Question(s) from Search: IIT

A straight line L through the point (3, −2) is inclined at an angle of 60° to the line $\sqrt{3}x+y=1$

. If the line L also intersects the X- axis then the equation of L is

a) $y+\sqrt{3}x+2-3\sqrt{3}=0$

b) $y-\sqrt{3}x+2+3\sqrt{3}=0$

c) $\sqrt{3}y-x+3+2\sqrt{3}=0$

d) $\sqrt{3}y+x-3+2\sqrt{3}=0$

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

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 n is a positive integer and 0 ≤ v < π then show that

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

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

 for every 0 < α, β < 2.

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

The value of  where x > 0 is

a) 0

b) – 1

c) 1

d) 2

The value of

a) 5050

b) 5051

c) 100

d) 101

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

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

Multiple choices

Let g (x) = x f (x), where   at x = 0

a) g is  but  is not continuous

b) g is  while f is not

c) f and g are both differentiable

d) g is  and  is continuous

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)

A five digit number divisible by 3 is formed using the numerals 0, 1, 2, 3, 4, and 5 without repetition. Total number of ways this can be done is

a) At least 30

b) At most 20

c) Exactly 25

d) None of these

A rectangle with sides (2m – 1) and (2n – 1) is divided into squares of unit length by drawing parallel lines. Then the number of rectangles possible with odd side lengths is

a) mn (m + 1)(n + 1)

b)

c)

d)

Find the values of a and b, so that the functions

Is continuous for 0 ≤ x ≤ π

a)

b)

c)

d)

Let α ε ℝ, then a function f : ℝ → ℝ is differentiable at α if and only if there is a function g : ℝ → ℝ which is continuous at α and satisfies f(x) – f(α) = g(x) (x – α) for all x ε ℝ.

a) True

b) False

If two functions f and g satisfy the given conditions  x, y ε ℝ, f(x – y) = f(x)g(y) – f(y)g(x) and g(x – y) = g(x) . g(y) + f(x) . f(y).

If the RHD at x = 0 exists for f(x) then find the derivative of g(x) at x = 0.

Let

be a real valued function. The set of points where f(x) is not differentiable are

a) {0}

b) {1}

c) {0, 1}

d) {∅}

Let f(x) = [x] where [.] denotes the greatest integer function. Then the domain of f is .  .  .  ., points of discontinuity of f are .  .  .  .

a) ∀ x ε I

b) ∀ x ε I − {0}

c) ∀ x ε I – {0, 1}

d) ∀ x ε I – {0, 1, 2}

PQ and PR are two infinite rays, QAR is an arc.

		U

Points lying in the shaded region excluding the boundary satisfies

a)   |z + 1| > 2; |arg(z + 1)| <

b)   |z + 1| < 2; |arg(z + 1)| <

c)  

d)  

If a continuous function f defined on the real line ℝ, assumes positive and negative values in ℝ then the equation f(x) = 0 has a root in ℝ. For example, it is known that if a continuous function f on ℝ is positive at some points and its minimum value is negative then the equation f(x) = 0 has a root in ℝ. Consider the function f(x) =  for all real x where k is a real constant.

The positive value of k for which  has only one root is

a)

b) 1

c) e

d) ln2

Let the complex numbers  are vertices of an equilateral triangle. If  be the circumcentre of the triangle, then prove that

Consider the following linear equations
ax + by + cz = 0
bx + cy + az = 0
cx + ay + bz = 0
Match the statements/expressions in column 1 with column 2