CAT 2024 Slot 3 | Past Year Question Paper

Question 1.

In a group of 250 students, the percentage of girls was at least 44% and at most 60%.The rest of the students were boys. Each student opted for either swimming or running or both. If 50% of the boys and 80% of the girls opted for swimming while 70%of the boys and 60% of the girls opted for running, then the minimum and maximum possible number of students who opted for both swimming and running, are

A

72 and 88, respectively

B

75 and 96, respectively

C

72 and 80, respectively

D

75 and 90, respectively

Question 2.

If $(a + b\sqrt{3})^2 = 52 + 30\sqrt{3}$ , where a and b are natural numbers, then $a + b$ equals

A

7

B

8

C

9

D

10

Question 3.

The average of three distinct real numbers is 28. If the smallest number is increased by 7 and the largest number is reduced by 10, the order of the numbers remains unchanged, and the new arithmetic mean becomes 2 more than the middle number, while the difference between the largest and the smallest numbers becomes 64.Then, the largest number in the original set of three numbers is

Question 4.

If $10^{68}$ is divided by 13, the remainder is

A

5

B

8

C

9

D

4

Question 5.

Sam can complete a job in 20 days when working alone. Mohit is twice as fast as Sam and thrice as fast as Ayna in the same job. They undertake a job with an arrangement where Sam and Mohit work together on the first day, Sam and Ayna on the second day, Mohit and Ayna on the third day, and this three-day pattern is repeated till the work gets completed. Then, the fraction of total work done by Sam is

A

\cfrac{1}{20}

B

\cfrac{3}{10}

C

\cfrac{1}{5}

D

\cfrac{3}{20}

Question 6.

A circular plot of land is divided into two regions by a chord of length $10\sqrt{3}$ meters such that the chord subtends an angle of 120° at the center. Then, the area, in square meters, of the smaller region is

A

20\left(\cfrac{4 \pi}{3} + \sqrt{3}\right)

B

25\left(\cfrac{4 \pi}{3} + \sqrt{3}\right)

C

20\left(\cfrac{4 \pi}{3} - \sqrt{3}\right)

D

25\left(\cfrac{4 \pi}{3} - \sqrt{3}\right)

Question 7.

Consider the sequence $t_1 = 1, t_2 = -1$ and $t_n = \left(\cfrac{n - 3}{n - 1}\right)t_{n - 2}$ for $n \geq 3$ . Then, the value of the sum $\cfrac{1}{t_2} + \cfrac{1}{t_4} + \cfrac{1}{t_6} + ....... +\cfrac{1}{t_{2022}} + \cfrac{1}{t_{2024}}$ , is

A

-1024144

B

-1022121

C

-1023132

D

-1026169

Question 8.

The number of distinct real values of x, satisfying the equation $max \left\{x, 2\right\} - min\left\{x, 2\right\} = \mid x + 2 \mid - \mid x - 2 \mid$ , is

Question 9.

Aman invests Rs 4000 in a bank at a certain rate of interest, compounded annually. If the ratio of the value of the investment after 3 years to the value of the investment after 5 years is 25 : 36, then the minimum number of years required for the value of the investment to exceed Rs 20000 is

Question 10.

The sum of all distinct real values of x that satisfy the equation $10^x + \cfrac{4}{10^x} = \cfrac{81}{2}$ , is

A

2 \log_{10}2

B

4 \log_{10}2

C

\log_{10}2

D

3 \log_{10}2

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cat 2024 Complete Paper Solution | Slot 3

Question 1.

Question 2.

If $(a + b\sqrt{3})^2 = 52 + 30\sqrt{3}$ , where a and b are natural numbers, then $a + b$ equals

Question 3.

Question 4.

If $10^{68}$ is divided by 13, the remainder is

Question 5.

Question 6.

A circular plot of land is divided into two regions by a chord of length $10\sqrt{3}$ meters such that the chord subtends an angle of 120° at the center. Then, the area, in square meters, of the smaller region is

Question 7.

Consider the sequence $t_1 = 1, t_2 = -1$ and $t_n = \left(\cfrac{n - 3}{n - 1}\right)t_{n - 2}$ for $n \geq 3$ . Then, the value of the sum $\cfrac{1}{t_2} + \cfrac{1}{t_4} + \cfrac{1}{t_6} + ....... +\cfrac{1}{t_{2022}} + \cfrac{1}{t_{2024}}$ , is

Question 8.

The number of distinct real values of x, satisfying the equation $max \left\{x, 2\right\} - min\left\{x, 2\right\} = \mid x + 2 \mid - \mid x - 2 \mid$ , is

Question 9.

Aman invests Rs 4000 in a bank at a certain rate of interest, compounded annually. If the ratio of the value of the investment after 3 years to the value of the investment after 5 years is 25 : 36, then the minimum number of years required for the value of the investment to exceed Rs 20000 is

Question 10.

The sum of all distinct real values of x that satisfy the equation $10^x + \cfrac{4}{10^x} = \cfrac{81}{2}$ , is

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cat 2024 Complete Paper Solution | Slot 3

Question 1.

Question 2.

If (a+b3)2=52+303(a + b\sqrt{3})^2 = 52 + 30\sqrt{3}(a+b3​)2=52+303​, where a and b are natural numbers, then a+ba + ba+b equals

Question 3.

Question 4.

If 106810^{68}1068 is divided by 13, the remainder is

Question 5.

Question 6.

A circular plot of land is divided into two regions by a chord of length 10310\sqrt{3}103​ meters such that the chord subtends an angle of 120° at the center. Then, the area, in square meters, of the smaller region is

Question 7.

Question 8.

The number of distinct real values of x, satisfying the equation max{x,2}−min{x,2}=∣x+2∣−∣x−2∣max \left\{x, 2\right\} - min\left\{x, 2\right\} = \mid x + 2 \mid - \mid x - 2 \midmax{x,2}−min{x,2}=∣x+2∣−∣x−2∣, is

Question 9.

Aman invests Rs 4000 in a bank at a certain rate of interest, compounded annually. If the ratio of the value of the investment after 3 years to the value of the investment after 5 years is 25 : 36, then the minimum number of years required for the value of the investment to exceed Rs 20000 is

Question 10.

The sum of all distinct real values of x that satisfy the equation 10x+410x=81210^x + \cfrac{4}{10^x} = \cfrac{81}{2}10x+10x4​=281​, is

If $(a + b\sqrt{3})^2 = 52 + 30\sqrt{3}$ , where a and b are natural numbers, then $a + b$ equals

If $10^{68}$ is divided by 13, the remainder is

A circular plot of land is divided into two regions by a chord of length $10\sqrt{3}$ meters such that the chord subtends an angle of 120° at the center. Then, the area, in square meters, of the smaller region is

The number of distinct real values of x, satisfying the equation $max \left\{x, 2\right\} - min\left\{x, 2\right\} = \mid x + 2 \mid - \mid x - 2 \mid$ , is

The sum of all distinct real values of x that satisfy the equation $10^x + \cfrac{4}{10^x} = \cfrac{81}{2}$ , is