It is given that the average of the 30 integers = 5Sum of the 30 integers = 30*5=150There are exactly 20 integers whose value is less than 5.To maximise the average of the 20 integers, we have to assign minimum value to each of the remaining 10 integersSo the sum of 10 integers = 10*6=60The sum of the 20 integers = 150-60= 90Average of 20 integers =&nbsp;$\frac{90}{20}$ = 4.5

The amount with Amal at the end of 1 year = 12000*1.08+10000*1.03*1.03=23569Interest received by Amal = 23569-22000=1569Let the amount invested by Bimal = 100bInterest received by Bimal = 100b*7.5*1/100=7.5bIt is given that the amount of interest received by both of them is the same&nbsp;7.5b=1569b=209.2Amount invested by Bimal = 100b = 20920

$\ \frac{\ n^2+3n+4n+12}{n^2-4n+3n-12}$=$\ \frac{\ n^{ }\left(n+3\right)+4\left(n+3\right)}{n^{ }\left(n-4\right)+3\left(n-4\right)}$=$\ \frac{\left(\ n+4\right)\left(n+3\right)}{\left(n-4\right)\left(n+3\right)}$=$\ \frac{\left(\ n+4\right)}{\left(n-4\right)}$=$\ \frac{\left(\ n-4\right)+8}{\left(n-4\right)}$=$\ 1+\frac{8}{\left(n-4\right)}$ which will be maximum when n-4 =8n=12D is the correct answer.

(n-m)(n+m) = 1*105, 3*35, 5*21, 7*15, 15*7, 21*5, 35*3, 105*1.n-m=1, n+m=105&nbsp;==&gt; n=53, m=52n-m=3, n+m=35 ==&gt; n=19, m=16n-m=5, n+m=21&nbsp;==&gt; n=13, m=8n-m=7, n+m=15 ==&gt; n=11, m=4n-m=15, n+m=7 ==&gt; n=11, m=-4n-m=21, n+m=5 ==&gt; n=13, m=-8n-m=35, n+m=3 ==&gt; n=19, m=-16n-m=105, n+m=1 ==&gt; n=53, m=-52Since only positive integer values of m and n are required. There are 4 possible solutions.

When A and B met for the first time at 10:00 AM, A covered 60% of the track.So B must have covered 40% of the track.It is given that A returns to P at 10:12 AM i.e A covers 40% of the track in 12 minutes60% of the track in 18 minutesB covers 40% of track when A covers 60% of the track.B covers 40% of the track in 18 minutes.B will cover the rest 60% in 27 minutes, hence it will return to B at 10:27 AM

$a_1 - a_2 + a_3 - a_4 + .... + (-1)^{n - 1} a_n = n$It is clear from the above equation that when n is odd, the co-efficient of a is positive otherwise negative.$a_1 - a_2 = 2$$a_1 = a_2 + 2$&nbsp;&nbsp;$a_1 - a_2 + a_3 = 3$On substituting the value of $a_1$ in the above equation, we get$a_3$ = 1$a_1 - a_2 + a_3 - a_4 = 4$On substituting the values of $a_1, a_3$ in the above equation, we get$a_4$ = -1$a_1 - a_2 + a_3 - a_4 +a_5 = 5$On substituting the values of $a_1, a_3, a_4$ in the above equation, we get$a_5$ = 1So we can conclude that $a_3, a_5, a_7....a_{n+1}$ = 1 and&nbsp;$a_2, a_4, a_6....a_{2n}$ = -1Now we have to find the value of&nbsp;$a_{51} + a_{52} + .... + a_{1023}$Number of terms = 1023=51+(n-1)1n=973There will be 486 even and 487 odd terms, so the value of&nbsp;$a_{51} + a_{52} + .... + a_{1023}$ = 486*-1+487*1=1

<img src="https://lightyellow-stork-291267.hostingersite.com/wp-content/uploads/2025/11/image_qn5jwvO.png">Let 'h' be the height of the triangle ABC, semiperimeter(S) $= \frac{4+4+r+4+4+r}{2} = 8+r$,&nbsp;$a=4+r, b=4+r, c=8$Area of triangle ABC&nbsp;$=\sqrt{\ s\cdot\left(s-a\right)\left(s-b\right)\left(s-c\right)}=$ $\sqrt{\left(\ 8+r\right)\times\ 4\times\ 4\times\ r}$ =&nbsp;$\ \frac{\ 1}{2}\times \left(4+4\right)\times height$Height (h) =&nbsp;$\sqrt{\left(8+r\right)r}$Now, $h+4 = 4 \longrightarrow \sqrt{8+r} + r = 4$ (Considering the height of the triangle)$\sqrt{\left(8+r\right)r}$=4-r16r=16r=1 Alternatively,<img src="https://lightyellow-stork-291267.hostingersite.com/wp-content/uploads/2025/11/image_8XcBgkc.png">$AE^@ + EC^2 = AC^2 \longrightarrow 4^2 + (4-r)^2 = (4+r)^2 \longrightarrow \longrightarrow \longrightarrow r=1$

The roots of&nbsp;$x^2 - 4x - log_{2}{A} = 0$ will be real and distinct if and only if the discriminant is greater than zero16+4*$log_{2}{A}$ &gt; 0$log_{2}{A}$ &gt; -4A&gt; 1/16

Given,The quadratic equation $x^2 + bx + c = 0$ has two roots 4a and 3a7a=-b12$a^2$ = cWe have to find the value of&nbsp;$b^2 + c = 49a^2+ 12a^2=61a^2$Now lets verify the options&nbsp;61$a^2$ = 3721 ==&gt; a= 7.8 which is not an integer61$a^2$ = 361 ==&gt; a= 2.42 which is not an integer61$a^2$ = 427 ==&gt; a= 2.64 which is not an integer61$a^2$ = 549 ==&gt; a= 3 which is an integer

It is given that the base of the pyramid is square and each of the four sides are equilateral triangles.Length of each side of the equilateral triangle = 20cmSince the side of the triangle will be common to the square as well, the side of the square = 20cm <img src="https://lightyellow-stork-291267.hostingersite.com/wp-content/uploads/2025/11/image_3RZCpyF.png">Let h be the vertical height of the pyramid ie OAOB = 10 since it is half the side of the square&nbsp;AB is the height of the equilateral triangle i.e 10$\sqrt{\ 3}$AOB is a right angle, so applying the Pythagorean formula, we get$OA^2+ OB^2= AB^2$$h^2$ + 100= 300h=10$\sqrt{\ 2}$

cat 2019 Complete Paper Solution | Slot 2

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cat 2019 Complete Paper Solution | Slot 2

Question 1.

<div><p>The average of 30 integers is 5. Among these 30 integers, there are exactly 20 which do not exceed 5. What is the highest possible value of the average of these 20 integers?</p></div>

Question 2.

Question 3.

<div><p>What is the largest positive integer n such that $\frac{n^2 + 7n + 12}{n^2 - n - 12}$ is also a positive integer?</p></div>

Question 4.

<div><p>How many pairs (m, n) of positive integers satisfy the equation $m^2 + 105 = n^2$?</p></div>

Question 5.

<div><p>Two ants A and B start from a point P on a circle at the same time, with A moving clock-wise and B moving anti-clockwise. They meet for the first time at 10:00 am when A has covered 60% of the track. If A returns to P at 10:12 am, then B returns to P at</p></div>

Question 6.

<div><p>Let $a_1, a_2, ...$ be integers such that $a_1 - a_2 + a_3 - a_4 + .... + (-1)^{n - 1} a_n = n,$ for all $n \geq 1.$ Then $a_{51} + a_{52} + .... + a_{1023}$ equals</p></div>

Question 7.

<div><p>Two circles, each of radius 4 cm, touch externally. Each of these two circles is touched externally by a third circle. If these three circles have a common tangent, then the radius of the third circle, in cm, is</p></div>

Question 8.

<div><p>Let A be a real number. Then the roots of the equation $x^2 - 4x - log_{2}{A} = 0$ are real and distinct if and only if</p></div>

Question 9.

<div><p>The quadratic equation $x^2 + bx + c = 0$ has two roots 4a and 3a, where a is an integer. Which of the following is a possible value of $b^2 + c$?</p></div>

Question 10.

<div><p>The base of a regular pyramid is a square and each of the other four sides is an equilateral triangle, length of each side being 20 cm. The vertical height of the pyramid, in cm, is</p></div>