Let S(K) = 1 +3+5+..... (2K-1) = 3+K^{2}. Then which of the following is true?
S(1) is correct
Principle of mathematical induction can be used to prove the formula
S(K) ≠S(K+1)
S(K)⇒ S(K+1)
D.
S(K)⇒ S(K+1)
S(K) = 1 + 3 + 5 + ...... + (2K - 1) = 3 + K^{2}
Put K = 1 in both sides
∴ L.H.S = 1 and R.H.S. = 3 + 1 = 4 ⇒ L.H.S. ≠ R.H.S.
Put (K + 1) on both sides in the place of K L.H.S. = 1 + 3 + 5 + .... + (2K - 1) + (2K + 1)
R.H.S. = 3 + (K + 1)2 = 3 + K2 + 2K + 1
Let L.H.S. = R.H.S.
1 + 3 + 5 + ....... + (2K - 1) + (2K + 1) = 3 + K^{2} + 2K + 1
⇒ 1 + 3 + 5 + ........ + (2K - 1) = 3 + K^{2} If S(K) is true, then S(K + 1) is also true. Hence, S(K) ⇒ S(K + 1)
The number of ways of distributing 8 identical balls in 3 distinct boxes so that none of the boxes is empty is
5
3^{8}
21
D.
21
The required number of ways
If z = x – i y and z^{1/3} = p+ iq , then is equal to
1
-2
2
-1
B.
-2
D.
-1
Let z, w be complex numbers such that z iw + = 0 and arg zw = π. Then arg z equals
π/4
5π/4
3π/4
π/2
C.
3π/4
Since z + iw = 0 ⇒ z = −iw
⇒ z = iw
⇒ w = -iz
Also arg(zw) = π
⇒ arg (-iz^{2}) = π
⇒ arg (-i) + 2 arg(z) = π
How many ways are there to arrange the letters in the word GARDEN with the vowels in alphabetical order?
120
480
360
240
C.
360
A total number of ways in which all letters can be arranged in alphabetical order = 6! There are two vowels in the word GARDEN. A total number of ways in which these two vowels can be arranged = 2!
∴ Total number of required ways
∴ Total number of required ways
If one root of the equation x^{2}+px+12 =0 is 4, while the equation x^{2} +px +q = 0 has equal roots, then the value of 'q' is
49/3
4
3
12
A.
49/3
Since 4 is one of the roots of equation x2 + px + 12 = 0. So it must satisfied the equation.
∴ 16 + 4p + 12 = 0
⇒ 4p = -28
⇒ p = -7
The other equation is x^{2} - 7x + q = 0 whose roots are equal. Let roots are α and α of above equation
⇒ 2α = 7 ⇒ α = 7/ 2 and product of roots α.α = q ⇒ α^{2} = q
(7/2)^{2} = q
q =49/4
If (1 – p) is a root of quadratic equation x^{2} +px + (1-p)=0 , then its roots are
0, 1
-1, 2
0, -1
-1, 1
C.
0, -1
Since (1 - p) is the root of quadratic equation
x2 + px + (1 - p) = 0 ........ (i)
So, (1 - p) satisfied the above equation
∴ (1 - p)^{2} + p(1 - p) + (1 - p) = 0
(1 - p)[1 - p + p + 1] = 0 (1 - p)(2) = 0
⇒ p = 1 On putting this value of p in equation (i)
x^{2} + x = 0
⇒ x(x + 1) = 0 ⇒ x = 0, -1
If |z^{2}-1|=|z|^{2}+1, then z lies on
the real axis
an ellipse
a circle
the imaginary axis
B.
an ellipse
Given that
|z^{2}- 1| = |z|^{2}+ 2
|z^{2} + (-1)| = |z^{2}| + |-1|
It shows that the origin, -1 and z^{2} lies on a line and z^{2} and -1 lies on one side of the origin, therefore
z2 is a negative number. Hence z will be purely imaginary. So we can say that z lies on y-axis.
If a_{1}, a_{2}, a_{3} , ....,a_{n} , .... are in G.P., then the value of the determinant is
0
-2
1
2
A.
0
Let two numbers have arithmetic mean 9 and geometric mean 4. Then these numbers are the roots of the quadratic equation
x^{2} + 18x +16 = 0
x^{2}-18x-16 = 0
x^{2}+18x-16 =0
x^{2}-18x +16 =0
D.
x^{2}-18x +16 =0
Let α and β be two numbers whose arithmetic mean is 9 and geometric mean is 4.
∴ α + β = 18 ........... (i)
and αβ =16 ........... (ii)
∴ Required equation is x2 - (α + β)x + (αβ) = 0 ⇒ x2 - 18x + 16 = 0 [using equation (i) and equation (ii)]
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