1. If x^2 - 20 x + 75 < 0 then
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Correct Ans:5 < x < 15
Explanation:
x^2 - 20 x + 75 < 0
=> (x – 15) (x – 5) < 0
=> x < 15, 5
=> 5 < x < 15
SHOW ANSWER
Correct Ans:-10 < x < 14
Explanation:
Given, |x-2| < 12
=> -12 < x -2 < 12
=> -12 + 2 < x < 12 + 2
=> -10 < x < 14
3. Evaluate the expression x^2 - 4x + 5 when x = 1.
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Correct Ans:2
Explanation:
Subs x = 1 in the given expression: x^2 - 4x + 5
=> (1)^2 – 4(1) + 5
= 1 – 4 + 5
= 2
4. Evaluate: x^3 - 6x^2 + 11x - 7 when x = 2
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Correct Ans:-1
Explanation:
Substitute x = 2 in the given eqn: x^3 - 6x^2 + 11x – 7
=> (2)^3 – 6*(2)^2 + 11*(2) – 7
= 8 – 24 + 22 – 7
= -1
5. Evaluate the expression x^3 + 3x^2 + 3x + 1, when x = 10
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Correct Ans:1331
Explanation:
Subs x = 10 in the given eqn, x^3 + 3x^2 + 3x + 1
=> (10)^3 + 3 * (10^2) + (3 * 10) + 1
= 1000 + 300 + 30 + 1
= 1331
6. If x^2 - 10 x + 16 > 0 then which of the following option is correct?
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Correct Ans:x > 8
Explanation:
Given, x^2 - 10 x + 16 > 0
=> (x -8) (x-2) > 0
=> X > 8 and x > 2
Substitute values randomly taken from the interval and mark that option which satisfies the question.
On substituting x = 7 (which is greater than 2) in the eqn:x^2 - 10 x + 16, we get, -5 as the answer which is not greater than "0".
On substituting x = 9 (which is greater than 8) in the eqn:x^2 - 10 x + 16, we get, 7 as the answer which is greater than "0".
Hence, the correct answer is x > 8.
7. If x^2 + x + 1 = 0 then the number of real solutions is
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Correct Ans:0
Explanation:
8. If x^2 - 7x + 12 < 0 then find the solution of x ?
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Correct Ans:3 < x < 4
Explanation:
9. Evaluate t^3 - 3t^2 + 3t - 2 when t = 4
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Correct Ans:26
Explanation:
10. If x^2 - 2x + 1 < 0 then which of the following is true ?
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Correct Ans:No solution
Explanation:
11. Compute 23^3 - 12^3 - 11^3
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Correct Ans:9108
Explanation:
23^3 = 12167
12^3 = 1728
11^3= 1331
Now, 23^3 - 12^3 - 11^3= 12167 – 1728 – 1331
= 9108
12. Given that x + 1/x = 3 then evaluate x^3 + 1/x^3 = ?
SHOW ANSWER
Correct Ans:18
Explanation:
13. If x + 1 /x = 2 then evaluate x^23 + 1/x^32
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Correct Ans:2
Explanation:
Given, x + (1 /x) = 2
=> (x^2 + 1) / x = 2
=> (x^2 + 1) = 2x
=> x^2 - 2x + 1 = 0
=> (x – 1) (x – 1) = 0
=> x = 1
Subs x = 1 in the eqn x^23 + 1/x^32
=> (1)^23 + (1/1^32)
= 1 + 1
= 2
Therefore, x^23 + 1/x^32 = 2
14. Evaluate x^2 - 27 x + 180 at x = 15
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Correct Ans:0
Explanation:
Subs x = 15 in the given eqn x^2 - 27 x + 180
=> (15)^2 – 27*(15) + 180
=> 225 – 405 + 180
=> 405 – 405
=> 0
So, Ans is 0
15. Evaluate the expression x^3 + 3x +1 at x =2
SHOW ANSWER
Correct Ans:15
Explanation:
Subs x = 2 in the expression: x^3 + 3x +1
=> 2^3 + 3*2 + 1
=> 8 + 6 + 1
=> 15
16. The average of the linear factors of x^2 - 14x + 48 = 0 is
SHOW ANSWER
Correct Ans:x - 7
Explanation:
Given, x^2 – 14x + 48
=> x^2 –2 * (7) x + 48
=> x^2 –2 * (7) x + 48 + (7^2) - (7^2)
=> {x^2 –2 * (7) x + (7^2)} + 48 - 49
By the formula: (a^2 – 2ab – b^2) = (a – b)^2
=> (x – 7)^2 – 1
=> (x – 7)^2 – 1^2
By the formula: (a^2) – (b^2) = (a-b)(a+b)
=> (x – 7 + 1) (x – 7 – 1)
=> (x - 6) (x – 8)
The two linear factors are (x - 6) and (x – 8)
Average = (x – 6 + x – 8) / 2
= (2x -14)/2
= 2(x – 7) / 2
= (x – 7)
Therefore, Average = x – 7
17. If x^2 - 9 < 0 then which of the following options are true?
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Correct Ans:-3 < x < 3
Explanation:
Given, x^2 - 9 < 0
=> x^2 < 9
=> x < sqrt(9)
=> x < ± 3
=> -3 < x < 3
18. If |x - 1| < 11 then which of the following options are true?
SHOW ANSWER
Correct Ans:-10 < x < 12
Explanation:
Given, | x - 1 | < 11
=> -11 < (x-1) < 11
=> (-11 + 1) < x < (11 + 1)
=> -10 < x < 12
19. Find the sum of the linear factors of x^2 - 12 x + 20
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Correct Ans:2(x-6)
Explanation:
20. The sum of the linear factors of x^2 - 10 x+ 21 = 0 is
SHOW ANSWER
Correct Ans:2(x-5)
Explanation:
x^2 - 10x + 21
= x^2 - 2 (5x) + 21
= x^2 - 2 (5) x + 21
= x^2 - 2(5) x + 5^2 + 21 - 25
=x^2 -2(5)(x) + 5^2 - 4
= (x-5)^2 - 4
= (x-5)^2 - 2^2
= ( (x-5) - 2 ) x ((x-5) + 2)
= (x - 5 - 2) x (x - 5 + 2)
=(x-7) x (x-3)
The two linear factors are x - 3 and x - 7
Sum is x - 3 + x - 7 = 2x - 10 = 2(x-5)
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