Mathematics - Theory Of Equations
Exam Duration: 45 Mins Total Questions : 30
If tn denotes nth term of an AP and \({ t }_{ p }=\frac { 1 }{ q } \) and \({ t }_{ q }=\frac { 1 }{ q } \) then which of the following is necessarily a root of the equation \((p+2q-3r){ x }^{ 2 }+(q+2r-3p)x+(r+2p-3q)=0\) is
- (a)
tp
- (b)
tq
- (c)
tpq
- (d)
tp+q
If x2-x-2 is a factor x4-λx2-μ ,then \(\sqrt { (\lambda ^{ 2 }-\mu ^{ 2 }) } \) equals
- (a)
1
- (b)
3
- (c)
5
- (d)
7
If ∝ and β are the roots of the equation x2-](x+1)-q=0, then the value of \(\frac { \alpha ^{ 2 }+2\alpha +1 }{ \alpha ^{ 2 }+2\alpha +q } +\frac { \beta ^{ 2 }+2\beta +1 }{ \beta ^{ 2 }+2\beta +q } \) is
- (a)
2
- (b)
1
- (c)
0
- (d)
none of these
If ∝,β are the roots of the equation 8x2-3x+27=0 then the value of \([(\alpha ^{ 2 }/\beta )^{ 1/3 }+(\beta ^{ 2 }/\alpha )^{ 1/3 }]\) is
- (a)
1/3
- (b)
1/4
- (c)
1/5
- (d)
1/6
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
If ∝,β are the roots of the equation x2+x+1=0 ,then the equation whose roots are ∝19 and β7 is
- (a)
x2+x+1=0
- (b)
x2-x-1=0
- (c)
x2+x+2=0
- (d)
x2+19x+7=0
The number of real solutions of \(1+|{ e }^{ x }-1|={ e }^{ x }({ e }^{ x }-2)\) is
- (a)
0
- (b)
1
- (c)
2
- (d)
4
If the equation ax2+2bx-3c=0 has non real roots and (3c?4)<(a+b);then c is always
- (a)
<0
- (b)
>0
- (c)
≥0
- (d)
zero
If x2 + px + 1 is a factor of the expression ax3 + bx2 + c, then
- (a)
a2+c2=-ab
- (b)
a2-c2=-ab
- (c)
a2-c2=-bc
- (d)
none of these
If ∝,β,⋎,δ are the roots of x4 + x2 + 1 = 0, then the equation whose roots are ∝2,β2,⋎2,δ2 is
- (a)
(x2-x+1)2=0
- (b)
(x2-x+1)2=0
- (c)
x4-x2+1=0
- (d)
x2-x+1=0
For a>0, the roots of the equation
\(a+log_{ x }{ a }^{ 2 }+log{ a }^{ 2 }xa^{ 3 }\) =0, are given by
- (a)
a-4/3
- (b)
a-3/4
- (c)
a-1/2
- (d)
a-1
The number of real solution of the equation \(\left( \frac { 9 }{ 10 } \right) ^{ x }=-3+x-{ x }^{ 2 }\) is
- (a)
none
- (b)
one
- (c)
two
- (d)
more than two
If 2a+3b+6c=0 (a,b,c \(\in \)R), then the quadratic equation ax2+bx+c=0 has
- (a)
at least one root in [0,1]
- (b)
at least one root in (-1,1]
- (c)
at least one root in [0.2]
- (d)
none of the above
A quadratic equation whose difference of roots is 3 and the sum of the squares of the roots is 29 is given by
- (a)
x2+9x+14=0
- (b)
x2+7x+10=0
- (c)
x2-7x-10=0
- (d)
x2-7x+10=0
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.
Which statement is correct
- (a)
least positive integer for set R is 2
- (b)
least positive integer for set R is 3
- (c)
greatest positive integer for set T is 3
- (d)
none of the above
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.
Which statement is correct
- (a)
P U Q=S U T
- (b)
P U Q=S U T ~{4,5}
- (c)
P=T
- (d)
none of the above
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
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
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,∝)
If 0<x<1000 and \(\left[ \frac { x }{ 2 } \right] +\left[ \frac { x }{ 3 } \right] +\left[ \frac { x }{ 5 } \right] =\frac { 31 }{ 30 } x\) , where [x] is the greatest integer less than or equal to x, the number of possible values of x is
- (a)
34
- (b)
33
- (c)
32
- (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 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:
Given that \({ \beta }_{ 1 },{ \beta }_{ 3 }\) be roots of the equation Ax2 - 4x + 1 = 0 and \({ \beta }_{ 2 },{ \beta }_{ 4 }\) the roots of the equation Bx2 - 6x + 1 = 0. If \({ \beta }_{ 1 },{ \beta }_{ 2 },{ { \beta }_{ 3 },\beta }_{ 4 }\) are in HP; then the integral values of A and B respectively are
- (a)
-3, 8
- (b)
-3. 16
- (c)
3,8
- (d)
3,16
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:
If a, b, c, d and x are distinct real numbers such that (a2 + b2 + c2 )x2 - 2(ab + bc +cd)x + (b2 + c2 + d2) \(\le\) 0, then a, b,c, d
- (a)
are in AP
- (b)
are in GP
- (c)
are in HP
- (d)
satisfy ab = cd
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:
The harmonic mean of the roots of the equation \(\left( 5+\sqrt { 2 } \right) { x }^{ 2 }-\left( 4+\sqrt { 5 } \right) x+8+2\sqrt { 5 } =0\) is
- (a)
2
- (b)
4
- (c)
6
- (d)
8
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 \(\alpha ,\beta \) be the roots of the equation (x - a)(x - b) + c = 0, c \(\neq\) 0, then the roots of the equation \(\left( x-\alpha \right) \left( x-\beta \right) =c\) are
- (a)
a, c
- (b)
b, c
- (c)
a, b
- (d)
a + c, b + c
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:
If \(\alpha ,\beta \) are the roots of the equation \(1!+2!+3!+...\left( x-1 \right) !+x!={ k }^{ 2 }\) and \(k\epsilon I\) , where \(\alpha <\beta \) and if \({ \alpha }_{ 1 },{ \alpha }_{ 2 },{ \alpha }_{ 3 },{ \alpha }_{ 4 }\) are the roots of the equation \(\left( a+\sqrt { b } \right) ^{ { x }^{ 2 }-\left[ 1+2\alpha +3{ \alpha }^{ 2 }+{ 4\alpha }^{ 3 }+{ 5\alpha }^{ 4 } \right] }+\left( a-\sqrt { b } \right) ^{ { x }^{ 2 }+\left[ -5\beta \right] }=2a\) where a2 - b = 1 and [.] denotes the greatest integer function, then the value of \(\left| { \alpha }_{ 1 }+{ \alpha }_{ 2 }+{ \alpha }_{ 3 }+{ \alpha }_{ 4 }-{ \alpha }_{ 1 }{ \alpha }_{ 2 }{ \alpha }_{ 3 }{ \alpha }_{ 4 } \right| \) is
- (a)
216
- (b)
221
- (c)
224
- (d)
209
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 :
If \(\left( \sqrt { \left( 49+20\sqrt { 6 } \right) } \right) ^{ \sqrt { a\sqrt { a\sqrt { a...\infty } } } }+\left( 5-2\sqrt { 6 } \right) ^{ { x }^{ 2 }+x-3-\sqrt { x\sqrt { x\sqrt { x...\infty } } } }=10\) where a = x3 - 3, then x is
- (a)
\(-\sqrt{2}\)
- (b)
\(\sqrt{2}\)
- (c)
- 2
- (d)
2
Let f(x)=ax2+bx+c; a,b,c \(\in \) R and a≠0, Suppose f(x)>0 for all x\(\in \)R. Let g(x)=f(x)+f'(x)+f''(x) Then
- (a)
g(x)>0 ∀ x \(\in \) R
- (b)
g(x)<0 ∀ x \(\in \) R
- (c)
g(x)=0 has non real complex roots
- (d)
g(x)=0 has real roots
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+3bx+c=0, then the value of \(\sum { (\alpha -\beta )^{ 2 }(\beta -\gamma ) } ^{ 2 }\) is equal to
- (a)
9b2
- (b)
27b2
- (c)
81b2
- (d)
243b2