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All questions of Algebra- Inequalities for A Level Exam
Find the value of x when x is a natural number and 24x< 100.- a){5,6,……..∞}
- b){1,2,3,4}
- c){1,2,3,4,5}
- d){0,1,2,3,4}
Correct answer is option 'B'. Can you explain this answer?
Find the value of x when x is a natural number and 24x< 100.
a)
{5,6,……..∞}
b)
{1,2,3,4}
c)
{1,2,3,4,5}
d)
{0,1,2,3,4}
|
Shreya Gupta answered |
We are given: 24x < 100
24x < 100
=> 24x /24 < 100/24 [Dividing both sides by positive number.]
=> x < 25/6
When x is a natural number, in this case, the following values of x make the statement true
x = 1, 2, 3, 4.
The solution set of the inequality is {1, 2, 3, 4}.
What is the region represented by x > 0 and y < 0?- a)2nd quadrant
- b)3rd quadrant
- c)4th quadrant
- d)1st quadrant
Correct answer is option 'C'. Can you explain this answer?
What is the region represented by x > 0 and y < 0?
a)
2nd quadrant
b)
3rd quadrant
c)
4th quadrant
d)
1st quadrant
|
|
Suresh Reddy answered |
If x >0 then x will be positive and y less than 0 means it gets negative value
that means it lies in 4th quadrant.
that means it lies in 4th quadrant.
The solution of the inequality 3(2-x)≥2(1-x) for real x is :- a)x < 4
- b)x > 4
- c)x ≤4
- d)x ≥ 4
Correct answer is option 'C'. Can you explain this answer?
The solution of the inequality 3(2-x)≥2(1-x) for real x is :
a)
x < 4
b)
x > 4
c)
x ≤4
d)
x ≥ 4
|
Mohit Rajpoot answered |
3(2 - x) ≥ 2(1 - x)
6 - 3x ≥ 2 - 2x
4 ≥ x
6 - 3x ≥ 2 - 2x
4 ≥ x
Pairs of consecutive even positive integers, both of which are larger than 7 such that their sum is less than 28, are- a)(8, 9), (9, 10), (10, 11)
- b)(8, 10), (10, 12)
- c)(8, 10), (10, 12), (12, 14)
- d)(8, 10), (10, 12), (12, 14), (14, 16)
Correct answer is option 'C'. Can you explain this answer?
Pairs of consecutive even positive integers, both of which are larger than 7 such that their sum is less than 28, are
a)
(8, 9), (9, 10), (10, 11)
b)
(8, 10), (10, 12)
c)
(8, 10), (10, 12), (12, 14)
d)
(8, 10), (10, 12), (12, 14), (14, 16)
|
Neha Joshi answered |
Let x be the smaller of the two consecutive even positive integers .
Then the other integer is x+2.
Since both the integers are larger than 7,x > 7 ....(1)
Also the sum of the two integers is less than 28.
x + (x + 2) < 28
⇒ 2x + 2 < 28
⇒ 2x < 28 − 2
⇒ 2x < 26
=> x < 13…...(2)
From(1) and (2), we get
7 < x <13
Since x is an even number, x can take the values 8, 10 and 12.
Thus the required possible pairs are (8,10),(10,12) and (12,8).
Then the other integer is x+2.
Since both the integers are larger than 7,x > 7 ....(1)
Also the sum of the two integers is less than 28.
x + (x + 2) < 28
⇒ 2x + 2 < 28
⇒ 2x < 28 − 2
⇒ 2x < 26
=> x < 13…...(2)
From(1) and (2), we get
7 < x <13
Since x is an even number, x can take the values 8, 10 and 12.
Thus the required possible pairs are (8,10),(10,12) and (12,8).
A solution of 10% boric acid is to be diluted by adding a 4% boric acid solution to it. The resulting mixture is to be more than 5% but less than 8% boric acid. If we have 750 litres of the 10% solution, then the quantity of the 4% solution that has to be added will lie between- a)370 litres and 3750 litres
- b)375 litres and 3750 litres
- c)320 litres and 1280 litres
- d)370 litres and 3700 litres
Correct answer is option 'B'. Can you explain this answer?
A solution of 10% boric acid is to be diluted by adding a 4% boric acid solution to it. The resulting mixture is to be more than 5% but less than 8% boric acid. If we have 750 litres of the 10% solution, then the quantity of the 4% solution that has to be added will lie between
a)
370 litres and 3750 litres
b)
375 litres and 3750 litres
c)
320 litres and 1280 litres
d)
370 litres and 3700 litres
|
Krishna Iyer answered |
Let x litres of 4% boric acid solution is required to be added
Then, total mixture = (x + 750)litres
This resulting mixture is to be more than 5% but less than 8% boric acid
Total amount of acid = 750 of 10% + x of 4%
ATQ
5/100(750 + x) < (750*10 + 4x)/100 < 8/100(750 + x)
3750 + 5x < 7500 + 4x ; 7500 + 4x < 6000 + 8x
x < 3750 ; x > 375
375 and 3750
Then, total mixture = (x + 750)litres
This resulting mixture is to be more than 5% but less than 8% boric acid
Total amount of acid = 750 of 10% + x of 4%
ATQ
5/100(750 + x) < (750*10 + 4x)/100 < 8/100(750 + x)
3750 + 5x < 7500 + 4x ; 7500 + 4x < 6000 + 8x
x < 3750 ; x > 375
375 and 3750
If -5x+2<7x -4, then x is- a)<3/2
- b)<1/2
- c)>1/2
- d)>3/2
Correct answer is option 'C'. Can you explain this answer?
If -5x+2<7x -4, then x is
a)
<3/2
b)
<1/2
c)
>1/2
d)
>3/2
|
Pooja Shah answered |
-5x + 2 < 7x - 4
6 < 12x
x > 1/2
6 < 12x
x > 1/2
Find the solution for the pair of solution x > 1 and x > -1- a)No solution
- b)-1 < x < 1
- c)x < -1
- d)x > 1
Correct answer is option 'D'. Can you explain this answer?
Find the solution for the pair of solution x > 1 and x > -1
a)
No solution
b)
-1 < x < 1
c)
x < -1
d)
x > 1
|
Riya Banerjee answered |
x > 1 and x > -1
Take two points 1,-1 on number line
Therefore x > 1
Take two points 1,-1 on number line
Therefore x > 1
IQ of a person is given by the formula 
, where M A is mental age and CA is chronological age. If 85 < IQ < 145 for a group of 12-year children, then the mental age will lie between- a)10.4 and 17.2
- b)10 and 18
- c)10 and 17
- d)10.2 and 17.4
Correct answer is option 'D'. Can you explain this answer?
IQ of a person is given by the formula , where M A is mental age and CA is chronological age. If 85 < IQ < 145 for a group of 12-year children, then the mental age will lie between
, where M A is mental age and CA is chronological age. If 85 < IQ < 145 for a group of 12-year children, then the mental age will lie betweena)
10.4 and 17.2
b)
10 and 18
c)
10 and 17
d)
10.2 and 17.4
|
Tejas Verma answered |
85≤MA/CA×100≤145
(85*12)/100≤MA/CA×100≤(145*12)/100
10.2≤MA≤17.4
range of mental age is[10.2,17.4].
(85*12)/100≤MA/CA×100≤(145*12)/100
10.2≤MA≤17.4
range of mental age is[10.2,17.4].
Solve: 30x < 200, when x is a natural number:
- a){2, 3, 4, 5, 6}
- b){1, 2, 3, 4, 5, 6}
- c){4, 5, 6, 7, 8, 9}
- d){1, 2, 3, 4, 5, 6, 7}
Correct answer is option 'B'. Can you explain this answer?
Solve: 30x < 200, when x is a natural number:
a)
{2, 3, 4, 5, 6}
b)
{1, 2, 3, 4, 5, 6}
c)
{4, 5, 6, 7, 8, 9}
d)
{1, 2, 3, 4, 5, 6, 7}
|
Learners Habitat answered |
We are given 30x < 200
When x is a natural number, in this case the following values of x make the statement true.
1, 2, 3, 4, 5, 6.
The solution set of the inequality is {1, 2, 3, 4, 5, 6}.
When x is a natural number, in this case the following values of x make the statement true.
1, 2, 3, 4, 5, 6.
The solution set of the inequality is {1, 2, 3, 4, 5, 6}.
The solution of 4x-2 > 6 is- a)x > 2
- b)x < 6
- c)x = 4
- d)x > 4
Correct answer is option 'A'. Can you explain this answer?
The solution of 4x-2 > 6 is
a)
x > 2
b)
x < 6
c)
x = 4
d)
x > 4
|
|
Gacha Girl's Channel answered |
A is the correct answer because 4 x- 2 = 6
Find all pairs of consecutive odd natural numbers, both of which are larger than 10, such that their sum is less than 40.- a)(11 , 13) , (13 , 15) , (15 , 17) , (17 , 21)
- b)(9 , 11) , (13 , 15) , (15 , 17) , (17 , 19)
- c)(11 , 13) , (13 , 15) , (17 , 19) , (19 , 21)
- d)(11 , 13) , (13 , 15) , (15 , 17) , (17 , 19)
Correct answer is option 'D'. Can you explain this answer?
Find all pairs of consecutive odd natural numbers, both of which are larger than 10, such that their sum is less than 40.
a)
(11 , 13) , (13 , 15) , (15 , 17) , (17 , 21)
b)
(9 , 11) , (13 , 15) , (15 , 17) , (17 , 19)
c)
(11 , 13) , (13 , 15) , (17 , 19) , (19 , 21)
d)
(11 , 13) , (13 , 15) , (15 , 17) , (17 , 19)
|
|
Dipika Patel answered |
**Explanation:**
To find the pairs of consecutive odd natural numbers that satisfy the given conditions, we need to consider the following:
1. The numbers should be consecutive odd natural numbers.
2. Both numbers should be larger than 10.
3. The sum of the two numbers should be less than 40.
Let's analyze each option to see if it satisfies these conditions:
**Option A: (11, 13), (13, 15), (15, 17), (17, 21)**
- (11, 13): The sum is 24, which is less than 40. This pair satisfies all conditions.
- (13, 15): The sum is 28, which is less than 40. This pair satisfies all conditions.
- (15, 17): The sum is 32, which is less than 40. This pair satisfies all conditions.
- (17, 21): The sum is 38, which is less than 40. This pair satisfies all conditions.
**Option B: (9, 11), (13, 15), (15, 17), (17, 19)**
- (9, 11): The numbers are not larger than 10. This pair does not satisfy the second condition.
- (13, 15): The sum is 28, which is less than 40. This pair satisfies all conditions.
- (15, 17): The sum is 32, which is less than 40. This pair satisfies all conditions.
- (17, 19): The sum is 36, which is not less than 40. This pair does not satisfy the third condition.
**Option C: (11, 13), (13, 15), (17, 19), (19, 21)**
- (11, 13): The sum is 24, which is less than 40. This pair satisfies all conditions.
- (13, 15): The sum is 28, which is less than 40. This pair satisfies all conditions.
- (17, 19): The sum is 36, which is not less than 40. This pair does not satisfy the third condition.
- (19, 21): The sum is 40, which is not less than 40. This pair does not satisfy the third condition.
**Option D: (11, 13), (13, 15), (15, 17), (17, 19)**
- (11, 13): The sum is 24, which is less than 40. This pair satisfies all conditions.
- (13, 15): The sum is 28, which is less than 40. This pair satisfies all conditions.
- (15, 17): The sum is 32, which is less than 40. This pair satisfies all conditions.
- (17, 19): The sum is 36, which is not less than 40. This pair does not satisfy the third condition.
Therefore, the correct answer is **Option D: (11, 13), (13, 15), (15, 17), (17, 19)** as it is the only option where all the pairs satisfy all the given conditions.
To find the pairs of consecutive odd natural numbers that satisfy the given conditions, we need to consider the following:
1. The numbers should be consecutive odd natural numbers.
2. Both numbers should be larger than 10.
3. The sum of the two numbers should be less than 40.
Let's analyze each option to see if it satisfies these conditions:
**Option A: (11, 13), (13, 15), (15, 17), (17, 21)**
- (11, 13): The sum is 24, which is less than 40. This pair satisfies all conditions.
- (13, 15): The sum is 28, which is less than 40. This pair satisfies all conditions.
- (15, 17): The sum is 32, which is less than 40. This pair satisfies all conditions.
- (17, 21): The sum is 38, which is less than 40. This pair satisfies all conditions.
**Option B: (9, 11), (13, 15), (15, 17), (17, 19)**
- (9, 11): The numbers are not larger than 10. This pair does not satisfy the second condition.
- (13, 15): The sum is 28, which is less than 40. This pair satisfies all conditions.
- (15, 17): The sum is 32, which is less than 40. This pair satisfies all conditions.
- (17, 19): The sum is 36, which is not less than 40. This pair does not satisfy the third condition.
**Option C: (11, 13), (13, 15), (17, 19), (19, 21)**
- (11, 13): The sum is 24, which is less than 40. This pair satisfies all conditions.
- (13, 15): The sum is 28, which is less than 40. This pair satisfies all conditions.
- (17, 19): The sum is 36, which is not less than 40. This pair does not satisfy the third condition.
- (19, 21): The sum is 40, which is not less than 40. This pair does not satisfy the third condition.
**Option D: (11, 13), (13, 15), (15, 17), (17, 19)**
- (11, 13): The sum is 24, which is less than 40. This pair satisfies all conditions.
- (13, 15): The sum is 28, which is less than 40. This pair satisfies all conditions.
- (15, 17): The sum is 32, which is less than 40. This pair satisfies all conditions.
- (17, 19): The sum is 36, which is not less than 40. This pair does not satisfy the third condition.
Therefore, the correct answer is **Option D: (11, 13), (13, 15), (15, 17), (17, 19)** as it is the only option where all the pairs satisfy all the given conditions.
The solution of inequality 4x + 3 < 5x + 7 when x is a real number is
- a)[4, ∞)
- b)(-∞, 4)
- c)(-4,∞)
- d)None of the these
Correct answer is option 'C'. Can you explain this answer?
The solution of inequality 4x + 3 < 5x + 7 when x is a real number is
a)
[4, ∞)
b)
(-∞, 4)
c)
(-4,∞)
d)
None of the these
|
Sanchita Reddy answered |
4x + 3 < 5x + 7
subtract 4 both sides,
4x + 3 - 3 < 5x + 7 - 3
⇒ 4x < 5x + 4
subtract ' 5x ' both sides ,
[ equal number may be subtracted from both sides of an inequality without affecting the sign of inequality]
4x - 5x < 5x + 4 - 5
-x < 4
now, multiple with (-1) then, sign of inequality change .
-x.(-1) > 4(-1)
x > -4
hence, x€ ( -4 , ∞)
subtract 4 both sides,
4x + 3 - 3 < 5x + 7 - 3
⇒ 4x < 5x + 4
subtract ' 5x ' both sides ,
[ equal number may be subtracted from both sides of an inequality without affecting the sign of inequality]
4x - 5x < 5x + 4 - 5
-x < 4
now, multiple with (-1) then, sign of inequality change .
-x.(-1) > 4(-1)
x > -4
hence, x€ ( -4 , ∞)
A solution is to be kept between 30∘C and 35∘C What is the range of temperature in degree Fahrenheit ?- a)Between 40∘F and 60∘F
- b)Between 30∘Fand 35∘F
- c)Between 86∘F and 95∘F
- d)Between 76∘F and 105∘F
Correct answer is option 'C'. Can you explain this answer?
A solution is to be kept between 30∘C and 35∘C What is the range of temperature in degree Fahrenheit ?
a)
Between 40∘F and 60∘F
b)
Between 30∘Fand 35∘F
c)
Between 86∘F and 95∘F
d)
Between 76∘F and 105∘F
|
Manoj Patel answered |
Explanation:
To convert Celsius to Fahrenheit, we use the formula:
F = (9/5)C + 32
where F is the temperature in Fahrenheit and C is the temperature in Celsius.
Let's find the Fahrenheit equivalents of the given temperature range:
Lower limit:
F = (9/5)30 + 32 = 86F
Upper limit:
F = (9/5)35 + 32 = 95F
Therefore, the range of temperature in degree Fahrenheit is between 86F and 95F.
Answer:
Option (c) is correct. The range of temperature in degree Fahrenheit is between 86F and 95F.
To convert Celsius to Fahrenheit, we use the formula:
F = (9/5)C + 32
where F is the temperature in Fahrenheit and C is the temperature in Celsius.
Let's find the Fahrenheit equivalents of the given temperature range:
Lower limit:
F = (9/5)30 + 32 = 86F
Upper limit:
F = (9/5)35 + 32 = 95F
Therefore, the range of temperature in degree Fahrenheit is between 86F and 95F.
Answer:
Option (c) is correct. The range of temperature in degree Fahrenheit is between 86F and 95F.
Identify solution set for [| 4 −− x | + 1 < 3?- a)(2 , 6)
- b)(3 , 6)
- c)(2 , 4)
- d)(2 , 3)
Correct answer is option 'A'. Can you explain this answer?
Identify solution set for [| 4 −− x | + 1 < 3?
a)
(2 , 6)
b)
(3 , 6)
c)
(2 , 4)
d)
(2 , 3)
|
|
Neha Joshi answered |
|4 − x| + 1 < 3
⇒ 4 − x + 1 < 3
Add −4 and −1 on both sides, we get
4 − x + 1 − 4 − 1 < 3 − 4 − 1
⇒ − x < −2
Multiply both sides by −1, we get
x > 2
Also,|4−x| + 1 < 3
⇒ −(4−x) + 1 < 3
⇒ − 4 + x + 1 < 3
Add 4 and −1 on both sides, we get
− 4 + x + 1 + 4 − 1 < 3 + 4 − 1
⇒ x < 6
Thus, x ∈ (2,6).
⇒ 4 − x + 1 < 3
Add −4 and −1 on both sides, we get
4 − x + 1 − 4 − 1 < 3 − 4 − 1
⇒ − x < −2
Multiply both sides by −1, we get
x > 2
Also,|4−x| + 1 < 3
⇒ −(4−x) + 1 < 3
⇒ − 4 + x + 1 < 3
Add 4 and −1 on both sides, we get
− 4 + x + 1 + 4 − 1 < 3 + 4 − 1
⇒ x < 6
Thus, x ∈ (2,6).
A connected planar graph having 6 vertices, 7 edges contains _____________ regions.- a)15
- b)2
- c)1
- d)11
Correct answer is option 'B'. Can you explain this answer?
A connected planar graph having 6 vertices, 7 edges contains _____________ regions.
a)
15
b)
2
c)
1
d)
11
|
Lavanya Menon answered |
By euler’s formula the relation between vertices(n), edges(q) and regions(r) is given by n-q+r=2.
A plumber can be paid under two schemes as given below:
I: Rs 600 and Rs 50 per hour.
II: Rs 170 per hour.
If the job takes n hours, then the values of n for which the scheme I will give the plumber better wages are- a)less than 4 hours
- b)less than 5 hours
- c)more than 5 hours
- d)4 hours
Correct answer is option 'B'. Can you explain this answer?
A plumber can be paid under two schemes as given below:
I: Rs 600 and Rs 50 per hour.
II: Rs 170 per hour.
If the job takes n hours, then the values of n for which the scheme I will give the plumber better wages are
I: Rs 600 and Rs 50 per hour.
II: Rs 170 per hour.
If the job takes n hours, then the values of n for which the scheme I will give the plumber better wages are
a)
less than 4 hours
b)
less than 5 hours
c)
more than 5 hours
d)
4 hours
|
|
Lavanya Menon answered |
Your question is not complete it should be,
I: &600 fix and &50 per hourNow solving the question,Let the work is done in n hour as per given , So After n hour according to first scheme he will get an amount
= 600+50n After n hour according to second scheme he will get an amount,
=170nNow we are asked for values of n such that scheme
I give better wages so 600+50n >170n600>120nor n <5 .SO if the numbers of hour are less than 5 then he will get better wages by first scheme.
I: &600 fix and &50 per hourNow solving the question,Let the work is done in n hour as per given , So After n hour according to first scheme he will get an amount
= 600+50n After n hour according to second scheme he will get an amount,
=170nNow we are asked for values of n such that scheme
I give better wages so 600+50n >170n600>120nor n <5 .SO if the numbers of hour are less than 5 then he will get better wages by first scheme.
Given that x is an integer, find the values of x which satisfy the simultaneous linear inequalities 2 + x < 6 and 2 −3x < − 1.- a)1 , 2 , 3
- b)2 , 3
- c)2 , 3, 4,
- d)1 , 2, 3, 4.
Correct answer is option 'B'. Can you explain this answer?
Given that x is an integer, find the values of x which satisfy the simultaneous linear inequalities 2 + x < 6 and 2 −3x < − 1.
a)
1 , 2 , 3
b)
2 , 3
c)
2 , 3, 4,
d)
1 , 2, 3, 4.
|
EduRev JEE answered |
Given Inequalities:
- 2 + x < 6
- 2 −3x < − 1
Step 1: Solve the first inequality 2 + x < 6
Subtract 2 from both sides:
x < 6 - 2
Simplifying:
x < 4
Thus, from the first inequality, we have:
x < 4
Step 2: Solve the second inequality 2 - 3x < -1
Subtract 2 from both sides:
- 3x < -1 - 2
Simplifying:
- 3x < -3
Now, divide both sides by -3, and remember to reverse the inequality sign when dividing by a negative number:
x > 1
Step 3: Combine the results
From the first inequality, we know x < 4.
From the second inequality, we know x > 1.
Thus, the solution for x must satisfy both:
1 < x < 4
The integer values of xxx that lie between 1 and 4 are:
x = 2, 3
The correct option is: B: 2, 3
Subtract 2 from both sides:
x < 6 - 2
Simplifying:
x < 4
Thus, from the first inequality, we have:
x < 4
Step 2: Solve the second inequality 2 - 3x < -1
Subtract 2 from both sides:
- 3x < -1 - 2
Simplifying:
- 3x < -3
Now, divide both sides by -3, and remember to reverse the inequality sign when dividing by a negative number:
x > 1
Step 3: Combine the results
From the first inequality, we know x < 4.
From the second inequality, we know x > 1.
Thus, the solution for x must satisfy both:
1 < x < 4
The integer values of xxx that lie between 1 and 4 are:
x = 2, 3
The correct option is: B: 2, 3
In a game a person wins a TV if in four throws of a dice he get sum greater than 20 .In three throws he got numbers as 5,3,6. What should be his fourth throw so that he wins a TV?- a)5 or 6
- b)never wins.
- c)5
- d)6
Correct answer is option 'B'. Can you explain this answer?
In a game a person wins a TV if in four throws of a dice he get sum greater than 20 .In three throws he got numbers as 5,3,6. What should be his fourth throw so that he wins a TV?
a)
5 or 6
b)
never wins.
c)
5
d)
6
|
|
Krithika Kulkarni answered |
Method to Solve :
Numbers obtained in three throws are 6, 5 and 4.
Let the number obtained in fourth throw be x.
Now, Sum > 20
6 + 5 + 4 + x > 20
15 + x > 20
x > 20 – 15
x > 5
He must get a 6 in the fourth throw to win the game.
Let the number obtained in fourth throw be x.
Now, Sum > 20
6 + 5 + 4 + x > 20
15 + x > 20
x > 20 – 15
x > 5
He must get a 6 in the fourth throw to win the game.
The cost and revenue functions of a product are given by C(x) = 2x + 400 and R(x) = 6x + 20 respectively, where x is the number of items produced by the manufacturer. The minimum number of items that the manufacturer must sell to realize some profit is- a)95
- b)96
- c)105
- d)100
Correct answer is option 'B'. Can you explain this answer?
The cost and revenue functions of a product are given by C(x) = 2x + 400 and R(x) = 6x + 20 respectively, where x is the number of items produced by the manufacturer. The minimum number of items that the manufacturer must sell to realize some profit is
a)
95
b)
96
c)
105
d)
100
|
|
Lavanya Menon answered |
Profit = revenue - cost
But given profit > 0
therefore R(x) - C(x) >0
therefore 4x - 380 >0
therefore 4x > 380
therefore x>95 manufacturer must sell 96 items.
But given profit > 0
therefore R(x) - C(x) >0
therefore 4x - 380 >0
therefore 4x > 380
therefore x>95 manufacturer must sell 96 items.
Solve the inequality 3 −− 2x ≤ 9- a)x ≥ − 6
- b)x ≥ − 3
- c)x ≤ − 3
- d)none of these.
Correct answer is option 'B'. Can you explain this answer?
Solve the inequality 3 −− 2x ≤ 9
a)
x ≥ − 6
b)
x ≥ − 3
c)
x ≤ − 3
d)
none of these.
|
|
Raghav Bansal answered |
3 − 2x ≤ 9
-2x ≤ 6
-x ≤ 3
-3 ≤ x
-2x ≤ 6
-x ≤ 3
-3 ≤ x
The solution to |3x – 1| + 1 < 3 is- a)2 < x < 3/4
- b)-1/3 < x < 1
- c)-1/3 < x < 1/4
- d)-3 < x < 3
Correct answer is option 'B'. Can you explain this answer?
The solution to |3x – 1| + 1 < 3 is
a)
2 < x < 3/4
b)
-1/3 < x < 1
c)
-1/3 < x < 1/4
d)
-3 < x < 3
|
|
Neha Joshi answered |
|3x - 1| + 1 < 3
|3x -1| < 2
Opening mod, we get
3x - 1 < 2, -3x + 1 > 2
3x < 3, -3x > 1
x < 1, x > -1/3
-1/3 < x < 1
|3x -1| < 2
Opening mod, we get
3x - 1 < 2, -3x + 1 > 2
3x < 3, -3x > 1
x < 1, x > -1/3
-1/3 < x < 1
Find the solution for the pair of inequations x > 1 and x < -1- a)no solution
- b)x < -1
- c)-1 < x < 1
- d)x > 1
Correct answer is option 'A'. Can you explain this answer?
Find the solution for the pair of inequations x > 1 and x < -1
a)
no solution
b)
x < -1
c)
-1 < x < 1
d)
x > 1
|
|
Pooja Shah answered |
x > 1 means x is greater than 1 in this equation the values of x can be 2,3,4,5.............∞
x < -1 means is less than -1 therefore values of x can be -2, -3, -4, -5............ -∞
So no common point for x
Hence there will be no solution
x < -1 means is less than -1 therefore values of x can be -2, -3, -4, -5............ -∞
So no common point for x
Hence there will be no solution
Which one of them is the solution for x, when x is integer and 12 x > 30?- a)-1
- b)3
- c)1
- d)0
Correct answer is option 'B'. Can you explain this answer?
Which one of them is the solution for x, when x is integer and 12 x > 30?
a)
-1
b)
3
c)
1
d)
0
|
Gaurav Kumar answered |
12x > 30
x > 30/12
x > 2.5
x is an integer. So, minimum value of x is 3.
x > 30/12
x > 2.5
x is an integer. So, minimum value of x is 3.
If 5x+6<2x-3, then- a)x>-3
- b)x<-3
- c)x>3
- d)x<3
Correct answer is option 'B'. Can you explain this answer?
If 5x+6<2x-3, then
a)
x>-3
b)
x<-3
c)
x>3
d)
x<3
|
|
Rohit Jain answered |
5x+6<2x-3
5x-2x < -3-6
3x < -9
x<-3
5x-2x < -3-6
3x < -9
x<-3
IQ of a person is given by the formula 
where MA is mental age and CA is the chronological age. If 
for a group of 12 years old children, find the range of their mental age.- a)9.6 to 14.8
- b)9.6 and 16.8
- c)9.6 to 16.8
- d)14.8 to 16.8
Correct answer is option 'C'. Can you explain this answer?
IQ of a person is given by the formula
where MA is mental age and CA is the chronological age. If for a group of 12 years old children, find the range of their mental age.
for a group of 12 years old children, find the range of their mental age.a)
9.6 to 14.8
b)
9.6 and 16.8
c)
9.6 to 16.8
d)
14.8 to 16.8
|
|
Raghav Bansal answered |
IQ = (MA*100)/CA
and 80 ≤ IQ ≤ 140
80 ≤ (MA*100)/CA ≤ 140
80 ≤ (MA*100)/12 ≤ 140
960 ≤ (MA*100) ≤ 1680
9.6 ≤ MA ≤ 16.8
and 80 ≤ IQ ≤ 140
80 ≤ (MA*100)/CA ≤ 140
80 ≤ (MA*100)/12 ≤ 140
960 ≤ (MA*100) ≤ 1680
9.6 ≤ MA ≤ 16.8
For a student to qualify for a certain course, the average of his marks in the permitted 3 attempts must be more than 60. His first two attempts yielded only 45 and 62 marks respectively. What is the minimum score required in the third attempt to qualify?- a)73
- b)91
- c)83
- d)89
Correct answer is option 'A'. Can you explain this answer?
For a student to qualify for a certain course, the average of his marks in the permitted 3 attempts must be more than 60. His first two attempts yielded only 45 and 62 marks respectively. What is the minimum score required in the third attempt to qualify?
a)
73
b)
91
c)
83
d)
89
|
|
Tejas Verma answered |
No of attempts = 3
Average = (45+62+x)/3=60
x=73
Average = (45+62+x)/3=60
x=73
Which of the following is correct ?- a)If 0 > -7 , then 0 < 7
- b)If 8 > 1 , then -8 > -1
- c)If -4 < 7 , then 4 < - 7
- d)If -2 < 5 , then 2 > 5
Correct answer is option 'A'. Can you explain this answer?
Which of the following is correct ?
a)
If 0 > -7 , then 0 < 7
b)
If 8 > 1 , then -8 > -1
c)
If -4 < 7 , then 4 < - 7
d)
If -2 < 5 , then 2 > 5
|
|
Aditi Nambiar answered |
The statement you provided is incomplete. Please provide the complete statement for accurate evaluation.
Find the value of x which satisfies 5x – 3 < 7, where x is a natural number.- a){1, 2}
- b)1
- c)(1,∞)
- d)[1,∞)
Correct answer is option 'B'. Can you explain this answer?
Find the value of x which satisfies 5x – 3 < 7, where x is a natural number.
a)
{1, 2}
b)
1
c)
(1,∞)
d)
[1,∞)
|
Vikas Saini answered |
From inequality it comes out to be x less than 2 but it is also a natural no. so 1,2 is the. only answer in this case
Which of the following is not a linear inequality?- a)ax2 + bx + c < 0
- b)ax + by + c ≥ 0
- c)ax + b < 0
- d)ax + by + c ≤ 0
Correct answer is option 'A'. Can you explain this answer?
Which of the following is not a linear inequality?
a)
ax2 + bx + c < 0
b)
ax + by + c ≥ 0
c)
ax + b < 0
d)
ax + by + c ≤ 0
|
Shubham Rajput answered |
A is a quadratic equation not a linear equality because square of a function can't be negative
In the first four papers each of 100 marks, Rishi got 95, 72, 73, 83 marks. If he wants an average of greater than or equal to 75 marks and less than 80 marks, find the range of marks he should score in the fifth paper .
- a)52 ≤ × < 77
- b)73 < × < 100
- c)25 < × < 75
- d)75 < × < 80
Correct answer is option 'A'. Can you explain this answer?
In the first four papers each of 100 marks, Rishi got 95, 72, 73, 83 marks. If he wants an average of greater than or equal to 75 marks and less than 80 marks, find the range of marks he should score in the fifth paper .
a)
52 ≤ × < 77
b)
73 < × < 100
c)
25 < × < 75
d)
75 < × < 80
|
Ayush Joshi answered |
Suppose Richard scores x in the fifth paper. Then,75
Solution of 2x + 2|x| ≥ 2√2- a)(-∞, log2(√2 + 1))
- b)(0, 8)
- c)(1/2, log2(√2 - 1))
- d)(-∞, log2(√2 - 1)) ∪ [1/2, ∞)
Correct answer is option 'D'. Can you explain this answer?
Solution of 2x + 2|x| ≥ 2√2
a)
(-∞, log2(√2 + 1))
b)
(0, 8)
c)
(1/2, log2(√2 - 1))
d)
(-∞, log2(√2 - 1)) ∪ [1/2, ∞)
|
Ruchi Basak answered |
Understanding the Inequality
To solve the inequality 2x + 2|x| ≥ 2√2, we need to consider two cases based on the definition of the absolute value.
Case 1: x ≥ 0
- Here, |x| = x. Thus, the inequality becomes:
- 2x + 2x ≥ 2√2
- 4x ≥ 2√2
- x ≥ √2/2
Case 2: x < />
- In this case, |x| = -x. The inequality becomes:
- 2x - 2x ≥ 2√2
- 0 ≥ 2√2, which is never true.
Hence, for x < 0,="" there="" are="" no="" solutions.="" />Combining Solutions
From Case 1, we have that x ≥ √2/2.
Now, we need to identify the range of values that this corresponds to in terms of logarithmic form. We can express √2/2 as log2(√2 - 1) to find the boundaries of the solution.
Final Solution Set
Thus, the solution to the inequality is:
- x is in the range of (-∞, log2(√2 - 1)) ∪ [√2/2, ∞).
This matches option 'D': (-∞, log2(√2 - 1)) ∪ [1/2, ∞).
Conclusion
- The solution involves understanding how to break the absolute value and analyze each case.
- The final answer confirms that there are no solutions for negative x, while positive x provides a clear range starting from √2/2.
This detailed breakdown explains why option 'D' is correct.
To solve the inequality 2x + 2|x| ≥ 2√2, we need to consider two cases based on the definition of the absolute value.
Case 1: x ≥ 0
- Here, |x| = x. Thus, the inequality becomes:
- 2x + 2x ≥ 2√2
- 4x ≥ 2√2
- x ≥ √2/2
Case 2: x < />
- In this case, |x| = -x. The inequality becomes:
- 2x - 2x ≥ 2√2
- 0 ≥ 2√2, which is never true.
Hence, for x < 0,="" there="" are="" no="" solutions.="" />Combining Solutions
From Case 1, we have that x ≥ √2/2.
Now, we need to identify the range of values that this corresponds to in terms of logarithmic form. We can express √2/2 as log2(√2 - 1) to find the boundaries of the solution.
Final Solution Set
Thus, the solution to the inequality is:
- x is in the range of (-∞, log2(√2 - 1)) ∪ [√2/2, ∞).
This matches option 'D': (-∞, log2(√2 - 1)) ∪ [1/2, ∞).
Conclusion
- The solution involves understanding how to break the absolute value and analyze each case.
- The final answer confirms that there are no solutions for negative x, while positive x provides a clear range starting from √2/2.
This detailed breakdown explains why option 'D' is correct.
The longest side of a triangle is three times the shortest side and the third side is 2cm shorter than the longest side if the perimeter of the triangles at least 61cm, find the minimum length of the shortest side.- a)16 cm
- b)11 cm
- c)61 cm
- d)9 cm
Correct answer is option 'D'. Can you explain this answer?
The longest side of a triangle is three times the shortest side and the third side is 2cm shorter than the longest side if the perimeter of the triangles at least 61cm, find the minimum length of the shortest side.
a)
16 cm
b)
11 cm
c)
61 cm
d)
9 cm
|
Manish Aggarwal answered |
Let the shortest side of the triangle be x cm.
Then, the longest side will be 3x and the third side will be 3x − 2.
∴ Perimeter of the triangle ≥ 61
⇒ x + 3x + 3x − 2 ≥ 61
⇒ 7x ≥ 61 + 2
Dividing throughout by ⇒ x ≥ 9( Dividing throughout by 7)
Hence, the minumum length of the shortest side is 9 cm.
Then, the longest side will be 3x and the third side will be 3x − 2.
∴ Perimeter of the triangle ≥ 61
⇒ x + 3x + 3x − 2 ≥ 61
⇒ 7x ≥ 61 + 2
Dividing throughout by ⇒ x ≥ 9( Dividing throughout by 7)
Hence, the minumum length of the shortest side is 9 cm.
The inequations -4x+1≥0 and 3-4x<0 have the common solutions given by
- a){}
- b){0,2}
- c){-2,2}
- d)None of these
Correct answer is option 'D'. Can you explain this answer?
The inequations -4x+1≥0 and 3-4x<0 have the common solutions given by
a)
{}
b)
{0,2}
c)
{-2,2}
d)
None of these
|
Abhishek Dasgupta answered |
-4x+1≥0 and 3-4x<0
1 ≥ 4x and 3 < 4x
1/4 ≥ x and 3/4 < x
so x ∈ (-∞,1/4] U (3/4 , ∞ )
A man wants to cut three lengths from a single piece of board of length 91cm. The second length is to be 3 cm longer than the shortest and third length is to be twice as long as the shortest. What are the possible lengths for the shortest board if the third piece is to be at least 5 cm longer than the second?- a)5 ≤ × ≤ 91
- b)3 ≤ × ≤ 5
- c)3 ≤ × ≤ 91
- d)8 ≤ × ≤ 22
Correct answer is option 'D'. Can you explain this answer?
A man wants to cut three lengths from a single piece of board of length 91cm. The second length is to be 3 cm longer than the shortest and third length is to be twice as long as the shortest. What are the possible lengths for the shortest board if the third piece is to be at least 5 cm longer than the second?
a)
5 ≤ × ≤ 91
b)
3 ≤ × ≤ 5
c)
3 ≤ × ≤ 91
d)
8 ≤ × ≤ 22
|
|
Ashwin Verma answered |
Let x be the length of the shortest board.
The second length is x+3.
The third length is 2x.
The third length is at least 5 cm longer than the second: 2x ≥ (x+3)+5
2x ≥ x+8
x ≥ 8
The possible lengths for the shortest board are x ≥ 8. Answer: \boxed{8}.
The second length is x+3.
The third length is 2x.
The third length is at least 5 cm longer than the second: 2x ≥ (x+3)+5
2x ≥ x+8
x ≥ 8
The possible lengths for the shortest board are x ≥ 8. Answer: \boxed{8}.
If a < b then -a ______ - b- a)–a < -b
- b)–a ≤ -b
- c)–a ≥ -b
- d)-a > -b
Correct answer is option 'D'. Can you explain this answer?
If a < b then -a ______ - b
a)
–a < -b
b)
–a ≤ -b
c)
–a ≥ -b
d)
-a > -b
|
|
Bhargavi Sen answered |
If a person ate only fruits and vegetables for a week, they may experience several potential benefits. These can include:
1. Increased nutrient intake: Fruits and vegetables are rich in essential vitamins, minerals, and antioxidants. Eating a variety of fruits and vegetables can provide a wide range of nutrients necessary for optimal health.
2. Improved digestion: Fruits and vegetables are high in fiber, which can promote healthy digestion and prevent constipation. The high water content in many fruits and vegetables can also help maintain hydration and promote regular bowel movements.
3. Enhanced immune function: Fruits and vegetables are packed with immune-boosting nutrients like vitamin C, beta-carotene, and antioxidants. These nutrients can support a strong immune system, helping to fight off infections and diseases.
4. Weight management: Fruits and vegetables are generally low in calories and high in fiber, which can help with weight management. The high fiber content can promote feelings of fullness and reduce overall calorie intake.
5. Improved skin health: Fruits and vegetables contain antioxidants that can help protect the skin from damage caused by free radicals. Additionally, the high water content in fruits and vegetables can contribute to hydrated and healthy-looking skin.
6. Reduced risk of chronic diseases: A diet rich in fruits and vegetables has been associated with a lower risk of chronic diseases such as heart disease, certain types of cancer, and diabetes. The various nutrients and antioxidants found in fruits and vegetables may help protect against these conditions.
However, it is important to note that a balanced diet should include a variety of food groups to ensure all essential nutrients are obtained. It is recommended to consult with a healthcare professional or registered dietitian before making any significant changes to your diet.
1. Increased nutrient intake: Fruits and vegetables are rich in essential vitamins, minerals, and antioxidants. Eating a variety of fruits and vegetables can provide a wide range of nutrients necessary for optimal health.
2. Improved digestion: Fruits and vegetables are high in fiber, which can promote healthy digestion and prevent constipation. The high water content in many fruits and vegetables can also help maintain hydration and promote regular bowel movements.
3. Enhanced immune function: Fruits and vegetables are packed with immune-boosting nutrients like vitamin C, beta-carotene, and antioxidants. These nutrients can support a strong immune system, helping to fight off infections and diseases.
4. Weight management: Fruits and vegetables are generally low in calories and high in fiber, which can help with weight management. The high fiber content can promote feelings of fullness and reduce overall calorie intake.
5. Improved skin health: Fruits and vegetables contain antioxidants that can help protect the skin from damage caused by free radicals. Additionally, the high water content in fruits and vegetables can contribute to hydrated and healthy-looking skin.
6. Reduced risk of chronic diseases: A diet rich in fruits and vegetables has been associated with a lower risk of chronic diseases such as heart disease, certain types of cancer, and diabetes. The various nutrients and antioxidants found in fruits and vegetables may help protect against these conditions.
However, it is important to note that a balanced diet should include a variety of food groups to ensure all essential nutrients are obtained. It is recommended to consult with a healthcare professional or registered dietitian before making any significant changes to your diet.
Given that x is an integer, find the values of x which satisfy both 2x + 3 > 7 and x + 4 < 10- a)4 , 5
- b)3
- c)4
- d)3 , 4 , 5
Correct answer is option 'D'. Can you explain this answer?
Given that x is an integer, find the values of x which satisfy both 2x + 3 > 7 and x + 4 < 10
a)
4 , 5
b)
3
c)
4
d)
3 , 4 , 5
|
|
Learners Habitat answered |
We are given two inequalities:
- 2x + 3 > 7
- x + 4 < 10
We need to find the integer values of x that satisfy both inequalities.
Step 1: Solve the first inequality 2x + 3 > 7
Subtract 3 from both sides:
2x > 7 - 3
Simplifying:
2x > 4
Divide both sides by 2:
x > 2
So, the first inequality gives us:
x > 2
Step 2: Solve the second inequality x + 4 < 10
Subtract 4 from both sides:
x < 10 - 4
Simplifying:
x < 6
Step 3: Combine the results
From the first inequality, we know x > 2.
From the second inequality, we know x < 6.
Thus, the solution for x must satisfy both:
2 < x < 6
The integer values of x that lie between 2 and 6 are:
x = 3, 4, 5
The correct option is: D: 3, 4, 5
Step 1: Solve the first inequality 2x + 3 > 7
Subtract 3 from both sides:
2x > 7 - 3
Simplifying:
2x > 4
Divide both sides by 2:
x > 2
So, the first inequality gives us:
x > 2
Step 2: Solve the second inequality x + 4 < 10
Subtract 4 from both sides:
x < 10 - 4
Simplifying:
x < 6
Step 3: Combine the results
From the first inequality, we know x > 2.
From the second inequality, we know x < 6.
Thus, the solution for x must satisfy both:
2 < x < 6
The integer values of x that lie between 2 and 6 are:
x = 3, 4, 5
The correct option is: D: 3, 4, 5
Identify the solution set for –( x – 3 ) + 5 – 2x- a)(−∞,−1)
- b)(−∞,−2)
- c)(−∞,−5)
- d)none of these
Correct answer is option 'B'. Can you explain this answer?
Identify the solution set for –( x – 3 ) + 5 – 2x
a)
(−∞,−1)
b)
(−∞,−2)
c)
(−∞,−5)
d)
none of these
|
|
Niharika Bose answered |
To solve the inequality 3(a - 6), we need to isolate the variable a.
First, distribute the 3 to both terms inside the parentheses:
3(a - 6) = 3a - 18
Now, we have the inequality 3a - 18.
We need to determine the range of values for a that satisfy this inequality.
To do this, we can add 18 to both sides of the inequality:
3a - 18 + 18 > 0 + 18
3a > 18
Next, divide both sides of the inequality by 3:
(3a)/3 > 18/3
a > 6
Therefore, the solution to the inequality 3(a - 6) is a > 6.
First, distribute the 3 to both terms inside the parentheses:
3(a - 6) = 3a - 18
Now, we have the inequality 3a - 18.
We need to determine the range of values for a that satisfy this inequality.
To do this, we can add 18 to both sides of the inequality:
3a - 18 + 18 > 0 + 18
3a > 18
Next, divide both sides of the inequality by 3:
(3a)/3 > 18/3
a > 6
Therefore, the solution to the inequality 3(a - 6) is a > 6.
If 3x + 22x ≥ 5x, then the solution set for x is:- a)(-∞, 2]
- b)[2, ∞)
- c)[0, 2]
- d){2}
Correct answer is option 'A'. Can you explain this answer?
If 3x + 22x ≥ 5x, then the solution set for x is:
a)
(-∞, 2]
b)
[2, ∞)
c)
[0, 2]
d)
{2}
|
KP Classes answered |
We have,
3x + 22x ≥ 5x
⇒ (3/5)x + (4/5)x ≥ 1
⇒ (3/5)x + (4/5)x ≥ (3/5)2 + (4/5)2
⇒ x ≤ 2 ⇒ x ∈ (-∞, 2]
[If ax + bx ≥ 1 and a2 + b2 = 1], then x ∈ (-∞, 2)
3x + 22x ≥ 5x
⇒ (3/5)x + (4/5)x ≥ 1
⇒ (3/5)x + (4/5)x ≥ (3/5)2 + (4/5)2
⇒ x ≤ 2 ⇒ x ∈ (-∞, 2]
[If ax + bx ≥ 1 and a2 + b2 = 1], then x ∈ (-∞, 2)
Find the pairs of consecutive even positive integers both of which are smaller than 10 and their sum of more than 11- a)(4, 8)
- b)(6, 8)
- c)(6, 8) and (4, 8)
- d)(6, 4)(4, 2)
Correct answer is option 'B'. Can you explain this answer?
Find the pairs of consecutive even positive integers both of which are smaller than 10 and their sum of more than 11
a)
(4, 8)
b)
(6, 8)
c)
(6, 8) and (4, 8)
d)
(6, 4)(4, 2)
|
Geetika Shah answered |
Let x be the smaller of the two consecutive even positive integers .
Then the other integer is x+2.
Since both the integers are smaller than 10,x<10 ....(1)
Also the sum of the two integers is more than 11.
x+(x+2)>11
⇒ 2x+2>11
⇒ 2x>11−2
⇒ 2x>9
⇒ x>9/2
⇒ x>4.5....(2)
From (1) and (2) we obtain 4.5>x>11
Since x is an even number, x can take the values 6,8 and 10.
Thus the required possible pairs are (6,8).
Then the other integer is x+2.
Since both the integers are smaller than 10,x<10 ....(1)
Also the sum of the two integers is more than 11.
x+(x+2)>11
⇒ 2x+2>11
⇒ 2x>11−2
⇒ 2x>9
⇒ x>9/2
⇒ x>4.5....(2)
From (1) and (2) we obtain 4.5>x>11
Since x is an even number, x can take the values 6,8 and 10.
Thus the required possible pairs are (6,8).
The longest side of a triangle is 4 times the shortest side and the third side is 4 cm shorter than the longest side. If the perimeter of the triangle is at least 59 cm, then the minimum length of the shortest side is- a)9 cm
- b)8 cm
- c)7 cm
- d)6 cm
Correct answer is option 'C'. Can you explain this answer?
The longest side of a triangle is 4 times the shortest side and the third side is 4 cm shorter than the longest side. If the perimeter of the triangle is at least 59 cm, then the minimum length of the shortest side is
a)
9 cm
b)
8 cm
c)
7 cm
d)
6 cm
|
|
Nitin Verma answered |
Problem Analysis:
Let the shortest side of the triangle be x cm.
According to the given information, the longest side is 4 times the shortest side, so it will be 4x cm.
The third side is 4 cm shorter than the longest side, so it will be (4x - 4) cm.
The perimeter of a triangle is the sum of the lengths of all three sides.
So, the perimeter of this triangle will be x + 4x + (4x - 4) = 9x - 4 cm.
Solution:
To find the minimum length of the shortest side, we need to find the minimum value of x for which the perimeter is at least 59 cm.
Let's set up the inequality:
9x - 4 ≥ 59
9x ≥ 63
x ≥ 7
Therefore, the minimum length of the shortest side is 7 cm.
Hence, the correct answer is option 'C'.
Let the shortest side of the triangle be x cm.
According to the given information, the longest side is 4 times the shortest side, so it will be 4x cm.
The third side is 4 cm shorter than the longest side, so it will be (4x - 4) cm.
The perimeter of a triangle is the sum of the lengths of all three sides.
So, the perimeter of this triangle will be x + 4x + (4x - 4) = 9x - 4 cm.
Solution:
To find the minimum length of the shortest side, we need to find the minimum value of x for which the perimeter is at least 59 cm.
Let's set up the inequality:
9x - 4 ≥ 59
9x ≥ 63
x ≥ 7
Therefore, the minimum length of the shortest side is 7 cm.
Hence, the correct answer is option 'C'.
The number of real values of parameter k for which (log16x)2−log16x+log16k=0 will have exactly one solution is- a)0
- b)2
- c)1
- d)4
Correct answer is option 'C'. Can you explain this answer?
The number of real values of parameter k for which (log16x)2−log16x+log16k=0 will have exactly one solution is
a)
0
b)
2
c)
1
d)
4
|
Nipuns Institute answered |
Let log16x=t
Hence, t2−t+log16k=0 and change k>0
The equation has exactly one solution. Hence, the discriminant must be zero.
Δ=1−4log16k
⇒0=1−4log16k
⇒log16k=1/4
⇒k=2
So, number of real values of k is 1 when k=2.
Hence, t2−t+log16k=0 and change k>0
The equation has exactly one solution. Hence, the discriminant must be zero.
Δ=1−4log16k
⇒0=1−4log16k
⇒log16k=1/4
⇒k=2
So, number of real values of k is 1 when k=2.
x2 ―3|x| + 2 < 0, then x belongs to- a)(1, 2)
- b) ( ― 2, ― 1)
- c) ( ―2, ― 1) ∪ (1,2)
- d)( ― 3, 5)
Correct answer is option 'C'. Can you explain this answer?
x2 ―3|x| + 2 < 0, then x belongs to
a)
(1, 2)
b)
( ― 2, ― 1)
c)
( ―2, ― 1) ∪ (1,2)
d)
( ― 3, 5)
|
|
Nipuns Institute answered |
x2 ― 3|x| + 2 < 0
⇒|x|
2 ― 3|x| + 2 < 0
⇒(|x| ― 1)(|x| ― 2) < 0
⇒1 < |x| < 2
⇒ ― 2 < x < ―1 or 1 < x < 2
∴ X E ( ―2, ― 1) ∪ (1,2)
⇒|x|
2 ― 3|x| + 2 < 0
⇒(|x| ― 1)(|x| ― 2) < 0
⇒1 < |x| < 2
⇒ ― 2 < x < ―1 or 1 < x < 2
∴ X E ( ―2, ― 1) ∪ (1,2)
If ab = 4 (a, b ∈ ℝ⁺), then- a)a + b ≤ 4
- b)a + b = 4
- c)a + b ≥ 4
- d)None of these
Correct answer is option 'C'. Can you explain this answer?
If ab = 4 (a, b ∈ ℝ⁺), then
a)
a + b ≤ 4
b)
a + b = 4
c)
a + b ≥ 4
d)
None of these
|
Bibek Desai answered |
Understanding the Problem
Given that ab = 4, we need to analyze the implications for the sum a + b. The variables a and b are positive real numbers, as indicated by the notation.
Using AM-GM Inequality
The Arithmetic Mean-Geometric Mean (AM-GM) inequality states that for any two non-negative numbers x and y, the following holds:
- (x + y) / 2 ≥ √(xy)
Applying this to our variables a and b:
- (a + b) / 2 ≥ √(ab)
Since ab = 4, we have:
- (a + b) / 2 ≥ √(4)
- (a + b) / 2 ≥ 2
Multiplying both sides by 2 gives:
- a + b ≥ 4
Implications of the Result
This implies that the sum a + b is always greater than or equal to 4 when ab = 4. Hence:
- Option C (a + b ≥ 4) is correct.
Why Other Options are Incorrect
- Option A (a + b ≤ 4): This suggests that the sum could be less than or equal to 4, which contradicts the AM-GM result.
- Option B (a + b = 4): While a + b can equal 4, this is not guaranteed for all positive values of a and b that satisfy ab = 4.
- Option D (None of these): Since we identified Option C as correct, this option is invalid.
Conclusion
Therefore, the correct answer is option C, a + b ≥ 4, as validated through mathematical reasoning using the AM-GM inequality. This ensures that the relationship between a and b when their product is fixed at 4 will always yield a sum that is at least 4.
Given that ab = 4, we need to analyze the implications for the sum a + b. The variables a and b are positive real numbers, as indicated by the notation.
Using AM-GM Inequality
The Arithmetic Mean-Geometric Mean (AM-GM) inequality states that for any two non-negative numbers x and y, the following holds:
- (x + y) / 2 ≥ √(xy)
Applying this to our variables a and b:
- (a + b) / 2 ≥ √(ab)
Since ab = 4, we have:
- (a + b) / 2 ≥ √(4)
- (a + b) / 2 ≥ 2
Multiplying both sides by 2 gives:
- a + b ≥ 4
Implications of the Result
This implies that the sum a + b is always greater than or equal to 4 when ab = 4. Hence:
- Option C (a + b ≥ 4) is correct.
Why Other Options are Incorrect
- Option A (a + b ≤ 4): This suggests that the sum could be less than or equal to 4, which contradicts the AM-GM result.
- Option B (a + b = 4): While a + b can equal 4, this is not guaranteed for all positive values of a and b that satisfy ab = 4.
- Option D (None of these): Since we identified Option C as correct, this option is invalid.
Conclusion
Therefore, the correct answer is option C, a + b ≥ 4, as validated through mathematical reasoning using the AM-GM inequality. This ensures that the relationship between a and b when their product is fixed at 4 will always yield a sum that is at least 4.
What is the solution set for 
- a)(2 , ∞)
- b)(0 , 2)
- c)(-∞ , - 2)
- d)none of these
Correct answer is option 'A'. Can you explain this answer?
What is the solution set for
a)
(2 , ∞)
b)
(0 , 2)
c)
(-∞ , - 2)
d)
none of these
|
Mani Kandan answered |
X=-2,2
x+2 ×-2 x+2/x-2
(-infinity,-2) (+) (-) ( - )
(-2,2) ( + ) ( - ) ( - )
(2,infinity ) ( + ) ( + ) ( +)
so answer is A=(2,infinity )
x+2 ×-2 x+2/x-2
(-infinity,-2) (+) (-) ( - )
(-2,2) ( + ) ( - ) ( - )
(2,infinity ) ( + ) ( + ) ( +)
so answer is A=(2,infinity )
The set of admissible values of x such that (2x + 3) / (2x - 9) < 0 is:- a)(-∞, -3/2) ∪ (9/2, ∞)
- b)(-∞, 0) ∪ (9/2, ∞)
- c)(-3/2, 0)
- d)(-3/2, 9/2)
Correct answer is option 'D'. Can you explain this answer?
The set of admissible values of x such that (2x + 3) / (2x - 9) < 0 is:
a)
(-∞, -3/2) ∪ (9/2, ∞)
b)
(-∞, 0) ∪ (9/2, ∞)
c)
(-3/2, 0)
d)
(-3/2, 9/2)
|
Samridhi Bajaj answered |
Understanding the Inequality
To solve the inequality (2x + 3) / (2x - 9) < 0,="" we="" need="" to="" determine="" when="" the="" expression="" is="" negative.="" />Finding Critical Points
1. Numerator: Set 2x + 3 = 0
- 2x = -3
- x = -3/2
2. Denominator: Set 2x - 9 = 0
- 2x = 9
- x = 9/2
These points, x = -3/2 and x = 9/2, help us to divide the number line into intervals.
Analyzing the Intervals
We will test values from the intervals created by the critical points:
1. Interval (-∞, -3/2): Choose x = -2
- (2(-2) + 3) / (2(-2) - 9) = (-4 + 3) / (-4 - 9) = -1 / -13 > 0 (not valid)
2. Interval (-3/2, 9/2): Choose x = 0
- (2(0) + 3) / (2(0) - 9) = (3) / (-9) = -1/3 < 0="" (valid)="" 3.="" />interval="" (9/2,="" ∞):="" choose="" x="5" -="" (2(5)="" +="" 3)="" (2(5)="" -="" 9)="(10" +="" 3)="" (10="" -="" 9)="13" 1=""> 0 (not valid)
Conclusion
The expression (2x + 3) / (2x - 9) is negative in the interval (-3/2, 9/2). Therefore, the set of admissible values of x such that (2x + 3) / (2x - 9) < 0="" is:="" />Final Answer
(-3/2, 9/2)
This corresponds to option D.
To solve the inequality (2x + 3) / (2x - 9) < 0,="" we="" need="" to="" determine="" when="" the="" expression="" is="" negative.="" />Finding Critical Points
1. Numerator: Set 2x + 3 = 0
- 2x = -3
- x = -3/2
2. Denominator: Set 2x - 9 = 0
- 2x = 9
- x = 9/2
These points, x = -3/2 and x = 9/2, help us to divide the number line into intervals.
Analyzing the Intervals
We will test values from the intervals created by the critical points:
1. Interval (-∞, -3/2): Choose x = -2
- (2(-2) + 3) / (2(-2) - 9) = (-4 + 3) / (-4 - 9) = -1 / -13 > 0 (not valid)
2. Interval (-3/2, 9/2): Choose x = 0
- (2(0) + 3) / (2(0) - 9) = (3) / (-9) = -1/3 < 0="" (valid)="" 3.="" />interval="" (9/2,="" ∞):="" choose="" x="5" -="" (2(5)="" +="" 3)="" (2(5)="" -="" 9)="(10" +="" 3)="" (10="" -="" 9)="13" 1=""> 0 (not valid)
Conclusion
The expression (2x + 3) / (2x - 9) is negative in the interval (-3/2, 9/2). Therefore, the set of admissible values of x such that (2x + 3) / (2x - 9) < 0="" is:="" />Final Answer
(-3/2, 9/2)
This corresponds to option D.
x - 1)(x² - 5x + 7) < (x - 1), then x belongs to- a)(1, 2) ∪ (3, ∞)
- b)(2, 3)
- c)(-∞, 1) ∪ (2, 3)
- d)None of these
Correct answer is option 'C'. Can you explain this answer?
x - 1)(x² - 5x + 7) < (x - 1), then x belongs to
a)
(1, 2) ∪ (3, ∞)
b)
(2, 3)
c)
(-∞, 1) ∪ (2, 3)
d)
None of these
|
|
KP Classes answered |
(x - 1)(x² - 5x + 7) < (x - 1)
⇒ (x - 1)(x² - 5x + 6) < 0
⇒ (x - 1)(x - 2)(x - 3) < 0
∴ x ∈ (-∞, 1) ∪ (2, 3)
⇒ (x - 1)(x² - 5x + 6) < 0
⇒ (x - 1)(x - 2)(x - 3) < 0
∴ x ∈ (-∞, 1) ∪ (2, 3)
What is the solution set for 
- a)(5 , 6)
- b)(- 6 , 6)
- c)(- 6 , 0)
- d)none of these
Correct answer is option 'C'. Can you explain this answer?
What is the solution set for
a)
(5 , 6)
b)
(- 6 , 6)
c)
(- 6 , 0)
d)
none of these
|
Tarun Kaushik answered |
Given Inequality:
Step 1: Break down the inequality
We need to solve both parts of the compound inequality:
Step 2: Solve each part separately
Step 2: Solve each part separately
Multiply both sides of the inequality by 2 to get rid of the denominator:
0 < -x
Now, multiply both sides by -1 (which reverses the inequality):
x < 0
Multiply both sides by 2 to eliminate the denominator:
-x < 6
Multiply both sides by -1 (which reverses the inequality):
x > -6
Step 3: Combine the results
From both inequalities, we now have:
-6 < x < 0
The solution set is x ∈ (−6, 0), which corresponds to Option C.
0 < -x
Now, multiply both sides by -1 (which reverses the inequality):
x < 0
Multiply both sides by 2 to eliminate the denominator:
-x < 6
Multiply both sides by -1 (which reverses the inequality):
x > -6
Step 3: Combine the results
From both inequalities, we now have:
-6 < x < 0
The solution set is x ∈ (−6, 0), which corresponds to Option C.
The graphical solution of -1 < x < 2 on number line is- a)
- b)
- c)
- d)
Correct answer is option 'D'. Can you explain this answer?
The graphical solution of -1 < x < 2 on number line is
a)
b)
c)
d)
|
|
Mohit Mittal answered |
D is correct as x between -1 and 2 and excluding -1 and 2
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