FINAL BOOSTER
RELATION AND FUNCTIONS
a) one one and into
b) onto but not one one
c) one but not onto
d) neither one one nor onto
2) Let (-1,1) ---> R be such that f(cos4θ)= 2/(2- sec²θ) for θ∈ (0, π/4) U (π/4,π/2). Then the value of f(1/3) is are
a) 1- √(3/2) b) 1+ √(3/2) c) 1- √(2/3) d) 1 + √(2/3)
3) Let f(x)= x² and g(x)= sinx for all x∈E. Then the set of all x satisfying (fogogof)(x= (gogof)(x), where (fog)(x)= f(g(x)), is
a) ±√(nπ), n ∈ {0,1,2,....}
b) ±√(nπ), n ∈ {1,2,....}
c) (π/2) + 2nπ, n ∈ {....-2,-1,0,1,2,....}
d) 2nπ), n ∈ {-2,-1,0,1,2,....}
4) Let f: (0,1)--> R be defined by f(x)= (b - x)/(1- bx), where b is a constant such that 0<b< 1 then
a) f is not invertible on (0,1(
b) f≠ f ⁻¹on (0,1) and f'(b)= 1/f'(0)
c) f= f ⁻¹on (0,1) and f'(b)= 1/f'(0)
d) f ⁻¹ is differentiable on (0,1).
INVERSE TRIGONOMETRIC FUNCTIONS
5) The value of cot[ ²³ₙ₌₁∑ cot⁻¹ {1+ ⁿₖ₌₁∑ 2k}] is
a) 23/25 b) 25/23 c) 23/24 d) 24/23
6) Match the column
Column A
P) [(1/y²){cos(tan⁻¹y) + y sin(tan⁻¹y}/{cot(sin⁻¹y) + tan(sin⁻¹y)}²+ y⁴]¹⁾⁴ take value
Q) If cosx + cos y + cos z =0 sinx + sin y + sin z then possible values of cos{(x -y)}/2 is
R) if cos(π/4 - x) cos2x + sin x sin2x secx = cosx sin2x secx + cos(π/4+ x) cos2x then possible value of secx is
S) If cot(sin⁻¹ √(1- x²))= sin(tan⁻¹(x√6)), x ≠ 0 then possible value of x is
Column B
1) (1/2) √(5/3)
2) √2
3) 1/2
4) 1
Codes:
P Q R S
a) 4 3 1 2
b) 4 3 2 1
c) 3 4 2 1
d) 3 4 1 2
7) Let f(θ)= sin[tan⁻¹{sinθ/√cos2θ)}], where -π/4< θ < π/4. Then the value of d/d(tanθ). (f(θ)) is
Matrices
8) For 3x 3 matrices M and N, which of the following statement/s is/are not correct ?
a) NᵀMN is symmetrical Or skew symmetric according as M is symmetric or skew Symmetric .
b) MN ~ NM is skew symmetric for all symmetric matrices M and N.
c) MN is symmetric for all symmetric matrices M and N
d) (adj M) (adj N)= adj(MN) invertible matrices M and N.
9) If P is a 3x3 matrices such that Pᵀ = 2P + I, where Pᵀ is the transpose of P and I is the 3x3 identity Matrix, there exists a column
X= x 0
y = 0
z 0 such that
a) PX = 0
0
0
b) PX= X c) PX= 2X d) PX= - X
10) Let M and N be two 3x3 nonsingular skew symmetric matrices such that MN= NM. If Pᵀ denotes the transpose of P, then M²N²(MᵀN)⁻¹ (MN⁻¹)ᵀ is equal to
a) M⅖ b) - N⅖ c) - M² d) MN
* Paragraph for Question No 11 to 13
Let a,b, and c be three real numbers satisfying the matrices
A= a b c & B= 1 9 7 & C= 0 0 0
8 2 7
7 3 7 with the relation AB = C
11) in the point P(a,b,c) with reference to (E), lies on the plane 2x + y + z =1, then the value of 7a+ b + c is
a) 0 b) 12 c) 7 d) 6
12) Let ω be a solution of x³-1=0 with Im(ω)> 0. If a=2 with b and c satisfying (E), then the value of 3/ωᵃ + 1/ωᵇ + 3/ωᶜ is equal to
a) -2 b) 2 c) 3 d) -3
13) Let b= 6, with a and c satisfying (E). If α and β are the roots of the quadratic equation ax²+ bx + c =0, then ∞ₙ₌₁∑(1/α + 1/β )ⁿ is
a) 6 b) 7 c)6/7 d) ∞
14) Let ω ≠ 1 be a cube root of unity and S be the set of all non-singular matrices of the form
1 a b
ω 1 c
ω² ω 1
Where each of a,b,c is either ω or ω². Then the number of disticnt matrices in the set S is
a) 2 b) 6 c) 4 d) 8
15) Let M be a 3x3 matrix
A= 0 & B= -1
1 2
0 3 satisfying MA= B,
M= 1 1
-1 = 1
0 -1 , and
MC = D where C= 1 & D= 0
1 0
1 12. then the sum of the diagonal entries of M is
Determinants
16) Let P=[aᵢⱼ] be a 3x3 matrix and let Q= [bᵢⱼ], where bᵢⱼ = 2ⁱ⁺ʲ aᵢⱼ for 1≤ i, j ≤ 3. If the determinants of Pis 2, then the determinants of the matrix Q is
a) 2¹⁰ b) 2¹¹ c) 2¹² d) 2¹³
17) If the adjoint of a 3x3 matrix
P= 1 4 4
2 1 7
1 2 3 then the possible value/s of the determinants of Pis/are
a) -2 b) -1 c) 1 d) 2
Application of Derir
18) The number of points in (-∞, ∞), for which x²- x sin x - cos x = 0 is
a) 6 b) 4 c) 2 d) 0
19) A rectangular sheet of fixed perimeter with sides having their lengths in the ratio 8:15 is converted into an open rectangular box by folding after removing squares of equal area from all four corners. If the total area of the removed square is 100, the resulting box has maximum volume. The lengths of the sides of the rectangular sheet are
a) 24 b) 32 c) 45 d) 60
20) Let f(x)= x sinπx, x> 0. Then for all natural numbers nf'(x) vanishes at
a) a unique point in the interval (n, n+ 1/2)
b) a unique point in the interval (n+ 1/2n+1)
c) a unique point in the interval (n, n+ 1)
d) 2 points in the interval (n, n+1)
21) The function f(x)= 2[x] + |x +2| - ||x+2 - 2|x|| has a local minimum or a local maximum at x=
a) -2 b) -2/3 c) 2 d) 2/3
Paragraph for Q. Nos 22 and 23
Let f:[0,1]---> R (the setup all real numbers) be function. Suppose the function f is twice differentiable f(0)= f(1)= 0 and satisfies f"(x) - 2f'(x)+ f(x)≥ eˣ,x ∈ [0,1]
22) Which of the following is true for 0< x <1 ?
a) 0< f(x) < ∞
b) -1/2< f(x) < 1/2
c) -1/4< f(x) < 1
d) ∞< f(x) < 0
23) if the function e⁻ˣ f(x) assumes its minimum in the interval [0,1] at x= 1/4, which of the following is true ?
a) f'(x) < f(x), 1/4 < x < 3/4
b) f'(x) > f(x), 0 < x < 1/4
c) f'(x) < f(x), 0 < x < 1/4
d) f'(x) < f(x), 3/4 < x < 1
24) let p(x) be a real polynomial of least degree which has a local maximum at x=1 and a local minimum at x=3. If P(1)=6 and p(3)= 2, then p'(1) is
a)
25) Let f: R---R> R be defined as f(x)= |x|+ |x²-1|. The total number of points at which f attains either a local maximum or a local minimum is
26) The number of disticnt real roots of x⁴- 4x³+ 12x²+ x -1=0 is
Integrals
27) Let f:[1/2, 1]---R (the set of all real numbers) be a positive, non-constant and differentiable function such that f'(x)< 2f(x) and f(1/2)= 1. Then the value of
¹₁/₂ ∫ f(x) dx lies in the interval
a) (2e -1, 2e)
b) (e -1, 2e -1)
c) ((e -1)/2, e -1)
d) (0, (e -1)/2)
28) The integral ∫ sec²x/(secx + tanx)⁹⁾² dx equals (for some arbitrary constant K)
a) - 1/(secx + tanx)¹¹⁾²{1/11 - (1/7) (secx + tanx)²}+ K
b) 1/(secx + tanx)¹¹⁾²{1/11 - (1/7) (secx + tanx)²}+ K
c) -1/(secx + tanx)¹¹⁾²{1/11 + (1/7) (secx + tanx)²}+ K
d) 1/(secx + tanx)¹¹⁾²{1/11 + (1/7) (secx + tanx)²}+ K
29) The value of the integral ∫ (x²+ ln{(π+x)/(π- x)}} cosx dx at (π/2,-π/2) is
a) 0 b) π²/2 - 4 c) π²/2 + 4 d) π²/2
Paragraph for Question No 30 and 31
Let f(x)= (1- x)² sun²x + x² for all x ∈R, and let g(x)= ∫[2(t -1)/(t +1) ln t] f(t) dt at (π,1) for all x ∈ [1, ∞).
30) Which of the following is true ?
a) g is increasing on (1, ∞)
b) g is decreasing on (1, ∞)
c) g is increasing on (1,2) and decreasing on (2, ∞)
d) g is decreasing on (1,2) and decreasing on (2, ∞)
31) Consider the statements :
P: there exists some x ∈ R such that f(x)+ 2x = 2(1+ x²)
Q: there exists some x ∈ R such that 2gpf(x) +1= 2x(1+ x). Then
a) both P and Q are true
b) P is true and Q is false
c) P is false and Q is true
d) both P and Q are false
32) If f(x)= ˣ₀∫ ₑt² (t -2)(t - 3) dt for all x ∈ (0, ∞), then
a) f has a local Maximum at x= 2
b) f is decreasing on (2,3)
c)? there exist some c ∈ (0, ∞) such that f"(c)=0
d) f has a local minimum at x=3
33) The value of ∫ (x sinx²)/(sinx²+ sin(ln 6 - x²)) dx at (√ln3, √ln2) is
a) (1/4) ln (3/2)
b) (1/2) ln (3/2)
c) ln (3/2)
d) (1/6) ln (3/2)
Application of Integrals
34) The area enclosed by the curves y= sinx + cosx and y= |cosx - sin x| over the interval [0,π/2] is
a) 4(√2 -1)
b) 2√2(√2 -1)
c) 2(√2 +1)
d) 2√2(√2 +1)
35) Let S be the area of the region enclosed by y=ₑ-x², y= 0, x= 0 and x =1. Then
a) S≥ 1/e b) S≥ 1- 1/e c) S≤ (1/4) (1+ 1/√e)
d) S≤ 1/√2 + (1/√e) (1- 1/√2)
36) Let the straight line x= b divide the area enclosed by y= (1- x²), y= 0 and x=0 into two parts R₁(0≤ x ≤ b) and R₂(b ≤ x ≤ 1) such that R₁- R₂= 1/4 then b equal
a) 3/4 b) 1/2 c) 1/3 d) 1/4
37) Let f: [-1,2]---> [0, ∞) be a continuous function such that f(x)= f(1- x) for all x∈ [-1,2].
Let R₁,= ²₋₁ ∫ x f(x) dx, and R₂ be the area of the region bounded by y= f(x), x = -1, x =2, and the x-axis. Then
a) R₁= 2R₂ b) R₁= 3R₂ c) 2R₁= R₂ d) 3R₁= R₂
Differential Equations
38) A curve passes through the point (1,π/6). Let the slope of the curve at each point (x, y) be y/x + sec(y/x), x> 0. then the equation of the curve is
a) sin(y/x)= logx + (1/2)
b) cosec(y/x)= logx + 2
c) sec(2y/x)= logx + 2
d) cos(2y/x)= logx + (1/2)
39) If y(x) satisfies the differential equation y' - y tan x = 2x secx and y(0)= 0, then
a) y(π/4)= π²/8√2
b) y'(π/4)= π²/18
c) y(π/3)= π²/9
d) y'(π/3)= 4π/3 + 2π²/3√3
40) Let f: [1, ∞)---> [2, ∞) be a differentiable function such that f(1)=2. If 6 ˣ₂ ∫f(t) dt = 3x f(x) - x³ for all x≥ 1, then the value of f(2) is
41) Let y'(x)+ y(x) g'(x)= g(x)g'(x), y(0)=0, x ∈R, where f'(x) denotes df(x)/dx and g(x) is a given non-constant differentiable function on R with g(0)= g(2)=0. Then the value of y(2) is
Vector Algebra
42) Let PR= 3i + j - 2k and SQ= i - 3j - 4k determine diagonals of a parallelogram PQRS and PT= i+ 2j + 3k be another vector . Then the volume of the parpalloid determined by the vectors PT, PQ and PS is
a) 5 b) 20 c) 10 d) 30
43) Consider the set of 8 vectors V={ai + bj + ck; a,b,c R[-1,1)}. Three non coplanar vectors can be chosen from V in 2ᵖ ways. Then p is
44) Two lines L₁: x =5, y/(3- α)= z/-2 and L₂: x = α, y/-1= z/(2- α) are coplanar. then α can take value/s
a) 1 b) 2 c) 3 d) 4
45) Match the Column
Column A
P) Volume of parpalloid determined by vector a,b and c is 2. Then the volume of the parallelopiped determined by vectors 2(ax b), 3(bxc) and (cxa) is
Q) Volume of parallelopiped determined vectors a,b and c is 5. Then the volume of the parallelopiped determined by vectors 3(a+ b), (b+ c), 2(c + a) is
R) Area of the triangle with adjacent sides determined by vectors and b is 20. Then the area of the triangle with adjacent sides determined by vectors (2a + 3b) and (a - b) is
S) Area of a parallelogram with adjacent sides determined by vectors a and b is 30. Then the area of the parallelogram with adjacent sides determined by vectors (a+ b) and a is
Column B
1) 100
2) 30
3) 24
d) 60
Codes
P Q R S
a) 4. 2 3 1
b) 2 3 1 4
c) 3 4 1 2
d) 1 4 3 2
46) If a, b and c are unit vectors satisfying |a - b|²+ |b - c|²+ |c - a|²= 9, then |2a + 5b + 5c| is
47) If a and b are vectors such that |a+ b|=√29 and ax (2i+ 3j + 4k)= (2i+ 3j + 4k) x b, then a possible value of (a+ b)(-7i + 2j + 3k) is
a) 0 b) 3 c) 4 d) 8
48) The vector/s which is/are coplanar with vectors i+ j + 2k and i+ 2j + k, and perpendicular to the vector i+ j + k is/are
a) j - k b) - i+ j c) i- j d) -j + k
Three Dimensional Geometry
49) Perapendicular are drawn from point on the line (x +2)/2= (y +1)/-1= z/3 to the plane x + y + z =3. The feet of perpendicular lie on the line
a) x/5 = (y -1)/8 = (z-2)/-13
b) x/2 = (y -1)/3 = (z-2)/-5
c) x/4 = (y -1)/3 = (z-2)/-7
d) x/2 = (y -1)/-7 = (z-2)/5
50) A line l passing through the origin is perpendicular to the lines
L₁: (3+ t)i + (-1+ 2t)j + (4+ 2t)k, - ∞<t< ∞
L₂: (3+ 2s)i + (3+ 2s)j + (2+ s)k, - ∞<t< ∞
then the coordinate/s of the point/s on L₂ at a distance of √17 from the point of intersection of l and L₁ is/are
a) (7/3,7/3, 5/3)
b) (-1,-1,0)
c) ( 1,1,1)
d) (7/9,7/9,8/9)
51) consider the lines L₁ : (x -1)/2= y/-1= (z +3)/1, L₂ : (x -4)/1= (y+3)/1= (z +3)/2 and the planes P₁: 7x + y + 2z =3, P₂= 3x + 5y - 6z =4. Let ax + by + cz = d the equation of the plane passing through the point of intersection of lines L₁ and L₂ and perpendicular of the planes P₁, P₂
Match the column and select the correct answer using the code given below the lists
List I
P) a=
Q) b=
R) c=
S) d=
List II
1) 13
2)-3
3)1
4)-2
Codes
P Q R S
a) 3 2 4 1
b) 1 3 4 2
c) 3 2 1 4
d) 2 4 1 3
52) The point P is the intersection of the straight line joining the points Q(2,3,5) and R(1,-4,4) with the plane 5x - 4y - z =1. If S is the foot of the perpendicular drawn from the point T(2,1,4) to QR , then the length of the line segment PS is
a) 1/√2 b) √2 c) 2 d) 2√2
53) The equation of a plane passing through the line of intersection of the plane x + 2y + 3z =2 and x - y + z=3 and at a distance 2/√3 from the point (3,1,-1) is
a) 5x - 11y + z=17
b) √2x + y =3 √2 -1
c) x + y + z=√3
d) x - √2y =1- √2
54) If the straight line (x -1)/2= (y+1)/k = z/2 and (x +1)/5 = (y+1)/2 = z/k are coplanar, then the plane/s containing these two lines is/are
a) y+ 2z = -1 b) y+ z = -1 c) y - z = -1 d) y - 2z = -1
55) Let a= i+ j + k, b= i- j + k and c= i- j - k be three vectors. A vector v in the plane of a and b, whose projection on c is 1/√3, is given by
a) i - 3j + 3k b) -3i- 3j - k c) 3i- j + 3k d) i + j - 3k
56) Let a= -i - k, b= -i + j c= i + j +3k be three the given vectors. if r is a vector such that rx b = c x b and r. a =0, then the value of r.b is
Probability
57) Four persons independently solve a certain problem correctly with probabilities 1/2, 3/4, 1/4, 1/8. Then the probability that the problem is solved correctly by at least one of them is
a) 235/256 b) 21/256 c) 3/256 d) 253/256
58) Of these three independent events, E₁, E₂ and E₃ the probability that only E₁ occurs is α, only E₂ occurs is β and only E₃ occurs is γ. Let the probability p that none of the events, E₁, E₂ or E₃ occurs satisfy the equation (α - 2β)p= αβ and (β- 3γ)p= 2βγ. All the given probabilities are assumed to lie in the interval (0,1). Then
Probability of occurrence of E₁/probability of occurrence of E₃ =
Paragraph for Question 59 and 60
A box B₁ contains one white ball, 3 red balls and 2 black balls. Another box B₂ contains two white balls, 3 red balls 4 Black balls. A third box B₃ contains 3 white balls, 4 red balls and 5 black balls.
59) If 1 ball is drawn from each of the boxes B₁, B₂ and B₃, the probability that all 3 drawn balls are of the same colour is
a) 82/648 b) 90/648 c) 558/648 d) 566/648
60) If 2 balls are drawn (without replacement) from a randomly selected box and one of the balls is white and the other ball is red, the probability that these two balls are drawn from box B₂, is
a) 116/181 b) 126/181 c) 65/181 d) 55/181
61) A ship is fitted with 3 engines E₁, E₂ and E₃. the engs function independently of each other with respective Probabilities 1/2,1/4 and 1/4. For the ship to be operational at least two of the engines must function. Let X denote the event that the ship is operational and let X₁, X₂ and X₃denote respectively the events that the engines E₁, E₂ and E₃ are functioning. Which of the following is/are true ?
a) P[Xᶜ₁/X]= 3/16
b) P [exactly two engines of the ship are functioning |X]= 7/8
c) P[X/X₂]= 5/16
d) P'[X/X₁]= 7/16
62) let X and Y be two events such that P(X/Y)= 1/2, P(Y/X)= 1/3 and P(X∩Y)= 1/6. Which of the following is/are correct ?
a) P(X U Y)= 2/3
b) X and Y are independent
x) X and Y are not independent
d) P(Xᶜ ∩Y)= 1/3
Paragraph for Question 63 and 64
Let U₁ and U₂ be two urns such that U₁ contain three white and 2 Red balls, and U₂ contains only 1white ball. A fair coin is tossed. if head appears then 1 ball is drawn at random from U₁ and put into U₂ . However, if tail appears then 2 balls are draw at random from U₁ and put into U₂. Now one ball is drawn at random from U₂.
63) The probability of the drawn ball from U₂ being white is
a) 13/30 b) 23/30 c) 19/30 d) 11/30
64) Given that the drawn ball from U₂ is white, the probability that head appeared on the coin is
a) 17/23 b) 11/23 c) 15/23 d) 12/23
65) Let E and F be two independed events . The probability that exactly one of the occurs is 11/25 and in the probability of none of them occuring is 2/25. if P(T) denotes the probability of occurrence of the event T, then
a) P(E)= 4/5, P(F)= 3/5
b) P(E)= 1/5, P(F)= 2/5
c) P(E)= 2/5, P(F)= 1/5
d) P(E)= 3/5, P(F)= 4/5
Miscellaneous
66) match the column
Column A
A) if a= j + √3 k, b= - j + √3 k and c= 2√3 k form a triangle, then the internal angle of the triangle between a and b is
B) If ᵇₐ ∫ (f(x) - 3x) dx = a²- b², then the value of f(π/6) is
C) The value of π²/ln 3 ∫ (sec(πx) dx at (5/6,7/6) is
D) The maximum value of|Arg{1/(1- z)}| for|z|= 1, z≠ 1 is given by
Column B
p) π/6
q) 2π/3
r) π/3
s) π
t) π/2
67) Match the column
Column A
A) The set [Re{2iz/(1- z²)}, z is a complex number |z| =1, z≠ ± 1]
B) The domain of the function f(x)= sin⁻¹ {8(3)ˣ⁻²/(1- 3²⁽ˣ⁻¹⁾} is
C) If the determinants
1 tanθ 1
f(θ)= -tanθ 1 tanθ
-1 -tanθ. 1 then the set {f(θ): 0 ≤θ < π/2} is
D) If f(x)= √x³(3x -10), x > 0, then f(x) is increasing in
Column B
p) (-∞, -1) U (1, ∞)
q) (-∞,0) U (0, ∞)
r) [2, ∞)
s) (-∞,-1] U [1, ∞)
t) (-∞,0) U [2, ∞)
BOOSTER - 3
1) If |x|< 1, then the coefficient of xⁿ in (1+ 2x + 3x²+ 4x³+.....∞)¹⁾² is
a) n b) n+1 c) 1 d) -1
2) The sum of infinite terms of the GP (√2+1)/(√2-1), 1/(2- √2), 1/2,.....∞is
a) 3+ 2√2 b) 4+ 3√2 c) 2+ 3√2 d) 4+ 2√2
3) The coefficient of xᵖ and xᑫ in the expansion of (1+ x)ᵖ⁺ᑫ are
a) equal
b) equal with opposite signs
c) reciprocal to each other d) none
4) The sum of the infinite series 1/2! + 1/4! + 1/6! +.....∞ is
a) (e²-2)/e b) (e²-1)/2 c) (e²-2)/2e d) (e-1)²/2e
5) If zᵣ = cos(π/2ʳ) + i sin(π/2ʳ), then the value of (z₁, z₂, z₃.....∞) is
a) -3 b) -2 c) -1 d) 1
6) The co-efficient of x³ in the expansion of 3ˣ is
a) (logₑ3)³/6
b) 3³/6
c) (logₑ3)³/3
d) (logₑ3)/2
7) The value of {(x-1)/(x+1) + (1/2) (x²-1)/(x+1)² + (1/3) (x²-1)/(x +1)³+......∞is
a) (1/2) logₑ(x+1)
b) logₑx
c) logₑ{x/(x+1)}
d) logₑ{(x+1)/x}
8) The positive integers just greater than (1 + 0.0001)¹⁰⁰⁰⁰ is
a) 4 b) 5 c) 3 d) 2
9) The value of ∞ᵣ₌₁∑ⁿCᵣ/ⁿPᵣ is
a) e b) e+1 c) e-2 d) e -1
10) The value of (1+ C₁/C₀)(1+ C₂/C₁)(1+ C₃/C₂).....(1+ Cₙ/Cₙ₋₁) is
a) (n+1)ⁿ/n!
b) (n +1)/n!
c) (n +1)ⁿ/(n -1)!
d) (n -1)ⁿ/n!
11) The natural number n for which the inequality 2ⁿ > 2n +1 is valid, is
a) n>3 b) n≥ 3 c) n ≥ 2 d) none
12) If (1+ x)¹⁵= a₀ + a₁x + a₂x²+ ....+ a₁₅x¹⁵, then the value of ¹⁵ᵣ₌₁∑ aᵣ/aᵣ₋₁ is
a) 110 b) 115 c) 120 d) 135
13) Two events A and B are such that P(A)= 1/4, P(B/A)= 1/2 and P(A/B)= 1/4; then the value of P(Aᶜ/Bᶜ) is
a) 1/4 b) 3/4 c) 1/2 d) 2/3
14) The probability that a regularly scheduled flight departs on time is 0.9, the probability that it arrives on time 0.8 and the probability that it departs and arrives on time is 0.7. then the probability that a plane arrives on time, given that it departs on time, is
a) 0.72 b) 8/9 c) 7/9 d) 0.56
15) A sample of 4 items is drawn at random from a lot of 10 items, containing three defectives. If x denotes the number of defective items in the sample then P(0<x<3) is equals to
a) 4/5 b) 3/10 c) 1/2 d) 1/6
16) A and B are two independent event such that P(A)= 1/2 and P(B)= 1/3. Then the value of P(Aᶜ ∩ Bᶜ) is
a) 2/3 b) 1/6 c) 5/6 d) 1/3
17) if n things are arranged at random in a row then the probability that m particular things are never together is
a) m!(n - m)!/n!
b) 1- m!(n - m)!/n!
c) 1- m!/n!
d) 1- m!(n - m+1)!/n!
18) If A= 3 5 & B= 1 17
2 0 0 -10 then |AB| is
a) 80 b) 100 c) -110 d) 92
19) The inverse of the matrix
5 -2
3 1 is
a) (1/13) -2 5
1 3
b) 1 2
-3 5
c) (1/11) -1 2
-3 5
d) 1 3
-2 5
20) If Aᵢ = aᶥ bᶥ
bᶥ aᶥ and|a|< 1, |b|< 1, then the value of ∞ᵢ₌₁∑ det(Aᵢ) is
a) (a²- b²)/{(1- a²)(1- b²)}
b) a²/(1- a²) - b²/(1- b²)
c) a²/(1- a²) + b²/(1- b²)
d) a²/(1+ a²) - b²/(1+ b²)
21) If A is a singular matrix of order n then A. (adj A) is equals to
a) a null matrix
b) a row matrix
c) a column matrix
d) none
22) If the determinant of the matrix
a₁ b₁ c₁
a₂ b₂ c₂
a₃ b₃ c₃ is denoted by D, then the determinant of the matrix
a₁+ 3b₁ - 4c₁ b₁ 4c₁
a₂ +3b₂ - 4c₂ b₂ 4c₂ will be
a₃+ 3b₃ - 4c₃ b₃ 4c₃
a) D b) 2D c) 3D d) 4D
23) If the determinant
x -2 2x -3 3x -4
x -4 2x-9 3x -16 = 0
x -8 2x -27 3x-64
Then the value of x is
a) -2 b) 3 c) 4 d) 0
24) If a, b, c, d, e and f are in GP then the value of
a² d² x
b² e² y
c² f² z
depends on
a) x and y
b) y and z
c) z and x
d) none of x, y and z
25) If a,b,c are respectively the p-th, q-th, r-th terms of an AP then the value of
a p 1
b q 1
c r 1 is
a) p+ q+ r b) 0 c) 1 d) pqr
26) If for a triangle ABC
1 a b
1 c a = 0
1 b c
then the value of sin²A + sin²B + sin²C is
a) 4/9 b) 9/4 c) 1 d) 3√3/4
27) The middle term in the expansion of (1+ x)²ⁿ is
a) (2n!)xⁿ/n!
b) (2n!)xⁿ⁺¹/n!(n-1)!
c) (2n!)xⁿ/(n!)²
d) (2n!)xⁿ/{(n+1)!(n-1)!}
28) If xₙ = cos(π/3ⁿ) + i sin(π/3ⁿ), then the value of (x₁x₂x₃.....∞) is
a) I b) - I c) 1 d) -1
29) The locus of a point which moves such that the difference of its distance from two fixed points is always a constant, is
a) a circle b) a straight line c) an ellipse d) a hyperbola
30) The equation of the directrix of the parabola x²- 4x - 8y - 12= 0 is
a) y=0 b) x=1 c) y=-1 d) x = -1
31) The curve a²y²= b²(a²- x²) is symmetrical about
a) x-axis b) y-axis c) both axes d) none
32) Which of the following points lies on the hyperbola x²= 4ay ?
a) (at²,2at) b) (at,at²) c) (2at², at) d) (2at,at²)
33) The foci of the ellipse x²/16 + y²/b² = 1 and the hyperabola x²/144 - y²/81 = 1/25 coincide. then the value of b² is
a) 9 b) 7 c) 5 d) 1
34) The distance from the major Axis of any point on the ellipse x²/a² + y²/b² = 1 and the distance of its corresponding point on the oxiliary circle are in the ratio
a) b/a b) a/b c) a²/b² d) b²/a²
35) For the ellipse 24x² + 9y² - 150x - 90y + 225 = 0, eccentricity is equals to
a) 2/5 b) 3/5 c) 4/5 d) 1/5
36) What is the difference of the focal distances of any point on a hyperbola ?
a) eccentricity
b) length of transverse axis
c) distance between the foci
d) length of semi-transverse axis
37) Equation of the circle passing through the intersection of ellipse x²/a² + y²/b² = 1 and x²/b² + y²/a² = 1 is
a) x² + y² = a²
b) x² + y² = b²
c) x² + y² = a²b²/(a²+ b²)
d) x² + y² = 2a²b²/(a²+ b²)
38) The focal distance of the point 't' on the Parabola y²= 4ax is
a) at² b) a(1+ t²) c) a(t + 1/t)² d) a/t²
39) A circle touches the x-axis and also touches the circle with centre at 0,3) and radius 2. Then the locus of the centre of the circle is
a) a parabola b) a hyperabola c) an ellipse d) a circle
40) Let P be the point (1,0) and Q a point on the Parabola y²= 8x; then the locus of midpoint of PQ is
a) x²+ 4y +2=0
b) x²+- 4y +2=0
c) y²- 4x +2=0
d) y² + 4x +2=0
BOOSTER - 2
1) Assuming that the sums and the products given below defined , which of the following is not true for matrices ?
a) AB= AC does not imply B= C
b) A+ B = B+ A
c) (AB)'= B' A'
d) AB= O implies A= O or, B= O
2) The sum of the cofficients in the expansion of (1+ x - 3x²)¹⁰⁰ is
a) 100 b) -100 c) 1 d) -1
3) A fair dice thrown till we get 6; then the probability of getting 6 exactly in even number of turns is
a) 11/36 b) 5/11 c) 6/11 d) 1/6
4) 10ⁿ+ 3.4ⁿ⁺²+5 is always divisible by (for all n∈N(
a) 9 b) 7 c) 5 d) 17
5) The value of 2¹⁾⁴. 4¹⁾⁸. 8¹⁾¹⁶......∞is
a) 1 b) 3/2 c) 2 d) 4
6) l, m, n are p-th, q-th and r-th terms of a GP, all positive, then the value of
Log l p 1
Logm q 1
Log n r 1 is
a) -1 b) 2 c) 1 d) 0
7) The value of 2/3! + 4/5!+ 6/7! + ....∞ is
a) e b) 1/e c) 2e d) e²
8) A and B are two events such that P(A U B)= 3/4, P(A ∩B)= 1/4, P(A's)= 2/3; then the value of P(A'∩ B) is
a) 5/12 b) 3/8 c) 5/8 d) 1/4
9) One root of the equation
x+ a b c
b x+ c a =0
c a x+ c is
a) -(a +b) b) -(b +c) c) -a d) -(a +b + c)
10) The first three terms in the expansion of (1+ ax)ⁿ are 1, 6x and 16x²; then the values of a and n are
a) 2,9 b) 2/3,9 c) 2,3 d) 3/2,6
11) If ω is a cube root of unity then the value of
1 ω ω²
ω ω² 1
ω² 1 ω is
a) 1 b) ω c) 0 d) ω²
12) How many terms are there in the expansion of (4x + 7y)¹⁰ + (4x - 7y)¹⁰ ?
a) 6 b) 5 c) 11 d) 22
13) If A and B are two events such that P(AUB)= 5/6, P(A∩B)= 1/3, then which one of the following is not correct ?
a) A and B are not independent
b) A and B' are independent
c) A's and B are Independent
d) A and B are independent
14) If A= 1 0 2
-1 1 -2
0 2 1 and
Adj A= 5 a -2
1 1 0
-2 -2 b , then the value of a and b are
a) -4, 1 b) -4,-1 c) 4, 1 d) 4, -1
15) The value of the infinite series (x - y)/x + 1/2)(x - y)/x)²+ (1/3) (x - y)/x)³+....∞ is
a) logₑ(y/x) b) logₑ(x/y) c) 2logₑ(x/y) d) (1/2) logₑ(y/x)
16) The coefficient of x³ in the expension (2- 3x)/(1+ x)³ is
a) 2 b) -2 c) 38 d) -38
17) If a= 1+2+4+......to n terms , b= 1+3+9+......to n terms and c= 1+5+25+....to n terms, then the value of
a 2b 4c
2 2 2
2ⁿ 3ⁿ 5ⁿ is
a) 30ⁿ b) 10ⁿ c) 0 d) 2ⁿ+ 3ⁿ+ 5ⁿ
18) The value of 1²/1! + 2²/2! + 3²/3! +......∞is
a) 2e b) 2e+1 c) 2e -1 d) 2(e -1)
19) The value of fourth term in the expansion of 1/³√(1- 3x²) is
a) -40x³/3 b) 40x³/3 c) 20x³/3 d) -20x³/3
20) A coin and a six faced die, both unbiased, are thrown simultaneously. The probability of getting a head on the coin and an odd number on the die is
a) 1/2 b) 3/4 c) 1/4 d) 2/3
21) A number is chosen at random among the first 120 natural numbers . What is the probability that the number chosen being a multiple of 5 or 15 ?
a) 1/5 b) 1/8 c) 1/15 d) 1/6
22) If A= -1 0
0 2 then the value of A³- A² is equal to
a) I b) A c) 2A d) 2I
23) If ω, ω² are the cube root of unity then the value of m for which the matrix
1 ω m
ω m 1
m 1 ω is singular is
a) 1 b) -1 c) ω d) ω²
24) If A= x y
z t then the transpose of Adj A is
a) t z b) t y c) t -z d) none
-y -x -z -x y -x
25) A dice is thrown. If it shows a six, we draw a ball from a bag containing 2 black balls and 6 white balls. if it does not show a six then we toss a coin. Then the number of event points in the sample space of this experiment is
a) 18 b) 14 c) 12 d) 10
26) The solutions of the equation
x 2 -1
2 5 x = 0
-1 2 x
a) -3,1 b) 3,-1 c) 3,1 d) -3,-1
27) The sum of the infinite series (1+ 3/2! + 7/3! + 15/4!+.... ∞) is
a) e(e -1) b) e(e +1) c) e(1- e) d) 3e
28) If A is a square matrix of the order 3x 3 and λ is a scalar, then adj(λA) is equals to
a) λ adjA b) λ² adjA c) λ³ adjA d) λ⁴ adjA
29) The equation of the parabola whose focus is (5,3) and directrix is 3x - 4y +1=0, is
a) (4x +3y)²- 256x - 142y + 849=0.
b) (4x -3y)²- 256x - 142y + 849=0.
c) (3x +4y)²- 142x - 256y + 849=0.
d) (3x -4y)²- 256x - 142y + 849=0.
30) The eccentricity of the conic 9x²+ 25y²= 225 is
a) 2/5 b) 4/5 c) 3/5 d) 3/4
31) The locus of the point P(x,y) satisfying the relation √{(x -3)²+ (y -1)²} + √{(x +3)²+ (y -1)²}= 6 is
a) a straight line
b) a hyperbola
c) a circle
d) an ellipse
32) The locus of the mid-point of the line segment joining the focus to a moving point on the Parabola y²= 4ax is another parabola with directrix
a) x = - a b) 2x = - a c) x = 0 d) 2x = a
33) If x₁, x₂, x₃ and y₁, y₂, y₃ are both in GP with the same common ratio, then the points (x₁, y₁),(x₂, y₂),(x₃, y₃) are
a) vertices of a triangle
b) on a circle
c) collinear
d) on an ellipse
34) The eccentricity of the hyperbola 25x²- 9y²= 144 is
a) √34/4 b)√34/3 c) 6/√34 d) 9/√34
35) The curve represented by the equation 4x²+ 16y²- 24x - 32y - 12=0 is
a) an ellipse with eccentricity 1/2
b) an ellipse with eccentricity √3/2
c) a hyperbola with eccentricity 2
d) a hyperabola with eccentricity 3/2
36) The equation of the Parabola with vertex at the origin and directrix is y= 2, is
a) y²= -8x b) y²= 8x c) x²= 8y d) x²= - 8y
37) If (0,6) and (0,3) are respectively the vertex and focus of a parabola, then its equation is
a) x²- 12y = 72 b) y²- 12x = 72 c) x² +12y = 72 d) y²+ 12x = 72
38) The equation of the director circle of the hyperbola x²- y/4 = 1 is
a) x² + y²= 16 b) x² + y²= 4 c) x² + y²= 20 d) x² + y²= 12
39) An equilateral triangle is inscribed in the parabola y²= x whose one vertex is the vertex of the parabola. Then the length of a side of the triangle is
a) √3 units b) 8 units c) 2√3 units d) 1/2 units
Raw-2
1) Let S={2⁰, 2¹, 2²,....,2¹⁰}. Consider all possible positive differences of elements of S. If M is the sum of all these differences, then the sum of the digits of M is
a) 24 b) 27 c) 30 d) 31
2) A fair coin is tossed 12 times. The probability of getting atleast 8 cosecutive heads is
a) 3/2⁷ b) 5/2⁷ c) 5/2⁸ d) 3/2⁸
3) If 1/1.2.3 + 1/2.3.4 + 1/3.4.5 + .... + 1/10.11.12 = m/n, reduced fraction, then the sum of the digits of (m + n) is
a) 13 b) 14 c) 15 d) 16
4) If α, β are the roots of x²+ ax - 1/2a² = 0, a≠ 0, then the minimum value of α⁴+β⁴ is
a) 2 b) 3+√2 c) 2+√2 d) 4-√2
5) If 0< c < 1 and ∫ sin⁻¹(c cosx) dx at (π/2,0)= c/a₁ + c³/a₂ + c⁵/a₃+...... then (a₁ + a₂ + a₃) is dividible by
a) 3 b) 5 c) 7 d) 11
6) Let A(x₁, y₁), B(x₂, y₂) and (x₃,y₃) be the vertices of a ∆ ABC. A parallelogram AFDE is drawn with D, E and F on the line segment BC, CA and AB respectively. Then, maximum area of such a parallelogram is
a) (1/2) (area of ∆ ABC)
b) (1/4) (area of ∆ ABC)
c) (1/6) (area of ∆ ABC)
d) (1/8) (area of ∆ ABC)
Consider the line r= - j + k + β(-2i+ 2j + k), and the points C(1,2,3) and D(2,0,0)
7) The distance of the point D from the plane through the line and the point C is
a) √3 b) √(5/2) c) √7 d) √10
8) The distance of the point D from the image of the point C in the line is
a) √15 b) √20 c) √29 d) √34
9) The values 15x = 8y and 3x = 10y contain points P and Q respectively. If the midpoint of PQ is (8,6), then the length of PQ= m/n reduced fraction, where (m - 8n) is
10) Match the column
Let S={1,2,3,.....,10}
Column A
a) The number of subsets {x,y,z} of S so that x,y,z are in AP.
b) The number of subsets {x,y,z} of S so that no two of them are consecutive.
c)The number of subsets {x,y} of S so that x³+ y³ is divisible by 3.
d) The number of subsets {x,y} of S so that x²- y² is divisible by 3.
Column B
i) 15
ii) 20
iii) 24
iv) 30
v) 56
BOOSTER - 1
1) Sum of infinite number of terms of GP is 20 and sum of their square is 100; then the common ratio of the GP is
a) 5 b) 3/5 c) c) 2/5 d) 1/5
2) If xₙ = cos(π/2ⁿ)+ t sin(π/2ⁿ), then the value of (x₁, x₃, x₅....∞)+ 1/(x₂x₄x₈...∞) is
a) 1 b) -1 c) 2 d) 0
3) If r> 1, n > 2 are positive integers in the co-efficients of (r+2)th and 3rd terms in the expansion of (1+ x)²ⁿ are equal, then n equals to
a) 3r b)!3r+1 c) 2r d) 2r+1
4) If a> 0 and discriminant of ax²+ 2bx + c=0 is negative, then value of
a b ax+ b
b c bx +c
ax+ b bx+c 0 is
a) positive
b) (ac - b²)(ax²+ bx + c)
c) negative d) 0
5) A problem in mathematics is given to three students A, B and C and their respective probability of solving the problems 1/2, 1/3 and 1/4. Then the probability that problem is solved is
a) 3/4 b) 1/2 c) 2/3 d) 7/8
6) The probability that a leap year will have 53 Tuesday or Saturday is
a) 2/7 b) 3/7 c) 4/7 d) 1/7
7) If y= x - x²+ x³- x⁴+....∞, then the value of x will be (-1< x <1)...
a) y+ 1/y b) y/1+y) c) y- 1/y d) y/(1-y)
8) The value of the determinant
1+ a 1 1
1 1+b 1
1 1 1+c is
a) 1+ abc + ab+ ca
b) abc(1+ 1/a + 1/b+ 1/c)
c) 4abc
d) abc(1/a+ 1/b+ 1/c)
9) If A= 2 -1
-1 2 and I is the unit matrix of order 2, then A² is equal
a) 4A - 3I b) 3A - 4I c) A - I d) A + I
10) Let n> 5 and b≠ 0, if in the binomial expansion of (a - b)ⁿ, the sum of 5th and 6th terms is zero, then the value of a/b.
a) 5/(n -4) b) 1/5(n -4) c) (n -5)/6 d) (n -4)/5
11) If P(A)= 2/3, P(B)= 1/2 and P(AUB)= 5/6, then the event A and B are
a) mutually exclusive
b) independent as well as mutually exclusive
c) independent d) none
12) The roots of the equation
x 3 7
2 x -2
7 8 x
a) -2,-7,5 b)- 2,-5, 7 c) 2, 5, -7 d) 2 5 7
13) If f(x)= sinx cosx tanx
x³ x² x
2x 1 1
Then the value of lim ₓ→₀ f(x)/x¹ is
a) -3 b) 3 c) -1 d) 1
14) If n be any integers, then n(nm+1)(2n +1) is
a) an odd number
b) divisible by 6
c) a perfect square
d) none
15) The sum of the infinite series 1/2! - 1/3! + 1/4! - ...... ∞is
a) e b) e¹⁾² c) e⁻² d) none
16) The multiplicative inverse of matrix
2 1
7 4 is
a) 4 -1 b) 4 -1 c) 4 -7 d) -4 -1
-7 -2 -7 2 7 2 7 -2
17) The probability that atleast one of the events A and B occurs is 3/5. If A and B occur simultaneously with probability 1/5, then the value of P(A's)+ P(B') is
a) 2/5 b) 4/5 c) 6/5 d) 7/5
18) If 0< y < 2¹⁾³ and x(y³-1)= 1 then the value of (2/x + 2/3x³+ 2/5x⁵+ ....∞) is
a) logₑ{y³/(2- y³)}
b) logₑ{y³/(1- y³)}
c) logₑ{2y³/(1- y³)}
d) logₑ{y³/(1- 2y³)}
19) For a real number α, let A(α) denote the Matrix
cosα sinα
-sinα cosα.
then for real numbers α₁ and α₂, the value of A( α₁) A(α₂) is
a) A(α₁,α₂) b) A(α₁+α₂) c) A(α₁ - α₂) d) A(α₂ -α₁)
20) If the system of equations x + 2y + 3z =1, 2x + ky + 5z =1, 3x + 4y + 7z =1 has no solutions, then
a) -1 b) 1 c) 3 d) 2
21) The probability that the same number appears on throwing three dice simultaneously is
a) 1/6 b) 1/36 c) 5/36 d) none
22) A is a square matrix such that A³= I; then A⁻¹ is equal
a) A² b) A c) A³ d) none
23) If D= 1 a a²-bc
1 b b²- ca
1 c c²- ab
Then D is
a) 0 b) independent of a c) independent of b d) independent of c
24) If |x|< 1/2, then the coefficient of xʳ in the expansion of (1+ 2x)/(1- 2x)² is
a) r. 2ʳ b) (2r -1)2ʳ c) r.w²ʳ⁺¹ d) (2r +1)2ʳ
25) The value of the infinite series (1+ 1/(3.2²) + 1/(5.2⁴) + 1/7.2⁶)+.....∞) is
a) logₑ3 b) (1/2) logₑ3 c) 1- logₑ3 d) 2 logₑ3
26) If the n-th term of an infinite series is n(n +4)/n! , then the sum of infinite terms of the series is
a) 6e+1 b) 6e c) 5e d) 6e -1
27) If in the expansion of (1+ x)ᵐ(1+ x)ⁿ the co-efficients of x and x² are 3 and (-6) respectively, then the value of n is
a) 7 b) 8 c) 9 d) 10
28) If y x 0
0 y x= 0
x 0 y
and x ≠ 0, then which one of the following is correct ?
a) x is one of the cube roots of 1
b) y is one of the cube root of 1
c) y/x is one of the cube root of 1
d) y/x is one of the cube root of (-1(
29) The Locus of a point whose difference of distances from point (3,0) and (-3,0) is 4, is
a) x²/4 - y²/5= 1
b) x²/5 - y²/4 = 1
c) x²/2 - y²/3 = 1
d) x²/3 - y²/2 = 1
30) If the equation of latus rectum of a Parabola is x+ y -8=0 and the equation of the tangent at the vertex is x + y -12=0, then the length of the latus rectum is
a) 4√2 b) 2√2 c) 8 d) 8√2
31) If B and B' are the ends of minor axis and S and S' are the foci of the ellipse x²/25 + y²/9 = 1, then the area of the rhombus SBS'B' formed will be
a) 12 square units
b) 48 square units
c) 24 square units
d) 36 square units
32) The length of the axis of the conic 9x²+ 4y²- 6x + 4y +1=0 are
a) 1/2, 9 b) 1, 2/3 c) 2/3,1 d) 3,2
33) If the angle between the lines joining the end points of minor axis of an ellipse with its one focus is π/2, then the eccentricity of the ellipse is
a) 1/√2 b) 1/2 c) √3/2 d) 1/2√2
34) Which one of the following is independent of α in the hyperbola accentu (0<α <π/2) x²/cos²α - y²/sin² α = 1 ?
a) eccentricity b) abscissa of a focus c) directrix d) vertex
35) if the distance of a point in the ellipse x²/9 + y²/4= 1 from its centre is 2, then the eccentric angle of the point is
a) π/4 b) π/2 c) 3π/4 d) π/3
36) The focus of the curve y²+ 4x - 6y +13=0 is at
a) (2,3) b) (2,-3) c) (-2,3) d) (-2,-3)
37) The distance between the directrices of the hyperbola x= 8secθ, y= 8 tan θ is
a) 8√2 b) 16√2 c) 4√2 d) 6√2
38) If a focal of the parabola y²= ax is 2x - y -8=0, then the equation of the directrix is
a) x -4=0 b) x + 4=0 c) y -4=0 d) y + 4=0
39) If a≠ 0 and the line 2bx + 3cy + 4d=0 passes through the points of intersection of the parabolas y²= 4ax and x²= 4ay, then
a) d²+ (2b - 3c)²= 0
b) d²+ (3b + 2c)²= 0
c) d²+ (2b + 3c)²= 0
d) d²+ (3b - 2c)²= 0
40) The eccentricity of an ellipse, with its centre at the origin, is 1/2. If one of the directrices is x= 4, then the equation of the ellipse is
a) 4x²+ 3y²=12
b) 3x²+ 4y²=1
c) 4x²+ 3y²=1
d) 3x²+ 4y²=12
BOOSTER - 2
1) Let f(x)= cosx, if x ≥ 0
x +k, if x< 0
Find the value of constant k, given that lim ₓ→₀f(x) exists.
2) Evaluate lim ₓ→₀(sinx°)/x.
3) Differentiate sinx³ w.r.t.x.
4) dy/dx 5(2³ˡᵒᵍ₂ˣ) with respect to x.
5) Which of the following is a statement (or proposition)? Give reasons for your answer.
a) x²+ 5x +6=0
b) There is no rain without clouds.
6) lim ₓ→∞ √(x²+1)/x.
7) Write down the negation of each of the following statements.
a) Deepak is smart and healthy
b) 2+ 4> 5 or 3+ 4< 6.
c) x = 2 => x²= 4.
d) ∆ ABC is isosceles, if and only if angle B= angle C.
8) Evaluate: lim ₓ→₂ (x⁵-32)/(x³-8).
9) Find dy/dx when y= (x² sinx)/(1- x).
10) If y= √{(1- cos2x)/(1+ cos2x)}, x ∈ +0, π/2) U (π/2,0), then find dy/dx.
11) Evaluate:
lim ₓ→₀ (8/x){(1- cos(x²/2) - cos(x²/4) + cos(x²/2) cos(x²/4)}.
12) Show:
lim ₓ→π/4 (tan³x - tanx)/cos(x +π/4) = -4
13) If y= {√(x + a) - √(x -a)}²/√(x²- a²) find dy/dx where x > a > 0.
14) Find d/dx (sinx)/x.
15) Show that the statement
p: if x is a real number such that x³+ 4x =0, then x is "0" is true by
i) direct method
ii) method of contradiction
iii) method of contrapositive
BOOSTER- 3
1) The area (in square units) of the region bounded by the curve x²= 4y, the line x = 2 and x-axis is
a) 1 b) 2/3 c) 4/3 d) 8/3
2) Let P(a secθ, b tanθ) and Q(a secβ, tanβ) where θ+ β = π/2, be two points on the hyperbola x¹/a² - y²/b²= 1. If (h,k) be the point of intersection of the normal at P and Q, then the value of k is
a) (a²+ b²)/a b) - (a²+ b²)/a c) (a²+ b²)/b d) - (a²+ b²)/a
3) The equation of the tangent to the curve (1+ x²)y = 2- x where it crosses the x-axis is
a) x+ 5y =2 b) x- 5y =2 c) 5x - y =2 d) 5x + y =2
4) The area (in square unit) bounded by the parabolas y²= 4ax and x⅖= 4ay is
a) 64a²/3 b) 32a²/3 c) 16a²/3 d) 8a²/3
5) Equation of the tangent and the normal drawn at the point (6,0) on the ellipse x²/36 + y²/9 = 1 respectively are
a) x=6, y = 0
b) x + y=6, y - x +6 = 0
c) x=0, y = 3
d) x= -6, y = 0
6) The area( in square units) of the figure bounded by the curve y= cosx and y = sin x and the ordinates x= 0, x=π/4 is
a) √2+ 1 b) √2- 1 c) 1/√2 d) (√2- 1)/√2
7) The straight line x + y = a will be a tangent to the ellipse x²/9 + y²/16 = 1 if the value of a is
a) 8 b) ±10 c) ±5 d) ±6
8) The equation of the tangent to the parabola y²= 8x which is perpendicular to the line x - 3y +8=0, is
a) 3x + y +2=0 b) 3x - y -1=0 c) 9x - 3 y +2=0 d) 9x + 3 y +2=0
9) The area (in square unit) bounded by the parabola y²= 8x and its latuce rectum is
a) 16/3 b) 25/3 c) 16√2/3 d) 32/3
10) if the curve y²= 4x and xy = k cut orthogonally, then the value of k² will be
a) 16 b) 32 c) 36 d) 8
11) The area (in square units) bounded by the curve -3y²= x -9 and the lines x = 0, y= 0 and y = 1 is
a) 8/3 b) 3/8 c) 8 d) 3
12) If the slope of the normal to the curve x³= 8a²y at P is (-2/3), then the co-ordinates of P are
a) (2a,a) b) (a,a) c) (2a, - a) d) none
13) if a > 2b > 0, then the positive value of m for which the line y= mx - b √(1+ m²) is a common tangent to the circle x² + y²= b² and (x - a)²+ y²= b² is
a) 2b/√(a²- 4b²)
b) √(a²- 4b²)/2b
c) 2b/(a - 2b)
d) b/(a - 2b)
14) The area(in square unit) of the region bounded by the line y= |x -1| and y= 3 - |x| is
a) 6 b) 2 c) 4 d) 3
15) The minimum value of f(x)= x²+ 250/x is
a) 55 b) 25 c) 50 d) 75
16) If f(x)= kx³- 9x²+ 9x +3 is an increasing function then
a) k < 3 b) k≤ 3 c) k > 3 d) k is indeterminate
17) If f(x)= 1/(4x²+ 2x +1), then its maximum value is
a) 2/3 b) 4/3 c) 3/4 d) 1
18) If f(x)= 1/(x +1) - log(1+ x), x > 0, then f(x) is
a) a decreasing function
b) an increasing function
c) neither increasing nor decreasing
d) increasing when x > 1
19) let α, β be the roots of x²+ (3 - λ)x - λ=0, then the value of λ for Which α²+ β² is minimum, is
a) 0 b) 1 c) 3 d) 2
20) The function f(x)= 2x³- 3x⅖- 12x +4 has
a) no maximum and minimum
b) one maximum and one minimum
c) 2 maximum
d) 2 minimum
21) The height of the cylinder of maximum volume that can be inscribed in a sphere of radius a, is
a) 3a/2 b) √2a/3 c) 2a/√3 d) a/√3
22) Maximum value of (logx)/x in [0,∞) is
a) (log 2)/2 b) 0 c) 1/e d) e
23) Let the function f: R---> R be defined f(x)= 2c + cos x, then f(x)=
a) has maximum value at x = 0
b) has minimum value at x =π
c) is a decreasing function
d) is an increasing function
24) The maximum distance from the origin of a point on the curve x = a sin t - b sin(at/b), y= a cos t - b cos(at/b), both a,b > 0, is
a) a - b b) a+ b c) √(a²+ b²) d) √(a²- b²)
25)
26) If the slope of the tangent at (x,y) to a curve passing through the point (2,1) is (x²+ y²)/2xy, then the equation of the curve is
a) 2(x²- y²)= 3x
b) 2(x²- y²)= 3y
c) x(x²- y²)= 6
d) x(x² + y²)= 6
27)
28)
29) If y= 3x²+ 2 and if x changes from 10 to 10.1, then the approximate change in y will be
a) 8 b) 6 c) 5 d) 4
30) The rate of change of surface area of a sphere of radius r when the radius is increasing at the rate of 2 cm/s is proportional to
a) 1/r² b) r² c) r d) 1/r
BOOSTER - 4
If the curve y= aˣ and y= bˣ intersect at an angle α, then the value of tanα is
a) (a - b)/(1+ ab)
b) (loga - logb)/(1+ loga logb)
c) (a + b)/(1 - ab)
d) (loga + logb)/(1 - loga logb)
2) If the straight line y= 4x - 5 touches the curve y²= px³+ q at (2,3), then the values of p and q are
a) p=2, q= -7 b) p=2, q= 7 c) p= -2, q= -7 d) p= -2, q= 7
3) The area (in square unit) of the figure bounded by y²= 12x, x = 0 and y = 6 is
a) 12 b) 16 c) 3 d) 6
4) The area (in square unit) of the region bounded by the curves y= x³ and y= 2x² is
a) 2/3 b) 3/4 c) 4/3 d) 1/3
5) The ratio of the area bounded by the curves y= cos x and y= cos2x between x = 0, x =π/3 and x-axis is
a) √2 : 1 b) 1 : 1 c) 1: 2 d) 2 : 1
6) The equation of the normal to the parabola y²= 4ax at the point (at², 2at) is
a) tx + y = 2at + at³
b) x + ty = 2at + at³
c) tx - y = at + 2at³
d) x - ty = at + 2at³
7) If the slop of the normal to the parabola 3y²+ 4y +2=x at a point on it is 8, then the co-ordinate of the point are
a) (1,-1) b) (6,-2) c) (9,1) d) ( 2,0)
8) if the line lx + my + n =0 is a tangent to the parabola y²= 4ax, then
a) an²= ml b) al²= mn c) am²= nl d) a¹m = nl
9) The area (in square unit ) in the first quadrant between y²= 4x, y²= 16x and the straight line x= 9 is
a) 36 b) 24 c) 18 d) 9
10) The equation of tangents to the hyperbola 3x²- 4y²= 12 which are inclined at an angle 60° to the x-axis are
a) y= √3 x ± 12
b) y= √3 x ± 10
c) y= √3 x ± 15
d) y= √3 x ± 3
11) The equation of tangent to the curve xy²= 4(4 - x) where it meets the line y= x is
a) x+ y +4=0
b) x+ y -4=0
c) x - y - 2=0
d) x+ y +2=0
12) The normal to the curve x = 3 cosθ - cos³θ, y= 3 sinθ - sin³ θ at θ=π/4
a) is at distance of two units from the origin
b) is a distance of 4 units from the origin.
c) passes through the origin
d) passes through the point (2,3)
13) The area( in square unit) bounded by the parabola y= x²- 6x +10 and the straight lines x=6 and y= 2 is
a) 20/3 b) 16/3 c) 8 d) 32/3
14) The point of the curve x²+ 2y = 10 at which the tangent to the curve is perpendicular to the line 2x - 4y = 7, is
a) (2,3) b) (-2,3) c) (4,-3) d) (-4,-3)
15) let x and y be two variables and x >0, xy= 1, then the minimum value of x + y is
a) 1 b) 5/2 c) 10/3 d) 2
16) The function f(x)= 1- x³- x⁵ is decreasing for
a) 1≤ x ≤5 b) all real values of x c) x≤ 3 d) x ≥ 5
17) The function y= a(1- cosx) is maximum when x is
a) π/2 b) -π/2 c) π d) π/3
18) Let f(x)= x³+ 6x² + px +2; if the largest possible interval in which f(x) is decreasing function is (-3,-1), then the value of p is
a) 3 b) 9 c) -2 d) none
19) In -4< x <4, the function f(x)= ˣ₋₁₀∫ (t⁴ -4)e⁻⁴ᵗ dt has
a) no extrema b) eno extremum c) two extrema d) 4 extrema
20) If a₁, a₂, a₃,.....aₙ are n positive real numbers whose product is a fixed number c, then the minimum value of a₁ + a₂+....aₙ₋₁ + 2aₙ is
a) n(2c)¹⁾ⁿ b) (n +1)c¹⁾ⁿ c) 2nc¹⁾ⁿ d) (n +1)(2c)¹⁾ⁿ
21) The length of the longest interval in which the function 3sinx - 4 sin³x is increasing, is
a) π/2 b) π c) 3π/2 d) π/3
22) The real number x when added to its inverse gives the minimum value the sum at x =
a) -2 b) 2 c) 1 d) -1
23) If minimum value of f(x)= x²+ 2bx + 2c² is greater than maximum value of g(x)= - x²- 2cx + b², then for real value of x
a) √2|c|> |b| b) |c|> √2 |b| c) 0< c <2b d) none
24) Let f(x)= x³+ bx²+ cx+ d, 0< b² < c. Then f(x)
a) has a local maximum
b) has a local minimum
c) is strictly decreasing
d) is strictly increasing
25) If v= 4πr³/3, then and the rate (in cubic unit) at which v is increasing when r= 10 and dr/dt= 0.01, is
a) 4π b) π c) 40π d) 4π/3
26) if the time rate of change of the radius of a sphere is 1/2π, then the rate of change of its surface area (in square cm), when the radius is 5cm is
a) 20 b) 10 c) 4 d) 5
27)
28)
29)
30)