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1

WB JEE 2022

MCQ (Single Correct Answer)
English
Bengali

Let f be a non-negative function defined in $$[0,\pi /2]$$, f' exists and be continuous for all x and $$\int\limits_0^x {\sqrt {1 - {{(f'(t))}^2}} dt = \int\limits_0^x {f(t)dt} } $$ and f (0) = 0. Then

A
$$f\left( {{1 \over 2}} \right) < {1 \over 2}$$ and $$f\left( {{1 \over 3}} \right) < {1 \over 3}$$
B
$$f\left( {{1 \over 2}} \right) > {1 \over 2}$$ and $$f\left( {{1 \over 3}} \right) < {1 \over 3}$$
C
$$f\left( {{4 \over 3}} \right) < {4 \over 3}$$ and $$f\left( {{2 \over 3}} \right) < {2 \over 3}$$
D
$$f\left( {{4 \over 3}} \right) > {4 \over 3}$$ and $$f\left( {{2 \over 3}} \right) > {2 \over 3}$$

$$[0,\pi /2]$$-তে অঋণাত্মক অপেক্ষক f এভাবে সঙ্গাত আছে যে f'-এর অস্তিত্ব আছে ও সকল x-এর জন্য সন্তত এবং $$\int\limits_0^x {\sqrt {1 - {{(f'(t))}^2}} dt = \int\limits_0^x {f(t)dt} } $$ এবং f (0) = 0 । সেক্ষেত্রে

A
$$f\left( {{1 \over 2}} \right) < {1 \over 2}$$ and $$f\left( {{1 \over 3}} \right) < {1 \over 3}$$
B
$$f\left( {{1 \over 2}} \right) > {1 \over 2}$$ and $$f\left( {{1 \over 3}} \right) < {1 \over 3}$$
C
$$f\left( {{4 \over 3}} \right) < {4 \over 3}$$ and $$f\left( {{2 \over 3}} \right) < {2 \over 3}$$
D
$$f\left( {{4 \over 3}} \right) > {4 \over 3}$$ and $$f\left( {{2 \over 3}} \right) > {2 \over 3}$$
2

WB JEE 2022

MCQ (Single Correct Answer)
English
Bengali

Area of the figure bounded by the parabola $${y^2} + 8x = 16$$ and $${y^2} - 24x = 48$$ is

A
$${{11} \over 9}$$ sq. unit
B
$${{32} \over 3}\sqrt 6 $$ sq. unit
C
$${{16} \over 3}$$ sq. unit
D
$${{24} \over 5}$$ sq. unit

অধিবৃত্তদ্বয় $${y^2} + 8x = 16$$ ও $${y^2} - 24x = 48$$ দ্বারা সীমাবদ্ধ অঞ্চলের ক্ষেত্রফল হল

A
$${{11} \over 9}$$ বর্গ একক
B
$${{32} \over 3}\sqrt 6 $$ বর্গ একক
C
$${{16} \over 3}$$ বর্গ একক
D
$${{24} \over 5}$$ বর্গ একক
3

WB JEE 2022

MCQ (Single Correct Answer)
English
Bengali

If $$x{{dy} \over {dx}} + y = x{{f(xy)} \over {f'(xy)}}$$, then $$|f(xy)|$$ is equal to

A
$$C{e^{{{{x^2}} \over 2}}}$$ (where C is the constant of integration)
B
$$C{e^{{x^2}}}$$ (where C is the constant of integration)
C
$$C{e^{2{x^2}}}$$ (where C is the constant of integration)
D
$$C{e^{{{{x^2}} \over 3}}}$$ (where C is the constant of integration)

Explanation

Given, $$x{{dy} \over {dx}} + y = x\,.\,{{f(xy)} \over {f'(xy)}}$$

Let, $$xy = t$$

Differentiating both sides with respect to x,

$$y + x{{dy} \over {dx}} = {{dt} \over {dx}}$$

$$\therefore$$ $${{dt} \over {dx}} = x\,.\,{{f(t)} \over {f'(t)}}$$

$$ \Rightarrow {{f'(t)} \over {f(t)}}dt = x\,dx$$

Integrating both sides, we get

$$\int {{{f'(t)} \over {f(t)}}dt = \int {x\,dx} } $$

$$ \Rightarrow \ln |f(t)| = {{{x^2}} \over 2} + C$$

$$ \Rightarrow |f(t)| = {e^{{{{x^2}} \over 2} + C}}$$

$$ \Rightarrow |f(xy)| = {e^{{{{x^2}} \over 2}}}.\,{e^2}$$

$$ \Rightarrow |f(xy)| = {e^{{{{x^2}} \over 2}}}\,.\,C$$ [ec = constant = C]

যদি $$x{{dy} \over {dx}} + y = x{{f(xy)} \over {f'(xy)}}$$, তবে $$|f(xy)|$$ হবে

A
$$C{e^{{{{x^2}} \over 2}}}$$ (যেখানে C সমাকলন ধ্রুবক)
B
$$C{e^{{x^2}}}$$ (যেখানে C সমাকলন ধ্রুবক)
C
$$C{e^{2{x^2}}}$$ (যেখানে C সমাকলন ধ্রুবক)
D
$$C{e^{{{{x^2}} \over 3}}}$$ (যেখানে C সমাকলন ধ্রুবক)
4

WB JEE 2021

MCQ (Single Correct Answer)
English
Bengali
The area bounded by the parabolas $$y = 4{x^2},y = {{{x^2}} \over 9}$$ and the straight line y = 2 is
A
$${{20\sqrt 2 } \over 3}$$ sq. unit
B
$$10\sqrt 5 $$ sq. unit
C
$${{10\sqrt 3 } \over 7}$$ sq. unit
D
$$10\sqrt 2 $$ sq. unit

Explanation

Given curve, $$y = 4{x^2},y = {{{x^2}} \over 9}$$ and y = 2

Graph of given curve


Area of shaded region

$$ = 2\int_0^2 {\left( {3\sqrt y - {{\sqrt y } \over 2}} \right)dy} $$

$$ = 2\int_2^0 {{{5\sqrt y } \over 2}dy} $$

$$ = 5 \times {2 \over 3}[{y^{3/2}}]_0^2$$

$$ = {{10} \over 3}[{2^{3/2}} - 0]$$

$$ = {{10} \over 3} \times 2\sqrt 2 $$

$$ = {{20} \over 3}\sqrt 2 $$ sq. unit
অধিবৃত্তদ্বয় $$y = 4{x^2} \,\text{ও} \,y = {{{x^2}} \over 9}$$ এবং y = 2 সরলরেখার মধ্যে সীমাবদ্ধ অঞ্চলের ক্ষেত্রফল হবে
A
$${{20\sqrt 2 } \over 3}$$ বর্গ একক
B
$$10\sqrt 5 $$ বর্গ একক
C
$${{10\sqrt 3 } \over 7}$$ বর্গ একক
D
$$10\sqrt 2 $$ বর্গ একক

Explanation

প্রদত্ত বক্ররেখা, $$y = 4{x^2},y = {{{x^2}} \over 9}$$ এবং y = 2

প্রদত্ত বক্ররেখার লেখচিত্র


ছায়াযুক্ত অঞ্চলের এলাকা

$$ = 2\int_0^2 {\left( {3\sqrt y - {{\sqrt y } \over 2}} \right)dy} $$

$$ = 2\int_2^0 {{{5\sqrt y } \over 2}dy} $$

$$ = 5 \times {2 \over 3}[{y^{3/2}}]_0^2$$

$$ = {{10} \over 3}[{2^{3/2}} - 0]$$

$$ = {{10} \over 3} \times 2\sqrt 2 $$

$$ = {{20} \over 3}\sqrt 2 $$ বর্গ একক

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