THEORY OF QUADRATIC EQUATIONS
An question of the form
ax²+ bx + c= 0 ......(1)
where a≠0, a, b, c ∈ C, the set of complex complete numbers , is called a quadratic equation. The numbers a,b,c are called coefficients of the equation.
A root of the quadratic equation (1) is a complex number α such that
aα²+ bα + c= 0
The quantity D= b²- 4ac is known as discriminant of the equation. (1).
The roots of (1) are given by the formula
x= (- b + √D)/2a
Nature of the roots
1) If a, b, c ∈ R and a≠ 0. Then the following hold good.
a) The equation (1) has real and distinct root if and only if D> 0.
b) The equation (1) has real and equal roots if and only if D= 0.
c) The equation (1) has complex roots with non zero imaginary parts if and only if D< 0.
d) p+ iq (p, q ∈ R, q≠ 0) is a root of (1) if and only if p - iq is a root of (1)
2) If a, b, c ∈ Q and D is a perfect square of a rational number, then (1) has rational roots .
3) If a, b, c ∈ Q and p+ √q (p, q ∈ Q) is an irrational root of (1) then p - √q is also a root of (1)
4) If a= 1, b, c ∈ I and the roots of (1) are rational numbers , then these roots must be integerr.
5) If (1) is satisfied by more than two distinct complex numbers, then (1) becomes an identity , that is a= b = c = 0.
RELATION BETWEEN ROOTS AND COEFFICIENTS
If α and β are roots of the quadratic equation (1), then
α +β = -b/a and αβ = c/a
Note that a quadratic equation whose roots are α and β is given by
(x - α)(x - β)= 0.
QUADRATIC EXPRESSION AND ITS GRAPH
Let f(x)= ax²+ bx + c, where a,b,c ∈R and a≠ 0
We have f(x)=a[x²+ (b/a)x + c/a]
= a[x²+ (b/a)x + b²/4a²+ c/a - b²/4a²]
= a[(x + b/2a)²+ (4ac - b²)/4a²]........(2)
When is the quadratic Expression always positive (negative)?
It follows from (1) that f(x)> 0 (<0) ∀ x∈ R if and only if a> 0(<0) and
D= b² - 4ac < 0
Also, it follows from (2) that f(x)≥ 0 (≤ 0) ∀ x∈ R if and only if a>0(<0) and
D= b² - 4ac = 0. In this case for f(x)> 0(<0 for each x ∈ R , x≠ - b/2a. In this case the graph of y= f(x) will touch the x-axis at x= - b/2a.
Sign of a Quadratic Expression
If D= b²- 4ac > 0, then (2) can be written as
f(x) = a[(x + b/2a)² - {√(b²- 4ac)/2a}²]
= a[ x + {b + √(b²- 4ac)/2a}{x + {b + √(b²- 4ac)/2a}}]
= a(x - α)(x - β).
Where α= {-b - √(b²- 4ac)/2a} and
β= {-b + √(b²- 4ac)/2a}
If D= b²- 4ac > 0 and a> 0 then
> 0 for x <α or x> β
f(x) {< 0 for α < x <β
=0 for x= α ,β
If D= b²- 4ac > 0 and a< 0 then
< 0 for x <β or x> α
f(x) { >0 0 for β < x < α
= 0 for x= α ,β
Note that if a> 0, then f(x) has a least value at x= -b/2a. This least value is given by
f(-b/2a)= (4ac - b²)/4a.
If a< 0, then f(x) has a greatest value at x= - b/2a. This greatest value is given by
f(-b/2a)= (4ac - b²)/4a.
POSITION OF ROOTS OF A QUADRATIC EQUATION
let f(x)= ax²+ bx + c, where a, b,c ∈ R be a quadratic expression and let k, k₁, k₂ be three real numbers such that k₁ < k₂.
Conditions for Both the Roots to be More Than a Real Number k.
If a> 0, then the parabola y= ax²+ bx + c open upwards and intersect the x-axis in α and β where
α ,β= {- b ± √(b²- 4ac)/2a}
In this case both the roots α and βwill more than k if k lies to the left of both α ,β
From the figure we note that both the roots are more than k if and only if
(i) D> 0 (ii) k< - b/2a (iii) f(k)> 9
In case a< 0, both the roots will be more than k
If and only if
(i) D> 0 (ii) k < - b/2a (iii) f(k) < 0.
Combining the above two sets, we get both the roots of ax²+ bx + c= 0 are more than a real number k if only if
(i) D> 0 (ii) k < - b/2a (iii) af(k) > 0.
Conditions for Both the Roots to be Less Than a Real Number k
Both the roots ax²+ bx + c= 0 are less than a real number k if and only if
(i) D> 0 (ii) k < - b/2a (iii) f(k) > 0.
CONDITIONS FOR A NUMBER k TO LIE BETWEEN THE ROOTS OF A QUADRATIC EQUATION
The real number k lies between the roots of the quadratic equation f(x)= ax²+ bx + c = 0 if and only if a and f(k) are of opposite signs, that is, if and only if
(i) a> 0 (ii) D> 0 (iii) f(k)< 0 or
(i) a< 0 (ii) D> 0 (iii) f(k)< 0
Combining , we may say k lies between the roots of f(x)= ax²+ bx + c = 0 if and only if
(i) D> 0 (ii) a f(k)< 0
CONDITIONS FOR EXACTLY EXACTLY ONE ROOT OF A QUADRATIC EQUATION TO LIE IN THE INTERVAL (k₁, k₂) WHERE k₁< k₂
If a> 0, then exactly one root of f(x)= ax²+ bx + c= 0 lies in the interval internal (k₁, k₂) if and only if f(k₁<0) and f(k₂). Also, exactly one root lies in the interval (k₁,k₂) if and only if f(k₁)< 0 and f(k₂)> 0
Thus, if a> 0, exactly one root of f(x)= ax²+ bx + c = 0 lies in the interval (k₁,k₂) if and only if f(k₁) f(k₂)< 0.
Similarly, if a< 0, exactly one of the roots of f(x)= ax²+ bx + c = 0 lies in the interval (k₁, k₂) if fk₁) f(k₂)< 0.
CONDITIONS FOR BOTH THE ROOTS OF A QUADRATIC EQUATION TO LIE IN THE INTERVAL (k₁, k₂) WHERE k₁< k₂.
If a> 0, both the roots of f(x) ax²+ bx + c = 0 lie in the interval (k₁, k₂) if and only if
(i) D> 0 (ii) k₁< - b/2a < k₂ (iii) f+k₁)> 0 and f(k₂)> 0
In case a< 0, the condition read as
(i) D> 0 (ii) k₁< -b/2a < k₂ (iii) f+k₁)< 0 and f(k₂)< 0
CONDITIONS FOR QUADRATIC EQUATION TO HAVE A REPEATED ROOT
The quadratic equation f(x)= ax²+ bx + c = 0, a≠ 0 has α as a repeated if and only if f(α)= 0 and f'(α)= 0. In this case f(x)= a(x - α)². In fact α = - b/2a.
CONDITION FOR TWO QUADRATIC EQUATION to HAVE A COMMON ROOT
Suppose that the quadratic equation ax²+ bx + c = 0,and a'x²+ b'x + c' = 0, (where a, a'≠ 0 and ab' - a'b ≠0) have a common root . Let this common root be α Then
aα² + bα + c= 0 and a'α²+ b'α+ c'= 0
solving the above equations , we get
α²/(bc' - b'c) = α/(a'c - ac')= 1/(ab' - a'b)
=> α²= (bc' - b'c)/(ab' - a'b) and α= (a'c - ac')/(ab' - a'b)
Eliminating α we get
(a'c - ac')²/(ab' - a'b)² = (bc' - b'c)/(ab' - a'b)
=> (a'c - ac')² = (bc' - b'c)(ab' - a'b)
This is the required condition for two quadratic equation to have a common root.
How to obtain common root ?
Make coefficients of x² in both the equation same and subtract one equation from the other to obtain a linear equation in x. Solve it for x to obtain the common root.
CONDITION FOR TWO QUADRIC EQUATIONS TO HAVE THE SAME ROOTS
In case a'b - ab' = 0, then
a/a' = b/b' = c/c'.
In this case both the equation have the same roots.
EQUATIONS OF A HIGHER DEGREE
The equation
f(x)= a₀xⁿ + a₁xⁿ⁻¹+ a₂xⁿ⁻² + ....+ aₙ₋₁x + aₙ = 0
Where a₀, a₁, ...., aₙ₋₁, aₙ∈ C, the set of complex numbers, and a₀ ≠ 0, is said to be an equation of degree n. An equation of degree n has exactly n roots. Let α₁, α₂, ...., αₙ ∈ C be the n roots of (1) . Then
f(x)= a₀(x - α₁)(x - α₂)....(x - αₙ)
Also α₁ + α₂ + ....+ αₙ = a₁/a₀ and
α₁α₂.....αₙ= (-1)ⁿ aₙ/a₀.
CUBIC AND BIQUADRATIC EQUATIONS
If α, β, γ are the roots of ax³ + bx² + cx + d = 0, then
α+ β + γ = -b/a, βγ + γα+ αβ = c/a and αβγ = -d/a.
Also α, β, γ, δ are the roots of the equation ax⁴+ bx³ + cx² + dx + e=0, then α+ β+ γ+δ = -b/a, (α+ β)(γ+ δ)+ αβ+ γδ = c/a.
αβ(γ+ δ)+ γδ(α+ β)= - d/a, αβγδ = e/a.
TRANSFORMATION OF EQUATIONS
We know list some of the rules to form an equation whose roots are given in terms of the roots of another equation.
Let given equation be
a₀xⁿ + a₁xⁿ⁻¹ + ......aₙ₋₁x + aₙ = 0
Rule 1: To form an equation whose roots are k(≠ 0) times roots of the equations in (1) , replace x by x/k in (1).
Rule 2: To form an equation whose roots are the negative of the roots in equation (1), replace x by - x in (1). Alternatively, change the sign of the coefficients of xⁿ⁻¹, xⁿ⁻³, xⁿ⁻⁵, ......etc in (1).
Rule 3: To form an equation whose roots are k more than the roots of equation in (1), replace x by x - k in (1).
Rule 4: To form an equation whose roots are reciprocals of the roots in equation (1), replace x by 1/x in (1) and then multiply both the sides by xⁿ.
Rule 5: To form an equation whose roots are square of the roots of the equation in (1) proceed as follows:
Step 1: Replace x by √x in (1)
Step 2: Collection all the terms involving √x on one side.
Step 3: Square both the sides and simplify
For instance, to form an equation whose roots are squares of the roots of x³ - 2x² - x +2=0, replace x by √x to obtain
x √x - 2x - √x +2=0
=> √x (x -1)= 2(x -1)
Squaring we get
x(x -1)²= 4(x -1)²
Or, (x -4)(x² - 2x +1)=0
Or, x³ - 6x² + 9x - 4= ó
Rule 6: To form an equation whose roots are cubes of the roots of the equation in (1) proceed as follows:
Step 1: Replace x by x¹⁾³.
Step 2: Collect all the terms involving x¹⁾³ and c²⁾³ on one side.
Step3: Cube both the sides and simplify.
DESCARTES RULE OF SIGNS FOR THE ROOTS OF A POLYNOMIAL
Rule 1: The maximum number of positive real roots of a polynomial equation
f(x)= a₀xⁿ + a₁xⁿ⁻¹ + a₂xⁿ⁻² + .....aₙ₋₁x + aₙ =0.
is the number of changes of the signs of coefficients from positive to negative and negative to positive. For instance, in the equation x³ + 3x² + 7x - 11= 0 the signs of coefficients are
+ + + -
As there is just one change of sign, the number of positive roots of x³ + 3x² + 7x - 11= 0 is almost 1.
Rule 2: The maximum number of negative roots of the polynomial equation f(x)= 0 is the number of changes from positive to negative and negative to positive in the signs of coefficients of the equation f(-X)= 0.
SOME HINTS FOR SOLVING POLYNOMIAL EQUATIONS
1) To Solve an equation of the form (x - a)⁴ + (x - b)⁴ = A
Put y= x - (a + b)/2
In general to solve an equation of the form
(x - a)²ⁿ + (x - b)²ⁿ = A
Where n is a positive integer, we put y= x - (a+ b)/2.
2) To Solve an equation of the form
a₀f(x)²ⁿ + a₁(f(x))ⁿ + a₂ = 0......(1)
We put (f(x))ⁿ = y and solve a₀y² = ₁y + a₂ = 0
to obtain its roots y₁ and y₂
Finally, to obtain solutions of (1) we solve,
(f(x)ⁿ= y₁ and
(f(x))ⁿ= y₂
3) An equation of the form
(ax² + bx+ c₁)(ax² + bx + c₂)......(ax² + bx + cₙ)= A Where c₁, c₂, cₙ,∈ R, can be solved by putting ax²+ bx = y.
4) An equation of the form
(x - a)+x - b)(x - c)(x - d)= Ax²
Where ab= cd, can be reduced to a product of two quadratic polynomial by putting y = x + ab/2.
5) An equation of the form
(x - a)(x - b)(x - c)(x - d)= A
Where a< b < c < d, b - a = d - c
can be solved by chnge of variable
y= {(x - a) + (x - b) + (x - c) + (x - d)}/4
= x - (1/4) (a+ b + c + d)
6) A polynomial f(x,y) is said to be symmetric if f(x,y)= f(y,x) cross ∨ x, y.
A symmetric polynomial can be represented as a function of x + y and xy.
Solving Equations Reducible to Quadratic
1) To Solve equation of the type
ax⁴ + bx² + c = 0,
Put x²= y.
2) To Solve equation of the type a(p(x))² + bp(x)+ c = 0
(Where p(x) is an expression of x) Put p(x)= y.
3) To Solve equation of the type
ap(x)+ b/p(x) + c = 0
Where p(x) is an expression in x, put p(x)= y
This reduces the equation to
ay² + cy + b=9.
4) To Solve equation of the form
a(x² + 1/x²)+ b(x + 1/x) + c = 0
Put x+ 1/x = y
And solve
a(x² + 1/x²)+ b(x - 1/x)+ c= 0
Put x - 1/x = y
5) To Solve reciprocal equation of the type
ax⁴ + bx³ + cx² + bx + a = 0, a≠ 0
We divide the equation by x², to obtain
a(x²+ 1/x²) + b(x + 1/x)+ c= 0
and then put
6) To Solve equation of the type
(x + a)+x + b)(x + c)+x+ d)+ k = 0
Where a+ b = c + d, put x² + (a+ b)x = y
7) To Solve equation of the type
√(ax + b)= cx + d
Or
√(ax²+ bx + c)= dx + e
Square both the sides and solve for x.
8) To Solve equation of the type
√(ax + b) ± √(cx + d)= e
Proceed as follows.
Step 1: Transfer one of the radical to the other side and square both the sides.
Step 2: Keep the expression with radical sign on one side and transfer the remaining expression on the other side.
Step 3: Now solve as in 7 above.
USE OF CONTINUITY AND DIFFERENTIABILITY TO FIND ROOTS OF A POLYNOMIAL EQUATION
Let f(x)= a₀xⁿ + a₁xⁿ⁻¹+ a₂xⁿ⁻² + .....+ aₙ₋₁x + aₙ, thn f is continuous on R.
Since f is continuous on R, we may use the intermediate value theorem to find whether or not f has a real root.
If there exists a and b such that a< b and f(a) f(b)< 0, then there exists at least one c∈ (a,b) such that f(c)= 0.
Also, if f(-∞) and f(a) are of opposite signs, then atleast one root of f(x)= 0 lies in (-∞, a). Also, if f(a) and f(∞) are opposite signs, then atleast one root of f(x)= 0 lies in (a, ∞).
Result 1: If d(x)= 0 has a root α of multiplicity r (where r> 1), thn we can write
f(x)= (x - α)ʳ g(x)
Where g(α)≠ 0.
Also, f'(x)= 0 has α as a root with multiplicity r -1.
Result 2: if f(x)= 0 has n real roots, then f'(x)= 0 has (n -1) real roots.
It follows immediately using Result 1 and Rolle's Theorem.
Result 3: if f(x)= 0 has n distinct real roots, we can write
f(x)= a₀(x - α₁)(x - α₂)......(x - αₙ)
Where α₁, α₂, ..... αₙ are n distinct roots of f(x)= 0.
We can also write
f'(x)/f(x)= ⁿₖ₌₁∑ 1/(x - αₖ)
α α, β, γ δ
∈
SAP-1
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
1c 2a 3c 4d 5c 6a 7b 8b 9b 10a 11c 12b 13d 14c 15d 16b 17c 18a 19a 20b 21b 22c 23a 24c 25c 26d 27c 28a 29c 30b 31a 32b 33a 34a 35b 36c 37a 38d 39a 40d 41b 42a 43c 44d 45b 46a 47c 48a 49b 50c 51a 52d 53a 54d 55d 56c 57b 58d 59a 60c 61c 62b 63c 64a 65a
SAP-2
1) If α, β are the roots are the roots of x²+ px + q= 0 and γ, δ are the roots of x²+ rx + s= 0, then the value of (α- γ)(α-δ)(β -γ)(β -δ) is
a) (r - p)² - (q- s)²
b) (r - p)² + (q- s)²
c) (r - p)² - (q- s)² - 2rp(r - p)(q - s) d) none
2) The number of real solutions of x²+ 5|x| + 4=0 is
a) 4 b) 2 c) 1 d) 0
3) The number of real solutions of x²- 3|x|+2=0 is
a) 4 b) 2 c) 1 d) 0
4) If 2+ i √5 is a root of x²- px + q= 0 where p and q are real, then the ordered pair (p,q) is equal to
a) (4,9) b) (9,4) c) (3,3) d) (2,3)
5) If the quadratic equation 2x²+ ax + b= 0 and 2x²+ bx + a = 0, (a≠ b) have a common root, the value of a+ b is
a) -3 b) -2 c) -1 d) 0
6) If a,b,c, d and p are distinct real numbers such that
(a²+ b²+ c²)p²+ 2(ab + bc+ cd)p + (b²+ c²+ d²)≤ 0, then a, b, c, d
a) are in AP
b) are in GP
c) are in HP d) none
7) The number of real roots of of the equation sin(eˣ)= 5ˣ + 5⁻ˣ is
a) 0 b) 1 c) 2 d) infinitely many
8) The value of a for which the equation x³+ ax +1=0 and x⁴+ ax²+ 1= 0 have common root is
a) 2 b) -2 c) 0 d) none
9) The roots of the equation |x²- x -6|= x +2 are
a) -2, 1, 4 b) 0, 2, 4 c) 0, 1, 4 d) -2, 2, 4
10) The number of real roots of the equation |x²| - 3|x|+ 2= 0
a) 4 b) 1 c) 2 d) infinite
11) If α, β are the roots of x²- P(x +1) - C = 0 then value of
(α+1)(β +1) is
a) C -1 b) 4+ C c) 1+ C d) 1- C
12) Let f(x) be quadratic expression such that f(x)< 0 , x∈ R. If g(x)= f(x)+ f'(x)+ f"(x) then for x∈ R.
a) g(x)< 0 g(x)≤ 0 c) g(x)> 0 d) g(x)≥ 0
13) If x+1 is a factor of x⁴+ (p -3)x³ - (3p -5)x²+ (2p+9)x + 12, then value of p is
a) -2 b) 2 c) 1 d) -1
14) The equation ₓ(3/4)(log₅x)²+ log₅x - 5/4 = √5 has
a) only one real solution
b) exactly two real solution
c) one irrational solution
d) infinite number of solutions
15) Let f and g be two real valued functions and S={x | f(x)=0} and T={x | f(x)=0}, then S∩T represent the set of roots of
a) f(x) g(x)= 0
b) f(x)²+ g(x)² = 0
c) f(x) + g(x)= 0
d) f(x)/g(x)= 0
16) The number of real roots of the equation (x +3)²+ (x +1)²+ (x -5)²+ (x -6)²= 0 is
a) 2 b) 1 c) 0 d) none
17) If a,b,c are in AP and if (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, b², c² are in GP
c) a², b², c² are in HP d) none
18) If a,b,c are positive real numbers , then the number of positive real roots of the equation ax²+ bx + c= 0 is
a) 0 b) 1 c) 2 d) infinite
19) If the roots of the equation x²+ p²= 8x + 6p are real, then p belongs to the interval
a) [2,8] b) [-8,2] c) [- 2,8] d) [-8, -2]
20) If sum of the roots of the equation (a+ 1)x²+ (2a +3)x + (3a+4)= 0 is -2, then the product of the roots is
a) 1 b) -1 c) 2 d) -2
21) If 3- 4i is a root of x²- px + q=0 where p, q ∈R, then value (2p - q)/(p+ q) is
a) -12/31 b) -13/31 c) -15/31 d) none
22) If x= 1+ i is a root of x³- ix + 1 - i = 0, then the equation whose roots are the remaining two roots of x³- ix + 1 - i = 0 is
a) x²+(1+ i)x + 1+ i= 0
b) x²- (1+ i)x + 1+ i= 0
c) x²+ 2(1+ i)x - 2= 0 d) none
23) If α and β be the roots of the equation x²+ px - 1/2p²= 0, where p∈ E. Then the minimum possible value of α⁴+ β⁴ is
a) 2 b) 2√2 c) 2+√2 d) none
24) The equation √{x +3 - 4√(x -1)}+ √{x +8 - 6√(x -1)}= 0 has
a) no solution
b) exactly one solution
c) exactly two solutions
d) more than two solutions
25) The equation | x- x² - 1|= |2x - 3 - x²| has
a) no solution b) exactly one solution
c) exactly two solutions
d) more than two solutions
26) If sinα, cosα are the roots of the equation ax²+ bx+ c= 0, (a≠ 0), then
a) a²- b²+ 2ac=0
b) a²+ b² - 2ac=0
c) (a - c)²= b²+ c² d) none
27) If x ∈R, and k= (x²- x+ 1)/(x²+ x+ 1), then
a) 1/3≤k ≤ 3 b) k≥ 5 c) k≤ 0 d) none
28) If the quadratic equation x²+ bx+ ca= 0 and x²+ cx+ ab = 0 have a common root and b≠ c, then their other roots will satisfy the equation
a) x²+ (b+ c)x+ bc= 0
b) x²+ ax+ bc= 0
c) x²+ ax+ bc= 0 d) none
29) If the inequality (mx²+ 3x+ 4)/(x²+ 2x+ 2) < 5 is satisfied for all x∈ R, then
a) m< 5 b) m > 5 c) m < 71/24 d) m > 71/24
30) If a, b, c are distinct real numbers, then the equation expression
{(x - b)(x - c)}/{(a- b)(a - c)} + {(x - c)(x - a)}/{(b- c)(b - a)} + {(x - a)(x - b)}/{(c- a)(c - b)} is identically equal to
a) 1 b) x c) x² d) none
31) If ax²+ bx+ c, a, b, c ∈ R, a≠ 0 has no real zeros and a- b + c< 0, then value of ac is
a) positive b) zero c) negative d) non negative
32) The number of real roots of the equation eˢᶦⁿˣ + e⁻ˢᶦⁿˣ = 4 is
a) zero b) one
c) more than one but finitely many
d) infinitely many
33) If α , β are the roots of the equation ax²+ bx+ c=0 and α + h, β+ h are the roots of the equation Ax²+ 2Bx+ C=0, then
a) (b²- ac)/(B²- AC) = a/A
b) (b²- ac)= (B²- AC)
c) h = (bA - aB)/Aa
d) h = (Ac + aC)/(Aa+ Bb)
34) The quadratic equation x²+ 7x= 14(q²+1), where q is an integer has
a) real and distinct roots
b) integral roots
c) imaginary roots d) none
35) Let a,b,c ∈R and a> 0. If the quadratic equation ax²+ bx+ c=0 has two real roots α and β such that α < -1 and β > 1, then value of c/a + |b/a| is
a) less than 2 b) less than 1 c) less than 0 d) less than -1
36) Let a, b, c ∈R and a≠ 0 be such that (a+ c)²< b², then the quadratic equation ax²+ bx + c= 0 has
a) imaginary roots
b) real roots
c) two real roots lying between (-1,1) d) none
37) The integral values of a for which the equation (x - a)(x - 10) +1= 0 has integral roots are
a) -1,3 b) 2,3 c) 12,8 d) -8,-12
38) The number of real solution of 4ˣ⁺¹·⁵ + 9ˣ⁺⁰·⁵ = (10)(6ˣ) is
a) zero b) one c) two d) infinite
39) The number of real solution of ₂sin²x + ₅(₂cos²x)= 7 is
a) zero b) 1 c) finitely many d) infinitely many
40) The number of values of k for which the equation x²- 2x + k= 0 has two distinct roots lying in the interval (0,1) is
a) 0 b) 1 c) 2 d) infinitely many
41) If roots of the equation ax² + bx + c = 0 are real and are of the form α/(α -1), (α+1)/α, then value of (a+ b + c)² is
a) 4ac - b² b) b²- 4ac c) c²+ a²- 2b² d) none
42) Let x be an integer and x²+ x +1 is divisible by 3. When x is divided by 3, it leaves remainder
a) 0 b) 1 c) 2 d) any of (a),(b),(c)
43) If α, β are the roots of the equation x²+ ax+ b=0, then maximum value of - x²+ ax+ b + (1/4) (α - β)² is
a) (1/4) (a²- 4b) b) (1/4) (b²- 4a) c) a²/2 d) none
44) If both the roots of the equation x²+ bx+ c =0 lie in the interval (0,1), then
a) b = -1 , c= 2
b) b > -2 , c< 1
c) b = -5 , c< 2 d) none
45) Let a, b, c ∈R be such that a+ b+ c< 0, a - b+ c< 0 and c> 0. If α and β are the roots of the equation ax²+ bx + c =0, then the value of [α] + [β] is
a) 2 b) 1 c) -1 d) 0
46) If roots of the equation x²- 2mx + m²-1= 0 lie in the interval (-2,4), then
a) - 1< m < 3 b) 1< m < 5 c) 1< m < 3 d) - 1< m < 5
47) The value of √[8 + 2√{8+ 2√(8+ 2√(8+.....is
a) 10 b) 6 d) 8 d) none
48) The number of solutions of the equation
Sin(πx/2√3)= x²- 2√3 x + 4 is
a) 1 b) 2 c) 0 d) infinite
(Hint: sin(πx/2√3)= (x- √3)² + 1≥ 1 => Sin(πx/2√3)≥ 1. But Sin(πx/2√3)≤ 1. Thus Sin(πx/2√3)= 1 . So, (x - √3)²+ 1= 1= x= √3 check x=√3 satisfies the given equation)
49) The number of solutions of |x +2|= 2(3- x) is
a) 1 b) 2 c) 3 d) 0
50) If α, β are the roots of the equation ax²+ bx + c=0, then the equation ax²+ bx(x -1)+ c(x -1)²= 0 has roots
a) α /(1- α), β/(1- β)
b) (1-α)/α, (1-β)/β
c) α /(1+ α), β/(1+ β) d) none
51) Two non integer roots of
(x²- 5x)² - 7(x²- 5x)+ 6= 0 are
a) (1/2) (5+ √29), (1/2) (5- √29)
b) (1/2) (-5+ √29), (1/2) (-5+ √29)
c) (1/2) (-5+ √14), (1/2) (-5- √41) d) none
52) The number of real roots of
{(x-1)/(x +1)}⁴ - 13{(x-1)/(x +1)}²+ 36=0 is
a) 0 b) 2 c) 3 d) 4
53) The number of negative roots of
9ˣ⁺² - 6(3ˣ⁺¹)+1= 0 is
a) 0 b) 1 c) 2 d) 4
54) The number of real roots of 81{(2x-5)/(3x +1)}⁴ - 45{(2x-5)/(3x +1)}²+ 4=0 , x≠ 1/3 is
a) 1 b) 2 c) 3 d) 4
55) The number of irrational roots of
(x²+ 3x +2)² - 8(x²+ 3x) -4=0 is
a) 0 b) 2 c) 3 d) 4
56) The number of roots of the equation
√{x/(x -3)} + √{(x -3)/x}= 5/2, x≠ 0, x≠ 3 is
a) 0 b) 2 c) 3 d) 4
57) The number of irrational roots of the equation
4(x - 1/x)²+ 8(x + 1/x)= 29 is
a) 0 b) 2 c) 4 d) infinite
58) Irrational roots of the equation
2x⁴+ 9x²+ 8x²+ 9x +2= 0 are
a) -2-√3, 2 +√3
b) 2-√3, 2 +√3
c) -2 +√3, -2 -√3 d) none
59) Sum of the roots of the equation
4(x - 1/x)² - 4(x - 1/x)+ 1=0 is
a) 5 b) 1 c) -5/2 d) -1
60) The number of irrational roots of the equation
(x -1)(x -2)(3x -2)(3x +1)= 21 is
a) 0 b) 2 c) 3 d) 4
61) Product of roots of the equation
x - √(3x -6)=2 is
a) 10 b) 5 c) 7 d) 24
62) Product of roots of the equation
2√(2x +1)= 2x -1 is
a) -3 b) -5 c) 5 d) 3
63) Product of roots of the equation
√(13 - x²)= x +5 is
a) -6 b) 7 c) 6 d) -7
64) The number of roots of the equation
√(x²-4) - (x -2)= √(x²-5x +6) is
a) 0 b) 1 c) 2 d) 3
65) The product of the roots of the equation
√(x²-4x +3) + √(x²- 7x + 12) = 3 √(x -3) is
a) 15 b) -15 c) 20 d) -20
66) If a≠ b and difference between the roots of x²+ ax + b= 0 is equal to difference between the roots of x²+ bx + a = 0, then
a) a+ b +4= 0 b) a+ b - 4= 0 c) a- b -4= 0 d) a - b +4= 0
67) α ≠ β but α²= 5α -3 and β²= 5β -3, then the equation whose roots are α/β and β/α is
a) 3x²+ 19x + 3= 0 b) x² - 5x + 3= 0 c) 3x²- 19x + 3= 0 d) none
68) If α ∈ (0,π/2), then the expression √(x²+ x) + tan²α/√(x²+ x) is always greater than or equal to
a) 2 tanα b) 2 c) 1 d) sec²α
69) If a, b ∈R, and the equation
x²+ (a - b)x - a - b + 1= 0 has real roots for all b ∈R, then a lies in the interval
a) (1, ∞) b) (0,∞) c) (-∞,1) d) (-1,1)
70) 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) none
1d 2d 3a 4a 5b 6b 7a 8b 9d 10a 11d 12a 13b 14c 15b 16c 17d 18a 19c 20b 21b 22d 23c 24d 25b 26a 27a 28c 29c 30a 31a 32a 33c 34a 35d 36c 37c 38c 39d 40a 41b 42b 43c 44d 45c 46a 47d 48a 49a 50c 51a 52d 53b 54c 55d 56b 57b 58c 59b 60d 61a 62a 63c 64d 65a 66a 67c 68a 69a 70b
COMPLEX NUMBER
SAP-1
1) If z₁ and z₂ (≠0) are two complex such that|(z₁ - z₂)/(z₁ + z₂)|= 1, then
a) z₂= kz₁, k∈ R b) z₂ = ikz₁, k ∈ R c) z₁ = z₂ d) none
2) For any complex number z, the minimum value of |z| + |z -2i| is
a) 0 b) 1 c) 2 d) none
3) The inequality |z - i|< |z + i| represents the region
a) Re(z)> 0 b) Re(z)< 0 c) Im(z)> 0 d) Im(z)< 0
4) If x= 2+ 5i, then the value of x³ - 5x² + 33x -19 is equal to
a) -5 b) -7 c) 7 d) 10
5) If z= x + it and ω = (1- iz)/(z - i), then |ω|= 1 implies, that, in the complex plane
a) z lies on the imaginary axis
b) z lies on the real axis
c) z lies on the unit circle d) none
6) The real part of z = 1/(cosθ + i sinθ) is
a) 1/(1- cosθ) b) 1/2 c) (1/2) tanθ d) 2
7) If the imaginary part of (2z +1)/(iz +1) is -4, then the locus of the point representing z in the complex plane is
a) a straight line b) a parabola c) a circle d) an ellipse
8) The area of the triangle whose vertices are the points represented by the complex number z, iz and z + iz is
a) |z|²/4 b) |z|²/8 c) |z|²/2 d) |z|/2
9) If ω is a complex cube root of unity, then a root of the equation of determinant
x+1 ω ω²
ω x+ ω² 1 = 0
ω² 1 x+ω
a) x= 1 b) x= ω c) x= ω² d) x= 0
10) If zₖ = cos(kπ/10) + i sin(kπ/10), then z₁z₂z₃z₄ is equal to
a) -1 b) 1 c) -2 d) 2
11) The complex number z₁, z₂, z₃ are the vertices of an equilateral triangle. If z₀ is the circumcentre of the triangle, then z₁² + z₂² + z₃³ is equal to
a) z₀² b) 3z₀² c) z₀³ d) 3z₀³
12) Let z₁ and z₂ be two nonzero complex numbers such that z₁/z₂ + z₂/z₁ = 1, then the origin and points represented by z₁ and z₂
a) lie on a straight line
b) form a right triangle
c) form an equilateral triangle d) none
13) If ω is an imaginary root of unity, then value of the expression
1(2- ω)(2- ω²) + 2(3- ω)(3- ω²)+ ....+ (n -1)(n - ω)(n - ω²) is
a) (1/4) n²(n +1)² - n
b) (1/4) n²(n +1)² + n
c) (1/4) n²(n +1) - n
d) (1/4) n(n +1)² - n
14) If (1+ x+ x²)ⁿ= a₀+ a₁x + a₂x² + ....+ a₂ₙx²ⁿ, then value of a₀+ a₃+ a₆+ ....is
a) 1 b) 2ⁿ c) 2ⁿ⁻¹ d) 3ⁿ⁻¹
15) If z⁵ = (z -1)⁵, then the roots are represented in the Argand plane by the points that are
a) collinear b) concyclic c) vertices of a parallelogram d) none
16) The region of the Argand plane defined by |z - i| + |z + i|≤ 4 is
a) interior of an ellipse
b) exterior of a circle
c) interior and boundary of an ellipse d) none
17) If z₁ and z₂ are two nonzero complex numbers such that |z₁ + z₂|= |z₁|+ |z₂|, then arg(z₁) - arg(z₂) is equal to
a) -π b) -π/2 c) π/2 d) 0
18) Let a and b be two real numbers lying between 0 and 1 such that the points z₁ = a+ i, z₂= 1+ bi and z₃= 0 form an equilateral triangle, then (a, b) is equal to
a) (√3/2,√3/2) b) (2-√3,√3/2) c) (√3/2, 2-√3) d) (2- √3, 2-√3)
19) If |z - 25i|≤ 15, then |maximum arg(z) - minimum arg(z)| equal
a) 2 cos⁻¹(3/5) b) 2 cos⁻¹(4/5) c) π/2 + cos⁻¹(3/5) d) sin⁻¹(3/5) - cos⁻¹(3/5)
20) If |z|= 3, the area of the triangle whose sides are z, ωz and z+ ωz (where ω is a complex cube root of unity) is
a) 9√3/4 b) 3√3/2 c) 5/2 d) 8√3/3
21) The greatest and the least value of |z₁ + z₂| if z₁ = 24+ 7i and |z₂|= 6 are respectively
a) 31,19 b) 25,19 c) 31,25 d) none
22) If z= (7- i)/(3- 4i), then z¹⁴ equals
a) 2⁷ b) 2⁷i c) -2⁷i d) none
23) If α, β are the roots of x² + px + q = 0, and ω is a cube root of unity, then value of (ωα + ω²β)(ω²α + ωβ) is
a) p² b) 3q c) p² - 2q d) p² - 3q
24) If z₁, z₂, z₃ are three complex numbers such that z₁² + z₂² + z₃² - z₂z₃ - z₃z₁ - z₁z₂ = 0 then
a) z₁, z₂, z₃ must necesrbe equal
b) atleast two of z₁, z₂, z₃ must be equal
c) one of z₁, z₂, z₃ must be zero d) none
25) Maximum distance from origin of the points z satisfying the relation |z + 1/z|= 1 is
a) (√5+1)/2 b) (√5-1)/2 c) 3-√5 d) (3+√5)/2
26) If |z₁|= |z₂|= |z₃|= 1 and z₁ + z₂ + z₃ = 0, then the area of the triangle whose vertices are z₁, z₂, z₃ is
a) 3√3/4 b) √3/4 c) 1 d) 2
27) An equation of straight line joining the complex numbes a and ib (where a, b∈E and a, b ≠ 0) is
a) z(1/a - u/b)+ con(z)(1/a + i/b) = 2
b) z(a - ib)+ con(z)+a+ ib)= 2(a²+ b²)
c) z(a + ib)+ coj(z)(a - ib)= 2ab d) none
28) Two non parallel lines meet the circle |z|= r in the points a, b and c, d respectively. The point of intersection of thsw lins is
a) (a⁻¹ + b⁻¹ + c⁻¹ + d⁻¹)/(a⁻¹b⁻¹ + c⁻¹d⁻¹)
b) (ab + cd)/(a+ b + c+ d)
c) (a⁻¹ + b⁻¹ - c⁻¹ - d⁻¹)/(a⁻¹b⁻¹ - c⁻¹d⁻¹) d) none
29) If α and β are two complex numbers, the value of |α - √(α²- β²)|+ |α + √(α²- β² | - {|α + β| + |α - β| is
a) (|α| + | β|) b) √|α²- β²| c) (|α| + | β|) + √|α²- β²| d) none
30) A, B, C, D and E are the points on the complex plane representing the complex numbers z₁, z₂, z₃, z₄ and z₅ respectively. If (z₃ - z₂)z₄ = (z₁ - z₂)z₅, then the triangles ABC and DOE (O being the origin) are
a) similar b) congruent c) equal in area d) none
31) If z₁, z₂, z₃ are the vertices of an isosceles triangle, right angled at the vertex z₂, then the value of (z₁ - z₂)²+ (z₂ - z₃)² is
a) -1 b) 0 c) (z₁ - z₃)² d) none
32) The roots z₁, z₂, z₃ of the equation x³ + 3ax² + 3bx + c = 0 (a, b, c ∈ C) form an equilateral triangle in the Argand plane if and only if
a) a²= b b) a= b² c) a= ± b d) |a|= |b|
33) Points A(z₁), B(z₂) and C(z₃) form a triangle with centroid z₀. Triangles BCX, CAY, ABZ similar to triangle ABC are drawn on the sides of ∆ ABC. Centroid of ∆ XYZ is
a) 3z₀ b) - z₀ c) z₀ d) none
34) If sinα + sinβ + sinγ = cosα + cosβ + cos γ= 0, then value of ∑[cos2α - cos(β + γ)] is
a) 3/2 b) 1/2 c) -1/2 d) none
35) If the roots of (z -1)²⁵ = 2ω²(z +1)²⁵ (where ω is a complex cube root of unity) are plotted in the Argand plane, they lie on
a) straight line b) a circle c) an ellipse d) none
36) If a, b, c,p,q,r are three nonzero complex numbers such that p/a + q/b + r/c = 1+ i and a/p + b/q + c/r = 0, then value of p²/a² + q²/b² + r²/c² is
a) 0 b) -1 c) 2i d) -2i
37) Let ω, ω² be complex cube roots of unity. Then the determinant
∆=ω ω² -(x -1)²
ω² ω x(x -2)
1+ω -ω² ω+1
is equal to
a) 0 b) 1 c) 1- ω d) 2(1- ω²)
38) Let the determinant
z= 1 1- 2i 3+5i
1+ 2i -5 10i
3- 5i. -10i 11
Then
a) z is purely imaginary
b) z is purely real
c) z= 0 d) none
39) If (x + it)¹⁾³ = a+ ib, then x/a + y/b equals to
a) 4(a²- b²) b) 2(a² - b²) c) 2(a² - b²) d) none
40) If z∈ C, the minimum value of |z| + |z -1| is attained at
a) exactly one point
b) exactly two points
c) infinite number of points d) none
41) For all complex numbers z₁, z₂ satisfying |z₁|= 12 and |z₂ -3 - 4i|= 5, the minimum value of |z₁ - z₂| is
a) 0 b) 2 c) 7 d) 17
42) if z lies on the circle |z -1|= 1, then (z -2)/z equal to
a) 0 b) 2 c) -1 d) none
43) If 1, ω,...., ωⁿ⁻¹ are the nth roots of unity, then value of 1/(2- ω) + 1/(2- ω²) + ....+ 1/(2- ωⁿ⁻¹) equal
a) 1/(2ⁿ -1) b) n(2ⁿ-1)/(2ⁿ+1) c) (n -2)2ⁿ⁻¹/(2ⁿ -1) d) none
44) If ω= cos(π/n) + i sin(π/n), then the value of 1+ ω+ ω²+....+ ωⁿ⁻¹ is
a) 1+ i b) 1+ i tan(p/n) c) 1+ i cot(π/2n) d) none
45) Let z₁ and z₂ be nth roots of unity which subtends a right angle at the origin, then n must be of the form
a) 4k +1 b)4k +2 c) 4k +3 d) 4k
46) The complex numbers z₁, z₂, z₃ satisfying (z₁ - z₃)/(z₂ - z₃) = (1- i √3)/2 are the vertices of a triangle which is
a) of area √3
b) right angled and isosceles
c) equilateral
d) obtuse angled and isosceles
47) If z₁, z₂, z₃ are complex numbers such that
|z₁|= |z₂|= |z₃|= |1/z₁ + 1/z₂ + 1/z₃|= 1, then |z₁ + z₂ + z₃| is
a) equal to 1 b) less than 1 c) greater than 3 d) equal to 3
48) The value of the sum ¹³ₙ₌₁∑ (iⁿ + iⁿ⁺¹) where i= √-1 equal to
a) i b) i -1 c) - i d) 0
49) The number of real roots of the equation
x¹² - x¹¹ + x¹⁰ - ..... - x +1=0 is
a) 2 b) 6 c) 12 d) none
50) The number of real roots of the equation z³ + iz -1= 0 is
a) 0 b) 1 c) 2 d) 3
1b 2c 3c 4d 5b 6b 7c 8c 9d 10a 11b 12c 13a 14d 15a 16c 17d 18d 19b 20a 21a 22c 23d 24d 25a 26a 27a 28c 29d 30a 31b 31a 32b 33c 34d 35b 36c 37a 38b 39a 40c 41b 42d 43d 44c 45d 46c 47a 48b 49d 50a
SAP- 2
1) The amplitude of (1+√3i))/(√3+ i) is
a) -π/6 b) π/6 c) -π/3 d) π/3
2) The maximum number of real roots of the equation x²ⁿ -1= 0 is
a) 2 b) 3 c) n d) 2n
3) If 1, ω, ω²,.....ωⁿ⁻¹ are n roots of unity, then
(1- ω)(1- ω²)+....(1- ωⁿ⁻¹) is equal to
a) n b) 1 c) 0 d) n²
4) The inequality |z - 4|< |z -2| represents the region given by
a) Re(z)> 1 b) Re(z)>2 c) Re(z)> 3 d) Re(z)> 4
5) The points representing complex number z for which |z - 3|= |z -5| lie on
a) a circle b) ellipse c) straight line d) none
6) If z≠ 0, then arg(conjugate of z)+ arg(z) is equal to
a) 0 b) π/2 c) π/4 d) π
7) If α₁, α₂, ....α₈ are the 8, 8th roots of unity, then
a) α₁³ + α₂³ +....+ α₈³ = 24
b) α₁+ α₂+....+α₈ = 1
c) α₁⁵⁰ + α₂⁵⁰ +....+ α₈⁵⁰ = 0
d) α₁⁴⁸ + α₂⁴⁸ + ....+ α₈⁴⁸ = 0
8) In a geometrical progression first term and common ratio are both equal to {(1/2)(-1+ √3 i)}. The modulus of the 100th term is
a) 99 b) 98 c) 98.5 d) none
9) The region of the Argand plane defined by |z - i| + |z + i|≤ 4 is
a) interior of an ellipse
b) exterior of a circle
c) interior and boundary of an ellipse
d) exterior and boundary of a circle
10) Suppose z₁, z₂, z₃ are the vertices of an equilateral triangle inscribed in the circle |z|= 2. If z₁= 1+ i √3 . Then z₂ and z₃ are given by
a) -2, 1- i√3 b) -2, -1 - i √3 c) 2, -1+ i √3 d) 2, -1- i √3
11) If z is a complex number lying in the fourth quadrant of the Argand plane and |kz/(k +1) + 2i |>√2 for all real values of k (k≠ -1) then range of arg(z) is
a) (-π/8,0) b) (-π/6,0) c) (-π/4,0) d) none
12) If arg(z)< 0, then arg(-z) - arg(z) equals
a) π b) -π c) -π/2 d) π/2
13) Value of 4+ 5(-1/2 + i√3/2)³³⁴ + 3(-1/2 + i. √3/2)³⁶⁵ is
a) 1- i √3 b) -4+ √3 i c) - 4 √3 i d) -4√3 i
14) Suppose z₁, z₂, z₃ represents the vertices A, B, C respectively of a ∆ ABC with centroid at G. If the midpoint of AG is the origin, then
a) z₁+ z₂ + z₃= 0
b) 2z₁+ z₂ + z₃= 0
c) z₁+ z₂ + 4z₃= 0
d) 4z₁+ z₂ + z₃= 0
15) For positive integers n₁ and n₂ the value of the expressions
(1+ i)ⁿ₁ + (1+ i³)ⁿ₁ + (1+ i⁵)ⁿ₂+ (1+ i⁷)ⁿ₂
Where i= √-1 is a real number if and only if
a) n₁= n₂ +1 b) n₁= n₂ -1 c) n₁= n₂ d) n₁>0, n₂ > 0
16) The three points z₁, z₂ , z₃ are connected by the relation az₁ + bz₂ + cz₃= 0, whose a+ b + c= 0, then the points are
a) vertices of a right triangle
b) vertices of an isosceles triangle
c) vertices of an equilateral triangle
d) collinear
17) If the complex number (z -1)/(z +1) is purely imaginary, then
a) |z|= 1 b) |z|< 1 c) |z|> 1 d) |z| ≥ 2
18) If z is a complex number such that -π/2≤ arg z ≤ π/2, then which of the following inequality is true.
a) |z - con(z)|≤ |z| |arg(z) - arg(conjugate of z))|
b) |z - con(z)|≤ |arg(z) - arg(conjugate of z))|
c) |z - con(z)|> |z| |arg(z) - arg(conjugate of z))| d) none
19) The value of {(1+ i tanx)(1+ i tan y)}/{(cot y + i)(cot v + i)} is
a) tan u tan v. eⁱ⁽ˣ⁺ʸ⁻ᵘ⁻ᵛ⁾
b) cot u cot v. e ⁱ⁽ˣ⁻ʸ⁺ᵘ⁻ᵛ⁾
c) tan u cot v. eⁱ⁽ˣ⁺ʸ⁻ᵘ⁻ᵛ⁾ d) none
20) If the complex numbers z₁, z₂, z₃ are the vertices of a parallelogram ABCD, then the fourth vertex is
a) (1/2) (z₁+ z₂)
b) (1/4) (z₁ + z₂ - z₃ - z₄)
c) (1/3) (z₁ + z₂ + z₃)
d) z₁+ z₃ - z₂.
21) If z= i(conjugate of z), then
a) z is purely real
b) z is purely imaginary
c) z= a(1+ i), a∈ R d) z= 0
22) If |z₁|= |z₂|= ....|zₙ|= 1, then the value of
|z₁+ z₂+.....+zₙ| - |1/z₁ + 1/z₂ + ....+ 1/zₙ| is
a) n b) 0 c) - n d) none
23) if z= (√3/2 + i/2)⁵ + (√3/2 - i/2)⁵, then
a) Re(z)= 0 b) Im(z)= 0 c) Re(z)> 0 , Im(z)>0 d) Re(z)> 0 , Im(z)< 0
24) The smallest positive integer n for which {(1+ i)/(1- i)}ⁿ = 1 is
a) 2 b) 3 c) 4 d) none
25) If ω is a complex cube root of unity, then roots of (x -1)³+ 8= 0 are
a) -1,1+2ω, 1+ 2ω²
b) -1,1-2ω, 1- 2ω²
c) -1,-1,-1 d) none
26) The complex number z which satisfyes the equation|(z - 5i)/(z +5i)|= 1 lies on
a) the real axis
b) the straight line Im(z)= 5
c) a circle through the origin d) none
27) The equation z²= conj of z has
a) no solution
b) two solutions
c) four solutions
d) infinite number of solutions
28) If zᵣ = sin(2πr/11) - i cos(2πr/11), then the value of ¹⁰ᵣ₌₁∑ zᵣ is
a) -1 b) 0 c) - i d) none
29) The points satisfying the equations |z -1|= |z + 2i| lie on
a) a circle
b) an ellipse
c) a straight line
d) a hyperbola
30) If triangle with vertices z₁, z₂, z₃ are z₁', z₂', z₃' are directly similar, then value of the determinant
z₁ 1 z₁'
z₂ 1 z₂'
z₃ 1 z₃'
is
a) 0 b) -1 c) 1 d) none of these
31) The triangle with vertices z₁, z₂ and (1- i)z₁ + iz₂ is
a) an isosceles triangle
b) a right angled triangle
c) an isosceles right angled triangle
d) an equilateral triangle
32) If θ is real and z₁, z₂ are connected by z₁² + z₂² + 2z₁z₂ cosθ= 0, then triangle with vertices 0, z₁, z₂ is
a) equilateral
b) right angled
c) isosceles
d) none
33) Radius of the circle passing through i and orthogonal to circles |z|= 1 and |z -1|= 4 is
a) 5 b) 5- √2 c) 5+√2 d) 7
34) If a,b,c and u, c, w are complex numbers representing the vertices of two triangles ABC and DEF are such that c= (1- r)a+ rb and ω= (1- r)u + rv, where r is a complex number (|r|≠1), then the two triangles.
a) have the same area
b) are similar
c) are congruent d) none
35) If z= x + iy and ω= (1- iz)/(z - i) , then |ω|= 1 implies that in the complex plane
a) z lies on the imaginary axis
b) z lies on the real axis
c) z lies on the unit circle d) none
36) If z lies on the circle |z - 2i|= 2√2 then arg{(z -2)/(z +2)} is equal to
a) π/3 b) π/4 c) π/6 d) π/8
37) The curve represented by |z|= Re(z)+ 2 is
a) a straight line b) a circle c) an ellipse d) none
38) If ω≠ 1 is a cube root of unity satisfying
1/(a+ ω) + 1/(b + ω) + 1/(c + ω)= 2ω² and
1/(a+ ω²) + 1/(b + ω²) + 1/(c + ω²)= 2ω then the value of
1/(a+ 1) + 1/(b + 1) + 1/(c + 1) is
a) 2 b) -2 c) - 1+ ω² d) none
39) if |z - i Re(z)|= |z - Im(z)|, then z lies on
a) Re(z)= 2 b)Im(z)= 2 c) Re(z)+ Im(z)= 2 d) none
40) If ω is a complex cube root of unity, then value of expression
Cos[{(1+ ω)(1- ω²)+....+(10- ω)(10- ω²)} π/900]
a) -1 b) 0 c) 1 d) √3/2
41) If roots of the equation z² + az + b=0 are purely imaginary then
a) (b - con(b))²+ (a + conj(a))(conj of (a).b + a. conj(b))= 0
b) (b - con(b))²+ (a - conj(a))²= 0
c) (b - con(b))²- (a - conj(a))²=0 d) none
42) The relation a+ bi > c + di will have a meaning if and only if
a) b=0, d= 0 b) a=0, c= 0 c) a=0, d= 0 d) b=0, c= 0
43) The system of equations |z+ 1 - i|=√2 and |z|= 3 has
a) no solution b) one solution c) two solutions d) infinite number of solutions
44) If 8iz³ + 12z² - 18z + 27i=0, then
a) |z|= 3/2 b) |z|= 2/3 c) |z|= 1 d) |z|= 3/4
45) If the complex number z lies on the boundary of the circle of radius 3 and centre at -4, then the greatest value of |z +1| is
a) 4 b) 5 c) 6 d) 9
46) If x+ iy = 3/(cosθ+ i sinθ+2), then 4x - x²- y² reduces to
a) 2 b) 3 c) 4 d) 5
47) If z= 5+ t+ i √(25- t²) (t ∈ R), then locus of z is a curve which passes through
a) 5+ 0i b) -2 + 3i c) 2+ 4i d) none
48) If α and β are the roots of the equation x²- 2x +4=0, then the value of α⁶+ β⁶ is
a) 64 b) 128 c) 256 d) none
49) The perpendicular distance of point c (c is a complex number) on the Argand plane from the line a(conjugate of z)+ conjugate of (a).z + b= 0 (a is complex number and b is real is
a) |(a. conjugate of (c)+ conjugate of (a)c + b)/2|a|
b) |(ac+ conjugate of (a). Conjugate of (c) + b)/2|a|
c) |(a. conjugate of (c)+ conjugate of (a)c + b)/|a| d) none
50)The equation of circle in complex form which touches iz + conjugate of z+ 1 + i= 0 and for which the lines (1- i)z= (1+ i) . Conjugate of z and (1+ i)z + (1- i). Conjugate of z - 4i= 0 are normal is
a) | z - (1+ i)|= √2
b) | z - (1- i)|= 1/√2
c) √2| z - (1+ i)|= 1 d) none
1b 2a 3a 4c 5c 6a 7c 8d 9c 10a 11c 12a 13c 14d 15d 16d 17a 18a 19d 20d 21c 22b 23b 24c 25b 26a 27c 28b 29c 30a 31c 32c 33d 34b 35b 36b 37d 38a 39d 40b 41a 42a 43a 44a 45c 46b 47c 48b 49a 50c
α β γδ²²²²⁶⁶
∈ ∞ω θθθθ
α β γδ
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 = _____
Trigonometry full. 12 mix
SAP-1
1) cos(540° - θ) - sin(630- θ) is equal to
a) 0 b) 2 cosθ c) 2 sinθ d) sinθ - cosθ
2) 2sec²x - sec⁴x -2 cosec²x + cosec⁴x = 15/4 if tanx is equal to
a) 1/√2 b) 1/2 c) 1/2√2 d) 1/4
3) If 2 sinθ/(1+ cosθ + sinθ)= x, then cosθ/(1+ sinθ) is equal to
a) 1/x b) x c) 1+ x d) 1- x
4) If cosx = (2 cos y -1)/(2- cos y) (0< x, y<π), x+ y=π then tan(x/2) is
a) ⁴√3 b) √3 c) 3 d) 3²
5) If tan25°= x, then (tan155° - tan 115°)/(1+ tan155° tan115°) is equal to
a) (1- x²)/2x b) (1+ x²)/2x c) (1+ x²)/(1- x²) d) (1- x²)/(1+ x²)
6) If sinx + cos y= a and cosx + sin y= b, then tan{(x - y)/2} is equal to
a) a+ b b) a- b c) (a+ b)/(a - b) d) (a - b)/(a+ b)
7) The value of the determinant
Sin²13 sin²77 tan135
Sin²77 tan135 sin²13
Tan135 sin²13 sin²77 is equal to
a) -1 b) 0 c) 1 d) 2
8) If A= 130° and x= sinA+ cosA, then
a) apx> 0 b) x < 0 c) x = 0 d) x ≥ 0
9) If tan²36° + k(sin18°+ cos36°)= 5, then the value of k is
a) 2 b) 2√5 c) 4 d) 4√5
10) sin3θ/cos2θ< 0 if θ lies in
a) +13π/48, 14π/48)
b) (14π/48,18π/48)
c) 18π/48,23π/48)
d) any of these intervals
11) If cosx + cos y= a, sin x + sin y= b and k is the arithmetic mean between x and y then sin2k + cos2k is equal to
a) (a+ b)²/(a²+ b²)
b) (a - b)²/(a²+ b²)
c) (a²- b²)/(a²+ b²). d) none
12) If (sinx)/a= (cosx)/b= (tanx)/c= k, then bc + 1/ck + ak/(1+ bk) is
a) k(a + 1/a) b) (1/k)(a + 1/a) c) 1/k² d) a/k
13) sin²x + cos²(x + y)+ 2 sinx sin y cos(x + y) is independent of
a) x b) y c) both x and y d) none
14) If uₙ = sin nθ secⁿθ, vₙ = cos nθ secⁿθ, n≠ 1, θ≠ pπ, n, p∈I,
then (vₙ - vₙ₋₁)/uₙ₋₁ + 1uₙ/nvₙ= 0 for
a) all values of n
b) finite numbers of values of n
c) infinite number of values of n
d) no values of n
15) If 1/cosx cos y + tanx tan y= tan z, 0< x, y< π then 1- tan²z< 0 for
a) all values of x and y
b) no values of x and y
c) finite number of values of x and y
d) infinite number of values of x and y.
16) tan203° + tan22°+ tan203°+ tan22°=
a) -1 b) 0 c) 1 d) 2
17) If sin32°= k and cosx = 1- 2k²; α, β are the values of x between 0° and 360° with α< β, then
a) α+β= 180 b) β - α = 200 c) β= 4α + 40 d) β = 5α - 20
18) The minimum value of 27 tan²θ+ 3 cot²θ is
a) 9 b) 18 c) 27 d) 30
19) The value of sin12 sin48 sin54 is
a) sin30 b) sin²30 c) sin³30 d) cos³30
20) tan⁶(π/9) - 33 tan⁴(π/9)+ 27 tan²(π/9)=
a) tan(π/3) b) tan²(π/3) c) tan(π/6) d) tan²(π/6)
21) if 3 sin y = sin(2x + y), then tan(x + y)- 2 tan x is
a) independent of x
b) independent of y
c) independent of both x and y
d) independent of none of them
22) Let n be a fixed positive integer such that sin(π/2n)+ cos(π/2n)=√n/2 then
a) n< 4 b) n> 8 c) n= 6 d) none
23) If A= sin²θ + cos⁴θ, then for all values of θ
a) 1≤ A ≤ 2 b) 3/4 ≤ A ≤ 1 c) 13/16≤ A ≤ 1 d) 3/4≤ A ≤ 13/16
24) If a= cosφ cosψ + sinφ sinψ cosδ
b= cosφ sinψ - sinφcosψ cosδ and
c= sinφ sinδ, then a²+ b²+ c²=
a) -1 b) 0 c) 1 d) none
25) In a triangle ABC, BP is drawn perpendicular to BC to meet CA in P, such that CA= AP, then BP/AB=
a) 2 sinA b) 2 sinB c) 2 sinC d) none
26) If x+y = z, then cos²x + cos² y + cos²z - 2 cosx cos y cos z is equal to
a) cos²z b) sin²z c) 0 d) 1
27) If sin2θ= k, then the value of tan³θ/(1+ tan²θ) + cot³θ/(1+ cot²θ) is equal to
a) (1- k²)/k b) (2- k²)/2 c) k²+1 d) 2- k²
28) If sin²A = x, then sinA sin2A sin3A sin4A is a polynomial in x, the sum of whose coefficients is
a) 0 b) 40 c) 168 d) 336
29) If sinA/sinB =√3/2 and cosA/cosB=√5/2, 0< A, B <π/2, then tanA+ tanB is equal to
a) √3/√5 b) √5/√3 c) 1 d) (√3+√5)/√5
30) If cosθ= cosx cos y, then tan{(θ+ x)/2} tan{(θ- x)/2} is equal to
a) tan²(x/2) b) tan²(y/2) c) tan²(θ/2) d) cot²(y/2)
31) If α, β, γ, δ are the smallest positive angles in ascending order of magnitude which have their sines equal to the positive quantity k, then the value of 4 sin(α/2)+ 3 sin(β/2)+ 2 sin(γ/2) + sin(δ/2) is equal to
a) 2√(1- k) b) 2√(1+ k) c) 2√k d) none
32) The equation a sinx + b cosx = c, where|c|> √(a²+ b²) has
a) a unique solution
b) infinite number of solutions
c) no solution d) none
33) If cotx equals to the integral solution of the inequality 4x²- 16x +15< 0 and sin y equals to the slope of the bisector of the first quadrant, then sin(x + y) sin(x - y) is equal to
a) -3/5 b) -4/5 c) 2/√5 d) 3
34) The value of cos(2π/7) + cos(4π/7)+ cos(6π/7) + cos(7π/7) is
a) 1 b) -1 c) 1/2 d) -3/2
35) The greatest value of f(x)= 2 sinx + sin2x on [0,3π/2], is given by
a) 9/2 b) 5/2 c) 3√3/2 d) 3/2
36) If x= a cos³θ sin²θ, y= a sin³θ cos²θ and (x²+ y²)ᵖ/(xy)ᑫ, (p,q ∈N) is independent of θ, then
a) 4p= 5q b) 4q= 5p c) p+ q= 9 d) pq= 20
37) If (a - b) sin(x + y)= (a+ b) sin(x - y) and a tan(x/2) - b tan(y/2)= c, then the value of sin y is equal to
a) 2ab/(a²- b²- c²) b) 2bc/(a²- b²- c²) c) 2bc/(a²- b²+ c²) d) 2ab/(a²- b²+ c²)
38) If (cosx - cos y)/(cosx - cos z)=( sin²y cos z)/(sin²z cos y) then cosx is equal to
a) (cos y - cos z)/(1+ cos y cos z)
b) (cos y - cos z)/(1- cos y cos z)
c) (cos y + cos z)/(1+ cos y cos z) d) none
39) If 0< x, y<π and cosx+ cos y - cos(x + y)= 3/2 then sinx + cos y is equal to
a) 0 b) 1 c) (√3+1)/2 d) √3
40) If sinθ, cosθ, tanθ are in GP, then cos⁹θ+ cos⁶θ+ 3 cos⁵θ-1 is equal to
a) -1 b) 0 c) 1 d) none
41) If sinα, sinβ , sinγ are in AP and cosα, cosβ, cosγ are in GP then (cos²α+ cos²γ - 4 cosα cosγ)/(1- sinα sinγ).
a) -2 b) -1 c) 0 d) 2
42) If cosα + cosβ = a and sinα + sinβ = b and α - β = 2θ, then cos3θ/cosθ=
a) a²+ b²-2 b) a²+ b²-3 c) 3- a²- b² d) (a²+ b²)/4
43) If cosA= 3/4, then the value of 16 cos²(A/2) - 32 sin(A/2) sin(5A/2) is
a) -4 b) -3 c) 3 d) 4
44) If D= 1 cosθ 1
-sinθ 1 -cosθ
-1 sinθ 1
Then D lies in the interval
a) [0,4] b) [2,4] c) [2-√2, 2+√2] d) [-2,2]
45) The value of θ lying between θ= 0 and θ= π/2 and satisfying the equation of determinant
1+ sin²θ cos²θ 4 sin4θ
sin²θ 1+ cos²θ 4sin4θ =0
sin²θ cos²θ 1+ 4sin4θ
is
a) 3π/24 b) 5π/24 c) 11π/24 d) π/24
46) If x= sin³p/cos²p , y= cos³p/sin²p; and sin p+ cos p= 1/2, then x + y=
a) 75/18 b) 44/9 c) 79/18 d) 48/9
47) If sinθ+ cosecθ= 2, then sinⁿθ+ cosecⁿθ=
a) 2ⁿ b) 2⁻ⁿ c) 2 d) 2n
48) If a cosA - b sinA = c, then a sinA + b cosA is equal to
a) ±√(a²+ b²- c²) b) ±√(apb²+ c²- a²) c) ±√(c²+ a²- b²) d) ±√(a²+ b² + c²)
49) The general solution of the trigonometrical equation sinx + cosx = 1 is
a) x= 2nπ, n∈ I
b) x= 2nπ +π/2, n∈ I
c) px= nπ + (-1)ⁿπ/4, -π/4, n∈ I d) none
50) The value of x between 0 and 2π which satisfy the equation sinx √(8cos²x)= 1 are in AP. The common difference of the AP is
a) π/8 b) π/4 c) 3π/8 d) 5π/8
51) Number of solutions of the equations tanx + secx = 2 cosx lying in the interval [0,2π] is
a) 0123
52) The number of all possible triplets (a₁, a₂, a₃) such that a₁+ a₂ cos2x + a₃ sin²x= 0 for all x is
a) 0 b) 1 c) 3 d) infinite
53) If tan(cotx)= cot(tanx), then the value of sin2x is
a) π/4
b) 4/(2n -1)π, n∈I - {-1,0}
c) 2/π
d) 4/(2n +1)π, n∈I , n> 7
54) sinx + 2 sin2x = 3+ sin3x, 0≤ x≤ 2π has
a) 2 solutions in I quadrant
b) one solutions in II quadrant
c) no solutions in any quadrant
d) one solutions in each quadrant
55) Let f(x)= (sin²θ)x² + (cos²θ)x + cos²θ, f(x)= 0 has no real roots, then cos²θ can be
a) 3/4 b) 1/4 c) 1/8 d) 1/16
56) The equation (cosp - 1)x² + (cos p)x + sin p= 0 where x is a variable, has real roots if p lies in the interval
a) (0,2π) b) (-π,0) c) (-π/2,π/2) d) (0,π)
57) The set of values of x for which
(tan3x - tan2x)/(1+ tan3x tan2x)= 1 is
a) φ b) {π/4} c) {nπ+π/4, n= 1,2,3...}
d) {2nπ +π/4, n= 1,2,3....}
58) The general solution of the equation
(1+ sinx+....+(-1)ⁿ sinⁿx +...)/(1+ sinx+ ....+ sinⁿx+....)= (1- cos2x)/(1+ cos2x), x≠(2x +1)π/2, n∈ U is
a) (-1)ⁿ(π/3)+ nπ
b) (-1)ⁿ(π/6)+ nπ
c) (-1)ⁿ⁺¹(π/6)+ nπ
d) (-1)ⁿ⁻¹(π/3)+ nπ, (n∈I)
59) The number of solutions of the equation
sin⁵x - cos⁵x = 1/cosx - 1/sinx (dinx≠ cosx) is
a) 0 b) 1 c) infinite d) none
60) General solution of the equation log₂ sinx - log₂cosx - log₂(1- tanx) - log₂(1+ tanx)+ 1=0 is
a) (2n+1)π/8 b) (16n+1)π/8 c) (2n+1)π/4 d) none
61) The smallest positive root of the equation tanx - x = 0 lies in
a) I quadrant b) II quadrant c) III quadrant IV quadrant
62) If sin⁴x + cos⁴y +2= 4 sinx cos y, 0≤x, y ≤π/2 then sinx + cos y=
a) - 2 b) 0 c) 2 d) none
63) tan(pπ/4)= cot(qπ/4) if
a) p+ q = 0 b) p+ q= 2n +1 c) p+ q = 2n d) p+ q= 2(2n+1)
64) If sinx = cos y, √6 sin y = tan z and 2 sin z = √3 cosx; u, v, w denotes respectively sin⅖x, sin²y , sin²z then the value of the triplet(u,v,w) is
a) (1,00) b) (0,1,0) c) (1/2,1/2,3/4) d) (1/2,3/4,1/2)
65) A solution (x,y) of the system of equation x - y = 1/3 and cos²(πx) - sin²(πy)= 1/2 is given by
a) (2/3,1/3) b) (7/6,1/6) c) (13/6,11/6) d) (1/6,5/6)
66) If x+ y + z= π, tanx tan y = 2, tanx + tany + tan z= 6, then the value of z is
a) nπ+π/4, n∈ I b) nπ + tan⁻¹2, n ∈ I c) nπ + tan⁻¹3, n ∈ I d) none
67) cos2x - 3 cosx +1= 1/{(cot2x - cotx) sin(x -π)} holds.
a) if cosx = 0 b) if cosx = 1 c) if cosx= 2/5 d) for no real value of x.
68) cos(x - y) - 2 sinx + 2 sin y= 3 if
a) sinx= sin y
b) x+ y= 2nπ, (x - y)= (2km-1)π/2
c) x= 2kπ -π/2, y= 2nπ +π/2
d) cos(x - y)= -1 (n,k∈ I )
69) The equation 8 cosx cos2x cos4x = sin6x/sinx has a solution given by
a) x= nπ b) x= nπ+π/4 c) x= (2n +1)π/14 d) x= (2n +1)π/7 (n ∈ I )
70) cos3θ/(2 cos2θ -1)= 1/2 if
a) θ= nπ+π/3 b) θ= 2nπ± π/3 c) θ= 2nπ± π/6 d) θ= nπ+π/6
71) If 6 cos2θ + 2 cos²(θ/2)+ 2 sin²θ= 0, -π<θ < π, then θ is equal to
a) π/3 b) π/3, cos⁻¹(3/5) c) cos⁻¹(3/5) d) π/3, π - cos⁻¹(3/5)
72) The number of integral values of a for which the equation cos2x + a sinx = 2a -7 possesses solution is
a) 2 b) 3 c) 4 d) 5
73) The least difference between the roots of the equation 4 cosx (2- 3 sin²x)+ (cos2x +1)= 0 (0≤ x≤π/2) is
a) π/6 b) π/4 c) π/3 d) π/2
74) The equation cos⁴x - (a+ 2) cos²x - (a +3)= 0 possesses a solution
a) a>-3 b) a< -2 c) -3≤ a ≤ -2 d) a is any positive integer
75) The solution of |cosx|= cosx -2 sinx is
a) x= nπ b) x= nπ+π/4 c) x= nπ+(-1)ⁿ(π/4) d) x= (2n +1)π+π/4
1a 2a 3d 4a 5a 6d 7b 8a 9c 10a 11d 12b 13a 14d 15a 16c 17c 18b 19c 20b 21c 22c 23b 24c 25c 26d 27b 28a 29d 30b 31b 32c 33b 34d 35c 36a 37b 38c 39c 40b 41a 42b 43c 44c 45c 46c 47c 48a 49c 50b 51c 52d 53b 54c 55a 56d 57a 58b 59a 60b 61c 62c 63d 64a 65c 66c 67d 68c 69c 70b 71d 72d 73a 74c 75d
SAP-2
1) If 2 tanx + cot y = tan y, then the value of tan(y - x) is
a) tanx b) cotx c) tan y d) cot y
2) If cos(x - y)= a cos(x + y), then cotx cot y is equal to
a) (a-1)/(a+1) b) (a+1)/(a-1) c) (a-1) d) (a+1)
3) sin²A + sin²(A+ B)+ 2 sinA cosB sin(B-A) is equal to
a) sin²A b) sin²B c) cos²A d) cos²B
4) If 3 sin2θ/(5+ 4 cos2θ)= 1, then the value of tanθ is equal to
a) 1 b) 1/3 c) 3 d) none
5) If x= a sec³θ tanθ, y= b tan³θ secθ, then sin²θ is equal to
a) x/a - y/b b) x/a + y/b c) xy/ab d) ay/bx
6) cotθ - cot3θ is equal to
a) 2 sinθ sin3θ
b) 2 cosθ cos3θ
c) 2 cosθ cosec3θ
d) 2 sinθ cosec3θ
7) If 0<x, y < 2π, the number of solutions of the system of equations sinx sin y = 3/4 and cosx cos y = 1/4 is
a) 0 b) 1 c) 2 d) infinite
8) If A and B be acute positive angles satisfying 3 sin²A+ 2 sin²B= 1, 3 sin2A - 2 sin2B = 0 then
a) B=π/4 - A/2 b) A =π/4 - 2B c) B=π/2 - A/4 d) A =π/4 - B/2
9) If tan x, tan y , tan zare the roots of the equation x³- px² - r= 0, then the value of (1+ tan²x)(1+ tan²y)(1+ tan²z) is equal to
a) (p - r)² b) 1+ (p -r)² c) 1- (p -r)² d) none
10) If A, B, C are the angles of triangle such that angle A is obtuse then
a) tanA tanB< 1
b) tanB tanC < 1
c) tanA tanC< 1
d) tanA tanB tanC< 1
11) If tan(x/2)= cosecx - sinx, then cos²(x/2) is equal to
a) sin18 b) cos36 c) sin36 d) cos18
12) If a sin²x + b cos²x = a cos²y + b sin²y= 1 and a tanx= b tany(a≠ b) then
a) a+ b= 2ab b) a- b= 2ab c) a - b+ 2ab = 0 d) a + b+ 2ab = 0
13) The acute angle of a rhombus whose side is a mean proportional between its diagonals is
a) 15 b) 20 c) 30 d) 80
14) Given the height h and the angle bisector l drawn from the vertex of the right angle of a triangle, then cosine of an acute angle of the triangle is given by
a) (h+ √(l²- h²))/√2h
b) (h - √(l²- h²))/√2h
c) h/l d) (h - √(l²- h²))/√2l
15) If 2 sin²(x +π/4)+ √3 cos2x > 0, then
a) cos(2x -π/6)> -1/2
b) sin(2x -π/6)< -1/2
c) sin(2x -π/6)> -1/2
d) cos(2x -π/6)< -1/2
16) The equation sin⁴x + cos⁴x = a has a real solution if
a) 0< a≤ 1 b) 1/2≤ a≤1 c) 1/4≤ a≤1/2 c) -1≤ a≤1
17) x= ∞ₙ₌₀∑ cos²ⁿx, y= ∞ₙ₌₀∑ sin²ⁿx, z= x= ∞ₙ₌₀∑ cos²ⁿx sin²ⁿ2x, |cosx|< 1, |sinx|< 1 then x+ y+ z is equal to
a) xy b) yz c) zx d) xyz
18) For n∈ I, the line x= nπ+ π/2 does not intersect the graph of
a) cot(x+π) b) cos(x+π) c) sinx d) tanx
19) The least positive value of x satisfying (sin²2x + 4 sin⁴x - 4 sin²x cos²x)/(4- sin²2x - 4 sin²x)= 1/9 is
a) π/3 b) π/6 c) 2π/3 d) 5π/6
20) In a triangle ABC right angled at C, sin²A/sin²B - cos²A/cos²B is equal to
a) (a²- b²)c²/a²b²
b) (a⁴ + b⁴)/a²b²
c) (b²- c²)a²/b²c²
d) (c²- a²)b²/c²a²
21) If (tanx)/2= (tany)/3= (tanz)/5 and x+ y+z=π, then the value of tan²x + tan²y + tan²z is
a) 38/3 b) 38 c) 114 d) none
22) If the angles A, B, C of a triangle are in AP such that sin(2A + B)= 1/2 then sin(B+ 2C)=
a) -1/2 b) 1/2 c) √3/2 d) 1/√2
23) cos7.5°=
a) √{(2+√2+√6)/8}
b) √{(4+√2+√6)/8}
c)(2√2+√3+1)/2√2
d) √{(4+√2+√6)/4}
24) If tanx+ tan y= a, cotx + cot y= b, x - y = m (≠0) then
a) ab<4 ab=4 c) ab>4 d) ab= 0
25) If x/y= cosA/cosB then (x tanA + y tanB)/(x + y)=
a) (sinA+ cosB)/(cosA+ sinB)
b) (sinA+ sinB)/(cosAcosB)
c) tan{(A+ B)/2}
d) cot{(A- B)/2}
26) If x= a(cosθ+ θ sinθ), y= a(sinθ - θ cosθ) then aθ=
a) x+ y= a b) √(x²+ y²- a²) c) √(x²& y²+a²) d) x - y + a
27) (1+ cos(π/8))(1+ cos(3π/8))(1+ cos(5π/8))(1+ cos(7π/8))=
a) 1/2 b) cos(π/8) c) 1/8 d) (1+√2)/2√2
28) If 2 cosx + 2 cos3x= cos y, 2 sinx + 2 sin3x= sin y then the value of cos2x is
a) -7/8 b) 1/8 c) -1/8 d) 7/8
29) (cosA)/3= (cosB)/4= 1/5, -π/2< 0, 0<B<π/2, then 3sunA + 4 sinB=
a) 0 b) -1 c) 24/5 d) 1
30) The value of log₃tan1° + log₃tan2°+.....+ log₃tan89° is
a) 3 b) 1 c) 2 d) 0
31) If x= X cosθ - Y sinθ, y= X sinθ+ Y cosθ and ax²+ 2bxy + cy²= AX²+ 2HXY+ BY², then
a) H= 0 if θ= 0 b) H= 0 if θ= π/2 c) A+ B= a+ c d) H= c - a if θ= π/4
32) If tan²((π/2) - θ)/sec²θ. Cot²θ/sec((π/2- θ)). sin((π/2- θ)/sin⁴θ= cotⁿθ then n=
a) 2 b) 4 c) 6 d) 8
33) If sin5θ= a sin⁵θ+ b sin³θ+ c sinθ+ d, then
a) a+ b+ c=0
b) a+ b+ c + d=0
c) 5a+ 3b - 4c=0
d) a- 3c+ d=0
34) The number of solutions of
Sinθ+ 2Sin2θ+ 3Sin3θ+ 4Sin4θ= 10, 0<θ<π is
a) 0 b) 1 c) 2 d) 4
35) If tanA - tanB= x and cotB - cotA= y, then the value of cot(A - B) is
a) (x - y)/xy b) 1/x²+ 1/y² c) (x+y)/xy d) xy
36) cos3θ/cos³θ + sin3θ/Sin³θ is equal to
a) 3cos2θcosecθ
b) 3cot2θ/Sec2θ
c) 12 cot2θcosec2θ
d) 12 tan2θSec2θ
37) (x tanθ + y cotθ) (x cotθ+ y tanθ) - 4xy cos²θ=
a) x²+ y² b) 4xy c) (x + y)² d) none
38) cos11- cos2x is
a) a positive integer
b) a negative integer
c) a positive rational number
d) a negative rational number
39) If sinA, cosA and tanA are in GP., then cot⁶A - cot²=
a) -1 b) 0 c) 1 d) none
40) If tanA tanB, tanC satisfy the equation 3tan³θ - 4 tan²θ+ 3 tanθ +1=0, then A+ B+ C=
a) 0 b) π/2 c) 3π/4 d) 2π
41) if x sinθ + ysin2θ+ z sinSin3θ = sin4θ, (θ≠ nπ) then 8 cos³θ - 4z cos²θ - 2(y +2) cosθ is equal to
a) x - y b) x - z c) y - z d) none
42) The number of values of sinx satisfying sin5x= 5 sinx is
a) 0 b) 1 c) 2 d) 3
43) If sinx, sin y are the roots of the equation
a sin²θ+ b sinθ + + c= 0 and sinx + 2 sin y= 1 then a²+ 2b²+ 3ab + ac=
a) -1 b) 0 c) 1 d) a+ b+ c
44) If sin(θ/2)= a, cos(θ/2)= b, then
(1+ sinθ)(3 sinθ + 4cosθ+5)=
a) (a+ b)²(a+ 3b)²
b) (a+ b)²(3a+ b)²
c) (a- b)²(a- 3b)²
d) (a- b)²(3a- b)²
45) if cosx - Sinx = 1/2, then tan2x=
a) √7/3 b) √7/4 c) 3/√7 d) 2/√7
46) Which of the following gives the least value of A
a) cos2A= sin3A
b) cos3A= sin7A
c) tanA= cot3A
d) cotA= tan2A
47) If A, B, C are acute positive angles such that A+ B + C=π and cotA cotB cotC = k, then
a) k≥3 b) k≤ 1/3√3 c) k≤√3 d) k≤ 1/3√3
48) If sinA= sinB and cosA= cosB; A≠ B, then
a) tan{(A- B)/2}= 0
b) cos(A+ B)= 1
c) tan{(A+ B)/2}= 0
d) sin(A- B)= 1/2
49) cos22+ cos78+ cos 80=
a) 4 sin11 sin39 sin40
b) 1+ 4 cos11 cos39 cos40
c) 1+ 4 sin11 sin39 sin40
d) 4 cos11 cos39 cos40
50) tanx + (1/2) tan(x/2)+ (1/2²) tan(x/2²) + .....+ (1/2ⁿ⁻¹) Tan(x/2ⁿ⁻¹) is equal to
a) 1/2ⁿ cot(x/2ⁿ) - 2 cot2x
b) (1/2ⁿ⁻¹) cot(x/2ⁿ⁻¹) - 2 cot2x
c) tan{(2ⁿ -1)x/2ⁿ⁻¹}
d) 2 cot2x - (1/2ⁿ⁻¹) cot(x/2ⁿ⁻¹)
51) If 4nx=π, then the value of tanx tan2x tan3x.....tan(2n -1)x is
a) -1 b) 0 c) 1 d) none
52) The value of
(3+ cot76 cot16)/(Cot76+ cot16) is
a) cot44 b) cot46 c) tan2 d) cot92
53) If x cosθ= y cos(θ+ 2π/3)= z cos(θ+ 4π/3) then xy+ yz + zx=
a) cos²θ b) sin²θ c) 1 d) 0
54) If A> 0, B> 0 and A+ B =π/3 then the maximum value of tanA tanB is
a) 1/√3 b) 1/3 c) √3 d) 3
55) If tanθ, 2 tanθ+2, 3 tanθ+ 3 are in GP, then the value of
(7- 5 cotθ)/(9- 4√(sec²θ-1)) is
a) 12/5 b) -33/28 c) 33/100 d) 12/13
56) If sinθ+ cosθ= a and cosθ- sinθ= b , then sinθ(sinθ - cosθ)+ sin²θ(sin²- cos²θ)+ sin³θ(sin³θ - cos³θ)+ ....is equal to
a) (1- ab)/(1+ ab)
b) (1- a²/(3- a²)
c) (1- ab)/(1+ ab) + (1- a²/(3- a²)
d) (1+ ab)/(1- ab) + (a²-1)/(3- a²)
57) If x> 0 and the determinant
x sinθ cosθ
- sinθ x. 1 = 0 then
cosθ 1. x
x<√2 b) x=√2 c) x>√2 d) none
58) If x₁, x₂, x₃,.....xₙ are in AP whose common difference is θ, then the value of sinθ(secx₁. secx₂+ secx₂ secx₃+....secxₙ₋₁ secxₙ) is
a) sin nθ/(cosx₁ cosxₙ)
b) sin(n -1)θ/cosx₁ cosxₙ
c) sin nθ cosx₁ cosxₙ
d) cos(n -1)θ/sinx₁sinxₙ
59) If xₙ₊₁ = √(1/2) (1+ xₙ), then cos[√(1- x₀²)/(x₁x₂x₃....to infinite)] (-1< x₀< 1) is equal to
a) -1 b) 1 c) x₀ d) 1/x₀
60) If (1+ √(1+ x) tanθ= (1- √(1- x)) then x=
a) sinθ b) sin2θ c) sin4θ d) cos4θ
61) If f(θ)= sinθ(sinθ+ sin3θ), then f(θ)
a) ≥ only when θ≥ 0
b) ≤ 0 for all real θ
c) ≥ 0 for all real θ
d) ≤ 0 only when θ≤ 0
62) In a right angled triangle, the hypotenuse is 2√2 times the length of the perpendicular drawn from the opposite vertex in its hypotenuse then the other two angles are
a) π/3 , π/6 b) π/4,π/4 c) π/8, 3π/8 d) π/2, 5π/12
63) √cos2x + √(1+ sin2x)= √(sinx + cosx) if
a) sinx+ cosx= 1
b) x=2nπ
c) x= nπ+π/4
d) sinx - cosx= 0
64) If cot(π/3) cos(2πx)=√3, the general solution of the equation
a) 2nπ± π/3 b) n±1/3 c) n±1/6 d) n±1/2
65) 2 cos²x + 4 cosx = 3 sin²x if
a) cosx= (-2+√14)/5
b) cosx= (-2+√19)/5
c) sinx= (-2+√14)/5
d) sinx= (-2+√19)/5
66) sinx + 2 sin2x= 3+ sin3x
a) if sinx + cos2x= 0
b) if sin2x -1=0
c) If cosx= 0
d) for no real value of x
67) 6 tan²x - 2 cos²x= cos2x if
a) cos2x= -1 b) cos2x= 1 c) cos3x = -1/2 d) cos2x = 1/2
68) The greatest value of cosθ for which cos5θ= 0 is
a) 0 b) (1+√5)/4 c) √{(5+√5)/8} d) √{(√5+1)/4}
69) If tanpθ= tan qθ, then the values of θ form an AP with common difference
a) π/(p+ q) b) π/p c) π/q d) π/(p - q)
70) The number of pairs (x, y) satisfying the equation sinx + sin y= sin(x + y) and|x|+ |y|= 1 is
a) 2 b) 4 c) 6 d) infinite
71) The equation ˣ₀∫ (t²- 8t +13) dy= x sin(a/x) has a solution if sin(a/6)=
a) 0 b) 1 c) 3 d) 6
72) The smallest positive root of the equation √sin(1- x)= √cosx is
a) 1/2+ π/4 b) 1/2+ 3π/4 c) 1/2+ 5π/4 d) 1/2+ 7π/4
73) The sum of the roots of the equation
a) 4 cos³x - 4 cos²x - cos(π+ x) - 1= 0 in the interval [0,315] is pπ, where p is equal to
a) 2500 b) 2550 c) 2600 d) 2651
74) A solution (x,y) of x²+ 2x sinxy +1= 0 is
a) (1,0) b) (1,7π/2) c) (-1,7π/2) d) (-1,0)
75) eˢᶦⁿˣ - e⁻ˢᶦⁿˣ= 4 for
a) all real values of x
b) some x∈ [0,π/2]
c) some x ∈ (-π/2,π/2)
d) some x ∈ (-π/2,π/2)
1d 2b 3b 4c 5d 6c 7c 8a 9b 10b 11b 12a 13c 14d 15a 16b 17d 18d 19b 20a 21a 22a 23b 24c 25c 26b 27c 28a 29a 30d 31c 32d 33c 34a 35c 36c 37c 38d 39c 40b 41b 42b 43b 44a 45c 46b 47b 48a 49c 50b 51c 52a 53d 54b 55c 56c 57d 58b 59c 60c 61c 62c 63b 64c 65b 66d 67d 68c 69d 70c 71b 72d 73b 74b 74d
θ
