Question(s) from Search: IIT

Using mathematical induction, prove that

 for n > 1

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)

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.

Tangents are drawn to the circle  from a point on the hyperbola . Find the locus of the midpoint of the chord of contact.

If x is not an integral multiple of 2π use mathematical induction to prove that
 

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)

Prove by induction that
P_n = 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, .  .  .,
P_n = (1 – P) P_{n – 1} + P (1 – P) P_{n – 2}  n ≥ 3,
and ,

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

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)

Find all solutions of

a)

b)

c)

d)

(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)

The smallest positive integer n for which  is

a) 8

b) 12

c) 12

d) None of these

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

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

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

The orthocenter of the triangle formed by the lines
  lies in the quadrant number . . . . .

Fill in the blank

The value of f (x) =  lies in the interval …………….

a)

b)

c)

d)

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.

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.

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

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

If sinA sinB sinC + cosA cosB = 1then the value of sinC 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.

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

On the interval [0, 1] the function  takes the maximum value at the point

a) 0

b)

c)

d)