reproduced your data : 
      Id Gender   Age Participate    Q1   Q10    Q2    Q3    Q4
*  <int>  <chr> <int>       <int> <chr> <chr> <chr> <chr> <chr>
1     16   Male    20           1     0     1     0     1     1
2     17   Male    40           1     1     0     0     0     0
3     18   Male    33           1     1     0     0     0     0
4     19   Male    18           1     1     0     0     0     0
5     20   Male    24           1     0     0     1     0     0
6     21 Female    42           1     0     0     1     0     0
7     22 Female    19           1     1     0     0     1     1
8     28 Female    49           1     0     1     1     0     0
9     29 Female    17           1     1     0     1     0     0
10    31   Male    18           1     1     0     1     0     0
first you need to convert Q1 and Q10 to numeric format since they're currently saved as character. 
Mutate_each in the Dplyr package allows you to apply one or more functions to one or more columns to where starts_with in the same package allow you to select variables based on their names.
So using a combination of both you can do the following :
library(dplyr)
data <- data %>% 
  mutate_each(funs(as.numeric), starts_with("Q"))
Look at the results :
str(data)
'data.frame':   10 obs. of  9 variables:
 $ Id         : int  16 17 18 19 20 21 22 28 23 31
 $ Gender     : Factor w/ 2 levels "Female","Male": 2 2 2 2 2 1 1 1 1 2
 $ Age        : int  20 40 33 18 24 42 19 49 17 18
 $ Participate: int  1 1 1 1 1 1 1 1 1 1
 $ Q1         : num  0 1 1 1 0 0 1 0 1 1
 $ Q10        : num  1 0 0 0 0 0 0 1 0 0
 $ Q2         : num  0 0 0 0 1 1 0 1 1 1
 $ Q3         : num  1 0 0 0 0 0 1 0 0 0
 $ Q4         : num  1 0 0 0 0 0 1 0 0 0
Your Q* variables are now numeric so you can treat them by selecting only the variables starting with "Q" using the dplyr::select verb and using rowSumns allows you to sum up all the columns of a given row so :
data %>% select(starts_with("Q")) %>% rowSums(.) -> data$Score
Where : 
- select(starts_with("Q"))= Select the columns starting with Q
 
- rowSums(.)= sum up the selected columns
 
- ->= assign the result to- data$Score
 
and then you can check the results :
   Id Gender Age Participate Q1 Q10 Q2 Q3 Q4 Score
1  16   Male  20           1  0   1  0  1  1     3
2  17   Male  40           1  1   0  0  0  0     1
3  18   Male  33           1  1   0  0  0  0     1
4  19   Male  18           1  1   0  0  0  0     1
5  20   Male  24           1  0   0  1  0  0     1
6  21 Female  42           1  0   0  1  0  0     1
7  22 Female  19           1  1   0  0  1  1     3
8  28 Female  49           1  0   1  1  0  0     2
9  23 Female  17           1  1   0  1  0  0     2
10 31   Male  18           1  1   0  1  0  0     2