Differential Calculus jee mains questions | JEE Main Maths Quadratic Equations Previous Year Question

Q 21 :
Let $f : R \to R$ be a thrice differentiable function such that $f (0) = 0, f (1) = 1, f (2) = - 1, f (3) = 2 and f (4) = - 2 .$ Then, the minimum number of zeros of $(3 f' f'' + f f''') (x)$ is _______ . [2024]

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(5)

$∵ f : R \to R and f (0) = 0, f (1) = 1, f (2) = - 1, f (3) = 2 and f (4) = - 2, then$

$∴ f (x) has at least 4 real roots.$

$Then f' (x) has atleast 3 real roots and f'' (x) has atleast 2 real roots.$

$\frac{d}{d x} (f^{3} \cdot f'') = 3 f^{2} \cdot f' \cdot f'' + f^{3} \cdot f''' = f^{2} (3 f' \cdot f'' + f \cdot f''') .$

$Hence, f^{3} . f'' has at least 6 roots.$

$Then its differentiation has at least 5 distinct roots.$

Q 22 :
Let $[t]$ denote the greatest integer less than or equal to $t .$ Let $f : [0, \infty) \to R$ be a function defined by $f (x) = [\frac{x}{2} + 3] - [\sqrt{x}] .$ Let $S$ be the set of all points in the interval [0, 8] at which $f$ is not continuous. Then $\sum_{a \in S} a$ is equal to ______ . [2024]

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(17)

Given, $f (x) = [\frac{x}{2} + 3] - [\sqrt{x}] = [\frac{x}{2}] - [\sqrt{x}] + 3$

$[\frac{x}{2}]$ $is discontinuous at 2,4, 6, 8 and [\sqrt{x}] discontinuous at 1, 4$

$∴ f (x) is discontinuous at 1, 2, 6, 8$

$[∵ f (4^{+}) = 3 = f (4^{-})]$

$∴ \sum_{a \in S} a = 1 + 2 + 6 + 8 = 17$

Q 23 :
Let $f : (0, π) \to R$ be a function given by

$f (x)$ $= {\begin{matrix} {(\frac{8}{7})}^{\frac{\tan 8 x}{\tan 7 x}}, & 0 < x < \frac{π}{2} \\ a - 8, & x = \frac{π}{2} \\ {(1 + | c o t x |)}^{\frac{b}{a} | \tan x |}, & \frac{π}{2} < x < π \end{matrix}$

$where a, b \in Z . If f is continuous at x = \frac{π}{2}, then a^{2} + b^{2} is equal to_____.$ [2024]

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(81)

$Since, f is continuous at x = \frac{π}{2}$

$∴ \lim_{x \to \frac{π^{-}}{2}} f (x) = f (\frac{π}{2}) = \lim_{x \to \frac{π^{+}}{2}} f (x)$

$Now, \lim_{x \to \frac{π^{-}}{2}} {(\frac{8}{7})}^{(\frac{\tan 8 x}{\tan 7 x})}$

$= \lim_{h \to 0} {(\frac{8}{7})}^{\frac{\tan (4 π - 8 h)}{\tan (3 π + \frac{π}{2} - 7 h)}} = \lim_{h \to 0} {(\frac{8}{7})}^{\frac{\tan (- 8 h)}{\cot (7 h)}} = {(\frac{8}{7})}^{0} = 1$

$\Rightarrow a - 8 = 1 \Rightarrow a = 9$

$Also, \lim_{π \to {\frac{π}{2}}^{+}} f (x) = \lim_{π \to \frac{π}{2}} {(1 + | \cot x |)}^{\frac{b}{a} | \tan x |}$

$= \lim_{h \to 0} {(1 - \tan h)}^{- \frac{b}{9} \cot h}$

$= \lim_{h \to 0} {(1 - \tan h)}^{(\frac{1}{\tan h}) \cdot (\tan h) \cdot (\frac{- b}{9} \cot h)} = e^{\frac{b}{9}} = 1 \Rightarrow b = 0$

Hence, $a^{2} + b^{2} = 81 + 0 = 81$

Q 24 :
For a differentiable function $f : R \to R,$ suppose $f' (x) = 3 f (x) + α,$ where $α \in R,$ $f (0) = 1$ and $\lim_{x \to - \infty} f (x) = 7 .$ Then $9 f (- \log_{e} 3)$ is equal to ______ . [2024]

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(61)

We have $f' (x) = 3 f (x) + α$

Let $f (x) = y$ and $\frac{d y}{d x} = f' (x) \Rightarrow \frac{d y}{d x} = 3 y + α \Rightarrow \frac{d y}{3 y + α} = d x$

$\Rightarrow \frac{1}{3} \log (3 y + α) = x + c ... (i) [On integrating]$

Now, $f (0) = 1,$ so $y (0) = 1$

$\Rightarrow \frac{1}{3} \log (3 + α) = c$

$\Rightarrow \frac{1}{3} \log (\frac{3 y + α}{3 + α}) = x$ [Putting value of $c$ in eq. (i)]

$\Rightarrow \frac{3 y + α}{3 + α} = e^{3 x} \Rightarrow y = \frac{e^{3 x} (3 + α) - α}{3} = f (x)$

Now, $\lim_{x \to - \infty} f (x) = 7$

$\Rightarrow \lim_{x \to - \infty} \frac{e^{3 x} (3 + α) - α}{3} = 7 \Rightarrow \frac{- α}{3} = 7$

$\Rightarrow α = - 21$

$∴ f (x) = 7 - 6 e^{3 x}$

So, $9 f (- \log_{e} 3) = 9 (7 - 6 e^{3 (- \log_{e} 3)})$

$= 9 (7 - \frac{6}{27}) = 63 - 2 = 61$

Q 25 :
If $y = \frac{(\sqrt{x} + 1) (x^{2} - \sqrt{x})}{x \sqrt{x} + x + \sqrt{x}} + \frac{1}{15} (3 \cos^{2} x - 5) \cos^{3} x,$ then $96 y' (\frac{π}{6})$ is equal to _______ . [2024]

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(105)

We have, $y = \frac{(\sqrt{x} + 1) (x^{2} - \sqrt{x})}{x (\sqrt{x} + 1) + \sqrt{x}} + \frac{1}{15} (3 \cos^{2} x - 5) \cos^{3} x$

$= \frac{\sqrt{x} (x - 1) (x + 1 + \sqrt{x})}{\sqrt{x} (x + \sqrt{x} + 1)} + \frac{1}{15} (3 \cos^{2} x - 5) \cos^{3} x$

$= x - 1 + \frac{1}{5} \cos^{5} x - \frac{1}{3} \cos^{3} x$

$\Rightarrow y' = 1 + \frac{1}{5} \cdot 5 \cos^{4} x \cdot (- \sin x) - \frac{1}{3} \cdot 3 \cdot \cos^{2} x \cdot (- \sin x)$

$= 1 - \sin x \cdot \cos^{4} x + \sin x \cdot \cos^{2} x$

$\Rightarrow 96 y' (\frac{π}{6}) = 96 [1 - \frac{1}{2} \cdot \frac{9}{16} + \frac{1}{2} \cdot \frac{3}{4}] = 105$

Q 26 :
$Let for a differentiable function f : (0, \infty) \to R, f (x) - f (y) \geq \log_{e} (\frac{x}{y}) + x - y, \forall x, y \in (0, \infty) .$

$Then \sum_{n = 1}^{20} f' (\frac{1}{n^{2}}) is equal to _____ .$ [2024]

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(2890)

$f (x) - f (y) \geq \log_{e} (\frac{x}{y}) + x - y \forall (x, y) \in (0, \infty)$ ...(i)

Now, $f' (x) = \lim_{h \to 0} \frac{f (x + h) - f (x)}{h} = \lim_{h \to 0} \log_{e} \frac{(\frac{x + h}{x}) + h}{h}$

$= \lim_{h \to 0} [\frac{\log_{e} (1 + \frac{h}{x})}{h} + 1] = \frac{1}{x} + 1$ ..(ii)

Now, $\sum_{n = 1}^{20} f' (\frac{1}{n^{2}}) = \sum_{n = 1}^{20} \frac{1}{\frac{1}{n^{2}}} + 1$ [Using (i)]

$= \sum_{n = 1}^{20} (n^{2} + 1) = {[\frac{n (n + 1) (2 n + 1)}{6} + n]}_{n = 20}$

$= \frac{20 \times 21 \times 41}{6} + 20 = 2870 + 20 = 2890$

Q 27 :
$Let f (x) = x^{3} + x^{2} f' (1) + x f'' (2) + f''' (3), x \in R . Then f' (10) is equal to _____ .$ [2024]

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(202)

$f (x) = x^{3} + x^{2} f' (1) + x f'' (2) + f''' (3), x \in R$

$f' (x) = 3 x^{2} + 2 x f' (1) + f'' (2),$

$f'' (x) = 6 x + 2 f' (1)$

$f''' (x) = 6 \Rightarrow f''' (3) = 6$

$f'' (2) = 6 (2) + 2 f' (1) = 12 + 2 f' (1),$

$f' (1) = 3 {(1)}^{2} + 2 (1) f' (1) + f'' (2) = 3 + 2 f' (1) + f'' (2)$

$\Rightarrow f' (1) = 3 + 2 f' (1) + 12 + 2 f' (1)$

$\Rightarrow 3 f' (1) = - 15 \Rightarrow f' (1) = - 5$

$∴ f'' (2) = 12 + 2 (- 5) = 2$

Now, $f' (10) = 3 {(10)}^{2} + 2 (10) f' (1) + f'' (2)$

$= 300 + 20 (- 5) + 2 = 202$

Q 28 :
If the function $f (x)$ $= {\begin{matrix} \frac{1}{| x |}, & | x | \geq 2 \\ a x^{2} + 2 b, & | x | < 2 \end{matrix}$ is differentiable on $R,$ then $48 (a + b)$ is equal to ______ . [2024]

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(15)

$f (x)$ $= {\begin{matrix} \frac{1}{| x |}, & | x | \geq 2 \\ a x^{2} + 2 b, & | x | < 2 \end{matrix}$

Since, $f$ is differentiable so $f$ must be continuity

$⇒R.H.L. at 2 = L.H.L. at 2$

$\Rightarrow \frac{1}{2} = 4 a + 2 b$ ...(i)

Also, $f$ is differentiable at $x = 2,$

Now, around 2, $f (x)$ $= {\begin{matrix} \frac{1}{x}, & x \geq 2 \\ a x^{2} + 2 b, & x < 2 \end{matrix}$

$\Rightarrow$ $f' (x)$ $= {\begin{matrix} \frac{- 1}{x^{2}}, & x \geq 2 \\ 2 a x, & x < 2 \end{matrix}$

Now, R.H.D. at $x = 2 =$ L.H.D. at $x = 2$

$\Rightarrow - \frac{1}{4} = 4 a \Rightarrow a = - \frac{1}{16} \Rightarrow 2 b = \frac{1}{2} + \frac{1}{4} = \frac{3}{4} [Using (i)]$

$\Rightarrow b = \frac{3}{8}$

So, $48 (a + b) = 48 (\frac{3}{8} - \frac{1}{16}) = \frac{48 \times 5}{16} = 15$

Q 29 :
Let $f (x) = | 2 x^{2} + 5 | x | - 3 |, x \in R .$ If $m$ and $n$ denote the number of points where $f$ is not continuous and not differentiable respectively, then $m + n$ is equal to [2024]

5
3
2
0

1

2

3

4

(2)

We have, $f (x) = | 2 x^{2} + 5 | x | - 3 |, x \in R$

$f (x)$ is continuous at every point.

So, $m = 0$

Now, from the graph, $f (x)$ is non-differentiable at $x = - \frac{1}{2}, 0, \frac{1}{2}$

So, $n = 3$

$∴$ $m + n = 3$

Q 30 :
Consider the function $f : (0, 2) \to R$ defined by $f (x) = \frac{x}{2} + \frac{2}{x}$ and the function $g (x)$ defined by

$g (x) =$ ${\begin{matrix} \min {f (t)}, & 0 < t \leq x and 0 < x \leq 1 \\ \frac{3}{2} + x, & 1 < x < 2 \end{matrix}$

Then, [2024]

$g$ is neither continuous nor differentiable at $x = 1$
$g$ is continuous and differentiable for all $x \in (0, 2)$
$g$ is continuous but not differentiable at $x = 1$
$g$ is not continuous for all $x \in (0, 2)$

1

2

3

4

(3)

Given, $f : (0, 2) \to R$

$f (x) = \frac{x}{2} + \frac{2}{x}$

and $g (x) =$ ${\begin{matrix} \min {f (t)}, & 0 < t \leq x and 0 < x \leq 1 \\ \frac{3}{2} + x, & 1 < x < 2 \end{matrix}$

$\Rightarrow$ $f' (x) = \frac{1}{2} - \frac{2}{x^{2}} = \frac{(x - 2) (x + 2)}{2 x^{2}}$

$\Rightarrow$ $f (x)$ is decreasing in (0, 2).

$\Rightarrow \min (f (t)) = f (x), 0 < t \leq x$

$\Rightarrow$ $g (x) =$ ${\begin{matrix} \frac{x}{2} + \frac{2}{x}, & 0 < x \leq 1 \\ x + \frac{3}{2}, & 1 < x < 2 \end{matrix}$

From the graph, we see that $g (x)$ is continuous but not differentiable at $x = 1 .$

Topic Question Set

Q 21 :
Let $f : R \to R$ be a thrice differentiable function such that $f (0) = 0, f (1) = 1, f (2) = - 1, f (3) = 2 and f (4) = - 2 .$ Then, the minimum number of zeros of $(3 f' f'' + f f''') (x)$ is _______ . [2024]

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Q 24 :
For a differentiable function $f : R \to R,$ suppose $f' (x) = 3 f (x) + α,$ where $α \in R,$ $f (0) = 1$ and $\lim_{x \to - \infty} f (x) = 7 .$ Then $9 f (- \log_{e} 3)$ is equal to ______ . [2024]

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Q 25 :
If $y = \frac{(\sqrt{x} + 1) (x^{2} - \sqrt{x})}{x \sqrt{x} + x + \sqrt{x}} + \frac{1}{15} (3 \cos^{2} x - 5) \cos^{3} x,$ then $96 y' (\frac{π}{6})$ is equal to _______ . [2024]

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Q 26 :
$Let for a differentiable function f : (0, \infty) \to R, f (x) - f (y) \geq \log_{e} (\frac{x}{y}) + x - y, \forall x, y \in (0, \infty) .$

$Then \sum_{n = 1}^{20} f' (\frac{1}{n^{2}}) is equal to _____ .$ [2024]

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Q 27 :
$Let f (x) = x^{3} + x^{2} f' (1) + x f'' (2) + f''' (3), x \in R . Then f' (10) is equal to _____ .$ [2024]

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Q 28 :
If the function $f (x)$ $= {\begin{matrix} \frac{1}{| x |}, & | x | \geq 2 \\ a x^{2} + 2 b, & | x | < 2 \end{matrix}$ is differentiable on $R,$ then $48 (a + b)$ is equal to ______ . [2024]

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Q 29 :
Let $f (x) = | 2 x^{2} + 5 | x | - 3 |, x \in R .$ If $m$ and $n$ denote the number of points where $f$ is not continuous and not differentiable respectively, then $m + n$ is equal to [2024]

Q 30 :
Consider the function $f : (0, 2) \to R$ defined by $f (x) = \frac{x}{2} + \frac{2}{x}$ and the function $g (x)$ defined by

$g (x) =$ ${\begin{matrix} \min {f (t)}, & 0 < t \leq x and 0 < x \leq 1 \\ \frac{3}{2} + x, & 1 < x < 2 \end{matrix}$

Then, [2024]

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Topic Question Set

Q 21 : Let f:R→R be a thrice differentiable function such that f(0)=0, f(1)=1, f(2)=-1, f(3)=2 and f(4)=-2. Then, the minimum number of zeros of (3f'f''+ff''')(x) is _______ . [2024]

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Q 22 : Let [t] denote the greatest integer less than or equal to t. Let f:[0,∞)→R be a function defined by f(x)=[x2+3]-[x]. Let S be the set of all points in the interval [0, 8] at which f is not continuous. Then ∑a∈Sa is equal to ______ . [2024]

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Q 23 : Let f:(0,π)→R be a function given by f(x)={(87)tan 8xtan 7x,0<x<π2a-8,x=π2(1+|cot x|)ba|tan x|,π2<x<π where a,b∈Z. If f is continuous at x=π2, then a2+b2 is equal to _____. [2024]

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Q 24 : For a differentiable function f:R→R, suppose f'(x)=3f(x)+α, where α∈R, f(0)=1 and limx→-∞f(x)=7. Then 9f(-loge3) is equal to ______ . [2024]

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Q 25 : If y=(x+1)(x2-x)xx+x+x+115(3cos2x-5)cos3x, then 96y'(π6) is equal to _______ . [2024]

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Q 26 : Let for a differentiable function f:(0,∞)→R, f(x)-f(y)≥loge(xy)+x-y, ∀ x, y∈(0,∞). Then ∑n=120f'(1n2) is equal to _____ . [2024]

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Q 27 : Let f(x)=x3+x2f'(1)+xf''(2)+f'''(3), x∈R. Then f'(10) is equal to _____ . [2024]

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Q 28 : If the function f(x)={1|x|,|x|≥2ax2+2b,|x|<2 is differentiable on R, then 48(a+b) is equal to ______ . [2024]

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Q 29 : Let f(x)=|2x2+5|x|-3|, x∈R. If m and n denote the number of points where f is not continuous and not differentiable respectively, then m+n is equal to [2024]

Q 30 : Consider the function f:(0,2)→R defined by f(x)=x2+2x and the function g(x) defined by g(x)={min{f(t)},0<t≤x and 0<x≤132+x,1<x<2 Then, [2024]

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Q 21 :
Let $f : R \to R$ be a thrice differentiable function such that $f (0) = 0, f (1) = 1, f (2) = - 1, f (3) = 2 and f (4) = - 2 .$ Then, the minimum number of zeros of $(3 f' f'' + f f''') (x)$ is _______ . [2024]

Q 24 :
For a differentiable function $f : R \to R,$ suppose $f' (x) = 3 f (x) + α,$ where $α \in R,$ $f (0) = 1$ and $\lim_{x \to - \infty} f (x) = 7 .$ Then $9 f (- \log_{e} 3)$ is equal to ______ . [2024]

Q 25 :
If $y = \frac{(\sqrt{x} + 1) (x^{2} - \sqrt{x})}{x \sqrt{x} + x + \sqrt{x}} + \frac{1}{15} (3 \cos^{2} x - 5) \cos^{3} x,$ then $96 y' (\frac{π}{6})$ is equal to _______ . [2024]

Q 26 :
$Let for a differentiable function f : (0, \infty) \to R, f (x) - f (y) \geq \log_{e} (\frac{x}{y}) + x - y, \forall x, y \in (0, \infty) .$

$Then \sum_{n = 1}^{20} f' (\frac{1}{n^{2}}) is equal to _____ .$ [2024]

Q 27 :
$Let f (x) = x^{3} + x^{2} f' (1) + x f'' (2) + f''' (3), x \in R . Then f' (10) is equal to _____ .$ [2024]

Q 28 :
If the function $f (x)$ $= {\begin{matrix} \frac{1}{| x |}, & | x | \geq 2 \\ a x^{2} + 2 b, & | x | < 2 \end{matrix}$ is differentiable on $R,$ then $48 (a + b)$ is equal to ______ . [2024]

Q 29 :
Let $f (x) = | 2 x^{2} + 5 | x | - 3 |, x \in R .$ If $m$ and $n$ denote the number of points where $f$ is not continuous and not differentiable respectively, then $m + n$ is equal to [2024]

Q 30 :
Consider the function $f : (0, 2) \to R$ defined by $f (x) = \frac{x}{2} + \frac{2}{x}$ and the function $g (x)$ defined by

$g (x) =$ ${\begin{matrix} \min {f (t)}, & 0 < t \leq x and 0 < x \leq 1 \\ \frac{3}{2} + x, & 1 < x < 2 \end{matrix}$

Then, [2024]