BOOSTER - A
1) Total number of integral values of a so that x²- (a +1)x + a -1=0 has integral roots is equal to
a) 1 b) 2 c) 4 d) none. a
2) If α, β, γ are such that α + β + γ = 2, α⅖+ β²+ γ²= 6, α²+ β³+ γ³= 8, then α⁴+ β⁴+ γ⁴ is
a) 18 b) 10 c) 15 d) 36 a
3) The number of integral values of a for which the quadratic equation (x + a)(x + 1991)+ 1= 0 has integral roots are
a) 3 b) 0 c) 1 d) 2. d
4) The number of integral values of x satisfying √(- x²+ 10x - 16)< (x -2) is
a) 0 b) 1 c) 2 d) 3. d
5) If x²+ ax - 3x - (a +2)=0 has minimum value of (a²+1)(a²+ 2) is
a) 1 b) 0 c) 1/2 d) 1/4 c
6) If α, β be the roots of x²- a(x -1)+ b =0, then the value of 1/(α²- aα) + 1/(β²- bβ) + 2/(a + b) is
a) 4/(a+ b)
b) 1/(a + b)
c) 0 d) none. c
7) If the roots of the equation, ax²+ bx + c=0, are the form α/(α -1) a(α+1)/α then the value of (a + b + c)² is
a) 2b²- ac b) b²- 2ac c) b²- 4ac d) 4b²- 2ac. c
8) If a, b, c ∈ R and 1 is a root equation ax²+ bx + c=0, then equation 4ax²+ 3bx + 2c = 0, c ≠ 0 has
a) imaginary roots
b) real and equal roots
c) real and unequal roots
d) rational roots. c
9) If a, b, c ∈ R then the equation has (x¹+ ax - 3b)(x²- cx + b)(x²- dx + 2b)= 0
a) real roots
b) two real roots
c) no real roots
d) atleast real roots. d
10) Let α and β be the real and distinct roots of the equation ax²+ bx + c = |c|, (a > 0) and p,q be the real and distinct roots of the equation ax²+ bx + c =0. Then
a) p and q lie between α and β
b) p and q do not lie between α and β
c) only p lies between α and β
d) Only Q lies between α and β. a
11) The equation x²+ a²x + b²= 0 has two roots each of which exceeds c, then
a) a⁴> 4b²
b) c²+ a²c + b²> 0
c) - a²/2 > c
d) all are correct. d
12) If 0< a < b < c and the roots α, β of the equation ax²+ bx + c = 0 are imaginary, then
a) |α|= |β|> 1
b) |α|= |β|< 1
c) |α|≠ |β| d) none. a
13) Let P, q, r, s ∈ R and pr = 2(q + s). Consider two quadratic equations x²+ px + q = 0 ayx²+ rx + s = 0. Then
a) both the equation have real and equal roots
b) both the equation have real and distinct roots
c) atleast one of the equation has real roots
d) none. c
14) if α, β be the roots of 4x²- 16x + λ = 0, λ∈ R. Such that 1 < α < 2 and 2 < β < 3, then the number of integral solutions of λ is
a) 5 b) 6 c) 2 d) 3. d
15) If α and β are the roots of the equation ax²+ bx + c=0 and α⁴ and β⁴ are the roots of the equation lx²+ mx + n = 0, then the roots of the equation a²lx²- 4aclx + 2c²l + a²m = 0 are
a) always positive
b) always non-real
c) opposite in sign
d) negative. c
16) If the inequality (mx²+ 3x +4)/(x²+ 2x +2) < 5 is satisfied for all x∈ R, then
a) 1< m <5 b) -1< m <1 c) -5< m <11/24 d) ±√(2abc). d
17) If ax² + by²+ cz²+ 2ayz + 2bzx + 2cxy can be resolved into rational factors, then a³+ b³+ c³=
a) abc b) 2abc c) 3abc d) ±√(2abc). c
18) 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
19) 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. b
20) If f(x)= x²+ 2bx + 2c⅖ and g(x)= - x²- 2cx + b², such that minimum f(x)> maxyg(x), then the relation between b and c is
a) no real value of b and c
b) 0< c < b √2
c) |c|< |b| √2
d) |c|> |b| √2. d
21) Let a, b, c are distinct positive numbers such that each of the quadratic ax²+ bx + c, bx²+ CX + a and cx²+ ax + b is non-negative for all x∈E. If P= (a²+ b²+ c²)/(ab + bc + ca), then
a) 1≤ P≤4
b) 1≤ P<4
c) 1< P≤4
d) 1< P<4. d
22) If min{x²+ (a - b)x + 1 - a - b }> max {- x²+ (a + b)x -(1+ a+ b)} then
a) a²+ b²< 2
b) a²+ b²< 4
c) a²+ b² > 4 d) none. b
1a 2a 3d 4d 5c 6c 7c 8c 9d 10a 11d 12a 13c 14d 15c 16d 17c 18d 19b 20d 21d 22b
BOOSTER - B
1) If one root of the equation x²+ px +12=0 is 4, while the equation x²+ px + q= 0 has equal roots, then the value of q is
a) 49/4 b) 4/49 c) 5 d) none
2) If cosα is a root of 25x²+ 5x - 12=0-1< x < 0. Then the value of sin2α is
a) 12/25 b) -12/25 c) -24/25 d) 20/25
3) Two students while solving a quadratic equation in x, one copied the constant term incorrectly aygor the roots 3 and 2 . The other copied the constant term, and coefficient of x² correctly and got the roots as -6 and 1 respectively. The correct roots are
a) 3,-2 b) -3,2 c) -6,-1 d) 6,-1
4) The number of positive integeral solutions of x²+ 9 < (x +3)²< 8x +25 is
a) 2 b) 3 c) 4 d) 5
5) If α, β are the roots of the equation x²+ px + q= 0 and α⁴, β¾ are the roots of x²- rx + s = 0, then the equation x²- 4qx + 2q²- r = 0 always (β are real numbers)
a) two real roots
b) two negative roots
c) two positive roots
d) one positive and one negative roots
6) If a, b∈ Q, a≠ 0, then roots of the equation 5ax²+ 3b - 5bx = 3ax are
a) imaginary b) irrational c) rational d) real and equal
7) If 5ˣ + (2√3)²ˣ ≥ 13ˣ then the solution set for x is
a) (2, +∞) b) {2} c) (-∞,2] d) [0,2]
1a 2c 3d 4d 5a 6c 7c
BOOSTER - C
1) If the roots of the equation 8x⅗- 14x²+ 7x -1=0 are in GP, then the roots are
a) 1,1/2,1/4 b) 2,4,8 c) 3,6,12 d) none
2) The roots of the given equation (p - Q)x²+ (q - r)x + (r - p)=0 are
a) (p- q)/(r - p),1
b) (q- r)/(p - p),1
c) (r - p)/(p - q),1
d) 1, (q -r)/(p -q)
3
BOOSTER- D
1) The coefficient of x in the equation x²+ px + q=0 was a wrongly written as 17 in place of 13 and the roots thus found was a - 2 and -15. Then the roots of the correct equation are
a) - 3, 10 b) -3, -10 c) 3, -10 d) none
2) If α, β be the non zero roots of ax²+ bx+ c=0 and α², β² be the roots of a²x²+ b²x + c²=0, then a,b, c are in
a) GP b) HP c) AP d) none
3) If the ratio of the roots of ax²+ 2bx + c=0 is same as the ratio of the px²+ 2qx + r=0, then
a) 2b/ac= q²/pr b) b/ac= q/pr c) b²/ac = q²/pr d) none
4) If α, β are real and α², β² are the roots of the equation a²x²+ x +1 - a²= 0 (a > 1), then β²=
a) a² b) 1- 1/a² c) 1- a² d) 1+ a²
5) The value of m for which one of the roots of x²- 3x + 2m =0 is double of one of the roots of x²- x + m =0 is
a) -2 b) 1 c) 2 d) none
6) If the equation ax²+ bx + c=0 and x³+ 3x²+ 3x +2=0 have two common roots, then
a) a= b= c b) a= b≠ c c) a=- b = c d) none
7) Number of values of a for which equations x³+ ax +1=0 and x⁴+ ax²+ 1=0 have a common root
a) 0 b) 1 c) 2 d) infinite
8) Let R, s and t be the roots of the equation, 8x³+ 1001x + 2008=0. The value of (r+ s)³+ (s + t)³ +(t + r)³ is
a) 251 b) 751 c) 735 d) 753
9) if roots of x²- (a -3)x + a =0 are such that at least one of them is greater than 2, then
a) a ∈[7,9] b) a ∈ [7,9) c) a∈ [9, ∞) d) a ∈ [7, 9)
10) The interval of a for which the equation tan²x - (a - 4) tanx + 4 - 2a =0 has atleast one solution and x ∈ [0, π/4]
a) a∈ (2,3) b) a ∈ [2,3) c) a ∈ (1,4) d) a∈ [1,4]
11) The range of a for which the equation x²+ ax - 4=0 has its smaller root in the interval (-1,2) is
a) (-∞, -3) b) (0,3) c) (0, ∞) d) (-∞, -3) U (0, ∞)
12) The number of positive integral solutions of x⁴- y⁴= 3789108 is
a) 0 b) 1 c) 2 d) 4
13) If α, β, γ are such that
α+ β+ γ=2, α²+ β²+ γ²=2, α³+ β³+ γ³=8 then α⁴+ β⁴+ γ⁴ is
a) 18 b) 10 c) 15 d) 36
14) The number of integral value of a for which the quadratic equation (x + a)(x + 1991)=0 has integral roots are
a) 3 b) 0 c) 1 d) 2
15) The number of real solutions of the equation (9/10)ˣ - 3 + x - x² is
a) 2 b) 0 c) 1 d) none
16) If the equation cot⁴x - 2 cosec²x + a²= 0 has at least one solution then, sum of all possible integral values of a is equal to
a) 4 b) 3 c) 2 d) 0
17) For x²- (a +3)|x|+ 4= 0 to have real solutions, the range of a is
a) (-∞, -7) U [1,∞) b) (-3,∞) c) (-∞,7) d) [1,-∞)
18) In the quadratic equation : 4x²- 2(a + c -1)x + ac - b =0 (a > b >c)
a) both roots are greater than a
b) both roots are less than c
c) both roots lie between c/2 and a/2.
d) exactly one of the roots lies between c/2 and a/2.
19) The equation 2²ˣ + (a -1)2ˣ⁺² + a= 0 has roots of opposite signs the exhausted set of values of a is
a) a∈ (-1,0) b) a < 0 c) a∈ (- ∞, 1/3) d) a∈ (-∞, 1/3)
20) The number of integral values of x satisfying √(x²+10x -16) < x -2 is
a) 0 b) 1 c) 2 d) 3
21) if the following figure shows the graph of f(x)= ax²+ bx + c, then
a) ac< 0 b) bc> 0 c) ab>0 d) x = 0
22) if the roots of the equation, x⅗+ px²+ qx -1=0 form an increasing GP, where p and q are real, then
a) p+q=0 b) p ∈ (-3, ∞)
c) one of the root is unity
d) one root is smaller than 1 and one root is greater than 1.
23) if every pair from among the equations x²+ ax + bc=0, x²+ bx + ca=0 and x²+ cx + ab = 0 has a common root, then a. the sum of the three roots is (-1/2) (a+ b+ c)
b) the sum of the three common roots is 2(a+ b+ c).
c) the product of the three common roots is abc
d) The product of the three common roots is a²b²c².
24) 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, such that α, β, γ and δ are HP, then
a) a=3 b) a=4 c) b= 2 d) b= 8
25) If the roots of the equation x²+ ax + b=0 are c, d, then roots of the equation x²+ (2c + a)x + c²+ ac + b=0 are
a) c b) d - c c) 2c d) 0
26) If a,b,c ∈ R and abc< 0, then the equation bcx²+ 2(b + c -a)x + a=0, has
a) both positive roots
b) both the negative roots
c) real roots
d) one positive and one negative root.
27) For the quadratic equation x²+2(a+1)x + 9a -5=0, which of the following is are true ?
a) if 2< a < 5, then roots are of opposite sign.
b) If a< 0, then roots are opposite signs.
c) if a> 7, then both roots and negative.
d) if 2≤ a ≤ 5, then roots are unreal.
28) Let f(x)= x²+ bx + c, where b, c ∈ R. If f(x) is a factor of both x⁴+ 6x²+ 25 and 3x⁴+ 4x²+ 28x +5, then the least value of f(x) is
a) 2 b) 3 c) 5/2 d) 4
29) Let f(x)= ax³+ bx²+ CX + d, a> 0, a, b, c, d ∈ R and f(x)= 0 has all roots of repeated nature, if
g(x)= f'(x) - f"(x)+ f"'(x) then and x ∈ R
a) g(x)> 0 b) g(x) ≥ 0 c) g(x)< 0 d) g(x)≤ 0
1b 2a 3c 4b 5a 6a 7b 8d 9c 10b 11a 12a 13a 14d 15b 16d 17d 18d 19c 20d 21a,b,d 22a,c,d 23a,c 24a,d 25b,d 26c,d 27c,d 28d 29a
BOOSTER - E
1) a, b, c ∈R and is root of equation ax²+ bx + c =0, then equation 4ax²+ 3bx + 2c=0, c≠ 0 has
a) imaginary roots
b) real and equal roots
c) real and unequal roots
d) rational roots
2) If the equation ax²+ 2bx - 3c =0 has non real roots and (3c/4)< (a+ b). Then c is always
a) < 0 b) >0 c) ≥0 d) none
3) if α, β are the roots of x²- 3x + a =0, a ∈ R and α <1<β then
aa) a ∈ (-∞,2) b) a ∈ (∞, 9/4] c) (-2,9/4] d) none
4) if the equation ax¹+ bx + c =0 and cx²+ bx + a=0, a≠ c have a negative common roots, then the value of a - b + c is
a) 0 b) 1 c) 2 d) none
5) The roots of the equation ax²+ bx+ c=0, Where a∈ R, are two consecutive odd positive integers, then
a) |b|≤ 4a b) |b|≥ 4a c) |b|= 2a d) none
6) The integral values of x for which x²+ 19x +92=0 is perfect square are
a) -8 and -11 b) - 8 and 11 c) 8 and -11 d) ±8 , ±18
7) The equation x²+ a²x + b²=0 has 2 root each of which exceeds a number c, then
a) a⁴> 4b² b) c²+ a²c + b²> 0 c) -a²/2 > c d) all are correct
8) If 0< a <b < c and the roots of α, β of the equation ax²+ bx + c =0 are imaginary, then
a) |α|= |β|> 1
b) |α|=|β|< 1
c) |α|≠|β| d) none
9) If the inequality (mx²+ 3x+4)/(x²+ 2x +2)< 5 is satisfied for all x∈ R, then
a) 1< m <5 b) -1<m <1 c) -5< m < 11/24 d) m < 71/24
10) let p, q ∈{1,2,3,4}. The number of equation of the form px²+ qx +1=0 having real roots is
a) 15 b) 9 c) 7 d) 8
11) 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
12) If f(x)= x²+ 2bx + 2c² and g(x)= - x²- 2cx + b², such that minimum f(x)> maximum g(x), then the relation between b and c, is
a) no real values of b and c
b) 0 < c < b√2
c) |c|< |b| √2
d) |c|> |b| √2
13) Equation πᵉ/(x - e) + ₑπ/(x - πl + (π^π+ eᵉ)/(x - π - e)= 0 has
a) one real roots in (e,π) and other in (π- e, e)
b) one real root in and other in
c) 2 real roots in
d) no real roots
14) The roots of ax²+ bx + c=0. where a≠ 0 are non-real complex and a+ c < b. than
a) 4a+ c> 2b b) 4a+ c < 2b c) a+ 4c > 2b d) a+ 4c < 2b
15) Which of the following is correct for the quadratic equation x²+2(a -1)x + a+ 5=0
a) The equation has positive roots, if a ∈ (-5,-1)
b) the equation has roots of opposite sign, if a∈(∞,-5)
c) The equation has negative roots, if a ∈[4, ∞)
d) none
16) Consider the quadratic equation x²- 2px + p²- 1=0 where p is parameter, then
a) both the roots of the equation are less than both 4 if p ∈(-∞,3)
b) both the roots of the equation are greater then -2 if p∈(-∞,-1)
c) exactly one root of the equation lies in the interval (-2,4) if p∈(-1,3)
d) 1 lies between the roots of the equation if p∈(0,2)
17) α is a real roots of the equation ax²+ bx + c=0 and β is the real root of the equation - ax²+ bx + c= 0, then the equation ax²/2 + bx + c=0 has
a) real roots b) non-real roots then
c) has a root lying between α and β d) none
18) If the equation x²+ ax+ b =0 has distinct real roots and x²+ a|x|+ b =0 has only one real roots, then
a) b= 0 b) a< 0 c) b > 0 d) a > 0
19) If the equation ax²+ bx + c =0 has distinct real roots and ax²+ b|x|+ c=0 also has two distinct real roots then
a) a> 0 b) c/a < 0 c) b/a < 0
d) x= 0 cannot be a root of the first equation
20) Let f(x)= ax²+ bx + c, a, b, c ∈ R and a≠ 0. If f(x)> 0 and x ∈ R then
a) 4a - 2b + c > 0
b) 2a+ b+ c> 0
c) 10a + 3b + c > 0
d) 2a + b + c > 0
21) The equation ₓ₊₁|kogₓ₊₁(3+2x - x²)= (x -3) |x| has
a) Unique solution
b) two solutions
c) no solution
d) more than two
22) The number of real solution of the equation 1+ |eˣ -1|= eˣ(eˣ -2) is
a) 0 b) 1 c) 2 d) infinity many
23) if x is a positive real number, where
[] is g.i.f then [x/2] + [(x+1)/2]=
a) [x + 1/2]
b) [x] c) [x + 1/4] d) [2x + 1/4]
24) the least integer a, for which log₅(x²+1)≤ log₅ (ax²+ 4x + a) is true for all x ∈ R is
a) 6 b) 7 c) 10 d) 1
25) If 1 lies between the roots of the equation y²- my +1=0 and [x] denotes the greatest integer ≤ x then [4[x]/{|x|²+16}ᵐ] is equal to
a) 0 b) 1 c) 2 d) none
26) If α be the number of solutions of the equation [sinx ]= [x] and β be the greatest value of cos(x²- [x²]) in the interval [-1,1] where [] denotes greatest integral function, then
a) α<β b) α> β c) α< β d) none
27) Let f(x) be a function defined by F(x)= x - [x], 0 ≠ x ∈R, where[x] is the greatest integer less than or equal to x. Then the number of solutions if F(x)+ F(1/x)=1 is
a) 0 b) infinite c) 1 d) 2
28) If 5{x}= x + [x] and [x] - {x}= 1/2 when {x} and [x] fractional and integral part of x, then x is
a) 1/2 b) 3/2 c) 5/2 d) 7/2
29) The equation ₂|x²-12| = √ₑ|x| log4) has
a) no real solution
b) only two real solution whose sum is zero
c) only two real solutions whose sum is non zero
d) four real solution whose sum is zero.
1c 2a 3a 4a 5b 6a 7d 8a 9d 10c 11b 12d 13b,c 14b,d 15a,b,c 16a,d 17a,c 18a,d 19b,d 20a,b,c,d 21c 22b 23b 24b 25a 26c 27b 28b 29d
BOOSTER - F
1) Suppose that (2F(n) +1))/2 for n= 1,2,3,....and F(1)= 2. Then, F(101) equals
a) 50 b) 52 c) 54 d) none
2) if the sum of n terms of an AP is cn(n -1), where c≠ 0, then sum of the squares of these terms is
a) c²n²(n +1)²
b) (2.3) c²n(n -1)(2n -1)
c) (2/3) c²n(n +1)(2n +1) d) none
3) Let a₁, a₂, a₃....are in AP, if
(a₁ + a₂+ ....aₚ)/(a₁ + a₂ + ...aq)= p²/q², p≠ q, then a₆/a₂₁ equals
a) 41/11 b) 7/2 c) 2/7 d) 11/41
4) If Sₙ denotes the sum of the first n terms of an AP, whose first terms is a and Sₙₓ/Sₓ is independent of x, then Sₚ =
a) p³ b) p²a c) pa² d) a³
5) If |a|< 1 and|b|< 1, then the sum of the series
1+ (1+ a)b + (1+ a + a²)b²+ ....is
a) 1/{(1- a)(1- b)}
b) 1/{(1- a)(1- ab)}
c) 1/{(1- b)(1- ab)}
d) 1/{(1- a)(1- b)(1- ab)}
6) If x, y,z are in GP and aˣ = vʸ = cᶻ, then
a) logᵥa = logₐc
b) log꜀b = logₐc
c) logᵥa= log꜀b d) none
7) The geometric mean between -9 and -16 is
a) 12 b) - 12 c) - 13 d) none
8) If a²+ b², ab + bc and b²+ c² are in GP then a,b,c are in
a) AP b) GP c) HP d) none
9) Let a= 1115 1110 resident number represented by RNA pyr district numbers are in GP then the combination of the GP is 123 135 234 124 rgp then the sum of 50 terms of the series 2500 250 76884 the sum of the series 14575005 769 dashama 0.2 0.004 0.006 28
Some of terms are terms of constant the valuable so that is stomach aquants perfect square upon odd integers 2 geometric series at the same first term the combination of the first series is greater than that of the second series and both are positive denote the sums of terms of the two series respectively then show that
No comments:
Post a Comment