Any non-zero real numbers x,y such that y ≠ 3 y\neq3 y  = 3 and x y < x + 3 y − 3 \frac{x}{y}<\frac{x+3}{y-3} y x < y − 3 x + 3 , Will satisfy the condition.

If y < 0 y<0 y < 0 , and − x < y -x<y − x < y

x y < y x \frac{x}{y}<\frac{y}{x} y x < x y

If y > 10 y>10 y > 1 0 , and − x > y -x>y − x > y

If x < 0 x<0 x < 0 , and − x < y -x<y − x < y

For a real number x, if 1 2 , log ⁡ 3 ( 2 x − 9 ) log ⁡ 3 4 \frac{1}{2}, \frac{\log_3(2^x - 9)}{\log_3 4} 2 1 , l o g 3 4 l o g 3 ( 2 x − 9 ) , and log ⁡ 5 ( 2 x + 17 2 ) log ⁡ 5 4 \frac{\log_5\left(2^x + \frac{17}{2}\right)}{\log_5 4} l o g 5 4 l o g 5 ( 2 x + 2 1 7 ) are in an arithmetic progression, then the common difference is

log ⁡ 4 ( 7 2 ) \log_4\left(\frac{7}{2}\right) lo g 4 ( 2 7 )

log ⁡ 4 ( 3 2 ) \log_4\left(\frac{3}{2}\right) lo g 4 ( 2 3 )

log ⁡ 4 ( 23 2 ) \log_4\left(\frac{23}{2}\right) lo g 4 ( 2 2 3 )

For a real number x the condition ∣ 3 x − 20 ∣ + ∣ 3 x − 40 ∣ = 20 \mid3x-20\mid+\mid3x-40\mid=20 ∣ 3 x − 2 0 ∣ + ∣ 3 x − 4 0 ∣ = 2 0 necessarily holds if

10 < x < 15 10<x<15 1 0 < x < 1 5

CAT Linear Equations Practice Questions With Solutions

CAT Quant Linear Equations Practice Questions

VARC DILR

Question 1.

If $x$ and $y$ are positive real numbers such that $\log_{x}(x^2 + 12) = 4$ and $3 \log_{y} x = 1$ , then $x + y$ equals

A

20

B

68

C

10

D

11

Question 2.

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

A

0

B

1

C

-1

D

2

Question 3.

Any non-zero real numbers x,y such that $y\neq3$ and $\frac{x}{y}<\frac{x+3}{y-3}$ , Will satisfy the condition.

A

$\frac{x}{y}<\frac{y}{x}$

B

If $y<0$ , and $-x<y$

C

If $y>10$ , and $-x>y$

D

If $x<0$ , and $-x<y$

Question 4.

Let both the series $a_{1},a_{2},a_{3}$ ... and $b_{1},b_{2},b_{3}$ ... be in arithmetic progression such that the common differences of both the series are prime numbers. If $a_{5}=b_{9},a_{19}=b_{19}$ and $b_{2}=0$ , then $a_{11}$ equals

A

86

B

79

C

83

D

84

Question 5.

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 6.

For a real number x, if $\frac{1}{2}, \frac{\log_3(2^x - 9)}{\log_3 4}$ , and $\frac{\log_5\left(2^x + \frac{17}{2}\right)}{\log_5 4}$ are in an arithmetic progression, then the common difference is

A

$\log_4\left(\frac{3}{2}\right)$

B

$\log_4 7$

C

$\log_4\left(\frac{23}{2}\right)$

D

$\log_4\left(\frac{7}{2}\right)$

Question 7.

Let $a_n = 46 + 8n$ and $b_n = 98 + 4n$ be two sequences for natural numbers $n \leq 100$ . Then, the sum of all terms common to both the sequences is

A

14900

B

14798

C

15000

D

14602

Question 8.

For any natural number n, suppose the sum of the first n terms of an arithmetic progression is $(n + 2n^2)$ . If the $n^{th}$ term of the progression is divisible by 9, then the smallest possible value of n is

A

9

B

4

C

7

D

8

Question 9.

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 10.

The largest real value of a for which the equation $\mid x + a \mid + \mid x - 1 \mid = 2$ has an infinite number of solutions for x is

A

-1

B

0

C

1

D

2

Question 1.

Consider the arithmetic progression 3, 7, 11, ... and let $A_n$ denote the sum of the first n terms of this progression. Then the value of $\frac{1}{25} \sum_{n=1}^{25} A_{n}$ is

A

455

B

442

C

415

D

404

Question 2.

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 3.

Let r be a real number and $f(x) = \begin{cases}2x -r & ifx \geq r\\ r &ifx < r\end{cases}$ . Then, the equation $f(x) = f(f(x))$ holds for all real values of $x$ where

A

$x > r$

B

$x \leq r$

C

$x \neq r$

D

$x \geq r$

Question 4.

Three positive integers x, y and z are in arithmetic progression. If $y-x>2$ and $xyz=5(x+y+z)$ , then z-x equals

A

8

B

12

C

14

D

10

Question 5.

For a real number x the condition $\mid3x-20\mid+\mid3x-40\mid=20$ necessarily holds if

A

$10<x<15$

B

$9<x<14$

C

$7<x<12$

D

$6<x<11$

Question 6.

Consider a sequence of real numbers, $x_{1},x_{2},x_{3},...$ such that $x_{n+1}=x_{n}+n-1$ for all $n\geq1$ . If $x_{1}=-1$ then $x_{100}$ is equal to

A

4849

B

4949

C

4950

D

4850

Question 7.

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 8.

If $3x+2\mid y\mid+y=7$ and $x+\mid x \mid+3y=1$ then $x+2y$ is:

A

$-\frac{4}{3}$

B

$\frac{8}{3}$

C

$0$

D

$1$

Question 9.

A batsman played n + 2 innings and got out on all occasions. His average score in these n + 2 innings was 29 runs and he scored 38 and 15 runs in the last two innings. The batsman scored less than 38 runs in each of the first n innings. In these n innings, his average score was 30 runs and lowest score was x runs. The smallest possible value of x is

A

4

B

3

C

2

D

1

Question 10.

Let m and n be natural numbers such that n is even and $0.2<\frac{m}{20},\frac{n}{m},\frac{n}{11}<0.5$ . Then $m-2n$ equals

A

3

B

1

C

2

D

4

CAT Linear Equations Practice Questions With Solutions

CAT Quant Linear Equations Practice Questions

Question 1.

If xxx and yyy are positive real numbers such that log⁡x(x2+12)=4\log_{x}(x^2 + 12) = 4logx​(x2+12)=4 and 3log⁡yx=13 \log_{y} x = 13logy​x=1, then x+yx + y x+y equals

Question 2.

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

Question 3.

Any non-zero real numbers x,y such that y≠3y\neq3y​=3 and xy<x+3y−3\frac{x}{y}<\frac{x+3}{y-3}yx​<y−3x+3​, Will satisfy the condition.

Question 4.

Question 5.

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 6.

Question 7.

Let an=46+8na_n = 46 + 8nan​=46+8n and bn=98+4nb_n = 98 + 4nbn​=98+4n be two sequences for natural numbers n≤100n \leq 100n≤100. Then, the sum of all terms common to both the sequences is

Question 8.

For any natural number n, suppose the sum of the first n terms of an arithmetic progression is (n+2n2)(n + 2n^2)(n+2n2). If the nthn^{th}nth term of the progression is divisible by 9, then the smallest possible value of n is

Question 9.

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 10.

The largest real value of a for which the equation ∣x+a∣+∣x−1∣=2\mid x + a \mid + \mid x - 1 \mid = 2∣x+a∣+∣x−1∣=2 has an infinite number of solutions for x is

Question 1.

Consider the arithmetic progression 3, 7, 11, ... and let AnA_nAn​ denote the sum of the first n terms of this progression. Then the value of 125∑n=125An\frac{1}{25} \sum_{n=1}^{25} A_{n}251​∑n=125​An​ is

Question 2.

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 3.

Let r be a real number and f(x)={2x−rifx≥rrifx<rf(x) = \begin{cases}2x -r & ifx \geq r\\ r &ifx < r\end{cases}f(x)={2x−rr​ifx≥rifx<r​. Then, the equation f(x)=f(f(x))f(x) = f(f(x))f(x)=f(f(x)) holds for all real values of xxx where

Question 4.

Three positive integers x, y and z are in arithmetic progression. If y−x>2y-x>2y−x>2 and xyz=5(x+y+z)xyz=5(x+y+z)xyz=5(x+y+z), then z-x equals

Question 5.

For a real number x the condition ∣3x−20∣+∣3x−40∣=20\mid3x-20\mid+\mid3x-40\mid=20∣3x−20∣+∣3x−40∣=20 necessarily holds if

Question 6.

Consider a sequence of real numbers, x1,x2,x3,...x_{1},x_{2},x_{3},...x1​,x2​,x3​,... such that xn+1=xn+n−1x_{n+1}=x_{n}+n-1xn+1​=xn​+n−1 for all n≥1n\geq1n≥1. If x1=−1x_{1}=-1x1​=−1 then x100x_{100}x100​ is equal to

Question 7.

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 8.

If 3x+2∣y∣+y=73x+2\mid y\mid+y=73x+2∣y∣+y=7 and x+∣x∣+3y=1x+\mid x \mid+3y=1x+∣x∣+3y=1 then x+2yx+2yx+2y is:

Question 9.

Question 10.

Let m and n be natural numbers such that n is even and 0.2<m20,nm,n11<0.50.2<\frac{m}{20},\frac{n}{m},\frac{n}{11}<0.50.2<20m​,mn​,11n​<0.5. Then m−2nm-2nm−2n equals

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If $x$ and $y$ are positive real numbers such that $\log_{x}(x^2 + 12) = 4$ and $3 \log_{y} x = 1$ , then $x + y$ equals

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

Any non-zero real numbers x,y such that $y\neq3$ and $\frac{x}{y}<\frac{x+3}{y-3}$ , Will satisfy the condition.

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 $a_n = 46 + 8n$ and $b_n = 98 + 4n$ be two sequences for natural numbers $n \leq 100$ . Then, the sum of all terms common to both the sequences is

For any natural number n, suppose the sum of the first n terms of an arithmetic progression is $(n + 2n^2)$ . If the $n^{th}$ term of the progression is divisible by 9, then the smallest possible value of n 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

The largest real value of a for which the equation $\mid x + a \mid + \mid x - 1 \mid = 2$ has an infinite number of solutions for x is

Consider the arithmetic progression 3, 7, 11, ... and let $A_n$ denote the sum of the first n terms of this progression. Then the value of $\frac{1}{25} \sum_{n=1}^{25} A_{n}$ is

Let r be a real number and $f(x) = \begin{cases}2x -r & ifx \geq r\\ r &ifx < r\end{cases}$ . Then, the equation $f(x) = f(f(x))$ holds for all real values of $x$ where

Three positive integers x, y and z are in arithmetic progression. If $y-x>2$ and $xyz=5(x+y+z)$ , then z-x equals

For a real number x the condition $\mid3x-20\mid+\mid3x-40\mid=20$ necessarily holds if

Consider a sequence of real numbers, $x_{1},x_{2},x_{3},...$ such that $x_{n+1}=x_{n}+n-1$ for all $n\geq1$ . If $x_{1}=-1$ then $x_{100}$ is equal to

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

If $3x+2\mid y\mid+y=7$ and $x+\mid x \mid+3y=1$ then $x+2y$ is:

Let m and n be natural numbers such that n is even and $0.2<\frac{m}{20},\frac{n}{m},\frac{n}{11}<0.5$ . Then $m-2n$ equals