CAT 2022 Slot 3 | Past Year Question Paper

Question 1.

Suppose k is any integer such that the equation $2 x^2+k x+5=0$ has no real roots and the equation $x^2+(k-5) x+1=0$ has two distinct real roots for x . Then, the number of possible values of k is

A

7

B

8

C

9

D

13

Question 2.

The minimum possible value of $\frac{x^2-6 x+10}{3-x}$ , for x<3, is

A

\frac{1}{2}

B

-\frac{1}{2}

C

2

D

-2

Question 3.

Bob can finish a job in 40 days, if he works alone. Alex is twice as fast as Bob and thrice as fast as Cole in the same job. Suppose Alex and Bob work together on the first day, Bob and Cole work together on the second day, Cole and Alex work together on the third day, and then, they continue the work by repeating this three-day roster, with Alex and Bob working together on the fourth day, and so on. Then, the total number of days Alex would have worked when the job gets finished, is

Question 4.

A glass contains 500 cc of milk and a cup contains 500 cc of water. From the glass, 150 cc of milk is transferred to the cup and mixed thoroughly. Next, 150 cc of this mixture is transferred from the cup to the glass. Now, the amount of water in the glass and the amount of milk in the cup are in the ratio

A

3 : 10

B

10 : 3

C

1 : 1

D

10 : 13

Question 5.

In an examination, the average marks of students in sections A and B are 32 and 60, respectively. The number of students in section A is 10 less than that in section B. If the average marks of all the students across both the sections combined is an integer, then the difference between the maximum and minimum possible number of students in section A is

Question 6.

Let r be a real number and $f(x)=\left\{\begin{array}{cl}2 x-r & \text { if } x \geq r \\ r & \text { if } x\lt r\end{array}\right.$ Then, the equation f(x) = f(f(x)) holds for all real values of x where

A

x ≤ r

B

x ≥ r

C

x > r

D

x ≠ r

Question 7.

Suppose the medians BD and CE of a triangle ABC intersect at a point O. If area of triangle ABC is 108 sq. cm., then, the area of the triangle EOD, in sq. cm., is

Question 8.

The arithmetic mean of all the distinct numbers that can be obtained by rearranging the digits in 1421, including itself, is

A

2442

B

2222

C

3333

D

2592

Question 9.

Nitu has an initial capital of ₹20,000 . Out of this, she invests ₹8,000 at 5.5% in bank A,₹5,000 at 5.6% in bank B and the remaining amount at x% in bank C , each rate being simple interest per annum. Her combined annual interest income from these investments is equal to 5% of the initial capital. If she had invested her entire initial capital in bank C alone, then her annual interest income, in rupees, would have been

A

900

B

700

C

1000

D

800

Question 10.

Two cars travel from different locations at constant speeds. To meet each other after starting at the same time, they take 1.5 hours if they travel towards each other, but 10.5 hours if they travel in the same direction. If the speed of the slower car is 60 km/hr, then the distance traveled, in km, by the slower car when it meets the other car while traveling towards each other, is

A

150

B

100

C

90

D

120

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cat 2022 Complete Paper Solution | Slot 3

Question 1.

Suppose k is any integer such that the equation $2 x^2+k x+5=0$ has no real roots and the equation $x^2+(k-5) x+1=0$ has two distinct real roots for x . Then, the number of possible values of k is

Question 2.

The minimum possible value of $\frac{x^2-6 x+10}{3-x}$ , for x<3, is

Question 3.

Question 4.

Question 5.

Question 6.

Let r be a real number and $f(x)=\left\{\begin{array}{cl}2 x-r & \text { if } x \geq r \\ r & \text { if } x\lt r\end{array}\right.$ Then, the equation f(x) = f(f(x)) holds for all real values of x where

Question 7.

Suppose the medians BD and CE of a triangle ABC intersect at a point O. If area of triangle ABC is 108 sq. cm., then, the area of the triangle EOD, in sq. cm., is

Question 8.

The arithmetic mean of all the distinct numbers that can be obtained by rearranging the digits in 1421, including itself, is

Question 9.

Question 10.

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cat 2022 Complete Paper Solution | Slot 3

Question 1.

Suppose k is any integer such that the equation 2x2+kx+5=02 x^2+k x+5=02x2+kx+5=0 has no real roots and the equation x2+(k−5)x+1=0x^2+(k-5) x+1=0x2+(k−5)x+1=0 has two distinct real roots for x . Then, the number of possible values of k is

Question 2.

The minimum possible value of x2−6x+103−x\frac{x^2-6 x+10}{3-x}3−xx2−6x+10​, for x<3, is

Question 3.

Question 4.

Question 5.

Question 6.

Let r be a real number and f(x)={2x−r if x≥rr if x<rf(x)=\left\{\begin{array}{cl}2 x-r & \text { if } x \geq r \\ r & \text { if } x\lt r\end{array}\right.f(x)={2x−rr​ if x≥r if x<r​ Then, the equation f(x) = f(f(x)) holds for all real values of x where

Question 7.

Suppose the medians BD and CE of a triangle ABC intersect at a point O. If area of triangle ABC is 108 sq. cm., then, the area of the triangle EOD, in sq. cm., is

Question 8.

The arithmetic mean of all the distinct numbers that can be obtained by rearranging the digits in 1421, including itself, is

Question 9.

Question 10.

Suppose k is any integer such that the equation $2 x^2+k x+5=0$ has no real roots and the equation $x^2+(k-5) x+1=0$ has two distinct real roots for x . Then, the number of possible values of k is

The minimum possible value of $\frac{x^2-6 x+10}{3-x}$ , for x<3, is

Let r be a real number and $f(x)=\left\{\begin{array}{cl}2 x-r & \text { if } x \geq r \\ r & \text { if } x\lt r\end{array}\right.$ Then, the equation f(x) = f(f(x)) holds for all real values of x where