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Question(s) from Search: IIT

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426

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 line y = x meets y =  for k ≤ 0 at

a) No point

b) One point

c) Two points

d) More than two points

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 line y = x meets y =  for k ≤ 0 at

a) No point

b) One point

c) Two points

d) More than two points

IIT 2007
02:08 min
427

(One or more than one correct answer)
Let  and  be complex numbers such that  and . If has positive real part and  has negative imaginary part, then  may be

a) Zero

b) Real and positive

c) Real and negative

d) None of these

(One or more than one correct answer)
Let  and  be complex numbers such that  and . If has positive real part and  has negative imaginary part, then  may be

a) Zero

b) Real and positive

c) Real and negative

d) None of these

IIT 1986
05:31 min
428

If then ab + bc + ca lies in the interval

a)  

b)  

c)  

d)  

If then ab + bc + ca lies in the interval

a)  

b)  

c)  

d)  

IIT 1984
02:29 min
429

Find the values of x and y for which the following equation is satisfied

a) x = y = −1

b) x = y = 3

c) x = 1, y = 3

d) x = 3, y = −1

Find the values of x and y for which the following equation is satisfied

a) x = y = −1

b) x = y = 3

c) x = 1, y = 3

d) x = 3, y = −1

IIT 1980
05:23 min
430

The equation of the directrix of the parabola y2 + 4y + 4x +2 = 0 is

a) x = − 1

b) x = 1

c)

d)

The equation of the directrix of the parabola y2 + 4y + 4x +2 = 0 is

a) x = − 1

b) x = 1

c)

d)

IIT 2001
01:51 min
431

Let α, β be roots of the equation (x – a) (x – b) = c, c ≠ 0. Then the roots of the equation (x – α) (x – β) + c = 0 are

a) a, c

b) b, c

c) a, b

d) a + c, b + c

Let α, β be roots of the equation (x – a) (x – b) = c, c ≠ 0. Then the roots of the equation (x – α) (x – β) + c = 0 are

a) a, c

b) b, c

c) a, b

d) a + c, b + c

IIT 1992
02:15 min
432

If = x + iy then

a) x = 3, y = 1

b) x = 1, y = 3

c) x = 0, y = 3

d) x = 0, y = 0

If = x + iy then

a) x = 3, y = 1

b) x = 1, y = 3

c) x = 0, y = 3

d) x = 0, y = 0

IIT 1998
01:25 min
433

It is given that n is an odd integer greater than 3 and not a multiple of 3. Prove that  is a factor of  

It is given that n is an odd integer greater than 3 and not a multiple of 3. Prove that  is a factor of  

IIT 1985
07:09 min
434

The focal chord of  is tangent to  then the possible value of the slope of this chord are

a)   

b)  

c)   

d)   

The focal chord of  is tangent to  then the possible value of the slope of this chord are

a)   

b)  

c)   

d)   

IIT 2003
02:51 min
435

If p, q ε {1, 2, 3, 4}. The number of equations of the form  having real roots is

a) 15

b) 9

c) 7

d) 8

If p, q ε {1, 2, 3, 4}. The number of equations of the form  having real roots is

a) 15

b) 9

c) 7

d) 8

IIT 1994
03:39 min
436

If A =  and B = then the value of α for which A2 = B is

a) 1

b) −1

c) 4

d) No real values

If A =  and B = then the value of α for which A2 = B is

a) 1

b) −1

c) 4

d) No real values

IIT 2003
01:17 min
437

If   then show that |z| = 1.

If   then show that |z| = 1.

IIT 1995
02:14 min
438

Suppose that the normals drawn at three different points on the

parabola  pass through the point (h, 0). Show that h > 2.

Suppose that the normals drawn at three different points on the

parabola  pass through the point (h, 0). Show that h > 2.

IIT 1981
03:52 min
439

Through the vertex O of the parabola  chords OP and OQ are drawn at right angles. Show that for all positions of P, PQ cuts the axis of the parabola at a fixed point. Also find the locus of the midpoint of PQ.

Through the vertex O of the parabola  chords OP and OQ are drawn at right angles. Show that for all positions of P, PQ cuts the axis of the parabola at a fixed point. Also find the locus of the midpoint of PQ.

IIT 1994
05:22 min
440

For all x ε ( 0, 1 )

a)

b) ln (1 + x) < x

c) sinx > x

d) lnx > x

For all x ε ( 0, 1 )

a)

b) ln (1 + x) < x

c) sinx > x

d) lnx > x

IIT 2000
02:40 min
441

Given x = cy + bz, y = az + cx, z = bx + ay where x, y, z are not all zero, prove that  a2 + b2 + c2 + 2abc = 1

Given x = cy + bz, y = az + cx, z = bx + ay where x, y, z are not all zero, prove that  a2 + b2 + c2 + 2abc = 1

IIT 1978
03:30 min
442

Let and  are two complex numbers such that  then prove that .

Let and  are two complex numbers such that  then prove that .

IIT 2003
04:08 min
443

The number of values of k for which the system of equations

(k + 1) x + 8y = 4k

kx + ( k + 3 ) y = 3k – 1

has infinitely many solutions is

a) 0

b) 1

c) 2

d) Infinity

The number of values of k for which the system of equations

(k + 1) x + 8y = 4k

kx + ( k + 3 ) y = 3k – 1

has infinitely many solutions is

a) 0

b) 1

c) 2

d) Infinity

IIT 2002
02:56 min
444

Without expanding a determinant at any stage show that
 = Ax + B

where A, B are non-zero constants

Without expanding a determinant at any stage show that
 = Ax + B

where A, B are non-zero constants

IIT 1982
04:06 min
445

True/False
If the complex numbers  represent the vertices of an equilateral triangle with  then .

a) True

b) False

True/False
If the complex numbers  represent the vertices of an equilateral triangle with  then .

a) True

b) False

IIT 1984
02:27 min
446

The order of the differential equation whose general solution is given by  is

a) 5

b) 4

c) 3

d) 2

The order of the differential equation whose general solution is given by  is

a) 5

b) 4

c) 3

d) 2

IIT 1998
03:42 min
447

If f (x) =

a) f (x) is a strictly increasing function

b) f (x) has a local maxima

c) f (x) is a strictly decreasing function

d) f (x) is bounded

If f (x) =

a) f (x) is a strictly increasing function

b) f (x) has a local maxima

c) f (x) is a strictly decreasing function

d) f (x) is bounded

IIT 2004
02:07 min
448

Let Δa =
Then show that  = c, a constant.

Let Δa =
Then show that  = c, a constant.

IIT 1989
05:34 min
449

For any two complex numbers  and any real numbers  is equal to .  .  .  .

a)  

b)  

c)  

d)  

For any two complex numbers  and any real numbers  is equal to .  .  .  .

a)  

b)  

c)  

d)  

IIT 1988
02:43 min
450

The locus of a variable point whose distance from  is  times its distance from the line  is

a) Ellipse

b) Parabola

c) Hyperbola

d) None of these

The locus of a variable point whose distance from  is  times its distance from the line  is

a) Ellipse

b) Parabola

c) Hyperbola

d) None of these

IIT 1994
02:40 min

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