1. Number of solutions of \[2x^{\left(x!+x\right)}=\left(x!+x\right)^{x},x\epsilon N\]

is

a) 0

b) 1

c) 2

d) infinite

Explanation: For x \[\epsilon\] N, we can write the given equation as

2. Let n = 2019. The least positive integer k
for which

\[k\left(n^{2}\right)\left(n^{2}-1^{2}\right)\left(n^{2}-2^{2}\right)\left(n^{2}-3^{2}\right)....\left(n^{2}-\left(n-1\right)^{2}\right)=r!\]

for some positive integer r is

a) 2019

b) 2018

c) 1

d) 2

Explanation: We can rewrite the given expression as

3. If \[0< r < s\leq n\] and \[^{n}P_{r}=^{n}P_{s}\] , then value of
r + s is

a) 2n – 2

b) 2n – 1

c) 2

d) 1

Explanation:

4. if \[\frac{1}{4}.\frac{2}{6}.\frac{3}{8}.\frac{4}{10}....\frac{30}{62}.\frac{31}{64}=8^{x}\]

then value of x is

a) -7

b) -9

c) -10

d) -12

Explanation:

5. The sum\[\sum_{i=0}^{m}\left(\begin{array}{c}10\\ i\end{array}\right)\left(\begin{array}{c}20\\ m-i\end{array}\right)\]
(where \[\left(\begin{array}{c}p\\ q\end{array}\right)=0\] if (p < q) is maximum where m is

a) 5

b) 10

c) 15

d) 20

Explanation:

6. Let \[T_{n}\] denote the number of triangles
which can be formed by using the vertices of a regular
polygon of n sides. If \[T_{n+1}-T_{n}=21\] , then n equals

a) 5

b) 7

c) 6

d) 4

Explanation: The number of triangles that can be formed by

7. An eight digits number divisible by 9 is
to be formed by using 8 digits out of the digits 0, 1, 2, 3, 4,
5, 6, 7, 8, 9 without replacement. The number of ways in
which this can be done is

a) 9!

b) 2(7!)

c) 4(7!)

d) (36) (7!)

Explanation: We have 0 + 1 + 2 + 3 … + 8 + 9 = 45

To obtain an eight digits number exactly divisible by 9, we must not use either (0, 9) or (1, 8) or (2, 7) or (3, 6) or (4, 5). [Sum of the remaining eight digits is 36 which is exactly divisible by 9].

When, we do not use (0, 9), then the number of required 8 digit number is 8!.

When one of (1, 8) or (2, 7) or (3, 6) or (4, 5) is not used, the remaining digits can be arranged in 8! – 7! ways as 0 cannot be at extreme left.

Hence, there are 8! + 4(8! – 7!) = (36) (7!) numbers in the desired category

8. The number of rational numbers lying in
the interval (2018, 2019) all whose digits after the decimal
point are non-zero and are in decreasing order is

a) \[\sum_{i=1}^{9}\] \[^{9}P_{i}\]

b) \[\sum_{i=1}^{10}\] \[^{9}P_{i}\]

c) \[2^{9}-1\]

d) \[2^{10}-1\]

Explanation: A rational number of the desired category

9. The number of functions f from the set A
= {0, 1, 2} in to the set B = {0, 1, 2, 3, 4, 5, 6, 7} such that
\[f\left(i\right)\leq f\left(j\right)\] for i < j and \[i,j \epsilon A\] is

a) \[^{8}C_{3}\]

b) \[^{8}C_{3}+2\left(^{8}C_{2}\right)\]

c) \[^{10}C_{3}\]

d) \[^{9}C_{3}\]

Explanation:

10. The number of positive integral solutions
of the equation \[x_{1}x_{2}x_{3}x_{4}x_{5}=1050\] is

a) 1800

b) 1675

c) 1400

d) 1875

Explanation: Using prime-factorization of 1050, we can write the given equation as