Topic - Quant Mixed Bag

Solved ? - Yes

Source - ]]>

Topic - Quant Mixed Bag

Solved ? - Yes

Source - ]]>

m + m = - Q / P .

2 m = - Q / P.

Q = -2Pm

m x m = R / P

m^2 = R / P .

R = Pm^2.

P + Q + R = 1.

P - 2Pm + Pm^2 = 1.

P( 1 - m)^2 = 1.

P = 1, m =2.

then Q = -4.

Sum of roots = 4. ]]>

pq = - (a + 1).

p^2 + q^2 = (p + q)^2 - 2pq

= (a - 2)^2 + 2(a + 1)

= a^2 - 4a + 4 + 2a + 2

= a^2 - 2a + 6.

= (a-1)^2 + 5.

if a = 1,

then p^2 + q^2 = 5.

Hence minimum value = 5

(k+3k)x^2 + (5k+11k)x + 7k = 0.

4k x^2 + 16k x +7k = 0.

4x^2 + 16x + 7 = 0.

4x^2 + 14x + 2x + 7 = 0.

2x(2x+7)+(2x+7) = 0.

(2x+7) (2x+1) = 0.

x = -7/2 , -1/2.

absolute difference = -1/2 + 7/2 = 3 ]]>

f '(x) = 2px + q.

put f '(x) = 0

x = - q / 2p.

function f(x) attains maximum value at x = -q / 2p.

2 = -q / 2p ... (i)

by putting f(x) = 0.

px^2 + qx + r = 0.

sum of roots = -q / p.

from equation (i)

-q / 2p = 2.

-q / p = 4. ]]>

(a) b = -31 / 12

(b) m = -2

(c) n = 14

(d) either a or b. ]]>

m x n = -14.

m = -14 / n.

4m + 3n = 13

4(-14 / n) + 3n = 13.

= > -56 + 3n^2 = 13n.

= > 3n^2 - 13n - 56 = 0.

= > 3n^2 - 21n + 8n - 56 = 0.

= > 3n(n - 7) + 8(n - 7) = 0.

= > n = 7 , -8/3.

if n = 7, m = -2 and b = -5.

n = -8/3, m = 42 / 8 and b = -31/12.

Hence option d. ]]>

m + n = 35.

35 is an odd number and it is sum of one odd and one even number.

2 is only even number which is prime.

if m = 2, n =33.

but 33 is not prime here.

Hence there is no value of k for which both the roots of equation are prime numbers ]]>

(a) b.

(b) b - 4; b > 0 for other value of k

(c) b > 0 for all values of k

(d) b > 0 for k. ]]>

[(x - 4 ) (x - k )] = ax^2 + bx + c.

= > [x^2 - (4+k)x + 4k] = ax^2 + bx + c.

a > 0, c > 0.

suppose a = 1.

= > x^2 - (k+4)x + 4k = x^2 + bx + c.

c = 4k.

b = -(k+4)

(k+4) > 0, any value of k.

Hence option a. ]]>

mn = 30.

(1+m+m^2) (1+n+n^2)

= (1+n+n^2)+(m+mn+mn^2)+(m^2+m^2n+(mn)^2)

= 1+(m+n)+(m^2 + n^2)+mn + (mn)^2 + mn (m+n)

= 1 + (m+n) + (m+n)^2 - 2mn + mn +(mn)^2 + mn(m+n)

= 1 + 10 + 10^2 - 2 x 30 + 30 + 30^2 + 30(10)

= 1281. ]]>

ab = 72.

c + d = 8.

cd = k.

a+b = cd.

(a,b)= (1,72),(2,36),(3,24),(4,18 ),(6,12),(8,9).

(c,d )= (1,7),(2,6),(3,5),(4,4).

a + b = 73,38,27,22,18,17.

cd = 7,12,15,8.

no real value of a+b+c+d. ]]>

using rationalisation

x = 4 + 2√3 / 4

x = 1 + √3/2.

x^2 = 1 + 3/4 + √3 = 7/4 + √3

x^3 = 7/4 + √3 + 7√3/8 + 3/2

= 13/4 + √3 + 7√3 /8.

x^4 = 49/16 + 3 + 7√3/2.

x^4 - 4x^3 + 7x^2 -6x + 7/4

= 97/3 + 7√3/2 -13 - 4√3 - 7√3/2 + 49/4 + 7√3 - 6 -3√3 + 7/4

= 17/16. ]]>

t^2 + 4t - 12 = 0.

= > t^2 +6t -2t -12 = 0.

= > (t + 6) (t - 2) = 0.

t = -6, 2.

x^3 = -6

x = -6, -6w , -6w^2.

x^3 = 2.

x = 2, 2w, 2w^2.

only 2 real solution ]]>

Ashu’s speed = 100/5 = 20 m/s.

Dishu’s speed = 25 m/s.

Prashant’s speed = 15m/s.

Relative speed = 25+15 = 40 m/s.

Time when Dishu and Prashant meet = 500 / 40 = 12.5 seconds.

Distance travelled by Ashu in given time = 12.5 x 20 = 250 meter. ]]>