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
6)
7) Show that if one root of ax²+ bx + c =0 be the square of the other, then b³+ a²c + ac² = 3abc.
8)
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
