FUNCTIONS - 1
1) Let A={1, 2,3}, B={2,3,4} , then which of the following is a function from A to B?
a) {(1,2),(1,3),(2,3),(3,3)}
b) {(1,3),( 2,4)}
c) {(1, 3),(2,2) ( 3,3}
d) {(1,2),(2,3),(3,2),(3,4)}
2) If f: Q--> Q is defined as f(x)= x², then f(9) is equal to
a) 3 b) -3 c) {-3,3} d) φ
3) Which one of the following is not a function?
a) {(x,y): x, y ∈ R, x²= y}
b) {(x,y): x, y ∈ R, x= y²}
c) {(x,y): x, y ∈ R, x= y³}
d) {(x,y): x, y ∈ R, x³= y}
4) If f(x)= cos(log x), then
f(x²) f(y²)- (1/2){ f(x²/y²}+ f(x²y²)} has the value
a) -2 b) -1 c) 1/2 d) none
5) If cos(log x), then f(x)f(xy) - (1/2){f(x/y)+ f(xy)} has the value
a) -1 b) 1/2 c) -2 d) none
6) Let f(x)= |x -1|. Then
a) f(x²)={f(x)}²
b) f(x+ y)=f(x)f(xy)
c) f(|x|)= |f(x)| d) none
7) The range of f(x)= cos[x], for -π/2< x<π/2 is
a) {-1,1,0} b) {cos 1, cos2, 1} c) {cos1, - cos 1, 1} d) [-1,1]
8) Which of the following are functions ?
a) {(x,y): y²= x, x, y ∈ R}
b) {(x,y): y = |x|, x, y ∈ R}
c) {(x,y): x²+ y²= xp1, x, y ∈ R}
d) {(x,y): x²- y²= 1, x, y ∈ R}
9) If f(x)= log{(1+ x)/(1- x)} and g(x)= (3x + x³)/(1+ 3x²), then f(g(x)) =
a) f(3x) b) f(x)}³ c) 3f(x) d) - f(x)
10) If A={1,2,3}, B={x,y}, then the number of functions that can be defined from A into B is
a) 12 b) 8 c) 6 d) 3
11) If f(x)= log{(1- x)/(1- x)}, then f{(2x)/(1+ x²)} =
a) (f(x))² b) (f(x))³ c) 2f(x) d) 3f(x)
12) If f(x)=cos(logx), then value of
f(x)f(4) - (1/2) {f(x/4)+ 4f(x)} is
a) 1 b) -1 c) 0 d) ±1
13) If f(x)= (2ˣ + 2⁻ˣ)/2, then f(x+ y)f(x - y) is equal to
a) (1/2) [f(2x)+ f(2y)]
b) (1/2) [f(2x)- f(2y)]
c) (1/4) [f(2x)+ f(2y)]
d) (1/4) [f(2x) - f(2y)]
14) If f(x) - 3 f(1/x)=x²(x ≠ 0), then f(2) is equal to
a) -7/4 b) 5/2 c) -1 d) none
15) Let f: R--> R be defined by f(x)= 2x + |x|. Then f(2x)+ f(- x)- f(x)=
a) 2x b) 2|x| c) -2x d) -2|x|
16) If f(x)=log{(1+ x)/(1- x)} thenf{2x/(1+ x²)} is equal to
a) (f(x))² b) (f(x))³ c) 2f(x) d) 3f(x)
17) If x≠ 1 and f(x)= (x+1)/(x-1) is real function, then f(f(f(2))) is
a) 1 b) 2 c) 3 d) 4
18) If f(x)= cos(logx), then f(1/x)f(1/y) - (1/2) {f(xy)+ f(x/y)} is equal to
a) cos(x - y) b) log(cos(x - y)) c) 1 d) cos(x + y)
19) Let f(x)=x, g(x)= 1/x and fph(x)= f(x)g(x). Then, h(x)= 1
a) x ∈R b) x ∈Q c) x ∈R - Q d) x ∈R, x ∈Q
20) If f(x)=(sin⁴x + cos²x)/(sin²x + cos⁴x) for x ∈R , then f(2002)=
a) 1 b) 2 c) 3 d) 4
21) The function f: R--> R is defined by f(x)=cos²x + sin⁴x. Then f(R)=
a) [3/4,1) b) (3/4,1] c) [3/4,1] d) (3/4,1)
22) Let A={x ∈R: x≠ 0, -4≤ x ≤4} and f: A ∈ R be defined by f(x)= |x|/x for x ∈ A. Then [1,-1] b) x: 0≤ x ≤4] c) [1] d) {x: -4≤ x≤0}
23) If f: R--> R and g: R-->R are defined by f(x)=2x +3 and g(x)= x²+7, Then the value of x such that g(f(x)) = 8 are
a) 1,2 b) -1,2 c) -1,-2 d) 1,-2
26/7/26
RELATIONS - 1
1) If A={1, 2,4}, B={2,4,5}, C={2,5}, the pn (A- B) x (B - C) is
a) {(1,2),(1,5),(2,5)} b) {(1,4)} c) (1,4) d) {(1,4)}
2) If R is a relation on the set A={1,2,3,4,5,6,7,8,9} given by x R y <=> y= 3x, then R=
a) {(3,1),(6,2),(8,2),(9,3)}
b) {(3,1),(6,2),(9,3)}
c) {(3,1),(2,6),(3,9)} d) none
3) Let A={1,2,3}, B={1,3,5}, if relation R from A and B is given by R={(1,3),(2,5),(3,3)}. Then, R⁻¹ is
a) {(3,3),(3,1),(5,2)}
b) {(1,3),(2,5),(3,3)}
c) {(1,3),(5,2)} d) none
4) If A={1,2,3}, B={1,4,6,9} and R is a relation from A to B defined by 'x' is greater than y. The range of R is
a) {1,4,6,9} b) {4,6,9} c) {1} d) none
5) If R={(x,y: x, y∈ Z, x²+ y²≤ 41} is a relation on Z, then domain of R is
a) {0,1,2} b) {0,-1,-2} c) {-2,-1,0,1,2} d) none
6) A relation R is defined from {2,3,4,5} to {3,6,7,10} by : x R y <=> x relatively prime to y. Then , domain of R is
a) {2, 3,5} b) {3,5} c) {2,3,4} d) {2,3,4,5}
7) a relation φ from C to R is refined by x φ y <=> |x|= y. Which one is correct ?
a) (2+ 3i) φ 13 b) 3φ (-3) c) (1+ i) φ 2 d) iφ 1.
8) Let R be a relation on N defined by x+2= 8. The domain of R is
a) {2,4,8} b) {2,4,6,8} c) {2,4,6} d) {1,2,3,4}
9) R is a relation from {11, 12, 13} to { 8,10,12} defined by y= x -3. Then, R ⁻¹ is
a) {(8,11),( 10, 13)}
b) {( 11, 8),( 13, 10)}
c) {10, 13),(8, 11),(12, 10)} d) none
10) if the set A has p elements , B has q elements, then the number of elements in A x B is
a) p+ q b) p+ q +1 c) pq d) p²
11) Let R be a relation from a set A to a set B, then
a) R= AUB b) R= A ∩B c) R ⊆ A x B d) R⊆ B x A
12) If R is a relation from a finite set A having m elements to a finite set B having n elements, then the number of relations from A to B is
a) 2ᵐⁿ b) 2ᵐⁿ -1 c) 2mn d) mⁿ
13) If R is a relation on a finite set having n elements, then the number of relations on A is
a) 2ⁿ b) ₂n² c) n² d) nⁿ.
19/7/26
COMPLEX NUMBER - 2
1) The value of (1+ i)(1+ i²)(1+ i³)(1+ i⁴) is
a) 2 b) 0 c) 1 d) i
2) If (3+ 2i sinθ)/(1- 2i sin θ) is a real number and 0 <θ< 2π, then θ =
a) π b) π/2 c) π/3 d) π/6
3) If (1+ i)(1+ 2i)(1+ 3i)......(1+ ni)= a + ib, then 2x 5x 10 x......x (1+ n²) is equal to
a) √(a²+ b²) b) √(a² - b²) c) a²+ b² d) a² - b² e) a+ b
4) If √(a + ib)= x + iy, then possible of √(a - ib) is
a) x² + y² b) √(x² + y²) c) x + iy d) x - iy e) √(x² - y²)
5) z= cos(π/4) + sin(π/6), then
a) |z|= 1, arg(z)=π/4
b) |z|= 1, arg(z)=π/6
c) |z|= √3/2, arg(z)=5π/24
d) |z|= √3/2, arg(z)= tan⁻¹(1/√2)
6) The polar form is (i²⁵)³ is
a) cos(π/2) + i sin (π/2)
b) cos π+ i sinπ
c) cos π- i sinπ
d) cos (π/2) - i sin(π/2)
7) If i² = -1, then the sum i + i² + i³ + ..... upto 1000 terms is equal to
a) 1 b) -1 c) i d) 0
8) If z= -2/(1+ i √3), then the value of arg(z) is
a) π b) π/3 c) 2π/3 d) π/4
9) If a= cosθ + i sinθ, then (1+ a)/(1- a) =
a) cot(θ/2) b) cot θ c) i cot(θ/2) d) i tan(θ/2)
10) If (1+ i)(1+ 2i)(1+ 3i)......(1+ ni)= a + ib, then 2,5,10,17....(1+ n²)=
a) a - ib b) a² - b² c) a² + b² d) none
11) If (a² +1)²/(2a - i) = x + it, then x² + y² is equal to
a) (a²+1)⁴/(4a² +1)
b) (a+1)²/(4a² +1)
c) (a² -1)²/(4a² - 1)² d) none
12) The principal value of the amplitude of (1+ i) is
a) π/4 b) π/12 c) 3π/4 d) π
13) The principal positive integer n such that {2i/(1+ i)}ⁿ is a positive integer, is
a) 16 b) 8 c) 4 d) 2
14) If z is a non-zero complex number, then | |conjugate z|²/z. Conjugate z| is equal to
a) |conjugate z/z|
b) |z| c) |conjugate z| d) none
15) If a= 1+ i, then a² equals
a) 1- I b) 2i c) (1+ i)(1- i) d) i - 1
16) If (x + iy)¹⁾³= a + ib , Then x/a + y/b =
a) 0 b) 1 c) -1 d) none
17) (√-2)(√-3) is equal to
a) √6 b) -√6 c) i √6 d) none
18) The argument of (1- i √3)/(1+ i √3) is
a) 60° b) 120° c) 210° d) 240°
18/7/26
SETS - 1
1) For any set A, (A'b)' is equal to
a) A'b b) A c) φ d) none
2) Let A and B be two sets in the same universal set. Then A - B=
a) A∩B b) A'∩B c) A∩B' d) none
3) The number of subsets of a set containing n elements is
a) n b) 2ⁿ -1 c) n² d) 2ⁿ
4) For any two sets A and B, A ∩(A U B)=
a) A b) B c) φ d) none
5) If A={1,3,5,B} and B={2,4}, then
a) 4 ∈ A b) {4} ⊂ A c) B ⊂ A d) none
6) The symmetric difference of A and B is
a) (A- B) ∩(B - A)
b) (A- B) U(B - A)
c) (A U B) - ( A∩B)
d) {(A U B) - A} U {(AUB) - B}
7) The symmetric difference of A={1,2,3} and B={3,4,5} is
a) {1,2} b) {1,2,4,5} c) {4,3} d) {2,5,1,4,3}
8) For any two sets A and B, (A- B) U (B - A)=
a) (A- B) U A b) (B- A) U B c) (AU B) - (A∩B )
9) Which of the following statements is false:
a) A- B = A ∩B' b) A- B= A - (A ∩ B)
c) A- B= A - B' d) A- B = (AU B) - B
10) For any three sets A, B and C
a) A ∩(B - C)= (A∩B) - (A ∩C)
b) A ∩(B - C)= (A ∩B) - C
c) AU(B - C)= (AU B) ∩ (A U C')
d) AU(B - C)= (AU B) - (A U C).
11) Let A={x: x ∈R, x≥ 4} and B={x ∈ R: x < 5}, then A ∩B=
a) (4,5) b) (4,-5) c) (-4,5) d) (-4,-5)
12) Let U be the universal set containing 700 elements. If A, B are subsets of U such that n(A)= 200, n(B)= 300 and n(A∩B)= 100. Then n(A' ∩ B')=
a) 400 b) 600 c) 300 d) none
12/7/26
Complex number
1) Simplify: 1+ i² + i⁴+ i⁶.
2) Show that: [i³⁷ - (1/i)⁴¹]³= 8i.
3) Find the conjugate of (2+ 3i)².
4) Find x and y if (3x -7)+ 5iy = 2y +3 -4(1- x)i.
5) Find the modulus of -12+ 5i.
6) Express the reciprocal of the complex number 3+ i √5 in the form of a+ ib.
7) Find the modulus of (1+ i)/(1- i) + (1- i)/(1+ i).
8) Find the solution of the equation |1- i|ˣ= 2ˣ.
9) Show that the points representing the complex numbers (3+ 3i), (-3-3i) and (-3√3+ 3√3 i) are the vertices of an equilateral triangle.
10) Express (3+ i)/(-5- 4i) in the standard form a+ ib.
11) Find the modulus of (2+ 3i)/(3+ 2i).
12) Prove that the representative points of the complex numbers 1+ 4i, 2+ 7i, 3+ 10i are collinear.
13) Find the modulus of
{(3+2i)(1+ i)(2+ 3i)}/{(3+ 4i)(4+ 5i)}.
14) If x+ iy = √{(a+ ib)/(c + id)}, show that (x²+ y²)²= (a²+ b²)/(c²+ d²).
Short Questions Answer Type
i) Find the conjugate of the complex number (2+ 3i)/(2 - 3i). -5/13 -12i/13
ii) Find the modulus. of (1 + i)³/(1 - i³). 2
iii) Find the amplitude of -3-3i. -3π/4
iv) Resolve into factors: a²+ ab + b². (a - bω)(a - bω²)
v) Find the square root of -i . ±(1/√2) (1- i)
vi) Evaluate: (1- ω²)(1- ω⁴)(1- ω⁸)(1- ω¹⁶). 9
vii) If z= x + iy and |z - 2| = |2z - 1|, prove that x²+ y²=1.
viii) Find the smallest positive integer n for which {(1+ i)/(1- i)}ⁿ=. 1. 4
ix) Find the square root of q+ √(q²- 1), 0< q < 1. ±(1/√2) {√(1+ q)+ i √(1- q)}
x) If x = a+ b, y= aω + bω², z= aω²+ bω , find xyz. a³+ b³
xi) Solve: |z| - z = 1+ 2i. (z = x+ iy). z= 3/2 - 2i
xii) The sum and the product of the two complex numbers are respectively 6 and 25. Find the numbers. 3+4i, 3 -4i
xiii) Determine the three cube roots of i. - i, (i+√3)/2, (i - √3)/2
Choose the correct option
i) The absolute value of (-i/2) is
A) i/2 B) 1/2 C) 1/√2 C) none
ii) The value of √-4 . √-9 is
A) 6 B) -6 C) -6 or 6 D) none
iii) The value of ω⁴+ ω⁸+ ω⁻¹. ω⁻² is
A) ω B) ω² C) -1 D) 0
iv) The argument of ia (a < 0), is
A) 0 B) π/2 C) 3π/2 D) π
v) The value of (1- ω+ω²)⁵ + (1+ ω- ω²)⁵.
A) -1 B) 1 C) -32 D) 32
vi) The amplitude of (a + ib)² is
A) tan⁻¹(b/a) B) 2tan⁻¹(b/a) C) 2 tan⁻¹(a/b) D) tan⁻¹(a/b)
vii) If z = x + iy, the value of (amp z + amp of Conj of z) is
A) 0 B) π/2 C) π D) none
viii) The quantity, whose cube root is (1/2) (√3 + i), is
A) -1 B) 1 C) - i D) i.
ix) The value of (1+ ω)(1+ω²)(1+ ω⁴)(1+ ω⁸)......2n factors, is
A) 1 B) 2ⁿ C) -1 D) ωⁿ
x) For any complex number z, the minimum value of |z|+|z -1| is
A) 0 B) 1/2 C) 1 D) 3/2
xi) The real part of (2- i)²/(2+ i) is
A) -2/5 B) -6/5 C) -11/5 D) none
i b ii b iii d iv c v d vi b vii a viii d ix a x c xi d
General Questions
1) Show that, (1- ω+ω²)(1-ω²+ ω⁴)(1- ω⁴ +ω⁸)+ ..... to 2n factors = 2²ⁿ.
2) If X + iY be one of the cube roots of x + iy, prove that, 4(X²- Y²)= x/X + y/Y.
3) If x= a + b, y= aα + bβ, z= aβ + bα, where α, β are complex cube roots of unity, show that, x³+ y³+ z³= 3(a³+ b³).
4) If x= ω²- ω -2, evaluate x⁴+ 3x³+ 2x² --11x -4. 3
5) If x = 2- i √3, find the value of K from the equation 2x⁴- 5x³- 3x²+ 41x + K=0. -35
6) If x+ iy =√{(a+ ib)/(c + id)}, prove that (x²+ y²)²= (a²+ b²)/(c²+ d²).
7) If a = cos α + i sin α, b= cosβ+ i sinβ, c= cosγ + i sinγ and a+ b +c=0, show that a²+ b²+ c²=0.
8) If z = x + iy and (z -i)/(z -1)= ib, prove that (x - 1/2)²+ (y - 1/2)²= 1/2.
9) If (1+ x + x²)ⁿ = a₀ + a₁x + a₂x² + a₃x³+ ......+ a₂ₙx²ⁿ , then show that a₀ + a₃ + a₆ + .......= 3ⁿ⁻¹.
10) Prove that (a+ bω+ cω²)³+ (a+ bω²+ cω)³ = (2a- b - c)(2b - c - a)(2c - a - b) and 27abc, if a+ b + c =0.
11) If (1+ i)(1+ 2i)(1+ 3i)......(1+ ni)= a+ ib, then show that 2.5.10......(1+ n²)= a²+ b².
12) If If cos k + i sin k and 1+ √(1- a²) = na, prove that (a/2n) (1+ nx)(1+ n/x) = 1+ a cos k.
13) If z = x + iy and arg{(z -1)/(z +1)}=π/4 , show that the locus of (x,y) is a circle.
14) If ω be a Complex cube root of unity, find the simplified value of (a+ bω + cω²)/(c + aω + bω²) + (a+ bω + cω²)/(b + cω + aω²) . -1
15) Prove that the expression x³ᵖ + x³ᑫ⁺¹ + x³ʳ⁺² , where p, q, r are integers, is divisible by x²+ x+1.
16) Show that the points 2+ 3i, 0 ai 1/(-2+ 3i) are collinear.
17) Express a+ ib in the form pω + qω²). (b/√3 - a)ω + (-b/√3 - a)ω² or (- b/√3 - a)ω + (b)√3 - a)ω²
18) Show that the sum and the product of two complex numbers are real if and only if they are conjugate of each other.
19) Solve: z²+ Conj of z=0. (z= x + iy). 0, -1, 1/2 ± i√3/2
20) If a²+ b²+ c²=1 and b + ic = (1+ a)z, then prove that (1+ iz)/(1- iz) = (a+ ib)/(1+ c).
11/7/26
1995
1) Find a quadratic equation whose one root is a square root of -47 + 8√-3.
2) Prove that the expression (x² - yz)³+ (y² - zx)³+ (z² - xy)³ - 3(x² - yz)(y² - zx)(z² - xy) is a perfect square and find its square root.
4) Prove that tan20 tan40 tan80=√3.
5)) Show that tan225 cot405 + tan765 cot675= 0
6) If (r, θ) denotes the polar coordinates then the equation r cos²(θ/2)= 1 represents
a) a circle b) a parabola c) an ellipse d) none
7) If x lies in the interval [0,1] then the minimum value of x²+ x+ 1 is
a) 3/4 b) 1 c) 3 d) none
10//7/26
1) If p= a+ b+ c, q= a+ wb+ w²c, r= w²b+ wc where w is a nonreal cube root of 1, show that p³+ q³+ r³ - 3pqr= 27abc.
2) Solve: x¹⁾³ + (2x -3)¹⁾³ = {12(x -1)}¹⁾³.
3) In an arithmetic progression of n terms (n is even) the two middle terms are p- q, p+ q respectively. Show that the sum of the squares of all the terms of the progression is n[p² + (n² -1)q²/3].
4) Show that 3[sin⁴(3π/2 - x) + sin⁴(3π+ x)] - 2[sin⁶(π/2+ x) + sin⁶(5π - x)] is independent of x.
.5) If 3ˣ - 3ˣ⁻² = 8, find the value of xˣ.
6) If cosα + cosβ = cos(3π/7) and sinα + sinβ= sin(3π/7) find cos²{(α -β)/2}.
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