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1201

If  are unit coplanar vectors then the scalar triple product  

a) 0

b) 1

c)

d)

If  are unit coplanar vectors then the scalar triple product  

a) 0

b) 1

c)

d)

IIT 2000
1202

One or more than one correct option

If the line x = α divides the area of the region R = {(x, y) ∈ ℝ2 : x3 ≤ y ≤ x, 0 ≤ x ≤ 1 into two equal parts then

a) 2α44α2+1=0

b) α4+4α21=0

c) 12<α<1

d) 0<α<12

One or more than one correct option

If the line x = α divides the area of the region R = {(x, y) ∈ ℝ2 : x3 ≤ y ≤ x, 0 ≤ x ≤ 1 into two equal parts then

a) 2α44α2+1=0

b) α4+4α21=0

c) 12<α<1

d) 0<α<12

IIT 2017
1203

The sides of a triangle inscribed in a given circle subtend angles α, β and γ at the centre. The minimum value of the Arithmetic mean of
 
 

The sides of a triangle inscribed in a given circle subtend angles α, β and γ at the centre. The minimum value of the Arithmetic mean of
 
 

IIT 1987
1204

The value of k=1131sin(π4+(k1)π6)sin(π4+6)

a) 33

b) 2(33)

c) 2(31)

d) 2(2+3)

The value of k=1131sin(π4+(k1)π6)sin(π4+6)

a) 33

b) 2(33)

c) 2(31)

d) 2(2+3)

IIT 2016
1205

Let y(x) be the solution of the differential equation (xlnx)dydx+y=2xlnx,(x1)

. Given that y = 1 when x = 1, then y(e) is equal to

a) e

b) 0

c) 2

d) 2e

Let y(x) be the solution of the differential equation (xlnx)dydx+y=2xlnx,(x1)

. Given that y = 1 when x = 1, then y(e) is equal to

a) e

b) 0

c) 2

d) 2e

IIT 2015
1206

If Cr stands for  then the sum of the series
 
where n is a positive integer, is equal to

a) 0

b) (−)n/2(n + 1)

c) (−)n/2 (n + 2)

d) None of these

If Cr stands for  then the sum of the series
 
where n is a positive integer, is equal to

a) 0

b) (−)n/2(n + 1)

c) (−)n/2 (n + 2)

d) None of these

IIT 1986
1207

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

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

IIT 2002
1208

One or more than one correct options

If y(x) satisfies the differential equation y′ − ytanx = 2xsecx and y(0) = 0, then

a) y(π4)=π282

b) y(π4)=π218

c) y(π3)=π29

d) y(π3)=4π3+2π233

One or more than one correct options

If y(x) satisfies the differential equation y′ − ytanx = 2xsecx and y(0) = 0, then

a) y(π4)=π282

b) y(π4)=π218

c) y(π3)=π29

d) y(π3)=4π3+2π233

IIT 2012
1209

The sum if p > q is maximum when m is

a) 5

b) 10

c) 15

d) 20

The sum if p > q is maximum when m is

a) 5

b) 10

c) 15

d) 20

IIT 2002
1210

At present a firm is manufacturing 2000 items. It is estimated that the rate of change of production P with respect to additional number of workers x is given by dPdx=10012x

. If the firm employs 25 more workers then the new level of production of items is

a) 2500

b) 3000

c) 3500

d) 4500

At present a firm is manufacturing 2000 items. It is estimated that the rate of change of production P with respect to additional number of workers x is given by dPdx=10012x

. If the firm employs 25 more workers then the new level of production of items is

a) 2500

b) 3000

c) 3500

d) 4500

IIT 2013
1211

If a, b, c; u, v, w are complex numbers representing the vertices of two triangles such that c = (1 − r)a + rb, w = (1 − r)u + rv where r is a complex number. The two triangles

a) have the same area

b) are similar

c) are congruent

d) none of these

If a, b, c; u, v, w are complex numbers representing the vertices of two triangles such that c = (1 − r)a + rb, w = (1 − r)u + rv where r is a complex number. The two triangles

a) have the same area

b) are similar

c) are congruent

d) none of these

IIT 1985
1212

Prove that

 

Prove that

 

IIT 1979
1213

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 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.

IIT 2007
1214

Let f(x) = (1 – x)2 sin2x + x2 and g(x)=1x(2(t1)t+1lnt)f(t)dt

Which of the following is true?

a) g is increasing on (1, ∞)

b) g is decreasing on (1, ∞)

c) g is increasing on (1, 2) and decreasing on (2, ∞)

d) g is decreasing on (1, 2) and increasing on (2, ∞)

Let f(x) = (1 – x)2 sin2x + x2 and g(x)=1x(2(t1)t+1lnt)f(t)dt

Which of the following is true?

a) g is increasing on (1, ∞)

b) g is decreasing on (1, ∞)

c) g is increasing on (1, 2) and decreasing on (2, ∞)

d) g is decreasing on (1, 2) and increasing on (2, ∞)

IIT 2013
1215

Use mathematical induction to prove: If n is an odd positive integer
then  is divisible by 24.

Use mathematical induction to prove: If n is an odd positive integer
then  is divisible by 24.

IIT 1983
1216

Let PS is the median of the triangle with vertices P(2, 2), Q(6, −1) and R(7, 3), then the equation of the line passing through (1, −1) and parallel to PS is

a) 4x – 7y – 11 = 0

b) 2x + 9y + 7 = 0

c) 4x + 7y + 3 = 0

d) 2x – 9y – 11 = 0

Let PS is the median of the triangle with vertices P(2, 2), Q(6, −1) and R(7, 3), then the equation of the line passing through (1, −1) and parallel to PS is

a) 4x – 7y – 11 = 0

b) 2x + 9y + 7 = 0

c) 4x + 7y + 3 = 0

d) 2x – 9y – 11 = 0

IIT 2014
1217

One or more than one correct option

For a > b > c > 0, the distance between (1, 1) and the point of intersection of the lines ax + by + c = 0 and bx + ay + c = 0 is less than 22

, then

a) a + b – c > 0

b) a − b + c < 0

c) a − b + c > 0

d) a + b – c < 0

One or more than one correct option

For a > b > c > 0, the distance between (1, 1) and the point of intersection of the lines ax + by + c = 0 and bx + ay + c = 0 is less than 22

, then

a) a + b – c > 0

b) a − b + c < 0

c) a − b + c > 0

d) a + b – c < 0

IIT 2014
1218

Using mathematical induction, prove that
 
m, n, k are positive integers and  for p < q

Using mathematical induction, prove that
 
m, n, k are positive integers and  for p < q

IIT 1989
1219

If one of the diameters of the circle, given by the equation x2 + y2 – 4x + 6y – 12 = 0 is a chord of a circle S whose centre is at (−3, 2), then the radius of S is

a) 52

b) 53

c) 5

d) 10

If one of the diameters of the circle, given by the equation x2 + y2 – 4x + 6y – 12 = 0 is a chord of a circle S whose centre is at (−3, 2), then the radius of S is

a) 52

b) 53

c) 5

d) 10

IIT 2016
1220

Let p ≥ 3 be an integer and α, β be the roots of x2 – (p + 1) x + 1 = 0. Using mathematical induction show that αn + βn
i) is an integer
ii) and is not divisible by p.

Let p ≥ 3 be an integer and α, β be the roots of x2 – (p + 1) x + 1 = 0. Using mathematical induction show that αn + βn
i) is an integer
ii) and is not divisible by p.

IIT 1992
1221

The function  is not differentiable at

a) – 1

b) 0

c) 1

d) 2

The function  is not differentiable at

a) – 1

b) 0

c) 1

d) 2

IIT 1999
1222

One or more than one correct option

Let RS be a diameter of the circle x2 + y2 = 1 where S is the point (1, 0). Let P be a variable point (other than R and S) on the circle and the tangents to the circle at S and P meet at the point Q. The normal to the circle at P intersect a line drawn through Q parallel to RS at a point E. Then the locus of E passes through the point(s)

a) (13,13)

b) (14,12)

c) (13,13)

d) (14,12)

One or more than one correct option

Let RS be a diameter of the circle x2 + y2 = 1 where S is the point (1, 0). Let P be a variable point (other than R and S) on the circle and the tangents to the circle at S and P meet at the point Q. The normal to the circle at P intersect a line drawn through Q parallel to RS at a point E. Then the locus of E passes through the point(s)

a) (13,13)

b) (14,12)

c) (13,13)

d) (14,12)

IIT 2016
1223

If x is not an integral multiple of 2π use mathematical induction to prove that
 

If x is not an integral multiple of 2π use mathematical induction to prove that
 

IIT 1994
1224

A circle passing through (1, −2) and touching the axis of X at (3, 0) also passes through the point

a) (−5, 2)

b) (2, −5)

c) (5, −2)

d) (−2, 5)

A circle passing through (1, −2) and touching the axis of X at (3, 0) also passes through the point

a) (−5, 2)

b) (2, −5)

c) (5, −2)

d) (−2, 5)

IIT 2013
1225

The circles  and  intersect each other in distinct points if

a) r < 2

b) r > 8

c) 2 < r < 8

d) 2 ≤ r ≤ 8

The circles  and  intersect each other in distinct points if

a) r < 2

b) r > 8

c) 2 < r < 8

d) 2 ≤ r ≤ 8

IIT 1994

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