1. If \[1^{2}+2^{2}+3^{2}+...+2003^{2}=\left(2003\right)\left(4007\right)\left(334\right)\]

and \[\left(1\right)\left(2003\right)+\left(2\right)\left(2002\right)+\left(3\right)\left(2001\right)+...+\left(2003\right)\left(1\right)=\left(2003\right)\left(334\right)\left(x\right)\]

then x equals

a) 2005

b) 2004

c) 2003

d) 2001

Explanation:

2. Sum of the series
\[\frac{1}{2^{2}-1}+\frac{1}{4^{2}-1}+\frac{1}{6^{2}-1}+....\] upto 2n terms is

a) \[\frac{n}{2n+1}\]

b) \[\frac{1}{2n+1}\]

c) \[\frac{1}{n+1}\]

d) \[\frac{4}{2n-1}\]

Explanation:

3. Let \[a_{1},a_{2},....,a_{10}\] be in A.P. and \[h_{1},h_{2},....,h_{10}\]
be in H.P. If \[a_{1}=h_{1}=2\] and \[a_{10}=h_{10}=3\] , then a_{5} h_{6} is

a) 2

b) 3

c) 5

d) 6

Explanation: Let d be the common difference of the A.P., then

4. Suppose a, b, c are in A.P. and \[a^{2},b^{2},c^{2}\]
are in G.P. If a < b < c and a + b + c = 3/2, then the value
of a is

a) \[\frac{1}{2\sqrt{2}}\]

b) \[\frac{1}{2\sqrt{3}}\]

c) \[\frac{1}{2}-\frac{1}{\sqrt{3}}\]

d) \[\frac{1}{2}-\frac{1}{\sqrt{2}}\]

Explanation:

5. Let \[S_{1},S_{2},....\] be squares such that for
each n ≥ 1, the length of a side of \[S_{n}\] equals the length
of a diagonal of \[S_{n+1}\] . If the length of a side of \[S_{1}\] is
10 cm, then the smallest value of n for which Area \[\left(S_{n}\right)< 1\] is

a) 7

b) 8

c) 9

d) 10

Explanation: Let a

_{n}denote the length of a side of S

_{n}. We are given:

6. If a, b, c are in G.P., and log a – log 2b,
log 2b – log 3c and 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.

Explanation: We have b

^{2}= ac and

7. If \[x_{1},x_{2}...x_{n}\] are n non-zero real numbers
such that \[\left(x_2^1+x_2^2+...+x_{n-1}^{2}\right)\left(x_2^2+x_3^2+...+x_{n}^{2}\right)\leq \left(x_{1}x_{2}+x_{2}x_{3}+...+x_{n-1}x_{n}\right)^{2}\]

then \[x_{1},x_{2},...,x_{n}\] are in

a) A.P

b) G.P

c) H.P

d) A.G.P

Explanation: We shall make use of the identity

8. If three positive real numbers a, b, c
(c > a) are in H.P., then log (a + c) + log (a – 2b + c) is
equal to

a) 2 log (c – b)

b) 2 log (a + c)

c) 2 log (c – a)

d) log a + log b + log c.

Explanation: a, b, c are in H.P., b = (2ac)/(a + c). We have

9. If \[a_{1},a_{2},...,a_{n}\] are in A.P. with common
difference \[d\neq 0\] , then the sum of the series

sin d [cosec a_{1} cosec a_{2} + cosec a_{2} cosec a_{3}+....+cosec a_{n-1}cosec a_{n}]
is

a) \[\sec a_{1}-\sec a_{n}\]

b) cosec a_{1} - cosec a_{n}

c) \[\cot a_{1}-\cot a_{n}\]

d) \[\tan a_{1}-\tan a_{n}\]

Explanation:

10. If sum of first n terms of the series
\[S=\sqrt{3}+\sqrt{75}+\sqrt{243}+\sqrt{507}+....\] is \[435\sqrt{3}\] , then n is
equal to

a) 14

b) 15

c) 19

d) 21

Explanation: