Question(s) from Search: IIT

Prove by mathematical induction that
 for every positive integer n.

The sides of a rhombus are along the lines x – y + 1 = 0 and 7x – y – 5 = 0. If its diagonals intersect at (−1, −2) then which one of the following is a vertex of the rhombus?

a) $(-\mathrm{3,}-9)$

b) $(-\mathrm{3,}-8)$

c) $\left(\frac{1}{3},-\frac{8}{3}\right)$

d) $\left(\frac{-10}{3},\frac{-7}{3}\right)$

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.

Prove that  is an integer for every positive integer.

If f is a continuous function with  as |x| → ∞ then show that every line y = mx intersects the curve .

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.

The C be a circle with the centre at (1, 1) and radius 1. If T is the circle centred at (0, k) passing through origin and touches the circle C externally, then the radius of T is equal to

a) $\frac{\sqrt{3}}{\sqrt{2}}$

b) $\frac{\sqrt{3}}{2}$

c) $\frac{1}{2}$

d) $\frac{1}{4}$

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 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 to one if f is one to one

c) Continuous if f is continuous

d) Differentiable if f is differentiable

Evaluate

a)

b)

c)

d)

One or more than one correct option

The circle C₁ : x² + y² = 3 with centre at O intersect the parabola x² = 2y at the point P in the first quadrant. Let the tangent to the circle C₁ at P touches other two circles C₂ and C₃ at R₂ and R₃ respectively. Suppose C₂ and C₃ have equal radii $2\sqrt{3}$

and centres Q₂ and Q₃ respectively. If Q₂ and Q₃ lie on the Y- axis, then

a) $Q_2{Q}_3=12$

b) $R_2{R}_3=4\sqrt{6}$

c) $\mathrm{area}\mathrm{of}△{}_2{R}_3\mathrm{is}R\sqrt{2}$

d) $\mathrm{area}\mathrm{of}△{\mathrm{PQ}}_2{Q}_3\mathrm{is}4\sqrt{2}$

Let f : ℝ → ℝ be a function defined by f (x) =  . The set of points where f (x) is not differentiable is

a) }

b)

c) {0, 1}

d)

The circle passing through the point (−1, 0) and touching the Y – axis at (0, 2) also passes through the point

a) $\left(\frac{-3}{2},0\right)$

b) $\left(\frac{-5}{2},0\right)$

c) $\left(\frac{-3}{2},\frac{5}{2}\right)$

d) $(-\mathrm{4,}0)$

Let a, b, c be positive real numbers such that b² – 4ac > 0 and let α₁ = c. Prove by induction that
 

Is well defined and  for n=1, 2, …

Here well defined means that the denominator in the expression of  is not zero.

Let O be the vertex and Q be any point on the parabola x² = 8y. If the point P divides the line segment $\widehat{\mathrm{OQ}}$

internally in the ratio 1 : 3 then the locus of P is

a) x² = y

b) y² = x

c) y² = 2x

d) x² = 2y

Solve the following equation for x
 

a) −1

b)

c) 0

d) −1 and

If f is a differentiable function satisfying  for all n ≥ 1,

n  I then

a)

b)

c)

d)  is not necessarily zero

Evaluate

Let S be the focus of the parabola y² = 8x and PQ be the common chord of the circle x² + y² – 2x – 4y = 0 and the given parabola. The area of △QPS is

a) 2 sq. units

b) 4 sq. units

c) 6 sq. units

d) 8 sq. units

Multiple choices

The function f (x) = 1 + |sinx| is

a) continuous nowhere

b) continuous everywhere

c) differentiable nowhere

d) not differentiable at x = 0

e) not differentiable at infinite number of points

Let a, r, s, t be non-zero real numbers. Let P(at², 2at), Q, R(ar², 2ar and S(as², 2as) be distinct points on the parabola y² = 4ax. Suppose PQ is the focal chord and QR and PK are parallel, where K is point (2a, 0)If st = 1 then the tangent at P and normal at S to the parabola meet at a point whose ordinate is

a) $\frac{{\left(t^2+1\right)}^2}{2t^3}$

b) $\frac{a{\left(t^2+1\right)}^2}{2t^3}$

c) $\frac{{a\left(t^2+1\right)}^2}{r^3}$

d) $\frac{{a\left(t^2+2\right)}^2}{r^3}$

The tangent PT and the normal PN of the parabola y² = 4ax at the point P on it meet its axis at the points T and N respectively. The locus of the centroid of the triangle PTM is a parabola whose

a) Vertex is $\left(\frac{2a}{3},0\right)$

b) Directrix is x = 0

c) Latus rectum is $\frac{2a}{3}$

d) Focus is (a, 0)

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 E = {1, 2, 3, 4} and F = {1, 2} then the number of onto functions from E to F is

a) 14

b) 16

c) 12

d) 8

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

a) 0

b)

c)

d)