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- Total Questions 20
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1. Find the maximum value of the expression

Correct Answers :

[C]

Explanation :

For the given expression to be a maximum, the denominator should be minimized. (Since, the
function in the denominator has imaginary roots and is always positive). x^{2} + 5x + 10 will be minimized at x = –2.5 and its minimum values at x = –2.5 is 3.75.

Hence, required answer = 1/3.75 = 4/15.

2. Find the maximum value of the expression (x^{2} + 8x + 20).

Correct Answers :

[D]

Explanation :

Has no maximum.

3. Find the minimum value of the expression (p +1/p); p > 0.

Correct Answers :

[C]

Explanation :

The minimum value of (p + 1/p) is at p = 1. The value is 2.

4. If the product of roots of the equation x^{2} – 3 (2a + 4) x + a^{2} + 18a + 81 = 0 is unity, then a can take the values as

Correct Answers :

[C]

Explanation :

The product of the roots is given by: (a^{2} + 18a + 81)/1.

Since product is unity we get: a^{2} + 18a + 81 = 1

Thus, a^{2} + 18a + 80 = 0

Solving, we get: a = –10 and a = – 8.

5. For the equation 2^{a + 3} = 4^{a + 2} – 48, the value of a will be

Correct Answers :

[D]

Explanation :

Solve through options. LHS = RHS for a = 1.

6. The expression a^{2} + ab + b^{2} is _________for a < 0, b < 0

Correct Answers :

[C]

Explanation :

For a, b negative the given expression will always be positive since, a^{2}, b^{2} and ab are all
positive.

7. If the roots of equation x^{2} + bx + c = 0 differ by 2, then which of the following is true?

Correct Answers :

[D]

Explanation :

To solve this take any expression whose roots differ by 2.

Thus, (x – 3) (x – 5) = 0

fi x^{2} – 8x + 15 = 0

In this case, a = 1, b = –8 and c = 15.

We can see that b^{2} = 4(c + 1).

8. If f(x) = (x + 2) and g(x) = (4x + 5), and h(x) is defined as h(x) = f(x) ◊ g(x), then sum of roots of h (x) will be

Correct Answers :

[C]

Explanation :

h (x) = 4x^{2} + 13x + 10.
Sum of roots – 13/4.

9. If equation x^{2} + bx + 12 = 0 gives 2 as its one of the roots and x^{2} + bx + q = 0 gives equal roots then the value of b is

Correct Answers :

[B]

Explanation :

x^{2} + bx + 12 = 0 has 2 as a root.
Thus, b = –8.

10. If the roots of the equation (a^{2} + b^{2})x^{2} – 2(ac + bd)x + (c^{2} + d^{2}) = 0 are equal then which of the following is true?

Correct Answers :

[B]

Explanation :

Solve this by assuming each option to be true and then check whether the given expression has equal roots for the option under check.

Thus, if we check for option (b).

ad = bc.

We assume a = 6, d = 4 b = 12 c = 2 (any set of values that satisfies ad = bc).

Then (a^{2} + b^{2}) x^{2} – 2(ac + bc) x + (c^{2} + d^{2}) = 0

180 x^{2} – 120 x + 20 = 0.

We can see that this has equal roots. Thus, option (b) is a possible answer. The same way if we
check for a, c and d we see that none of them gives us equal roots and can be rejected.

11. For what value of c the quadratic equation x^{2} – (c + 6) x + 2(2c – 1) = 0 has sum of the roots as half of their product?

Correct Answers :

[C]

Explanation :

(c + 6) = 1/2 × 2(2c – 1) fi c + 6 = 2c – 1 fi c = 7

12. Two numbers a and b are such that the quadratic equation ax^{2} + 3x + 2b = 0 has – 6 as the sum and the product of the roots. Find a + b.

Correct Answers :

[B]

Explanation :

–3/a = –6 fi a = 1/2,

2b/a = –6 and a = 1/2

Gives us b = –1.5.

a + b = –1.

13. If a and b are the roots of the Quadratic equation 5y^{2} – 7y + 1 = 0 then find the value of 1/α + 1/β

Correct Answers :

[D]

Explanation :

1/a + 1/b = (a + b)/ab

= (7/5)/(1/5) = 7.

14. Find the value of the expression

Correct Answers :

[B]

Explanation :

Solving quadratically, we have option (b) as the root of this equation.

15. If a = what will be the value of

Correct Answers :

[B]

Explanation :

The approximate value of a = √13.92 = 3.6(approx)

a + 1/a = 3.6 + 1/3.6 is closest to 4.

16. If the roots of the equation (a^{2} + b^{2}) x^{2} – 2b(a + c) x + (b^{2} + c^{2}) = 0 are equal then a, b, c, are in

Correct Answers :

[B]

Explanation :

Solve by assuming values of a, b, and c in AP, GP and HP to check which satisfies the condition.

17. If a and b are the roots of the equation ax2 + bx + c = 0 then the equation whose roots are a + 1/β and b + 1/α

Correct Answers :

[C]

Explanation :

Assume any equation:

Say x^{2} – 5x + 6 = 0

The roots are 2, 3.

We are now looking for the equation, whose roots are:

(2 + 1/3) = 2.33 and (3 + 1/2) = 3.5.

Also a = 1, b = –5 and c = 6.

Put these values in each option to see which gives 2.33 and 3.5 as its roots.

18. If x^{2} + ax + b leaves the same remainder 5 when divided by x – 1 or x + 1 then the values of a and b are respectively

Correct Answers :

[A]

Explanation :

Remainder when x^{2} + ax + b is divided by x – 1 is got by putting x = 1 in the expression. Thus, we get.

a + b + 1 = 5 and

b – a + 1 = 5

fib = 4 and a = 0

19. Find all the values of b for which the equation x^{2} – bx + 1 = 0 does not possess real roots.

Correct Answers :

[C]

Explanation :

b^{2} – 4 < 0 fi – 2 < b < 2

20. Find the number of solutions of a^{3} + 2^{a+1} = a^{4}, given that n is a natural number less than 100.

Correct Answers :

[B]

Explanation :

In order to think of this situation, you need to think of the fact that “the cube of a number + a power of two” (LHS of the equation) should add up to the fourth power of the same number.
The only in which situation this happens is for 8 + 8 = 16 where a = 2 giving us 2^{3} + 2^{3} = 2^{4}.
Hence, Option (b) is the correct answer.

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