Class 11 Mathematics Limits And Derivatives MCQs Set 10

Question. Let \( f(x) = \lim_{n \to \infty} \frac{(x^2 + 2x + 3 + \sin \pi x)^n - 1}{(x^2 + 2x + 3 + \sin \pi x)^n + 1} \), then
(a) \( f(x) \) is continuous and differentiable for all \( x \in R \)
(b) \( f(x) \) is continuous but not differentiable for all \( x \in R \)
(c) \( f(x) \) is discontinuous at infinite number of points.
(d) \( f(x) \) is discontinuous at finite number of points.
Answer: (a) \( f(x) \) is continuous and differentiable for all \( x \in R \)

Question. Let \( f(x) = \lim_{h \to 0} \frac{(\sin(x+h))^{\ln(x+h)} - (\sin x)^{\ln x}}{h} \), then \( f\left(\frac{\pi}{2}\right) \) is
(a) equal to 0
(b) equal to 1
(c) \( \ln \frac{\pi}{2} \)
(d) non existent
Answer: (a) equal to 0

Question. Let \( g(x) = \begin{cases} \frac{x^2 + x \tan x - x \tan 2x}{ax + \tan x - \tan 3x} & ; x \neq 0 \\ 0 & ; x = 0 \end{cases} \). If \( g'(0) \) exists and is equal to non zero value \( b \), then \( \frac{b}{a} \) is equal to
(a) \( \frac{7}{13} \)
(b) \( \frac{7}{26} \)
(c) \( \frac{7}{52} \)
(d) \( \frac{5}{52} \)
Answer: (c) \( \frac{7}{52} \)

Question. If \( \lim_{n \to \infty} \frac{n \cdot 3^n}{n(x-2)^n + n \cdot 3^{n+1} - 3^n} = \frac{1}{3} \) where \( n \in N \), then the number of integer(s) in the range 'x' is
(a) 3
(b) 4
(c) 5
(d) infinite
Answer: (c) 5

Question. Let \( f(x) = \frac{e^{\tan x} - e^x + \ln(\sec x + \tan x) - x}{\tan x - x} \) be a continuous function at \( x = 0 \). The value of \( f(0) \) equals
(a) \( \frac{1}{2} \)
(b) \( \frac{2}{3} \)
(c) \( \frac{3}{2} \)
(d) 2
Answer: (c) \( \frac{3}{2} \)

Question. \( f(x) = \begin{cases} \frac{\cos^{-1}(1-\{x\}^2)\sin^{-1}(1-\{x\})}{\sqrt{2(\{x\}-\{x\}^3)}} & , \text{for } x \neq 0 \\ \frac{\pi}{2} & , \text{for } x = 0 \end{cases} \) where \(\{.\}\) denotes fractional part of \( x \), then, at \( x = 0 \), \( f(x) \), is
(a) continuous
(b) only left continuous
(c) only right continuous
(d) neither left continuous nor right continuous
Answer: (d) neither left continuous nor right continuous

Question. The value of \( \lim_{x \to \frac{\pi}{2}} \left( \frac{1}{2^{\cos^2 x}} + \frac{1}{3^{\cos^2 x}} + \frac{1}{4^{\cos^2 x}} + \frac{1}{5^{\cos^2 x}} + \frac{1}{6^{\cos^2 x}} \right)^{2 \text{ cos}^2 x} \) is
(a) 1
(b) 6
(c) 36
(d) \( \frac{1}{36} \)
Answer: (d) \( \frac{1}{36} \)

Question. If \( f(x) \) is continuous function \( \forall x \in R \) and the range of \( f(x) \) is \( (2, \sqrt{26}) \) and \( g(x) = \left[ \frac{f(x)}{c} \right] \) is continuous \( \forall x \in R \), then the least positive integral value of \( c \) is (where [.] denotes the greatest integer function)
(a) 2
(b) 3
(c) 5
(d) 6
Answer: (d) 6

Question. The function \( f(x) \) defined by \( f(x) = \begin{cases} \log_{(4x-3)}(x^2 - 2x + 5) & , \frac{3}{4} < x < 1 \text{ and } x > 1 \\ 4 & , x = 1 \end{cases} \)
(a) is continuous at \( x = 1 \)
(b) is discontinuous at \( x = 1 \) since \( f(1^+) \) does not exist though \( f(1^-) \) exists.
(c) is discontinuous at \( x = 1 \) since \( f(1^-) \) does not exist though \( f(1^+) \) exists.
(d) is discontinuous at \( x = 1 \) since neither \( f(1^+) \) nor \( f(1^-) \) exists.
Answer: (c) is discontinuous at \( x = 1 \) since \( f(1^-) \) does not exist though \( f(1^+) \) exists.

Question. \( f(x) = \max \left\{ \frac{x}{n}, |\sin \pi x| \right\}, n \in N \) has maximum points of non-differentiability for \( x \in (0, 4) \), then
(a) maximum value of \( n \) in more than 4.5
(b) least value of \( n \) is more than 3.5
(c) maximum value of \( n \) is less than 4.5
(d) least value of \( n \) is less than 3.5
Answer: (c) maximum value of \( n \) is less than 4.5

Question. \( \lim_{x \to \infty} \frac{\log_{10}(x+2) + [x] + 2}{x + \sum_{r=0}^{\infty} \left( \frac{1}{2} \right)^r} \) (where [.] is G.I.F) =
(a) 1
(b) 0
(c) 2
(d) does not exist
Answer: (a) 1

Question. If \( xy > 0 \) and \( \frac{x}{2} + \frac{y}{2} = \frac{-3}{2} \), \( x + y + \frac{y}{x} = \frac{1}{2} \), then \( \lim_{n \to \infty} \sum_{r=0}^{n} \left( \frac{x}{y} \right)^n = \)
(a) 1
(b) 3
(c) 0
(d) \( \frac{3}{2} \)
Answer: (c) 0

Question. Let \( f: R \to R \) be a continuous into function satisfying \( f(x) + f(-x) = 0, \forall x \in R \). If \( f(-3) = 2 \) and \( f(5) = 4 \) in \( [-5, 5] \), then the equation \( f(x) = 0 \) has
(a) exactly three real roots
(b) exactly two real roots
(c) atleast five real roots
(d) atleast three real roots
Answer: (d) atleast three real roots

Question. Let \( x_n \) be defined as \( \left( 1 + \frac{1}{n} \right)^{n+x_n} = e \), then \( \lim_{n \to \infty} x_n \) equals
(a) 1
(b) \( \frac{1}{2} \)
(c) \( \frac{1}{e} \)
(d) 0
Answer: (b) \( \frac{1}{2} \)

Question. Let \( f(x) \) be continuous for all \( x \in R \) except at \( x = 0 \) and \( f'(x) < 0, \forall x \in (-\infty, 0) \); \( f'(x) > 0, \forall x \in (0, \infty) \). Let \( \lim_{x \to 0^+} f(x) = 3, \lim_{x \to 0^-} f(x) = 4, f(0) = 5 \). Then the image of \( (0, 1) \) about the line \( y \lim_{x \to 0} f(\cos^3 x - \cos^2 x) = x \lim_{x \to 0} f(\sin^2 x - \sin^3 x) \) is
(a) \( \left( \frac{12}{25}, -\frac{9}{25} \right) \)
(b) \( \left( \frac{11}{25}, -\frac{9}{25} \right) \)
(c) \( \left( \frac{16}{25}, -\frac{8}{25} \right) \)
(d) \( \left( \frac{24}{25}, -\frac{7}{25} \right) \)
Answer: (c) \( \left( \frac{16}{25}, -\frac{8}{25} \right) \)

Question. Which of the following statements is/are true?
(a) If 'f' is differentiable at \( x = c \) then \( \lim_{h \to 0} \frac{f(c+h) - f(c-h)}{2h} \) exists and equals \( f'(c) \)
(b) Given a function 'f' and a point 'c' in the domain of f, if \( \lim_{h \to 0} \frac{f(c+h) - f(c-h)}{h} \) exists then the function is differentiable at \( x = c \)
(c) Let \( g(x) = \begin{cases} x^2 \sin \frac{1}{x^2}, & x \neq 0 \\ 0, & x = 0 \end{cases} \) then \( g' \) exists
(d) Let \( g(x) = \begin{cases} x \sin \frac{1}{x^2}, & x \neq 0 \\ 0, & x = 0 \end{cases} \) then \( g' \) exists and is continuous.
Answer: (a) If 'f' is differentiable at \( x = c \) then \( \lim_{h \to 0} \frac{f(c+h) - f(c-h)}{2h} \) exists and equals \( f'(c) \)

Question. Which of the following function(s) not defined at \( x = 0 \) has/have removable discontinuity at \( x = 0 \)?
(a) \( f(x) = \frac{1}{1 + 2^{\cot x}} \)
(b) \( f(x) = \cos \left( \frac{\sin x}{x} \right) \)
(c) \( f(x) = x \sin \left( \frac{\pi}{x} \right) \)
(d) \( f(x) = \frac{1}{\ln |x|} \)
Answer: (b) \( f(x) = \cos \left( \frac{\sin x}{x} \right) \)

Question. \( f(x) = \min \{1, \cos x, 1 - \sin x\}, -\pi \le x \le \pi \), then
(a) \( f(x) \) is not differentiable at 0
(b) \( f(x) \) is differentiable at \( \pi/2 \)
(c) \( f(x) \) has local maxima at 0
(d) \( f(x) \) local maximum at \( x = \pi/2 \)
Answer: (c) \( f(x) \) has local maxima at 0

Question. If \( f(x) = \min \{\tan x, \cot x\} \), then
(a) \( f(x) \) is discontinuous at \( x = 0, \frac{\pi}{4}, \frac{5\pi}{4} \)
(b) \( f(x) \) is continuous at \( x = 0, \frac{\pi}{2}, \frac{3\pi}{2} \)
(c) Range of \( f(x) \) is \( (-\infty, -1] \cup [0, 1] \)
(d) \( f(x) \) is periodic with period \( \pi \)
Answer: (d) \( f(x) \) is periodic with period \( \pi \)

Question. The function \( f(x) = ||e^x - 1| - 1| \) is
(a) continuous for all \( x \)
(b) differentiable for all \( x \)
(c) not continuous at \( x = 0, \ln 2 \)
(d) not differentiable at \( x = 0, \ln 2 \)
Answer: (d) not differentiable at \( x = 0, \ln 2 \)

Question. Given a real valued function \( f \) such that \( f(x) = \begin{cases} \frac{\tan^2 \{x\}}{(x^2 - [x]^2)} & \text{for } x > 0 \\ 1 & \text{for } x = 0 \\ \sqrt{\{x\} \cot \{x\}} & \text{for } x < 0 \end{cases} \), where [x] is the integral part and {x} is the fractional part of x, then
(a) \( \lim_{x \to 0^+} f(x) = 1 \)
(b) \( \lim_{x \to 0^-} f(x) = \cot 1 \)
(c) \( \cot^{-1} (\lim_{x \to 0^-} f(x))^2 = 1 \)
(d) \( \tan^{-1} (\lim_{x \to 0^+} f(x)) = \frac{\pi}{4} \)
Answer: (d) \( \tan^{-1} (\lim_{x \to 0^+} f(x)) = \frac{\pi}{4} \)

Question. If \( f(x) = \text{sgn}(x^2 - ax + 1) \) has maximum number of points of discontinuity, then
(a) \( a \in (2, \infty) \)
(b) \( a \in (-\infty, 2) \)
(c) \( a \in (-2, 2) \)
(d) \( a \in [-2, 2] \)
Answer: (a) \( a \in (2, \infty) \)

Question. If \( f(x) = \begin{cases} x^2 (\text{sgn}[x] + \{x\}), & 0 \le x < 2 \\ \sin x + |x-3|, & 2 \le x < 4 \end{cases} \), where [] and {} represent the greatest integer and the fractional part function, respectively.
(a) \( f(x) \) is differentiable at \( x = 1 \)
(b) \( f(x) \) is continuous but non-differentiable at \( x = 1 \)
(c) \( f(x) \) is non-differentiable at \( x = 2 \)
(d) \( f(x) \) is discontinuous at \( x = 2 \).
Answer: (d) \( f(x) \) is discontinuous at \( x = 2 \).

Question. Given \( f(x) = \sum_{r=1}^{n} (x^r + x^{-r})^2 ; x \neq \pm 1 \) and \( g(x) = \begin{cases} \lim_{n \to \infty} \left( \frac{(f(x) - 2n)x^{2n-2}(1-x^2)}{-1} \right) & \text{for } x \neq \pm 1 \\ -1 & \text{for } x = \pm 1 \end{cases} \), then \( g(x) \)
(a) is discontinuous at \( x = -1 \)
(b) is continuous at \( x = 2 \)
(c) has a removable discontinuity at \( x = 1 \)
(d) has an irremovable discontinuity at \( x = 1 \)
Answer: (d) has an irremovable discontinuity at \( x = 1 \)

PASSAGE : 1

\( f(x) = \begin{cases} x+a & x < 0 \\ |x-1| & x \ge 0 \end{cases} \), \( g(x) = \begin{cases} x+1 & x < 0 \\ (x-1)^2 + b & x \ge 0 \end{cases} \) Where a and b are non-negative real numbers

Question. The value of a, if \( (g \circ f)(x) \) is continuous for all real x, is
(a) -1
(b) 0
(c) 1
(d) 2
Answer: (c) 1

Question. The value of b, if \( (g \circ f)(x) \) is continuous for all real x, is
(a) -1
(b) 0
(c) 1
(d) 2
Answer: (c) 1

Question. For these values of a and b, \( (g \circ f)(x) \) for all \( x \in (-1, 1) \) is
(a) Even
(b) Odd
(c) neither even nor odd
(d) symmetrical about x-axis
Answer: (a) Even

PASSAGE : 2

Two functions \( f(x) \) & \( g(x) \) are defined as,
\( f(x) = \begin{cases} [h(x)] - \left\{ \frac{h(x)}{2} \right\} & , \text{for } x \in \text{domain of } h \\ 0 & , \text{for } x \notin \text{domain of } h \end{cases} \)
\( g(x) = \begin{cases} \text{sgn } h(x) & , \text{for } x \in \text{domain of } h \\ 0 & , \text{for } x \notin \text{domain of } h \end{cases} \)
Where, \( h(x) = \frac{1}{\sqrt{b-a}} \frac{\sqrt{\frac{b-a}{a}} \sin 2x}{1 + \left( \sqrt{\frac{b-a}{a}} \sin x \right)^2} \cdot \sqrt{a + b \tan^2 x} \)
for \( b > a > 0 \) & [.] denotes G.I.F & {.} denotes fractional part of x.

Question. For \( h(x) \) which one of the following is true.
(a) \( h(x) \) is continuous at \( x = 0 \) and \( x = \frac{\pi}{2} \)
(b) \( h(x) \) is continuous at \( x = 0 \) but discontinuous at \( x = \frac{\pi}{2} \)
(c) \( h(x) \) is discontinuous at \( x = 0 \) but continuous at \( x = \frac{\pi}{2} \)
(d) \( h(x) \) is discontinuous at \( x = 0 \) & \( x = \frac{\pi}{2} \)
Answer: (b) \( h(x) \) is continuous at \( x = 0 \) but discontinuous at \( x = \frac{\pi}{2} \)

Question. For \( f(x) \), which of the following is true
(a) \( f(x) \) is continuous at \( x = 0 \) and \( x = \frac{\pi}{2} \)
(b) \( f(x) \) is continuous at \( x = 0 \) but discontinuous at \( x = \frac{\pi}{2} \)
(c) \( f(x) \) is discontinuous at \( x = 0 \) but continuous at \( x = \frac{\pi}{2} \)
(d) \( f(x) \) is discontinuous at \( x = 0 \) & \( x = \frac{\pi}{2} \)
Answer: (b) \( f(x) \) is continuous at \( x = 0 \) but discontinuous at \( x = \frac{\pi}{2} \)

Question. \( g(\pi/4) + g(2\pi/3) + g(0) + g(\pi/2) = \)
(a) 0
(b) 1
(c) -1
(d) 2
Answer: (d) 2

PASSAGE : 3

Let \( f(x) = \begin{cases} x+2, & 0 \le x < 2 \\ 6-x, & x \ge 2 \end{cases} \), \( g(x) = \begin{cases} 1 + \tan x, & 0 \le x < \frac{\pi}{4} \\ 3 - \cot x, & \frac{\pi}{4} \le x < \pi \end{cases} \)

Question. \( f(g(x)) \) is
(a) discontinuous at \( x = \frac{\pi}{4} \)
(b) differentiable at \( x = \frac{\pi}{4} \)
(c) continuous but non-differentiable at \( x = \frac{\pi}{4} \)
(d) differentiable at \( x = \frac{\pi}{4} \), but derivative is not continuous.
Answer: (c) continuous but non-differentiable at \( x = \frac{\pi}{4} \)

Question. The number of points on non-differentiability of \( h(x) = |f(g(x))| \) is
(a) 1
(b) 2
(c) 3
(d) 4
Answer: (b) 2

Question. The range of \( h(x) = f(g(x)) \) is
(a) \( (-\infty, \infty) \)
(b) \( (4, \infty) \)
(c) \( (-\infty, 4) \)
(d) [ -4, 5 ]
Answer: (b) \( (4, \infty) \)

PASSAGE : 4

Let \( f(x) = 2 + |x-1| \) and \( g(x) = \min(f(t)) \) where \( x \le t \le x^2 + x + 1 \) then answer the following:

Question. Number of points of discontinuity of \( g(x) \) is
(a) 1
(b) 2
(c) 3
(d) 0
Answer: (d) 0

Question. Number of points where the function \( g(x) \) is not differentiable is
(a) 1
(b) 2
(c) 3
(d) 0
Answer: (d) 0

Question. Range of \( g(x) \) is
(a) [2, \(\infty\))
(b) [2, 2)
(c) [0, 2]
(d) \( (-\infty, \infty) \)
Answer: (a) [2, \(\infty\))

PASSAGE : 5

Let A be a \( n \times n \) matrix given by \( A = [a_{ij}] \) such that each horizontal row is an arithmetic progression and each vertical column is a geometrical progression. It is known that each column in geometric progression have the same common ratio. Given that \( a_{24} = 1, a_{42} = 1/8 \) and \( a_{43} = 3/16 \).

Question. Let \( S_n = \sum_{j=1}^{n} a_{4j} \), then \( \lim_{n \to \infty} \frac{S_n}{n^2} \) is equal to
(a) \( \frac{1}{4} \)
(b) \( \frac{1}{8} \)
(c) \( \frac{1}{16} \)
(d) \( \frac{1}{32} \)
Answer: (c) \( \frac{1}{16} \)

Question. Let \( d_i \) be the common difference of the elements in i-th row, then \( \sum_{i=1}^{n} d_i \)
(a) n
(b) \( \frac{1}{2} - \frac{1}{2^{n+1}} \)
(c) \( 1 - \frac{1}{2^n} \)
(d) \( \frac{n+1}{2^n} \)
Answer: (b) \( \frac{1}{2} - \frac{1}{2^{n+1}} \)

Question. The value of \( \lim_{n \to \infty} \sum_{i=1}^{n} a_{ii} \) is equal to
(a) \( \frac{1}{4} \)
(b) \( \frac{1}{2} \)
(c) 1
(d) 2
Answer: (b) \( \frac{1}{2} \)

PASSAGE : 6

If \( f(x) = x^2 - 2|x| \)
\( g(x) = \begin{cases} \text{minimum } \{f(t) : -2 \le t \le x\}, & -2 \le x < 0 \\ \text{minimum } \{f(t) : 0 \le t \le x\}, & 0 \le x \le 3 \end{cases} \)

Question. The function \( y = |f(x)| \) is differentiable for
(a) \( x \in R \)
(b) \( x \in R - \{0\} \)
(c) \( x \in R - \{0, 2\} \)
(d) ( -2, 2 )
Answer: (c) \( x \in R - \{0, 2\} \)

Question. The function \( g(x) \) is differentiable for
(a) \( x \in [-2, 3] \)
(b) \( x \in [-2, 3] - \{-1, 0, 1\} \)
(c) \( x \in [-2, 3] - \{0, 1\} \)
(d) \( x \in [-2, 3] - \{-1, 0, 2\} \)
Answer: (c) \( x \in [-2, 3] - \{0, 1\} \)

Question. \( g(x) = \dots (x \in [-1, 0]) \)
(a) 0
(b) \( -1/2 \)
(c) -1
(d) 7 / 3
Answer: (a) 0

PASSAGE : 7

Suppose f, g and h be three real valued function defined on R. Let \( f(x) = 2x + |x| \), \( g(x) = \frac{1}{3}(2x - |x|) \) and \( h(x) = f(g(x)) \)

Question. The value of a, if \( (g \circ f)(x) \) is continuous for all real x, is
(a) -1
(b) 0
(c) 1
(d) 2
Answer: (c) 1

Question. The value of b, if \( (g \circ f)(x) \) is continuous for all real x, is
(a) -1
(b) 0
(c) 1
(d) 2
Answer: (c) 1

Question. For these values of a and b, \( (g \circ f)(x) \) for all \( x \in (-1, 1) \) is
(a) Even
(b) Odd
(c) neither even nor odd
(d) symmetrical about x-axis
Answer: (a) Even

Question. The range of the function \( k(x) = 1 + \frac{1}{\pi} \left( \cos^{-1}(h(x)) + \cot^{-1}(h(x)) \right) \) is equal to
(a) \( \left[ \frac{1}{4}, \frac{7}{4} \right] \)
(b) \( \left[ \frac{5}{4}, \frac{11}{4} \right] \)
(c) \( \left[ \frac{1}{5}, \frac{4}{4} \right] \)
(d) \( \left[ \frac{7}{4}, \frac{11}{4} \right] \)
Answer: (b) \( \left[ \frac{5}{4}, \frac{11}{4} \right] \)

Question. The domain of definition of the function \( l(x) = \sin^{-1}(f(x) - g(x)) \) is equal to
(a) \( \left[ \frac{3}{8}, \infty \right) \)
(b) \( (-\infty, 1] \)
(c) \( [-1, 1] \)
(d) \( \left( -\infty, \frac{3}{8} \right] \)
Answer: (d) \( \left( -\infty, \frac{3}{8} \right] \)

Question. The function \( T(x) = f(g(f(x))) + g(f(g(x))) \) is
(a) continuous and differentiable in \( (-\infty, \infty) \)
(b) continuous but not derivable \( \forall x \in R \)
(c) neither continuous nor derivable \( \forall x \in R \)
(d) an odd function
Answer: (b) continuous but not derivable \( \forall x \in R \)

PASSAGES : 8

Consider the function \( f : R \to R \), defined as \( f(x) = \begin{cases} x^2 - x + 3, & x \in (-\infty, 3) \cap \mathbb{Q} \\ x + a, & x \in (-\infty, 2) - \mathbb{Q} \\ 2^x + 1, & x \in (2, 3) - \mathbb{Q} \\ 9 \tan\left(\frac{\pi x}{12}\right), & x \in [3, 6) \end{cases} \)

Question. If \( f(x) \) is continuous at \( x = 2 \) then the value of \( a \) is
(a) 1
(b) 2
(c) 3
(d) indeterminate
Answer: (c) 3

Question. The function \( f(x) \) at \( x = 3 \)
(a) has non-removable discontinuity
(b) has removable discontinuity
(c) is differentiable
(d) is continuous but not differentiable
Answer: (d) is continuous but not differentiable

Question. \( f'(4) \) is equal to
(a) \( \pi \)
(b) \( 3\pi \)
(c) \( \frac{3\pi}{2} \)
(d) \( \frac{3\pi}{16} \)
Answer: (b) \( 3\pi \)

MATRIX MATCHING QUESTIONS

Question. Match the following Column I with Column II:
Column I
(A) \( f(x) = \begin{cases} \frac{1}{|x|} & \text{for } |x| \ge 1 \\ ax^2 + b & \text{for } |x| < 1 \end{cases} \) is differentiable everywhere and \( k = |a+b| \), then value of k is
(B) If \( f(x) = \text{sgn}(x^2 - ax + 1) \) has exactly one point of discontinuity, then the value of \( a \) can be
(C) \( f(x) = [2 + 3n|\sin x|], n \in N, x \in (0, \pi) \) has exactly 11 points of discontinuity, then the value of \( n \) is
(D) \( f(x) = ||x| - 2 + a| \) has exactly three points of non-differentiability, then the value of \( a \) is
Column II
(P) 2
(Q) -2
(R) 1
(S) -1
Answer: A-R, S; B-P, Q; C-P, Q; D-P, R

Question. Match the following Column I with Column II:
Column I
(A) If \( f(x) = \frac{1 - \cos(1 - \cos x)}{x^4} \) is continuous everywhere \( f(0) = \frac{1}{\lambda} \) and let \( \mu \) be the number of points \( g(x) = \max\{x^2 - 1, 4, x^2 + 1\} \) is non-differentiable, then \( \lambda + \mu \) is divisible by
(B) \( f' \) be a continuous function on R. If \( f\left(\frac{1}{3^n}\right) = (\cos e^n) 3^{-n} + \frac{n^2}{n^2 + n + 1} \) and \( f(0) = \frac{1}{\lambda} \) and let \( \mu \) be the number of points if \( g(x) = \begin{cases} \min\{x, x^2\}, & x \ge 0 \\ \min\{2x, x^2 - 1\}, & x < 0 \end{cases} \) is non-differentiable in \([-2, 2]\), then \( \lambda + \mu \) is divisible by
(C) If \( f(x) = \frac{\sqrt{2} \cos x - 1}{\cot x - 1} \) is continuous at \( x = \frac{\pi}{4} \) and \( f\left(\frac{\pi}{4}\right) = \frac{1}{\lambda} \) and \( \mu \) be the jump of the function at the point of discontinuity of the function \( g(x) = \frac{1 - k^{1/x}}{1 + k^{1/x}} (k > 0) \), then \( |\lambda - \mu| \) is divisible by
Column II
(P) 2
(Q) 3
(R) 4
(S) 5
Answer: A-PS; B-PR, C-PQRS

Question. Let \( f(x) \) be a real valued function defined by \( f(x) = x^2 - 2|x| \) and \( g(x) = \begin{cases} \min\{f(t) : -2 \le t \le x\}, & x \in [-2, 0) \\ \max\{f(t) : 0 \le t \le x\}, & x \in [0, 3] \end{cases} \). Match Column I with Column II:
Column I
(A) \( g(x) \) is not continuous at \( x \) equal to
(B) \( g(x) \) is not derivable at \( x \) equal to
(C) Number of integral critical points of \( g(x) \) is equal to
(D) Absolute maximum value of \( g(x) \) is equal to
Column II
(P) -2
(Q) 0
(R) 1
(S) 2
(T) 3
Answer: A-Q, B-Q, S, C-T, D-T

Question. Match Column I with Column II for \( L = \lim_{x \to 0} \left[ \frac{n_1 \sin x}{x} \right], M = \lim_{x \to 0} \left[ \frac{n_2 x}{\sin x} \right], N = \lim_{x \to 0} \left[ \frac{n_3 \tan x}{x} \right] \) and \( P = \lim_{x \to 0} \left[ \frac{n_4 x}{\tan x} \right] \) where [.] denotes the greatest integer function and \( n_1, n_2, n_3, n_4 \in N \):
Column I
(A) For \( n_1 = 3, n_4 = 9 \) then \( P/L \) is divisible by
(B) For \( n_2 = 12, n_3 = 2 \), then \( M/N \) is divisible by
(C) For \( n_2 = 3, n_4 = 10 \) then \( P + M \) is divisible by
(D) For \( n_1 = 4, n_2 = 5, n_3 = 6, n_4 = 7 \) then \( L + N + P + M \) is divisible by
Column II
(P) 2
(Q) 3
(R) 4
(S) 5
(T) 6
Answer: A-PR; B-PQT; C-P, Q, R, T; D-P, S