TEST PER ON ALGEBRA(XI)

1) Find the smallest positive integer n, for which {(1- i)/(1+ i)}ⁿ is purely real. 

2) If ˣ√a = ʸ√b= ᵖ√c and a,b,c are in GP., show that x,y,p are in AP.        

3) If the roots of ax²+ bx+ c = 0 are both positive, then

A) a>0, b>0, c<0

B) a>0, b<0, c>0

C) a<0, b>0, c<0

D) a<0, b<0, c>0

4) The sum of n terms of a GP is 1/3 (2²ⁿ⁺¹ -2); find its common ratio.

5) Of the numbers 2, x, y, 12 (x>0, y>0), if three are in GP and last three in AP, find x and y.

6) If A be the sum and R the sum of the reciprocals of n terms of a GP, whose first term is 4 and last term 36, find the value of S/R.

7) Find the amplitude of (1+ sinx+ i cosx).

8) Express (1+ ix)(1+ iy)(1- ix)(1- iy) as sum of two squares.

9) If z= x + iy be a complex number, then |z²| equals to

A) z² B) conjugate of z² c) z. conjugate of z d) none

10) For what value of k, does the equation 8x²+ 2x + k=0 have its one root square of the other?

11) In an examination a candidate has to pass in each of the four subjects. The number of ways he can fail in the examination is

A) 1 b) 15 c) 16 d) 24

12) Which term in term in (x + 1/x²)⁹ is independent of x ?

a) 2nd b) 3rd c) 4th d) 5th

13) Show that the sum of the coefficient of xʳ and xʳ⁻¹ in the expansion of (1+ x)ⁿ is equal to the coefficient of xʳ in (1+ x)ⁿ⁺¹.

14) If p, q are the roots of equation ax²+ bx + c=0, evaluate 1/(ap + b) + 1/(aq + b).

15) For what value of p, the expression x²+ 4x +4 and x²+ px + 6 have a common factor?

16) For what values of m, the roots of x²- (m -1)x + m + 1/4= 0 are real and equal?

17) For what value of k, the roots of kx(1- x)= 1 will not be real?

18) In a GP, sum of n terms is 255, the last term is 128 and the common ratio is 2. Find n.

19) Find the square root of 9i.

20) The sum of n terms of a series is n/3 (4n²-1). The 5th term of the series is

a) 81 b) 84 c) 121 d) 165

21) At an election, a voter may vote for any number of candidates not greater than the number to be chosen. There are 6 candidates and 4 of them are to be chosen. Find the number of ways in which a voter may vote.

22) Find the middle terms in the expansion of (1+ x)⁹?

23) The middle term of the progression 3,7,11,......147 is

a) 71 b) 73 c) 75 d) 77

24) If p,q are the roots of x²- 2x +2=0, the least integer n (>0) for which pⁿ/qⁿ =1 , is

a) 2 b) 3  c) 4  d) none

25) A polygon has 44 diagonals; the number of its sides is

a) 8 b) 7 c) 11  d) none