β)6(\alpha+\beta) equals
Not attempted
Your answer: —
Correct answer: 7
52
Let f:R→Rf:\mathbb{R}\to\mathbb{R}, g:R→Rg:\mathbb{R}\to\mathbb{R} and h:R→Rh:\mathbb{R}\to\mathbb{R} be differentiable functions such that f(x)=x3+3x+2f(x)=x^3+3x+2, g(f(x))=xg(f(x))=x and h(g(g(x)))=xh(g(g(x)))=x for all x∈Rx\in\mathbb{R}. Then
Not attempted
A
g′(2)=115g'(2)=\dfrac1{15}
B
h′(1)=666h'(1)=666
C
h(0)=16h(0)=16
D
h(g(3))=36h(g(3))=36
53
The least value of α∈R\alpha\in\mathbb{R} for which 4αx2+1x≥14\alpha x^2+\dfrac1x\ge1, for all x>0x>0, is
Not attempted
A
164\dfrac1{64}
B
132\dfrac1{32}
C
127\dfrac1{27}
D
125\dfrac1{25}
54
Setup (shared with Q.60)\textbf{Setup (shared with Q.60)}: Let F:R→RF:\mathbb R\to\mathbb R be a thrice differentiable function. Suppose that F(1)=0, F(3)=−4F(1)=0,\ F(3)=-4 and F′(x)<0F'(x)<0 for all x∈(12,3)x\in\left(\dfrac12,3\right). Let f(x)=xF(x)f(x)=xF(x) for all x∈Rx\in\mathbb R. The correct statement(s) is(are)
Not attempted
A
f′(1)<0f'(1)<0
B
f(2)<0f(2)<0
C
f′(x)≠0f'(x)\ne0 for any x∈(1,3)x\in(1,3)
D
f′(x)=0f'(x)=0 for some x∈(1,3)x\in(1,3)
55
Let f,g:[−1,2]→Rf,g:[-1,2]\to\mathbb R be continuous functions which are twice differentiable on the interval (−1,2)(-1,2). Let the values of ff and gg at the points −1,0-1,0 and 22 be given in the following table: x=−1x=0x=2f(x)360g(x)01−1\displaystyle\begin{array}{|c|c|c|c|}\hline & x=-1 & x=0 & x=2\\\hline f(x) & 3 & 6 & 0\\\hline g(x) & 0 & 1 & -1\\\hline\end{array}. In each of the intervals (−1,0)(-1,0) and (0,2)(0,2) the function (f−3g)′′(f-3g)'' never vanishes. Then the correct statement(s) is(are)
Not attempted
A
f′(x)−3g′(x)=0f'(x)-3g'(x)=0 has exactly three solutions in (−1,0)∪(0,2)(-1,0)\cup(0,2)
B
f′(x)−3g′(x)=0f'(x)-3g'(x)=0 has exactly one solution in (−1,0)(-1,0)
C
f′(x)−3g′(x)=0f'(x)-3g'(x)=0 has exactly one solution in (0,2)(0,2)
D
f′(x)−3g′(x)=0f'(x)-3g'(x)=0 has exactly two solutions in (−1,0)(-1,0) and exactly two solutions in (0,2)(0,2)
56
Let mm and nn be two positive integers greater than 11. If lim⁡α→0(ecos⁡(αn)−eαm)=−e2\displaystyle\lim_{\alpha\to0}\left(\dfrac{e^{\cos(\alpha^n)}-e}{\alpha^m}\right)=-\dfrac e2, then the value of mn\dfrac mn is
Not attempted
Your answer: —
Correct answer: 2
57
This question is part of a Column-I to Column-II matching set (JEE Advanced 2015 Section 3, Q.59). Column II lists the values 1,2,3,4,51,2,3,4,5; select every value that correctly matches this Column-I entry. Entry (B)\textbf{Entry (B)}: Let aa and bb be real numbers such that the function f(x)={−3ax2−2,x<1bx+a2,x≥1f(x)=\begin{cases}-3ax^2-2, & x<1\\ bx+a^2, & x\ge1\end{cases} is differentiable for all x∈Rx\in\mathbb{R}. Then possible value(s) of aa is (are)
Not attempted
A
11
B
22
C
33
D
44
E
55
58
Let g:R→Rg:\mathbb{R}\to\mathbb{R} be a differentiable function with g(0)=0, g′(0)=0g(0)=0,\ g'(0)=0 and g′(1)≠0g'(1)\ne0. Let f(x)={x∣x∣g(x),x≠00,x=0f(x)=\begin{cases}\dfrac{x}{|x|}g(x), & x\ne0\\ 0, & x=0\end{cases} and h(x)=e∣x∣h(x)=e^{|x|} for all x∈Rx\in\mathbb{R}. Let (f∘h)(x)(f\circ h)(x) denote f(h(x))f(h(x)) and (h∘f)(x)(h\circ f)(x) denote h(f(x))h(f(x)). Then which of the following is (are) true?
Not attempted
A
ff is differentiable at x=0x=0
B
hh is differentiable at x=0x=0
C
f∘hf\circ h is differentiable at x=0x=0
D
h∘fh\circ f is differentiable at x=0x=0
59
Let f(x)=sin⁡(π6sin⁡(π2sin⁡x))f(x)=\sin\left(\dfrac{\pi}{6}\sin\left(\dfrac{\pi}{2}\sin x\right)\right) for all x∈Rx\in\mathbb{R} and g(x)=π2sin⁡xg(x)=\dfrac{\pi}{2}\sin x for all x∈Rx\in\mathbb{R}. Let (f∘g)(x)(f\circ g)(x) denote f(g(x))f(g(x)) and (g∘f)(x)(g\circ f)(x) denote g(f(x))g(f(x)). Then which of the following is (are) true?
Not attempted
A
Range of ff is [−12,12]\left[-\dfrac12,\dfrac12\right]
B
Range of f∘gf\circ g is [−12,12]\left[-\dfrac12,\dfrac12\right]
C
lim⁡x→0f(x)g(x)=π6\displaystyle\lim_{x\to0}\dfrac{f(x)}{g(x)}=\dfrac{\pi}{6}
D
There is an x∈Rx\in\mathbb{R} such that (g∘f)(x)=1(g\circ f)(x)=1
60
A cylindrical container is to be made from certain solid material with the following constraints: It has a fixed inner volume of V mm3V\ \text{mm}^3, has a 22 mm thick solid wall and is open at the top. The bottom of the container is a solid circular disc of thickness 22 mm and is of radius equal to the outer radius of the container. If the volume of the material used to make the container is minimum when the inner radius of the container is 1010 mm, then the value of V250π\dfrac{V}{250\pi} is
Not attempted
Your answer: —
Correct answer: 4
61
Match each entry of List I\textbf{List I} to its value in List II\textbf{List II}. List I\textbf{List I} P.\textbf{P.} Let y(x)=cos⁡(3cos⁡−1x)y(x)=\cos(3\cos^{-1}x), x∈[−1,1]x\in[-1,1], x≠±32x\ne\pm\dfrac{\sqrt3}{2}. Then 1y(x){(x2−1)d2y(x)dx2+xdy(x)dx}\dfrac{1}{y(x)}\left\{(x^2-1)\dfrac{d^2y(x)}{dx^2}+x\dfrac{dy(x)}{dx}\right\} equals Q.\textbf{Q.} Let A1,A2,…,AnA_1,A_2,\dots,A_n (n>2n>2) be the vertices of a regular polygon of nn sides with centre at the origin. Let a⃗k\vec{a}_k be the position vector of AkA_k. If ∣∑k=1n−1(a⃗k×a⃗k+1)∣=∣∑k=1n−1(a⃗k⋅a⃗k+1)∣\left|\displaystyle\sum_{k=1}^{n-1}(\vec a_k\times\vec a_{k+1})\right|=\left|\displaystyle\sum_{k=1}^{n-1}(\vec a_k\cdot\vec a_{k+1})\right|, the minimum value of nn is R.\textbf{R.} If the normal from the point P(h,1)P(h,1) on the ellipse x26+y23=1\dfrac{x^2}{6}+\dfrac{y^2}{3}=1 is perpendicular to the line x+y=8x+y=8, then the value of hh is S.\textbf{S.} Number of positive solutions satisfying the equation tan⁡−1 ⁣(12x+1)+tan⁡−1 ⁣(14x+1)=tan⁡−1 ⁣(2x2)\tan^{-1}\!\left(\dfrac{1}{2x+1}\right)+\tan^{-1}\!\left(\dfrac{1}{4x+1}\right)=\tan^{-1}\!\left(\dfrac{2}{x^2}\right) is List II\textbf{List II} 1.\textbf{1.} 11 2.\textbf{2.} 22 3.\textbf{3.} 88 4.\textbf{4.} 99 Which of the following is the correct combination?
Not attempted
A
P→4, Q→3, R→2, S→1P{\to}4,\ Q{\to}3,\ R{\to}2,\ S{\to}1
B
P→2, Q→4, R→3, S→1P{\to}2,\ Q{\to}4,\ R{\to}3,\ S{\to}1
C
P→4, Q→3, R→1, S→2P{\to}4,\ Q{\to}3,\ R{\to}1,\ S{\to}2
D
P→2, Q→4, R→1, S→3P{\to}2,\ Q{\to}4,\ R{\to}1,\ S{\to}3
62
For every pair of continuous functions f,g:[0,1]→Rf, g : [0,1] \to \mathbb{R} such that max⁡{f(x):x∈[0,1]}=max⁡{g(x):x∈[0,1]}\max\{f(x) : x \in [0,1]\} = \max\{g(x): x \in [0,1]\}, the correct statement(s) is(are):
Not attempted
A
(f(c))2+3f(c)=(g(c))2+3g(c)(f(c))^2+3f(c) = (g(c))^2+3g(c) for some c∈[0,1]c \in [0,1].
B
(f(c))2+f(c)=(g(c))2+3g(c)(f(c))^2+f(c) = (g(c))^2+3g(c) for some c∈[0,1]c \in [0,1].
C
(f(c))2+3f(c)=(g(c))2+g(c)(f(c))^2+3f(c) = (g(c))^2+g(c) for some c∈[0,1]c \in [0,1].
D
(f(c))2=(g(c))2(f(c))^2 = (g(c))^2 for some c∈[0,1]c \in [0,1].
63
Let a∈Ra \in \mathbb{R} and let f:R→Rf:\mathbb{R}\to\mathbb{R} be given by f(x)=x5−5x+af(x)=x^5-5x+a. Then:
Not attempted
A
f(x)f(x) has three real roots if a>4a > 4.
B
f(x)f(x) has only one real root if a>4a > 4.
C
f(x)f(x) has three real roots if a<−4a < -4.
D
f(x)f(x) has three real roots if −4<a<4-4 < a < 4.
64
Let f:R→Rf:\mathbb{R}\to\mathbb{R} and g:R→Rg:\mathbb{R}\to\mathbb{R} be respectively given by f(x)=∣x∣+1f(x)=|x|+1 and g(x)=x2+1g(x)=x^2+1. Define h:R→Rh:\mathbb{R}\to\mathbb{R} by h(x)={max⁡{f(x),g(x)}if x≤0,min⁡{f(x),g(x)}if x>0.h(x)=\begin{cases}\max\{f(x),g(x)\} & \text{if } x\leq 0,\\ \min\{f(x),g(x)\} & \text{if } x> 0.\end{cases} The number of points at which h(x)h(x) is not differentiable is
Not attempted
Your answer: —
Correct answer: 3
65
The slope of the tangent to the curve (y−x5)2=x(1+x2)2(y-x^5)^2=x(1+x^2)^2 at the point (1,3)(1,3) is
Not attempted
Your answer: —
Correct answer: 8
66
The largest value of the nonnegative integer aa for which lim⁡x→1{−ax+sin⁡(x−1)+ax+sin⁡(x−1)−1}1−x1−x=14\displaystyle\lim_{x\to1}\left\{\frac{-ax+\sin(x-1)+a}{x+\sin(x-1)-1}\right\}^{\frac{1-x}{1-\sqrt x}}=\frac14 is
Not attempted
Your answer: —
Correct answer: 0
67
The function f(x)=2∣x∣+∣x+2∣−∣∣x+2∣−2∣x∣∣f(x)=2|x|+|x+2|-\big||x+2|-2|x|\big| has a local minimum or a local maximum at x=x=
Not attempted
A
−2-2
B
−23-\dfrac{2}{3}
C
22
D
23\dfrac{2}{3}
68
Let f:[0,1]→Rf:[0,1]\to\mathbb{R} (the set of all real numbers) be a function. Suppose the function ff is twice differentiable, f(0)=f(1)=0f(0)=f(1)=0 and satisfies f′′(x)−2f′(x)+f(x)≥ex, x∈[0,1]f''(x)-2f'(x)+f(x)\geq e^x,\ x\in[0,1]. Which of the following is true for 0<x<10<x<1?
Not attempted
A
0<f(x)<∞0<f(x)<\infty
B
−12<f(x)<12-\dfrac12<f(x)<\dfrac12
C
−14<f(x)<1-\dfrac14<f(x)<1
D
−∞<f(x)<0-\infty<f(x)<0
69
Let f:[0,1]→Rf:[0,1]\to\mathbb{R} (the set of all real numbers) be a function. Suppose the function ff is twice differentiable, f(0)=f(1)=0f(0)=f(1)=0 and satisfies f′′(x)−2f′(x)+f(x)≥ex, x∈[0,1]f''(x)-2f'(x)+f(x)\geq e^x,\ x\in[0,1]. If the function e−xf(x)e^{-x}f(x) assumes its minimum in the interval [0,1][0,1] at x=14x=\dfrac14, which of the following is true?
Not attempted
A
f′(x)<f(x), 14<x<34f'(x)<f(x),\ \dfrac14<x<\dfrac34
B
f′(x)>f(x), 0<x<14f'(x)>f(x),\ 0<x<\dfrac14
C
f′(x)<f(x), 0<x<14f'(x)<f(x),\ 0<x<\dfrac14
D
f′(x)<f(x), 34<x<1f'(x)<f(x),\ \dfrac34<x<1
70
A line L:y=mx+3L: y=mx+3 meets the yy-axis at E(0,3)E(0,3) and the arc of the parabola y2=16x, 0≤y≤6y^2=16x,\ 0\le y\le6 at the point F(x0,y0)F(x_0,y_0). The tangent to the parabola at F(x0,y0)F(x_0,y_0) intersects the yy-axis at G(0,y1)G(0,y_1). The slope mm of the line LL is chosen such that the area of the triangle EFGEFG has a local maximum. Match List I with List II and select the correct answer using the code given below the lists: List I: P. m=m= Q. Maximum area of △EFG\triangle EFG is R. y0=y_0= S. y1=y_1= List II: 1. 12\dfrac12 2. 44 3. 22 4. 11 Codes:
Not attempted
A
(A) P→4, Q→1, R→2, S→3P\to4,\ Q\to1,\ R\to2,\ S\to3
B
(B) P→3, Q→4, R→1, S→2P\to3,\ Q\to4,\ R\to1,\ S\to2
C
(C) P→1, Q→3, R→2, S→4P\to1,\ Q\to3,\ R\to2,\ S\to4
D
(D) P→1, Q→3, R→4, S→2P\to1,\ Q\to3,\ R\to4,\ S\to2
71
Let f(x)=xsin⁡πxf(x)=x\sin\pi x, x>0x>0. Then for all natural numbers nn, f′(x)f'(x) vanishes at
Not attempted
A
a unique point in the interval (n,n+12)\left(n,n+\dfrac12\right)
B
a unique point in the interval (n+12,n+1)\left(n+\dfrac12,n+1\right)
C
a unique point in the interval (n,n+1)(n,n+1)
D
two points in the interval (n,n+1)(n,n+1)
72
A rectangular sheet of fixed perimeter with sides having their lengths in the ratio 8:158:15 is converted into an open rectangular box by folding after removing squares of equal area from all four corners. If the total area of removed squares is 100100, the resulting box has maximum volume. Then the lengths of the sides of the rectangular sheet are
Not attempted
A
24
B
32
C
45
D
60
73
The number of points in (−∞,∞)(-\infty,\infty), for which x2−xsin⁡x−cos⁡x=0x^2 - x\sin x - \cos x = 0, is
Not attempted
A
6
B
4
C
2
D
0
Show f(x)=x2−xsin⁡x−cos⁡xf(x)=x^2-x\sin x-\cos x is even, then use f′(x)=x(2−cos⁡x)f'(x)=x(2-\cos x) together with Rolle's theorem / strict monotonicity on (0,∞)(0,\infty) to count zeros.
Deconstruction\textbf{Deconstruction}: Count real roots of a transcendental-looking equation without solving it explicitly. Strategy\textbf{Strategy}: Exploit symmetry (the function is even) and use the sign of the derivative to establish strict monotonicity, which caps the number of zeros on each half-line. Execution\textbf{Execution}: Let f(x)=x2−xsin⁡x−cos⁡xf(x)=x^2-x\sin x-\cos x. - f(−x)=x2−(−x)sin⁡(−x)−cos⁡(−x)=x2−xsin⁡x−cos⁡x=f(x)f(-x)=x^2-(-x)\sin(-x)-\cos(-x)=x^2-x\sin x-\cos x=f(x), so ff is even. - f(0)=0−0−1=−1<0f(0)=0-0-1=-1<0. - As x→∞x\to\infty, f(x)∼x2→∞f(x)\sim x^2\to\infty, so by the Intermediate Value Theorem ff has at least one zero in (0,∞)(0,\infty). - f′(x)=2x−sin⁡x−xcos⁡x+sin⁡x=2x−xcos⁡x=x(2−cos⁡x)f'(x)=2x-\sin x-x\cos x+\sin x=2x-x\cos x=x(2-\cos x). - For x>0x>0, 2−cos⁡x>02-\cos x>0 always (since cos⁡x≤1<2\cos x\le1<2), so f′(x)>0f'(x)>0 for all x>0x>0: ff is strictly increasing on (0,∞)(0,\infty). - A strictly increasing continuous function going from f(0)=−1f(0)=-1 to +∞+\infty crosses zero exactly once on (0,∞)(0,\infty). - By evenness, there is exactly one more zero on (−∞,0)(-\infty,0). Total zeros =2=2. Verification\textbf{Verification}: An independent web-sourced commentary derives the identical intermediate result ff has precisely one zero in (0,∞)(0,\infty) and hence precisely two zeros in (−∞,∞)(-\infty,\infty) in its own worked derivation, but its printed answer-choice label was inconsistent with this conclusion (a labeling slip, since the paper's own options list 22 under choice (C) and 00 under (D), and the source's shown work unambiguously computes 22). The from-scratch derivation above and the exam's own official answer key both independently agree on 22. Final Answer\textbf{Final Answer}: (C) 2