1226 |
Using mathematical induction, prove that for n > 1
Using mathematical induction, prove that for n > 1
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IIT 1986 |
|
1227 |
If A and B are two independent events such that P (A) > 0 and P (B) ≠ 1 then is equal to a) b) c) d)
If A and B are two independent events such that P (A) > 0 and P (B) ≠ 1 then is equal to a) b) c) d)
|
IIT 1980 |
|
1228 |
Let p ≥ 3 be an integer and α, β be the roots of x2 – (p + 1) x + 1 = 0. Using mathematical induction show that αn + βn i) is an integer ii) and is not divisible by p.
Let p ≥ 3 be an integer and α, β be the roots of x2 – (p + 1) x + 1 = 0. Using mathematical induction show that αn + βn i) is an integer ii) and is not divisible by p.
|
IIT 1992 |
|
1229 |
Tangents are drawn to the circle from a point on the hyperbola . Find the locus of the midpoint of the chord of contact.
Tangents are drawn to the circle from a point on the hyperbola . Find the locus of the midpoint of the chord of contact.
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IIT 2005 |
|
1230 |
If x is not an integral multiple of 2π use mathematical induction to prove that
If x is not an integral multiple of 2π use mathematical induction to prove that
|
IIT 1994 |
|
1231 |
Let P (x1, y1) and Q (x2, y2), y1 < 0, y2 < 0 be the end points of the latus rectum of the ellipse x2 + 4y2 = 4. The equations of the parabolas with latus rectum PQ are a) b) c) d)
Let P (x1, y1) and Q (x2, y2), y1 < 0, y2 < 0 be the end points of the latus rectum of the ellipse x2 + 4y2 = 4. The equations of the parabolas with latus rectum PQ are a) b) c) d)
|
IIT 2008 |
|
1232 |
Prove by induction that Pn = Aαn + Bβn for all n ≥ 1 Where α and β are roots of the quadratic equation x2 – (1 – P) x – P (1 – P) = 0, P1 = 1, P2 = 1 – P2, . . ., Pn = (1 – P) Pn – 1 + P (1 – P) Pn – 2 n ≥ 3, and ,
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IIT 2000 |
|
1233 |
Consider the points P: (−sin (β – α), cosβ) Q: (cos (β – α), sinβ) R: (−cos{(β – α) + θ}, sin (β – θ)) where 0 < α, β, θ < then a) P lies on the line segment RQ b) Q lies on the line segment PR c) R lies on the line segment QP d) P, Q, R are non–collinear
Consider the points P: (−sin (β – α), cosβ) Q: (cos (β – α), sinβ) R: (−cos{(β – α) + θ}, sin (β – θ)) where 0 < α, β, θ < then a) P lies on the line segment RQ b) Q lies on the line segment PR c) R lies on the line segment QP d) P, Q, R are non–collinear
|
IIT 2008 |
|
1234 |
If the integers m and n are chosen at random between 1 and 100 then the probability that a number of form is divisible by 5, equals a) b) c) d)
If the integers m and n are chosen at random between 1 and 100 then the probability that a number of form is divisible by 5, equals a) b) c) d)
|
IIT 1999 |
|
1235 |
Find all solutions of a) b) c) d)
Find all solutions of a) b) c) d)
|
IIT 1983 |
|
1236 |
(One or more correct answers) Let 0 < P (A) < 1, 0 < P (B) < 1 and P (A ∪ B) = P (A) + P (B) – P (A ∩ B) then a) P (B/A) = P (B) – P (A) b) P (Aʹ – Bʹ) = P (Aʹ) – P (Bʹ) c) P (A U B)ʹ = P (Aʹ) P (Bʹ) d) P (A/B) = P (A)
(One or more correct answers) Let 0 < P (A) < 1, 0 < P (B) < 1 and P (A ∪ B) = P (A) + P (B) – P (A ∩ B) then a) P (B/A) = P (B) – P (A) b) P (Aʹ – Bʹ) = P (Aʹ) – P (Bʹ) c) P (A U B)ʹ = P (Aʹ) P (Bʹ) d) P (A/B) = P (A)
|
IIT 1995 |
|
1237 |
The smallest positive integer n for which is a) 8 b) 12 c) 12 d) None of these
The smallest positive integer n for which is a) 8 b) 12 c) 12 d) None of these
|
IIT 1980 |
|
1238 |
In a triangle ABC, let ∠ C = . If r is the inradius and R is the circumradius of the triangle then 2(r+R) = …………. a) a+b b) b+c c) c+a d) a+b+c
In a triangle ABC, let ∠ C = . If r is the inradius and R is the circumradius of the triangle then 2(r+R) = …………. a) a+b b) b+c c) c+a d) a+b+c
|
IIT 2000 |
|
1239 |
If z = x + iy and ω = then |ω| =1 implies that in the complex plane a) z lies on the imaginary axis b) z lies on the real axis c) z lies on unit circle d) none of these
If z = x + iy and ω = then |ω| =1 implies that in the complex plane a) z lies on the imaginary axis b) z lies on the real axis c) z lies on unit circle d) none of these
|
IIT 1983 |
|
1240 |
Show that the value of wherever defined a) always lies between and 3 b) never lies between and 3 c) depends on the value of x
Show that the value of wherever defined a) always lies between and 3 b) never lies between and 3 c) depends on the value of x
|
IIT 1992 |
|
1241 |
The orthocenter of the triangle formed by the lines lies in the quadrant number . . . . .
The orthocenter of the triangle formed by the lines lies in the quadrant number . . . . .
|
IIT 1985 |
|
1242 |
Fill in the blank The value of f (x) = lies in the interval ……………. a) b) c) d)
Fill in the blank The value of f (x) = lies in the interval ……………. a) b) c) d)
|
IIT 1983 |
|
1243 |
Let and intersect the line at P and Q respectively. Bisector of the acute angle between L1 and L2 intersects L3 in R. Statement 1 – The ratio PR : RQ equals because Statement 2 – In any triangle, bisector of an angle divides the triangle into two similar triangles. The question contains Statement 1(assertion) and Statement 2(reason). Of these statements, mark correct choice if a) Statement 1 and 2 are true. Statement 2 is a correct explanation for statement 1. b) Statement 1 and 2 are 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.
Let and intersect the line at P and Q respectively. Bisector of the acute angle between L1 and L2 intersects L3 in R. Statement 1 – The ratio PR : RQ equals because Statement 2 – In any triangle, bisector of an angle divides the triangle into two similar triangles. The question contains Statement 1(assertion) and Statement 2(reason). Of these statements, mark correct choice if a) Statement 1 and 2 are true. Statement 2 is a correct explanation for statement 1. b) Statement 1 and 2 are 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.
|
IIT 2007 |
|
1244 |
Show, by vector method, that the angular bisectors of a triangle are concurrent and find an expression for the position vector of point of concurrency in terms of position vectors of the vertices.
Show, by vector method, that the angular bisectors of a triangle are concurrent and find an expression for the position vector of point of concurrency in terms of position vectors of the vertices.
|
IIT 2001 |
|
1245 |
If AB is a diameter of a circle and C is any point on the circumference of the circle then a) The area of ΔABC is maximum when it is isosceles b) The area of ΔABC is minimum when it is isosceles c) The perimeter of ΔABC is minimum when it is isosceles d) None of these
If AB is a diameter of a circle and C is any point on the circumference of the circle then a) The area of ΔABC is maximum when it is isosceles b) The area of ΔABC is minimum when it is isosceles c) The perimeter of ΔABC is minimum when it is isosceles d) None of these
|
IIT 1983 |
|
1246 |
Let a > 0, b > 0, c > 0 then both the roots of the equation a) are real and positive b) have negative real parts c) have positive real parts d) none of these
Let a > 0, b > 0, c > 0 then both the roots of the equation a) are real and positive b) have negative real parts c) have positive real parts d) none of these
|
IIT 1979 |
|
1247 |
If sinA sinB sinC + cosA cosB = 1then the value of sinC is
If sinA sinB sinC + cosA cosB = 1then the value of sinC is
|
IIT 2006 |
|
1248 |
A plane passes through (1, −2, 1) and is perpendicular to the two planes and The distance of the plane from the point (1, 2, 2) is.
A plane passes through (1, −2, 1) and is perpendicular to the two planes and The distance of the plane from the point (1, 2, 2) is.
|
IIT 2006 |
|
1249 |
Let f and g be increasing and decreasing functions, respectively from [0, ∞) to [0, ∞). Let h(x) =f(g(x)). If h(0) = 0 then h(x) – h(t) is a) Always zero b) Always negative c) Always positive d) Strictly increasing e) None of these
Let f and g be increasing and decreasing functions, respectively from [0, ∞) to [0, ∞). Let h(x) =f(g(x)). If h(0) = 0 then h(x) – h(t) is a) Always zero b) Always negative c) Always positive d) Strictly increasing e) None of these
|
IIT 1988 |
|
1250 |
On the interval [0, 1] the function takes the maximum value at the point a) 0 b) c) d)
On the interval [0, 1] the function takes the maximum value at the point a) 0 b) c) d)
|
IIT 1995 |
|