Practice Questions

Q64.The sum 12 −2. 32 + 3. 52 −4. 72 + 5. 92 −… . . +15. 292 is _____ . , is

Q64.Let the digits a, b, c be in A.P. Nine-digit numbers are to be formed using each of these three digits thrice such that three consecutive digits are in A.P. at least once. How many such numbers can be formed? = p1 p2 p3 . . . pm , where

Q64.Five digit numbers are formed using the digits 1, 2, 3, 5, 7 with repetitions and are written in descending order with serial numbers. For example, the number 77777 has serial number 1 . Then the serial number of 35337 is

Q64.The sum to 20 terms of the series 2 ⋅22 −32 + 2 ⋅42 −52 + 2 ⋅62−. . . . . . . . . . . . is equal to __________.

Q64.The total number of 4 -digit numbers whose greatest common divisor with 54 is 2 , is

Q64.The total number of six digit numbers, formed using the digits 4, 5, 9 only and divisible by 6, is _____ .

Q65.Let a1 = b1 = 1 and an = an−1 + (n −1), bn = bn−1 + an−1, ∀n ≥2. If S = ∑10n=1( 2nbn ) and T = ∑8n=1 2n−1n then 27(2S −T) is equal to

Q65.Let 0 < z < y < x be three real numbers such that x1 , 1y , 1z are in an arithmetic progression and x, √2y, z are in a geometric progression. If xy + yz + zx = 3 xyz, then 3(x + y + z)2 is equal to √2

Q65.The largest natural number n such that 3n divides 66! is _______

Q65.The coefficient of x−6 , in the expansion of ( 4x5 + 2x25 ) 9 5 9 x 2 4 is −84 and the coefficient of x−3l is 2αβ where 2 − xl

Q65.The 8th common term of the series S1 = 3 + 7 + 11 + 15 + 19 + … S2 = 1 + 6 + 11 + 16 + 21 + … . is + y = + [t] denotes the greatest integer ≤t, then

Q65.Let A1, A2, A3 be the three A.P. with the same common difference d and having their first terms as A, A + 1, A + 2, respectively. Let a, b, c be the 7th , 9th , 17th terms of A1, A2, A3 , respectively such that a 7 1 2b 17 1 + 70 = 0 . If a = 29, then the sum of first 20 terms of an AP whose first term is c −a −b and c 17 1 common difference is d , is equal to _____ . 12 JEE Main 2023 (25 Jan Shift 1) JEE Main Previous Year Paper ar ) is equal to

Q66.The sum of the common terms of the following three arithmetic progressions. 3, 7, 11, 15, … … … … , 399 2, 5, 8, 11, . . . . . . . . . 359 and 2, 7, 12, 17, … … , 197 , is equal to _____ .

Q66.For the two positive numbers a, b, if a, b and 181 are in a geometric progression, while a1 , 10 and 1b are in an arithmetic progression, then, 16a + 12b is equal to _____ . Q67. ∑6k=0 51−kC3 is equal to (1) 51C4 −45C4 (2) 51C3 −45C3 (3) 52C4 −45C4 (4) 52C3 −45C3

Q66.Let {ak} and {bk}, k ∈N , be two G.P.s with common ratio r1 and r2 respectively such that a1 = b1 = 4 and r1 < r2 . Let ck = ak + bk, k ∈N . If c2 = 5 and c3 = 134 then ∑∞k=1 ck −(12a6 + 8 b4) is equal to

Q66.Let α be the constant term in the binomial expansion of (√x − x 32 ) , n ≤15. If the sum of the coefficients of the remaining terms in the expansion is 649 and the coefficient of x−n is λα, then λ is equal to ________.

Q66.If the constant term in the binomial expansion of ( ) β < 0 is an odd number, then |αl −β| is equal to _____ .

Q66.If (20)19 + 2(21)(20)18 + 3(21)2(20)17+. . . +20(21)19 = k(20)19 , then k is equal to _____. 11 are equal, then −

Q67.If the term without x in the expansion of 23 + 22 (x x3α ) is 7315 , then |α| is equal to _____ . m 21 . + 5√2(x−2) log2 3) powers of 2(x−2) log2 3 , be

Q67.If 2n+1P n−1 : 2n−1P n = 11 : 21 , then n2 + n + 15 is equal to :

Q67.The constant term in the expansion of 5 + x71 + 3x2) is _____ . (2x

Q68.Let [t] denote the greatest integer ≤t. if the constant term in the expansion of (3x2 − 2x51 ) 7 equal to _____ JEE Main 2023 (08 Apr Shift 1) JEE Main Previous Year Paper

Q68.Let the sixth term in the binomial expansion of (√2log2(10−3x) If the binomial coefficients of the second, third and fourth terms in the expansion are respectively the first, third and fifth terms of an A.P., then the sum of the squares of all possible values of x is _____ .

Q68.Let P(a1, b1) and Q(a2, b2) be two distinct points on a circle with center C(√2, √3). Let and OC be perpendicular to both CP and CQ. If the area of the triangle OCP is √35 , then a21 + a22 + b21 + b22 2 is equal to __________

Q69.Let m1 and m2 be the slopes of the tangents drawn from the point P(4, 1) to the hyperbola H : 25y2 −x216 = 1 If Q is the point from which the tangents drawn to H have slopes |m1| and |m2| and they make positive (PQ)2 intercepts α and β on the x− axis, then αβ is equal to _______.