Model Question Paper 1

Other integer = sum of two integers − (the given integer)

= −21 − (−10)

= −21 + 10

= −(21 − 10)

= −11.

Hence, option 3 is the correct option.

The smallest natural number is 1.

The greatest 2-digit number is 99.

The natural numbers strictly between 1 and 99 are:

2, 3, 4, ..., 98.

Counting them gives 98 − 2 + 1 = 97 natural numbers.

Hence, option 2 is the correct option.

We rearrange the factors using the commutative and associative properties of multiplication so as to group the numbers whose product is easy to compute.

Note that 25 × 8 = 200, which is easy to handle.

25 × 37 × 8 × 6 = (25 × 8) × (37 × 6)

= 200 × 222

= 44400.

Hence, 25 × 37 × 8 × 6 = 44400.

Four consecutive integers preceding (i.e. just before) −97 are obtained by repeatedly subtracting 1 from −97.

−97 − 1 = −98

−98 − 1 = −99

−99 − 1 = −100

−100 − 1 = −101

Listing them in order from smallest to largest, the four consecutive integers preceding −97 are −101, −100, −99 and −98.

Hence, the four consecutive integers preceding −97 are −101, −100, −99 and −98.

A 4-digit number has four places: thousands, hundreds, tens and ones. Since 5 is fixed at the ten's place, the number has the form:

Thousands	Hundreds	Tens	Ones
?	?	5	?

The remaining three digits must be different from each other and different from 5.

Greatest 4-digit number: To make the number greatest, we put the largest available digits at the higher places (left side).

• Thousands place: 9 (the largest digit)
• Hundreds place: 8 (next largest, ≠ 9 and ≠ 5)
• Tens place: 5 (fixed)
• Ones place: 7 (next largest, ≠ 9, ≠ 8 and ≠ 5)

So the greatest such number is 9857.

Smallest 4-digit number: To make the number smallest, we put the smallest available digits at the higher places. However, the thousands digit cannot be 0 (otherwise it will not remain a 4-digit number).

• Thousands place: 1 (smallest non-zero digit)
• Hundreds place: 0 (smallest digit, ≠ 1 and ≠ 5)
• Tens place: 5 (fixed)
• Ones place: 2 (smallest digit ≠ 0, ≠ 1 and ≠ 5)

So the smallest such number is 1052.

Hence, the greatest 4-digit number is 9857 and the smallest 4-digit number is 1052.

We have to form natural numbers using each of the digits 7, 0 and 3 exactly once.

The one-digit numbers that can be formed using 7, 0, 3 are 7, 0, 3.

2-digit numbers that can be formed using 7, 0, 3 are:

30, 37, 70, 73 (0 cannot be on the tens place because then the number will become 1-digit)

Now we are forming 3-digit numbers.

The hundreds digit cannot be 0, because then the number would not be a 3-digit number. So the hundreds digit must be either 7 or 3.

Case I: Hundreds digit is 7.

The remaining digits 0 and 3 can be arranged at the tens and ones places in 2 ways:

• 7 0 3 → 703
• 7 3 0 → 730

Case II: Hundreds digit is 3.

The remaining digits 0 and 7 can be arranged at the tens and ones places in 2 ways:

• 3 0 7 → 307
• 3 7 0 → 370

Hence, all the possible natural numbers formed by the digits 7, 0 and 3 (without repetition) are 3, 7, 30, 37, 70, 73, 307, 370, 703 and 730.

We first change the sign of each integer being subtracted and then add. After that, we group the positive and negative integers separately.

−237 − (−328) + (−205) − 76 + 89

= −237 + 328 − 205 − 76 + 89

Now group the positive and negative integers:

Sum of positive integers = 328 + 89 = 417.

Sum of absolute values of negative integers = 237 + 205 + 76 = 518.

Therefore,

−237 + 328 − 205 − 76 + 89 = 417 − 518

= −(518 − 417)

= −101.

Hence, −237 − (−328) + (−205) − 76 + 89 = −101.

On that day, Abhijeet covered the following distances one after another:

(1) From home to school (in the morning) = 3 km 520 m.

(2) From school, on his way back, until he saw the woman = 1 km 370 m.

(3) From that place to the nearest hospital = 2 km 775 m.

(4) From the hospital back to his home = 4 km 565 m.

Total distance covered = 3 km 520 m + 1 km 370 m + 2 km 775 m + 4 km 565 m.

Converting each distance into metres:

3 km 520 m = 3520 m

1 km 370 m = 1370 m

2 km 775 m = 2775 m

4 km 565 m = 4565 m

Adding:

3520 + 1370 = 4890 m

2775 + 4565 = 7340 m

4890 + 7340 = 12230 m

Now 12230 m = 12000 m + 230 m = 12 km 230 m.

Hence, the total distance covered by Abhijeet on that day is 12 km 230 m (i.e. 12230 m).

Descending order means arranging from the greatest to the smallest.

The given integers are −353, 207, −289, 702, −335, 0, −77.

The positive integers are 207 and 702. In decreasing order: 702 > 207.

Then comes 0, which is greater than every negative integer.

The negative integers are −353, −289, −335 and −77. Their absolute values are 353, 289, 335 and 77.

Arranging the absolute values in decreasing order: 353 > 335 > 289 > 77. So, for these negative integers, the one with bigger absolute value is smaller. Hence the negatives in decreasing order are −77 > −289 > −335 > −353.

Combining: 702 > 207 > 0 > −77 > −289 > −335 > −353.

Hence, the given integers in descending order are 702, 207, 0, −77, −289, −335, −353.

The smallest five-digit number is 10000.

On dividing 10000 by 254, we get:

254 × 39 = 9906

254 × 40 = 10160

So 10000 = 254 × 39 + 94, i.e. the remainder is 94.

This means 10000 is not exactly divisible by 254. To get the smallest five-digit number which is exactly divisible by 254, we add (254 − 94) = 160 to 10000 (so that the next multiple of 254 is reached):

10000 + 160 = 10160.

Check: 10160 ÷ 254 = 40, and 254 × 40 = 10160. So 10160 is a 5-digit number and is exactly divisible by 254.

Hence, the smallest five-digit number which is exactly divisible by 254 is 10160.