Question(s) from Search: IIT

Let  and  where O, A and B are non-collinear points. Let p denote the area of the quadrilateral OABC and let q denote the area of the quadrilateral with OA and OC as adjacent sides. If p = kq then k = .  .  .  .  .

Prove that  = 2[cosx + cos3x + cos5x + … + cos(2k−1)x] for any positive integer k. Hence prove that  =

The function
f(x) =|px – q| + r |x|, x ε (−, )
where p > 0, q > 0, r > 0 assumes minimum value on one point if

a) p ≠ q

b) r = q

c) r ≠ p

d) r = p = q

Let f : R → R be any function defined g : R → R by g (x) = |f (x)| for all x. Then g is

a) onto if f is onto

b) one to one if f is one to one

c) continuous if f is continuous

d) differentiable if f is differentiable

If f : [ 1,  → [ 2, ] is given by f (x) = x +  then ( x ) is given by

a)

b)

c)

d) 1 +

The function of f : R → R be defined by f (x) = 2x + sinx for x ε R . Then f is

a) one-one and onto

b) one-one but not onto

c) onto but not one-one

d) neither one-one nor onto

Multiple choice

There exists a triangle ABC satisfying the conditions

a) bsinA = a, A <

b) bsinA > a, A >

c) bsinA > a, A <

d) bsinA < a, A <, b > a

e) bsinA < a, A >, b = a

With usual notation if in a triangle ABC,  then

 .

a) True

b) False

If in a triangle ABC, cosA cosB + sinA sinB sin C = 1 then show that  a : b : c = 1 : 1 :

a) True

b) False

If the lines  and  intersect then the value of k is

a)

b)

c)

d)

The area of a triangle whose vertices are
 is

The parameter on which the value of the determinant
Δ =
does not depend upon is

a) a

b) p

c) d

d) x

Consider the lines

 ;

 
The unit vector perpendicular to both L₁ and L₂ is

a)

b)

c)

d)

If b > a then the equation ( x – a ) ( x – b )1 = 0 has

a) Both roots in [ a, b ]

b) Both roots in ( , a )

c) Both roots in (  )

d) One root in ( , a ) and other in ( )

For what value of m does the system of equations 3x + my = m, 2x − 5y = 20 have a solution satisfying the condition x > 0, y > 0.

a) m  (−∞, ∞)

b) m  (−∞, −15) ∪ (30, ∞)

c)  

d)  

If α is a repeated root of a quadratic equation f(x) = 0 and A(x), B(x), C(x) be polynomials of degree 3, 4, 5 respectively, Then show that
 

is divisible by f(x) where prime denotes the derivatives.

The number of ordered pairs satisfying the equations
 is

a) 4

b) 2

c) 0

d) 1

Let O (0, 0), A(2, 0) and  be the vertices of a triangle. Let R be the region consisting of all those points P inside ΔOAB which satisfies d(P, OA) ≤ d(P, OB) . d(P, AB), where d denotes the distance from the point to the corresponding line. Sketch the region R and find its area.

a)

b)

c)

d)

Let f(x) be a continuous function given by
 

Find the area of the region in the third quadrant bounded by the curve x = − 2y² and y = f(x) lying on the left of the line 8x + 1 = 0.

a) 192

b) 320

c) 761/192

d) 320/761

Let d be the perpendicular distance from the centre of the ellipse  to the tangent at a point P on the ellipse. Let F₁ and F₂ be the two focii of the ellipse, then show that

Find the area of the region bounded by the curves y = x², y = |2 – x²| and y = 2 which lies to the right of the line x = 1.

a)

b)

c)

d)

Prove that in an ellipse the perpendicular from a focus upon a tangent and the line joining the centre of the ellipse to the point of contact meet on the corresponding directrix.

A curve passing through the point  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. Determine the equation of the curve.

Let f : ℝ → ℝ be any function. Define g : ℝ → ℝ by g(x) = |f(x)| for all x. Then g is

a) Onto if f is onto

b) One–one if f is one–one

c) Continuous if f is continuous

d) Differentiable if f is differentiable

f(x) is a differentiable function and g(x) is a double differentiable function such that  
If  prove that there exists some c ε (−3, 3) such that .