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1

WB JEE 2022

MCQ (Single Correct Answer)
English
Bengali

If a, b, c are in G.P. and log a $$-$$ log 2b, log 2b $$-$$ log 3c, log 3c $$-$$ log a are in A.P., then a, b, c are the lengths of the sides of a triangle which is

A
acute angled
B
obtuse angled
C
right angled
D
equilateral

যদি a, b, c গুণোত্তর প্রগতিতে থাকে এবং log a $$-$$ log 2b, log 2b $$-$$ log 3c, log 3c $$-$$ log a সমান্তর প্রগতিতে থাকে, তবে a, b ও c যে ত্রিভুজের তিনটি বাহুর দৈর্ঘ্য হবে সে ত্রিভুজটি হবে

A
সূক্ষ্মকোণী ত্রিভুজ
B
স্থুলকোণী ত্রিভুজ
C
সমকোণী ত্রিভুজ
D
সমবাহু ত্রিভুজ
2

WB JEE 2021

MCQ (Single Correct Answer)
English
Bengali
Three unequal positive numbers a, b, c are such that a, b, c are in G.P. while $$\log \left( {{{5c} \over {2a}}} \right),\log \left( {{{7b} \over {5c}}} \right),\log \left( {{{2a} \over {7b}}} \right)$$ are in A.P. Then a, b, c are the lengths of the sides of
A
an isosceles triangle
B
an equilateral triangle
C
a scalene triangle
D
a right-angled triangle

Explanation

Given, a, b, c are in G.P. and $$\log \left( {{{5c} \over {2a}}} \right),\log \left( {{{7b} \over {5c}}} \right),\log \left( {{{2a} \over {7b}}} \right)$$ and A.P.

$$\therefore$$ b2 = ac

and

$$2\log \left( {{{7b} \over {5c}}} \right) = \log \left( {{{5c} \over {2a}}} \right) + \log \left( {{{2a} \over {7b}}} \right)$$

$$ \Rightarrow {{49{b^2}} \over {25{c^2}}} = {{5c} \over {7b}}$$

$$ \Rightarrow {(7b)^3} = {(5c)^3}$$

$$ \Rightarrow 7b = 5c \Rightarrow c = {7 \over 5}b$$

and $${b^2} = ac$$

$$ \Rightarrow {b^2} = a\left( {{7 \over 5}b} \right)$$

$$ \Rightarrow b = {7 \over 5}a$$

Sides are $${{5b} \over 7},b,{7 \over 5}b$$

$$\therefore$$ a, b, c are the length of the sides of scalene triangle.
তিনটি অ-সম ধনাত্মক সংখ্যা a, b, c গুণােত্তর প্রগতিতে আছে। $$\log \left( {{{5c} \over {2a}}} \right),\log \left( {{{7b} \over {5c}}} \right),\log \left( {{{2a} \over {7b}}} \right)$$ সমান্তর প্রগতিতে আছে। সেক্ষেত্রে a, b, c যে ত্রিভুজের তিনটি বাহুর দৈর্ঘ্য সূচিত করবে, সেই ত্রিভুজটি
A
সমদ্বিবাহু
B
সমবাহু
C
বিষমবাহু
D
সমকোণী

Explanation

দেওয়া হয়েছে, a, b, c আছে G.P-তে। এবং $$\log \left( {{{5c} \over {2a}}} \right),\log \left( {{{7b} \over {5c}}} \right),\log \left( {{{2a} \over {7b}}} \right)$$ A.P-তে.

$$\therefore$$ b2 = ac

এবং

$$2\log \left( {{{7b} \over {5c}}} \right) = \log \left( {{{5c} \over {2a}}} \right) + \log \left( {{{2a} \over {7b}}} \right)$$

$$ \Rightarrow {{49{b^2}} \over {25{c^2}}} = {{5c} \over {7b}}$$

$$ \Rightarrow {(7b)^3} = {(5c)^3}$$

$$ \Rightarrow 7b = 5c \Rightarrow c = {7 \over 5}b$$

এবং $${b^2} = ac$$

$$ \Rightarrow {b^2} = a\left( {{7 \over 5}b} \right)$$

$$ \Rightarrow b = {7 \over 5}a$$

বাহু গুলি হল $${{5b} \over 7},b,{7 \over 5}b$$

$$\therefore$$ a, b, c হল স্কেলিন ত্রিভুজের বাহুর দৈর্ঘ্য।
3

WB JEE 2021

MCQ (Single Correct Answer)
English
Bengali
The digit in the unit's place of the number 1! + 2! + 3! + .... + 99! is
A
3
B
0
C
1
D
7

Explanation

We have,

1! + 2! + 3! + 4! + 5! + .... + 99!

Since unit digit of 5!, 6!, 7!, ..... 99! are zero.

$$\therefore$$ 1! + 2! + 3! + 4! + 5! + ... + 99!

= 1 + 2 + 6 + 24 = 33

$$\therefore$$ Unit digit = 3
1! + 2! + 3! + .... + 99! সংখ্যার এককের ঘরের অঙ্ক হল
A
3
B
0
C
1
D
7

Explanation

আমাদের কাছে,

1! + 2! + 3! + 4! + 5! + .... + 99!

যেহেতু একক সংখ্যা 5!, 6!, 7!, ..... 99! শূন্য।

$$\therefore$$ 1! + 2! + 3! + 4! + 5! + ... + 99!

= 1 + 2 + 6 + 24 = 33

$$\therefore$$ একক সংখ্যা = 3
4

WB JEE 2021

MCQ (Single Correct Answer)
English
Bengali
Consider the real valued function h : {0, 1, 2, ...... 100} $$\to$$ R such that h(0) = 5, h(100) = 20 and satisfying h(p) = $${1 \over 2}$$ {h(p + 1) + h(p $$-$$ 1)} for every p = 1, 2 ..... 99. Then the value of h(1) is
A
5.15
B
5.5
C
6
D
6.15

Explanation

Given, h(p) = $${1 \over 2}$$ {h(p + 1) + h(p $$-$$ 1)} for every p = 1, 2, ....., 99

2h(p) = h(p + 1) + h(p $$-$$ 1)

$$\Rightarrow$$ h(p $$-$$ 1), h(p), h(p + 1) are in AP.

Now, h(100) = 20

$$\Rightarrow$$ h(0) + 99d = 20

$$\Rightarrow$$ 5 + 99d = 20 ($$\because$$ h(0) = 5)

$$\Rightarrow$$ d = $${{15} \over {99}} = {5 \over {33}}$$

$$\therefore$$ h(1) = h(0) + d

= 5 + $${5 \over {33}}$$ = 5 + 0.15 = 5.15
বাস্তব মানবিশিষ্ট অপেক্ষক h : {0, 1, 2, ...... 100} $$\to$$ R এরূপ যে h(0) = 5, h(100) = 20 ও h(p) = $${1 \over 2}$$ {h(p + 1) + h(p $$-$$ 1)} for every p = 1, 2 ..... 99 । h (1) এর মান হবে ?
A
5.15
B
5.5
C
6
D
6.15

Explanation

প্রদত্ত, প্রতেক p = 1, 2, ....., 99 এর জন্য h(p) = $${1 \over 2}$$ {h(p + 1) + h(p $$-$$ 1)}

2h(p) = h(p + 1) + h(p $$-$$ 1)

$$\Rightarrow$$ h(p $$-$$ 1), h(p), h(p + 1) are in AP.

এখন, h(100) = 20

$$\Rightarrow$$ h(0) + 99d = 20

$$\Rightarrow$$ 5 + 99d = 20 ($$\because$$ h(0) = 5)

$$\Rightarrow$$ d = $${{15} \over {99}} = {5 \over {33}}$$

$$\therefore$$ h(1) = h(0) + d

= 5 + $${5 \over {33}}$$ = 5 + 0.15 = 5.15

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