CAT 2022 Question Paper Slot 1 with Solutions, Download PDF

Instructions

For the following questions answer them individually

Question 51

Ankita buys 4 kg cashews, 14 kg peanuts and 6 kg almonds when the cost of 7 kg cashews is the same as that of 30 kg peanuts or 9 kg almonds. She mixes all the three nuts and marks a price for the mixture in order to make a profit of ₹1752. She sells 4 kg of the mixture at this marked price and the remaining at a 20% discount on the marked price, thus making a total profit of ₹744. Then the amount, in rupees, that she had spent in buying almonds is

Video Solution

Solution

It is given,

7C = 30P = 9A and Ankita bought 4C, 14P and 6A.

Let 7C = 30P = 9A = 630k

C = 90k, P = 21k, and A = 70k

Cost price of 4C, 14P and 6A = 4(90k)+14(21k)+6(70k) = 1074k

Marked up price = 1074k + 1752

S.P = $$\frac{1}{6}\left(1074k+1752\right)+\left(\frac{4}{5}\right)\left(\frac{5}{6}\right)\left(1074k+1752\right)$$ = $$\frac{5}{6}\left(1074k+1752\right)$$

S.P - C.P = profit

$$1460-\frac{1074k}{6}=744$$

$$\frac{1074k}{6}=716$$

k = 4

Money spent on buying almonds = 420k = 420*4 = Rs 1680

The answer is option A.

Question 52

Let A be the largest positive integer that divides all the numbers of the form $$3^k + 4^k + 5^k$$, and B be the largest positive integer that divides all the numbers of the form $$4^k + 3(4^k) + 4^{k + 2}$$ , where k is any positive integer. Then (A + B) equals

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Video Solution

Solution

A is the HCF of $$3^k+4^k+5^k$$ for different values of k.

For k = 1, value is 12

For k = 2, value is 50

For k = 3, value is 216

HCF is 2. Therefore, A = 2

$$4^k+3\left(4^k\right)+4^{k+2}=4^k\left(1+3\right)+4^{k+2}=4^{k+1}+4^{k+2}=4^{k+1}\left(1+4\right)=5\cdot4^{k+1}$$

HCF of the values is when k = 1, i.e. 5*16 = 80

Therefore, B = 80

A + B = 82

Question 53

Let a, b, c be non-zero real numbers such that $$b^2 < 4ac$$, and $$f(x) = ax^2 + bx + c$$. If the set S consists of all integers m such that f(m) < 0, then the set S must necessarily be

Video Solution

Solution

$$b^2 < 4ac$$ means that the discriminant is less than 0. Therefore, f(x)>0 for all x if the coefficient of $$x^2$$ is positive, and f(x)<0 for all x if the coefficient of $$x^2$$ is negative.

We are given that f(m)<0 and m is an integer.

So the set containing values of m will either be empty if the coefficient of $$x^2$$ is positive, or it will be a set of all integers if the coefficient of $$x^2$$ is negative.

Question 54

The number of ways of distributing 20 identical balloons among 4 children such that each child gets some balloons but no child gets an odd number of balloons, is

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Solution

Let the number of balloons each child received be 2a, 2b, 2c and 2d

2a + 2b + 2c + 2d = 20

a + b + c + d = 10

Each of them should get more than zero balloons.

Therefore, total number of ways = $$(n-1)_{C_{r-1}}=(10-1)_{C_{4-1}}=9_{C_3}=84$$

Question 55

Let a and b be natural numbers. If $$a^2 + ab + a = 14$$ and $$b^2 + ab + b = 28$$, then $$(2a + b)$$ equals

Video Solution

Solution

a(a + b + 1) = 14 ...... (1)

b(a + b + 1) = 28 ...... (2)

$$\frac{a}{b}=\frac{1}{2}$$

b = 2a

Substituting in (1), we get

a(3a + 1) = 14

$$3a^2+a-14=0$$

$$3a^2-6a+7a-14=0$$

$$3a\left(a-2\right)+7\left(a-2\right)=0$$

Given, a and b are natural numbers.

Therefore, a = 2 and b = 4

2a + b = 2(2) + 4 = 8

The answer is option A.

Question 56

Amal buys 110 kg of syrup and 120 kg of juice, syrup being 20% less costly than juice, per kg. He sells 10 kg of syrup at 10% profit and 20 kg of juice at 20% profit. Mixing the remaining juice and syrup, Amal sells the mixture at ₹ 308.32 per kg and makes an overall profit of 64%. Then, Amal’s cost price for syrup, in rupees per kg, is

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Solution

Total syrup - 110 kg

Total juice - 120 kg

It is given, cost price of syrup is 20% less than the cost price of juice.

Let the cost price of juice per kg be 10CP

Cost price of syrup per kg is 8CP

10kg syrup -> cost price = 80CP

It is given, 10kg syrup is sold at 10% profit. This implies selling price = 1.1*80CP = 88CP

20kg juice -> cost price = 200CP

It is given, 20kg juice is sold at 20% profit. This implies selling price = 1.2*200CP = 240CP

It is given, Mixing the remaining juice and syrup, Amal sells the mixture at ₹ 308.32 per kg

Selling price of the remaining mixture = 308.32*200 = Rs 61664

Total S.P = 61664 + 328CP

Total C.P = 880CP + 1200CP = 2080CP

Overall profit = 64%

$$61664+328CP=\frac{164}{100}\left(2080CP\right)$$

Solving, we get CP = 20

Cost price for syrup per kg = 8CP = 8*20 = Rs 160

Question 57

All the vertices of a rectangle lie on a circle of radius R. If the perimeter of the rectangle is P, then the area of the rectangle is

Video Solution

Solution

$$A=lb$$

$$l^2+b^2=4r^2$$

$$P\ =2\left(l+b\right)$$

$$\frac{P}{2}=l+b$$

Squaring on both the sides, we get

$$\frac{P^2}{4}=l^2+b^2+2lb$$

$$\frac{P^2}{4}=4r^2+2lb$$

$$\frac{P^2}{8}-2r^2=lb$$

The answer is option B.

Question 58

The average of three integers is 13. When a natural number n is included, the average of these four integers remains an odd integer. The minimum possible value of n is

Video Solution

Solution

It is given that average of three numbers is 13.

Sum = 3*13 = 39

It is given, $$\ \frac{\ 39+n}{4}$$ is a odd number.

Minimum value $$\ \frac{\ 39+n}{4}$$ can take such that n is a natural number is 11

$$\ \frac{\ 39+n}{4}=11$$

n = 5

The answer is option C.

Question 59

A mixture contains lemon juice and sugar syrup in equal proportion. If a new mixture is created by adding this mixture and sugar syrup in the ratio 1 : 3, then the ratio of lemon juice and sugar syrup in the new mixture is

Video Solution

Solution

Lemon juice : sugar syrup in the mixture is 1:1, i.e. 50% Lemon juice and 50% sugar syrup.

In sugar syrup, 100% is sugar syrup.

These two are mixed in the ratio 1:3.

Lemon juice = $$\ \frac{\ 1\left(50\%\right)}{1+3}$$

Sugar syrup = $$\ \frac{\ 1\left(50\%\right)+3\left(100\%\right)}{1+3}=\frac{350}{4}$$

Required ratio = 50:350 = 1:7

The answer is option D.

Question 60

The largest real value of a for which the equation $$\mid x + a \mid + \mid x - 1 \mid = 2$$ has an infinite number of solutions for x is

Video Solution

Solution

In the question, it is given that the equation $$\mid x + a \mid + \mid x - 1 \mid = 2$$ has an infinite number of solutions for any value of x. This is possible when x in |x+a| and x in |x-1| cancels out.

Case 1:

x + a < 0, x - 1 $$\ge\ $$ 0

- a - x + x - 1 = 2

a = -3

Case 2:

x + a $$\ge\ $$ 0 and x - 1 < 0

x + a - x + 1 = 2

a = 1

Largest value of a is 1.

The answer is option C.

CAT Content

CAT 2022 Question Paper Slot 1

Let A be the largest positive integer that divides all the numbers of the form $$3^k + 4^k + 5^k$$, and B be the largest positive integer that divides all the numbers of the form $$4^k + 3(4^k) + 4^{k + 2}$$ , where k is any positive integer. Then (A + B) equals

Let a, b, c be non-zero real numbers such that $$b^2 < 4ac$$, and $$f(x) = ax^2 + bx + c$$. If the set S consists of all integers m such that f(m) < 0, then the set S must necessarily be

The number of ways of distributing 20 identical balloons among 4 children such that each child gets some balloons but no child gets an odd number of balloons, is

Let a and b be natural numbers. If $$a^2 + ab + a = 14$$ and $$b^2 + ab + b = 28$$, then $$(2a + b)$$ equals

All the vertices of a rectangle lie on a circle of radius R. If the perimeter of the rectangle is P, then the area of the rectangle is

The average of three integers is 13. When a natural number n is included, the average of these four integers remains an odd integer. The minimum possible value of n is

A mixture contains lemon juice and sugar syrup in equal proportion. If a new mixture is created by adding this mixture and sugar syrup in the ratio 1 : 3, then the ratio of lemon juice and sugar syrup in the new mixture is

The largest real value of a for which the equation $$\mid x + a \mid + \mid x - 1 \mid = 2$$ has an infinite number of solutions for x is

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CAT Content

CAT 2022 Question Paper Slot 1

Let A be the largest positive integer that divides all the numbers of the form $$3^k + 4^k + 5^k$$, and B be the largest positive integer that divides all the numbers of the form $$4^k + 3(4^k) + 4^{k + 2}$$ , where k is any positive integer. Then (A + B) equals

Let a, b, c be non-zero real numbers such that $$b^2 < 4ac$$, and $$f(x) = ax^2 + bx + c$$. If the set S consists of all integers m such that f(m) < 0, then the set S must necessarily be

The number of ways of distributing 20 identical balloons among 4 children such that each child gets some balloons but no child gets an odd number of balloons, is

Let a and b be natural numbers. If $$a^2 + ab + a = 14$$ and $$b^2 + ab + b = 28$$, then $$(2a + b)$$ equals

All the vertices of a rectangle lie on a circle of radius R. If the perimeter of the rectangle is P, then the area of the rectangle is

The average of three integers is 13. When a natural number n is included, the average of these four integers remains an odd integer. The minimum possible value of n is

A mixture contains lemon juice and sugar syrup in equal proportion. If a new mixture is created by adding this mixture and sugar syrup in the ratio 1 : 3, then the ratio of lemon juice and sugar syrup in the new mixture is

The largest real value of a for which the equation $$\mid x + a \mid + \mid x - 1 \mid = 2$$ has an infinite number of solutions for x is

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