CAT 2023 Slot 1 | Past Year Question Paper

Question 1.

If x and y are positive real numbers such that $\log _x\left(x^2+12\right)=4$ and $3 \log _y x=1$ , then x+y equals

A

20

B

11

C

68

D

10

Question 2.

If x and y are real numbers such that $x^2+(x-2 y-1)^2=-4 y(x+y)$ , then the value x−2y is

A

0

B

1

C

2

D

-1

Question 3.

If $\sqrt{5 x+9}+\sqrt{5 x-9}=3(2+\sqrt{2})$ , then $\sqrt{10 x+9}$ is equal to

A

4 \sqrt{5}

B

2 \sqrt{7}

C

3 \sqrt{31}

D

3 \sqrt{7}

Question 4.

Let n be the least positive integer such that 168 is a factor of $1134^n$ . If m is the least positive integer such that $1134^n$ is a factor of $168^m$ , then m+n equals

A

12

B

9

C

15

D

24

Question 5.

The number of integer solutions of equation $2|x|\left(x^2+1\right)=5 x^2$ is

Question 6.

Let α and β be the two distinct roots of the equation $2 x^2-6 x+k=0$ , such that (α+β) and αβ are the distinct roots of the equation $x^2+p x+p=0$ . Then, the value of 8(k−p) is

Question 7.

The equation $x^3+(2 r+1) x^2+(4 r-1) x+2=0$ has -2 as one of the roots. If the other two roots are real, then the minimum possible non-negative integer value of r is

Question 8.

Brishti went on an 8-hour trip in a car. Before the trip, the car had travelled a total of x km till then, where x is a whole number and is palindromic, i.e., x remains unchanged when its digits are reversed. At the end of the trip, the car had travelled a total of 26862 km till then, this number again being palindromic. If Brishti never drove at more than 110 km/h , then the greatest possible average speed at which she drove during the trip, in km/h , was

A

90

B

100

C

80

D

110

Question 9.

The minor angle between the hours hand and minutes hand of a clock was observed at 8:48am . The minimum duration, in minutes, after 8.48 am when this angle increases by 50% is

A

\frac{36}{11}

B

\frac{24}{11}

C

2

D

4

Question 10.

In an examination, the average marks of 4 girls and 6 boys is 24. Each of the girls has the same marks while each of the boys has the same marks. If the marks of any girl is at most double the marks of any boy, but not less than the marks of any boy, then the number of possible distinct integer values of the total marks of 2 girls and 6 boys is

A

19

B

21

C

20

D

22

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cat 2023 Complete Paper Solution | Slot 1

Question 1.

If x and y are positive real numbers such that $\log _x\left(x^2+12\right)=4$ and $3 \log _y x=1$ , then x+y equals

Question 2.

If x and y are real numbers such that $x^2+(x-2 y-1)^2=-4 y(x+y)$ , then the value x−2y is

Question 3.

If $\sqrt{5 x+9}+\sqrt{5 x-9}=3(2+\sqrt{2})$ , then $\sqrt{10 x+9}$ is equal to

Question 4.

Let n be the least positive integer such that 168 is a factor of $1134^n$ . If m is the least positive integer such that $1134^n$ is a factor of $168^m$ , then m+n equals

Question 5.

The number of integer solutions of equation $2|x|\left(x^2+1\right)=5 x^2$ is

Question 6.

Let α and β be the two distinct roots of the equation $2 x^2-6 x+k=0$ , such that (α+β) and αβ are the distinct roots of the equation $x^2+p x+p=0$ . Then, the value of 8(k−p) is

Question 7.

The equation $x^3+(2 r+1) x^2+(4 r-1) x+2=0$ has -2 as one of the roots. If the other two roots are real, then the minimum possible non-negative integer value of r is

Question 8.

Question 9.

The minor angle between the hours hand and minutes hand of a clock was observed at 8:48am . The minimum duration, in minutes, after 8.48 am when this angle increases by 50% is

Question 10.

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cat 2023 Complete Paper Solution | Slot 1

Question 1.

If x and y are positive real numbers such that log⁡x(x2+12)=4\log _x\left(x^2+12\right)=4logx​(x2+12)=4 and 3log⁡yx=13 \log _y x=13logy​x=1 , then x+y equals

Question 2.

If x and y are real numbers such that x2+(x−2y−1)2=−4y(x+y)x^2+(x-2 y-1)^2=-4 y(x+y)x2+(x−2y−1)2=−4y(x+y), then the value x−2y is

Question 3.

If 5x+9+5x−9=3(2+2)\sqrt{5 x+9}+\sqrt{5 x-9}=3(2+\sqrt{2})5x+9​+5x−9​=3(2+2​), then 10x+9\sqrt{10 x+9}10x+9​ is equal to

Question 4.

Let n be the least positive integer such that 168 is a factor of 1134n1134^n1134n. If m is the least positive integer such that 1134n1134^n1134n is a factor of 168m168^m168m, then m+n equals

Question 5.

The number of integer solutions of equation 2∣x∣(x2+1)=5x22|x|\left(x^2+1\right)=5 x^22∣x∣(x2+1)=5x2 is

Question 6.

Let α and β be the two distinct roots of the equation 2x2−6x+k=02 x^2-6 x+k=02x2−6x+k=0, such that (α+β) and αβ are the distinct roots of the equation x2+px+p=0x^2+p x+p=0x2+px+p=0. Then, the value of 8(k−p) is

Question 7.

The equation x3+(2r+1)x2+(4r−1)x+2=0x^3+(2 r+1) x^2+(4 r-1) x+2=0x3+(2r+1)x2+(4r−1)x+2=0 has -2 as one of the roots. If the other two roots are real, then the minimum possible non-negative integer value of r is

Question 8.

Question 9.

The minor angle between the hours hand and minutes hand of a clock was observed at 8:48am . The minimum duration, in minutes, after 8.48 am when this angle increases by 50% is

Question 10.

If x and y are positive real numbers such that $\log _x\left(x^2+12\right)=4$ and $3 \log _y x=1$ , then x+y equals

If x and y are real numbers such that $x^2+(x-2 y-1)^2=-4 y(x+y)$ , then the value x−2y is

If $\sqrt{5 x+9}+\sqrt{5 x-9}=3(2+\sqrt{2})$ , then $\sqrt{10 x+9}$ is equal to

Let n be the least positive integer such that 168 is a factor of $1134^n$ . If m is the least positive integer such that $1134^n$ is a factor of $168^m$ , then m+n equals

The number of integer solutions of equation $2|x|\left(x^2+1\right)=5 x^2$ is

Let α and β be the two distinct roots of the equation $2 x^2-6 x+k=0$ , such that (α+β) and αβ are the distinct roots of the equation $x^2+p x+p=0$ . Then, the value of 8(k−p) is

The equation $x^3+(2 r+1) x^2+(4 r-1) x+2=0$ has -2 as one of the roots. If the other two roots are real, then the minimum possible non-negative integer value of r is