A body weighing 13 kg is suspended by two strings 5 m and 12 m long, their other ends being fastened to the extremities of a rod 13 m long. If the rod be so held that the body hangs immediately below the middle point. The tensions in the strings are
12 kg and 13 kg
5 kg and 5 kg
5 kg and 12 kg
5 kg and 13 kg
C.
5 kg and 12 kg
If the roots of the quadratic equation x^{2} + px + q = 0 are tan30° and tan15°, respectively then the value of 2 + q − p is
2
3
0
1
B.
3
x^{2} + px + q = 0
tan 30° + tan 15° = − p
tan 30° ⋅ tan 15° = q
⇒ − p = 1 − q
⇒ q − p = 1
∴ 2 + q − p = 3
If both the roots of the quadratic equation x^{2} – 2kx + k^{2} + k – 5 = 0 are less than 5, then k lies in the interval
(5, 6]
(6, ∞)
(-∞, 4)
[4, 5]
C.
(-∞, 4)
If z_{1} and z_{2} are two non-zero complex numbers such that |z_{1} + z_{2}| = |z_{1}| + |z2| then argz_{1} – argz_{2} is equal to
π/2
-π
0
-π/2
C.
0
z_{1} + z_{2}| = |z_{1}| + |z_{2}| ⇒ z_{1} and z_{2} are collinear and are to the same side of origin; hence arg z_{1} – arg z_{2} = 0
If the coefficient of x^{7} in equals the coefficient of x^{-7} inthen a and b satisfy the relation
a – b = 1
a + b = 1
a/b =1
ab =1
D.
ab =1
If the equation a_{n}x^{n} +a_{n-1}x^{n-1} +....... +a_{1}x =0, a_{1} ≠ 0, n≥2, has a positive root x = α, then the equation na_{n}x^{n-1} + (n-1)a_{n-1}x_{n-2} +......+a_{1} = 0 has a positive root, which is
greater than α
smaller than α
greater than or equal to α
equal to α
B.
smaller than α
f(0) = 0, f(α) = 0
⇒ f′(k) = 0 for some k∈(0, α).
If x is so small that x^{3} and higher powers of x may be neglected, then may be approximated as
C.
All the values of m for which both roots of the equations x^{2} − 2mx + m^{2} − 1 = 0 are greater than −2 but less than 4, lie in the interval
−2 < m < 0
m > 3
−1 < m < 3
1 < m < 4
C.
−1 < m < 3
Equation x^{2} − 2mx + m^{2} − 1 = 0
(x − m)^{2} − 1 = 0
(x − m + 1) (x − m − 1) = 0
x = m − 1, m + 1 − 2 < m − 1 and m + 1 < 4
m > − 1 and m < 3 − 1 < m < 3.
If z^{2} + z + 1 = 0, where z is a complex number, then the value of
18
54
6
E.
12
z^{2} + z + 1 = 0 ⇒ z = ω or ω^{2}