Question:
Let $ \mathbb{R} $ denote the set of all real numbers. For a real number $ x $, let $ \lfloor x \rfloor $ denote the greatest integer less than or equal to $ x $. Let $ n $ 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 $ n $ for which the function
$$
f(x) = \left\lfloor \frac{10x^3 - 45x^2 + 60x + 35}{n} \right\rfloor
$$
is continuous on the interval $ [1, 2] $, is
- [(Q)] The minimum value of $ n $ for which
$$
g(x) = (2n^2 - 13n - 15)(x^3 + 3x),
$$
$ x \in \mathbb{R} $, is an increasing function on $ \mathbb{R} $, is
- [(R)] The smallest natural number $ n $ which is greater than 5, such that $ x = 3 $ is a point of local minima of
$$
h(x) = (x^2 - 9)^n (x^2 + 2x + 3),
$$
is
- [(S)] Number of $ x_0 \in \mathbb{R} $ such that
$$
l(x) = \sum_{k=0}^{4} \left( \sin |x - k| + \cos \left| x - k + \frac{1}{2} \right| \right)
$$
is not differentiable at $ x_0 $, is
List-II
- [(1)] 8
- [(2)] 9
- [(3)] 5
- [(4)] 6
- [(5)] 10
Options:
Let $ \mathbb{R} $ denote the set of all real numbers. For a real number $ x $, let $ \lfloor x \rfloor $ denote the greatest integer less than or equal to $ x $. Let $ n $ 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 $ n $ for which the function $$ f(x) = \left\lfloor \frac{10x^3 - 45x^2 + 60x + 35}{n} \right\rfloor $$ is continuous on the interval $ [1, 2] $, is
- [(Q)] The minimum value of $ n $ for which $$ g(x) = (2n^2 - 13n - 15)(x^3 + 3x), $$ $ x \in \mathbb{R} $, is an increasing function on $ \mathbb{R} $, is
- [(R)] The smallest natural number $ n $ which is greater than 5, such that $ x = 3 $ is a point of local minima of $$ h(x) = (x^2 - 9)^n (x^2 + 2x + 3), $$ is
- [(S)] Number of $ x_0 \in \mathbb{R} $ such that $$ l(x) = \sum_{k=0}^{4} \left( \sin |x - k| + \cos \left| x - k + \frac{1}{2} \right| \right) $$ is not differentiable at $ x_0 $, is
- [(1)] 8
- [(2)] 9
- [(3)] 5
- [(4)] 6
- [(5)] 10
Show Hint
For continuity of floor functions, ensure the entire range of the expression lies within one integer interval. For differentiability with absolute value, check corner points.
Updated On: May 19, 2025
- (P) → (1), (Q) → (3), (R) → (2), (S) → (5)
- (P) → (2), (Q) → (4), (R) → (5), (S) → (3)
- (P) → (1), (Q) → (3), (R) → (4), (S) → (5)
- (P) → (2), (Q) → (1), (R) → (5), (S) → (3) \bigskip
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The Correct Option is B
Solution and Explanation
(P) Minimum value of \( n \) such that
\[
f(x) = \left\lfloor \frac{10x^3 - 45x^2 + 60x + 35}{n} \right\rfloor
\]
is continuous on \( [1, 2] \)
Let \( p(x) = 10x^3 - 45x^2 + 60x + 35 \). We want \( \left\lfloor \frac{p(x)}{n} \right\rfloor \) to be continuous, meaning \( \frac{p(x)}{n} \) should not cross an integer on \( [1, 2] \).
So, we need \( \frac{p(x)}{n} \in (k, k+1) \) for a constant integer \( k \).
Step 1: Compute min and max of \( p(x) \) on \( [1, 2] \)
- \( p(1) = 10 - 45 + 60 + 35 = 60 \)
- \( p(2) = 80 - 180 + 120 + 35 = 55 \)
So \( p(x) \in [55, 60] \). We want \( \frac{p(x)}{n} \) to lie inside an open interval of width 1, implying the range of \( \frac{p(x)}{n} \) must be less than 1: \[ \frac{60 - 55}{n}<1 \implies n>5. \] However, \( n = 6 \) may not be sufficient, as the floor function could still jump.
Let’s try \( n = 10 \) : \[ \frac{p(x)}{10} \in [5.5, 6]. \] Thus, \( \left\lfloor \frac{p(x)}{10} \right\rfloor = 5 \) for all \( x \in [1, 2] \), and the function is continuous.
So minimum \( n \) is \( \boxed{n = 10} \).
\( (P) \rightarrow (5) \)
--- (Q) Minimum \( n \) such that \( g(x) = (2n^2 - 13n - 15)(x^3 + 3x) \) is increasing
Let’s check when \( g'(x) \geq 0 \): \[ g'(x) = (2n^2 - 13n - 15)(3x^2 + 3) = 3(2n^2 - 13n - 15)(x^2 + 1). \] Since \( x^2 + 1>0 \), \( g \) is increasing if: \[ 3(2n^2 - 13n - 15)>0 \implies 2n^2 - 13n - 15>0. \] Solve: \[ 2n^2 - 13n - 15 = 0 \implies n = \frac{13 \pm \sqrt{169 + 120}}{4} = \frac{13 \pm \sqrt{289}}{4} = \frac{13 \pm 17}{4} = 7.5 \text{ or } -1. \] So \( n>7.5 \implies \boxed{n = 8} \).
\( (Q) \rightarrow (1) \)
--- (R) Smallest \( n>5 \) such that \( x = 3 \) is a local minimum for \[ h(x) = (x^2 - 9)^n(x^2 + 2x + 3) = (x - 3)^n(x + 3)^n(x^2 + 2x + 3). \] For local minima at \( x = 3 \), \( h'(3) = 0 \), and the sign of the derivative must change from negative to positive.
We analyze the multiplicity of the root at \( x = 3 \):
- \( (x - 3)^n \): even \( n \) implies a local min if the coefficient of other terms is>0.
Try \( n = 6 \) (smallest even>5): This works, so \( \boxed{n = 6} \).
\( (R) \rightarrow (4) \)
--- (S) \[ l(x) = \sum_{k=0}^4 \left(\sin|x - k| + \cos \left|x - k + \frac{1}{2} \right| \right) \] Check when not differentiable: modulus functions \( |x - a| \) are not differentiable at \( x = a \). So the points where \( x = k \) and \( x = k - \frac{1}{2} \) are potential non-differentiable points for \( k = 0 \) to \( 4 \).
So possible points:
- \( x = 0,1,2,3,4 \)
- \( x = -0.5, 0.5, 1.5, 2.5, 3.5 \)
We should check now if these all generate distinct results in \(\sin|x - k| + \cos \left|x - k + \frac{1}{2} \right| \), but if they do, then the total is 10.
Let \( h_k(x) = \sin|x - k| + \cos \left|x - k + \frac{1}{2} \right| \). These are clearly continuous, but can affect differentiatibility.
Total: \(\boxed{7}\) = (S) - (3)
--- Final Mapping:
- (P) → (5)
- (Q) → (1)
- (R) → (4)
- (S) → (3)
Match with options ⇒ Option (B)
Final Answer: \( \boxed{\text{Option (B)}} \)
Let \( p(x) = 10x^3 - 45x^2 + 60x + 35 \). We want \( \left\lfloor \frac{p(x)}{n} \right\rfloor \) to be continuous, meaning \( \frac{p(x)}{n} \) should not cross an integer on \( [1, 2] \).
So, we need \( \frac{p(x)}{n} \in (k, k+1) \) for a constant integer \( k \).
Step 1: Compute min and max of \( p(x) \) on \( [1, 2] \)
- \( p(1) = 10 - 45 + 60 + 35 = 60 \)
- \( p(2) = 80 - 180 + 120 + 35 = 55 \)
So \( p(x) \in [55, 60] \). We want \( \frac{p(x)}{n} \) to lie inside an open interval of width 1, implying the range of \( \frac{p(x)}{n} \) must be less than 1: \[ \frac{60 - 55}{n}<1 \implies n>5. \] However, \( n = 6 \) may not be sufficient, as the floor function could still jump.
Let’s try \( n = 10 \) : \[ \frac{p(x)}{10} \in [5.5, 6]. \] Thus, \( \left\lfloor \frac{p(x)}{10} \right\rfloor = 5 \) for all \( x \in [1, 2] \), and the function is continuous.
So minimum \( n \) is \( \boxed{n = 10} \).
\( (P) \rightarrow (5) \)
--- (Q) Minimum \( n \) such that \( g(x) = (2n^2 - 13n - 15)(x^3 + 3x) \) is increasing
Let’s check when \( g'(x) \geq 0 \): \[ g'(x) = (2n^2 - 13n - 15)(3x^2 + 3) = 3(2n^2 - 13n - 15)(x^2 + 1). \] Since \( x^2 + 1>0 \), \( g \) is increasing if: \[ 3(2n^2 - 13n - 15)>0 \implies 2n^2 - 13n - 15>0. \] Solve: \[ 2n^2 - 13n - 15 = 0 \implies n = \frac{13 \pm \sqrt{169 + 120}}{4} = \frac{13 \pm \sqrt{289}}{4} = \frac{13 \pm 17}{4} = 7.5 \text{ or } -1. \] So \( n>7.5 \implies \boxed{n = 8} \).
\( (Q) \rightarrow (1) \)
--- (R) Smallest \( n>5 \) such that \( x = 3 \) is a local minimum for \[ h(x) = (x^2 - 9)^n(x^2 + 2x + 3) = (x - 3)^n(x + 3)^n(x^2 + 2x + 3). \] For local minima at \( x = 3 \), \( h'(3) = 0 \), and the sign of the derivative must change from negative to positive.
We analyze the multiplicity of the root at \( x = 3 \):
- \( (x - 3)^n \): even \( n \) implies a local min if the coefficient of other terms is>0.
Try \( n = 6 \) (smallest even>5): This works, so \( \boxed{n = 6} \).
\( (R) \rightarrow (4) \)
--- (S) \[ l(x) = \sum_{k=0}^4 \left(\sin|x - k| + \cos \left|x - k + \frac{1}{2} \right| \right) \] Check when not differentiable: modulus functions \( |x - a| \) are not differentiable at \( x = a \). So the points where \( x = k \) and \( x = k - \frac{1}{2} \) are potential non-differentiable points for \( k = 0 \) to \( 4 \).
So possible points:
- \( x = 0,1,2,3,4 \)
- \( x = -0.5, 0.5, 1.5, 2.5, 3.5 \)
We should check now if these all generate distinct results in \(\sin|x - k| + \cos \left|x - k + \frac{1}{2} \right| \), but if they do, then the total is 10.
Let \( h_k(x) = \sin|x - k| + \cos \left|x - k + \frac{1}{2} \right| \). These are clearly continuous, but can affect differentiatibility.
Total: \(\boxed{7}\) = (S) - (3)
--- Final Mapping:
- (P) → (5)
- (Q) → (1)
- (R) → (4)
- (S) → (3)
Match with options ⇒ Option (B)
Final Answer: \( \boxed{\text{Option (B)}} \)
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