Practice problems
MCQ questions can have more than 1 correct answer.
Q-1) If \(a_1,a_2,…a_n\) are in HP, then \(\frac{a_1}{a_2+a_3+…a_n},\frac{a_2}{a_1+a_3+…a_n},…,\frac{a_n}{a_1+a_2+…a_{n-1}}\) are in
a) A.P b) G.P c) H.P d) None of these
Q-2) Find the value of \(\sum\limits_{r=0}^n(a+r+ar)(-a)^r\)
Q-3) The value of \(S=\frac{4}{19}+\frac{44}{19^2}+\frac{444}{19^3}+…\infty\) is
a) \(\frac{40}{9}\) b) \(\frac{38}{81}\) c) \(\frac{36}{171}\) d) None of these
Q-4) Let \(a,b,c,d \in \mathbb{R^+}\) with \(a,b,c\) in A.P and \(b,c,d\) in H.P, then
a) \(ab=cd\) b) \(ac=bd\) c) \(bc=ad\) d) None of these
Q-5) If the (p+q)th term of a G.P is \(a\) and the (p-q)th term is \(b\), then find the pth term.
Q-6) Define \(S_n=1^2-2^2+3^2-4^2+5^2+…n\)terms then
a) \(S_{40}=-820\) b) \(S_{2n}>S_{2n+2}\) c) \(S_{51}=-1326\) d) \(S_{2n+1}>S_{2n-1}\)
Q-7) For \(m,n\in \mathbb{N}\), the value(s) for \(m\) for which \(n^m+1\) divides \(a=1+n+n^2+…n^{63}\) is(are):
a) \(8\) b) \(16\) c) \(32\) d) \(64\)
Q-8) Let \(E=\sum\limits_{i=1}^\infty\frac{1}{i^2}\), then
a) \(E<3\) b) \(E>\frac{3}{2}\) c) \(E>2\) d) \(E<2\)
Q-9) Which of the following can be terms (not necessarily consecutive) of any A.P.:
a) \(1,6,19\) b) \(\sqrt{2}, \sqrt{50}, \sqrt{98}\) c) \(\ln 2, \ln 16, \ln 128\) d) \(\sqrt{2},\sqrt{3},\sqrt{7}\)
Q-10) Consider the sequence \(\left\{ a_n\right\}\) with \(a_1=2\) and \(a_n=\frac{a_{n-1}^2}{a_{n-2}}\) for all \(n \ge 3\). Also given that \(a_2,a_5\in \mathbb{N}\) with \(a_5 \le 162\). What can be the possible values for \(a_5\)
a) \(162\) b) \(64\) c) \(32\) d) \(2\)
Passage for Questions 11-13
Sum of certain consecutive odd positive integers is \(57^2-13^2\)
Q-11) Number of integers are:
a) \(40\) b) \(37\) c) \(44\) d) \(51\)
Q-12) Least value of these integers is
a) \(22\) b) \(27\) c) \(31\) d) \(43\)
Q-13) Greatest of these integers is
a) divisible by 7 b) divisible by 9 c) divisible by 11 d) None of these
Passage for Questions 14 & 15
Consider the sequence in the form of groups : \((1), (2,2),(3,3,3),(4,4,4,4)…\)
Q-14) The sum of first \(2000\) terms is:
a) \(84336\) b) \(96324\) c) \(78466\) d) None of these
Q-15) The 2000th term of the sequence is not divisible by
a) \(3\) b) \(9\) c) \(7\) d) None of these
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