For all real values of x, the range of the function f ( x ) = x 2 + 2 x + 4 2 x 2 + 4 x + 9 f(x)=\frac{x^{2}+2x+4}{2x^{2}+4x+9} f ( x ) = 2 x 2 + 4 x + 9 x 2 + 2 x + 4 is:

[ 3 7 , 1 2 ) [\frac{3}{7},\frac{1}{2}) [ 7 3 , 2 1 )

[ 4 9 , 8 9 ] [\frac{4}{9},\frac{8}{9}] [ 9 4 , 9 8 ]

[ 3 7 , 8 9 ) [\frac{3}{7},\frac{8}{9}) [ 7 3 , 9 8 )

( 3 7 , 1 2 ) (\frac{3}{7},\frac{1}{2}) ( 7 3 , 2 1 )

For real x, the maximum possible value of x 1 + x 4 \frac{x}{\sqrt{1+x^{4}}} 1 + x 4 <path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"> x is

1 2 \frac{1}{\sqrt{2}} 2 <path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"> 1

1 3 \frac{1}{\sqrt{3}} 3 <path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"> 1

CAT Maxima & Minima Practice Questions With Solutions

CAT Quant Maxima & Minima Practice Questions

VARC DILR

Question 1.

The price of a precious stone is directly proportional to the square of its weight. Sita has a precious stone weighing 18 units. If she breaks it into four pieces with each piece having distinct integer weight, then the difference between the highest and lowest possible values of the total price of the four pieces will be 288000. Then, the price of the original precious stone is

A

1944000

B

972000

C

1620000

D

1296000

Question 2.

If $p^{2}+q^{2}-29=2pq-20=52-2pq$ , then the difference between the maximum and minimum possible value of $(p^{3}-q^{3})$

A

243

B

486

C

378

D

189

Question 3.

For some real numbers a and b, the system of equations $x + y = 4$ and $(a+5)x+(b^2-15)y=8b$ has infinitely many solutions for x and y. Then, the maximum possible value of ab is

A

33

B

25

C

15

D

55

Question 4.

Let $0 \leq a \leq x \leq 100$ and $f(x) = \mid x - a \mid + \mid x - 100 \mid + \mid x - a - 50\mid$ . Then the maximum value of f(x) becomes 100 when a is equal to

A

25

B

100

C

50

D

0

Question 5.

If a and b are non-negative real numbers such that a+ 2b = 6, then the average of the maximum and minimum possible values of (a+ b) is

A

3

B

4

C

3.5

D

4.5

Question 6.

If $c=\frac{16x}{y}+\frac{49y}{x}$ for some non-zero real numbers x and y, then c cannot take the value

A

$60$

B

$-50$

C

$-70$

D

$-60$

Question 7.

Consider six distinct natural numbers such that the average of the two smallest numbers is 14, and the average of the two largest numbers is 28. Then, the maximum possible value of the average of these six numbers is

A

23

B

24

C

23.5

D

22.5

Question 8.

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

A

$-\frac{1}{2}$

B

2

C

$\frac{1}{2}$

D

-2

Question 9.

For all real values of x, the range of the function $f(x)=\frac{x^{2}+2x+4}{2x^{2}+4x+9}$ is:

A

$[\frac{4}{9},\frac{8}{9}]$

B

$[\frac{3}{7},\frac{8}{9})$

C

$(\frac{3}{7},\frac{1}{2})$

D

$[\frac{3}{7},\frac{1}{2})$

Question 10.

If $f(x)=x^{2}-7x$ and $g(x)=x+3$ , then the minimum value of $f(g(x))-3x$ is:

A

-20

B

-12

C

-15

D

-16

Question 1.

Among 100 students, $x_1$ have birthdays in January, $X_2$ have birthdays in February, and so on. If $x_0=max(x_1,x_2,....,x_{12})$ , then the smallest possible value of $x_0$ is

A

8

B

9

C

10

D

12

Question 2.

Let $f(x)=x^{2}+ax+b$ and $g(x)=f(x+1)-f(x-1)$ . If $f(x)\geq0$ for all real x, and $g(20)=72$ . then the smallest possible value of b is

A

16

B

4

C

1

D

0

Question 3.

For real x, the maximum possible value of $\frac{x}{\sqrt{1+x^{4}}}$ is

A

$\frac{1}{2}$

B

$1$

C

$\frac{1}{\sqrt{3}}$

D

$\frac{1}{\sqrt{2}}$

Question 4.

Let m and n be positive integers, If $x^{2}+mx+2n=0$ and $x^{2}+2nx+m=0$ have real roots, then the smallest possible value of $m+n$ is

A

7

B

6

C

8

D

5

Question 5.

Let $\ a_{1},a_{2}...a_{52}\$ be positive integers such that $\ a_{1}$ < $a_{2}$ < ... < $a_{52}\$ . Suppose, their arithmetic mean is one less than arithmetic mean of $a_{2}$ , $a_{3}$ , .... $a_{52}$ . If $a_{52}$ = 100, then the largest possible value of $a_{1}$ is

A

48

B

20

C

23

D

45

Question 6.

The minimum possible value of the sum of the squares of the roots of the equation $x^2+(a+3)x-(a+5)=0$ is

A

1

B

2

C

3

D

4

CAT Maxima & Minima Practice Questions With Solutions

CAT Quant Maxima & Minima Practice Questions

Question 1.

Question 2.

If p2+q2−29=2pq−20=52−2pqp^{2}+q^{2}-29=2pq-20=52-2pqp2+q2−29=2pq−20=52−2pq, then the difference between the maximum and minimum possible value of (p3−q3)(p^{3}-q^{3})(p3−q3)

Question 3.

For some real numbers a and b, the system of equations x+y=4x + y = 4x+y=4 and (a+5)x+(b2−15)y=8b(a+5)x+(b^2-15)y=8b(a+5)x+(b2−15)y=8b has infinitely many solutions for x and y. Then, the maximum possible value of ab is

Question 4.

Let 0≤a≤x≤1000 \leq a \leq x \leq 1000≤a≤x≤100 and f(x)=∣x−a∣+∣x−100∣+∣x−a−50∣f(x) = \mid x - a \mid + \mid x - 100 \mid + \mid x - a - 50\midf(x)=∣x−a∣+∣x−100∣+∣x−a−50∣. Then the maximum value of f(x) becomes 100 when a is equal to

Question 5.

If a and b are non-negative real numbers such that a+ 2b = 6, then the average of the maximum and minimum possible values of (a+ b) is

Question 6.

If c=16xy+49yxc=\frac{16x}{y}+\frac{49y}{x}c=y16x​+x49y​ for some non-zero real numbers x and y, then c cannot take the value

Question 7.

Consider six distinct natural numbers such that the average of the two smallest numbers is 14, and the average of the two largest numbers is 28. Then, the maximum possible value of the average of these six numbers is

Question 8.

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

Question 9.

For all real values of x, the range of the function f(x)=x2+2x+42x2+4x+9f(x)=\frac{x^{2}+2x+4}{2x^{2}+4x+9}f(x)=2x2+4x+9x2+2x+4​ is:

Question 10.

If f(x)=x2−7xf(x)=x^{2}-7xf(x)=x2−7x and g(x)=x+3g(x)=x+3g(x)=x+3, then the minimum value of f(g(x))−3xf(g(x))-3xf(g(x))−3x is:

Question 1.

Among 100 students, x1x_1x1​ have birthdays in January, X2X_2X2​ have birthdays in February, and so on. If x0=max(x1,x2,....,x12)x_0=max(x_1,x_2,....,x_{12})x0​=max(x1​,x2​,....,x12​), then the smallest possible value of x0x_0x0​ is

Question 2.

Let f(x)=x2+ax+bf(x)=x^{2}+ax+bf(x)=x2+ax+b and g(x)=f(x+1)−f(x−1)g(x)=f(x+1)-f(x-1)g(x)=f(x+1)−f(x−1). If f(x)≥0f(x)\geq0f(x)≥0 for all real x, and g(20)=72g(20)=72g(20)=72. then the smallest possible value of b is

Question 3.

For real x, the maximum possible value of x1+x4\frac{x}{\sqrt{1+x^{4}}}1+x4​x​ is

Question 4.

Let m and n be positive integers, If x2+mx+2n=0x^{2}+mx+2n=0x2+mx+2n=0 and x2+2nx+m=0x^{2}+2nx+m=0x2+2nx+m=0 have real roots, then the smallest possible value of m+nm+nm+n is

Question 5.

Question 6.

The minimum possible value of the sum of the squares of the roots of the equation x2+(a+3)x−(a+5)=0x^2+(a+3)x-(a+5)=0 x2+(a+3)x−(a+5)=0 is

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If $p^{2}+q^{2}-29=2pq-20=52-2pq$ , then the difference between the maximum and minimum possible value of $(p^{3}-q^{3})$

For some real numbers a and b, the system of equations $x + y = 4$ and $(a+5)x+(b^2-15)y=8b$ has infinitely many solutions for x and y. Then, the maximum possible value of ab is

Let $0 \leq a \leq x \leq 100$ and $f(x) = \mid x - a \mid + \mid x - 100 \mid + \mid x - a - 50\mid$ . Then the maximum value of f(x) becomes 100 when a is equal to

If $c=\frac{16x}{y}+\frac{49y}{x}$ for some non-zero real numbers x and y, then c cannot take the value

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

For all real values of x, the range of the function $f(x)=\frac{x^{2}+2x+4}{2x^{2}+4x+9}$ is:

If $f(x)=x^{2}-7x$ and $g(x)=x+3$ , then the minimum value of $f(g(x))-3x$ is:

Among 100 students, $x_1$ have birthdays in January, $X_2$ have birthdays in February, and so on. If $x_0=max(x_1,x_2,....,x_{12})$ , then the smallest possible value of $x_0$ is

Let $f(x)=x^{2}+ax+b$ and $g(x)=f(x+1)-f(x-1)$ . If $f(x)\geq0$ for all real x, and $g(20)=72$ . then the smallest possible value of b is

For real x, the maximum possible value of $\frac{x}{\sqrt{1+x^{4}}}$ is

Let m and n be positive integers, If $x^{2}+mx+2n=0$ and $x^{2}+2nx+m=0$ have real roots, then the smallest possible value of $m+n$ is

The minimum possible value of the sum of the squares of the roots of the equation $x^2+(a+3)x-(a+5)=0$ is