1. If three distinct numbers are chosen randomly from the first 100 natural numbers, then the probability that all three of them are divisible by both 2 and 3 is
a. 4/33 b. 2/33 c. 4/25 d. 4/35 e. 4/1155
2.If a, b and c are three integers such that at least two of them are unequal, and ω (≠1) is a cube root of unity, then the least value of the expression |a + bω + cω²| is
a. 1
b. 0
c. √3/2
d. 1/2
3. If y = y(x) is a function of x satisfying the relation x cos y + y cos x = π, then y"(0) equals
a. 1
b. -1
c. -π
d. π
4. The area bounded by the parabolas y = (x+1)² and y = (x-1)² and the line y = 1/4 is
a. 1/6 sq. units
b. 4/3 sq. units
c. 1/3 sq. units
d. 4 sq units
5. A unbiased cubical die is rolled until one appears. The probability that an even number of trials is required is
a. 5/6
b. 6/11
c. 5/11
d. 1/6
Sunday, November 29, 2009
IIT JEE Mathematics - Revision Problem Set 3
1. One Indian couple (wife and husband) and four Americal couples are to be seated randomly around a circular table. Then the conditional probability that the Indian man is seated adjacent to his wife given that each American man is seated adjacent to his wife is
a. 1/2 b. 1/3 c. 1/5 d. 1/8 e.2/5
2. The number of values of x in the interval [0,5π] satisfying the equation 3 sin²x 7 sin x+2 = 0 is
a. 0
b. 5
c. 6
d. 10
3. Which of the following n umber(s) is/(are) rational?
a. sin 15°
b. cos 15°
c. sin 15° cos 15°
d. sin 15° cos 75°
4. If the vertices P,Q,R of a triangle are rational points, which of the following points of the triangle PQR is(are) always rational point(s)?
a. centroid
b. incentre
c. circumcentre
d. orthocentre.
5. The number of values of x where the function f(x) = cos x + cos [(√2)x] attains its maximum is
a. 0
b. 1
c. 2
d. infinite
Questions 2 to 5 are from JEE 1998 paper
a. 1/2 b. 1/3 c. 1/5 d. 1/8 e.2/5
2. The number of values of x in the interval [0,5π] satisfying the equation 3 sin²x 7 sin x+2 = 0 is
a. 0
b. 5
c. 6
d. 10
3. Which of the following n umber(s) is/(are) rational?
a. sin 15°
b. cos 15°
c. sin 15° cos 15°
d. sin 15° cos 75°
4. If the vertices P,Q,R of a triangle are rational points, which of the following points of the triangle PQR is(are) always rational point(s)?
a. centroid
b. incentre
c. circumcentre
d. orthocentre.
5. The number of values of x where the function f(x) = cos x + cos [(√2)x] attains its maximum is
a. 0
b. 1
c. 2
d. infinite
Questions 2 to 5 are from JEE 1998 paper
Monday, August 24, 2009
IIT JEE Mathematics - Revision Problem Set 2 - Past JEE Questions
1. Find the equation of the normal to the curve
y = (1+x)y + sin-1(sin2) at x = 0.
2. Find the coordinates of the point at which the circles x²+y²-4x-2y = -4 and x²+y²-12x-8y = -36 touch each other.
3. In a triangle ABC, D and E are points on BC and AC respectively, such that BD = 2DC ad AE = 3EC. Let P be the point of intersection of AD and BE. Find BP/PE. (The original problem said using vector methods).
4. Determine te smallest positive value of x (in degrees) for which
tan (x+100°) = tan (x+50°) tan(x)tan(x-50°).
5. Numbers are selected at random, one at a time, from the two-digit numbers 00,01,02...,99 with replacement. An event E occurs if and only if the product of the two digits of a selected number is 18. If four numbers are selected, find the probability that the event E occurs at least 3 times.
y = (1+x)y + sin-1(sin2) at x = 0.
2. Find the coordinates of the point at which the circles x²+y²-4x-2y = -4 and x²+y²-12x-8y = -36 touch each other.
3. In a triangle ABC, D and E are points on BC and AC respectively, such that BD = 2DC ad AE = 3EC. Let P be the point of intersection of AD and BE. Find BP/PE. (The original problem said using vector methods).
4. Determine te smallest positive value of x (in degrees) for which
tan (x+100°) = tan (x+50°) tan(x)tan(x-50°).
5. Numbers are selected at random, one at a time, from the two-digit numbers 00,01,02...,99 with replacement. An event E occurs if and only if the product of the two digits of a selected number is 18. If four numbers are selected, find the probability that the event E occurs at least 3 times.
Saturday, August 22, 2009
Past IIT JEE Problems - Questions - Collection 1 (For JEE 2010)
1. The value of (tan x)/tan 3x, wherver defined never lies between 1/3 and 3. State whether the statement is true or false.
2. Determine a positive integer n≤5, such that
∫01 ex(x-1)ndx = 16 - 6e
3. Three circles touch one another externally. The tangents at their points of contact meet at a point whose distance from a point of contact is 4. Find the ratio of the product of the radii to the sum of th radii of the circles.
4. A lot contains 50 defective and 50 nondefective bulbs. Two bulbs are drawn at random, one at a time, with replacement. The events A,B, and C are defined as follows:
A = {the first bulb is defective}
B = {the second bulb is nondefective}
C = {the two bulbs are both defective or both nondefective}
Are A,B and C are pairwise independent?
5. Determine all values of α for which the point (α, α2) lies inside the triange formed by the lines
2x + 3y -1 = 0
x +2y - 3 = 0
5x - 6y - 1 = 0
(Source : 1992 JEE paper)
2. Determine a positive integer n≤5, such that
∫01 ex(x-1)ndx = 16 - 6e
3. Three circles touch one another externally. The tangents at their points of contact meet at a point whose distance from a point of contact is 4. Find the ratio of the product of the radii to the sum of th radii of the circles.
4. A lot contains 50 defective and 50 nondefective bulbs. Two bulbs are drawn at random, one at a time, with replacement. The events A,B, and C are defined as follows:
A = {the first bulb is defective}
B = {the second bulb is nondefective}
C = {the two bulbs are both defective or both nondefective}
Are A,B and C are pairwise independent?
5. Determine all values of α for which the point (α, α2) lies inside the triange formed by the lines
2x + 3y -1 = 0
x +2y - 3 = 0
5x - 6y - 1 = 0
(Source : 1992 JEE paper)
Tuesday, May 5, 2009
Indefinite Integration - Revision facilitator -Substitution Method
Integration by substitution
Write the substitution expression used for these integrations
1. Integrals of the form [f '(x)/f(x)]dx
2. Integrals of the functional form 1/(x²±a²)
3. Integrals of the form [1/(ax²+bx+c)]dx
4. Integrals of the form [1/√(ax²+bx+c)]dx
5. Integrals of the form [(px+q)/(ax²+bx+c)]dx
6. Integrals of the form [(px+q)/√(ax²+bx+c)]dx
7. Integrals of the functional form [P(x)/(ax²+bx+c)]dx
8. Integrals of √(a²±x²) and √(x²-a²)
9. Integrals of the functions of the form √(ax²+bx+c)dx
10. Integrals of the functions of the form (px+q)[√(ax²+bx+c)]dx
11. Integration of [(x²+1)/(x^4+λx²+1)]dx
12. Integration of Function [G(x)/(P√Q)]dx
13. Integrals of the form sin ^m x cos ^n x dx
14. Integrals of the functional form [1/(a sin²x + b cos²x +c)]dx
15. Integrals of the functional form [1/(a sin x + b cos x +c)]dx
16. Integrals of the functional form [(a sin x + b cos x)/(c sin x + d cos x)]dx
17. Integrals of [(a sin x+b cos x +c)/(p sin x + q cos x +r)] dx
Write the substitution expression used for these integrations
1. Integrals of the form [f '(x)/f(x)]dx
2. Integrals of the functional form 1/(x²±a²)
3. Integrals of the form [1/(ax²+bx+c)]dx
4. Integrals of the form [1/√(ax²+bx+c)]dx
5. Integrals of the form [(px+q)/(ax²+bx+c)]dx
6. Integrals of the form [(px+q)/√(ax²+bx+c)]dx
7. Integrals of the functional form [P(x)/(ax²+bx+c)]dx
8. Integrals of √(a²±x²) and √(x²-a²)
9. Integrals of the functions of the form √(ax²+bx+c)dx
10. Integrals of the functions of the form (px+q)[√(ax²+bx+c)]dx
11. Integration of [(x²+1)/(x^4+λx²+1)]dx
12. Integration of Function [G(x)/(P√Q)]dx
13. Integrals of the form sin ^m x cos ^n x dx
14. Integrals of the functional form [1/(a sin²x + b cos²x +c)]dx
15. Integrals of the functional form [1/(a sin x + b cos x +c)]dx
16. Integrals of the functional form [(a sin x + b cos x)/(c sin x + d cos x)]dx
17. Integrals of [(a sin x+b cos x +c)/(p sin x + q cos x +r)] dx
Sunday, February 15, 2009
Practice Problems - February - 2009 - XI Portion
Chapter: Elementary Trigonometry
Multiple Answer Multiple Choice
Which of the following statements are correct?
a. sin 1 > sin 1°
b. tan 2 <0
c. tan 1 > tan 2
d. tan 2 < tan 1 <0
See for discussion orkut community topic
http://www.orkut.co.in/Main#CommMsgs.aspx?cmm=39291603&tid=5302403071366110401
Chapter: Solutions of Triangles
If in a triangle ABC, sin A, sin B, and sin C are in arithmetic progression, then
a. the altitudes are in A.P.
b. the altitudes in G.P
c. altitudes are in H.P.
d. the altitudes are equal
Multiple Answer Multiple Choice
Which of the following statements are correct?
a. sin 1 > sin 1°
b. tan 2 <0
c. tan 1 > tan 2
d. tan 2 < tan 1 <0
See for discussion orkut community topic
http://www.orkut.co.in/Main#CommMsgs.aspx?cmm=39291603&tid=5302403071366110401
Chapter: Solutions of Triangles
If in a triangle ABC, sin A, sin B, and sin C are in arithmetic progression, then
a. the altitudes are in A.P.
b. the altitudes in G.P
c. altitudes are in H.P.
d. the altitudes are equal
Friday, February 13, 2009
A Problem in elementary Trigonometry
Multiple Answer Multiple Choice
Which of the following statements are correct?
a. sin 1 > sin 1°
b. tan 2 <0
c. tan 1 > tan 2
d. tan 2 < tan 1 <0
See for discussion orkut community topic
http://www.orkut.co.in/Main#CommMsgs.aspx?cmm=39291603&tid=5302403071366110401
Which of the following statements are correct?
a. sin 1 > sin 1°
b. tan 2 <0
c. tan 1 > tan 2
d. tan 2 < tan 1 <0
See for discussion orkut community topic
http://www.orkut.co.in/Main#CommMsgs.aspx?cmm=39291603&tid=5302403071366110401
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