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cat 2021 Complete Paper Solution | Quant Slot 1

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Question 1.

f(x) = $\frac{x^{2}+2 x-15}{x^{2}-7 x-18}$ is negative if and only if

-5 < x < -2 or 3 < x < 9

-2 < x < 3 or x > 9

x < -5 or 3 < x < 9

x < -5 or -2 < x < 3

Question 2.

The number of integers n that satisfy the inequalities |n - 60| < |n - 100| < |n - 20| is

Question 3.

Identical chocolate pieces are sold in boxes of two sizes, small and large. The large box is sold for twice the price of the small box. If the selling price per gram of chocolate in the large box is 12% less than that in the small box, then the percentage by which the weight of chocolate in the large box exceeds that in the small box is nearest to

127

135

124

144

Question 4.

If 5 - $\log _{10} \sqrt{1+x}$ + 4 $\log _{10} \sqrt{1-x}$ = $\log _{10} \frac{1}{\sqrt{1-x^{2}}}$ , then 100 x equals

Question 5.

A basket of 2 apples, 4 oranges and 6 mangoes costs the same as a basket of 1 apple, 4 oranges and 8 mangoes, or a basket of 8 oranges and 7 mangoes. Then the number of mangoes in a basket of mangoes that has the same cost as the other baskets is

Question 6.

If the area of a regular hexagon is equal to the area of an equilateral triangle of side 12 cm, then the length, in cm, of each side of the hexagon is

4 √6

6 √6

2 √6

√6

Question 7.

Onion is sold for 5 consecutive months at the rate of Rs 10, 20, 25, 25, and 50 per kg, respectively. A family spends a fixed amount of money on onion for each of the first three months, and then spends half that amount on onion for each of the next two months. The average expense for onion, in rupees per kg, for the family over these 5 months is closest to

Question 8.

If r is a constant such that | $x^{2}$ - 4 x - 13| = r has exactly three distinct real roots, then the value of r is

Question 9.

Amal purchases some pens at ₹ 8 each. To sell these, he hires an employee at a fixed wage. He sells 100 of these pens at ₹ 12 each. If the remaining pens are sold at ₹ 11 each, then he makes a net profit of ₹ 300, while he makes a net loss of ₹ 300 if the remaining pens are sold at ₹ 9 each. The wage of the employee, in INR, is

Question 10.

Anil invests some money at a fixed rate of interest, compounded annually. If the interests accrued during the second and third year are ₹ 806.25 and ₹ 866.72, respectively, the interest accrued, in INR, during the fourth year is nearest to

931.72

926.84

929.48

934.65

Question 11.

The strength of an indigo solution in percentage is equal to the amount of indigo in grams per 100 cc of water. Two 800 cc bottles are filled with indigo solutions of strengths 33% and 17%, respectively. A part of the solution from the first bottle is thrown away and replaced by an equal volume of the solution from the second bottle. If the strength of the indigo solution in the first bottle has now changed to 21% then the volume, in cc, of the solution left in the second bottle is

Question 12.

Amar, Akbar and Anthony are working on a project. Working together Amar and Akbar can complete the project in 1 year, Akbar and Anthony can complete in 16 months, Anthony and Amar can complete in 2 years. If the person who is neither the fastest nor the slowest works alone, the time in months he will take to complete the project is

Question 13.

The number of groups of three or more distinct numbers that can be chosen from 1, 2, 3, 4, 5, 6, 7 and 8 so that the groups always include 3 and 5, while 7 and 8 are never included together is

Question 14.

A circle of diameter 8 inches is inscribed in a triangle ABC where ∠ABC = 90°. If BC = 10 inches then the area of the triangle in square inches is

Question 15.

Suppose the length of each side of a regular hexagon ABCDEF is 2 cm. It T is the mid point of CD, then the length of AT, in cm, is

√15

√13x

√12

√14

Question 16.

Suppose hospital A admitted 21 less Covid infected patients than hospital B, and all eventually recovered. The sum of recovery days for patients in hospitals A and B were 200 and 152, respectively. If the average recovery days for patients admitted in hospital A was 3 more than the average in hospital B then the number admitted in hospital A was

Question 17.

The amount Neeta and Geeta together earn in a day equals what Sita alone earns in 6 days. The amount Sita and Neeta together earn in a day equals what Geeta alone earns in 2 days. The ratio of the daily earnings of the one who earns the most to that of the one who earns the least is

7 : 3

11 : 3

11 : 7

3 : 2

Question 18.

The natural numbers are divided into groups as (1), (2, 3, 4), (5, 6, 7, 8, 9), ….. and so on. Then, the sum of the numbers in the 15th group is equal to

6119

7471

4941

6090

Question 19.

Anu, Vinu and Manu can complete a work alone in 15 days, 12 days and 20 days, respectively. Vinu works everyday. Anu works only on alternate days starting from the first day while Manu works only on alternate days starting from the second day. Then, the number of days needed to complete the work is

Question 20.

How many three-digit numbers are greater than 100 and increase by 198 when the three digits are arranged in the reverse order?

Question 21.

Two trains cross each other in 14 seconds when running in opposite directions along parallel tracks. The faster train is 160 m long and crosses a lamp post in 12 seconds. If the speed of the other train is 6 km/hr less than the faster one, its length, in m, is

184

180

190

192

Question 22.

If $x_{0}$ = 1, $x_{1}$ = 2, and $x_{n + 2}$ = $\frac{1+x_{n+1}}{x_{n}}$ , n = 0, 1, 2, 3,..., then $x_{2021}$ is equal to?

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cat 2021 Complete Paper Solution | Quant Slot 1

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Question 1.

f(x) = x2+2x−15x2−7x−18\frac{x^{2}+2 x-15}{x^{2}-7 x-18}x2−7x−18x2+2x−15​ is negative if and only if

Question 2.

The number of integers n that satisfy the inequalities |n - 60| < |n - 100| < |n - 20| is

Question 3.

Question 4.

If 5 - log⁡101+x\log _{10} \sqrt{1+x}log10​1+x​ + 4log⁡101−x\log _{10} \sqrt{1-x}log10​1−x​ = log⁡1011−x2\log _{10} \frac{1}{\sqrt{1-x^{2}}}log10​1−x2​1​, then 100 x equals

Question 5.

A basket of 2 apples, 4 oranges and 6 mangoes costs the same as a basket of 1 apple, 4 oranges and 8 mangoes, or a basket of 8 oranges and 7 mangoes. Then the number of mangoes in a basket of mangoes that has the same cost as the other baskets is

Question 6.

If the area of a regular hexagon is equal to the area of an equilateral triangle of side 12 cm, then the length, in cm, of each side of the hexagon is

Question 7.

Question 8.

If r is a constant such that |x2x^{2}x2 - 4 x - 13| = r has exactly three distinct real roots, then the value of r is

Question 9.

Question 10.

Anil invests some money at a fixed rate of interest, compounded annually. If the interests accrued during the second and third year are ₹ 806.25 and ₹ 866.72, respectively, the interest accrued, in INR, during the fourth year is nearest to

Question 11.

Question 12.

Question 13.

The number of groups of three or more distinct numbers that can be chosen from 1, 2, 3, 4, 5, 6, 7 and 8 so that the groups always include 3 and 5, while 7 and 8 are never included together is

Question 14.

A circle of diameter 8 inches is inscribed in a triangle ABC where ∠ABC = 90°. If BC = 10 inches then the area of the triangle in square inches is

Question 15.

Suppose the length of each side of a regular hexagon ABCDEF is 2 cm. It T is the mid point of CD, then the length of AT, in cm, is

Question 16.

Question 17.

The amount Neeta and Geeta together earn in a day equals what Sita alone earns in 6 days. The amount Sita and Neeta together earn in a day equals what Geeta alone earns in 2 days. The ratio of the daily earnings of the one who earns the most to that of the one who earns the least is

Question 18.

The natural numbers are divided into groups as (1), (2, 3, 4), (5, 6, 7, 8, 9), ….. and so on. Then, the sum of the numbers in the 15th group is equal to

Question 19.

Anu, Vinu and Manu can complete a work alone in 15 days, 12 days and 20 days, respectively. Vinu works everyday. Anu works only on alternate days starting from the first day while Manu works only on alternate days starting from the second day. Then, the number of days needed to complete the work is

Question 20.

How many three-digit numbers are greater than 100 and increase by 198 when the three digits are arranged in the reverse order?

Question 21.

Two trains cross each other in 14 seconds when running in opposite directions along parallel tracks. The faster train is 160 m long and crosses a lamp post in 12 seconds. If the speed of the other train is 6 km/hr less than the faster one, its length, in m, is

Question 22.

If x0x_{0}x0​ = 1, x1x_{1}x1​ = 2, and xn+2x_{n + 2}xn+2​ = 1+xn+1xn\frac{1+x_{n+1}}{x_{n}}xn​1+xn+1​​ , n = 0, 1, 2, 3,..., then x2021x_{2021}x2021​ is equal to?

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f(x) = $\frac{x^{2}+2 x-15}{x^{2}-7 x-18}$ is negative if and only if

If 5 - $\log _{10} \sqrt{1+x}$ + 4 $\log _{10} \sqrt{1-x}$ = $\log _{10} \frac{1}{\sqrt{1-x^{2}}}$ , then 100 x equals

If r is a constant such that | $x^{2}$ - 4 x - 13| = r has exactly three distinct real roots, then the value of r is

If $x_{0}$ = 1, $x_{1}$ = 2, and $x_{n + 2}$ = $\frac{1+x_{n+1}}{x_{n}}$ , n = 0, 1, 2, 3,..., then $x_{2021}$ is equal to?