Mathematics - 2018 - Paper-1

If 573xy is divisible by 90, it should be divisible by 9 and 10 both.
Since it is divisible by 10, the last digit, i.e. y is 0.
Since the number is divisible by 9, sum of its all digits must be 9 or its multiple.
Sum of digits of 573xy = 15 + x + y = 15 + x
Therefore, the value of x is 3.
Thus, the value of x + y is 3 + 0 = 3.

For the numbers to be in standard form, the negative sign must be with the numerator, and not with the denominator, and both numerator and denominator should have no common factor other than 1.
Therefore, -49 / 71 is in standard form as the negative sign is with numerator and both numerator and denominator have only 1 as the common factor.

0.25(4f - 3) = 0.05(10f - 9)
0.25 × 4f - 3 × 0.25 = 0.05 × 10f - 9 × 0.05
f - 0.75 = 0.5f - 0.45
0.5f = 0.30
f = 0.6

Let the number be 10a + b.
a = tens digit
b = units digit
a = b + 4
a + b = 1/7 (10a + b)
7a + 7b = 10a + b
6b = 3a
a = 2b
But a = b + 4
So, 2b = b + 4
b = 4
So, a = 8
Thus, the number is 84.

In a class of 17 students, six boys failed in a test.
Those who passed scored 12, 15, 17, 15, 16, 15, 19, 17, 18, 18 and 19 marks.
Arranging the marks in ascending order:
Marks of 11 students who passed = 12, 15, 15,15, 16, 17, 17, 18, 18, 19, 19
Marks of students who fail in the test will have scored marks less than 12.
So, median score of 17 students will be the 9^th term when arranged in ascending order (independent of marks of students who failed).
Median = 9^th term in the series of 17 students = 3^rd term in the series of 11 students who passed = 15

Let x be the number of student in the class.
16 × x + 40 = 17(x + 1)
x = 40 - 17 = 23

The square root 6600 will not be a natural number as √6600 = 81.24...
Alternately,
6561 < 6600 < 6724
81² < 6600 < 82²
So, 6600 is a not perfect square.

The square number closest to and less than 176 is 169.
Therefore, 7 must be subtracted from it.
176 - 7 = 169
And √169 = 13

Total amount to be paid before discount to purchase a pair of jeans and two shirts = Rs. 1450 + (Rs. 850 × 2) = Rs. 3150
Now, 10% discount was allowed.
So, net price to be paid = 3150 - 10% of 3150
= 3150 - 315
Therefore, Rs. 2835 is the amount that the customer would have to pay.

Let CP of 10 greeting card be Rs. 1000.
CP of 1 greeting card = Rs. 100
SP of 8 greeting card = Rs. 1000
SP of 1 greeting card = Rs. 125
Gain = 125 - 100 = Rs. 25
Gain % = 25 / 100 × 100 = 25%

B does the job in 12 days.

So, he does 1/12 of the job in 1 day.

So, he will do 9/12 = 3/4 of the job in 9 days.

Remaining work = 1 - 3/4 = 4 – 3 / 4 = 1/4

A does the job in 20 days.

So, A does 1/20 of the job in 1 day.

To do 1/4 of the job, he will take 1/4 × 20 = 5 days.

Speed = 60 km/hr
Time = 2 hours
Distance = Speed × Time = 120 km
Now, when the car is travelling at a speed of 80 km/hr,
Time = 120 / 80 = 1.5 hr or 1 hr 30 min

82² - 18²
= (82 + 18) × (82 - 18) {Using a² - b² = (a + b)(a - b)}
= 100 × 64
= 6400

A square pyramid has 8 edges, 5 faces and 5 vertices.

Let the sides be x, 2x and 3x.
Surface area = 2(lb + bh + hl)
2(2x² + 6x² + 3x²) = 88
11x² = 44
x = 2
Thus, the sides are 2 cm, 4 cm and 6 cm.
Volume = 2 × 4 × 6 = 48 cm³

Area of trapezium = 1/2 (l + b)h

Let the lengths of the two parallel sides be 4x and 3x.

630 =1/2 × (4x + 3x) × 12

42x = 630
Solving the above equation, we get:
x = 15
Thus, length of the shorter side = 3x = 3 × 15 = 45 cm

Let the length of base be b and height be h.
According to the question:

h = b/4

Area = 1/2 × b × b/4 = 50

b² = 400
Thus, b = 20 m

4^3.5 : 2⁵ = (2²)^3.5 : 2⁵
= 2⁷ : 2⁵
= 2² : 1
= 4 : 1

n the number 67y19, y can be any digit from 0 to 9. So, we can check all the possibilities one by one:

Now, 67019 is not divisible by 11.
67119 is not divisible by 11.
67219 is not divisible by 11.
67319 is not divisible by 11.
67419 is divisible by 11 giving 6129 as the quotient.
67519 is not divisible by 11.
67619 is not divisible by 11.
67719 is not divisible by 11.
67819 is not divisible by 11.
67919 is not divisible by 11.

So, the value of y is 4.

We take, a = 1, b = 2 and c = 3
Given, a + b + c = a × b × c
1 + 2 + 3 = 2 × 3
6 = 6 (True)
Therefore, a² + b² + c² = 1² + 2² + 3² = 1 + 4 + 9 = 14

3 + 23y - 8y²
= 3 + 24y - y - 8y²
= 3(1 + 8y) - y(1 + 8y)
= (3 - y)(1 + 8y)

The motor car travelling at 70 km/hr will cover 70 × 2 = 140 km in 2 hours.
For the next 2 hours, the speed will be 80 km/hr. So, the car will cover 80 × 2 = 160 km in that time.
Thus, the car covers 300 km in 4 hours.
Remaining distance = 345 - 300 = 45 km
This distance will be covered at a speed of 90 km/hr.
Time taken to cover 45 km at 90 km/hr = 45/90 = 0.5 hr or 1/2  hr
So, the entire distance of 345 km will be covered in 4.5 hours or 4 1/2 hrs.

It is given that 1200 soldiers in a fort had enough food supplies for 28 days.
Let the number of soldiers transferred to a different fort be x.
After 4 days, remaining number of soldiers = (1200 - x)
The remaining food supplies lasted 32 days for these soldiers.
Hence, 1200 × 24 = (1200 - x) × 32
900 = (1200 - x)
x = 1200 - 900 = 300
So, 300 soldiers were transferred to a different fort.

Let ABC be a right-angled triangle with AB = BC = a and AC = b.
Using a² + a² = b², we get b = a√2.
Given, perimeter of ∆ABC = 6 + 3√2
Thus, 2a + b = 6 + 3√2 = 3(2 + √2)
2a + a√2 = 3(2 + √2)
= a(2 + √2) = 3(2 + √2)
a = 3
Hence, area of ∆ABC = 1/2 a² 
= 1/2 (3)²
= 4.5 m²