Question(s) from Search: IIT

Let a, b, c and d be non-zero real numbers. If the point of intersection of lines 4ax + 2ay + c = 0 and 5bx + 2by + d = 0 lie in the fourth quadrants and is equidistant from the two axes, then

a) 2bc – 3ad = 0

b) 2bc + 3ad = 0

c) 2ad – 3bc = 0

d) 3bc + 2ad = 0

One or more than one correct option

Let α, λ, μ ∈ ℝ. Consider the system of linear equations αx + 2y = λ 3x – 2y = μWhich of the following statements is/are correct?

a) If α = −3, then the system has infinitely many solutions for all values of λ and μ

b) If α ≠ −3, then the system of equations has a unique solution for all values of λ and μ

c) If λ + μ = 0, then the system has infinitely many solutions for α = −3

d) If λ + μ ≠ 0, then the system has no solution for α = −3

One or more than one correct option

Circle(s) touching X – axis at a distance 3 from the origin and having an intercept of length $2\sqrt{7}$

on Y – axis is/are

a) x² + y² – 6x + 8y + 9 = 0

b) x² + y² – 6x + 7y + 9 = 0

c) x² + y² – 6x − 8y + 9 = 0

d) x² + y² – 6x − 7y + 9 = 0

One or more than one correct option

A circle S passes through the point (0, 1) and is orthogonal to the circles (x – 1)² + y² = 16 and x² + y² = 1, then

a) Radius of S is 8

b) Radius of S is 7

c) Centre of S is (−7, 1)

d) Centre of S is (−8, 1)

A tangent PT is drawn to the circle x² + y² = 4 at the point $P\left(\sqrt{3},1\right)$

. A straight line L, perpendicular to PT is tangent to the circle (x – 3)² + y² = 1A possible equation of L is

a) $x-\sqrt{3}y=1$

b) $x+\sqrt{3}y=1$

c) $x-\sqrt{3}y=-1$

d) $x+\sqrt{3}y=5$

The centre of those circles which touch the circle x² + y² – 8x – 8y = 0, externally and also touch the X- axis, lie on

a) A circle

b) An ellipse which is not a circle

c) A hyperbola

d) A parabola

Let (x, y) be any point on the parabola y² = 4x. Let P be the point that divides the line segment from (0, 0) to (x, y) in the ratio of 1 : 3. Then the locus of P is

a) x² = y

b) y² = 2x

c) y² = x

d) x² = 2y

Let the curve C be the mirror image of the parabola y² = 4x with respect to the line x + y + 4 = 0. If A and B are points of intersection of C with the line y = −5 then the distance between A and B is . . .?

Consider the parabola y² = 8x. Let △₁ be the area of the triangle formed by the end points of its latus rectum and the point $P\left(\frac{1}{2},2\right)$

on the parabola and △₂ be the area of the triangle formed by drawing tangent at P and the end points of the latus rectum. Then $\frac{{△}_1}{{△}_2}$ is

Determine the equation of the curve passing through origin in the form  which satisfies the differential equation

The integral $\underset{2}{\overset{4}{\int }}\frac{{\mathrm{logx}}^2}{{\mathrm{logx}}^2+\log \left(x^2-12x+36\right)}\mathrm{dx}$

is equal to

a) 2

b) 4

c) 1

d) 6

The value of the integral $\underset{\sqrt{\log 2}}{\overset{\sqrt{\log 3}}{\int }}\frac{x\sin x^2}{\sin x^2+\sin \left(\log 6-x^2\right)}\mathrm{dx}$

is equal to

a) $\frac{1}{4}\log \frac{3}{2}$

b) $\frac{1}{2}\log \frac{3}{2}$

c) $\log \frac{3}{2}$

d) $\frac{1}{6}\log \frac{3}{2}$

One or more than one correct options

Let F : ℝ → (0, 1) be a continuous function. Then which of the following function(s) has (have) the value zero at some point in the interval (0, 1)?

a) $e^x-\underset{0}{\overset{x}{\int }}f\left(t\right)\mathrm{sint}\mathrm{dt}$

b) $f\left(x\right)+\underset{0}{\overset{\frac{\pi }{2}}{\int }}f\left(t\right)\mathrm{sint}\mathrm{dt}$

c) $x-\underset{0}{\overset{\frac{\pi }{2}-x}{\int }}f\left(t\right)\mathrm{cost}\mathrm{dt}$

d) $x^9-f(x)$

One or more than one correct options

The value(s) of $\underset{0}{\overset{1}{\int }}\frac{x^4{(1-x)}^4}{{1+x}^2}\mathrm{dx}$

is (are)

a) $\frac{22}{7}-\pi$

b) $\frac{2}{105}$

c) $0$

d) $\frac{71}{15}-\frac{3\pi }{2}$

$\underset{-2}{\overset{2}{\int }}\frac{{3x}^2}{1+e^x}\mathrm{dx}$

equals

a) 8

b) 2

c) 4

d) 0

The value of $\underset{0}{\overset{1}{\int }}{4x}^3\left[\frac{d^2}{{\mathrm{dx}}^2}{\left(1-x^2\right)}^5\right]\mathrm{dx}$

is

a) 4

b) 0

c) 2

d) 6

Let f be a non-negative function defined on the interval [0, 1]. If $\underset{0}{\overset{x}{\int }}\sqrt{1-{\left(f^{\prime }(t)\right)}^2}\mathrm{dt}=\underset{0}{\overset{x}{\int }}f\left(t\right)\mathrm{dt},0\le x\le 1$

and f(0) = 0, then

a) $f\left(\frac{1}{2}\right)<\frac{1}{2}\wedge f\left(\frac{1}{3}\right)>\frac{1}{3}$

b) $f\left(\frac{1}{2}\right)>\frac{1}{2}\wedge f\left(\frac{1}{3}\right)>\frac{1}{3}$

c) $f\left(\frac{1}{2}\right)<\frac{1}{2}\wedge f\left(\frac{1}{3}\right)<\frac{1}{3}$

d) $f\left(\frac{1}{2}\right)>\frac{1}{2}\wedge f\left(\frac{1}{3}\right)<\frac{1}{3}$

One or more than one correct answer

Let P and Q be distinct points on the parabola y² = 2x such that the circle with PQ as diameter passes through the vertex O of the parabola. If P lies in the first quadrant and the area of triangle OPQ is $3\sqrt{2}$

then which of the following is/are the coordinates of P?

a) $\left(\mathrm{4,}2\sqrt{2}\right)$

b) $\left(\mathrm{9,}3\sqrt{2}\right)$

c) $\left(\frac{1}{4},\frac{1}{\sqrt{2}}\right)$

d) $\left(\mathrm{1,}\sqrt{2}\right)$

The area (in square units) of the region described by A = {(x, y) : x² + y² ≤ 1 and y² ≤ 1 – x} is

a) $\frac{\pi }{2}+\frac{4}{3}$

b) $\frac{\pi }{2}-\frac{4}{3}$

c) $\frac{\pi }{2}-\frac{2}{3}$

d) $\frac{\pi }{2}+\frac{2}{3}$

If the straight line x = b divides the area enclosed by y = (1 – x)² , y = 0 and x = 0 into two parts R₁ (0 ≤ x ≤ b) and R₂ (b ≤x ≤ 1) such that $R_1-R_2=\frac{1}{4}$

then b equals

a) $\frac{3}{4}$

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

c) $\frac{1}{3}$

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

If y = y(x) satisfies the differential equation $8\sqrt{x}\sqrt{9+\sqrt{x}}\mathrm{dy}={\left(\sqrt{4+\sqrt{9+\sqrt{x}}}\right)}^{-1}\mathrm{dx},x>0$

and $y\left(0\right)=\sqrt{7}$ Then y(256) =

a) 16

b) 3

c) 9

d) 80

Let T > 0 be a fixed real number. Suppose f is a continuous function such that for all x  ℝ, f(x + T) = f(x). If  then the value of  is

a)

b)

c) 3I

d) 6I

The question contains Statement – 1(assertion) and Statement – 2 (reason). Let f (x) = 2 + cosx for all real x.

Statement 1: For each real t, there exists a point c in [t, t + π] such that  because

Statement 2: f (t) = f[t, t + 2π] for each real t

a) Statement 1 and 2 are true. Statement 2 is a correct explanation of Statement 1.

b) Statement 1 and 2 are true. Statement 2 is not a correct explanation of Statement 1.

c) Statement 1 is true and Statement 2 is false.

d) Statement 1 is false. Statement 2 is true.

The function  (where [y] is the greatest integer less than or equal to y) is discontinuous at

a) All integers

b) All integers except 0 and 1

c) All integers except 0

d) All integers except 1

The integer n, for which  is a finite

non–zero number is

a) 1

b) 2

c) 3

d) 4