Assignments Class 11 Mathematics Complex Numbers and Quadratic Equations

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Complex Numbers and Quadratic Equations Assignments Class 11 Mathematics

Question. The period of f(x) = x −[x] if it is periodic, is:   
(a) f(x) is not periodic
(b) 1/2
(c) 1
(d) 2

Question. If f(x) = 2x-3/x-2 then [ f{f(x)}] equals:   
(a) x
(b) –x
(c) x/2
(d) -1/x

Question. If f : R → R, f(x) = 2x − 1 and g : R → R, 2 g(x) = x then (gof )(x) equals?   
(a) 2x²-1
(b) (2x − 1)²
(c) 4x² – 2x -1
(d) x²+2x-1

Question. For every integer n, let n a and n b be real numbers. Let function f : R→ R be given by 

If f is continuous, then which of the following hold(s) for all n?
(a) a_n-1 -b_n-1 =0
(b) a_n -b_n =0
(c) a_n -b_n-1 =0
(d) a_n-1 -b_n =0

Question. Suppose that g(x) =1+√x and f (g(x)) = 3+2√x+x, then f(x) is:   
(a) 1+2x²
(b) 2+x²
(c) 1+x
(d) 2+x

Question. Let f: (-Π/2 , Π/2) → R be given byf(x)=[log(sec x+tan x)]³.   
Then:
(a) f(x) is an odd function
(b) f(x) is a one-one function
(c) f(x) is an onto function
(d) f(x) is an even function

Question. Let g(x) be a function defined on [–1,1]. If the area of the equilateral triangle with two of its vertices at (0, 0) and [x, g(x)] is √3 / 4, then the function g(x) is:   
(a) g(x) = ± √1− x
(b) g(x) = √1− x
(c) g(x) = − √1− x
(d) g(x) = √1+ x

Question. If f : R→ R is given by f (x) = 3x − 5, then f(x)−1 ?   
(a) Is given by 1/3x-5
(b) Is given by x+5/3
(c) Does not exist because f is not one-one
(d) Does not exist because f is not onto

Question: Let f : R → R be defined by f (x) = 3x − 4, thenf (x) −1 is:   
(a) 3x + 4
(b) 1/3x-4
(c) 1/3(x+4)
(d) 1/3(x-4)

Question. The period of the function f(x) = sin²x is:   
(a) π/2
(b) π
(c) 2π
(d) 3π

Question. If y =f(x) = x+2/x-1 , then:   
(a) x = f(y)
(b) f(1) = 3
(c) y increases will x for x < 1
(d) f is a rational function of x

Question. If g(x)x² +x – 2 and 1/2 (gof x) = 2x² – 5x + 2, then f (x ) is equal to:   
(a) 2x − 3
(b) 2x + 3
(c) 2x² +3x+1
(d) 2x² -3x-1

Question. Let L be the set of all straight lines in the Euclidean plane.   
Two lines l1 and l2 are said to be related by the relation R
if l1 is parallel to l2. Then the relation R is:
(a) Reflexive
(c) Symmetric
(b) Transitive
(d) Equivalence

Question. If S is the set of all real x such that 2x-1/2x³+3x²+x is positive, then S contains:   
(a) (-∞, -3/2)
(b) (-3/2 ,-1/4)
(c) (-1/4 ,-1/2)
(d) (1/2 ,3)

Question. If 2 2 f (x) = cos[π²]x + cos[−π²]x, where [x] stands for the greatest integer function, then:   
(a) f (π / 2) = −1
(b) f (π ) = 1
(c) f (−π ) = 0
(d) f (π / 4) = 1

Question. The period of f(x) = sin (πx/n-1) + cos (πx/n) n ∈ Z, n > 2 is:   
(a) 2πn(n − 1)
(b) 4 n (n − 1)
(c) 2n(n −1)
(d) None of these

Question. Let f : (0,1)→ R be defined by f(x) = b-x/1-bx where b is a constant such that 0 < b <1.Then:   
(a) f is not invertible on (0, 1)
(b) f ≠ f^-1 one (0, 1) and f(b) =1/f'(0)
(c) f = f^-1 one (0, 1) and f(b) =1/f'(0)
(d) 1 f − is differentiable on (0, 1)

Question. For every pair of continuous function f , g : [0,1]→ R such that max { f (x) : x∈[0,1]} = max{g(x) : x∈[0,1]}.   
The correct statement: (s) is (are)
(a) [ f (c)]² + 3 f (c) = [g(c)]² + 3g(c) for some c∈[0,1]
(b) [ f (c)]² + f (c) = [g(c)]² + 3g(c) for some c∈[0,1]
(c) [ f (c)]² + 3 f (c) = [g(c)]² + g(c) for some c∈[0,1]
(d) [ f (c)]² = [g(c)]² for some c∈[0,1]

Assertion and Reason

Note: Read the Assertion (A) and Reason (R) carefully to mark the correct option out of the options given below:
(a) If both assertion and reason are true and the reason is the correct explanation of the assertion.
(b) If both assertion and reason are true but reason is not the correct explanation of the assertion.
(c) If assertion is true but reason is false.
(d) If the assertion and reason both are false.
e. If assertion is false but reason is true.

Question. Consider the following relation R on the set of real square matrices of order 3. 1 R {(A,B) : A P BP = = − for some invertible matrix P}   
Assertion: R is an equivalence relation.
Reason: For any two invertible 3×3 matrices M and N.

Question. Assertion: The curve y = x2/2 + x+ 1 is symmetric with respect to the line x = 1.   
Reason: A parabola is symmetric about its axis.

Question. Assertion: Let A{2, 3, 7, 9}and B = {4, 9, 49, 81} f:A →   
B is a function defined as f(x) = x2. Then is a bijection from A to (b)
Reason: A function f from a set A to a set B is a bijection
if f(A) = B and f(x1) ≠ f(x2) if x1 ≠ x2 for all x1, x2 ∈A and n(A) = n(B).

Question. Consider the following relations. R= {(x,y)| x,y}are real numbers and x=xy for some rational number w} s = [(m/n , p/q)] m,n,p,q are integer such that n.q ≠ 0 and qm = pn}
Assertion: S is an equivalence relation but R is not an equivalence relation.   
Reason: R and S both are symmetri(c)

Question. Let R be a relation on the set N of natural numbers defined by n Rm ⇔ n is a factor of m (i.e., n | m) : 
Assertion: R is not an equivalence relation
Reason: R is not symmetric

Question. Let A = {1,2,3}and B = {3,8}?   
Assertion: (A∪B) × (A∩B) = {(1,3), (2,3), (3,3)(8,3)}
Reason: (A×B) ∩ (B×A) = {(3,3)} .

80. Let f(x) = sinx + cosx.g(x) = sin x/1-cosx ?   
Assertion: f is neither an odd function nor an even function.
Reason: g is an odd function.

Question. Assertion: f:R→R is a function definead by f (x) = 5x + 3.   
If 1, g = f-1 then g(x) = x-3/5.
Reason: If f : A→B is a bijection and g : B→ A is the inverse of f, then fog is the identity function on a.   

Question. Let F(x) be an indefinite integral of sin2 x:   
Assertion: The function F(x) satisfies F(x + π) = F (x) for all real x.
Reason: sin2 (x + π) = sin2 x for all real x.

Question. Let f be a function defined by f (x) = (x −1)2 +1, (x ≥1)   
Assertion: The set {x : f(x) = f-1(x)} = {1,2}
Reason: f is a bijection and f-1(x) = 1+√x-1 ,x ≥ 1.

Question. Let X and Y be two sets:   
Assertion: X ∩(Y ∩ X ) ‘ =φ
Reason: If X∪Y has m elements and X∩Y has n elements then symmetric difference X Y has m − n elements.

Question. Let f(x) = 2 + cos x for all real x: 
Assertion: For each real t, there exists a point c in [t,t +π] such that f’(c) = 0.
Reason: f’(t) = f (t + 2π) for each real t.

Question. Assertion: A function f:R→R satisfied the equation f (x) – f(y)   
= x – y∀x, y∈R and f (3) = 2, then f (xy) = xy −1
Reason: f (x) = f (1/ x)∀x∈R, x ≠ 0, and f (2) = 7 / 3 if f(x) = x²+x+1/x²-x+1.

Comprehension Based

Paragraph-I

Let f be a function satisfying f(x) = ax /ax +√a = ge(x)(a>0)

Question. Let f(x) = g9(x), then the value of 

is: (where [.] denotes the greatest integer function)   
(a) 995
(b) 996
(c) 997
(d) 998

Question. If the value of (Imge 18) then the value of n is:   
(a) 493
(b) 494
(c) 987
(d) 988

Question. Let f(x) = g4(x) ,then (Imge 18 )   
(a) zero
(b) even
(c) odd
(d) none of these

Question. The value of (Imge 18 )   
(a) 0
(b) 2n–1
(c) 2n
(d) none of these

Question. The value of g5 (x) + g5 (1–x) is:   
(a) 1
(b) 5
(c) 10
(d) none of these

Paragraph-II

Let F(x) = f (x) + g(x),G(x) = f (x) − g(x) and H(x) = f(x)/g(x) , where 2 f (x) = 1− 2sin x and g(x) = cos 2x,∀f : R →[−1,1] and g : R →[−1,1].

Question. If F: R → [–2, 2], then:   
(a) F (x) is one – one function
(b) F (x) is onto function
(c) F (x) is into function
(d) none of the above

Question. If the solutions of F (x) – G (x) = 0 are x₁,x₂,x₃,…xn where x∈[0, 5π], then: 
(a) x1,x2,x3,…xn are in AP with common difference π/4
(b) the number of solution of F (x) – G (x) = 0 is 10, ∀x∈[0, 5π].
(c) the sum of all solutions of F(x) −G(x) = 0,∀x∈[0,5π ] is 25π
(d) (b) and (c) are correct

Question. Which statement is correct?   
(a) period of f(x), g(x) and F(x) makes are AP with common difference π/3
(b) period of f(x), g(x) and F(x) are same and is equal to 2π
(c) sum of periods of f(x), g(x) and F(x) is 3π
(d) sum of periods of f(x), g(x) and F(x) is 6π

Question. Which statement is correct?   
(a) the domain of G(x) and H(x) are same
(b) the rang of G(x) and H(x) are same
(c) the union of domain of G(x) and H(x) are all real
(d) the union of domain of G(x) and H(x) are rational numbers

Question. Domain and range of H (x) are respectively: 
(a) R and {1}
(b) R and {0, 1}
(c) (Imge 18)

Question. Let the functions defined in Column I have domain (–π/2, π/2) and range (–∞,∞)? 

Question. Let f(X) = x²-6+5/x²-5x+6 ? 

Question. Let f1 : R→R, f2 :[0,∞]→R, f1 : R→Rand 4 f : R →[0,∞) be 

 

Question. √2i equals?     
a. 1 + i
b. 1 – i
c. − √2i
d. None of these

Question. The reflection of the complex number 2-i/3+i ) (where i = −1) in the straight line z(1+ i) = z̅ (i −1) is:       
a. −1-i/2
b. −1+i/2
c. i(i + 1)/2
d. −1/1+i

Question. In the arg and diagram, if O ,P and Q represents respectively the origin, the complex numbers z and z + iz, then the angle ∠OPQ is   
a. π/4
b. π/3
c. π/2
d. 2π/3

Question. Real part of e^eθ is     
a. e^cosθ [cos(sinθ)]
b. e^cosθ [cos(cosθ)]
c. e^cosθ [sin(cosθ)]
d. e^cosθ [sin(sinθ)]

Question. iⁱ is equal to:    
a. e^π/2       
b. e^-π/2
c.-π/2
d. None of these

Question. If in the diagram, A and B represent complex number z1 and z2 respectively, then C represents: 

a. z¹ + z²
b. z¹ – z²
c. z¹. z²
d. z¹ / z²

Question. If (1-i/1+i)100 = a+ib , then:   
a. a = 2, b = –1
b. a = 1, b = 0
c. a = 0, b = 1
d. a = –1, b = 2

Question. If the complex number z1,z2 and the origin form an equilateral triangle then z₁² + z₂² =      
a. z₁ + z₂
b. z₁z̅₂
c. z̅₂z₁
d. |z₁|² =  |z₂|²

Question. The points 1+3i, 5+i and 3 + 2i in the complex plane are     
a. Vertices of a right angled triangle
b. Collinear
c. Vertices of an obtuse angled triangle
d. Vertices of an equilateral triangle

Question. The amplitude of  e^eθ is equal to:     
a. sinθ       
b. − sinθ
c. e^cosθ
d. e^sinθ 

Question. Let z1,z2 be two complex numbers represented by points on the circle |z| = 1and |z| = 2 respectively, then :     
a. max |2z1 + z2| = 4
b. min |z1 – z2| =1
c. |z2 + 1/z1| ≥ 3
d. none of these

Question. If z = x + iy, then area of the triangle whose vertices are points z, iz and z + iz is     
a. 2|z|²
b. 1/2 |z|²
c. |z|²
d. 3/2 |z|²

Question. The locus of the points z which satisfy the condition arg (z–1 /z+1 ) = π/3  is:   
a. A straight line
b. A circle
c. A parabola
d. None of these

Question. The locus of z satisfying the inequality log1/3 |z+1| > log1/3 |z – 1| is:     
a. R(z) < 0
b. R(z) > 0
c. I (z) < 0
d. None of these

Question. If 1 α + iβ = tan− (z), z = x + iy and α is constant, the locus of ‘z’ is:     
a. x² + y² + 2xcot 2α = 1
b. cot 2α (x²+y²) = 1+ x
c. x² + y² + 2xsin 2α = 1
d. x² + y² + 2ytan 2α = 1

Read the following passage and answer the questions. Let A, B, C be three sets of complex number as defined below
A = {z : Imz ≥1},B={z:| z−2 − i |= 3 |}
C = {Z;Re((l − i)z) + √2}

Question. Let z be any point in A∩B∩C .The | z +1− i |² + | z − 5 − i |² lies between:   
a. 25 and 29
b. 30 and 34
c. 35 and 39
d. 40 and 44

Question. The number of elements in the set A∩B∩C is:     
a. 0
b. 1
c. 2
d.∞

Question. Let z be any point in A∩B∩C and let w be any point satisfying | w− 2 − i |< 3. Then, | z | − | w | +3 lies between:     
a. −6 and 3
b. −3 and 6
c. −6 and 6
d. −3 and 9

Question. If a,b,c, are distinct integers and ω ≠ 1 is a cube root of unity and if minimum value of | a + bω + cω² | + | a + bω² + 2ω | 1/ 4 = n then the value of n must be equal to:     

Question. If the equation of all the circles which are orthogonal to |z|     
= 1and | z +1|= 4 is | z + 7 − ib | = √(√+b²) , i = √−1 and b∈R, then the value of λ must be equal to :

Question.

 (where I =√-1 then the value of λ must be equal to: