SAP- 1
1) If α and β are the roots of ax²+ bx + c= 0, find the values of following:
a) 1/(aα + b) + 1/(aβ + b). b/ac
b) β/(aα + b) + α/(aβ + b). -2/a
c) (aα + b)⁻³ + (aβ + b)⁻³. (b³- 3abc)/a³c³
d) (aα + b)⁻²+ (aβ + b)⁻². (b²- 2ac)/a²c²
2) If α and β are the roots of the equation ax² + bx + c=0, find the equation whose roots are as given below:
a) 1/(α + β), 1/α + 1/β. bcx²+ (b²+ ac)x + ab=0
b) α/β, β/α. acx² - (b²+ 2ac)x + ac=0
c) α + 1/β, β + 1/α. acx²+ b(a+ c)x + (a+ c)²=0
d) α²+ β², 1/α² + 1/β². a²c²x²- (b²- 2ac)(a²+ c²)x + (b²- 2ac)²=0
e) 1/(aα + b), 1/(aβ + b). acx²- bx +1= 0
3) α ≠ β, but α²= 5α -3, β²= 5β- 3, find the equation whose roots are α/β and β/α. 3x²- 19x +3=0
4) If α, β are the roots of x²+ ax + b = 0, then show that α/β is a root of the equation bx² + (2b - a²)x + b = 0.
5) If α and β are the roots of x² - p(x +1) - c = 0, show that (α +1)(β+1)= 1+ c. Hence show that (α²+ 2α +1)/(α²+ 2α +c) + (β² + 2β +1)/(β² + 2β +c) = 1.
6) In a triangle PQR, angle R=π/2. If tan(P/2) and tan(Q/2) are the roots of the equation ax² + bx + c = 0, (a≠0), the pn
a) a+ b = c b) b+ c = a c) a+ c = b d) b= c. a
7) If sinθ and cosθ are the roots of the equation lx²+ mx + n = 0, then show that l²- m²+ 2ln = 0
8) If one root of the equation ix² - 2(i + 1)x + (2- i)= 0 is 2- i, then show that the other root is - i.
9) Find the roots of the equation 8sec²x - 6 secx + 1= 0. No roots, because 2 or 4 not possible
10) Find the roots of the equations a(b - 2c)x² + b(c - 2a)x + c(a - 2b)= 0 if ab + bc + ca = 0. 1, c(a - 2b)/a(b - 2c)
8a) If the roots of the equation (x - a)(x - b) - k = 0 be c and d, then show that the roots of the equation
(x - c)(x - d)+ k = 0 are a and b.
b) If α, β are the roots of the equation (x - a)(x - b)+ c= 0
Find the roots of the equation
(x - α)(x - β)= c.
9) a) Ramesh and Mahesh solve an equation. In solving Ramesh commits a mistake in constant term and finds the roots 8 and 2. Mahesh commits a mistake in the coefficient of x and finds the roots -9 and -1. Find the correct roots. 9,1
b) Two candidates attempt to solve a quadratic of the form x² + px + q = 0. One starts with wrong value of p and finds the roots to be 2 and 6. The other starts with a wrong value of q and finds the roots to be 2, -9. Find the correct roots. -3,-4
c) The coefficient of x in the quadratic equation x² + px + q=0 was taken as 17 in place of 13, its roots were found to be -2 and -15. Find the roots of the original equation. -10, -3
10a) Show that A. M of the roots of x² - 2ax + b² = 0 is equal to the geometric mean of the roots of the equation x² - 2bx + a²= 0, and vice versa.
b) Let p and q be roots of the equation x²- 2x + A=0 and let r and s be the roots of the equation x²- 18x + B=0.
If p< q < r < s are in arithmetic progression, then A= ..... and B= ... -1,3,7,11
11) a) Given that α, γ are roots of the equation Ax²- 4x +1= 0 and β, δ the roots of the equation Bx²- 6x +1=0, find the values of A and B such that α, β γ and δ are in HP. 3,8
b) The number of quadratic equations which are unchanged by squaring their roots is
a) 2 b) 4 c) 6 d) none. 4
12a) If α and β are the roots of the equation, 2x²- 3x - 6=0, find the equation whose roots are α²+2, β²+2. 4x²- 49x + 118=0
b) The roots of the equation 8x²- 10x +3 =0 are α and β² where β²> 1/2 then the equation whose roots are ( α + iβ)¹⁰⁰ and ( α - iβ)¹⁰⁰ is
a) x²- x +1=0 b) x²+ x +1=0 c) x²- x -1=0 d) x² + x -1=0 b
13) If α and β are the roots of the equation x²- 2x +3=0, find the equation whose roots are
a) α+2, β+2. x²- 6x+11=0
b) (α-1)/(α+1) , (β-1)/(β+1). 3x²- 2x +1=0
14)a) If α be a root of the equation, 4x²+ 2x -1=0, show that 4α³- 3α is the other root.
b) Form a quadratic equation whose roots are a/(√(a) ± √(a - b)). x²- 3x +2=0
c) α, β are the roots of the equation x²- 2x +3=0. Determine the equation whose roots are P= α²- 3α²+ 5α -2 and Q= β²- β²+ β+5. x²- 3x +2=0
15) Let a, b, c be real numbers with a≠ 0 and let α, β be the roots of the equation ax²+ bx + c=0.
Express the roots of a³x²+ abcx + c³= 0 in terms of α, β. α'= α²β, β'= αβ²
16 a) Let α, β be the roots of ax²+ bx + c=0 and γ, of lx²+ mx + n=0, then find the equation whose roots are αγ+ βδ and αδ + βγ. x²- Sx + P= 0
b) If α + β= 3 and α³+ β³=7, then α and β are the roots of 9x²- 27x +20=0.
c) Find a quadratic equation whose roots α and β are connected by relation α +β= 2 and (1- α)/(1+β) + (1- β)/(1+ α) = 2{(4λ²+15)/(4λ²-1)}. x²- 2x - (4λ²+11)/4 = 0.
17) a) If α, β are the roots of the equation x²- px + q=0, then find the equation the roots of which are (α²- β²)(α³- β³) and α³β²+ α²β³. t²- St + P= 0
b) If α, β are the roots of the equation x²- bx + c=0 then find the equation whose roots are (α²+ β²)(α³+ β³) and α⁵ β³+ α³ β⁵- 2 α⁴β⁴. x²- Sx+ P= 0, where S= p+ q and P= pq
c) Find the equation whose roots are (α+ β)² and (α-β)², where α and β are the roots of 2x²+ 2(m + n)x + (m²+ n²)= 0. x²- 4mnx - (m²- n²)²= 0.
18) a) If α, β be the roots of x²- px + q=0 and α', β' be those of x²- p'x + q' =0. Find the value of
(α - α')²+ (β - α')² + (α - β')¹+ (β -β')². 2[p²- 2q + p'²- 2q' - pp']
b) If α, β are the roots of the equation 6x²- 6x +1=0 then show that (1/2) (p+ qα + rα²+ sα³)+ (1/2) (p + qβ+ rβ²+ sβ³) is p/1 + q/2 + r/3 + s/4.
19) a) If α, β be the roots of ax²+ 2bx + c = 0 and α+ δ, β+ δ be those of Ax²+ 2Bx + C= 0, then show that (b²- ac)/(B²- AC)= (a/A)².
Another form
The two quadratic equation ax²+ bx + c= 0 and lx²+ mx + n = 0 have roots α, β and δ, γ respectively. If α, β, δ, γ be in AP and ∆₁ and ∆₂ be the discriminants of these quadratics, then show that ∆₁/∆₂ = a²/l².
b) The ratio of the roots of the equation ax²+ bx + c=0 is same as the ratio of the roots of the equation Ax²+ Bx + C=0. If D₁ and D₂ are the discriminants of ax² + bx + c= 0 and Ax² + Bx + C=0 respectively, then show that D₁ : D₂ = b² : B².
20) Let α, β are the roots of x²- x + p=0 and δ, γ be the roots of x²- 4x + q=0. If α β, δ, γ are in GP then the integral values of p and q respectively, are
a) -2,-32 b) -2,3 c) -6,3 d) -6,-32 a
21) a) If α, β be the roots of x²+ px + q=0 and γ, δ the roots of x²+ px + r=0, show that (α - γ) (α- δ)= (β- γ)(β- δ)= q+ r.
b) If If α, β be the roots of x²+ px + 1=0 and γ, δ the roots of x²+ qx + 1 =0, show that (α - γ) (β- γ)(α + δ)(β+ δ)= q² - p²
22)a) If the roots of the equation px²+ qx +2=0 are reciprocals of each other, then
a) p=0 b) p= -2 c) p= ±2 d) p= 2. d
b) Let P, Q, R be defined as
P= a²b + ab² - a²c - ac²,
Q= b²c + bc² - a²b - ab²
R= a²c + c²a - c²b - cb²
Where a, b, c are all +ve and the equation Px²+ Qx + R=0 has equal roots then a, b, c are in
a) AP b) GP c) HP d) none. c
23) Find the condition that the roots of the equation ax²+ bx + c be such that
a) one root is n times the other. nb²= ac(n +1)²
b) one root is three times the other. 3b²= 16ac
c) both roots are equal. b²= 4ac
24) If the roots of the equation ax²+ bx + c=0 are of the form (k +1)/k and (k+2)/(k+1), show that (a+ b + c)² - 4ac.
25) Let a, b, c, d are real numbers in GP. If u, v, w satisfy the system of equations u+ 2v + 3w = 6, 4u + 5v + 6w = 12, 6u + 9v = 4
Then show that the roots of the equation (1/u + 1/v + 1/w)x²+ [(b - c)²+ (c - a)²+ (d - b)²]x + u+ v + w = 0 and 20x² + 10(a - d)²x - 9=0 are reciprocals of each other.
26) a) If one root of the equation ax²+ bx + c =0 be the square of the other, then show that b³+ ac²+ a²c = 3abc.
b) If one of the equation x²+ px + q=0, is square of the other then show that p³ - q(3p -1) + q²= 0.
c) For the equation 3x²+ px + 3 =0, p> 0, if one of the roots is square of the other, then p is equal to
a) 1/3 b) 1 c) 3 d) 2/3 c
27)a) If x= 2+ 2²⁾³ + 2¹⁾³, then the value of x³- 6x² + 6x is ____ 2
b) If one root of the equation ax²+ bx + c=0 is equal to the nth power of the root, then show that
(acⁿ)¹/⁽ⁿ⁺¹⁾ + (aⁿc)¹/⁽ⁿ⁺¹⁾ + b =0
28a) If the roots of the equation 1/(x + p) + 1/(x + q) = 1/r are equal in magnitude but opposite in sign show that p+ q = 2r and that the product of the roots is equal to -(1/2) (p²+ q²).
b) If the roots of the equation 3x²+ 2(k²+1)x + (k²- 3k +2)= 0 be of opposite signs, then show that 1< k < 2.
29) a) Solve the equation:
{(x - b)(x - c)}/{(a- b)(a - c)} + {(x - c)(x - a)}/{(b- c)(b - a)} + {(x - a)(x - b)}/{(c- a)(c - b)} = 1. -(1/2)(p²+ q²)
b) If the equation (k²- 5k + 6)x²+ (k²- 3k +2)x + (k²- 4)= 0 is satisfied by more than two values of x, then determine the value of k. 2
30) a) If the sum of the roots of ax²+ bx + c =0 be equal to sum of their squares show that 2ac = ab+ b².
b) If the sum of the roots of the equation ax²+ bx + c= 0 is equal to sum of the squares of their reciprocals, then show that bc², ca², ab² are in AP or c/b, b/a, a/c are in HP.
31) a) α, β are the roots of the equation λ(x²- x)+ x + 5 = 0. If λ₁ and λ₂ are the two values of λ for which the roots α, β are connected by the relation α/β + β/α = 4/5, find the value of λ₁/λ₂ + λ₂/λ₁ . 254
b) If α, β be the roots of the equation λ²(x²- x)+ 2λx +3=0 and λ₁, λ₂ be the two values of λ for which α and β are connected by the relation α/β + β/α = 4/3 then find the equation whose roots are λ₁²/λ₂ and λ₂²/λ₁. λ²- 4λ- 6=0, (λ≠ 0)
32) a) If the ratio of the roots of the equation x²+ px + q=0 be equal to ratio of the roots of x²+ lx + m =0, then show that p²m = l²q.
b) If the ratio of the roots of a₁x²+ b₁x + c₁= 0 be equal to the ratio of the roots of a₂x²+ b₂x + c₂ = 0, then show that a₁/a₂, b₁/b₂, c₁/c₂ are in GP.
33) a) If α, β are the roots of the equation x +1= λx(1- λx) and λ₁, λ₂ be the two values of λ determined from the equation α/β + β/α = π -2, show that λ₁²/λ₂²+ λ₂²/λ₁²+2= 4{(π+1)/(π-1)}².
b) If the ratio of the roots of the equation lx²+ nx + n =0 be p: q, then show that √(p/q) + √(q/p) + √(n/l)= 0
34) a) Find the value of p for which x+1 is a factor of x⁴+ (p -3)x³ - (3p -5)x²+ (2p -9)x + 6. Find the remaining factors for this value of p. 4, (x+1)(x-1)(x+3)(x-2)
b) If x²- 3x +2 is a factor of x⁴- px² + q=0. Prove p=5, q= 4.
35) a) The roots x₁ and x₂ of the equation x² + px + 12=0 possesses the property x₁ - x₂ = 1. Find the value of p. ±7
b) Knowing that 2 and 3 are the roots of the equation 2x³+ mx²- 13x + n=0, determine m and n and find third root of the equation. -5,30,
36) a) If x²+ x²/(x +1)²= 3 and x be real, then show that x= (1±√5)/4.
b) If x²+ x+ 1 is a factor of ax³+ bx²+ cx + d, then show that the real root of ax³+ bx²+ cx + d=0 is -d/a.
37) If x= 2+ i √3 then find the value of
a) 4x²+ 8x +35. 55+24√3 i
b) If 2+ i √3 is a root of x²+ px + q=0 where p, q are real then (p,q)= (.....). -4,7
38) a) If x= 1+ 2i then show that x³+ 7x²- 13x +16= - 29.
b) Find the equation one of whose roots is 2+ √3 and hence find the value of expression x³- 7x²+ 13x -2 for x= 2+ √3.
c) Find all the roots of the equation 4x⁴- 24x³+ 57x²+ 18x - 45=0 if one of them is 3+ i √6. 3± i√6, ±(√3/2)
39)a) Let α + iβ, α, β ∈ R, be a root of the equation x³+ qx + r=0, q, r ∈R. Find a real cubic equation, independent of α and β, whose one root is 2α. t³+ qt - r =0
b) Find a quadratic equation whose one root is square root of -47 + 8 √3. x²±2x +49=0
40) Solve:
a) x³- 13x²+ 15x +189= 0 if one root exceeds other by 2. 7,9,-3
b) x⁴- 2x³+ 4x²+ 6x -21=0 if two of its roots are equal in magnitude but opposite in sign. ±√3, 1± i √6
c) If the sum of two roots of the equation 4x⁴- 8x³- 13x²+ 2x +3=0 is zero, find all its roots. ±1/2,3,-1
41) Solve: x⁴- 2x²+ 8x - 3=0. -1±√2, -1±√2 i
42) If 1, a₁, a₂, .....aₙ₋₁ are the n, nth roots of unity, then show that
(1- a₁)+1- a₂)+1- a₃).....(1- aₙ₋₁)= n.
43) a) If α and β be the roots of the equation x² - ax + b=0 and Vₙ = αⁿ + βⁿ, then show that Vₙ₊₁ = aVₙ - bVₙ₋₁. Hence obtain the value of α⁵ + β⁵. a⁵- 5a³b + 5ab²
b) If α, β are the roots of x²+ px + q=0 and also of x²ⁿ + pⁿxⁿ + qⁿ =0 if α/β, β/α are the roots of xⁿ + 1+ (x +1)ⁿ= 0, then show that n must be an even integer.
44) Let f(x)= Ax² + Bx + C where A, B, C are real numbers. Show that if f(x) is an integer whenever x is an integer, then the numbers 2A, A+ B and C are all integers. Conversely, show that if the numbers 2A, A+ B and C are all integers then f(x) is an integer whenever x is an integer.
45)a) Let a, b, c be real. If ax²+ bx + c=0 has two real roots α and β, where α< -1 and β> 1, then show that
1+ c/d + |b/a|< 0
b) If the roots of the equation x²- 2ax + a²+ a -3=0 are real and less than 3, then
a) a< 2 b) 2≤ a ≤3 c) 3< a ≤ 4 d) a > 4. a
c) Find the values of real parameter 'a' for which the equation
(tan²θ+1)²+ 4a(tan²θ +1)tanθ + 16 tan²θ =0 has four distinct roots in (0,π/2). (-5/2,-2)
46) a) if a+ b+ c=0, then the equation 3ax²+ 2bx + c=0 has atleast one root in (0,1). 3ax²+ 2bx + c
b) If 2a+ 3b + 6c=0 (a,b, c ∈R) then show that the equation ax²+ bx + c =0 has atleast one root in [0,1].
47) a) If b> a, then the equation (x - a)(x - b) -1=0, has
a) both roots in [a,b]
b) both roots in (-∞,a)
c) both roots in (b, + ∞)
d) one root in (-∞, a) and other in (b, +∞). d
b) If α and β (α < β) are the roots of the equation x²+ bx + c=0, where c< 0 < b, then
a) 0<α<β b) α<0< β |α| c) α< β< 0 d) α<0<|α|<β
48) a) show that the value of λ for which 2x² - 2(2λ+1)x + λ(λ+1)= 0 may have one root less than λ and other root greater than λ are given by λ > 0 or λ < -1.
b) For the equation
x²- (k +1)x + (k²+ k -8)= 0 if one root is greater than 2 and other is less than 2. Then show that k lies between -2 and 3.
49) a) If a,b,c are real numbers, a≠ 0. If α is a root of a²x²+ bx+ c= 0, β is a root of a²x²- bx - c= 0 and 0<α<β, then the equation a²x²+ 2bx + 2c= 0 has a root γ that always lies between α and β.
b) If 1 lies between the roots of the equation 3x²- 3 sinαx - 2 cos²α = 0 then α lies in the interval
a) (0,π/2) b) (π/12,π/2) c) (π/6,5π/6) d) (π/6,π/2) U (π/2,5π/6)
50) Let -1≤ p ≤ 1. Show that the equation 4x³- 3x - p= 0 has a unique root in the interval [1/2,1] and identify it.
Sap-2
1) Find the condition that in the equations ax²+ bx + c=0 and a'x²+ b'x + c'= 0
a) one root be common. (ca' - ac')/(ab' - a'b)
b) A root of first be reciprocal of a root of the second. (cc' - aa')²= (ba' - cb')(ab' - bc')
c) both have the same pair of roots. a/a' = b/b' = c/c'
2) a) If the equations x²+ px + q=0 and x²+ p'x + q' =0 have a common root show that it must be equal to (pq' - p'q)/(q - q') or (q - q')/(p' - p)
b) If the equations f(x) ≅ ax³ + 3bx² + 3cx + d= 0 and g(x)≅ ax² + 2bx + c=0 have a common root, then show that (bc - ad)² = 4(ac - b²)(bd - c²).
3a) Find k if the equations 4x² - 11x + 2k = 0 and x² - 3x - k = 0 have a common root and obtain the common root for this value of k. 17/6
b) Determine a such that x²- 11x + a and x²- 14x + 2a may have a common factor. 0,24
4a) Find the value of a solid that the equations (2a -5)x²- 4x - 15= 0 and (3a - 8)x² - 5x - 21= 0 have a common root. 4,8
b) If the equations x² - x - p= 0 and x² + 2xp - 12=0 have common root, find it. 2
c) Find the condition on the complex constant α, β if z²+ αz + β=0 has real roots. (β - conj β)²= (conj of α - α)(α. Conj of β - conj of α.β)
5) If a, b, c are in GP, then the equations ax² + 2bx + c=0 and dx² + 2ex + f =0 have a common root if d/a, e/b, f/c are in
a) AP b) GP c) HP d) none a
Another form:
If ax² + 2bx + c=0 and px² + 2qx + r=0 have a common root a/p, b/q, c/r in AP, then show that p, q, r are in GP
6) a) If α, β are the roots of x² + px + q=0 and δγ are the roots of x² + rx + s=0, evaluate (α-γ)(α-δ)(β-γ)(β- δ) in terms of p, q, r, s.
Deduce the condition that the equations have a common root. (q - s)²= (p - r)(qr - ps)
b) Eliminate x from the equations a+ c = (b/x) - dx , a - c = (d/x) - bx.
7) α and β are the roots of ax² + bx + c= 0 and f(x)= a₁x² + b₁x + c₁ then show that f(α). f(β)= 1/a² [(c₁a - ca₁)² - (ab₁ - a₁b)(bc₁ - b₁c)]²
8) If α, β are the roots of ax²+ bx + c =0. α₁, - β are the roots of a₁x² + b₁x + c₁ =0, show that α, α₁ are the roots of x²/{b/a + b₁/a₁)} + x + 1/{b/c + b₁/c₁} =0
9)a) If the equations x² + bx + ca =0 and x² + cx + ab =0 have a common root, then their other roots are the roots of the equation, x² + ax + bc=0.
b) If the equations x² + abc + c=0 and x² + acx + b =0 have a common root, then establish that their other roots are the roots of the equation x² - a(b + c)x + a²bc = 0.
10) If one root of the equation x² + ax + b=0 is a root of the equation x² + cx + d= show that the other roots satisfy the equation
x² + x(2a - c)+ (a² - ac + d)= 0.
11) a) If each pair of the three equations x² + p₁x + q₁= 0, x² + p₂x + q₂ = 0 and x² + p₃x + q₃ = 0 have a common root, then show that
p₁² + p₂² + p₃² + 4(q₁ + q₂ + q₃)= 2(p₁p₂ + p₂p₃ + p₃p₁).
b) If every pair of equations x² + ax + bc=0, x² + bx + ca=0, x² + cx + ab=0 have a common root, then find the sum and product of these common roots. -(1/2) ∑ a, abc
c) If the three equations x² + ax + 12=0, x² + bx + 15=0 and x² + (a+ b)x + 36=0 have a common possible root then find a and b and the roots. 7,-7 and 8,-8
12) if each pair of equations a₁x² + b₁x + c₁ = 0, a₂x² + b₂x + c₂ = 0 and a₃x² + b₃x + c₃ =0 have a common root, then show that
i) (c₁a₂ + c₂a₁)/(c₁a₂ - c₂a₁) + a₁a₂a₃/a₃+a₁b₂ - a₂b₁) = 0
ii) {(a₁b₂ - a₂b₁)/(a₁c₂ - a₂c₁)}² = a₁a₂c₃/c₁c₂a₃.
13) If the equations ax² + 2bx + c= 0, a'x² + 2b'x + c'=0 have a common root, show that the equation
(b² - ac)x² + (2bb' - ac' - a'c)x + (b'² - a'c')= 0 has equal roots.
14) a) If the equations x² + ax + b = 0 and x²+ bx + a=0 have a common root, then the numerical value of a+ b is ....-1
b) If the equation x² + bx + c =0 and bx² + cx +1=0 have a common root, then either b+ c+1=0 or b² + c² +1= bc + b + c
15) If the equations ax² + bx + c=0 and x³ + 3x² + 3x +2=0 have two common roots, then show that a= b= c.
16) a) If the roots of the equation, (c² - ab)x² - 2(a² - bc)x + (b² - ac)=0 be equal show that either a=0 or a³+ b³ + c³ = 3abc.
b) For what values of m the roots of the equation
x² - 2x(1+ 3m)+ 7(3+ 2m)=0 will be equal? 2, -10/9
17) a) If the two roots of the equation (λ-1)(x² + x +1)² - (λ+1)(x⁴ + x² +1)=0 are real and distinct then show that λ lies in the interval (-∞, -2) U (2,∞).
b) The equation (6- x)⁴ + (8- x)⁴= 16 has
i) sum of roots 28.
ii) product of roots 2688
iii) two real roots
iv) two imaginary roots.
c) Determine the nature of the roots of the equation
(x - a)³+ (x - b)³+ (x - c)³= 0. Ono real root and two imaginary roots
18) a) Determine the values of m for which the equation
5x²- 4x + 2+ m(4x² - 2x -1)=0 will have
i) equal roots. 1,-6/5
ii) product of roots as 2. -8/9
iii) sum of the roots as 6. -13/11
b) The product of the roots of the equation x² - 3λx + 2e²ˡᵒᵍλ -1=0. If the roots be real, then λ= 2.
19) a) If the roots of the equation, (b - c)x² + (c - a)x + (a - b)= 0 be equal, then show that a, b, c are in arithmetic progression.
b) If a(b - c)x² + b(c - a)x + c(a - b)=0 has equal roots, show that a, b, c are in harmonical progression.
20) a) Show that the roots of (x - a)(x - b)+ (x - b)+x - c)+ (x - c)+x - a)=0 are always real and they will be equal if and only if a= b = c.
b) Examine the nature of the roots of the equation (b - x)² - 4(a - x)(c - x)=0 where a,b, c are real. Real
21) a) The equation ax² + bx + c=0 where a, b, c are real numbers connected by the relation 4a+ 2b + c=0 and ab > 0 has real roots.
b) If a, b, c are positive and are in AP. Show that the roots of the equation ax² + bx + c =0 are real for|c/a -7|≥4√3.
22) a) Show that the roots of
(a² + b²)x² - 2b(a+ c)x + (b² + c²)=0 will be real if a, b, c are in GP and in this case these roots will be equal as well.
b) If a, b, c be distinct real and +ve numbers which are in HP, then the roots of the equation ax² + 2bx + c=0 are real and distinct. Is this statement true or not? No it is imaginary
23) a) Discuss the nature of the roots of the equation 4ax² + 3bx + 2c = 0 where a, b, c ∈ R and are connected by the relation a+ b + c= 0. Real
b) If the roots of the equation
[a²+ 2(1- b)]x²+ 2a(1+ b)x + 2b(b -1)+ a²=0
be equal then show that a²= 4b.
24)a) If the roots of the equation (a² + b²)x² -2(bc + ad)x + (c² + d²)= 0, be real, then show that they will be equal as well and then a/b = d/c.
b) Show that if the roots of the equation (a⁴ + b⁴)x² + 4abcdx + (c⁴ + d⁴)=0 are real then they can't be unequal.
c) If a, b, c be real then show that the roots of the equation 1/(x + a) + 1/(x + b) + 1/(x + c)= 3/4, x≠ 0 are real.
25) Discuss the nature of the roots of the equations:
a) (a+ c - b)x² + 2cx + (b + c - a)=0.
b) (b + c)x² - (a+ b + c)x + a=0.
c) 2(a² + b²)x² + 2(a+ b)x +1=0.
26) a)Show that the roots of the equation
(b + c - a)x² + (c + a - b)x + (a+ b - c)=0 are rational if a, b, c be all rationals such that a+ b+ c=0.
b) Show that the roots of the equation (a+ 2b - 3c)x² + (b + 2c - 3a)x + (c + 2a - 3b)=0 are rational, if a, b, c are rational. Hence determine the roots of the equation.
c) If (ax²+ bx + c)y + (a'x² + b'x + c')=0 and x is a rational function of y then show that (ac' - a'c')²= (ab' - a'b)(bc' - b'c)
27) If the roots of the equation x² - 2cx + ab=0 be real and unequal, then show that the roots of x² -2(a+ b)x + (a² + b²)+ 2c² =0 will be imaginary.
b) Consider the equations ax² + 2bx + c= 0 and (a+ c)(ax² + 2bx + c) - 2(ac - b²)(x² +1)=0.
If the roots of one are real (complex) then the roots of the other are complex (real).
28) a) If the roots of the equation x² - ax + b =0 are real and differ by a quantity which is less than c(c> 0), then b lies between (a² - c²)/4 and a²/4.
b) If the roots of the equation
(a-1)(x² + x +1)² = (a+1)(x⁴ + x² +1) are real and distinct then show that a² - 4> 0.
29) a) Show that if p,q,r,s are real numbers and pr= 2(q+ s), then atleast one of the equation x² + px + q=0, x² + rx + s =0 has real roots.
b) If P(x)= ax² + bx + c and
Q(x)= - ax² + dx + c where ac≠ 0 then P(x) Q(x)=0 has atleast two real roots.
30) If a< b < c < d, then the roots of the equation (x - a)(x - c)+ 2(x - b)(x - d)= 0 are real and distinct. T/F
31) a) The equation 2(log₃x)² - |log₃x|+ k = 0 has four solutions, determine the interval in which k lies. (0,1/8)
b) If α and β are the roots of x² + px + q =0 and α⁴, β⁴ are the roots of x² - rx + s=0, then the equation x² - 4qx + 2q² - r=0 has always two real roots.
32) a) If the roots of the equation x² + a² = 8x + 6a be real then show that a lies between -2 and 8.
b) Show that if the roots of 9x² + 4ax +4=0 are imaginary, then a must lie between -3 and 3.
c) The equation x² +2(m -1)x + m +5=0 has atleast one+ve root. Determine the range for m. (-∞, -1]
33) a) If a≠ 1 and a≠ -2, show that the roots of the equation
(a² + a -2)x² + (2a² + a +3)x + a² -1=0 are rational. Hence solve the equation. -(a-1)/(a+2), -(a +1)/(a -1)
b) Let A, B, C be three angles such that A=π/4 and tanB tanC= p. Find all possible values of p such that A, B, C are the angles of a triangle.
34) a) If p, q, r are real and p≠ q then roots of the equation
(p- q)x² + 5(p+ q)x - 2(p - q)= 0 are
a) real and equal b) complex c) real and unequal d) none. c
b) The roots of px² + 2qx + r=0 and qx² - 2√(pr) x + q=0 are simultaneously real then
i) p=q, r≠0 ii) p/q= q/r iii) 2q= ± √(pr) iv) none. ii
35) a) If the roots α, β if ax² + bx + c=0 be real then establish the relation between the coefficients under the following conditions:
i) roots are equal and opposite
ii) roots are of opposite signs.
iii) roots are both -ve
iv) roots are both+ve.
b) Let a> 0, b> 0, then both roots of the equation ax²+ bx + c=0
i) are real and negative
ii) have negative real parts
iii) none. i, ii
36) a) Show that the roots of
(a- b)²x² + 2(a + b - 2c)x +1= 0 are real or imaginary according as c does not or does lie between a and b, a< b.
b) If the roots of the equation
(m -3)x² - 2mx + 5m=0 are real and+ve then show that m∈ ]3,15/4]
c) If the equation x² +2(a +1)x + 9a -5=0 has only negative roots then show that a≥ 6.
d) If both the roots of the equation x² - 6ax + 2 - 2a + 9a² =0 exceed 3, then show that a> 11/9.
37) Find for what real values of x the expressions
a) x² - 2x -3. -1,3
b) 2x² - 5x -3 are positive or negative. -3,1/2
c) If x² + 2ax + 10 -3a > 0, for all x∈ R then
i) a< -5 ii) -5< a < 2 iii) a> 5 iv) 2< a <5. ii
38) a) For real values of x, show that the value of the expression (11x² +12x +6)/(x² + 4x +2) cannot lie between -5 and 3.
b) x² + (a - b)x + (1- a - b)=0, a, b∈ R.
Find the condition on a, for which both roots of the equation are real and unequal.
c) Determine the values of x which satisfies the inequalities
x² - 3x +2> 0 and x² -3x -4≤ 0.
39)a) If x be real show that the expression (x +2)/(2x² + 3x +6) takes all values in the interval [-1/13, 1/3].
b) Show that the value of
tanx/tan3x or (sinx cos3x)/(cosx sin3x)
Whenver defined never lies between 1/3 and 3.
40) a) Show that for real values of x the expression (x-1)+x+3)/(x -2)(x+4) cannot lie between 4/9 and 1
b) If x is real, the experiment(x² +2x -11)/(x -3) takes all values which do not lie between 4 and 12.
41) a) If x is real, find the maximum and minimum values of (x² + 14x +9)/(x² + 2x +3).
b) Show that for any real values of x the expression (x + a)/(x² + bx + c²) will have any value if b²> 4c² and a² + c²< ab.
42) a) If x is real show that (x² - x +1)/(x² + x +1) takes values from 1/3 to 3.
b) Show that (x² - 3x +4)/(x² + 3x +4) can never be greater than 7 nor less than 1/7 for real values of x.
c) Find out the range in which the value of the function (x² + 34x -71)/(x² +2x -7) lies for all real values of x. Justify your answer.
43) a) Find the values of x for which the following inequality holds
(8x² + 16x -51)/{(2x -3)(x +4)}> 3.
b) Find the values of x, which satisfy the inequality
(x -2)/(x +2) > (2x -3)/(4x -1).
44) Find the set of all x for which 2x/(2x²+ 5x +2)> 1/(x+1).
45) a) Show that if x is real, the expression (x² - bc)/(2x - b - c) has no real value ps between b and c.
Sap-4
1) Two non integer roots of the equation
(x²+ 3x)² - (x²+ 3x) -6=0 are
a) (1/2) (-3+ √11), (1/2) (-3- √11)
b) (1/2) (-3+ √7), (1/2) (-3- √7)
c) (1/2) (-3+ √21), (1/2) (-3- √21) d) none
2) Two non integer roots of
{(3x -1)/(2x +3)}⁴ - 5{(3x -1)/(2x +3)}⅖ +4= 0 are
a) -5/7,-2/5 b) -2/4,7/5 c) 5/7,7/5 d) -2/5, 3/5
3) Sum of the roots of the equation
4ˣ - 3. 2ˣ⁺³ + 128=0 are
a) 5 b) 6 c) 7 d) 8
4) The only value of x satisfying the equation is
6√{x/(x +4)} - 2√{(x +4)/x} = 11 where x ∈ R
a) 4/35 b) -4/35 c) 16/3 d) none
5) The number of real values of x satisfying the equation
2(x²+ 1/x²) - 9(x + 1/x) + 14= 0
a) 1 b) 2 c) 3 d) 4
6) The non integer roots of x⁴- 3x³- 2x²+ 3x +1= 0 are
a) (1/2)(3+ √13), (1/2)(3 - √13)
b) (1/2)(3- √13), (1/2)(-3 - √13)
c) (1/2)(3+ √17), (1/2)(3 - √17) d) none
7) The number of real solution of
1/(x +1) + 1/(x +5) = 1/(x +2) + 1/(x +4) is
a) 0 b) 1 c) 2 d) 3
8) Number of real solutions of
(x -1)(x +1)(2x +1)+2x -3)= 15 is
a) 0 b) 2 c) 3 d) 4
9) The number of solutions of the equation
√[2x √(2x +4)]= 4 is
a) 0 b) 1 c) 2 d) 4
10) The number of solutions of
√(3x²+ x +5)= x -3 is
a) 0 b) 1 c) 2 d) 4
11) The number of solutions of
√(4- x) + √(x +9)= 5 is
a) 0 b) 1 c) 2 d) 3
12) The number of real solutions of
√(x²-4x +3) + √(x²-9)= √(4x²- 14x +6) is
a) 0 b) 1 c) 2 d) 4
13) The value of a for which one root of the equation
(a²- 5a +3)x² + (3a -1)x +2=0 is twice as large as other, is
a) -2/3 b) 1/3 c) -1/3 d) 2/3
14) Eange of the function f(x)= (x²+ x +2)/(x²+ x +1), x ∈ R is
a) (1, ∞) b) (1,3/2) c) (1,7/3] d) 1,7/5]
15) If f(x)= x²+ 2bx + 2c² and g(x)= - x²- 2cx + b² are such that minimum f(x)> maximum g(x), then relation between b and c, is
a) no relation b) 0< c<b/2 c) |c|< |b|/√2 d) |c|> √2 |b|
16) if a, b are the roots of x²+ px +1= 0, and c, d are the roots of x²+ qx +1= 0, the value of
E= (a - c)(b - c)(a + d)(b + d) is
a) p²- q² b) q²- p² c) q² + p² d) none
17) If 4ˣ - 3ˣ⁻¹⁾² = 3ˣ⁺¹⁾² - 2²ˣ⁻¹ , then the value of x is
a) 5/2 b) 2 c) 3/2 d) 1
18) For a> 0, a≠ 1, the number of values of x satisfying the equation
2logₓa + logₐₓa + 3 logₐ²ₓ a= 0 is
a) 2 b) 3 c) 4 d) infinite
19) The number of solutions of
√(x+1 - √(x -1)= 1 (x ∈R)
a) 1 b) 2 c) 4 d) infinite
20) If a, b, c are real and a≠ b, then the roots of the equation
2(a - b)x² - 11(a+ b + c)x -3(a - b)= 0 are
a) real and equal
b) real and unequal
c) purely imaginary d) none
21) Let a> 0, b> 0 and c> 0. Then both the roots of the equation
2ax²+ 3bx + 5c= 0
a) are negative
b) have real parts
c) have positive real parts d) none
22) If a, b, c are real, then both the roots of the equation
(x - b)(x - c)+ (x - c)(x - a)+ (x - a)(x - b)= 0 are always
a) positive b) negative c) real d) none
23) The equation
2x - 3/(x -2) = 4 - 3/(x -2) has
a) no root b) one root c) two equal roots d) none
24) If a, b,c are positive real numbers which are in GP , then the equation ax²+ 2bx + c= 0 and dx² + 2ex + f= 0 have common root if a/d, b/e, c/f are in
a) AP b) GP c) HP d) none
25) If P(x)= ax²+ bx + c and Q(x)= - ax²+ dx + c, where ac ≠ 0, then P(x) Q(x)= 0 has
a) no real root
b) exactly two real roots
c) atleast two distinct real roots d) none
26) If the product of the roots of the equation
x²+ 5kx + 2e⁴ˡⁿᵏ -1=0 is 31, then sum of the root is
a) -10 b) 5 c) -8 d) none
27) The number of real roots of
(7+ 4 √3)|ˣ|⁻⁸ + (7- 4 √3)|ˣ|⁻⁸ = 14 is
a) 0 b) 2 c) 4 d) none
28) Sum of all the values of x satisfying the equation
log₁₇log₁₁(√(x +11) + √x)= 0 is
a) 25 b) 36 c) 171 d) 0
29) let α, β be the roots of the equation (x - a)(x - b)= c with c≠ 0. then the roots of the equation (x - α)(x - β)+ c= 0 are
a) a,c b) b,c c) a, b d) a+ c, b+ c
30) If p,q are roots of x²+ px + q= 0, then
a) p=1 b) p= 1 or 0 c) p= -2 d) p= -2 or 0
31) The equation √(x +1) - √(x -1)= √(4x -1), (x ∈R)
a) no solution b) one solution c) two solution d) more than two solutions
32) The sum of all the real roots of the equation
|x -2|²+ |x -2| - 2= 0 is
a) 7 b) 4 c) 1 d) none
33) Let p and q be the roots of x²- 2x + A= 0 and r and s be the roots of x²- 18x + B= 0. If p< q < r< s are in AP, then ordered pair (A, B) is equal to
a) (-3,77) b) (77,-3) c) (-3,-77) d) none
34) In a triangle PQR, angle R= π/2. If tan(P/2) and tan(Q/2) are the roots of the equation ax² + bx + c=0 where a≠ 0, then
a) a+ b= c b) b+ c= a c) a+ c= b d) b= c
35) If α, β (α> β) are the roots of the equation x²+ bx + c= 0 c< 0< b, then
a) 0< α < β b) 0< α < β<|α| c) α < β<0 d) α < 0< |α|< β
36) For the equation 3x²+ px + 3= 0 , p> 0, if one of the roots is square of the other, than p is equals to
a) 1/3 b) 1 c) 3 d) 2/3
37) If the roots are the equation x² - 2ax + a²- 3 = 0 are real and less than 3, then
a) a<2 b) 2≤ a≤3 c) 3<a ≤4 d) a> 4
38) If b> a, then the equation (x - a)+x - b) -1= 0 has
a) both roots in [a, b]
b) both roots in (- ∞, a)
c) both roots in (b, ∞)
d) one root in (-∞,a) and other in (b, ∞).
39) Let α, β be the roots of x² - x + p = 0 and γ, δ be the roots of x² - 4x + q = 0 . If α, β, γ, δ are in GP then the integral value of p and q respectively, are
a) -2,- 32 b) -2,3 c) -6,3 d) -6, -32
40) If a, b, c are not all equal and α and β be the roots of the equation ax² + bx + c = 0, then value of (1+ α+ α²)(1+ β+ β²) is
a) 0 b) positive c) negative d) non negative
41) If a,b,c are in AP and if the equations
(b - c)x²+ (c - a)x + (a - b)= 0 and
2(c + a)x²+ (b + c)x = 0 have a common root, then
a) a², b², c² are in AP
b) a², c², b² are in AP
c) c², a², b² are in AP d) none
42) Value of
x= √[6+ √{6+ √{6+....up to
a) 3 b) 2 c) 1 d) none
43) two complex numbers α and β are such that α + β = 2 and α⁴+ β⁴= 272, then the quadratic equation whose roots are α and β is
a) x²-2x -16= 0
b) x²-2x + 12= 0
c) x²-2x -8= 0 d) none
44) The equation (cos p -1)x² + (cos p)x + sin p = 0 in variable x has real roots, if p belongs to the interval
a) 0,2π) b) (-π,0) c) (-π/2,π/2) d) (0,π)
45) If the roots of the equation
1/(x + a) + 1/(x + b) = 1/c are equal in magnitude but opposite in sign, then their product is
a) (1/2) (a²+ b²) b) - (1/2) (a²+ b²) c) ab/2 d) -ab/2
46) If the quadratic equations x²-11x + a= 0 and x²-14x + 2a= 0 have common root, then the values of a are
a) 0, 24 b) 0,-24 c) 1,-1 d) -2,1
47) If α, β are the roots of the equation ax² + bx + c = 0, then the value of α³+ β³ is
a) (3abc+ b³)/a³
b) (a³+ b³)/3abc
c) (3abc- b³)/a³
d) - (3abc+ b³)/a³
48) If the sum of the roots of the quadratic equations ax² + bx + c = 0 is equal to the sum of the squares of their reciprocals, then
a) ab², ca², bc² are in AP
b) ab², bc², ca² are in AP
c) ab², bc², ca² are in AP d) none
49) If the ratio of the roots of the equation x² + bx + c = 0 is the same as that of the ratio of the roots of x² + qx + r = 0, then
a) br²= qc² b) cq²= rb² c) q²c²= b²r² d) Bpbq= rc
50) If a, b are the non zero distinct roots of x² + ax + b = 0, then the least value of x² + ax + b is
a) 2/3 b) 9/4 c) -9/4 d) 1
51) If a+ b+ c= 0, then the quadratic equation 3ax² + 2bx + c = 0 has
a) at least one root in [0,1]
b) one root in [2,3] and other is [-2, -1]
c) imaginary roots d) none
52) If a< b < c< d, then the equation 3(x - a)(x - c)+ 5(x - b)(x - d)=0
a) real and distinct rootes
b) real and equal roots
c) purely imaginary roots d) none
53) For real x, the function (x - a)(x - c)/(x - b) will assume all real values provided
a) a< b < c b) b< c < a c) c< a < cpb d) none
54) Let a,b,c ∈R and a≠ 0. If α is a root of a²x²+ bx + c= 0, β is a root of a²x²- bx - c= 0 and 0< α < β, then the equation a²x²+ 2bx + 2c= 0 has a root γ that always satisfies
a) γ= (1/2) (α + β) b) γ= α + β/2 c) γ= (α + β) d) α <γ< β
55) Suppose p,q,r,s ∈R and α, β be the roots of x²+ px + q= 0 and α⁴, β⁴ be the roots of x²- rx + s= 0, then the equation x²- 4qx + 2q² - r= 0 has always
a) two imaginary roots
b) two positive roots
c) two negative roots
d) one positive and one negative root
56) The equation
x⁽³/⁴⁾⁽ˡᵒᵍ₂ˣ⁾^²⁺ ˡᵒᵍ₂ˣ ⁻ ⁵/⁴= √2 has
a) exactly two real roots
b) no real root
c) one irrational root d) none
57) Let f(x) be a quadratic expression which is positive for all x, if g(x)= f(x)+ f'(x) then for all real x,
a) g(x)< 0 b) g(x)> 0 c)g(x) = 0 d) g(x)≥ 0
58) If α, β are the roots of ax²+ bx + c= 0, then the quadric equation whose roots are 2α+3 and 2β+3 is
a) 4ax² - 3bx + c= 0
b) 6a¹x² - 4abx + 6c= 0
c) ax² +2(b- 3a)x + 9a+ 2b= 0 d) none
59) If α, β are the roots of the equation ax²+ bx + c= 0, then the equation whose roots are α³, β³ is
a) a³y²+ (b³- 3abc)y+ c³= 0
b) a³y²+ (3abc - b³)y- c³= 0
c) a²y²+ 2aby+ c³= 0 d) none
60) If sinα and cosα are the roots of 25x²+ 5x -12= 0, then value of sin2α is
a) 12/25 b) -12/25 c) -24/25 d) 4/5
61) Let P(x) be a polynomial with integral coefficients . If there exist two integers a and b such that P(a) - P(b)= 1, then
a) both a and b must be even
b) both a and b must be odd
c) a and b are two consecutive integers d) none
62) Let am b, c be non zero real such that
¹₀∫ (1+ cos⁸x)+ax²+ bx+ c) dx
²₀∫ (1+ cos⁸x)+ax²+ bx+ c) dx
Then the quadratic equation has ac¹+ bx + c= 0 has
a) no root in (0,2)
b) at least one root in (1,2)
c) a double root +0,2) d) none
63) If a, b, c are distinct real numbers, then the expression
f(x)= a²{(x - b)(x - c)}/{(a- b)(a - c)} + b²{(x - c)(x - a)}/{(b- c)(b - a)} + c²{(x - a)(x - b)}/{(c- a)(c - b)} is identically equal to
a) x²- (a + b + c)x + abc
b) x² + x - abc c) x² d) none
64) The number of real solutions of the equation
27¹⁾ˣ + 12¹⁾ˣ = 2(8¹⁾ˣ) is
a) 0 b) 1 c) infinite d) none
65) If 0< a < b< c < d, then the quadratic equations ax²+ {1+ a(b + c)}x + abc - d = 0 has
a) real and distinct roots out of which one lies between c and d
b) real and distinct roots out of which one lies between a and b
c) real and distinct roots out of which one lies between b and c
d) nonreal roots
TRIGONOMETRICAL RATIOS AND IDENTITIES
SAP- 1
1) 2(sin⁶x + cos⁶x) - 3(sin⁴x+ cos⁴x)+ 1=0
2) 3[sin⁴(3π/2 - x) + sin⁴(3π+ x)] - 2[sin⁶(π/2+ x) + sin⁶(5π- x)] is equal to
a) 0 b) 1 c) 3 d) sin4x + sin6x e) none
3) sin⁶x + cos⁶x + 3sin²x cos²x = 1
4) 3(sinx - cosx)⁴ + 6(sinx + cosx)² + 4(sin⁶x + cos⁶x) is independent of x.
5) (sin⁸x - cos⁸x)= (sin²x - cos²x)(1- 2 sin²x cos²x).
6) (3+ cos4x) cos2x= 4(cos⁸x - sin⁸x).
7) If sinx+ cosx= a, then find the values of|sinx - cosx| and cos⁴x + sin⁴x.
8) If sinx + cosecx = 2, then sin²x + cosec²x is equal to 2. T/F
9) f(x)= cos²x + sec²x≥ 2. T/F
Or minimum value of f(x) is 2.
10) Given A= sin²x + cos⁴x, then for all real x.
a) 1≤ A≤2 b) 3/4≤A ≤1 c) 13/16 ≤A ≤1 d) 3/4 ≤A ≤ 13/16
11) Let A= sin⁸x + cos¹⁴x, then for all real x
a) A≥ 1 b) 0< A ≤1 c) 1/2< A ≤ 3/2 d) none
12) If x, y are acute, sinx= 1/2, cos y= 1/3, then (x + y) belong to
a) (π/3,π/2) b) (π/2,2π/3) c) (2π/3,5π/6) d) (5π/6,π)
13) (tanx + cot x)²= sec²x + cosec²x = sec²x cosec²x.
14) (1+ tan x tan y)² + (tanx - tan y)² = sec²x sec²y.
15) (secx - tan x)/(sec x + tan x)= 1- 2 secx tanx + 2 tan²x.
16) 1/(secx - tan x) - 1/cosx = 1/cosx - 1/(secx + tanx).
17) (secx + tan x -1)(secx - tanx +1) - 2 tan x= 0
18) If (secx + tanx)(sec y + tan y)(sec z + tan z)= (secx - tan x)(sec y - tan y)(sec z - tan z) show that each of the side is equal to ±1.
19) If (1+ sinx)(1+ sin y)(1+ sin z)= (1- sin x)(1- sin y)(1- sin z), show that each side is equal to ± cosx cos y cos z.
20) Let f(x)= sinx (sinx + sin3x). Then f(x).
a) ≥ 0 only when x≥ 0
b) ≤ 0 for all real x
c) ≥0 for all real x
d) ≤ 0 only when x ≤ 0
21) The maximum value of (cosx₁). (cosx₂)......(cosxₙ), under the restriction 0≤ x₁, x₂, .....xₙ≤ π/2 and (cotx₁).(Cotx₂).....(cotxₙ)= 1 is
a) 1/2ⁿ⁾² b) 1/2ⁿ c) 1/2n d) 1
22) √{(1- sinx)/(1+ sinx)}= secx - tan x.
23) √{(1+ cosx)/(1 - cosx)}= cosecx + cotx.
24) If sinx + sin²x= 1, then show that cos¹²x + 3 cos¹⁰x + 3 cos⁸x + cos⁶x -1= 0
25) If sinx+ sin²x + sin³x = 1, then cos⁶x - 4cos⁴x + 8cos²x = _____.
26) sec⁴x (1- sin⁴x) - 2 tan²x = 1.
27) tan²x - sin²x = sin⁴x sec²x= tan²x sin²x.
28) (cotx + tant)/(cot y + tanx)= cotx tan y.
29) (sinx + cosx)(tanx + cotx)= secx + cosecx
30) (cosx cosecx - sinx secx)/(cosx + sinx)= cosecx - secx.
31) (1+ cotx - cosecx)(1+ tanx + secx)= 2
32) (cosecx - sinx)(secx - cosx)(tanx + cotx)= 1
33) (tanx + secx -1)/(tanx - secx +1)= (1+ sinx)/cosx.
34) cot²x(secx -1)/(1+ sinx) = sec²x. (1- sinx)/(1+ secx).
35) (secx +1- tanx)/(tanx - secx +1)= (1+ cosx)/sinx.
36) cosx/(1- tanx) + sinx/(1- cotx)= sinx + cosx.
37) tₙ= sinⁿx + cosⁿx, then (t₃ - t₅)/t₁ = (t₅ - t₇)/t₃.
38) tanx/(1- cotx) + cotx/(1- tanx)= secx cosecx +1.
39) (sinx + cosecx)²+ (cosx + secx)²= tan²x + cot²x +7.
40) (1+ cotx + tanx)(sinx - cosx)= secx/cosec²x - cosecx/sec²x.
41) (secx - cosecx)(1+ tanx + cotx)= tanx secx - cotx cosecx.
42) {2sinx tanx(1- tanx)+ 2 sinx sec²x}/(1+ tanx)²= 2sinx/(1+ tanx).
43) (tanx + cosec y)²+ (cot y - secx)²= 2 tanx cot y(cosecx + sec y).
44) {(1+ sinx - cosx)/(1+ sinx + cosx)}²= (1- cosx)/(1+ cosx).
45) If 2sinx/(1+ cosx + sinx)= y, then (1- cosx + sinx)/(1+ sinx) is also y.
46) {1/(sec²x - cos²x) + 1/(cosec²x - sin²x)}. sin²x cos²x = (1- sin²x cos²x)/(2+ sin²x cos²x).
47) (cosecx - secx)(cotx - tanx)= (cosecx + secx)(secx cosecx -2).
48) If tanx+ sinx = m and tanx - sinx = n, then show that m²- n² = 4√(mn).
49) Eliminate x from the relations
a secx = 1- b tan x and a² sec²x = 5+ b² tan²x.
50) If cosecx - sinx = m, secx - cosx = n, eliminate x.
51) If cosecx - sinx = a³, secx - cosx= b³, then a²b²(a² + b²)= 1.
52) If cotx + tanx = a, secx - cosx = b eliminate x.
53) If c cos³x + 3c cosx sin²x = m, c sin³x + 3c cos²x sinx = n, then show that (m + n)²⁾³ + (m - n)²⁾³= 2c²⁾³.
54) If cosx + sinx= √2 cosx, show that cosx - sinx =√2 sinx.
55) If 3 sinx + 5 cosx = 5, show that 5 sinx - 3 cosx = ±3.
56) If a cosx + b sin x = p, a sinx - b cosx = q, show that a² + b² = p² + q².
57) If a cosx - b sin x = c, show that a sinx + b cosx = ±√(a² + b² + c²).
58) If a sinx + b cosx = c, then show that (a - b tanx)/(b + a tanx)= ±√(a² + b² + c²)/c.
59) If tan²x = (1- e²), show that secx + tan³x cosecx = (2- e²)³⁾².
60) If ax/cosθ + by/sinθ = (a²- b²) and (ax sinθ)/cos²θ - (by cosθ)/sin²θ = 0, show that (ax)²⁾³ + (by)²⁾³= (a² - b²)²⁾³.
61) If sinθ = (m² - n²)/(m²+ n²), determine the values of tanθ, secθ, cosecθ.
62) If tanθ = 2x(x+1)/(2x +1), determine sinθ and cosθ.
63) If cosθ = 2x/(1+ x²), find the values of tanθ and cosecθ.
64) If secx = p + 1/4p, then secx + tanx = 2p or 1/p
65) If secθ + tanθ = p, obtain the values of secθ, tanθ, sinθ in terms of p.
66) If cosx/cos y = a, sinx/sin y = b, then (a² - b²)sin²y= a² -1
67) If tanθ = p/q, show that (p sinθ - q cosθ)/(p sinθ + q cosθ) = (p² - q²)/(p² + q²).
68) Is the equation sec²θ= 4xy/(x + y)² possible for real values of x and y ?
If not, then find out a relation between x and y so that it may be possible.
69) If m² + m'² + 2mm' cosθ = 1,
n² + n'² + 2nn' cosθ = 1 and mn + m'n' + (mn' + m'n) cosθ = 0 show that m² + n² = cosec²θ.
SAP-2
1) The value of sin⁶θ + cos⁶θ + 3 sin²θ cos²θ is
a) 0 b) 1 c) 2 d) 3
2) The least value of 2sin²θ+ 3 cos²θ is
a) 1 b) 2 c) 3 d) 5
3) The greatest value of sin⁴θ + cos⁴θ is
a) 1/2 b) 1 c) 2 d) 3
4) The value of sin²θ cos²θ(sec²θ+ cosec²θ) is
a) 2 b) 4 c) 1 d) 0
5) If sinθ + cosecθ = 2, then sin²θ + cosec²θ is equal to
a) 1 b) 4 c) 2 d) none
6) For how many values of x between 0 and 2π is the equation
2cosec2x cotx - cot²x = 1 valid ?
a) 0 b) 2 c) 1 d) none
7) Incorrect statement is
a) sinθ= -1/5 b) cosθ= 1 c) secθ= 1/2 d) tanθ= 20
TRUE OR FALSE
8) sec²θ= 4xy/(x + y)² is true if and only if
a) x+ y≠ 0 b) x= y, x≠ 0 c) x= y d) x≠ 0, y≠ 0
9) If x= a cos²θ sinθ and y= a sin²θ cosθ, then (x² + y²)³/(x²y²( is independent of θ.
10) The inequality ₂sin²θ + ₂cos²θ≥ 2√2 holds for all real θ.
11) The equation sinθ = x + 1/x holds true for all real θ.
FILL IN THE BLANK
12) The least value of tan²θ + cot²θ is _____
13) The value of sinθ cosθ(tanθ + cotθ) is ____
14) If for real x, the equation x+ 1/x = 2 cosθ holds, then cosθ= ____
15) If cosecθ - cotθ = q, then the value of cosecθ = _____
SET THEORY
1) Set: A set is a collection of well defined and well distinguished objects of our perception or thought.
The words 'well defined objects ' imply that we must be given a rule with the help of which we should readily be able to say whether a particular object 'belongs to' the set of not. The words 'well distinguished objects ' imply that if the objects of the collection be named, then in doing so, the number of objects will not increase.
The set are usually denoted by capital letters of English alphabet viz, A, B, CA
2) ELEMENTS
The objects, which constitute the set, are said to be elements of the set.
These are also known as members or points of the set. The elements are usually denoted by small letters of English alphabet viz, a, b, c,....
i) If a is an element of the set A, we write it as a ∈A and is ready as " a belongs to A"
ii) If a is not an element of the set A, we write it as a∉ A and is ready as " a does not belong to A"
3) REPRESENTATION OF SETS
There are two methods to represent a set.
a) Roster /Tabulation Method. In this method, the set is represented by listing all its elements, saparating the elements by commas and enclosing them in curvilinear brackets.
b) Defining Property Method. In this method, the set is represented by specifying the common property of the elements.
Thus the set A is represented by A= {a: P(A) is true}.
Here 'a' stands for 'an arbitrary elements' of the set' and (:) stands for 'such that' and P(A) stands for 'common property '
4) FINITE AND INFINITE SETS
a) finite set. If it has finite number of elements.
b) Infinite set. If it has an infinite number of elements.
Order of a finite set is the number of elements it contains.
The order of a finite set A is denoted by O(A).
5) EMPTY SET
A set having no element is....
It is also called Null set or Void set.
6) SINGLETON SET
A set having only one element is....
7) SUB-SETS
a) Subset. Let A and B be two sets. Then the set A is said to be a subset of the set B if each element of A is also an element of B.
Symbolically, we write it as A⊆ B.
Here B is superset of A and is written as B⊇A.
b) Proper Subset. A set A is a proper subset of B if and only if each element of A is in B and there is atleast one element in B, which is not in A.
Symbolically, if A is a proper subset of B, then A ⊂B and A≠ B or A ⊂ B
8) COMPARABLE SETS
Two sets are said to be comparable iff either A ⊂B or B⊃A.
9) EQUAL AND EQUIVALENT SETS
a) Equal Sets. Two sets A and B are said to be equal (written as A= B) iff A ⊂B and B ⊂A.
Two sets A and B are said to be equal if they have exactly same elements.
b) Equivalent Sets , Two sets are said to be equivalent if they have same number of elements.
10) FAMILY OF SETS
A set said to be family of sets if it's elements are also sets.
This is also known as set of sets.
If A= {a,b}, then S={φ, {a}, {b}, {a,b} is the set of sets.
11) POWER SET
The set of all possible subsets of a set A is said to be the power set of A and is denoted by P(A).
If a= P{a,b,c}, then P(A)={φ, {a}{b}, {c}, {a,b}, {b,c}, {a,c}, {a,b,c}}.
12) UNIVERSAL SET
The main set under discussion or the set containing all possible values in the given frame of reference is said to be universal set and is denoted by U or E or X.
13) OPERATIONS IN SETS
a) Union of Sets.
(I) Let A and B be two sets. The union of A and B (denoted by A U B) is the set of all those elements which are either in A or in B or in both.
Symbolically, A∪B= x: x ∈A or x ∈ B}.
II) Let A₁, A₂,..., Aₙ be n(≥2) sets. Then the union of these (denoted by ⁿᵢ₌₁U Aᵢ) is the set of all those elements which are in Aᵢ (1≤ i≤ n) for atleast one value of i.
b) Intersection of sets.
I) Let A and B be two sets. The intersection of A and B (denoted by A∩B) is the set of all those elements which are in both A and B.
II) Let A₁, A₂, .....,Aₙ be n(≥2) sets. Then the intersection of these (denoted by ⁿᵢ₌₁ ∩Aᵢ) is the set of all those elements which are in Aᵢ(1≤ i≤ n) for each i.
14) FUNDAMENTAL RESULTS
i) Identity Laws, A ∪φ, A ∩φ= φ
ii) Idempotent Laws. A ∪A= A, A ∩A = A
iii) Commutative Laws. A ∪ B= B ∪ A, A ∩ B= B∩ A
iv) Associative Laws. A∪(B ∪C)= (A∪B) ∪ C
A∩(B∩C)= (A∩B)∩C
c) Distributive Laws, A∪ +B∩C)= (A∪B)∩(A ∪ C)
A∩(B∪C)= (A∩B)∪(A∩C).
15) Disjoit Sets
If and only if they have no common element.
Let A and B be two sets
Here A ∩B= φ.
16) DIFFERENCE OF SETS
Let A and B be two sets. Then (A - B) is the set of those elements of the set A which are not in the set B.
Symbolically. A - B= {x: x ∈A and x ∉B}
Similarly, B - A ={x: x ∈B and x ∉A}
17) SYMMETRIC DIFFERENCE OF SETS
Let A and B be two sets. Then their symmetric difference is the union of the sets A- B and B- A. This is denoted by A ∆B.
Symbolically
A∆B ={x: x ∈A - B or x ∈B - A}
18) COMPLEMENT OF A SET
Let X be the universal set and A be any set. Then the complement of the set A is the set of all those elements of X, which are not in the set A.
This is denoted by Aᶜ or A' or X - A
Symbolically, Aᶜ={x: x∈X and x ∉A}.
19) FUNDAMENTAL RESULTS
a) Xᶜ= φ, φᶜ= X
b) (Aᶜ)ᶜ= A
c) If A ⊆ B, then B⊆Aᶜ
d) A U Aᶜ= X and A∩Aᶜ = φ
e) De-morgan's Laws. (AUB)ᶜ = Aᶜ∩ Bᶜ
(A∩ B)ᶜ= AᶜU Bᶜ
20) USE OF SETS IN PRACTICAL PROBLEMS
If A, B, C are finite sets of n elements each, then
a) n(AUB)= n(A)+ n(B); if A, B are disjoint
b) n(AUB)= n(A)+ n(B) - n(A∩B); if A, B are not disjoint
c) n(A∩Bᶜ)= n(A) - n(A∩B)
d) n(B∩Aᶜ)= n(B)- n(A∩B)
e) n(AUB)= n(A∩Bᶜ)+ n(B∩Aᶜ)+ n(A∩B).
f) n(AUBUC)= n(A)+ n(B)+ n(C) - n(A∩B) - n(B ∩C) - n(A∩C)+ n(A∩B∩C).
21) ORDERED PAIR
An ordered pair is a pair of entries in the specified order.
In the ordered pair (a,b), a is the first element and b the second element.
22) CARTESIAN PRODUCT/DIRECT PRODUCT OF SETS
a) The set of all ordered pairs of elements (a,b); a ∈A, b ∈ B is called Cartesian product of two sets A and B and is denoted by A x B
Symbolically, A x B={(a,b): a ∈A, b∈B}.
b) The Cartesian product of n(>2) sets A₁, A₂,......Aₙ is the set of all ordered n-triples (a₁, a₂, .......aₙᵢ), where aᵢ∈ Aᵢ (1≤ i≤ a) and is denoted by A₁x A₂ x......x Aₙ or ⁿᵢ₌₁ΠAᵢ.
Symbolically, ⁿᵢ₌₁ΠAᵢ= {(a₁, a₂,......,aₙ): aₙ ∈ Aₙ, 1≤ i≤ n}.
23) FUNDAMENTAL RESULTS
a) Ax B≠ B x A
b) A x φ= φ x = φ
c) n(Ax B)= n(B x A)= n(A) x n(B)
d) If A ⊆B, C ⊆ D, then A x C ⊆ B x D
e) n(A₁x A₂ x......x Aₙ)= n(A₁) x n(A₂)......xXₙ (Aₙ)
IN FOCUS
ⱼ
φΠ ᵢ
φφ
ᶜᶜᶜᶜφ
∪∪∪∪
ⱼ∉∆
₁₂ₙᵢⱼ∉∆
⊆⊇⊂∩∪∀≺∩⊃⊄⊅¹²₁₂ₙⁿᵢ₌₁ᵢ₁₂₌₁ᵢ₁₂ₙᵢⱼ∉∆ₙⁿᵢ
1) Let N be the set of non-negative integers, I the set of integers Nₚ the set of non positive integers, E the set of even integers and P the set of prime numbers. Then
a) I - N= Nₚ
b) N ₚ∩ Nₚ = φ
c) E ∩C = φ
d) N ∆Nₚ= I - {0}
2) Let A and B be two sets, then (AUB)ᶜ U(Aᶜ∩B) equals
a) Aᶜ b) Bᶜ c) A d) none
3) If A and B are two sets , then A∩(AUB)ᶜ equals
a) A B) B C) φ d) none
4) The set (A∩Bᶜ)ᶜ U (B∩C) equals
a) Aᶜ U Bᶜ b) Aᶜ U Cᶜ c) Aᶜ U B U C d) none
5) Let U be the universal set and AU BU C= U Then [(A - B) U(B - C) U(C A)]ᶜ equals
a) AUBUC b) A∩B∩C c) AU(B∩C) d) A∩(B UC)
6) The set (AUBUC)∩(A∩Bᶜ∩Cᶜ)∩Cᶜ equals
a) A∩C b) B U Cᶜ c) B ∩Cᶜ d) none
7) If A and B are two sets, then A ∩(AUB) equal
a) A b) B c) Aᶜ d) Bᶜ
8) If A={1,3,5,7,9,11,13,15,17}, B={2,3,.....18} and N is the universal set, then Aᶜ U ((A U B)∩ Bᶜ is
a) A B) N c) B d) none
9) If A and B are disjoint nonempty sets, then A - (A - B) equals
a) A B) B C) φ d) A U B
10) Which of the following is empty set?
a) {x: x is a real number and x² -1=0}
b) {x: x is a real number and x² +1=0}
c) {x: x is a real number and x² -9 =0}
d) {x: x is a real number and x²= x +2}
11) Which of the following is a singleton set?
a) {x: |x|=5, x ∈ I}
b) {x: |x|=6, x ∈ N}
c) {x: x²=5, x ∈ N}
d) {x: x²+ 3x +2=0, x ∈ N}
12) Which of the following does not have a proper subset:
a) {x: x ∈ N, 4< x<5}
b) {x: x ∈ Q}
c) {x: x ∈ WpQ, 4< x<5} d) none
13) If A, B and C are any three sets, then A - (BUC) equals
a) (A-- B) U(A - C)
b) (A- B) ∩(A - C)
c) (A-- B) U C
d) (A - B) ∩ C
14) If A, B, C are three sets, then A∩(B U C) equals
a) (AUB) ∩(AUC)
b) (A∩B) U(A∩C)
c) (AUB) U (AU C) d) none
15) If Q={x: x= 1/y, where x ∈N} then:
a) 0∈Q b) I∈ Q c) 2∈ Q d) 2/3 ∈Q
16) If A={x: x∈I, -2≤ x ≤ 2}, B==x: x∈ I, 0≤ x ≤ 3}
C={x: x ∈N, 1≤ x ≤ 2}
D={(x,y): (x,y) ∈N x N, x+ y= 8}, then
a) n(BUC)= 5
b) n(D)= 6
c) n(AU(BUC)(= 5 d) none
17) If for α∈ N, αN= {αx: x ∈ N}, then the set 8N ∩6N is
a) 8N b) 12N c) 24N d) 48N
18) Let n(A)= 3 and n(B)= 6 and A B. Then the number of elements in A ∩B is
a) 3 b) 9 c) 6 d) none
19) Sets A and B have 3 and 6 elements respectively. What can be the minimum number of elements in AUB ?
a) 3 b) 6 c) 9 d) 18
20) Two finite sets have m and n elements. The total number of subsets of the first set is 56 more than the total number of subsets of second set. The values of m and n are
a) 7,6 b) 6,3 c) 5,1 d) 8,7
21) If A, B and C are any three sets, then Ax (BUC) is
a) (AxB) U (Ax C)
b) (A UB) x (AU C)
c) (Ax B) ∩(Ax C)
d) none
22) If A, B and C are any three sets, then A x (B ∩C) is
a) (Ax B) U(Ax C)
b) (Ax B)∩(Ax C)
c) (AUB) x (AUC)
d) (A∩B) x (A∩C)
23) If S₁={1,2,3,.....20}, S₂= {a,b,c,d}, S₃={b,d,e,f}. The number of elements of (S₁x S₂) U(S₁x S₃) is
a) 100 b) 120 c) 100 d) 40
24) If A={1,2,3,6,11,18,21}, B={5,7,9} and N is the universal set, then Aᶜ U(A U B) ∩ Bᶜ equals
a) A b) B C) N d) N - A
25) The set (AUB)∩(A ∩Bᶜ∩Cᶜ)ᶜ ∩Cᶜ equals
a) A∩C b) B∩Cᶜ c) Bᶜ∩Cᶜ d) none
26) Suppose A₁, A₂, ......A₃₀ are thirty sets, each having 5 elements and B₁, B₂, .....Bₙ are n sets, each element of S belongs to exactly 10 if the Aᵢ's and exactly 9 of Bᵢ's. Then n is equal to
a) 15 b) 3 c) 45 d) none
27) Consider the set of all determinants of order 3 with entries 0 or 1 only. Let B be the subset of A consisting of all determinants with value 1. Let C be the subset of the set of all determinants with value -1. Then
a) C is empty
b) B has as many elements as C has
c) A= B U C
d) B has twice as many elements as C has
28) If (1,3),(2,5) and (3,3) are the elements of Ax B and the total number of elements in A x B is 6, then the remaining elements of Ax B are
a) (1,5),(2,3),(3,5)
b) (5,1),(3,2),(5,3)
c) (1,5),(2,3),(5,3) d) none
29) If A, B, C be three sets such that AUB = AUC and A∩B= A∩C, then
a) A= B b) B= c c) A= C d) A= B = C
30) Let A={x,y): y= eˣ, x∈ R}, B= [(x,y): y= e⁻ˣ, x ∈ R}. Then
a) A∩B= φ b) A∩B≠ φ c) AUB= R d) none
31) Let A={(x,y): y= eˣ, x ∈R}, B={(x,y): y= x, x∈ R}. Then
a) B⊆ A B) A ⊆ B C) A∩B= ∪φ d) AU B = A
32) If X={4ⁿ - 3n -1: n∈ N} and Y={9(n -1); n∈ N} then X U Y is
a) X b) Y c) N d) none
33) If X= {8ⁿ - 7n -1: n∈ N} and Y= {49(n -1)| n ∈ N}, thn
a) X⊂ Y b) Y⊂ X c) X= Y d) none
34) If the sets A and B are defined as:
A={(x,y): y= 1/x, 0 ≠ x R}
B={(x,y): y= - x, x ∈ R}, then:
a) A∩B= A b) A∩B= B c) A∩B= φ d) none
35) If A={φ, {φ}}, then the power set of A is
a) A B) {φ, {φ}, A} c) {φ, {φ}, {{φ}}A} d) none
36) In a class of 100 students, 55 students have passed in mathematics and 67 students have passed in Physics. Then the number of students who have passed in Physics only is
a) 22 b) 33 c) 10 d) 45
37) Out of 800 boys in a school, 224 played cricket, 240 played hockey and 336 played basketball. Of the total, 64 played both basketball and hockey, 80 played cricket and hockey; 24 played all the three games. The number of boys which did not play any game is
a) 128 b) 216 c) 240 d) 160
38) From 50 students taking examinations in mathematics, physics and chemistry, 37 passed mathematics, 24 physics and 43 chemistry. Atmost 19 passed mathematics and physics, atmost 29 mathematics and chemistry and atmost 20 physics and chemistry. The largest possible number that could have passed all three examination is
a) 9 b) 10 c) 12 d) none
39) Of the members of three athletic teams in a school 21 are in the cricket team, 26 are in the hockey team and 29 are in the football team. Among them, 14 play hockey and cricket, 15 play hockey and football, and 12 play football and cricket. Eight play all the three games. The total number of members in the three athletic teams is
a) 43 b) 49 c) 76 d) none
40) In a college of 300 students, every student reads 5 newspaper and every newspaper is ready by 60 students. The number of newspaper is
a) atleast 30
b) atmost 20
c) exactly 25 d) none
