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Question(s) from Search: IIT

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251

If the system of equations

x + ay = 0

az + y = 0

ax + z = 0

has infinite solutions then the value of a is

a) −1

b) 1

c) 0

d) No real values

If the system of equations

x + ay = 0

az + y = 0

ax + z = 0

has infinite solutions then the value of a is

a) −1

b) 1

c) 0

d) No real values

IIT 2003
04:39 min
252

Let z and ω be two complex numbers such that |z| ≤ 1 and |w| ≤ 1 then show that .

Let z and ω be two complex numbers such that |z| ≤ 1 and |w| ≤ 1 then show that .

IIT 1995
06:01 min
253

A is a point on the parabola . The normal at A cuts the parabola again at B. If AB subtends a right angle at the vertex of the parabola, find the slope of AB.

A is a point on the parabola . The normal at A cuts the parabola again at B. If AB subtends a right angle at the vertex of the parabola, find the slope of AB.

IIT 1982
06:08 min
254

If a, b, c, d are positive real numbers such that a + b + c + d = 2 then M = ( a + b ) ( c + d ) satisfies

a) 0 ≤ M ≤ 1

b) 1 ≤ M ≤ 2

c) 2 ≤ M ≤ 3

d) 3 ≤ M ≤ 4

If a, b, c, d are positive real numbers such that a + b + c + d = 2 then M = ( a + b ) ( c + d ) satisfies

a) 0 ≤ M ≤ 1

b) 1 ≤ M ≤ 2

c) 2 ≤ M ≤ 3

d) 3 ≤ M ≤ 4

IIT 2000
01:54 min
255

Show that the locus of a point that divides a chord of slope 2 of the parabola  internally in the ratio 1:2 is a parabola. Find its vertex.

Show that the locus of a point that divides a chord of slope 2 of the parabola  internally in the ratio 1:2 is a parabola. Find its vertex.

IIT 1995
06:25 min
256

Let α, β be the roots of  and γ, δ roots of . If α, β, γ, δ are in geometric progression then the integral values of p and q respectively are

a) −2, −32

b) −2, 3

c) −6, 3

d) −6, −32

Let α, β be the roots of  and γ, δ roots of . If α, β, γ, δ are in geometric progression then the integral values of p and q respectively are

a) −2, −32

b) −2, 3

c) −6, 3

d) −6, −32

IIT 2001
05:16 min
257

For what values of k does the following system of equations possess a non-trivial solution over the set of rationals? Find all the solutions.

x + y – 2z = 0

2x – 3y + z = 0

x – 5y + 4z = k

For what values of k does the following system of equations possess a non-trivial solution over the set of rationals? Find all the solutions.

x + y – 2z = 0

2x – 3y + z = 0

x – 5y + 4z = k

IIT 1979
05:23 min
258

Prove that there exists no complex number z such that  and .

Prove that there exists no complex number z such that  and .

IIT 2003
04:27 min
259

Three normals with slopes  are drawn from a point P not on the axis of the parabola . If  results in the locus of P being a part of the parabola, find the value of α.

Three normals with slopes  are drawn from a point P not on the axis of the parabola . If  results in the locus of P being a part of the parabola, find the value of α.

IIT 2003
05:59 min
260

Find the value of the expression

1.(2−ω)(2−+ 2.(3−ω)(3−+ … (n−1).(n−ω)(n−

where ω is an imaginary cube root of unity.

a) n(n−1)(+3n+4)

b) n(n+1)(+3n+4)

c) n(n−1)(+n+1)

d) n(n+1)(+n+1)

Find the value of the expression

1.(2−ω)(2−+ 2.(3−ω)(3−+ … (n−1).(n−ω)(n−

where ω is an imaginary cube root of unity.

a) n(n−1)(+3n+4)

b) n(n+1)(+3n+4)

c) n(n−1)(+n+1)

d) n(n+1)(+n+1)

IIT 1996
05:00 min
261

If  are positive real numbers whose product is a fixed number c then the minimum value of  is

a)

b)

c)

d)

If  are positive real numbers whose product is a fixed number c then the minimum value of  is

a)

b)

c)

d)

IIT 2002
02:06 min
262

If three complex numbers are in arithmetic progression then they lie on a circle in the complex plane.

a) True

b) False

If three complex numbers are in arithmetic progression then they lie on a circle in the complex plane.

a) True

b) False

IIT 1985
01:13 min
263

A solution of the differential equation  is

a)  y = 2

b)  y = 2x

c)  

d)  2

A solution of the differential equation  is

a)  y = 2

b)  y = 2x

c)  

d)  2

IIT 1999
01:47 min
264

For all x,  then the interval in which a lies is

a) a <

b)

c)

d)

For all x,  then the interval in which a lies is

a) a <

b)

c)

d)

IIT 2004
01:45 min
265

Let the three digit numbers A28, 3B9 and 62C where A, B, C are integers between 0 and 9, be divisible by a fixed number k. Show that the determinant
 

is divisible by k.

Let the three digit numbers A28, 3B9 and 62C where A, B, C are integers between 0 and 9, be divisible by a fixed number k. Show that the determinant
 

is divisible by k.

IIT 1990
04:45 min
266

If a and b are real numbers between 0 and 1 such that the points  form an equilateral triangle then a is equal to .  .  .  .

a)  

b)  

c)  

d)  

If a and b are real numbers between 0 and 1 such that the points  form an equilateral triangle then a is equal to .  .  .  .

a)  

b)  

c)  

d)  

IIT 1989
03:07 min
267

Let E be the ellipse  and C be the circle . Let P and Q be the points (1, 2) and (2, 1) respectively. Then

a) Q lies inside C but outside E

b) Q lies outside both C and E

c) P lies inside both C and E

d) P lies inside C but outside E

Let E be the ellipse  and C be the circle . Let P and Q be the points (1, 2) and (2, 1) respectively. Then

a) Q lies inside C but outside E

b) Q lies outside both C and E

c) P lies inside both C and E

d) P lies inside C but outside E

IIT 1994
04:15 min
268

Let a, b, c be the sides of a triangle where a ≠ c and λ ε R. If roots of the equation  are real then

a)

b)

c)

d)

Let a, b, c be the sides of a triangle where a ≠ c and λ ε R. If roots of the equation  are real then

a)

b)

c)

d)

IIT 2006
04:47 min
269

Find the value of the determinant

where a, b, c are respectively pth, qth and rth term of a harmonic

progression.

a) 0

b) 1

c) ½

d) None of the above

Find the value of the determinant

where a, b, c are respectively pth, qth and rth term of a harmonic

progression.

a) 0

b) 1

c) ½

d) None of the above

IIT 1997
04:23 min
270

If tangents are drawn to the ellipse  then the locus of the mid-points of the intercepts made by the tangents between the coordinate axes is

a)

b)

c)

d)

If tangents are drawn to the ellipse  then the locus of the mid-points of the intercepts made by the tangents between the coordinate axes is

a)

b)

c)

d)

IIT 2004
03:11 min
271

Let S is the set of all real x, such that  is positive, then S contains

a)

b)

c)

d)

Let S is the set of all real x, such that  is positive, then S contains

a)

b)

c)

d)

IIT 1986
04:28 min
272

Let pλ4 + qλ3 + rλ2 + sλ + t =  be an identity in λ where p, q, r, s, t are constants. Find the value of t.

a) 0

b) +1

c) –1

d) ±1

Let pλ4 + qλ3 + rλ2 + sλ + t =  be an identity in λ where p, q, r, s, t are constants. Find the value of t.

a) 0

b) +1

c) –1

d) ±1

IIT 1981
02:38 min
273

Let P be a variable point on the ellipse  with foci F1 and F2. . If A is the area of  then the maximum value of A is  . . . . .

Let P be a variable point on the ellipse  with foci F1 and F2. . If A is the area of  then the maximum value of A is  . . . . .

IIT 1994
02:27 min
274

A spherical rain drop evaporates at a rate proportional to its surface area at any instant. The differential equation giving the rate of change of the radius vector of the rain drop is . . . . .

A spherical rain drop evaporates at a rate proportional to its surface area at any instant. The differential equation giving the rate of change of the radius vector of the rain drop is . . . . .

IIT 1997
01:37 min
275

The value of the determinant  is …………

a) 0

b) 1

c) a2 + b2 + c2 – abc

d) a2 + b2 + c2 – 3abc

The value of the determinant  is …………

a) 0

b) 1

c) a2 + b2 + c2 – abc

d) a2 + b2 + c2 – 3abc

IIT 1988
02:49 min

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