`(-3, -2] cup [2, 3)`
`(- 3, 3)`
`[- 3, -2] cup [2, 3]`
None of the above
`p^2 m = l^2q`
`m^2p = l^2q`
`m^2p = q^2l`
`m^2p^2 = l^2q`
Both the roots are real
One root is real and the other is complex
Both the roots are complex
Cannot say
`(c - 1)/b`
`(1 - c)/b`
`b/(c - 1)`
`b/(1 - c)`
AP
GP
HP
None of the above
`- 2 < k < 2 `
`-5 < k < 3`
`-3 < k < 5`
`-1 < k < 3`
Only 1
Only 2
Both 1 and 2
Neither 1 nor 2
Only 1
Only 2
Both 1 and 2
Neither 1 nor 2
` |P| < 4`
` |P| <= 4`
` |P| > 4`
` |P| >= 4`
One
Two
Three
Four
` - 9/8`
`9/8`
`- 8/9`
`8/9`
`x^ 2 - 10 x + 9 = 0`
`x^ 2 + 10 x + 9 = 0`
`x^ 2 - 10 x + 16 = 0`
`x^ 2 - 8 x - 9 = 0`
`4`
`3`
`2`
`1`
`6`
`7`
`10`
`20`
`- 3/4`
`3/4`
` - 4/3`
` -4/5`
exactly one real root
at least one real root
at least two real roots
at most two real roots
`4096`
`2048`
`1024`
`512`
`x^2 - 6ax + 9a^2 - b = 0`
`3ax^2 + x - sqrt(b) = 0`
`x^2 + 3ax + sqrt(b) = 0`
` sqrt(b)x^2 + x - 3a = 0`
One
Two
Four
No real root
`-1`
`0`
`1`
`2`
are imaginary
are distinct and real
are equal and real
Cannot be determined
`2`
`3`
`5`
`8`
Only `b <= -4`
Only `b >= 4`
`-4 < b < 4`
`b <= - 4, b >= 4`
`1, 2`
`1, 1`
`1, 0`
`2, 2`
`x^2 - (b^2 - 2ac) x + c = 0`
`a^2x^2 - (b^2 - 2ac)x + c = 0`
`ax^2 - (b^2 - 2ac)x + c^2 = 0`
`a^2x^2 - (b^2 - 2ac)x + c^2 = 0`
`8 ac = 25b`
`8ac = 9b^2`
`8b^2 = 9ac`
`8b^2 = 25ac`
one real root
two real roots
two imaginary roots
four real roots
`0`
`1`
`2`
`3`
`2x^2 - x + 3= 0 `
`x^2 - 3x + 2 = 0`
`x^2 + 3x + 2 = 0`
`x^2 - 3x - 2 = 0`
` - b/a`
`b/c`
`c/b`
` - c/b`
`x^2 + 2 mx + m^2 - mn + n^2 = 0`
`x^2 + 2 mx + (m - n)^2 = 0`
`x^2 - 2 mx + m^2 - n^2 = 0`
`x^2 + 2 mx + m^2 - n^2 = 0`
`q + r`
`p + q`
`q + r`
`p + q`
`p = 25//12`
`p < 25//12`
`p >25//12`
`p <= 25//12`
`170`
`180`
`190`
`290`
`b^2 = a(a + 4c)`
`a^2 = b(b + 4c)`
`a^2 = c(a + 4c)`
`b^2 = a(b + 4c)`
`-1`
`1`
`2`
`-2`
`1 - r`
`q- r`
`1 + r`
`q + r`
`3 : 1`
`1 : 2`
`1 : 3`
`3: 2`
`bc = a^2`
`bc = 36 a^2`
`bc = 72 a^2`
`bc = 108 a^2`
`(a, c)`
`(b, c)`
`(a, b)`
`(a + b, a + c)`
`aq = 2(b + p)`
`aq = b + p`
`ap = 2(b + q)`
`ap= b + q`
`2`
`5`
`2 + 5i`
`2 - 5i`
` - ( b (c - a))/(a(b - c))`
` ( b (c - a))/(a(b - c))`
` ( c (a - b))/(a(b - c))`
` - ( c (a - b))/(a(b - c))`
`-7//2,2`
`-3//2,4`
`-5//3,3`
`3//2,4`
`x^2 - x - 1 = 0`
`x^2 - x + 1 = 0`
`x^2 + x - 1 = 0`
`x^2 + x + 1 = 0`
10
8
6
4
`2`
`5//12`
`12//25`
`25//12`
`47//49`
`49//47`
`-47 //49`
`- 49//47`
complex
pure imaginary
irrational
rational
`2 - sqrt(3)`
`2 + sqrt(3)`
`7 - 4sqrt(3)`
`4`
`2b = a + c`
`b^2 = ac`
`b + c = 2a`
`b = ac`
`16`
`- 16`
`8`
`-8`
`1 ± i`
`2 ± i `
`1 ± sqrt(2)`
`2 ± i sqrt(2)`
`alpha^7` and `beta^(13)`
`alpha^(13)` and `beta^7`
`alpha^(20)` and `beta^(20)`
None of these
4 or 8
5 or 10
6 or 12
3 or 6
6, - 4, 1
4, 6, - 1
3,- 2, 1
6, 4, 1
`(alpha^4 - beta^4 )` is real
`2(alpha^5 + beta^5 ) = (alpha beta )^5`
`(alpha^6 - beta^6) = 0`
`(alpha^8 + beta^8) = (alpha beta)^8`
`x^2 - 2px - (p^2 - q) = 0`
`x^2 - 2px + (p^2 - q) = 0`
`x^2 + 2px - (p^2 - q) = 0`
`x^2 + 2px + (p^2 - q) = 0`
`12`
`15`
`16`
`18`
`- 3 < b < 3`
`- 2 < b < 2`
`b > 2`
`b < - 2`
1, 0
0, 1
-2, 0
-2, 1
`a^2 + b^2 = 2ac`
`b^2 - c^2 = 2ab`
`b^2 - a^2 = 2ac`
`b^2 + c^2 = 2ab`
always complex
always real
always purely imaginary
None of the above
no roots
one root
two equal roots
infinite roots
real
imaginary
positive
negative
-2 only
1 only
-2 and 1
-2 and -1
3, 8
-3, -8
3, -8
-3, 8
` a = (m - q)//(l -p)(l != p)`
`a = (m + q)//(l + p)(l != p)`
`l = (m- q)//(a- p)(a != p)`
`p = (m - q)//(a- l)(a != l)`
` (c-a)/(b - c)`
` (a - b)/(b - c)`
` ( b - c)/(a - b)`
`(c -a)/(a - b)`
`2n^2`
`2n^4`
`2`
`n^2`
`((B^2 - 4AC))/A^2`
`((B^2 - 4AC))/A^2`
`((2AC - B^2))/A^2`
`B^2 - 2C`
not necessarily real, if the coefficients are real
always imaginary
always real
real, if the coefficients are real
`p^2 + q^2 -2pr = 0`
`p^2 - q^2 + 2pr = 0`
`(p + r)^2 = 2(p^2 + r^2 )`
`(p- r)^2 = q^2 + r^2`
`1`
`2`
`-2`
`3`
`p^2 - 4q`
`(p^2 - 4q)/2`
`(p^2 - 4q)/q^2`
`(p^2 - 2q)/q^2`
` - 2/3`
`2/3`
`4`
`8`
`2`
`3`
`4`
`9`
`-1 <= x <= 4`
`2 <= x <= 4`
`-1 < x <= 1`
`-1 <= x <= 1` or `2 < x <= 4`
`5`
`sqrt5`
`1`
`(5)^(1//4)`
`q = -1`
`q = 1`
`q = 0`
`q = 1/2`
`1` unit
`2` units
`3` units
`4` units
`k^2`
`1/k^2`
`2k^2`
`1/(2k^2)`
If `b^ 2 - 4ac > 0`, then `f^(-1)(0)` does not contain `0`
If `b2^ - 4ac < 0`. then `f^(-1)(0)` must contain `0`
If `b^ 2 - 4ac > 0`, then `f^(-1)(0)` may contain `0`
If `b2^ - 4ac < 0`. then `f^(-1)(0)` may contain `0`
`6`
`3 sqrt(2)`
`4 sqrt(2)`
`12`
`10`
`15`
`20`
`30`
` sqrt(alpha/beta ) + sqrt(beta/alpha) - sqrt(m/l) = 0`
` sqrt(alpha/beta ) + sqrt(beta/alpha) + sqrt(m/l) = 0`
` sqrt( alpha + beta)/(alpha beta) - sqrt(m/l) = 0`
The arithmetic mean of `alpha` and `beta` is the same as their geometric mean.
k = 0 only
k = - 3 only
k = 0 or k = 3
k = 0 or k = - 3
`beta , 1/alpha`
`alpha , 1/beta`
` - alpha , - beta`
`1/alpha , 1/beta`
Signs of a and c should be like
Signs of b and c should be like
Signs of a and b should be like
None of the above
` - 1/3`
`- 1/2`
`0`
`1`
`(b/B)^2`
`(a/A)^2`
`(a^2b^2)/(A^2B^2)`
`(ab)/(AB)`
`(b/B)^2`
`(a/A)^2`
`(a^2b^2)/(A^2B^2)`
`(ab)/(AB)`
`a/(bc)`
`b/(ac)`
`(-b)/(ac)`
`(-a)/(bc)`
`-2`
`0`
`30`
`34`
`a = 2 , b = 4`
`a = 2 ,b = - 4`
`a = 1, b = 1/2`
`a = -1, b = - 1/2`
`x in (-1, 4)`
`x in [-1, 4]`
`x in (-oo, -1) cup (4, oo)`
`x in (-oo , 1) cup [4, oo)`