Class notes Mathematics , physics, chemistry NCERT

Basic Mathematics and Logarithm


Single Correct Type Questions 6. The number of integral solutions x of
2
1. Let a, b, c and d be positive real numbers such that a + b  x−7  [11 April, 2023 (Shift-I)]
log  7   ≥ 0 is
+ c + d = 11. If the maximum value of a5b3c2d is 3750b,  x+   2x − 3 
 2
then the value of b is [11 April, 2023 (Shift-II)] (a) 6 (b) 8 (c) 5 (d) 7
(a) 90 (b) 110 (c) 55 (d) 108  π
7. If for x ∈  0,  log10 sin x + log10 cos x = –1 and
 2
{
2. Let A = x ∈  :  x + 3 ]+[ x + 4  ≤ 3 } 1
log10(sin x + cos= x) (log10 n − 1), n > 0 , then the value
2
 x 3 
x −3

−3 x  of n is equal to : [16th March, 2021 (Shift-I)]
 x  : 3  ∑ r =1 r  < 3  , where [t] denotes
∞
B =∈
  10   (a) 16 (b) 12
greatest integer function. Then,
(c) 9 (d) 20
 [6 April, 2023 (Shift-I)] 8. The inverse of y = 5log x is: [17 March, 2021 (Shift-I)]
(a) A ∩ B = φ 1

(a) x = ylog 25 (b) x = y log5
(b) A = B
1
(c) B ⊂ C, A ≠ B (c) x = 5log y (d) x = 5 log y
(d) A ⊂ B, A ≠ B 9. The sum of the roots of the equation,
3. The integer ‘k’. for which the inequality x – 2(3k – 1) x
2 [31 Aug, 2021 (Shift-II)
+ 8k2 – 7 > 0 is valid for every x in R is: ( ) (
x + 1 - 2log 2 3 + 2 x + 2log 4 10 - 2 - x = 0 , is:
)
 [25 Feb, 2021 (Shift-I)]
(a) 3 (b) 2 (a) log2 12 (b) log2 13
(c) 4 (d) 0 (c) log2 11 (d) log2 14
4. If {p} denotes the fractional part of the number p, then Integer Type Questions
 3200  10. Let a, b, c be three distinct positive real numbers such that
  , is equal to: [6 Sep, 2020 (Shift-I)]
 8  (2a)logea = (bc)logeb and bloge2 = alogec . Then 6a + 5bc is equal
to ________. [10 April, 2023 (Shift-I)]
5 1 7 3
(a) (b) (c) (d) 11. If the sum of all the roots of the equation e2x – 11ex – 45e–x
8 8 8 8 81
5. The number of real roots of the equation 5 + |2x – 1| = 2x + = 0 is loge p, then p is equal to _________ .
2
(2x – 2) is [10 April, 2019 (Shift-II)] [27 June, 2022 (Shift-I)]
(a) 3 (b) 2 (c) 4 (d) 1 12. The number of solutions of the equation log4(x – 1) =
log2 (x – 3) is [26 Feb, 2021 (Shift-I)]

1 JEE PYQs Mathematics P
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, equal to _________ [6 April, 2023 (Shift-I)]
13. If 3(cos 2 x) =
( 3 − 1) cos x + 1 , the number of solutions
 x
\ b2 + b – a2 = 4 – 1 = 3
of the given equation when x ∈ 0,  is
 2
[26 Feb, 2021 (Shift-I)]
18. If a + b + c = 1, ab + bc + ca = 2 and abc = 3, then the
14. The number of solutions of the equation log(x+1) (2x2 + 7x
+ 5) + log(2x+5) (x+1)2 – 4 = 0, x > 0, is. value of a4 + b4 + c4 is equal to....
 [20 July 2021 (Shift-II)]  [25 July, 2021 (Shift-II)]

15. The number of distinct solutions of the equation, 19. The missing value in the following figure is
log1/2|sin x| = 2 – log1/2 |cos x| in the interval [0, 2p], is  [18 Mar, 2021 (Shift-I)]
__________ [9 Jan, 2020 (Shift-I)]

16. Let m be the minimum possible value of log3 (3y1 + 3y2 + 2 3
3y3), where y1, y2, y3 are real numbers for which y1 + y2 +
y3 = 9. Let M be the maximum possible value of (log3x1 + 1 5
log3x2 + log3x3), where x1, x2, x3 are positive real numbers 1 ?
for which x1 + x2 + x3 = 9. Then the value of log2 (m3) + 4
24 6
3
log3(M2) is _____. [JEE Adv 2020] 12 4
17. Let the point (p, p + 1) lie inside the region
8 7
=E { 2
}
( x, y ) : 3 − x ≤ y ≤ 9 − x ,0 ≤ x ≤ 3 .If the set of
all values of p is the interval (a, b). then b2 + b – a2 is Use the logic which gives answer in single digit. (2021)




2 JEE PYQs Mathematics P
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