JEE Main Mathematics - Theory Of Equations
Exam Duration: 60 Mins Total Questions : 30
In a triangle PQR, \(\angle R=\frac { \pi }{ 2 } \).If \(tan\left( \frac { P }{ 2 } \right) \) and \(tan\left( \frac { Q }{ 2 } \right) \)are the roots of the equation \({ ax }^{ 2 }+bx+c=0(a\neq 0)\) then
- (a)
a+b=c
- (b)
b+c=a
- (c)
a+c=b
- (d)
b=c
The curve y=(λ+1)x2+2 intersects the curve y=λx+3 in exact one point if λ equals
- (a)
{-2}
- (b)
{2}
- (c)
{-2,2}
- (d)
{1}
If the equation ax2+bx+c=0 and x3+3x2+3x+2=0 have two common roots,then
- (a)
a=b≠c
- (b)
a≠b=c
- (c)
a=b=c
- (d)
a=-b=c
If λ be an integer and α,β be the roots of x2-16x+λ=0 such that 1<∝<2 and 2<β3,then possible values of λ are
- (a)
{60,64,68}
- (b)
{61,62,63}
- (c)
{49,50,..62,63}
- (d)
{62,65,68,71,75}
If the roots of the equation ax2+bx+c=0 are of the form ∝/(∝-1) and (∝+1)/∝ then the value of (a+b+c)2 is
- (a)
2b2-ac
- (b)
b2-2ac
- (c)
b2-4ac
- (d)
4b2-2 ac
The solution of x-1=(x-[x])(x-{x}) (where [x]and {x} are the integral and fractional part of x) is
- (a)
x∊R
- (b)
x∈R-[1,2)
- (c)
x∈ [1,2)
- (d)
x∈ R-[1,2]
For a≠b, if the equation x2+ax+b=0 and x2+bx+b=0 have a common root,then the value of the (a+b) is
- (a)
-1
- (b)
0
- (c)
1
- (d)
2
The value of p for which both the roots of the equation 4x2-20px++(25p2+15p-66)=0 are less than 2 lies in
- (a)
(4/5,2)
- (b)
(2,∝)
- (c)
(-1,-4/5)
- (d)
(-∝,-1)
Let a,b,c ∈ R and a≠ 0 .If ∝ is a root of a2x2+bx+c=0,β is a root of a2x2+bx-c=0 and 0<∝<β then the equation,a2x2+2bx+2c=0 has a root Υ that always satisfies
- (a)
γ=∝
- (b)
γ=β
- (c)
γ=(∝+β)/2
- (d)
∝<γ<β
If a < 0, the positive root of the equation x2 - 2a I x - a I - 3a2 = 0 is
- (a)
\(a(-1-\sqrt { 6 } )\)
- (b)
\(a(1-\sqrt { 2 } )\)
- (c)
\(a(-1+\sqrt { 6 } )\)
- (d)
\(a(1+\sqrt { 2 } )\)
If 0<a<b<c, and the roots ∝,β of the equation ax2+bx+c=0 are non real complex roots, then
- (a)
|∝|=|β
- (b)
|∝|>1
- (c)
|β|<1
- (d)
none of these
If a+3b+9c=0, ac<0 and one root of the equation ax2+bx+c=0 is square of the other, then
- (a)
a and b have same sign
- (b)
band c have opposite sign
- (c)
both roots are rational
- (d)
a, b, c are irrational
Let P, Q, R, Sand T are five sets about the quadratic equation (a - 5)x2-2ax+(a - 4) = 0, a≠-5 such that
P: All values of a for which the product of roots of given quadratic equation is positive.
Q: All values of a for which the product of roots of given quadratic equation is negative.
R: All values of a for which the product of real roots of given quadratic equation is positive.
S: All values of a for which the roots of given quadratic equation are real.
T: All values of a for which the given quadratic equation has complex roots.
If coefficient of x2 and constant term changes to each other, then
- (a)
set P remain same
- (b)
set Q changes to set P but set P not change to set
- (c)
set P changes to set Q and set Q changes to set P
- (d)
set P changes to set Q but set Q not changes to set R
Let F (x) be a' function defined by \(F(x)=x-[x],0\neq x\epsilon R\), where [x] is the greatest integer less than or equal to x. Then the number of solutions of F(x) + F(1/x) = 1 is/are
- (a)
0
- (b)
infinite
- (c)
1
- (d)
2
The largest interval in which x12- x9 + X4 - X + 1 > 0 is
- (a)
[0,∞)
- (b)
(-∞,0]
- (c)
(-∞,∞)
- (d)
none of these
The system of equation "x-1|+3y=4,x-|y-1|=2 has
- (a)
no solution
- (b)
a unique solution
- (c)
two solutions
- (d)
more than two solutions
If 5 {x}=x+[x] and [x]-{x}= \(\frac { 1 }{ 2 } \) where {x} and [x] are fractional; and integral part of x then x is
- (a)
1/2
- (b)
3/2
- (c)
5/2
- (d)
7/2
If ∝,β be the roots of x2-x-1=0 and An =∝n +βn ,then AM of An-1 and An is
- (a)
2An+1
- (b)
(1/2)An+1
- (c)
2An-1
- (d)
none of these
Let S be the set of values of 'a' for which 2 lie between the roots of the quadratic equation x2 + (a + 2) x - (a + 3) = 0, then S is given by
- (a)
(-∝,-5)
- (b)
(5,∝)
- (c)
(-∝,-5)
- (d)
[5,∝)
The values of a for which the equation 2 (log3x)2 -llog3 X I + a = a possess four real solutions
- (a)
-2<a<0
- (b)
\(0<a<\frac { 1 }{ 8 } \)
- (c)
0<a<5
- (d)
none of these
The solution set of (x)2 + (x + 1)2 = 25, where (x) is the nearest integer greater than or equal to x, is
- (a)
(2,4)
- (b)
[-5,-4] U [2,3)
- (c)
[-4,-3) U [3,4)
- (d)
none of these
Let consider quadratic equation ax2 + bx + c = 0 .... (i)
where \(a,b,c\epsilon R\) and \(a\neq 0\). If Eq. (i) has roots, \(\alpha ,\beta \)
\(\therefore \quad \alpha +\beta =-\frac { b }{ a } ,\alpha \beta =\frac { c }{ a } \) and Eq. (i) can be written as ax2 + bx + c = a(x - \(\alpha \))(x - \(\beta \)).
Also, if a 1 , a 2 , a3, a 4 , .... are in AP, then \({ a }_{ 2 }-{ a }_{ 1 }={ a }_{ 3 }-{ a }_{ 2 }={ a }_{ 4 }-{ a }_{ 3 }=....\neq 0\) and if b 1 , b 2 , b 3 , b 4 , ... are in GP, then \(\frac { { b }_{ 2 } }{ { b }_{ 1 } } =\frac { { b }_{ 3 } }{ { b }_{ 2 } } =\frac { { b }_{ 4 } }{ { b }_{ 3 } } =...\neq 1\) Now, if c 1 , c 2 , c 3 , c 4 , .... are in HP, then \(\frac { 1 }{ { c }_{ 2 } } -\frac { 1 }{ { c }_{ 1 } } =\frac { 1 }{ { c }_{ 3 } } -\frac { 1 }{ { c }_{ 2 } } =\frac { 1 }{ { c }_{ 4 } } -\frac { 1 }{ { c }_{ 3 } } =...\neq 0\)
On the basis of above information, answer the following questions:
Let \({ \alpha }_{ 1 },{ \alpha }_{ 2 }\) be the roots of x2 - x + p = 0 and \({ \alpha }_{ 3 },{ \alpha }_{ 4 }\) be the roots of x2 - 4x + q = 0. If \({ \alpha }_{ 1 },{ \alpha }_{ 2 },{ \alpha }_{ 3 },{ \alpha }_{ 4 }\) are in GP, then the integral values, of p and q respectively are
- (a)
- 2, - 32
- (b)
- 2, 3
- (c)
- 6, 3
- (d)
- 6, - 32
Let \(\left( a+\sqrt { b } \right) ^{ Q(x) }+\left( a-\sqrt { b } \right) ^{ Q(x)-2\lambda }=A,\) where \(\lambda \epsilon N,A\varepsilon R\) and a2 - b = 1
\(\therefore \quad \left( a+\sqrt { b } \right) \left( a-\sqrt { b } \right) =1\quad \Rightarrow \quad \left( a+\sqrt { b } \right) =\left( a-\sqrt { b } \right) ^{ -1 }\quad and\quad \left( a-\sqrt { b } \right) =\left( a+\sqrt { b } \right) ^{ -1 }\)
ie, \(\left( a\pm \sqrt { b } \right) =\left( a+\sqrt { b } \right) ^{ \pm 1 }\quad or\quad \left( a-\sqrt { b } \right) ^{ \pm 1 }\)
On the basis of above information, answer the following questions:
The number of real solutions of the equation \(\left( 15+4\sqrt { 14 } \right) ^{ t }+\left( 15-4\sqrt { 14 } \right) ^{ t }=30\) are where \(t={ x }^{ 2 }-2\left| x \right| \)
- (a)
0
- (b)
2
- (c)
4
- (d)
6
The number of solutions of |[x] - 2x|= 4, where [x] denotes the greatest integer ≤ x, is
- (a)
infinite
- (b)
4
- (c)
3
- (d)
2
The solution set of equation (x+1)2+[x-1]=(x-1)2+[x-1]2 ,where [x] a(x) are the greatest integer and nearest integer to x, is
- (a)
x∈ R
- (b)
x ∈ N
- (c)
x ∈ I
- (d)
x ∈ Q
If ∝ and β are the roots of the equation x2+pa+q=0 and ∝4,β4 are the roots of x2-rx+s=0, then the equation x2-4qx+2q2-r=0 has always
- (a)
two real roots
- (b)
two positive roots
- (c)
two negative roots
- (d)
one positive and one negative root
Let consider the quadratic equation (1+m)x2-2(1+3m)x+(1+8m)=0, where m\(\in \)R〜({-1}
The number of integral values of m such that given quadratic equation has imaginary roots are
- (a)
0
- (b)
1
- (c)
2
- (d)
3
If ∝,β,\(\gamma \) be the roots of the equation ax3+bx2+cx+d=0. To obtain the equation whose are f(∝),f(β),f(\(\gamma \)), where f is a function, we put y=f(∝) and obtain ∝=f-1(y)
Now, ∝ is a root of the equation ax3+bx2+cx+d=0, then we obtain the desired equation which is a {f-1(y)}3+b{f-1(y)}2+c{f-1(y)}+d=0
For example, if ∝,β,\(\gamma \) are the roots of ax3+bx2+cx+d=0. To find equation whose are ∝2,β2,\(\gamma \)2, we put y=∝2
⇒ ∝=\(\sqrt { y } \)
As ∝ is a root of ax3+bx2+cx+d=0
we get ay3/2+by+c\(\sqrt { y } \)+d=0
or \(\sqrt { y } \)(ay+c)=-(by+d)
On squaring both sides, then y(a2y2+2acy+c2)=b2y2+2bdy+d2 or a2y3+(2ac-b2)y2+(c2-2bd)y-d2=0 This is desired equation
If ∝,β,\(\gamma \) are the roots of the equation x3-x-1=0, then the value of \(\prod { \left( \frac { 1+\alpha }{ 1-\alpha } \right) } \) is equal to
- (a)
-7
- (b)
-5
- (c)
-3
- (d)
-1