BINOMIAL THEOREM
1) Expand (2/x - x/2)⁵, x≠ 0.
2) Using binomial theorem , write the value of (a+ b)ⁿ + (a - b)ⁿ and hence find the value of (√3+ √2)⁶ - (√3 - √2)⁶.
3) Find the 9th term in the expansion of (3x - 1/2x)⁸, x≠ 0.
4) Find the term independent of x in (2x²- 1/x)¹².
5) Find the middle terms in the expansion of (3- x³/6)⁷.
6) Use binomial theorem to evaluate (10.1)⁵.
7) Examine whether or not there is any term containing x⁹ in the expansion of (2x¹ - 1/x)²⁰.
8) In the binomial expansion of (a - b)ⁿ, n≥ 5, The sum of 5th and 6th terms is zero, then a/b equals
a) (n -5)/6 b) (n -4)/5 c) 5/(n -4) d) 6/(n -5)
9) If the coefficient of rth and (r +4)th terms are equal in the expansion of (1+ x)³⁰, then the value of r will be
a) 7 b) 8 c) 9 d) 10
10) If the coefficient of x² and x³ in the expansion of (3+ ax)⁹ be same, then the value of a is.
a) 3/7 b) 7/3 c) 7/9 d) 9/7
11) Using binomial theorem, the value of (0.999)³ correct to 3 decimal places is
a) 0.999 b) 0.998 c) 0.997 d) 0.995
QUADRATIC EQUATION
1) 5ˣ⁺¹ + 5²⁻ˣ = 5³ +1.
2) √{x/(1- x)} + √{(1- x)/x} = 13/6.
3) (x +1)(x +2)(x +3)(x +4)= 120.
4) Prove that both the roots of the equation x²- x -3=0 are irrational.
5) For what values of m will the equation x¹- 2mx + 7m -11= 0 have
a) equal roots.
b) reciprocal roots ?
6) if the roots of 2x²- 5x + k =0 be double the other, find the value of k.
7) If α, β be the roots of the equation x¹- x -1=0, determine the value of
a) α²+ β².
b) α³+ β³.
8) If the roots of the equation ax²+ bx + c=0 be in the ratio 3:4, show that 12b²= 49ac.
9) if x is real, prove that the quadratic expression
a) (x -2)(x +3)+ 7 is always
b) 4x - 3x²- 2 is always negative.
10) What is the minimum value of x²- 4x +3=0.
11) For what real values of a, will the expression x²- ax +1 - 2a², for the real x, be always positive ?
12) If x be real, prove that the value of (2x⅖- 2x +4)/(x²- 4x +3) cannot lie between -7 and 1.
13) if the roots of the equation qx²+ 2px + 2q=0 are real and unequal, prove that the roots of the equation (p + q)x²+ 2qx + (p - q)= 0 are imaginary.
14) If α, β be the roots of x²- px + q=0, find the value of α⁵β⁷+ α⁷β⁵ in terms of p and q.
15) If the difference between the roots of the equation x¹+ ax +1=0 is less than √5, then the set of possible value q of a is
a) (3,∞) b) (-∞,-3) c) (-3,3) d) (-3,∞).
16) let α, β be the roots of the equation x²- px + r=0 and α/2, 2β be the roots of the equation x²- qx + r=0, then the value of r is
a) (2/9) (p - q)(2q - p)
b) (2/9) (q- p)(2p - q)
c) (2/9) (q - 2p)(2q - p)
d) (2/9) (2p - q)(2q - p).
17) α, β are the roots of ax²+ 2bx + c=0 and α + β, β + δ are the roots of Ax²+ 2Bx + C=0, then what is (b¹- ac)/(b²- ac) equal to ?
a) (b/B)² b) (a/A)² c) (a²b²/A²B²) d) (ad/AB) e) none
18) If α, β are the roots of the equation x²- 2x -1=0, then what is the value of α²β⁻² + α⁻²β² ?
a) -2 b) 0 c) 30 d) 34
19) If the roots of the equation x²+ px + q=0 are thn30° and tan15°, then value of 2+ q - p is
a) 1 b) 2 c) 3 d) 0
20) If the roots of the quadratic equation x²- 2kx + k²- 5 =0 are less than 5, then k lies in the interval
a) (5,6) b) (6,∞) c) (- ∞,4) d) [4,5].
21) If α and β are the roots of ax²+ bx + c =0 and if px²+ qx + r=0 has roots (1- α)/α and (1- β)/β then r=
a) a+ 2b b) a+ b + c c) ab+ bc+ ca d) abc.
22) The equation x²- 6x + a=0 and x²- cx + 6 =0 have one root in common. The other roots of the first and second equations are integers in the ratio 4:3. then the common root is
a) 1 b) 4 c) 3 d) 2
23) If α, β are the roots of the equation λ(x²- x) + x + 5=0 and λ₁ and λ₂ are two values of λ obtained from α/β + β/α = 4/5, then λ₁/λ₂² + λ₂/λ₁² equals
a) 4192 b) 4144 c) 4096 d) 4048
24) If α, βbe the roots of x² - a(x -1)+ b =0, then value of
1/(α² - aα) + 1/(β² - aβ) + 2/(a+ b) is
a) 4/(a+ b) b) 1/(a+ b) c) 0 d) -1
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