Mathematics - Polynomials
Exam Duration: 45 Mins Total Questions : 30
If the quotient is 3x2 - 2x + 1, remainder is 2x - 5 and divisor is x + 2, what is the dividend?
- (a)
3x3- 4x2+ x- 3
- (b)
3x3- 4x2- x + 3
- (c)
3x3 + 4x2- x+ 3
- (d)
3x3 + 4x2- x- 3
Given P = Product of x2y and \(\frac { x }{ y } \) and Q = Quotient obtained when x2 is divided by D. If P = Q,what is the value of D?
- (a)
0
- (b)
1
- (c)
x
- (d)
\(\frac { 1 }{ x } \)
Identify one of the factors of \({ x }^{ 2 }+\frac { 1 }{ { x }^{ 2 } } +2-2x-\frac { 2 }{ x } \) from the following
- (a)
\(x-\frac { 1 }{ x } \)
- (b)
\(x+\frac { 1 }{ x } -1\)
- (c)
\(x+\frac { 1 }{ x } \)
- (d)
\({ x }^{ 2 }+\frac { 1 }{ { x }^{ 2 } } \)
If (x2+ 3x + 5) (x2- 3x + 5) = m2 - n2,find m.
- (a)
x2-3x
- (b)
3x+5
- (c)
x2+5
- (d)
x2-5
If x + Y + z = 0, what is the value of x3+y3+z3?
- (a)
xyz
- (b)
2xyz
- (c)
3xyz
- (d)
0
If \({ x }^{ \frac { 1 }{ 3 } }+{ y }^{ \frac { 1 }{ 3 } }+{ z }^{ \frac { 1 }{ 3 } }=0\) which of the following is true?
- (a)
x3 + y3 + z3= 0
- (b)
x + y + z = 27xyz
- (c)
(x + y + z)3 = 27xyz
- (d)
x3 + y3 + z3= 27xyz
If y - 2 and y - \(\frac { 1 }{ 2 } \) are the factors of py2 + 5y + r, which of the following holds good?
- (a)
p > r
- (b)
p = r
- (c)
p < r
- (d)
Both (A) and (C)
Which of the following polynomials has -5 as a zero?
- (a)
(p - 5)
- (b)
x2- 25
- (c)
p2 - 5p
- (d)
x2+ 5
Factorise: x2- z2+y2 - p2 + 2pz - 2xy
- (a)
(x-y-p-z)(x-y-p+z)
- (b)
(x-y+p-z)(x-y-p+z)
- (c)
(x+y+p-z)(x+y+p+z)
- (d)
(x+y-p+z)(x-y-p+z)
Find the zeroes of the polynomial p(z) = (4z + \(\pi\) ) (z - 4\(\pi\)).
- (a)
\(4\pi ,-\frac { \pi }{ 4 } \)
- (b)
\(-4\pi ,\frac { \pi }{ 4 } \)
- (c)
\(4\pi ,\frac { \pi }{ 4 } \)
- (d)
\(-4\pi ,-\frac { \pi }{ 4 } \)
If (x - 1) is a factor of polynomial f(x) but not of g(x), it must be a factor of which of the following polynomials?
- (a)
f (x) g (x)
- (b)
-f(x) + g(x)
- (c)
f'(x) - g(x)
- (d)
{f(x) + g(x)}g(x)
If (x - a) (x - b) are factors of polynomial g(x), which of the following statements is correct?
- (a)
g(a) = 0, g(b)\(\neq \) 0
- (b)
g(a) =0, g(b) =0
- (c)
g(a) \(\neq \) 0, g(b) \(\neq \)0
- (d)
g(a) \(\neq \) 0, g(b) = 0
Factorise:4a2+ 9b2+ c2 + 12ab + 4ac + 6bc
- (a)
(2a + 3b + c)2
- (b)
(a+3b+2c)2
- (c)
(a-3b+2c)2
- (d)
(2a+3b-c)2
What are the factors of \({ x }^{ 3 }+{ x }^{ 2 }-\frac { 1 }{ { x }^{ 2 } } +\frac { 1 }{ { x }^{ 3 } } \)
- (a)
\(\left( { x }^{ 2 }+1 \right) \left( x+\frac { 1 }{ x } -1+\frac { 1 }{ { x }^{ 2 } } \right) \)
- (b)
\(\left( x+1 \right) \left( { x }^{ 2 }+\frac { 1 }{ { x }^{ 2 } } -1+\frac { 1 }{ x } -x \right) \)
- (c)
\(\left( x+\frac { 1 }{ x } \right) \left( { x }^{ 2 }+x-1-\frac { 1 }{ x } +\frac { 1 }{ { x }^{ 2 } } \right) \)
- (d)
\(\left( { x }^{ 2 }+\frac { 1 }{ { x }^{ 2 } } \right) \left( x+\frac { 1 }{ x } -1 \right) \)
Given that (1 - x) (1 + x + x2 + x3 + x4) is \(\frac { 31 }{ 32 } \) and x is a rational number, what is 1 + x + x2+ x3 + x4+ x5?
- (a)
\(\frac { 31 }{ 64 } \)
- (b)
\(\frac { 63 }{ 32 } \)
- (c)
\(\frac { 63 }{ 64 } \)
- (d)
\(\frac { 31 }{ 32 } \)
If p(x) = 4x3 - 3x2 + 2x + 1, q(x) = x3- x2+ x + 1 and r(x) = x2- 2x + 1 find the value of 3p(x) + 7q(x) + r(x).
- (a)
19x3 - 15x2 + 11 x + 11
- (b)
- 19x3 - 15x2 + 11 x - 11
- (c)
19x3 - 15x2 - 11x + 11
- (d)
19x3 - 15x2 - 11 x - 11
The remainder when x4-y4 is divided by x-y is ______
- (a)
0
- (b)
x+y
- (c)
x2-y2
- (d)
2y4
When p(x) = x3 + ax2 + 2x + a is divided by (x + a), the remainder is _______
- (a)
0
- (b)
a
- (c)
-a
- (d)
2a
x12 - y12 =_____
- (a)
(x - y)(x2 + xy + y2)(x + y)(x2 - xy + y2) (x2 + y2)(x4 - x2y2 + y4)
- (b)
(x+y)(x2-xy+y2)(x+y)(x2-xy+y2)(x2+y2)(x4-x2y2+y4)
- (c)
(x+y)(x2+xy-y2)(x+y)(x2-xy+y2)(x2+y2)(x4-x2y2+y4)
- (d)
(x-y)(x2-xy+y2)(x+y)(x2-xy+y2)(x2+y2)(x4-x2y2+y4)
Given that x = 2 is a solution of x3-7x+6=0 The other solutions are ________
- (a)
-1,3
- (b)
1,-3
- (c)
1,-2
- (d)
-1,-2
The product (a + b) (a - b) (a2 - ab + b2) (a2 + ab + b2) is equal to ______
- (a)
a6+b6
- (b)
a6-b6
- (c)
a3-b3
- (d)
a3+b3
Find the remainder when the expression 3x3 + 8x2 - 6x + 1 is divided by x + 3.
- (a)
1
- (b)
10
- (c)
6
- (d)
0
If x2- 1 is a factor of ax4+ bx3 + cx2 + dx + e, then
- (a)
a + b + e = c + d
- (b)
a + b + c = d + e
- (c)
b + c + d = a + e
- (d)
None of these
If a, b, c are all non-zeroes and a + b + c = 0, then \(\frac { { a }^{ 2 } }{ bc } +\frac { { b }^{ 2 } }{ ca } +\frac { c^{ 2 } }{ ab } \) = _______
- (a)
0
- (b)
1
- (c)
2
- (d)
3
Length, breadth and height of a cuboidal tank are (x - 3y)m, (x + 3y)m and (x2 + 9y2)m respectively. Find the volume of the tank.
- (a)
(x3 + 3xy + 27y3)m3
- (b)
(x4 + 2x2y2+ 81y4)m3
- (c)
(x4- 81y4)m3
- (d)
(x4+ 81y4)m3
A rectangular field has an area (35x2 + 13x - 12)m2. What could be the possible expression for length and breadth of the field?
- (a)
(5x + 4)m and (7x - 3)m
- (b)
(3x + 9)m and (7x-12)m
- (c)
Both (A) and (B)
- (d)
None of these
Area of a rectangular field is (2x3 - 11x2 - 4x + 5) sq. units and side of a square field is (2x2 + 4) units. Find the difference between their areas (in sq. units).
- (a)
4x2 - 2x3 - 27x2 - 4x + 11
- (b)
4x4 - 2x3 + 27x2 + 4x + 11
- (c)
4x4 + 27x2 + 4x-11
- (d)
4x4 + 2x3 + 27x2 + 4x + 11
If (5x2 + 14x + 2)2 - (4x2 - 5x + 7)2 is divided by (x2 + x + 1), then quotient 'q' and remainder 'r' respectively, are ______
- (a)
(x2 + 19x - 5), 0
- (b)
9(x2 + 19x- 5), 0
- (c)
(x2 + 19x- 5), 1
- (d)
9(x2 + 19x- 5), 1
Match the following:
Column-I | Column-II |
(P) If f(x) = x3 - 6x2 + 11x- 6 then f(-1)=_____ |
(i) -210 |
(Q) If f(x)=2x3-13x2+17x+12, then f(-3) = ______ |
(ii) 1 |
(R) If x=\(\frac{4}{3}\)is a root of f(x)=6x3-11x2+kx-20, then k= _____ |
(iii) -24 |
(S) If x=-1 is aroot of f(x)=x100+2x99+k, then k= _____ |
(iv) 19 |
- (a)
(P) ⟶ (iii); (Q) ⟶ (iv); (R) ⟶ (i); (S) ⟶ (ii)
- (b)
(P) ⟶ (ii); (Q) ⟶ (iv); (R) ⟶ (i); (S) ⟶ (iii)
- (c)
(P) ⟶ (iii); (Q) ⟶ (i); (R) ⟶ (iv); (S) ⟶ (ii)
- (d)
(P)~ (iii); (Q) ~ (ii); (R) ~ (i); (S) ~ (iv)
Study the given statements.
Statement-I :
\(\frac { ({ a }^{ 2 }-{ b }^{ 2 })^{ 3 }+({ b }^{ 2 }-{ c }^{ 2 })^{ 3 }+({ c }^{ 2 }-{ a }^{ 2 })^{ 3 } }{ (a+b)^{ 3 }+(b+c)^{ 3 }+(c+a)^{ 3 } } \) =(a+b)(b+c)(c+a)
Statement-II: a2 + b2 + c2 - ab - bc - ca
=\(\frac{1}{2}\)[(a-b)2+(b-c)2+(c-a)2]
Which of the following options holds?
- (a)
Both Statement-I and Statement-II are true.
- (b)
Statement-I is true but Statement-II is false
- (c)
Statement-I is false but Statement-II is true.
- (d)
Both Statement-I and Statement-II are false.