1) A die is rolled 4 times and the outcomes are listed. The probability that the list contains only 3 different numbers is
a) 7/9 b) 5/9 c) 4/9 d) 7/18
2) Let a and b be integers, 0< b < a. The image of the point A(a,b) in the line y= x is B, the image of B in the y-axis is C, the image of C in x-axis in D and the image of D in the y-axis is E. If the area of the Pentagon is 451, then a- b=
a) 1 b) 2 c) 3 d) 5
3) ABCD is a square of side 10. EAB is an isosceles triangle with base AB. If the area common to the triangle EAB and the square ABCD is 80, the area of ∆ EAB is
a) 175 b) 100 c) 125 d) 150
4) The area common to the curve x²+ 3y²= 3 and 3x²+ y²= 3 is
a) π/√3 b) 2π/√3 c) √3π d) √3π/2
5) 4 cos²x sinx - 2 sin²x = 3 sinx if x=
a) nπ b) (4n -1)π/2 ± π/3 c) nπ+ (-1)ⁿπ/10 d) nπ/2
COMPREHENSION
Consider 5 digit numbers formed with the digit 0,1,2,3,4,5 without repetition.
6) The number of even numbers is
a) 108 b) 144 c) 216 d) 312
7) The number of numbers divisible by 3 is
a) 108 b) 144 c) 216 d) 240
8) The number of numbers divisible by 4 is
a) 108 b) 120 c) 144 d) 160
MATRIX MATCH
9) Match the following columns
COLUMN I
a) f(x)= xᵅ sin(1/x), x≠ 0,
f(0)= 0, then f'(x) is continuous for α.
b) If the curve y= ax²+ bx + c passes through the point (1,2) and is tangent to the line x= y at the origin, then y(-1)=
c) If f(x)= sinx + ∫ sinx cost dt at (π/2,0),
Then ∫f(x) dx at (π/2,0)=
d) The area bounded by the common tangents to the curve x² = 2(y² +1) and y² = 2(x² +1) is
COLUMN II
p) 0
q) 1
r) 2
s) 3
t) 4
INTEGER MATCH
10) In a triangle ABC, the points D, E, F lie on the sides BC, CA, AB respectively. If the lines AD, BE, CF are concurrent at O. And AO/OD + BO/OE + CO/OF = 7, then the value of AO/OD. BO/OE . CO/OF is
CBSE - XII
Section -A (1x10= 10)
1) If A, B, C are three non-zero square matrices of same order, find the condition on A such that AB= AC=> B = C.
2) If a= i + j; b = j + k; c= k + i, find a unit vector in the direction of a + b + c.
3) Find a, for which f(x)= a(x + sinx)+ a is increasing for all x ∈ R.
4) Evaluate ¹·⁵₀∫ [x] dx, (where [.] is a greatest integer function).
5) If A is a square matrix of order 3 such that |adjacent A|= 64. Find |A|.
6) Find the principal value of tan⁻¹{tan(4π/3)}.
7) R a relation on set A={2,4,7} is given by
R={(2,2),(2,4),(4,4),(7,7)}. Can you say that relation R is reflexive? Give reasons.
8) Find the angle between vectors a and b with magnitude √3 and 2 respectively and having a.b =√6.
9) Find the distance of the point P(2,1,-1) from the plane x - 2y + 4z = 9.
10) For what value of k, the matrix
2. k
3 4 has no inverse.
Section - B (4 marks each)
Section - B (4 marks each)
11) Let the relation R={(x,y)}∈ W x W: y= 2x - 4}. If (a,2) and (4 b²) belong to relation R, find the values of a and b.
OR
Let '*' be a binary operation defined on Q, given by a * b = ab/5 for a, b ∈ Q. is '*' commutative ? Is '*' associative ?
12) Prove that sec²(tan⁻¹2)+ cosec²(cot⁻¹3)= 15.
13) For what value of k, is the function
f(x)= { (1- vos4x)/8x², x≠ 0
k, x= 0
continuous at x= 0 ?
OR
Differentiate (sinx + x²)/cos2x with respect to x.
14) Using properties of determinants , show that
| a b ax + by
b c bx + cy
ax + by bx + cy 0
= (b²- ac)(ax²+ 2bxy + cy²).
15) Evaluate : (x²+1)/(x⁴+ x²+1) dx.
OR
Evaluate : ∫ {(sin⁻¹√x - cos⁻¹√x)/(sin⁻¹√x + cos⁻¹√x)} dx.
16) The edge of a variable cube is increasing at the rate of 3 cm/sec. How fast is the volume of the cube increasing when the edge is 10cm long ?
17) Solve the differential equation:
√(1+ x²+ y²+ x²y²) + dy/dx = 0.
18) Solve the differential equation dy/dx + 2y/x = 1/x² satisfying the condition y(2)= -1.
19) Let a= i + 4j + 2k, b= 3i -2j + 7k and c= 2i - j + 4k. Find a vector which is perpendicular to both a and b and c.d= 15.
20) Find the equation of the points (2,2,1) and (9,3,6) and perpendicular to the line 2x + 6y + 6z = 1.
21) If x= θsinθ + cosθ and y= θ cosθ - sinθ, find d²y/dx².
22) An urn contains 10 white and 3 black balls. Another urn contains 3 white and 5 black balls . 2 balls are drawn from first without noticing their colour and put into the second urn and then a ball is drawn from the second urn. Find the probability that it is a white ball .
SECTION- C(6 Marks each)
23) Show that semi-vertical angle of a right circular cone of given surface area and maximum volume is sin⁻¹(1/3).
OR
A window is in the shape of a rectangle surmounted by a semi-circle. If its perimeter is 30m. Then find the dimensions of the window so that it may admit maximum light.
24) Find the area bounded by the curve y= 6x - x² and y= x² - 2x.
25) Evaluate ∫ x/(1+ sinx) dx at (π,0).
Using properties of definite integrals Evaluate ³₁∫ (2x²+ 5) dx, as limit of sums.
26) The probability that a bulb produced in a factory will fuse after 150 days of use 0.05. Find the probability that out of 5 such bulbs
a) none
b) not more than one
c) more than one
d) at least one will fuse after 150 days.
27) Find the inverse of the matrix
A= 1 2 -2
-1 3 0
0 -2 1
28) Show that the lines (x -1)/2 = (y -2)/3 = (z -3)/4 and (x -4)/5= (y - 1)/2 = z intersect. Find their point of intersection.
29) If a young man rides his motorcycle at 25 km/hr, he had to spend Rs 2 per km on petrol . If he rides at a faster speed of 40 kmph . The petrol cost increases at Rs 5 per km. He had Rs 100 to spend on petrol and wishes to find what is the maximum distance he can travel in one hour. Express this is an LPP and solve it graphically.
REGIONAL MATHEMATICAL OLYMPIAD
1) ABCD be a triangle. Let D, E, F be points respectively on the segments BC, CA, AB such that AD , BE, CF concur at the point K. Suppose BD/DC = BF/FA and angle ADB= ang AFC. prove that angle ABE= angle CAD.
2) Let {a₁, a₂, a₃, ..., a₂₀₁₁} be a permutation (that is rearrangement) of the number 1,2,3,....2011. show that there exist two numbers j, k such that 1≤ j < k ≤ 2011 and |aⱼ - j|= |aₖ - k|.
3) A natural number n is choosen strictly between two consecutive perfect squares . The smaller of these two squares is obtained by subtracting k from n and the larger one is obtained by adding l to n. Show that n - kl is a perfect square.
3) A natural number n is choosen strictly between two consecutive perfect squares . The smaller of these two squares is obtained by subtracting k from n and the larger one is obtained by adding l to n. Show that n - kl is a perfect square.
4) Consider a 20-sided convex polygon K, with vertices A₁, A₂, A₃, ..., A₂₀ in that order. Find the number of ways in which three sides of K be chosen so that every pair among them has at least two sides of K between them. ( for example A₁A₂, A₄A₅, A₁₁A₁₂) is an admissible triple while (A₁A₂, A₄A₅, A₁₉A₂₀) is not.)
5) Let ABC be a triangle and BB₁, CC₁ be respectively the bisectors of Angle A, B with B₁ on AC and C₁ on AB. Let E, F be the feet of perpendiculars drawn from A on BB₁, CC₁ respectively . Suppose D is the point at which the incircle of ABC touches AB. Prove that AD= EF.
6) Find all pairs (x,y) of real numbers such that ₁₆x²+ y ₊ ₁₆x+ y²= 1.
IIT- JEE
1) f(x)= {|x|, if 0|x|≤ 2
1, if x= 0
has at x= 0
a) Local Maxima
b) local minima
c) tangent d) none
2) Rolle's theorem holds in [1,2] for the function f(x)= x³+ bx²+ cx at the point 4/3. The value of b, c are respectively
a) 8,-5 b) -5,8 c) 5 -8 d) -5,-8
3) Point on the line x - y = 3 which is nearest to the curve x²= 4y is
a) (0,3) b) (3,0) c) (2,-1) d) none
4) ∫ (sin²9x)/sinx dx at (π/2,0) =
a) 1 + 1/2 + 1/3 +...1/9
b) 1/2 + 1/4 +....+ 1/18
c) 1+ 1/3 + 1/5+.....+ 1/19
d) 1+ 1/3 + 1/5+....+ 1/17
5) ∫ dx/(1+ cosx) dx at (π,0)=
a) 0 b) π c) 1 d) does not exist
6) Let I₁ =∫ (x² √x)/(1+ x)⁶ dx at (∞,0) , I₂= ∫ (x √x)/(1+ x)⁶ at (∞,0), then
a) I₁ = 2I₂ b) I₂ = 2I₁ c) I₁ = I₂² d) I₁ = I₂.
7) sinx sin2x sin3x sin4x dx at (π/2,0)=
a) π/4 b) π/8 c) π/16 d) π/32
8) ³₀∫ dx/{(x -1)(x -2)}=
a) 0 b) log(1/4) c) 1 d) does not exist
9) ∫ dx/(sun⁶x + cos⁶x)= cot⁻¹(f(x))+ c then [f(π/6)]= (where [ . ] denotes G.I.F).
10) If I₁ = ¹₀∫ {x⁵/²(1- x)⁷/²}/12 dx, I₂= {x⁵/²(1- x)⁷/²}/(3+ x)⁸ dx then
a) I₂ = 144 √3 I₁
b) I₁ = 864 √3 I₂
c) I₁ = I₂.
d) I₁I₂ = 864 √3.
11) The area bounded by the curve f(x)= x + sinx and its inverse function between the ordinate x= 0 to x= 2π is
a) 4π square units
b) 8π square units
c) 4 square units
d) 8 square units
12) Area the region bounded by the curves y= x² and y= 2/(1+ x²) is
a) π+ 2/3 b) π- 2/3 c) π/2 + 1/3 d) 2π- 2/3
13) The solution of (1+ x²) dy/dx + y = ₑtan⁻¹x is
a) 2y ₑtan⁻¹x = ₑ2tan⁻¹x + c
b) 2y ₑ-tan⁻¹x = ₑ3tan⁻¹x + c
c) 2y ₑ-tan⁻¹x = ₑtan⁻¹x + c
d) y ₑ-tan⁻¹x = ₑ2tan⁻¹x + c
14) The degree of the differential equation satisfying √(1+ x²) + √(1+ y²)= A{x √(1+ y²) - y √(1+ x²)} is
a) 2 b) 3 c) 4 d) 1
15) Solution of the differential equation (x + 2y³) dy/dx = y is
a) x= y²(c + y²)
b) x= y(c - y²)
c) x= 2y(c - y²)
d) x= y(c + y²)
16) Solution of the differential equation
dy/dx = (3x²y⁴+ 2xy)/(x²- 2x³y³) is
a) x²y² + x²/y = c
b) x³y² + x²/y = c
c) x³y² + x²/y = c
d) x²y² + x²/y = c
17) The area of the smaller portion enclosed by the curves x²+ y²= 9 and y²= 8x is
a) √2/3 + 9π/4 - (9/2) sin⁻¹(1/3)
b) 2(√2/3 + 9π/4 - (9/2) sin⁻¹(1/3))
c) 2(√2/3 + 9π/4 + (9/2) sin⁻¹(1/3))
d) √2/3 + 9π/4 + (9/2) sin⁻¹(1/3)
18) The area of the loop of the curve y²= x⁴(x +2) is [in square units]
a) 32√2/105 b) 64√2/105 c) 128√2/105 d) 256√2/105
19) if the x intercept of any tangent to a curve is 3 times the x coordinate of the point of tangency, then the equation of the curve, given that it passes through (1,1) is
a) y= √x b) y= 1/x² c) y= 1/√x d) y= 1/√y³
20) The differential equation of all circle which pass through the origin and whose centres lie on y-axis.
a) (x²- y²) dy/dx - 2xy =0
b) (x²- y²) dy/dx + 2xy =0
c) (x²- y²) dy/dx - xy =0
d) (x²- y²) dy/dx + xy =0
SECTION II
Multiple Correct Answer W
21) Which of the following functions f: R --> R are are bijective ?
a) f(x)= x sinx
b) f(x)= x - sin²x
c) f(x)= x + √x²
d) f(x)= - x + cos²x.
22) The points on the ellipse 4x²+ 9y²= 13 such that the rate of decrease of ordinate is equal to rate of increase in abscissa are
a) (3/2,2/3) b) (-3/2,-2/3) c) (3/2, -2/3) d) (-3/2,2/3)
23) If I= ∫ x/(π²- cos²x) dx at (π,0) then
a) 0< I < 1 b) 0< I < 2 c) 2 < I < 3 d) I= π/2√(π²-1).
24) Let f(x)= (4+ 5 cosx)/(5+ 4 cosx) (x ∈(0,π/2)) and
∫ dx/(5+ 4 cosx)²= (5/27) g(x) - (4/27) h(x) + c then
a) g(x)= tan⁻¹(f(x))
b) h(x)= √(1+ f²(x))
c) g(x)= cos⁻¹(f(x))
d) h(x)= √(1- f²(x))
25) Area bounded by the curve y= logₑx, y = 0 and x= 3 is
a) (logₑ9 -2) square.unit
b) (logₑ27 + 2) square.unit
c) (logₑ+27/e²)) square.unit
d) greater than 1 square unit
26) The solution of the differential equation (y²dx - 2xydx)= x³y³dy + x²y⁴ dx is
a) log(x/y²)= (xy)²/2+ c
b) x/cy²= ₑ(xy)²/2
c) x/y²= 2 log(xy) + c
d) x/y²= ₑ2(xy) + c.
27) The function f(θ)= d/dθ ∫ᶿ₀ dx/{(1- cos2 θ)(1+ cos2x)} satisfies the differential equation
a) df/dθ + 4 f(θ) cot2θ= 0
b) df/dθ + 2 f(θ) tan2θ= 4 cosec³2θ
c) f(θ)= cosec²2θ
d) f(θ)= -(cosec²θ)(cot²2θ)
28) If the area enclosed by y²= 4ax and line y= ax is 1/3 sq.units, then the roots of the equation x²+ 2x = a are
a) -4 b) 2 c) -2 d) 8
29) Solution of the differential equation {x + y dy/dx}/{y - x dy/dx} = {x sin²(x²+ y²)}/y³ is
a) - cot(x²+ y²)= (x/y)²+ c.
b) y²/(x²+ y²c)= tan²(x²+ y²)
c) - cot(x²+ y²)= (y/x)²+ c.
d) (y²x²c)/x⅖= - tan²(x²+ y²)
30) A solution of the differential equation (dy/dx)² - dy/dx (eˣ + e⁻ˣ)+ 1= 0 are given by
a) y+ e⁻ˣ = c b) y- e⁻ˣ= c c) y+ eˣ = c d) y - eˣ = c
SECTION III
Comprehension Type
Paragraph for Question 31 to 33
f(x) = (tan⁻¹x)² + 2/√(1+ x²)
31) f increases in the region
a) (0, ∞) b) R c) (-∞,0) d) none
32) Maximum value of f is
a) π²+1 b) π²/4 c) 1 d) doesn't exist
33) Number of points on local extreme of f is
a) 0 b) 1 c) 2 d) none
Paragraph for Question 34 to 36
Two real valued differentiable function f(x) satisfy the following conditions:
i) f'(x)= {f(x) - g(x)}/3 ii) g'(x)= 2{g(x) - f(x)}/3 iii) f(0)=5 d) g(0)= 1
Then
34) ∫ 1/f(x) dx=
a) x - log(f(x))+ c
b) (1/2)( x - log(f(x))+ c
c) (1/3)( x - log(f(x))+ c
d) 1/( x - log(f(x))+ c
35) ∫ 1/g(x) dx=
a) x - log(g(x))+ c
b) (1/2)( x - log(g(x))+ c
c) (1/3)( x - log(g(x))+ c
d) 1/( x - log(g(x))+ ca
36) ∫ g(x)/f(x) dx=
a) x + 3log(f(x))+ c
b) ( x - 3log(f(x))+ c
c) 3log(f(x)) - x + c
d) 2( x - log(f(x))+ c
Paragraph for Question 37 to 39
An equation of the form dy/dx + P(x) y = Q(x), is called a linear differential equation of the first order. Its solution is given as ᵧₑ -∫ P dx ₌ Q. ₑ∫ P dx + k, k being the constant of integration.
A number of differential equations can be reduced to the linear differential equation and then one can solve them.
37) The general solution of the differential equation
dy/dx = eˣ⁻ʸ(eˣ - eʸ) is, (c being parameter)
a) eʸ = eˣ - 1 + ꜀ₑ-(e)ˣ.
b) eʸ = 1- eˣ + ꜀ₑ-(e)ˣ.
c) eˣ.= eʸ - 1 + ꜀ₑ-(e)ˣ.
d) eˣ = 1- eʸ + ꜀ₑ-(e)ˣ.
38) The general solution of the differential equation dy/dx + (y log y)/x = (y/x²) (log y)² is, (c being the parameter)
a) x = (1- cx²) log y
b) x = (1- 2cx²) log y
c) 2x = (1+ 2cx²) log y
d) x = (1 + cx²) log y
39) The general solution of the differential equation (y²+ e²ˣ) dy - y³ dx = 0, (c being the parameter), is
a) y²e⁻²ˣ + 2 log y = c
b) y²e⁻²ˣ - 2 log y = c
c) y²e⁻ˣ - 2 log y = c
d) y²e⁻ˣ + 2 log y = c
SECTION- IV
Matrix match Type
40) Match the following the columns :
COLUMN I
A) The value of a for which x³+ 3(a -7)x² + 3(a²-9)x -2 has a positive point of Maxima .
B) The minimum value of μ for which x³ - λx² + μx - 6 = 0 has real and positives roots .
C) The maximum value of f(x)= sinx + cos2x for x ∈ [0,2π]
D) The set of values of x for which log(1+ x)≤ x
Column II
p) 9/8
q) (-∞, -3) U (3, 29/7)
r) (3(6)²⁾³, ∞)
s) (-1,∞)
41) Match the following columns:
COLUMN I
(Curve)
A) y²= x & x²= y : x> 0
B) y²= 4x & xy= 16
C) y= sinx & y = cosx
D) x³- 3xy²+ 2=0 & 3x²y - y³= 2
COLUMN II
p) tan⁻¹3
q) tan⁻¹2 √2
r) π/2
s) tan⁻¹(3/4)
SECTION - V
Integer Type
42) If f: R---> R is a monotonic, differentiable real valued function, a, b are two real numbers and then ᵇₐ ∫ (f(x)+ f(a))(f(x) - f(a)) dx = k ᶠ⁽ᵇ⁾f(ₐ)∫ x (b - f⁻¹(x)) dx, then the value of k is
43) Let f be twice differentiable such that f''(x)= - f(x) and f"(x)= g(x). If h(x)= (f(x))²+ (g(x))², where h(5)=9. Then the value of h(10) is
44) If the equation of the tangent to the curve y(x -2)(x -3) - x +7 =0 at the point where it cuts the y-axis is ax+ by + 42=0, then find the value of |a - b|
45) For x∈ (0,π/2), if ∫ √(secx + tan x) + √(secx - tanx) dx = √2 sin⁻¹(1- g(x))+ c then [g(π/6)]= (where [.] denotes GIF)
46) For x∈ (0,π/2), if ∫ (1+ sin2x)/{√tanx + √cotx} dx = 1/(2√2) [f(x) √sin2x+ sin⁻¹(f(x))] + c then [f(3π/4)]= (where [.] denotes GIF)
47) For x∈ (0,π/2), if ∫ tanx √secx (1+cos² x)/√(cosx + sin²x) dx = 2√(f(x)+ 1)+ c then [f(π/4)]= (where [.] denotes GIF).
48) For y< 0 if y is differentiable function of x such that y(x + y)= x and ∫ dx/(x + 2y) = - log(k - y)+ cwhere k∈N then k=
49) Let f: R---> R be defined by f(x)= ³₁∫ dt/(1+ |t - x|). If ³₁∫ f(t) dt = (6 logk) - (k +1), then k=
50) The minimum area bounded by the function y= f(x) and y= αx + 9(α ∈ R) where f satisf the relation f(x + y)= f(x)+ f(y)+ y √f(x) ∀ x, y∈ R and f'(0)= 0 is 9A, value of A is
51) If the area bounded by the curve y= - x²+ 6x -5, y= - x²+ 4x -3 and the line y= 3x - 15 is 73/λ, where x > 1 then the value of λ is
52) Let C be the curve passing through the point (1,1) has the property that the perpendicular distance of the origin from the normal at any point P of the curve is equal to the distance of P from the x-axis, if the area bounded by the curve C and x-axis in the first quadrant is kπ/2 square units, then the value of k is
53) If f: R - {-1} --> R and f (which is not a constant function and f(x)≠ x) is differentiable function satisfies f(x)+ f(y) + x f(y))= y + f(x)+ yf(x) ∀ x, y ∈ R - {-} then the value of (2010)[1+ f(2009)] is
53) If the area of the curve ³√x² + ³√y²= 16 lying in a first quadrant is (48π) k. Then the value of k is
55) The curve represented by the differential equation xdy - y dx = ydy intersects the yaxis at A(0,1) and the line y= e at (a,b) then the value of (a + b) is
56) Let y= f(x) passing through (e, eᵉ) which satisfy the differential equation
(2ny + xy logx)dx - x logx dy = 0, x> 0, y>0.
If g(x)= lim ₓ→∞ f(x), then ᵉ₁/ₑ∫ fpg(x)dx =
CIRCLES
Part A
1) Prove that the two circles, which pass through (0,a) and (0,-a) and touch the line y= mx+ c, will cut orthogonally if c²= a²(2+ m²).
2) Find the radius of smaller circle which touches the straight line 3x - y = 6 at (1,-3) and also touches the line y= x.
3) A circle of constant radius r passes through the origine I, and cuts the axes at A and B. Show that the locus of the foot of the perpendicular from O to AB is (x²+ y²)(1/x² + 1/y²)= 4r².
4) Find the locus of the mid-point of the chord of the circle x²+ y²= a² which subtend a right angle at the point (p,q).
5) Find the condition on a,b, c such that two chords of the circle x²+ y² - 2ax - 2by + a²+ b² - c²= 0 passing through the point (a, b+ c) are bisected by the line y= x.
6) The circle x²+ y²= 1 cuts the x-axis at P and Q, another circle with centre at Q and variable radius intersects the first circle at R above x-axis and the line segment PQ at S. If the maximum area of the triangle QSR is ∆. then find the value of 4√3/∆.
7) Suppose the line y= 2x + a does not intersect or touch the circle x²+ y² - 2x - 2y + 1= 0 and x²+ y² - 16x - 2y + 61=0 and let the circles lie on opposite side of the line, then a must lie in the interval (2√5 - 5k, -√5-1); then find the numerical quantity k.
8) Let 2x²+ y² - 3xy=0 be the equations of a pair of tangents drawn from the origin O to a circle of radius 3 with centre in the first quadrant. If A is one of the points of contact, then the length of OA is 3(3+ √(2λ) then find the numerically λ.
PART B
1) The radius of the circle passing through the points (1,-2), (3,-4) and (5,-6) is
a) 10 b) 11 c) 12 d) 15
2) the cartesian equation of the curve given by x = 5+ 3 cosθ , y = 7+ 3 sinθ is
a) (x -5)²+ (y -7)²= 9
b) (x -5)²+ (y -7)²= 6
c) (x -3)²+ (x - y)²= 7
d) (x -7)²+ (y -5)²= 9
3) The point where the line x + y= 5 touches the circle x²+ y² - 2x - 4y + 3=0 is
a) (2,3) b) (3,2) c) (2,2) d) (3,3)
4) The equation of the tangent to the circle x²+ y² + 4x - 4y + 4 =0 which makes equal intercepts on the positive coordinate axes is
a) x+ y= 2 b) x + y=2√2 c) x+ y=4 d) x+ y= 8
5) The equation of the circle whose diameter is the common chord of the circle x²+ y² + 3x +2y + 1=0 and x²+ y² +3x + 4y + 2 =0 is
a) x²+ y²+ 8x + 10y + 2=0
b) x²+ y² - 5x + 4y + 7=0
c) 2x²+ 2y² + 6x +2y + 1 =0
d) 2x²+ 2y² +6x +2y -1 =0
6) If the lines a₁x + b₁y + c₁ = 0 and a₂x + b₂x + c₂ = 0 cut the coordinates axes in co cyclic points, then
a) a₁a₂ = b₁b₂ b) a₁b₁ = a₂b₂ c) a₁b₂ = a₂a₁ d) none
7) The length of the tangents from any point on the circle 15x²+ 15y² - 48x + 64y =0 to the two circles 5x²+ 5y² - 24x + 32y +75=0, 5x²+ 5y² - 48x + 64y+ 300 =0 are in the ratio
a) 1:2 b) 2:3 c) 3: 4 d) 2:1
8) The two circles x²+ y² = ax and x²+ y² = c² (with c> 0) touch each other if
a) c=|a| b) 2a=|c| c) 2c = a d) none
9) If a circle passes through the point (a,b) and cuts the circle x²+ y² = p² orthogonally, then the equation of the locus of its centre is
a) 2ax + 2by - (a²+ b²+ p²)= 0.
b) 2ax + 2by - (a²- b²+ p²)= 0.
c) x² + y² - 3ax - 4by + (a²+ b²- p²)= 0.
d) x² + y² - 2ax - 3by + (a²- b²- p²)= 0.
10) the locus of the centre of the circle which cuts the circle x²+ y² + 2g₁x + 2f₁y + c₁ = 0 and x²+ y²+ 2g₂x + 2f₂y + c₂= 0 orthogonally , is
a) an ellipse
b) the radical axis of the given circle
c) a conic d) another circle
11) The locus of the mid-point of the chords of the circle x²+ y² = 4 which suspendeds a right angle at the origin is
a) x+ y = 2 b) x²+ y² = 1 c) x²+ y² = 2 d) x+ y = 1
12) Tangents drawn from the point (4, 3) to the circle x²+ y² -2x - 4y = 0 are inclined at an angle
a) π/6 b) π/4 c) π/3 d) π/2
13) Let AB be a chord of the circle x²+ y² = r² subtending right angle at the centre, then the locus of the centroid of the triangle PAB as P moves on the circle is
a) a parabola b) a circle c) an ellipse d) a pair of straight lines
14) The equation of locus of the points of intersection of the tangents to the circle x= r cosα, y= r sinα at points whose parametric angles differ by π/3, is
a) x²+ y² = r² b) x²+ y² = 4r²/3 c) x²+ y² = 2r² d) x²+ y² = r²/3
15) If the curves ax²+ 2hxy + by² + 2gx + 2fy + c= 0 and Ax²+ 2Hxy + By²+ 2Gx + 2Fy+ C = 0 intersect at four concyclic points then (a+ b)/h is equal to
a) (A - B)/H b) (A- B)/2H c) (A- B)/3H d) (A- B)/4H.
16) Let S= x²+ y² + 2gx + 2fy + c= 0 be a given circle. Then the locus of the foot of the perpendicular drawn from the origin upon any chord of S which subtends right angle at the origin, is
a) x²+ y²+ gx + fy + c/2 = 0
b) x²+ y² = g
c) x²+ y² = f
d) x²+ y² + g= 0
17) Circles are drawn passing through the origin O to intersect the coordinates axes at point P and Q such that m. OP + n. OQ = k, then the fixed point satisfying all of them is given by
a) (m,n) b) (m²/k, n²/k) c) (mk/(m²+ n²), nk/(m²+ n²)) d) (k,k)
18) Consider a family of circles passing through the intersection of the lines √3(y -1)= x -1 and y -1= √3(x -1) and having its centre on the acute angle bisector of the given lines. Then the common chords of each member of the family and the circle x²+ y² + 4x - 6y + 5 = 0 are concurrent at
a) (1/2,1/2) b) (1/2,3/2) c) (3/2,3/2) d) (-1/2,-1/2)
19) Let PQ and RS be tangents at the extremities of the diameter PR of a circle of radius r. If PS and RQ intersect at a point X on the circumference of the circle, then 2r equals
a) √(PQ. RS) b) (PQ + RS)/2 c) (2PQ. RS)/(PQ + RS) d) √{(PQ²+ RS²)/2}
20) Tangents drawn from the point P(1,8) to the circle x²+ y² - 6x - 4y - 11= 0 touch the circle at the point A and B. The equation of the circumcircle of the triangle PAB is
a) x²+ y² + 4x - 6y +19= 0
b) x²+ y² - 4x - 10y +19= 0
c) x²+ y² - 2x + 6y -29= 0
d) x²+ y² - 6x - 4y +19= 0
Multiple Questions
21) let L₁ be a straight line passing through the origin and L₂ be the straight line x+ y = 1. if the intercepts made by the circle x²+ y² - x + 3y= 0 on L₁ and L₂ are equal, then which of the following equations can represent L₁ ?
a) x+ y = 0 b) x - y=0 c) 7y + x =0 d) x - 7y =0
22) The circle x²+ y² - 4x - 4y + 4= 0 is inscribed in a triangle which have a two of its sides along the co-ordinate axes. If the locus of the circumcircle of the triangle is x + y + xy + k √(x²+ y²)= 0, then k is equal to
a) 1 b) -1 c) 2 d) none
23) Equation of the circles concentric with the circle x²+ y² - 2x - 4y = 0 and touching the circle x²+ y² + 2x - 1= 0, must be
a) x²+ y² - 2x - 4y= 0
b) x²+ y² - 2x - 4y + 3= 0
c) x²+ y² - 2x - 4y - 13= 0
d) x²+ y² - 2x - 4y - 1= 0
24) If tangents are drawn from origin to the circle x²+ y² - 2x - 4y + 4= 0, then
a) slope of one tangent is 3/4
b) angle between tangents is tan⁻¹(4/3)
c) angle between tangents is tan⁻¹(3/4)
d) slope of one of the tangent is not defined
25) All those circles which pass through (2,0) and (-2,0) are ortogonal to the circle/s
a) x²+ y² - 5x + 4= 0,
b) x²+ y² +13x + 4= 0,
c) x²+ y² - 12x + 4= 0,
d) x²+ y² - 10x + 4= 0,
26) If θ is the angle subtended by the circle S= x²+ y² + 2gx + 2fy + c at the point P(x₁, y₁) and S= x₁² + y₁²+ 2gx₁ + 2fy₁ + c, then
a) cotθ= 2 √S₁/√(g²+ f²- c)
b) cot(θ/2)= √S₁/√(g²+ f²- c)
c) θ = 2 tan⁻¹ √{(g²+ f²- c)/S₁}
d) cosθ = (S₁+ c - g²- f²)/(S₁ - c + g²+ f²)
27) The range of values of a such that angle θ between the pair of tangents drawn from (a,0) to the circle x²+ y² = 1 satisfies π/2< θ< π, lies in
a) (1,2) b) (1,√2) c) (-√2, -1) d) (-√2, -1) U (1, √2)
28) A and B be any points on the circle x²+ y² = 4 such that AB is diameter of the circle. If b₁, b₂ are the length of perpendicular from A and B on x+ y = 1, then
a) max. b₁b₂ = 7/2 b) max. b₁b₂= 4
c) maximum value is attained by one diameter
d) maximum value is attained by two diameters .
29) Point M moved on the circle (x -4)²+ (y-8)² = 20. Then it broke away from it and moving along a tangent to the circle, cuts x-axis of the point (-2,0). The coordinates of a point on the circle at which the moving point broke away is
a) (-3/5,46/5) b) (-2/5,44/5) c) (6,4) d) (3,5)
30) A line parallel to the line x - 3y = 2 touches the circle x²+ y² - 4x -5 = 0 at the point
a) (1,-4) b) (1,2) c) (3,-4) d) (3,2)
Assertion Reason Type
Direction: Question number 31 and 32 are Assertion - Reason type questions . Each of these questions contents two statements:
Statement-1( Assertion ) and Statement- 2 (Reason). Each of these questions also has four alternative choices, only one of which is the correct answer . You have to select the correct choice.
a) Statement- 1 is true, Statement- 2 is true, Statement- 2 is the correct explanation for Statement- 1.
b) Statement- 1 is true, Statement- 2 is true, Statement- 2 is not the correct explaynation for Statement- 1.
c) Statement- 1 is true, Statement- 2 is false.
d) Statement- 1 is false, Statement- 2 is true
31) Statement -1: The equation x²+ y² - 2x -2ay - 8 = 0 represents for different values of a, a system of circles passing through two fixed points lying on the x-axis.
Statement- 2: S= 0 is a circle & L=0 is a straight line, then S + IL =0 represents the family of circles passing through the points of intersection of circle and straight line. where (I is arbitrary parameter).
32) Statement- 1: The least and greatest distances of the point P(10, 7) from the circle x²+ y² - 4x - 2y - 20 = 0 are 5 and 15 units respectively.
Statement- 2: A point (x₁, y₁) lies outside a circle
S= x² + y² + 2gx + 2fy + c=0 if S₁> 0, where S₁ = x₁²+ y₁²+ 2gx₁ + 2fy₁ + c.
Comprehension Type
Paragraph for Question 33 to 35
Let C₁ = x² + y² = r₁², C₂ = x²+ y²= r₂² (r₁< r₂) be two circles. Let A be fixed point (r₁, 0) on C₁ and B be a variable point on C₂. Let the line BA meets the circle C₂ again at E.
33) The maximum and minimum values of BE are given by
a) r₂, √(r₂² - r₁²) b) 2r₂, √(r₂² - r₁²) c) 2r₂, 2 √(r₂² - r₁²) d) r₂, 2 √r₂² - r₁²)
34) If O is the origin, then the best possible lower and upper bounds for OA²+ OB² + BE² are
a) 5r₂² - 3r₁², 5r₂²+ r₁²
b) 5r₂² - r₁², 5r₂²+ 3r₁²
c) r₁², r₂²
d) r₁² - r₂², r₁²+ r₂²
35) The locus of midpoint of AB must be
a) (x - r₂/2)² + y²= r₁²/4
b) (x - r₁/2)²+ y²= r₁²/4
c) (x - r₁/2)²+ y²= r₂²/4
d) (x - r₁)²+ y²= r₂².
Matrix Match Type
36) There are two circles C₁ : x²+ y²= 4 and C₂ : x²+ y²- 2x - 4y + 4=0, then
COLUMN I
A) Length of the common chord
B) Length of their common tangent
C) The sum of areas of two circles
D) The angle of intersection of C₁ and C₂.
COLUMN II
p) π/2
q) 4/√5
r) 2
s) 5π
37) Match the following columns:
COLUMN I
A) Locus of points of intersection of lines x= at², y= 2at
B) Locus of points of intersection of perpendicular tangents to the circle x²+ y²= a²
C) Locus of the points of the intersection of lines x cosθ = y cotθ= a
D) The locus of the midpoints of the chords x²+ y²- 2ax = 0 passing through the origin.
COLUMN II
p) x²+ y²= 2a²
q) y²= 4ax
r) x²+ y²= ax
s) x²+ y²= a
Integer Answer Type
38) let A(-2,2) and B(2,-2) be two points and AB subtends an angle of 45° at any points P in the plane in such a way that area of ∆ APB is 8 square units, then number of possible position of P is
39) The sum of the square of the length of the chords intercepted by the line x + y = n, n ∈ N on the circle x²+ y²= 4 is p, then p/11=
40) If (α, β) is a point on the circle whose centre is on the x-axis and which touches the line at x+ y =0 at (2,-2), then |α|= where [.] denotes the GIF
41) The number of such points (a+ 1, √3 a), where a is any integers lying inside the region bounded by the circles x²+ y²- 2x - 3= 0 and x²+ y²- 2x - 15 = 0, is
42) If the line 3x - 4y - k =0 touches the circle x²+ y²- 4x - 8y - 5 = 0 at (a,b), then the positive integral value of (k + a+ b)/5=
43) The difference between the radii of the largest and the smallest circles which have their centres on the circumference of the circle x²+ y² + 2x + 4y - 4= 0 and passes through the point (a,b) lying outside the given circle, is
44) Tangents are drawn to the circle x²+ y² = 12 at the points where it is meet by the circle x²+ y²- 5x + 3y -2 = 0, then the x-coordinate of the point of intersection of these tangents is
PROBABILITY
1) A team of 8 couples, (husband and wife) attend a lucky draw in which 4 persons picked up for a prize. The probability that there is at least one couple is
a) 11/39 b) 15/39 c) 14/39 d) 12/39
2) Five boys and 3 girls are seated at the random in a row . The probability that no boy sits between two girls is
a) 1/56 b) 1/8 c) 3/28 d) 3/56
3) in a convex hexagon two diagonals are drawn at random. The probability that the diagonals intersected at an interior point of the hexagon is
a) 5/12 b) 7/12 c) 2/5 d) 1/12
4) From 4 children, 2 women and 4 men, 4 are selected. The probability that there are exactly 2 children among the selected is
a) 11/21 b) 9/21 c) 10/21 d) 8/25
5) If A and B are two events such that P(A)> 0 and P(B)≠ 1, then P(A'/B') is equal to
a) 1- P(A/B) b) 1- P(A'/B') c) P(A')/P(B') d) {1- P(AUB)}/P(B')
6) India and Pakistan play a 5 match test series of hockey , the probably that India wins atleast 3 matches is
a) 1/23/54/51/3
7) A bag contains 3 red and 3 white balls. Two balls are drawn one by one. The probability that they are of different colours is
a) 3/10 b) 2/5 c) 3/5 d) 1/5
8) A bag contains 5 brown and 4 white socks. A man pulls out 2 socks. The probability that they are of the same colour is
a) 5/108 b) 1/6 c) 5/18 d) 4/9
9) If P(AUB)= 2/3, P(A∩B)= 1/6 and P(A)= 1/3, then
a) A and B are independent events
b) A and B are disjoint events
c) A and B are dependent events d) none
10) if a person throws 3 dice, the probability of getting sum of digit exactly 15 is
a) 5/72 b) 5/108 c) 5/36 d) 1/72
11) if the letters of INTERMEDIATE are arranged, in the probability that no two E's occur together is
a) 6/11 b) 5/11 c) 2/11 d) 3/11
12) The probability that an anticraft gun can hit an enemy plane at the first, second and third shot are 0.6, 0.7 and 0.1 respectively. The probability that the gun hits the plane is
a) 0.108 b) 0.892 c) 0.14 d) 0.91
13) One hundred identical coins, each with probability, p, of showing up heads are tossed once. If 0< p < 1 and the probability of heads showing on 50 coins is equals to that of heads showing on 51 coins. Then the value of p is
a) 1/2 b) 49/101 c) 50/101 d) 51/101
14) A natural number is chosen at random from the first 100 natural numbers. The probability that {(x -20)(x -40)}/(x -30)< 0 is
a) 1/50 b) 3/50 c) 3/25 d) 7/25
15) If A and B are two events such P(AUB)= 5/6, P(A)= 1/3, P(B)= 3/4, then A and B are
a) mutually exclusive b) dependent c) independent d) none
16) If (1+ 3p)/3, (1- p)/2 and (1- 2p)/2 are the probabilities of three mutually exclusive events, then the set of all values of p is
a) φ b) [1/2,1/3] c) [0,1] d) [1/3,2/3]
17) A bag contains a large number of white and black marbles in equal proportions . Two samples of 5 marbles are selected (with replacement) at random . The probability that the first sample contains exactly 1 black marble, and the second sample contains exactly 3 black marbles is,
a) 25/51215/3215/102435/256
18) If two events A and B are such that P(A')= 0.3, P(B)= 0.4 and P(A∩B')= 0.5, then P{B/(A UB')}=
a) 1/4 b) 1/5 c) 3/5 d) 2/5
19) A is set containing n elements. A subject P₁ of A is chosen at random. The set A is reconstructed by replacing the elements of P₁. A subset P₂ is again chosen at random. The probability that P₁ U P₂ contains exactly one elements, is
a) 3n/4ⁿ b) 3ⁿ/4ⁿ c) 3/4 d) 3/4ⁿ
20) The probability that in a group of N (<365) people, atleast two will have the same birthday is
a) 1- (365)!/{(365- N)!(365)!}
b) (365)ᴺ(365)!/{(365- N)!(365)! -1
c) 1- {(365)ᴺ(365)!}/{(365+ N)!
d) 1- {(365)!}/=(365 - N)!{(365)ᴺ}
21) Let E and F between two independent events such that P(E)> P(F). The probability that both E and F happen is 1/2 and the probability that neither E nor F happens is 1/2, then
a) P(E)=1/3, P(F)= 1/4
b) P(E)=1/2, P(F)= 1/6
c) P(E)=1, P(F)= 1/12
d) P(E)=1/3, P(F)= 1/2
22) A company has two plants to manufacture televisions. Plant I manufacture 70% televisions and Plant II manufacture 30%. At plant I, 80% of the televisions are rated as of standard quality and at plant II, 90% of the televisions are rated as of standard quality. A television is chosen at random and is found to be of standard quality. The probability that it has come from plant II is
a) 17/50 b) 27/83 c) 3/5 d) 9.83
23) x₁, x₂, x₃, ....x₅₀ are fifty real numbers such that xᵣ < xᵣ₊₁ for r= 1,2,3,....49. Five numbera out of these are picked up at random. The probability that the five numbers have x₂₀ as the middle number is
a) (²⁰C₂ x ³⁰C₂)/⁵⁰C₅
b) (³⁰C₂ x ¹⁹C₂)/⁵⁰C₅
c) (¹⁹C₂ x ³¹C₃)/⁵⁰C₅
d) (²⁰C₂ x ³⁰C₂)/⁴⁹C₅
24) If the integers m and n are chosen at random from 1 to 100, then the probability that a number of the form 7ⁿ + 7ᵐ is divisible by 5 equals
a) 1/4 b) 1/2 c) 1/8 d) 1/3
25) The probability that a man can hit a target is 3/4. He tries 5 times. The probability that he will hit the target atleast three times is
a) 291/364 b) 371/461 c) 471/502 d) 459/512
26) 15 coupons are numbered 1, 2, ...., 15, respectively. 7 coupons are selected at random one at a time with replacement. The probability that the largest number appearing on a selected coupon is
a) (9/16)⁶ b) (8/15)⁷ c) (3/5)⁷ - (8/15)⁷ d) (7/15)⁷
27) India plays two matches each with West Indies and Australia. In any match the probabilities of India getting points 0, 1 and 2 are 0.45, 0.05 and 0.50 respectively . Assuming that the outcomes are independent the probability of India getting atleast 7 points is
0.8750 b) 0.0875 c) 0.0625 d) 0.0250
28) Two numbers are selected at random from 40 consecutive natural numbers. The probability that the sum of the selected members is odd will be
a) 14/29 b) 20/39 c) 1/2 d) 19/39
29) If all letters of the word MISSISSIPPI are rearranged then the probability that all S come together will be
a) 1/165 b) 4/165 c) 8/165 d) 2/165
30) The probability of having at least one tail in 4 throws with a coin is
a) 15/16 b) 1/16 c) 1/4 d) 1
Multiple Choice Questions
31) Cards are drawn one by one without replacement until two aces are drawn . Let P(m) be the probability that the event occurs in exactly m trials, then P(m) must be zeros at
a) m=2 b) m=50 c) m= 51 d) m= 52
32) Let p be the probability that in a pack of playing cards two kings are adjacent and Q be the probability thatno to kings are together, then
a) p= q b) p< q c) p+ q= 1 d) q= (48 x 47 x 46)/(52 x 51 x 50)
33) Which of the following statements are true ?
a) The probability that birthday of twelve people will fall in 12 calendar months = 12/12¹²
b) The probability that birthday of six people will fall in exactly two calendar months is ¹²C₂(2⁶ -2)/6¹²
c) The probability that birthday of six people will fall in exactly two calendar months is ¹²C₂(2⁶ -2)/12⁶
d) The probability that birthday of n (n≤ 365) people are different is ³⁶⁵Pₙ/(365)ⁿ
34) A bag contains N tickets numbered 1,2,3,....N. if r tickets are drawn one by one with replacement then the probability that all different numbers are drawn is
a) ᴺCᵣ/ᴺPᵣ
b) {N(N -1)(N -2).....(N - r+1)}/Nʳ
c) ᴺPᵣ/Nʳ d) 1/r!
35) A die is thrown twice. Let X₁ and X₂ be the outcomes of these trials respectively. Consider the following events
A₁= {X₁ is divisible by 2, X₂ is divisible by 3}
A₂={X₁ is divisible by 3, X₂ is divisible by 2}
A₃={X₁ is divisible by X₂}
A₄={X₂ is divisible by X₁}
A₅={X₁ + X₂ is divisible by 2}
A₆={X₁ + X₂ is divisible by 3}, Then
a) A₁ and A₂ are independent
b) A₁ and A₅ are independent
c) A₃ and A₄ are independent
d) P(A₁∩₂)= 1/36
36) If M and N are any two events, the probability that exactly one of them occur is
a) P(M)+ P(N) - 2P(M ∩N)
b) P(M)+ P(N) - P(M ∩N)
c) P(M')+ P(N') - P(M' ∩N')
d) P(M ∩N')+ P(M' ∩N)
37) Let 0< P(A)< 1, 0< P(B)< 1 and P(A UB)= P(A)+ P(B) - P(A) P(B), then
a) P(B/A)= P(B) - P(A).
b) P(A'UB')= P(A') - P(B')
c) P(A' U B')= P(A') P(B')
d) P(A/B)= P(A)
38) Each of n bags contains a white and b black balls. One ball is is transferred from first bag to the second bag then one ball is transferred from second bag to the third bag and so on. Let Pₙ be the probability that ball transferred from nᵗʰ bag is white, then
a) P₁= a/(a+ b)
b) P₂= a/(a+ b)
c) P₃= a/(a+ b)
d) P₄= a/(a+ b)
39) In a single cast with two dice the odds against drawing 7 is
a) 1/6 b) 1215/243 c) 5:1 d) 1:5
40) Given that x∈ [0,1] and y∈ [0,1]. Let A be the event of (x,y) satisfying y²≤ x and B be the events of (x,y) satisfying x²≤ y. Then
a) P(A∩B)= 1/3
b) A, B are not exhaustive
c) A, B are mutually exclusive
d) A, B are not independent
41) Let x= 33ⁿ. The index n is given a positive integral value at random. The probability that the value of x will have 3 in the units place is
a) 1/4 b) 1/2 c) 289/1156 d) 2/3
42) Three dies are thrown simultaneously . The probability of getting a sum of 15 is
a) 5/3 x (1/6)² b) 5/36 c) 5/72 d) 5/108
43) The probability of getting a sum of 12 in four throws of an ordinary dice is
a) (1/6)(5/6)³ b) (1/6)(5/6)⁴ c) (1/36)(5/6)² d) (5/6)³
44) Three different numbers are selected at random from the set A={1,2,3,...10}. The probability that the product of two of the numbers is equal to the third is
a) 3/4 b) 1/40 c) 23/920 d) 1/4
45) If A and B are two events such that P(A U B) ≥ 3/4 and 1/8≤ P(A ∩B≤ 3/8, then
a) P(A)+ P(B)≤ 11/8
b) P(A) P(B)≤ 3/8
c) P(A)+ P(B)≥ 7/8
d) P(A) P(B)≤ 1/8
46) A second order determinant is written down at random using the numbers 1, -1 as elements. The probability that the value of the determinant is non-zero is
a) 1/2 b) 3/8 c) 5/8 d) ⁹C₅/¹⁰C₅.
47) Numberr 1,2, 3,....100 are written down on each of the cards A, B and C. One number is selected at random from each of the cards. The probability that the numbers so selected can be the measure (in cm) of three sides of right angled triangles no two of which are similar, is
a) 4/100³ b) (3/500)² c) 3!/100³ d) (1/100)(3/50)²
48) The probabilities that a student passes in mathematics, physics and chemistry are m, p and c respectively. OF these subjects , a student has a 75% chance of passing in at least one, a 50% chance of passing in atleast two, and a 40% chance of passing in exactly two subjects. Which of the following relations are true ?
a) p+ m+ c = 19/20
b) p+ m + c= 27/20
c) pmc= 1/10 d) pmc= 1/4
49) For any two events A and B
a) P(A∩B)≥ P(A)+ P(B) -1
b) P(A∩B)≥ P(A)+ P(B)
c) P(A∩B)= P(A)+ P(B) - P(AUB)
d) P(A∩B)= P(A)+ P(B) + P(AUB)
50) A coin is tossed repeatedly. A and B call alternatively for winning a prize of Rs 30. One who calls correctly first wins the prize. A starts the call. Then expectation of
a) A is Rs 10 b) B is Rs 10 c) A is Rs 20 d) B is Rs 20
51) For two events A and B, let P(A)= 1/3; P(B)= 1/4 and P(AUB)= 1/2, then the correct statement is
a) P(A/B)= 1/3 b) P(A' ∩B')= 1/2
c) A and B are independent
d) P(B/A)= 1.6
Assertion Reason Type
Directions : Question numbers 52 to 56 are Assertion - Reason type questions . Each of these questions contains two statements:
Statement- I (Assertion ) and Statement- 2 (Reason ). Each of these questions also has four alternative choice.
a) Statement -1 is true Statement- 2 is true, Statement- 2 is the correct explanation for Statement- 1
b) Statement- 1 is true, Statement- 2 is true, Statement- 2 is not the correct explanation for Statement- 1.
c) Statement -1 is true, Statement- 2 is false
d) Statement -1 is false, Statement- 2 is true
52) Let A and B be two independent events of a random experiment.
Statement-1: P(A' ∩B)= P(A') P(B)
Statement- 2: P(A' ∩B')= 1- P(A) P(B)
53) A fair dice is rolled once.
Statement- 1: The probability of a getting a composite number is 1/3.
Statement- 2: There are 3 possibilities for the obtained number (i) the number is a prime number (ii) the number is a complete number (iii) the number is 1, and hence probability of each possibility is 1/3.
54) Let A and B are two events such that P(A)= 3/5 and P(B)= 2/3, then
Statement- 1: 4/15≤ P(A∩B)≤ 3/5
Statement- 2: 2/5≤ P(A/B)≤ 9/10
55) Consider the system of equations ax+ by= 0, cx+ dy = 0, where a,b,c,d ∈ [0,1]
Statement- 1: The probability that the system of equations has a unique solutions is 3/8.
Statement- 2: The probability that the system of equations has a solution is 1.
56) Statement- 1: if two events E₁ and E₂ are independent then E'₁ and E'₂ are also independent.
Statement- 2: P(E'₁∩ E'₂)= P(E₁ U E₂)'
= 1- P(E₁ U E₂)= 1- P(E₁) P(E₂)
Comprehension Type
Paragraph for Question 57 to 59
The probability of happening of an event in one trial being known, then the probability of its happening exactly x times in n trials is given by ⁿCₓqⁿ⁻ˣ.pˣ where
p= probability of happening the event
q= probability of not happening the event= 1- p.
Now ⁿCₓqⁿ⁻ˣoˣ is (x +1)ᵗʰ term in the expansion of (q + p)ⁿ whose expansion gives the happening of the event 0, 1, 2,...., n times respectively.
57) In four throws with the a fair of die, the chance of throwing doublets atleast twice is
a) 19/144 b) 125/144 c) 17/144 d) 18/144
58) A man takes a forward step with probability (0.8) and backwards step with probability (0.2). What is the probability that at the end of 9 steps he is exactly 3 steps away from the starting point
a) 69888/5⁸ b) 5377/5⁸ c) 5378/5⁸ d) 5376/5⁸
59) Unbiased coin is tossed 6 times. The probability of getting atmost 4 heads is
a) 7/64 b) 57/64 c) 21/32 d) 11/32
Paragraph for Question 60 to 62
A commander of an army battalion is punishing two of his soldiers X and Y. He arranged a duel between them. The rules of the duel are that they are pick up their guns and shoot at each other simultaneously .
If one or both hit, then the duel is over. If both shoot miss then they repeat the process . Suppose that the results of the shots are independent and that each shot of X will hit Y with probability 0.4 and each shot of Y will hit X with probability 0.2. Now answer the following questions .
60) The probability that the duel ends after first round is
a) 11/25 b) 12/25 c) 13/25 d) 2/25
61) The probability that X is not hit, is
a) 3/25 b) 7/25 c) 5/13 d) 8/13
62) The probability that both the soldiers are hit, is
a) 5/13 b) 2/13 c) 8/13 d) 1/13
Paragraph for Question 63 to 65
Let E₁, E₂, E₃,....Eₙ be a set of mutually exclusive and exhaustive events and A be an event. Then
P(A)= ⁿᵢ₌₁∑ P(Eᵢ) . P(A/Eᵢ) and P(Eⱼ/A)= {P(Eⱼ). P(A/Eⱼ)}/P(A) for j= 1,2,....n, where P(E/G) denote, the probability of occurring the event F given that G has already occurred. There are two bags of red and yellow colours . Red bag contains 4 fair coins and 3 biased coins and yellow bag contains 5 fair coins and 7 biased coins. Biased coin has tail on both sides . 2 coins are transferred from red bag to yellow bag and then a coin is taken from yellow bag and tossed .
63) Probability that both coins, transferred from red bag to yellow bag were fair, is
a) 1/7 b) 3/7 c) 4/7 d) 2/7
64) Probability that both coins, transferred from red bag to yellow bag , were of mixed type, is
a) 3/7 b) 4/7 c) 2/7 d) 1/7
65) If both coins transferred from red bag to yellow bag were biased, then the probability that tossing of coin results in head, is
a) 23/28 b) 9/28 c) 5/28 d) 19/28
Matrix Match Type
66) A is set containing n elements. A subset P of A is chosen at random. The set A is reconstructed by replacing the elements of the subset P. A subset Q of A is again chosen at random. The probability that
Column I COLUMN II
A) P∩Q= ∅ p) n(3ⁿ⁻¹)/4ⁿ
B) P ∩Q is a singleton q) (3/4)ⁿ
C) P∩Q contains r) ²⁰Cₙ/4ⁿ
2 elements
D) |P|= |Q|, where |X| s) 3ⁿ⁻²{n(n-1)/2(4ⁿ)
number of elements in X
67) A player tosses a coin and is to score one point for every head and two points for every tail turned up. He is to play until his score reaches or passes n. Pₙ is the chance of obtaining exactly a score of n, then
COLUMN I COLUMN II
A) P₁ equals to p) 5/8
B) P₂ equals to q) 1/2
C) P₃ equals to. r) 3/4
D) P₄ equals to. s) 11/16
68) Two dice are thrown . Let A be the event that sum of the points on the two dice is odd and B be the event that at least one 3 is there, then match the following:
COLUMN I. COLUMN II
A) P(AUB) p) 12/36
B) P(A∩B) q) 6/36
C) P(A∩B) r) 23/36
D) P(B) s) 11.36
Part B
1) A has three shares in a lottery containing 3 prizes and 9 blabka. B has two shares in a lottery containing 2 prizes and 6 blanks. Compare their chances of success.
2) A coin is tossed m+ n times (m> n). Show that the probability of atleast m consecutive heads come up is (n +2)/2ᵐ⁺¹.
3) If m things are distributed among 'a' men and 'b' women, show that the probability that the number of things received by the men is odd, is (1/2) [{(b+ a)ᵐ - (b - a)ᵐ}/(b + a)ᵐ].
4) An artillery target maybe either at point A with probability 8/9 or at point B with probability 1/9. We have 21 shells each of which can be fixed either at point A or B. Each shell may hit the target independently of the other shell with probability 1/2. How many shells must be fired at point A to hit the target with maximum probability ?
5) Let A and B be two independent witness in a case. The probability that A will speak the truth is x and the probability that B will speak the truth is y. A and B agree in a certain statement. Show that the probability that the statement is true is (xy)/(1- x - y + 2xy).
6) Two teams A and B play a tournament. The first one to win (n +1) games, win the series. The probability that A wins a game is p and that B wins a game is q (no ties). Find the probability that A wins the series. Hence or otherwise prove that ⁿᵣ₌₀∑ ⁿ⁺ʳCᵣ . 1/2ⁿ⁺ʳ = 1.
7) If a pair of fair dice is rolled 5 times, then find out the probability that 3 times we get sum more than 9.
8) A bag contains 10 fair coins and 25 coins having heads on both sides. A coin is selected at random and tossed. if it gives head, then find out the probability that it was a fair coin.
9) 3 players A, B and C toss a coin cyclic in that order (that is A,B,C,A,B,C,A,B,....) till a head shows. Let P be the probability that the coin shows a head. Let α, β and γ be respectively the probabilities that A, B and C gets the first head. Prove that β = (1- p)α. Determine α, β and γ (in terms of p).
10) A coin has probability p of showing head when tossed it is tossed n times. Let pₙ denotes the probability that no two (or more) consecutive heads occur. Prove that p₁ = 1. p₂ = 1- p² and pₙ = (1- p)pₙ₋₁ + p(1- p)pₙ₋₂, for all n≥ 3.
11) For a student to quality, he must pass at least two out of three exams. The probability that he will pass the 1st exam is p. If he fails in one of the exams then the probability of his passing in the next exam is p/2 otherwise. it remains the same. Find of probability that he will qualify .
12) A is targeting to B, B and C are targeting to A. Probability of hitting the target A, B and C are 2/3, 1/2, and 1/3 respectively. if A hit then find the probability that B hits the target and C does not.
13) P₁, P₂, ......Pₙ are eight players participating in a tournament . If i< j, then Pᵢ will win the match against Pⱼ. Players are paired up randomly for first round and winners of this round again paired up for the second round and so on. Find the probability P₄ reaches in the final.
Mock Test Paper AIEEE 2012
1) A function f(x) is defined as follows for real x
1- x², for x< 1
f(x)= 0, for x= 1
1+ x², for x≥ 1
Then
a) f(x) is not continuous at x= 1.
b) f(x) is continuous but not differentiable at x= 1.
c) f(x) is both continuous and differentiable at x = 1.
d) f(x) is continuous everywhere but differentiable nowhere.
2) Solution of the equation (1+ x²) d²y/dx½ = 1, when y(0)= y'(0)= 0 is
a) y= tan⁻¹x - log(1+ x²)
b) y= x tan⁻¹x - logg(1+ x²)
c) y= x tan⁻¹x - (1/2) log(1+ x²)
d) y= x tan⁻¹x + (1/2) log(1+ x²)
3) If a b ax - b
b c bx - c = 0
2 1 0 then the value of x is
a) -1 b) 1 c) -1/2 d) 1/2
4) The number of ways in which the letters of the word COMBINE can be arranged so that the words begin and end with vowel, is
a) 30 b) 504 c) 360 d) 720
5) If α, β are the roots of ax²+ 2bx + c= 0 and γ, δ are the roots of px²+ 2qx + r= 0. If α, β, γ, δ are in AP , then (b²- ac)/(q²- pr)
a) a²/p² b) b²/q² c) c²/r² d) none
6) In a triangle ABC, if (2cosA)/a + (cosB)/b + (2cosC)/c = a/bc + b/ca, then angle A is
a) 30 b) 60 c) 45 d) 90
7) The value of ∫ sinx/(1+ cosx + sinx) dx at (π/2,0)
a) π/4 b) π/4 + log√2 c) π/4 - log√2 d) π/4 - log2.
8) If y= (1+ x)(1+ x²)(1+ x⁴)....(1+ x²ⁿ) Then the value of (dy/dx)ₓ₌₀ is
a) 0 b) -1 c) 1 d) 2
9) An equilateral triangle is inscribed to the parabola y²= 4x whose vertex is at the vertex of the parabola. Then the length of its side is
a) 4/3 units b) 8/√3 units c) 16√3 units d) none
10) If (1- x + x²)ⁿ = a₀ + a₁x + a₂x² + ....+ a₂ₙx²ⁿ, then the value of a₀ + a₂ + a₄ + ....+ a₂ₙ is
a) 3ⁿ + 1/2 b) 3ⁿ - 1/2 c) (3ⁿ -1)/2 d) (3ⁿ+ 1)/2
11) Three integers are chosen at random from first 20 positive integer. The probability that their product is een, is
a) 12/19 b) 3/19 c) 17/19 d) 4/19
12) If the tangent to the curve y²= x³ at the point (m², m³) is also normal to the curve at the point(M², M³), then the value of mM is
a) - 1/9 b) -2/9 c) -1/3 d) -4/9
13) Locus of a point which devides chord at a distance 1 unit from the centre of the circle x²+ y²= 10 in the ratio 1:2.
a) x²+ y²=2 b) x²+ y²= 4 c) x²+ y²= 8 d) x²+ y²= 16
14) The value of lim ₓ→∞ {(x +6)/(x +2)}ˣ
a) e b) e⁻¹ c) e⁻⁶ d) e⁵
15) The base of an equilateral triangle is along x + y = 2. if one of the vertices is P(2, -1), the area of the triangle is
a) 4 square unit
b) 2√3square unit
c) 1/2√3 square unit
d) 2/√3 square unit
16) Sum of n terms of the series 2/3 + 8/9 + 26/27 + 80/81+ ... is
a) n+ (1/3) (3ⁿ -1)
b) n - (1/3) (3ⁿ -1)
c) n- (1/2) (3⁻ⁿ -1)
d) n- (1/2) (1- 3⁻ⁿ)
17) The area of the region bounded by y= |sinx|, x-axis and |x|= π is
a) 2 square unit
b) 4 square unit
c) 8 square unit d) none
18) If a= 3i - 5j, b= 6i + 3j and c = a x b then|a| |b| |c| =
a) √34:√45:√39
b) √34:√45:39
c) 34:45:39
d) 39 :35:34
19) If a line with direction ratios 2:2:1 intersects the lines (x -7)/3= (y -5)/2= (z -3)/1 and (x -1)/2 = (y +1)/4 = (z+1)/3 at A and B, then AB=
a) √2 b) 2 c) √3 d) 3
20) Let the position vectors of points A, B and C of triangle ABC respectively be i+ j + 2k, i + 2j + k and 2i+ j + k, Let I₁, I₂, I₃ be the length of perpendicular drawn from the orthocentre O on the sides AB, BC and CA respectively, then I₁+ I₂+ I₃ equals
a) √6/3 b) 1/√6 c) 2/√6 d) √6/2
21) The equation of common tangents to the parabola y²= 16x and x²+ y²= 8 are
a) y= x+ 4, y= - x - 4
b) y= 2x+ 4, 2y= - x - 9
c) y= x+ 9, y= - x - 4 d) none
22) A plane passes through a fixed point (a, b, c) and cuts the axes in A, B, C. The locus of a point equidistant from origin A, B and C must be
a) x/a + y/b + z/c = 2
b) a/x + b/y + c/z = 2
c) a/x + b/y + c/z = 1
d) a/x + b/y + c/z = 1/2
23) If standard deviation of 1, 2, 3, ...k is 2, then k=
a) 3 b) 4 c) 7 d) none
24) The variance of binomial distribution
a) exceeds π/4
b) cannot exceeds π/4
c) is exactly π/4 d) none
25) If z be complex number, the expression (z -1) (conjugate z -1) can be written as
a) |z²|+ 1 b) |z²| - 1 c) |z -1|² d) |z + 1|²
Directions : Question numbers 26 to 30 are Assertion - Reason type questions. Each of these questions contains two statements:
Statement- 1 (Assertion) and statement - 2(reason) each of these questions also has four alternative choices , only one of which is the correct answer. You have to select the correct choice.
a) Statement- 1 is true, Statement- 2 is true, Statement- 2 is correct explanation for Statement- 1.
b) Statement -1 is true, Statement- 2 is true, Statement- 2 is not a correct explanation for Statement- 1.
c) Statement -1 is true, Statement- 2 is false
d) Statement- 1 is false, Statement- 2 is true.
26) Statement - 1 if x₁ and x₂ are the means of two distributions such that x₁ < x₂ and mean x is the combined mean of the distribution, then x₁< x< x₂.
Statement- 2: if n₁ & n₂ be the number of observation of two groups whose means are x₁ & x₂ respectively, then x= (n₁x₁ + n₂x₂)/(n₁+ n₂).
27) Statement -1: If A and B are two events such that P(A)= 2/5, P(B)= 3/4 then 1/20≤ P(A ∩B) ≤ 2/5
Statement- 2: P(AUB)≤ max {P(A), P(B)} and P(A ∩B)≥ min{P(A), P(B)}
28) Statement-1: if the system of equation ax+ at + az =0, bx + y + bz= 0 and cx+ cy+ z= 0 where a,b,c are non-zero and non-unity, has a non trivial solution, then value of a/(1- a) + b/(1- b) + c/(1- c) is 1
Statement- 2: for non travel solution determinant of co-efficient matrix is zero.
29) Statement -1: The equation 2(sin⁻¹x)² = 5(sin⁻¹x) +2=0
Statement- 2: sin⁻¹(sinx)= x if x ∈ [-1.57, 1.57]
30) Statement- 1: ¹⁰⁰₀∫(x - [x]) dx = 50, where [.] denotes greatest integer function.
Statement- 2: if f(x) is a periodic function with period T, then ⁿ₀∫ d(x) ddx = n ᵀ₀∫ f(x) dx.
For various Engineering Entrance Exams
1) If p be the length of the perpendicular drawn from the focus S of an ellipse x²/a² + y²/b²= 1 to the tangent at Q and e₁ be the eccentricity of x²/p² - y²/b² = 1, then e₁½
a) > 2a/SQ b) < 2a/SQ c) = 2a/SQ d) none
2) In ∆ ABC, Π (1+ (b - c)/a)ᵃ always belongs to
a) (0,2) b) (0,1] c) [1/2,1] d) none
3) If for n > 1, aₙ₋₁(aₙ - aₙ₋₁)= 1, lim ₓ→₀⁺ [1+ (sinx)¹⁾ˣ]= a₁ and a²₄₀₂ ∈ (p,q), then p+ q =
a) 2013 b) 1212 c) 2017 d) none
4) The area bounded by y= e⁻ˣ, y= eˣ and directrix of y² - 4x - 4y +12= 0 is
a) ∞ᵣ₌₁∑ 1/(2r)!
b)
∞ᵣ₌₀ 2∑ 1/(2r)!
c) ∞ᵣ₌₀ ∑ 1/(2r)! d) none
5) If x+ a= y + α²/{α²(α² -4)+ 4} be a common chord of two parabola y²= 4ax and x²= 4ay , then their length of latus rectum cannot exceed
a) 1 b) 2 c) 3/2 d) none
6) In arrangement of numbers (1), (3,5,7), (9,11,13,15,17),.... Which is the midpoint terms of nth bracket?
a) 2n²- 2n b) 2n +1 c) 2n²-1 d) none
7) Find the least positive integral value of x for which 2, |x +1| and 2 - |x -3| may be in AP
a) 1 b) -1 c) 3 d) none
8) If f(x)= ²⁰¹²ᵣ₌₀∑ xʳ and A= [0 9
0 0] then f(A)=
a) I(unit matrix) b) A c) 2012 A d) none
9) If {cos⁻¹(x - 1)}²> {sin⁻¹(x -1)}², then [x]= ?, where [.] denotes the greatest integer function
a) {-1,0} b) {-1,0,1] c) [9,1] d) [0,1,2]
10) In ∆ ABC, if p= tan{(B - C)/2} cot{B+ C)/2}
q= tan{(C - A)/2} cot{C+ A)/2} and r= tan{(A- B)/2} cot{A+B)/2} then ∑p=?
a) 1 b)∑pq c) - pqr d) none
11) If ⅗√(x + iy)= cos²θ + i sin²θ , then minimum value of sin4θ is
a) 2√(xy) b) √(xy) c) x²y² d) none
12) If √{x + (1+ √12x)a} - √x = 1/√3 has infinitely many solutions (real (, then 2a + 3a²/2 + 4a³/3+ ....to ∞= ?
a) (1/2) - log(3/2)
b) (1/2) - log(1/3)
c) (1/2) + log(3/2) d) none
13) If x²/a² - y²/b²= 1 has eccentricity e θ is the angle between their asymptotes ,then
tan³(θ/2) cosec(θ/2)+ sec(θ/2)=?
a) e³ b) ²√e³ c) e d) none
More than One Option Correct
14) If normal at 't' to the parabola y²= 4ax, (a> 0) meets the parabola again at Q and QN be perpendicular to axis, then least value of QN will be
a) radius of (x -2)²+ (y + 1/2)²= 32a²
b) perpendicular distance of x - y + 8a =0 from (√2,√2)
c) radius of the director circle of x²/21a² + y²/11a² = 1
d) all of these
15) If m(> 0) be the slope of tangent to x²/a²+ y²/b²= 1, then area of triangle formed by tangent with axes may be
a) 2ab/3 b) ab c) 2ab d) 3ab
16) If xᵢ (i= 1,2,3,....n) be n positive real numbers such that logₓ₁[1+ logₓ₂{1+ logₓ₃(1+ ....to(n -1) terms}]= 0 then xₙ is
a) independent of x₁, x₂, ...,xₙ₋₁
b) always 1
c) product of x₁, x₂,...xₙ
d) none
17) In ∆ ABC, if angle C = 90°, then solution of sin⁻¹x + cos⁻¹(ax/c) + cos⁻¹(bx/c)=π are
a) 0 b) 1 c) -1 d) √3
Mock Test Paper WB- JEE
1) The area bounded by the curve y= 1+ logx, the x-axis and straight line x= e is equal to (in square unit) is
a) 3e -2 b) e c) e - 1/2 d) e + 1/2
2) The value of ⁴₀∫ |x - 1| dx is
a) 5/2 b) 5 c) 4 d) 1
3) If ∫ f(x)/(log(cosx) dx = - log(log cosx)+ C, then f((x) is equal to
a) tanx b) - sinx c) - cosx d) - tanx
4) If [x] denotes the greatest integer less than or equal to x, then the value of
²₀∫(|x - 2| + [x]) dx is equal to
a) 2 b) 3 c) 1 d) 3/2
5) The value of ∫ - (xeˣ)/(x +1)² dx is equal to
a) -eˣ/(x +1)² + c
b) eˣ/(x +1)²+ c
c) eˣ/(x +1) + c
d) -eˣ/(x +1) + c
6) lim ₙ→∞ {(n +1)/(n²+ 1²) + (n +2)/(n² + 2²)+ ....+ 1/n}=
a) π/4 - log2
b) π/4 - (1/2) log2
c) π/4 + log2
d) π/4 - (1/2) log2
7) If the area between y= mx² and x= my², (m> 0) is 1/4 sq.units then the value of m is
a) ±3√2 b) ±2/√3 c) √2 d) √3
8) The value of ∫ 1/(e²ˣ + e⁻²ˣ) dx is equal to
a) 2 tan⁻¹(2²ˣ)+ c
b) tan⁻¹(2²ˣ)+ c
c) (1/2) tan⁻¹(2²ˣ)+ c
d) 1/(e²ˣ + e⁻²ˣ) + c
9) ¹₀∫ xe⁻⁵ˣ dx is equal to
a) 1/25 - 6e⁻⁵/25
b) 1/25 + 6e⁻⁵/25
c) - 1/25 - 6e⁻⁵/25
d) 1/25 - e⁻⁵/5
10) An integrating factor of the differential equation (1+ x²) dy/dx + xy = x is
a) x/(1+ x²)
b) (1/2) log(1+ x²)
c) √(1+ x²) d) x
11) If I₁ = ∫ f(sin2x) sinx dx at (π/2,0) and
I₂= ∫ f(cos2x) cosx dx at (π/4,0), then I₁/I₂=
a) 1 b) √2/3 c) √2/1 d) 2
12) If m and n are degree and order of (1+ y₁²)²⁾³ = y₂, then the value of (m + n)/(m - n) is
a) 3 b) 4 c) 5 d) 2
13) The solution of dy/ddx = x + y, y(0)= 0 is
a) y= eˣ + x - 1
b) eˣ - x - 1
c) e⁻ˣ - x - 1
d) e⁻ˣ - x + 1
14) If Iₙ = ∫ tanⁿx dx, where n is a positive integer, than I₁₀ + I₈ is
a) 1/9 b) 1/8 c) 1/7 d) 9
15) ³₋₃∫ [f(x)+ f(-x)][g(x) - g(-x)] dx is equal to
a) 0 b) 2 ³₋₃∫ f(x) dx c) 2 ³₀∫ f(x) g(x) dx d) 2 ³₀∫ [f(x) - g(x)] dx
16) The general solution of the differential equation d²y/dx² = e²ˣ + e⁻ˣ is
a) e²ˣ + e⁻ˣ + c₁x + c₂
b) (1/4) e²ˣ - e⁻ˣ + c
c) (1/4) e²ˣ + e⁻ˣ + c₁x + c₂
d) (1/4) e⁻²ˣ + e⁻ˣ + c₁x + c₂
17) The solution of the differential equation x dy/dx + y = (1/x²) at (1,2) is
a) x²y + 1= 3x
b) x²y + 1= 0
c) xy + 1= 3x
d) x²(y + 1)= 3x
18) When x> 0, then ∫ cos⁻¹{(1- x²)/(1+ x²) dx is
a) 2[x tan⁻¹x - log(1+ x²)]+ c
b) 2[x tan⁻¹x + log(1+ x²)]+ c
c) 2x tan⁻¹x + log(1+ x²) + c
d) 2x tan⁻¹x - log(1+ x²) + c
19) ∫ dx/{(x +1) √x} dx=
a) tan⁻¹√x + c
b) 2tan⁻¹x + c
c) 2tan⁻¹√x + c
d) 2tan⁻¹ √x³ + c
20) If ∫ dx/(5+ 4 cosx) = P tan⁻¹(Q tan(x/2))+ c, then the value of P equal to
a) 1/3 b) 1/6 c) 2/3 d) 2/9
21) ⁸₄∫ √x/(√x + √(12-x)) dx =
a) 4 b) 2 c) 1 d) 1/2
22) ∫ √{(1+ cos2x)/2} dx at (π,0) =
a) 4 b) 2 c) -2 d) 0
23) ∫ [sinx + [4x/π]] dx at (π,3π/4)= ? Where [.] denotes greatest integer function
a) 0 b) 5π/4 squnits c) 7π/4 squnits d) 3π/4 squnits
24) The value of ∫ (x sinx²)/(sinx² + sin(log 6 - x²)) dx is
a) (1/4) log(3/2)
b) (1/2) log(3/2)
c) log(3/2)
d) (1/6) log(3/2)
25) Evaluate ∫ (x +1)/{(x +2) √(x +3)} dx
a) 2√(x +3) - log[{√(x +3) -1}/{√(x +3)+1}] + c
b) 2√(x +3) + log[{√(x +3) -1}/{√(x +3) - 1}] + c
c) 2√(x + 3) - log[{√(x -3) +1}/{√(x -3)-1}] + c
d) √(x +3) - log[{√(x -3) +1}/{√(x -3)-1}] + c
Practice Paper BITSAT
1) A variable line is drawn through the origin to cut two fixed lines L₁, L₂ at A₁, A₂. A point P is taken on the variable line such that (α + β)/OP = α/OA₁ + β/OA₂. The locus of P is
a) a straight line forming an isosceles triangle with L₁, L₂
b) a straight line forming a scalene triangle with L₁, L₂
c) a straight line concurrent with L₁, L₂
d) not a straight line.
2) If P and Q are the points of intersection of the circles x²+ y²+ 3x + 7y + 2p - 5=0 and x²+ y²+ 2x + 2y - p² =0 then there is a circle through P, Q and (1,1) for
a) all values of p
b) all except one value of p
c) all except two values of p
d) exactly one value of p
3) If 0< θ, φ< π/2 and sinθ = 1/2, cosφ= 1/3, then (θ+ φ) belongs to
a) (π/3, π/2) b) (π/2, 2π/3) c) (2π/3, 5π/6) d) (5π/6, π)
4) 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 zero
b) right angled isosceles
c) equilateral
d) obtuse angled isosceles
5) If (b - c)x² + (c - a)xy + (a - b)y² is a perfect square, then a,b,c are in
a) AP b) GP c) HP d) none
6) Any point on the parabola with focus (0, 1) and directrix x +2=0 is
a) (t²+1, 2t -1) b) (t²+1, 2t + 1) c) (t², 2t) d) (t²-1, 2t +1)
7) The domain of f(x) sin⁻¹log₂(x²/2) is
a) (0, ∞) b) (0,√2) c) (-1,0) U (0,1) d) [-2,-1] U[1,2]
8) If the curve y= ax²+ bx+ c passes through the point (1,2) and is to be tangent to the line x= y at the origin then, y(-1)=
a) 0 b) 2 c) -2 d) -1
9) The length of the longest interval in which f(x)= 3 sinx - 4 sin³x is increasing, is
a) π/3 b) π/2 c) 3π/2 d) π
10) A pair of tangents are drawn from the point (-1,1) to x² - 4x + 4y²= 0. If θ is the angle between them, then θ is
a) 2/5 b) 4/5 c) 6/5 d) 3√3/5
11) If the equation 2 sinx + 3 cos ax = 5 has a solution, then a is
a) rational b) irrational c) odd integer d) even integer
12) The normal at any point P(x,y) on a curve meets the x-axis at N. If OP= PN, where I is the origin, then the curve is a/an
a) circle b) parabola c) ellipse d) none
13) Let (-1,1)--> B, f(x)= tan⁻¹{2x/(1- x²)} is the one-one and onto. Then B=
a) (0,π/2) b) (0, - π/2) c) (-π/2,π/2) d) (-π/2, -π/2)
14) A chord of the hyperbola 4x²- 9y²= 36 is bisected at the point (3,5). The distance of the origin from a chord is
a) 63/√241 b) 50/√241 c) 13/√19 d) 13/√190
15) The number log₂7 is
a) an integer b) prime c) rational d) irrational
16) Triangles are formed with vertices of a regular polygonal 20 sides. The probability that no side of the polygon is a side of the triangle, is
a) 25/57 b) 30/57 c) 35/57 d) 40/57
17) 2 2 1
If A= 1 3 1
1 2 2
Then A³- 7A²+ 10A=
a) 5I + A b) 5I - A c) A- 5I d) 6I
18) The lines x= ay + b, z= cy + d and x= a'y + b', z= c'y + d' are perpendicular. then
a) a/a' + c/c'= - 1
b) a/a' + c/c'= 1
c) aa' + cc'= - 1
d) aa' + cc'= 1
19) In a triangle ABC , if (2cosA)/a + (cosB)/b + (cosC)/c = a/bc + b/ca, then A=
a) 30° b) 45° c) 60° d) 90°
20) Let a,b,c be unit vectors and α, β, γ be the angles between b.c, c.a, a.b respectively. If |a+ b+ c|= 1, then cosα + cosβ + cosγ =
a) 0 b) -1 c) 2 d) -3/4
21) In an AP of which 1 is the first trem, if the second, tenth and thirty fourth terms form a GP, then the 4th term of the AP is
a) 1/2 b) 1 c) 3/2 d) 2
22) The angle between the lines 2x = 3y = -z and 6x = - y = 4z is
a) π/6 b) 0 c) π/3 d) π/2
23) Three of the six vertices of a regular Hexagon are chosen at random. The probability that the triangle with three vertices is equilateral, is
a) 1/2 b) 1/5 c) 1/10 d) 1/20
24) b²+ c² a² a²
If b² c²+a² b² = λa²b²c²
c² c² a²+b²
Then λ=
a) 2 b) -2 c) 4 d) -4
25) ∫ dx/{x²(x⁴+ 1)³⁾⁴ =
a) (x⁴+ 1)¹⁾⁴/x + c
b) - (x⁴+ 1)¹⁾⁴/x + c
c) √(x⁴+ 1)/x + c
d) - √(x⁴+ 1)/x + c
26) The volume of the greatest right circular cone, that can be described by the revolution about a side of a right angled triangle of hypotenuse 1 unit is
a) 2π/3 b) 2π/3 √3 c) 2π/9√3 d) 4π/9√3
27) If tanA= (x sinB)/(1- x cosB), tanB = (y sinA)/(1- y cosA), then sinA/sinB=
a) x+ y b) x/y c) xy d) x - y
28) The number of values of x such that 5 tan⁻¹x + 3 cot⁻¹x = 2π is
a) 0 b) 1 c) 2 d) -2
29) If ff(x) is differentiable, f(1)= 0 and lim h→₀ f(1+ h)/h = 5, then f'(1)=
a) 4 b) 3 c) 6 d) 5
30) Let a,b,c,d be such that (a x b) x (c x d)= 0. Let p₁ and p₂ be the planes determined by the pairs of vector, a, b and c, d respectively. The angle between the planes p₁, p₂ is
a) 0 b) π/4 c) π/3 d) π/2
31) Let f(x)= [x] sin{π/[x+1]}. The points of discontinuity are
a) Z b) Z - {0} c) Z - {1} d) Z - {-1}
32) ∫ (sinx + cosx)²/√(1+ sin2x) dx at (π/2,0)=
a) 0 b) 1 c) 2 d) 3
33) The area bounded by the curve y= 6+ 4x - x² and the line 2x - y =2 is
a) 12 b) 24 c) 36 d) 48
34) The entire graph of the equation y= x²+ kx - x + 9 lies above the x-axis, then
a) k> 7 b) -5< k < 7 d) k> -5 d) none
35) The value least number of times a fair coin must be tossed so that the probability of getting at least one head is greater than 0.99, is
a) 7 b) 8 c) 9 d) 6
36) The number of integral values of k for which the equation 7 cosx + 5 sinx = 2k +1 has a solution is
a) 4 bb) 8 c) 12 d) 10
37) The eccentricity of the ellipse 16x²+ 7y²= 112 is
a) 7/16 b) 3/√17 c) 3/4 d) 2/7
38) The function f(x)= (tan(|x - π|π)/(1+ [x]²) is
a) discontinuous at some x
b) continuous at all x, but f'(x) does not exist for some x.
c) f'(x) exist for all x, but f"(x) does not exist for some x.
d) f"(x) exists for all x.
39) The curve described parametrically by x= t²+ t + 1, y= t²- t +1 represents
a) a pair of lines
b) an ellipse
c) a parabola
a) A hyperbola
40 The area bounded by the curve|x + y| + |x - y|= 1 is
a) 1/2 b) 1 c) 2 d) 4
41) If the equations k(6x²+ 3)+ rx+ 2x² -1=0 and 6k(2x²+1)+ px + 4x² -2=0 have both the roots common, then 2r - p=0
a) 2 b) 1 c) 0 d) k
42) If sin⁻¹x + sin⁻¹y + sin⁻¹z =3π/2, then (x + y + z)²
a) 0 b) 1 c) 4 dd) 9
43) If dy/dx = (x - y)/(x + y), y(1)= 1, then (y(0))²=
a) 1 b) 2 c) 3 d) 4
44) There are 3 copies each of 4 different books. The number of ways they can be arranged in a shelf is
a) 369600 b) 400400 c) 420600 d) 440720
45) The coefficient of x⁴ in (x/2 - 3/x²)¹⁰ is
a) 405/256 b) 504/259 c) 450/263 d) none
Sample Paper IIT-JEE (2012)
Single Correct Choice Type
1) In the sequence 1, 2, 2, 4, 4,4,4, 8, 8, 8, 8, 8, 8,8,8......., where n consecutive terms have the value n, the 1025th term is
a) 2⁹ b) 2¹⁰ c) 2¹¹ d) 2⁸
2) If aₙ> 1 for all n ∈N then log₂a₁ + log₃a₂+ .....logₐₙaₙ₋₁ + logₐ₁aₙ has the minimum value
a) 1 b) 2 c) 0 d) none
3) If |z|= maximum [|z -1|, |z +1] then
a) |z + conjugate of z|= 1/2
b) z + conjugate of z= 1
c) |z + conjugate of z|= 1 d) none
4) If θ ∈ [0,5π] and r ∈ R such that 2 sinθ = r⁴ - 2r²+ 3 then the maximum number of values of the pair (r, θ) is
a) 8 b) 10 c) 6 d) none
5) The number of real solutions of tan⁻¹√{x(x +1)} + sin⁻¹√(x²+ x+1)=π/2 is
a) zero b) one c) two d) infinite
6) In ∆ ABC, cosB, cosC+ sinB. sinC. sin²A = 1. Then the triangle is
a) right angled isosceles
b) isosceles whose equal angles are greater than π/4
c) equilateral d) none
Multiple Correct Choice Type
7) From a point p, the chord of contact to the ellipse
x²/a + y²/b = (a + b) ....(1)
Touches the ellipse x²/a + y²/b = 1 ....(2)
Then locus of the point P is
a) director circle of (1)
b) auxiliary circle of (2)
c) x² + y² = (a + b)²
d) x² + y² = (a² + b²)
8) On the interval I=[-2,2] the function
f(x)={(ₓ₊₁)ₑ-(1/|x| + 1/x), x ≠ 0
0 x, 0
a) is continuous for all x ∈ I -{0}
b) is continuous for all x ∈ I
c) assumes all intermediate values for f(-2) to f(2)
d) has a maximum value equal to 3/e
9) The point P(α, α+1) will lie inside the ∆ ABC whose vertices are A(0,3), B(-2,0) and C(6,1) if
a) α= -1 b) α= -1/2 c) α= 1/2 d) -6/7<α < 3/2
10) If x²+ mx+1=0 and (b - c)x²+ (c - a)x + (a - b)= 0 have both roots common then
a) m= -2 b) m=-1 c) a,b,c are in AP d) a,b,c are in HP
Assertion - Reason Type
This section contains 4 questions numbered 11 to 14. Each question contains Statement- I(Assertion) and Statement- 2(Reason). Examine the statement carefully and answer the questions according to the instructions given below :
a) If both Statement- 1 and statement -2 are correct and Statement- 2 is the proper reason of Statement- 1
b) If both Statement- 1 and Statement- 2 are correct and Statement- 2 is not the proper reason of Statement- 1.
c) If Statement- 1 is correct and Statement- 2 is wrong
d) if Statement- 1 is wrong and Statement -2 is correct.
11) Given two straight lines whose equations are:
(x -3)/1= (y -5)/-2 = (z -7)/1 and (x +1)/7 = (y +1)/-6 = (z +1)/1
Statement- 1: The line of shortest distance between the given lines is perpendicular to the plane x + 3y + 5z= 0
Statement- 2: The direction ratios of the normal to the plane ax + by+ cz+ d=0 are a/d, b/d, c/d.
12) Statement- 1: The probability of getting a tail most of the time in 10 tosses of a unbiased coin is(1/2) {1- 10!/(2¹⁰ 5!5!)}
Statement- 2: ²ⁿC₀ + ²ⁿC₁ + ²ⁿC₂ + ....+ ²ⁿCₙ = 2²ⁿ⁻¹, n ∈ N
13) Let a, b, C be real such that ax²+ bx + c=0 and x²+ x+ 1=0 have a common root.
Statement- 1: a= b = c
Statement- 2 Two quadratic equation with real co-efficients cannot have one imaginary root common.
14) Let ∫ cos√t dt at (x²,1),
Statement- 1: F'(x)= cosx
Statement- 2: If ˣₐ∫ φ(t) dt, then f'(x)= φ(x)
Paragraph Type
This section consists 3 paragraph, Based upon each of the paragraph 3 multiple choice questions have to be answered. Each question has 4 choices a, b, c, d out of which only one is correct:
Paragraph for Question 15 to 17
Perpendicular are drawn from the focus S of the parabola y= ax²+ bx + c upon the tangents to the parabola at the point A(-1,0) and B (1,2) meeting them at the point C(-1/4, 9/4) and D(3/3,9/4) respectively.
15) The coordinates of the focus are
a) (1/2,9/4) b) (1/2, 2) c) (0,9/4) d) (1,2)
16) The normal at A and B intersect at a point P. The foot of the third normal through the point P is
a) (1/2,9/4) b) (-1/2, 5/4) c) (0,2) d) (3/2,5/4)
17) Area of the region bounded by the parabola and the x-axis is
a) 5/4 b) 5 c) 5/2 d) 9/2
Paragraph for Question 18 to 20
Let an ellipse having major axis and minor axis parallel to the x-axis and y-axis respectively. Its two foci S and S' are (2,1), (4,1) and a line x + y = 9 is a tangent to this ellipse at point P.
18) Eccentricity of the ellipse is
a) 1/√12 b) 1√13 c) 1/2 d) none
19) Length of major axis
a) √13 b) 2√11 c) √52 d) 2√12
20) The latus rectum of ellipse
a) 12/13 b) 12/√13 c) 24/√13 d) 25/√13
Paragraph for Question 21 to 23
The function sin⁻¹x, cos⁻¹x, tan⁻¹x, cot⁻¹x, cosec⁻¹x and sec⁻¹x, are called inverse circular on inverse trigonometrical functions which are defined as follows:
sin⁻¹x -1≤x ≤ 1 π/2 ≤ sin⁻¹x ≤ 3π/2
Cos⁻¹x -1≤x ≤ 1 -π ≤ cos⁻¹x ≤ 0
tan⁻¹x x ∈R -π/2 ≤ tan⁻¹x ≤ π/2
cosec⁻¹x |x| ≥ 1 π/2 ≤ cosec⁻¹x ≤ 3π/2 ≠π
Sec⁻¹x |x|≥ 1 π ≤ sec⁻¹x 0 ≠ -π/2
Cot⁻¹x x ∈R 0 < cot⁻¹x < π
21) For x∈ [0,1], sin⁻¹x is equal to
a) Cos⁻¹√(1- x²)+ π
b) Cos⁻¹√(1- x²)+ π/2
c) Cos⁻¹√(1- x²)
d) none
Single Correct Choice Type
This section contains 9 multiple choice questions. Each question has 4 choices a, b, c, d out of which only one is correct:
1) Two circles have the equations x²+ y²- 4x - 6y - 8=0 and x²+ y²- 2x - 3 =0 then
a) they cut each other.
b) they touch each other.
c) One circle lies inside the other
d) one circle lies wholly outside the other
2) Let the r-th term, tᵣ of a series is given by tᵣ = r/(1+ r²+ r⁴). Then lim ₙ→∞ ⁿᵣ₌₁∑ tᵣ is
a) 1/4 b) 1 c) 1/2 d) none
3) In a ∆ ABC, angle B= 90° and b + a= 4. The area of the triangle is the maximum when angle C is
a) π/4 b) π/6 c) π/3 d) none
4) There are 4 white and 3 black balls in a bo. In another box there are 3 white and 4 black balls. An unbiased dice is rolled. if it shows a number less than or equal to 3 then a ball is drawn from the first box but , if it shows a number more than 3 then a ball is drawn from the second box. If the ball drawn is black then the probability that the ball was drawn from the first box is
a) 1/2 b) 6/7 c) 4/7 d) 3/7
5) The number of non-integral solutions of [|4x - x²| -1]= 3 is
a) 4 b) 2 c) 3 d) none
6) If the normal at any point P of an ellipse meets the major and minor axis at G and G' and OF be the perpendicular drawn from centre O to this normal then PF, PG must be equal to
a) b² b) a² c) ab d) none
7) The normal at an end of latus rectum of the ellipse x²/a² + y²/b²= 1 passes through an end of the manor axis is
a) e⁴+ e²= 1 b) e³+ e²= 1 c) e+ e²= 1 d) e³+ e= 1
8) lim ₙ→∞ {(3n +8)/(3n +5)}⁵ⁿ⁺⁹ is equal to
a) 3e⁵ b) e⁵ c) e³ d) none
9) Let AB = 3i + j - k and AC= i - j + 3k. If the point P on the line segment BC is equidistant from AB and AC then AP is
a) 2i - k b) i - 2k c) 2i + k d) none
Assertion -Reason Type
This section contains for questions numbered 10 to 13. Each question contains Statement- 1(Assertion) and Statement -2(Reason). Examine the statements carefully and answer the questions according to the instructions given below:
a)If both Statement- 1 and Statement- 2 are correct answer and statement -2 is the proper reason of Statement- 1
b) If both Statement -1 and Statement- 2 are correct and Statement- 2 is not the proper reason of Statement- 1
c) If Statement- 1 is correct and Statement- 2 is wrong
d) If Statement- 1 is wrong and Statement -2 is correct.
10) let (1+ x)³⁶= a₀ + a₁x + a₂x² +....+ a₃₆x³⁶
Statement- 1: a₀ + a₃ + a₆+...+a₃₆ = (2/3) (2³⁵+ 1)
Statement- 2: a₀ + a₁ + a₂ +...+ a₃₆= 2³⁶ and
a₀ + a₂ + a₄ +....+a₃₆ = 2³⁵
11) x is unimodular complex number
Statement- 1: arg(z²+ conjugate z) arg z
Statement- 2: conjugate of z= cos(arg z) - i sin(arg z)
12) a,b,c are three unequal positive numbers.
Statement- 1: The product of their sum and the sum of their reciprocals exceeds 9.
Statement- 2: AM of n positive numbers exceeds their HM
13) Statement -1: 1. 3. 5. ....(2n -1)> nⁿ, n ∈N
Statement- 2: The sum of the first n odd natural numbers is equal to n².
Paragraph Type
This section consists of 2 paragraph. Based upon each of the paragraph 3 multiple choice questions have to be answered . Each question has 4 choices a, b, c, d , out of which only one is correct.
Paragraph for Question 14 to 16
If z₁, z₂ be complex number representing two points A and B, then we define the complex slope of the line AB as μ = (z₁ - z₂)/(z₁- z₂), it can be noted that | μ|= 1 and μ remains same for any two points on the line AB, since if z₃ - z₄ be complex numbers representing some other points the same line, then
μ'= (z₃ - z₄)/(z₃ - z₄) =λ(z₁ - z₂)/λ(z₁ - z₂) (as z₃ - z₄) = λ(z₁ - z₂), λ real) = (z₁ - z₂)/ (z₁ - z₂)= μ
14) The complex slope of the line conjugate a.z + a. Conjugate z + b=0, where a is complex and b is real is
a) a/2 b) - a/a c) conjugate of a/a d) - conjugate of a/a
15) If the complex slope of a line which is not parallel to y-axis is cosθ + i sin θ, then the line makes an angle θ with x-axis, φ must be
a) 2θ b) 90° - θ c) θ/2 d) θ
16) If μ and μ' be complex slope of two perpendicular lines, then
a) μμ' =1 b) μμ' = - 1 c) μ + μ' =0 d) none
Paragraph for Question 17 and 19
Let f: [2, ∞) --> [1, ∞) defined by f(x)= ₂(x⁴- 4x²) and g: [π/2, π] --> A defined by g(x)= (sinx +4)/(sinx -2) be two invertible functions, then
17) f⁻¹(x) is equal to
a) - √[2+ √(4+ log₂x)]
b) √[2+ √(4+ log₂x)]
c) √[2- √(4+ log₂x)] d) none
18) The set A is equal to
a) [-5,-2] b) [2,5] c) [-5,2] d) [-5,-2]
19) The domain of f⁻¹g⁻¹(x) is
a) [-5, sin 1]
b) [-5, (sin 1)/(2- sin 1)]
c) [-5, (4+ sin 1)/(2- sin 1)]
d) [(4+ sin 1)/(2- sun 1), 2]
Matrix Match Type
The section 3 questions. Each question contains Statements given in two columns, which have to be matched . The statements in Column- I labelled A, B, C, D, while the statements in Column- II are labelled p,q,r and s. Anygiven statement in column I can have correct matching with ONE or MORE statement/s in Column- II. The appropriate bubbles corresponding to the answers to these questions have to be darkened as illustrated in the following example. If the correct matches are (A)- (p), (s), B- (q) and (r); (C)- (p) and (q) and D- (s), then the correct darkening bubbles will look like the following:
20) COLUMN I
A) Let f: R---> R be a periodic function such that
f(T+ x)= 1+ [1 -3f(x)+ 3(f(x))² - (f(x))³]¹⁾³ where T is fixed positive number, then period of f(x) is AT, where A=
B) The area between the curve y= 2x⁴- x², the x-axis and the ordinates of two minima of the curve is B/120, where B is
C) ⁴₀∫ f(x)/(f(x) + f(4- x)) dx =
D) f(x)= (1- sinx)/(π- 2x²) . (Log sinx)/Log(1+π²- 4πx + 4x²) is continuous at x=π/2 then -1/k equals to
COLUMN II
p) 3
q) 2
r) 7
s) 64
21) The parabola y²= 4ax has a chord AB joining points A(at₁², 2at₁) and B(at₂², 2at₂). Then match the following:
Column I
A) AB is a normal chord
B) AB is a focal chord
C) AB subtends 90° at point (0,0)
D) AB inclined at 45° to the axis of parabola
Column II
p) t₂ = - t₁ - 1/2
q) ₂ = -4/t₁
r) t₂= -1/t₁
s) t₂ = -t₁ - 2/t₁
22) Match the Column
Column I
A) If |x²- x|≥ x²+ x, then x
B) |x + y|> x - y, where x > 0, then y=
C) If log₂x≥ log₃x², then x=
D) [x]+ 2≥ [x] (where [.] denotes the greatest integer function
COLUMN II
p) [0, ∞)
q) (-∞,0)
r) [-1, ∞)
s) (0,1]
climate 45
∈ ²⁻ ²⁴²⁻ ⁻¹²²²²² θ θθθθθ ²²⁻¹²²∫³₁∫² ∫ ∞ μ λ α β γ δ ₁₂₃₄ₓₐˣ⁻ˣᵅⁿₙ φ ₁₂₃ ₙ ₓ ₐ
