MATH- TEST PAPERS

Friday, 21 August 2026

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Choose the correct option

i) The sum of the reciprocals of the roots of 4x²+ 3x+7 =0 is

A) 7/4 B) -7/4 C) -3/7 D) 3/7

ii) If one root of 5x²-6x+K =0 be reciprocal of other, then

A) K= 6 B) K= 5 C) K= -5 D) 1/5

iii) If x be real, the maximum value of 5+ 4x - 4x² will be

A) 5 B) 6 C) 1 D) 2

iv) The roots of x²+ 2(3m+5)x+ 2(9m²+ 25) =0 will be the real if

A) m> 5/3 B) m= 5/3 C) m<5/3 D) m=0

v) The equation (4- n)x²+ (2n+4)x+ 8n+1 =0 has equal integral roots, if

A) n =0 B) n = 1 C) n= 3 D) none

vi)) The equation whose roots are the reciprocals of the roots of ax²+ bx+c =0, is

A) bx²+ cx+a =0 B) cx²+ bx+a =0

C) bx²+ ax+c =0 D) cx²+ ax+b =0

vii) The value of the expression (ax)²+ bx+c, for any real x, will be always positive, if

A) b²- 4ac > 0 B) b²- 4ac < 0

C) b²- 4a²c > 0 D) b²- 4a²c < 0

viii) The value of m for which equation x² - x+ m² =0 has no real roots, can satisfy

A) m> 1/2 B) m > -1/2 C) m <-1/2 D) m< 1/2

ix) If x be real and a> 0 , the least value of ax²+ bx+c will be

A) -b/a B) - b/2a C) -(b²- 4ac)/2a D) -(b² - 4ac)/4a

x) The roots of ax²+ bx+c =0 will be both negative, if

A) a>0, b>0, c< 0 B) a>0, c>0, b <0

C) a>0 , b>0 , c>0 D) b>0, c>0, a < 0

xi) α,β are the roots of x²- 2x+ 2 =0, the least integer n(>0) or which αⁿ/βⁿ=q is

A) 2 B) 3 C) 4 D) none

I c ii b iii b iv b v c vi b vii d viii and c ix d x c xi c

General Questions

1) If roots of 2x²+ x+ 1 =0 are p and q, form an equation whose roots are p²/q and q²/p. 4x²- 5x+2 =0

2) The equation x²- cx+d =0 and x²- ax+ b =0 have one root common and the second equation has equal roots. Prove that ac = 2(b + d).

3) If the roots of x²+ 3x+ 4=0 are α and β, form an equation whose roots are (α - β)² and (α + β)². x²- 2x -63=0

4) if the roots of x²- px +q =0 are in the ratio 2: 3, show that 6p²= 25.

5) If the roots of ax²+ bx + c =0 are m, n, form an equation whose roots are 1/(m+ n) and 1/m + 1/n. bcx²+(ac+b²)x+ ab=0

7) Show that if one root of ax²+ bx + c =0 be the square of the other, then b³+ a²c + ac² = 3abc.

9) If the ratio of the roots of ax² + cx + c =0 be p: q, show that, √(p/q) + √(q/p) + √(c/a) = 0.

10) If m be a root of the equation 4x²+ 2x - 1 =0, prove that its other root is 4m²- 3m.

11) If the sum of the roots of 1/(x+ p) + 1/(x+ q) =1/r be equal to zero, show that the product of the roots is (1/2) (p²+ q²).

12)

13)

14) if one root of the equation ax²+ bx + c =0 be the cube of the other, show that ac(a+ c)²= (b²- 2ac)².

15) if m²= 5m.- 3, n²= 5n - 3 but m ≠ n, then find an equation whose roots are m/n and n/m. 3x²- 19x +3=0

16) The coefficient of x in x²+ px + q =0 is misprinted 7 for 13 and the roots are therefore obtain (-2) and ( - 15). Find the roots of the original equation. -3, -10

17) If b³+ a²c + ac² = 3abc, then what relation may exist between the roots of the equation ax²+ bx + c =0 ? One root is the square of the other

18) Find the maximum and minimum values of: x/(x²- 5x +9). 1, -1/11

19) If m, n are the roots of ax²+ 2bx + c =0, form an equation, whose roots are mω+ bω² and mω²+ nω. (ax - b)²= 3(ac - b²).

20) If √m+ √n denote the roots of x²- px + q =0, show that the equation, whose roots are m± n is (4x - p²)²= (p²- 4q)².

21) Prove that for all real values of x, the value of p²/(1+ x) - q²/(1- x) is real.

22) if x be real, prove that 4(a- x){x - a+ √(a²+ b²)} can never be greater than (a²+ b²).

23) if the quadratic x²+ px + q =0 and x²+ px + p =0 have a common root, Prove that their other roots will satisfy the equation. x²+ x + pq =0

24) Show that that if a, b, c are real, the roots of the equation (b - c)x²+ (c - a)x + (a - b) = 0 are real and they are equal if a, b, c are in AP.

25) If the roots of the equation ax²+ 2bx + b =0 are complex, show that the roots of the equation bx²+ (b - c)x - (a+ c -b) = 0 real.

26) Prove that the roots of the equation (x - a)(x - b)+ (x - b)(x - c)+ (x - c)(x - a)= 0 are always real and cannot be equal unless a= b = c.

27) If a, b, c are real, show that the roots of the equation 1/(x + a) + 1/(x + b)+ 1/(x + c) = 3/x are real.

28) Show that the equation (b- c)x²+ (c - a) x + (a - b) =0, (c - a)x²+ (a - b)x + (b - c)=0 have a common root. Find it and the remaining roots of the equations. 1, (a - b)/(b - c) and (b - c) (c - a)

29) Prove that the roots of the equation (a - b)²x²+ 2( a+ b - 2c)x + 1=0 are real or complex according as c does not or does lie between a and b.

30) Prove that if the equation ax²+ bx + c =0 and bx²+ cx + a=0 have a common root, then either a+ b + c=0 or a= b= c.

31) if the equation ax + by =1 and cx²+ dy² =1 have only one solution, prove that, a²/c + b²/d =1 and x= a/c, y = b/d.

32) if (a- λ)x²+ (b - λ)y² + (c - λ)z²+ 2fyz + 2gzx + 2hxy is a perfect square, show that prove that a - gh/f = b - hf/g = c - fg/h = λ.

33) Prove that x²+ y²+ z²+ 2ayz + 2bzx + 2cxy can be resolved into two rational factors if a²+ b²+ c²- 2abc =1.

34) If a, b are the roots of x²- px + q =0 and Uₙ= aⁿ + bⁿ, then prove that Uₙ₊₁ = p Uₙ - q Uₙ₋₁ Hence evaluate a⁵+ b⁵. p⁵- 5p³q + 5pq²

35) If x is be real, prove that the expression (x²- 2x cos 2k +1)/(x²- 2x cos 2m +1) lies between sin²k/sin²m and cos²k/cos²m.

36) Find K so that the value of x given by K/2x = a/(x + c) + b/(x - c) may be equal.

If K₁ , K₂ are the two values of K and x₁ , x₂ the corresponding values of x, show that K₁K₂ = (a - b)² and x₁ x₂ = c².

37) If the equation x²- p₁x + q₁ =0, x²- p₂x + q₂ =0, x² - p₃x + q₃ =0 have each pair a common root, prove that, p₁²+ p₂²+ p₃²+ 4(q₁ + q₂ + q₃) = 2(p₁p₂ + p₂p₃ + p₃p₁).

COMPLEX NUMBER

Useful factor results:

1. Any number of the form x + iy, where x, y real is called and i=√-1, is called a Complex number.

e.g. 0 = 0 + i0 is a complex number as it is the form of x + iy.

The complex number x + iy is usually denoted by z, where x is called the real part and y imaginary part of z.

a) z is purely real if y=0.

b) z is purely imaginary if x =0.

2. The conjugate of the complex number z= x+ iy is x - iy and is denoted by z .

3. The Modulus or Absolute value of z is written as mod z or | z |.

If z = x+ iy, then mod z =|z| = |x + iy| = +√(x²+ y²).

Note. z. Conjugate of z= (x+ iy)(x - iy) = x²+ y² = |z|² or |Conj of z|².

4. The Amplitude or Argument of z is written as amp z or arg z.

If z= x+ iy, then amp z or arg z = θ = tan⁻¹(y/x).

The value of θ satisfying cos θ = x/√(x²+ y²) and sin θ = y/√(x²+ y²) , - π< θ ≤π is called the θ

Note: z= x + iy = r(cosθ + i sin θ) where r= mode z & θ= amp z.

5. a) if a + ib=0, then a= 0, b =0.

b) if a + ib=c + id, , then a= c, b =d.

c) If √+a + ib) = x + iy, then √(a - ib) = x - iy.

6. If z₁ = x₁ + iy₁ and z₂ = x₂ + iy₂ be two complex numbers, then

a) |z₁ z₂|= |z₁|. |z₂|.

b) amp(z₁ z₂)= amp z₁ + amp z₂.

c) |z₁/z₂| =|z₁|/|z₂|

d) amp z₁/z₂ = amp z₁ - amp z₂.

e) |z₁ + z₂| ≤ |z₁| + |z₂|.

f) |z₁ - z₂| ≤ |z₁| + |z₂|.

g) |z₁ - z₂| ≥ |z₁| - |z₂|.

7. The cube roots of Unity i.e. the roots of x³= 1 are 1, 1/2(-1+√-3), 1/2(-1-√-3) of which the first is real and the last two are Complex. If any one of the complex roots be denoted by ω, the other root is ω². So the cube roots of unity are 1, ω, ω².

Important relations :

a) ω³= 1

b) ω³ⁿ = 1, n is an integer.

c) 1+ ω +ω²= 0.

Note.

If α, βare complex to root of 1, then

a) α³= 1, β³=1

b) αβ =1

c) α + β =-1.

8. Square root of a complex quantity (by inspection ):

For finding the square root of a+ ib, the imaginary part b, provided b is a multiple of 2, must be expressed as 2. x, y such that the difference of squares of x and y is a

Note:

If b is not a multiple of 2, then a+ ib = 1/2(2a + i 2b).

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).

PROGRESSION (AP, GP, HP)

Short Questions Answer Type

i) The first and the last term of an AP are 9 and 96. If its sum be 1575, then find its number of terms. 30

ii) The product of three numbers in GP is 64/125. Find the middle number. 4/5

iii) The nth term of an AP is p. show that the sum of the first (2n-1) terms is (2n -1)p.

iv) If p-th term of an AP is q and q-th term is p, then find n-th and (p+q)th term of the AP. p+q- n, 0

v) For What value of m, the sum of first m terms of the two series 3+ 10+ 17+... and 63 + 65 +67+... are equal ? 25

vi) The 5th term and the sum of the first 5 terms of an AP are 30 and 100 respectively. Find the sum of the first 10 terms. 325

vii) If a, b, c, d be in AP and x, y, z in GP, prove that, xᵇ⁻ᶜ. yᶜ⁻ᵃ. zᵃ⁻ᵇ =1

viii) Find the sum of all two digit natural numbers. 4905

ix) Find the sum of 1 + 11 + 111+...to n terms. (10/81)(10ⁿ -1) - n/9

x) The sum of the first n terms of an AP is 3n²+ 4n. What will be its nth term ? Which term of the AP is 121 ? 6n+1, 20

xi) If a²+ 3a+2, 3a²+ 2a+5, 4a²+ 5a+4 are in AP, then find a. 2

xii) Find the coefficient of x⁹⁹ in the polynomial (x-1)(x-2)(x-3)....(x-100). - 5050

xiii) Let Sₙ denote the sum of the first n terms of a GP. If S₂ₙ = 4Sₙ, then find the ratio S₃ₙ/Sₙ. 13

Choose the correct option

i) If a, b, c are in AP, then 2ᵃ , 2ᵇ , 2ᶜ will be in

a) AP b) GP c) HP d) none on them

ii) The third term of a GP is 4. The product of the first five terms is

a) 4² b) 4³ c) 4⁴ d) 4⁵

iii) The sum of first n terms of an AP is n²; the common difference is

a) 1 b) 2 c) 3 d) none

iv) If x, 2x+ 1, 4x+5, ......are in GP, then its 4th term is

a) 8 b) 18 c) 27 d) none of these

v) The sum of the first n terms 1 + 2 + 4 + 8 + .... is 255. The value of n will be

a) 7 b) 8 c) 9 d) none of these

vi) The p-th, q-th, r-th terms of a GP will be in GP, if p, q, r are in

a). AP b) GP c) HP d) no sequence

vii) The AM and GM between two positive numbers p and q are equal, then

a) p = q b) p > q c) p <q d) pq =1

viii) The number of terms of the series 96+ 48 +24 +...3/16 is

a) 9 b) 10 c) 8 d) none

ix) Let Sₙ denote the sum of the first n terms of an AP. If S₂ₙ = 3Sₙ, then the ratio S₃ₙ/Sₙ, is

a) 4 b) 6 c) 8 d) none

i.b ii d iii b iv c v b vi a vii a viii b ix b

GENERAL QUESTIONS

1) The arithmetic mean of two positive numbers is 15 and their geometric mean is 9. Find the numbers. 3,27

2) Show that the sum of an AP, whose first term is a, second term is b and last term c, is equal to {(a+ c)(b+ c - 2a)}/2(b -a).

3) Divide 26 into 3 parts such that they form a GP, and their product is 216. 2,6,18 or 18,6,2

4) if a,b,c are in AP prove that a(1/b + 1/c), b(1/c + 1/a), c(1/a + 1/b) are also in AP.

5) Find the nth term and the sum of the first n terms of the series 1 + 3 + 7 + 15 +.... 2ⁿ -1, 2ⁿ⁺¹ - 2 - n

6) The sum of n arithmetic means between two numbers is S. Show that the sum of m arithmetic means between the same two numbers is mS/n.

7) The sum the first p terms of an AP is q and the sum of the first q terms is p. Find the sum of first (p+q) terms. -(p+q)

8) If Sₙ denotes the sum of first n terms of an AP, Show that Sₙ₊₃ - 3Sₙ₊₂ + 3Sₙ₊₁ - Sₙ=0.

9) If the sum of the first 3 terms of an AP, of n terms is p and the sum of the last 3 terms is q, show that, Sₙ= n(p+q)/6.

10) If P be the continued product of n terms of a GP., whose first term is a and nth term is b, then prove that P²= (ab)ⁿ.

11) If S be sum, P the product and R the sum of the reciprocals of n terms of a GP., show that P²= (S/R)ⁿ.

12) If the sum of first P terms of an AP is equal to the sum of first Q terms, show that the sum of first (P+ Q) terms will be zero.

13) If u₁, u₂, u₃,.....are in AP and Sₘ= u₁+ u₂+....+uₘ. If uₘ= 4, u₄ₘ = 24 and S₄ₘ= 44Sₘ, find u₁ and m. -2,10

14) If a be the AM between b and c and G₁ , G₂ be two GMs between b and c, show that G₁³+ G₂³ =2abc.

15) If the sum p terms of an AP is to the sum of the q terms as p²: q², Show that p-th term/q-th term = (2p-1)/(2q -1).

16) If a₁, a₂, a₃,.......aₙ be in AP, show that 1/(a₁a₂) + 1/(a₂a₃) +.....+ 1/(aₙ₋₁aₙ)= (n -1)/(a₁aₙ).

17) If G be the GM and p and q be two AMs between two given quantities, prove that G²= (2p - q)(2q -p).

18) In a set of four numbers , the first three are in GP and the last three are in AP, with common difference 6. If first is the same as the fourth , find the four numbers. 8,-4,2,8

19) If Sₙ = 1+ 1/2 + 1/2² + .....+ 1/ⁿ⁻¹, then calculate the least value of n such that 2 - Sₙ < 1/100. 8

20) n geometric means are inserted between 5 and 320. The sum of the numbers and the means is 635. Find n. 5

21) Find the sum fall odd numbers which are perfect square between 90 and 890. 4330

22) The side of a right angled triangle are in AP and the smallest side is 12 cms. Find the largest side. 20cm

23) if a,b,c are in AP prove that 1/(√b + √c), 1/(√c + √a), 1/(√a + √b) are also in AP.

24) If a², b², c² are in AP, prove that 1/(b + c) , 1/(c + a), 1/(a + b) are also in AP.

25) The angles of a polygon are in AP. The least angle is 120° and the common difference is 5°. Find the number of sides of the polygon. 9

26) Solve : (x+1)+ (x+4) + (x+7)+........+ (x+121)= 2542. 1

27) Three fractions a/x , b/y, c/z are in GP. The three numbers are a,b,c are in AP with common difference d and the three denominators x,y,z are also in an AP with common difference d'. prove that d/d' = ± b/y.

28) If a₁, a₂, a₃, .....a₂ₖ are in AP, show that a₁² - a₂² + a₃² - a₄² + ....+ a²₂ₖ₋₁ - a²₂ₖ = k(a²₁ - a²₂ₖ/(2k -1).

29) The sum of the first n terms of a GP of 3n terms is S₁ , the sum of the next n terms is S₂ and the sum of the last n terms is S₃. Show that S₁ , S₂, S₃ are in GP.

30) Solve: 1+ a + a²+ a³+......+aˣ⁻¹+ aˣ = (1+ a)(1+ a²)(1+ a⁴)(1+ a⁸). 15

31) If a, r and Sₜ denote respectively the first term, common ratio and sum of the first t terms of a GP and if

Uₙ = S₁ + S₂ + S₃ +....+ Sₙ, then show that rSₙ+ (1- r)Uₙ = na.

32) If S₁, S₂, S₃,.....Sₙ are the sums of n terms of n geometrical progression, whose first terms are each 1 and common ratios are 1,2,3,....n, show that

S₁+ S₂ + 2S₃ + 3S₄+......+(n -1)Sₙ = 1ⁿ + 2ⁿ+ 3ⁿ+......+nⁿ.

33) The series of natural numbers is divided into groups:

(1); (2,3,4); (5,6,7,8,9);..... and so on. Show that the sum of the numbers in the n-th group is (n -1)³ + n³.

MULTIPLE AND SUBMULTIPLE

Short Questions:

i) If sinx + cosx=p, find sin2x. p²-1

ii) If sinx + cosx=. √2, find cos2x. 0

iii) If tanx= 1/3, find sin2x, cos2x, tan2x. 3/5,4/5,3/4

iv) Show cos2x + tanx sin2x =1.

v) If cos2x = 4/5, find tanx and cotx. ±1/3,±3

vi) Evaluate: (cos³x - cos3x)secx + (sin³x + sin3x)cosecx. 3

vii) If 0≤ x ≤ π/4 and sin2x= 4/5, find tanx. 1/2

viii) If 5sin²x + 3cos²x=4, find cos2x, sin2x. 0,1

ix) Show: 2 sin(π/8)= √{2 - √2}.

x) If sinx = - 4/5, find cos(x/2), sin(x/2), where x is an angle of third quadrant. -1/√5,2/√5

xi) If cosx=4/5, find cos2x, sin2x. 7/25, ±24/25

xii) If cotx - tanx = 2, find cot2x. 1

xiii) Show, 4(cos³10+ sin³20)= 3(cos10+ sin20).

xiv) If tanx tan 3x=1, find tan2x. ±1

xv) Show, 1/(sin10) - √3/(cos 10)= 4.

xvi) Find the maximum value of cos³x sinx - sin³x cosx. 1/4

xvii) If tan²β = (1- sinα)/(1+ sinα), Show α + 2β =π/2.

xviii) If cosx = 3/4, show 32sin(x/2) sin(5x/2)= 11.

xix) If cosx + sinx =√2 cosx, show, sin4x =1.

xx) Simplify: cos²(π/2 - x) - sin(2π/3 - x) sin(x - π/3). 3/4

xxi) If cos⁶x + sin⁶x + k sin²2x=1, find the value of k. 3/4

xxii) Show, (1+ cos(π/8))(1+ cos(3π/8))(1+ cos(5π/8))(1+ cos(7π/8)) = 1/8.

xxiii) Show, (2sinx)/(sin3x) + (tanx)/(tan3x)= 1.

Choose the Correct Option:

i) The value of 6 sin20- 8 sin³20 is

a) -1 b) 1 c) -√3 d) √3

ii) If tanx= b/a, then a cos2x + b sin2x equal to

a) a b) b c) a+ b d) none

iii) If tan(x/2)= 7, the value of 4sinx - 3 cosx is

a) 5 b) 4 c) 3 d) none

iv) If sinx - cosx =√2 sinx, the value of cot2x is

a) -1 b) 0 c) 1 d) √2

v) If cosx = -4/5 and sin2x = 24/25, the quadrant in which x lies, is

a) 1st quadrant b) 2nd quadrant c) 3rd quadrant d) 4th quadrant

vi) If π/2 < x < π and 3 sin²x + 5 cos²x = 4, the value of sin2x is

a) 1 b) 0 c) -1 d) none

vii) The value of sin (π/10) sin (13π/10) is

a) -1/4 b) 1/4 c) -√5/4 d) √5/4

viii) If 2 sec2x = tany + cot y, then one of the values of x + y is

a) π b) π/2 c) π/4 d) none

ix) If tan(x/2)= 2, then tanx will be

a) -5/3 b) -4/3 c) -3/5 d) 4/5

x) If 180°< x < 270° and sinx = - 3/5, then tan(x/2) is

a) -3 b) -1/3 c) -3 or -1/3 d) none

i d ii a iii b iv c v c vi c vii a viii c ix b x a

GENERAL QUESTIONS

1) Show, secx + tanx = tan(π/4 + x/2).

2) If tan²x = 1+ 2 tan²y, then show that

a) cos2y = 1+ 2 cos2x.

b) cos2x + sin²y = 0.

3) Show, tanx + 2 tan2x + 4 tan4x + 8 cot8x = cotx.

4) Show, (3+ cos4x)/(1- cos4x)= (1/2) (cot²x + tan²x).

5) Evaluate:

a) cos(π/5) cos(3π/5). -1/4

b) cos(π/5) - cos(2π/5). 1/2

6) Show, cos²(A - 60)+ cos²A+ cos²(A+ 60)= 3/2.

7) If x, y are acute angles and cos2x = (3 cos2y -1)/(3- cos2y), then show that, tanx = √2 tany.

8) Show, cos²40° cos²80° + cos²80° cos²20+ cos²20° cos²40° = 9/16.

9) If tanx= (tany + tanz)/(1+ tany tanz) show that sin2x= (sin2y + sin2z)/(1+ sin2y sin2z).

10) If tanx = 1/7 and tan y= 1/3, then show that x + 2y =π/4.

11) Show: tan 9 - tan 27 - tan 63 + tan 81=4.

12) Show cos⁴(π/16) + cos⁴(3π/16) + cos⁴(5π/16) + cos⁴(7π/16) =3/2

13) Evaluate: sin²73+ sin²47- sin73 sin47. 3/4

14) Show: 1+ cos56+ cos58 - cos66 = 4 cos28 cos29 sin33.

15) If sinx + sin2x= m and cosx + cos 2x = n, then show that (m²+ n²)(m²+ n² -3)= 2n.

16) If sec(x + a)+ sec(x - a) = 2secx, show that cosx =√2 cos(a/2).

17) If a= π/(2ⁿ +1), show that 2ⁿ cosa cos2a cos4a.....cos2ⁿ⁻¹a=1.

18) If tanx = a/b, show that, a cosec(x/3) - b sec(x/3)= 2√(a²+ b²).

19) Show that, cos2a = 2 sin²b+ 4 cos(a + b) sina sinx + cos2(a+ b).

20) If tan a/tanb = (1+ cos²a)/(1+ sin²a), show that, sin(3a+ b)= 7sin(a - b).

21) Show: sin²18+ sin² 24+sin²36+sin²42= 1+ sin²6+ sin²12.

22) Show that cot(15/2) = √2+ √3+ √4 + √6.

23) If cosx + cosy = a and sinx + siny= b, find sin(x + y) and cos(x - y) in terms of a, b. 2ab/(a²+ b²), (a²+ b²-2)/2

24) Evaluate: sin 36 sin 72 sin108 sin 144. 5/16

25) If tan(x/2)= √{(1- e)/(1+ e)} tan(y/2) show that cost = (cosx -e)/(1- e cosx).

26) If tan(x + y - z)/tan(x - y + z)= tany/tanz, show that sin(y - z)=0, or sin2x + sin2y + sin2z=0.

27) If a cosx + b sinx = c and b cosecx - a secx = c, show that, tan2x = 2ab/(a²- b²+ c²).

28) If (1+ √(1+ a))tanx = 1+ √(1- a), show that, sin4x= a.

29) show that: cosec (π/7) = cosec (2π/7) + cosec(3π/7).

30) Show: cos(π/32) = (1/2) √[2+ √{2+ √(2+ √2)}]. Hence or otherwise evaluate sin(π/32). (1/2) √[2- √{2+ √(2+ √2)}].

31) Show: cot70+ 4 cos70=√3.

32) show: 4 sin10+ √3 tan10= 1.

33) If tanx = (sina sinb)/(cosa+ cosb), show that one of the values of tan(x/2) is tan(a/2) tan(b/2)

34) if tan(A+ B)= 3 tanA, show sin(2A+ 2B)+ sin2A = 2 sin2B.

35) If tanx=√{(a-b)/(a+b)} tan(A/2), and cosy= (acos A +b)/(a+ bcosA), then show that, y= 2x.

36) If x=a(cosy+ siny sin 2y), and y= a(siny + cosy sin 2y), show that, (x+y)²⁾³+(x-y)²⁾³= 2a²⁾³

37) If cos a= cosx cosy, and cos b= cos m cosy, and tan(a/2)= tan(b/2)= tan(y/2) then prove sin²y= (secx -1)((sec m -1)

38) If tan(x+y), Tanz, tan(x-y) are in G. P. Prove that tan(x+z), tanx, tan(x-z) are also in G. P.

39) If tanx tany= √{(p-q)/(p+q)} prove (p -q cos 2x)(p-q cos 2y)= p² - q²

40) If xy+ y z+ z x= 1, show that x/(1-x²) + y/(1-y²) + z/(1-z²) = 4xyz/{(1-x²)(1-y²)(1-z²)}

41) If tanx= n(secx -1)², then prove that, cot³(x/2) - cot(x/2)= 2n

42) sin(π/14) sin(3π/14) sin(5π/14)= 1/8.

43) If x, y are two values of z satisfying atanz + b sec z= c, show tan(x+y)= 2ac/(a²- c²)

44) Show: tan 10+ tan 70 - tan 50=√3

45) If 3 sin²x + 2sin²y= 1 and 3sin 2x - 2 sin 2y=0, where x, y are positive acute angles, then prove that, x + 2y=π/2.

46) If cos³x/cos (y-3x) = sin³x/sin(y- 3x) = K, then show that, 2K² - K cosy -1= 0.

47) Show: cos(π/7) cos(3π/7)+ cos(3π/7) cos(5π/7)+ cos (5π/7) cos(π/7) = - 1/2

48) show: cos(π/11) + cos(3π/11)+ cos(5π/11) + cos(7π/11) + cos(9π/11) = 1/2.

49) If tan(x+y)= a+b and tan(x-y)= a - b, then show that, a tanx - b tany = a² - b².

50) If the equation acos 2x + b sin 2x= c has y and z as its solutions, Prove that,

A) tany+ tan z= 2b/(c+a)

B) tan y tanz= (c-a)/(c+a)

51) Show : sec²(π/16)+ sec²(3π/16)+ sec²(5π/16)+ sec²(7π/16)= 32 Hence, or otherwise find the value of tan²(π/16)+ tan²(3π/16) + tan²(5π/16)+ tan²(7π/16). 28

52) If {sin²(x+y)}/{sin²(z+y)}= sin 2x/sin 2z, show tanx tanz= tan²y

53) Find the range of the value of (3cosx + 4 sinx) sinx. Mx9/2, mn -1/2

54) Show: cos(3π/7) cos(4π/7) cos(6π/7)= 1/8.

55) Show: cos(2π/7) + cos(4π/7)+ cos(6π/7)= -1/2

56) sin(2π/7) + sin(4π/7) + sin(8π/7)= √7/2

Friday, 10 July 2026

XI test 26/27

MATHS- 2

(Attempt all questions from section A and all questions Either from section B or Section C

Section - A (80 Marks)

1) (10 x 2 =20)

a) Let T{x: (x+5)/(x -7) - 5 = (4x -40)/(13- x)}. is T an empty set?

b) Find the domain and the range of the function

f(x)= 1/√(5- x).

c) Show: tan50= tan40+ 2 tan 10.

d) In a triangle ABC, a= 1, b=√3 and c= 7/6. Find the other two angles.

e) If (2+ 3i)/(3- 4i)= a + ib, find the value of a and b.

f) If m,n be the roots of the equation ax²+ bx + c= 0 form an equation whose roots are 1/m. 1/n.

g) Find n if ⁿC₅ = ⁿC₇.

Or cos2A/(1+ sin2A)= tan(π/4 - A).

h) Find dy/dx of (xeˣ + x⁷+ (x +1)√x)/x.

i) Evaluate: lim ₓ→₀ (sinx - 2 sin3x + sin5x)/x.

j) While shuffling a pack of 52 cards, 2 cards are accidentally dropped. Find the probability that the missing cards are of different colours.

2) a) If f(x)= y= (ax - b)/(cx - a), then show that f(y)= x where x≠ a/c.

b) Find the term containing x¹⁰ in expansion of (2x²- 3/x)¹¹. (2+2)

3) a) Show that √{(1+ sinx)/(1- sinx)}+ tan(π/4+ x/2).

b) Solve for x: 4 sin⁴x + cos⁴x = 1. (2+2)

4) Using mathematical induction show that 5ⁿ⁺¹ + 4. 6ⁿ - 9 is divisible by 20 and n ∈ N.

If (z -i)/(z -1) is purely imaginary show that the point z lies on the circle whose centre is the point (1/2) (1+ i) and radius is 1/√2. (4)

5) 7 candidates are to be examined, 2 in mathematics and the remaining in different subjects. In how many ways can they be seated in a row so that the two examinees in mathematics may not sit together.

A bag contains six white marbles and five red marbles. Find the number of ways in which four marbles can be drawn from the bag if---

i) They can be of any colour

ii) two must be white and two must be red. (4)

6) Determine whether the expansion of (x²- 2/x)¹⁸ will contain a term containing x¹² ? If it contains also find out the term. (4)

7) Find the equation of the lines through the point (3,2) which make and angle of 45° with the line x - 2y = 3. (4)

8) Find the equation of the circle passing through the point (7,3) having 3 units and whose centre lies on the line y= x -1. (4)

Find the equation of tangents to the circle x²+ y²- 2x - 4y - 20 =0 through the point P(8,1).

9) Differentiate the function (x³+ 2x) by first principal of differentiation. (4)

10)a) If the equation 3x²+ px +1=0 and 2x²+ ax +1=0 have a common root, show that 2p²+ 3q²- 5pq +1= 0.

b) Show that for all real values of x the expression (x²- 2x +4)/(x²+ 2x +4) has the greatest value 3 and least value 1/3. (3+3)

11)a) 1/(x + y), 1/2y, 1/z are in AP, then show that y is the geometric mean between x and z.

Find the sum of the series to the n terms 1 +3 +7+ 15 +31+... to n terms. 6

12) The diameter of circle (in mm) drawn in a design are given below:

Diameter (in mm) no of circles

33-36 15

37-40 17

41-44 21

45-48 22

49-52 25

Calculate the standard deviation and mean diameter of the circles . 6

Section - B

13) (3x2=6)

a) Find the equation of the parabola with focus (6,0) and directrix x= -6.

b) Construct the table: ~ (p v ~ q)

c) Using contrapositive method proved that is n² is an even integer the x is also an even integer the x is also an even integer .

14) a) Find the equation of the ellipse if its foci are (±2,0) and the length of the latus rectum is 10/3.

Find the equation of the hyponovaloids whose foci are (0,±13) and length of the conjugate axis is 24. (4)

15) Find the coordinates of the point P which is five-sixth of the way from A(-2,0,6) to B(10,-6,-12).

The centroid of ∆ ABC is at the point (1,1,1). if the co-ordinates of A and B are (3,-5,7) and (-1,7,-6) respectively , find the co-ordinate of the point C. 6

16) A double olimate of the parabola y²= 4ax is the length 8a. Prove that the lines from the vertex to its ends are at right angles.

Section - C. (20 Marks)

16)a) Let mean monthly salary paid to all employees of a company is Rs 8300. The mean monthly salary paid to male and female employees was Rs 8000 and Rs 9000 respectively. Determine the percentage of males and females employed by the company. (2)

b) Find the values of a and b from the following data: (4)

Marks. No of students

00-05 7

05-10 a

10-15 25

15-20 30

20-25 b

Total. 100 given that the third decile is 1.1 (4)

Calculate the mode of the following data:

C-I frequency

05-15 6

15-25 11

25-35 21

35-45 23

45-55 14

55-65 5

17) a) If in a sample of n observation given that ∑ d²= 55 and rank correlation r= 2/3. Then find the value of n. (2)

b) The mathematical aptitude score of 10 computer programmers with their job performance is given below: (4)

Person math(sc) job rating

A 7 8

B 5 16

C 1 8

D 4 9

E 3 5

F 0 4

G 2 3

H 6 8

I 8 17

J 9 12

Calculate spearman's rank correlation.

Find Karl Perason's coefficient of correlation between X and Y for the following data:

X: 5 4 3 2 1

Y: 4 2 10 8 6

18) Find the consumer price index for 2007 on the basis of 2005 from the following data using weighted average of price relative method. (4)

Items: Food Rent cloth fuel

Price ('05): 200 100 150 50

Price ('07): 280 200 120 100

Weight: 30 20 20 10

19) Obtain the three year moving average for the following series of observations.

Year: Sales(Rs '0000)

1995 3.6

1996 4.3

1997 4.3

1998 3.4

1999 4.4

2000 5.4

2001 3.4

2002 2.4

Represent these graphically. (4)

H. W-1

1) If the pth, qth and rth terms of an AP are respectively a⁻¹, b⁻¹, and c⁻¹ show that, (q- r)bc + (r - p)ca + (p - q)ab =0.

2) How many terms of the series 1/2+ 1/3+ 1/6+ ..... must be taken so that the sum may be (-3/2) ?

3) Find the sum of 1- 3 + 5 - 7 + 9 - 11+ .....to n terms.

4) How many even numbers are there between 15 and 115 ? Find the sum of all these numbers.

5) Find the sum of all the numbers between 200 and 300 which are multiples of 7.

6) If (p +1)th term of an AP be a, find the sum of first (2p +1) terms of the AP.

7) If the 11th term of an AP be 25, find the sum of first 21 terms of the AP.

8) There are (2n +1) terms in an AP. Show that the ratio of the sum of odd terms and the sum of even terms is (n +1) : n.

9) Find the 99th term of the series 2+ 7+14+23+34+....

10) How many terms are there in the series 1+ 3+6+10+15+21+...+5050?

11) The sum of four numbers in AP is 20 and the sum of their squares is 120, find the numbers.

12) The sum of six numbers in AP is 345 and the difference between the first and the sixth is 55, find the numbers.

13) The fourth term of an AP is thrice the first term and the seventh term exceeds twice the third term by 2. Find the sum of first ten terms of the AP.

14) If the sum of first n terms of an AP is 40, the common difference is 2 and the last term is 13, find the value of n.

15) The sum of first n terms of two AP's are in ratio (3n +1) : (3n -1). Find the ratio of their tenth terms.

16) Prove that the sum of n arithmetic means between two numbers is n times the arithmetic mean of those two numbers.

____&&&______________&&&&&&_____

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).