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Equations and Inequations Questions

281. If x^2 -7x + 10 < 0, then which of the following option satisfy the inequation ? 




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Correct Ans:Both (a) and (b)
Explanation:
x^2 -7x + 10
=> x^2 - 5x -2x + 10 = 0
=> x( x - 5) - 2 ( x -5) = 0
=> ( x - 2) ( x - 5) = 0
Thus x => 2 , 5
=>( x- 2 ) = 0
=> x = 2
=> ( x - 5) = 0
=> x = 5
=> x = 2 , 5

Thus both a & b gives value of x.
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282. If x + 1/x = 21, then find the value of x^2 + 1/x^2 = ?




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Correct Ans:439
Explanation:
Given x + (1 / x) = 21
Squaring on both sides, we get
=> [x + (1 / x)]^2 = 21 * 21
By Formula: (a + b)^2 = a^2 + b^2 + 2ab
=> x^2 + (1/x)^2 + 2 * x * (1/x) = 441
=> x^2 + (1/x)^2 + 2 = 441
=> x^2 + (1/x)^2 = 441 - 2
=> x^2 + 1/x^2 = 439
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283. Which of the following values satisify the condition x^2 - 5x + 6 < 0




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Correct Ans:2, -3
Explanation:
x^ 2 - 5x + 6 = 0
= x^2 - 2x -3x + 6 = 0
= x( x -2) + 3 ( x - 2)
= (x + 3) + ( x - 2 ) = 0
=>(x + 3) = 0
=> x = -3
=> ( x - 2 ) = 0
=> x = 2
=> x = 2 , -3
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284. Find the solution for the equation x^2 - 49x + 600 = 0 




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Correct Ans:x = 25
Explanation:
On solving this eqn by using the formula for the eqn: ax2+bx+c= 0
=> X = [-b ± sqrt(b^2 – 4ac)] / 2a
Given equation: x^2 - 49x + 600 = 0
Here a = 1, b = -49, c = 600
=> X = [49 ± sqrt (49^2 – 4*1*600)] / 2
= [49 ± sqrt (2401 – 2400)] / 2
= [49 ± sqrt (1)] / 2
= (49 ± 1) / 2
= (49 + 1) / 2, (49 - 1) / 2
= 50 / 2, 48/ 2
=> X = 25, 24
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285. Sum of two digits in a given two digit number is 7. The number formed by reversing the digits exceed the original number by 27. Find the original number.




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Correct Ans:25
Explanation:
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286. Given x^2 - 16 < 0, then which of the following options are true? 




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Correct Ans:x < 4
Explanation:
x^2 - 16 < 0
=> x^2 < 16
=> x < sqrt(16)
=> x < ± 4
=> -4 < x < +4
Thus among given options, x < 4 is the correct answer.
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287. Find the possible values of x which satisfies the equation x^2 - 4 < 0 




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Correct Ans:either (a) or (c)
Explanation:
Given, x^2 - 4 < 0
=> x^2 < 4
=> x < sqrt(4)
=> x <± 2
=> -2 < x < 2
=> x > -2
OR x < 2
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288. Given | x - 7 | < 1 , then find the values what x can take?




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Correct Ans:6 < x < 8
Explanation:
Given | x - 7 | < 1
=> -1 < x-7 < 1
=> -1 + 7 < x < 1 + 7
=> 6 < x < 8
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289. Find the value obtained when we substitute x=8 in the equation x^2 - 13x + 42 




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Correct Ans:2
Explanation:
Given equation, x^2 - 13 x + 42
Sub x= 8, we get 8^2 - 13(8) + 42 = 64 - 104 + 42 = 106 - 104 = 2
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290. Which of the following options are possibly the solution for the equations x^2 - 26 x + 120 = 0  




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Correct Ans:None of the above 
Explanation:
x^2 - 26 x + 120 = 0
x ^ 2 + 4 x - 30 x = 0
x ^ 2 + 4x - 30 x = 0
x ( x + 4 ) - 30 ( x + 4 ) = 0
( x - 30 ) ( x + 4) = 0
x = 30 , x = -4.
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291. In a group of cows and hens, the number of legs are 14 more than twice the number of heads. Find the number of cows.




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Correct Ans:7
Explanation:
Let the number of cows be C and the number of hens be H.
Given, Number of legs = 14 + Number of heads
We know that, Cow has 4 legs, whereas Hen has 2 legs
=> 4C + 2H = 14 + 2(C + H)
=> 4C + 2H = 14 + 2C + 2H
=> 4C – 2C + 2H – 2H = 14
=> 2C = 14
=> C = 7
Therefore, the number of Cows = 7
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292. In a group of cows and hens, the number of legs are 12 more than twice the number of heads. Find the number of cows




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Correct Ans:6
Explanation:
Let the number of cows be x and the number of hens by y,
Then, 4x + 2y = 2(x+y) + 12
Solving the above equations,we get number of cows = 6
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293. A student got twice as many sums wrong as he got right. If he attempted 48 sums in all, how many did he solve correctly ?




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Correct Ans:16
Explanation:

Suppose the boy got x sums right and 2x sums wrong.
Then, x + 2x = 48
3x = 48
x = 16.

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294. A student got twice as many sums wrong as he got right. If he attempted 75 sums in all, how many did he solve correctly?




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Correct Ans:25
Explanation:

Suppose the boy got "x"sums right.
Given,A student got twice as many sums wrong as he got right
=> He got "2x" sums wrong.

Given,he attempted 75 sums in all
=> Right sums + wrong sums = 75 sums
=> x + 2x = 75
=> 3x = 75
=> x = 75 / 3
=> x = 25

Therefore,the boy got "25"sums right.

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295. Given expression x^3 -3x^2 + 2x, evaluate for x= 2 ?




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Correct Ans:0
Explanation:
x^3 - 3x^2 + 2x
Sub, x = 2, we get
2^3 - 3(2^2) + 2(2) = 8 - 12 + 4 = 0
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296. If x=2, then find the value of x^3  - 3x^2 +3x-6 




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Correct Ans:-4
Explanation:
x^3 - 3x^2 + 3x --6
Substituting x=2
=> 2^3 - 3(2^2) + 3(2) - 6
= 8 - 12 + 6 - 6
= -4
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297. If x + (1/x) = 3, then what will be the value of  x2 + (1/x2) = ? 




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Correct Ans:7
Explanation:
Given x + (1 / x) = 3
Squaring on both sides, we get
=> [x + (1 / x)]^2 = 3 * 3
By Formula: (a + b)^2 = a^2 + b^2 + 2ab
=> x^2 + (1/x)^2 + 2 * x * (1/x) = 9
=> x^2 + (1/x)^2 + 2 = 9
=> x^2 + (1/x)^2 = 9 - 2
=> x^2 + 1/x^2 = 7
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298. Given that |x-2| < 6, then find which of the following possible values x can take?




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Correct Ans:-4 <  x < 8
Explanation:
Given |x-2| < 6
=> -6 < x - 2 < 6
=> -6 + 2 < x < 6 + 2
=> -4 < x < 8
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299. Given that |x-2| < 3, then find which of the following possible values x can take?




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Correct Ans:-1 <  x < 5 
Explanation:
Given |x-2| < 3
=> -3 < x - 2 < 3
=> -3 + 2 < x < 3 + 2=> -1 < x < 5
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300. Given that |x-2| < 7, then find which of the following possible values x can take ?




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Correct Ans:-5 <  x < 9 
Explanation:
Given |x-2| < 7
=> -7 < x - 2 < 7
=> -7 + 2 < x < 7 + 2=> -5 < x < 9
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