Sample Paper II for IIT-Mathematics

Apr 11 • Engineering Sample Papers • 1789 Views • 1 Comment on Sample Paper II for IIT-Mathematics

  IIT SAMPLE PAPER

This is the sample paper for the maths section of IIT exam.

Ques 1:- which of the following is a contradiction?

a)pv(~p^q)        b) (p^q) ~(pvq)    c) (p=>q)=>p  d) none of these

Ques 2:- Let f(x) satisfy the requirements of Lagrange’s Mean Value theorem in [0,2]. If f(0)=0 and {f(x)<=1/2 for all x in [0,2],then

a)f(x)<=2       b) {f(x)}<=1    c) f(x)=2x    d) f(x)=3 for at least one x in [0,2]

Ques 3:- The ends A and B of a rod of length √5 are sliding along the curve y= 2x²   .Let  X_a  and X _b be the x-coordinate of the ends. At the moment when A is at (0,0) and B is at (1,2) the derivative dx_b /dx_a has the value equal to

a)1/3                 b) 1/5           c) 1/8            d) 1/9

Ques 4:- The value of f{0} so that f{x}=(4^x-1)³/sin(x/4)log(1+x² /3) is continuous  everywhere ,is equal to

a)3(log 4)³         b) (log  4)³     c) 12(log 4)³    d) 15(log 4)³

Ques 5:- the area of the region bounded by the curves   y=│x-1│ and y=3-│x│is

a)6sq unit          b) 2 sq unit    c) 3  sq unit   d) 4 sq unit

Ques 6:-The equation of the curve ,whose slope at any point different from origin is y+y/x, is

a)y=xe^x               b) xy=e^x        c) y+xe^x =c        d) y=c  xe ^x  , c≠0

Ques 7:- The value of ∫e^x .x² +1/(x+1)² dx is

a)e^x (x-1)/(x+1)+c      b) e^x (x+1)/(x-1)+c     c) none of these

d) e^x .x+c

Ques 8:- The sum of three terms of a strictly increasing GP is (a)S and sum of the squares of these terms is S²  so where thus a² lies

a)(1/3,2)      b) (1/3,3)        c) (1,2)   d) none of these

Ques 9:- The sum of three terms of a strictly increasing GP is (a)S and sum of the squares of these terms is S²  then if S=10√3 then the greatest value of the middle term is

a)5               b) 5√3              c) 10          d) 10√3

Ques 10:- The focal chord to y²=16x is tangent to (x-6)² +y² =2,then the possible values of the slope of this chord are

a){-2,2}        b) 2,1/2}      c {-1,1}        d) {2,-1/2}

Ques 11:-The number of ordered pairs (α,β)where α,βϵ(-∏,∏) satisfying cos(α-β) =1 and cos( α+,β)=1/e is

a)0           b) 1           c) 2          d) 4

Ques12:- If y ≠ y(x) and 2+sin x/y+1(dx/dy)=-cos x y(0)=1,then y(∏/2)equals

a)1/3           b) 2/3      c) -1/3      d) 1

Ques 13:-The Cartesian equation of plane r=(1+∏-β)i +(2-∏)j+(3-2∏=2 β)k is

a)2x+y=5       b) 2x-y=5    c) 2x+z=5    d) 2x-z=5

Ques 14:-If α, β are the roots of the equation  x²=p(x+1)-q=0 then the value of (α²+2 α+1/ α²+2 α+q)+( (β²+2V+1/ β²+2V+q) is

a)2            b) 0          c) 1            d) none of these

Ques 15:- let a relation R be defined by B={(4,5).(1,4),(4,6),(7,6),(3,7)} then R^-1 OR is

a){(1,5),(1,6),(3,6)}           b {(1,1),(4,4),(7,7),(3,3)}    c) none of these    d) {(1,1),(4,4),(4,7),(7,4),(7,7),(3,3)}

Ques 16:-The sum of n terms of the following series  1+(1+x)+(1+x+x²)+…… will be

a)1-xⁿ/1-x           b)x(1-xⁿ)1-x

c)n(1-x)-x(1-xⁿ)/(1-x)²

d)none of theses

Ques17:-if a>b then f(x)=√x-a/b-x is continuous on

a)(b,a)         b)[b,a]      c)[b,a)   d)(b,a]

Ques18:-Each of these questions contains two statement I (Assertion) and statement II (reason)thus Statement I is The line 2x+y =5 is a diameter of the circle x²+ y² -6x+2y=0, Statement II normal of a circle always pass through center of circle.  Each of these question also has four alternatives choices , only one of which is the correct answer you have to select one of the codes a,b,c,d  given below:

A) Statement I is true, Statement II is true; Statement II is a correct explanation for Statement I

B) Statement I is true, Statement II is true; Statement II is a not correct explanation for Statement I

C) Statement I is true, Statement II is false

D) Statement I is false, Statement II is false

Ques19:- Each of these questions contains two statement I (Assertion) and staementII (reason)thus Statement I for a=-1/√3 the volume of the parallelepiped formed by vectors I+aj, ai+j+k and j+ak is maximum, Statement II the volume of the parallelepiped having three coterminous edges  vector a, vector b and vector c=│[a bc]│. Each of these question also has four alternatives choice, only one of which is the correct answer you have to select one of the codes a,b,c,d  given below:

A) Statement I is true, Statement II is true; Statement II is a correct explanation for Statement I

B) Statement I is true, Statement II is true; Statement II is a not correct explanation for Statement

C) Statement I is true, Statement II is false

D) Statement I is false, Statement II is false

Ques 20:- Each of these questions contains two statement I (Assertion) and statement II (reason) thus Statement I The domains of f(x)√cos(sin x) and g(x)=√sin (cos x) are same ,Statement II since ,-1<+cos(sin x)<=1 and -1<+sin (cos x)<=1. Each of these question also has four alternatives choice ,only one of which is the correct answer you have to select one of the codes a,b,c,d  given below:

A) Statement I is true, Statement II is true; Statement II is a correct explanation for Statement I

B) Statement I is true, Statement II is true; Statement II is a not correct explanation for Statement I

C) Statement I is true, Statement II is false;

D) Statement I is false, Statement II is false

Ques21:-If a,b,c  are sides of a triangle and matrices is  a²        b²       c²                                 

(a+1)²  (b+1)²(c+1)²

(a-1) ²  (b-1) ²  (c-1) ²

=0 is then

a)    ∆ABC  is equilateral

b)   ∆ABC   is right angled isosceles

c)    ∆ABC   is isosceles

d)   None of these

Ques 22:-The ends of a rod of length L move on two mutually perpendicular lines. The locus of the point on the rod which divides it  in the ratio 1:2 is

a)36x²+9y²=l ²             b)36x²+9y²=4l ²         c)9x²+36y²=4l ²      d)none of these

Ques 23:-The position vectors of points A and B are  i-j+3k and 3i+3j+3k respectively . The equation of a plane is vector r .(5i+2j-7k)+9=0 . The points A and B

a)lie on the plane           b)are on the same side of the plane

c)are on the opposite sides of the plane

d)none of these

Ques 24:- A die is tossed thrice .A success is getting 1 or 6 on a toss .The mean and the variance of number of successes is

a)a=1,b²=2/3                b)a=2/3, b²=1         c)a=2, b²=2/3          d)none of these

Ques25:- The integral ∫_-1/2^1/2[[x]+log (1+x)/(1_x)]dx is equal to

a)-1/2               b)0              c)1            d)2log ½

Ques 26:- P and Q be two points on the upper half of the ellipse x²/a²+y²/b²=1 . The center of the ellipse is at the origin O and PQ is parallel to the x-axis such that the triangle OPQ has the maximum possible areas.Point is randomly selected from inside of the upper half of the ellipse . The probability the it lies outside the triangle is

a)∏-1/∏              b)2∏-1/2∏             c) ∏-1/2∏          d) ∏-1/4∏

Ques 27:- A square OABX is formed by line pairs xy=0 and xy+1=x+y, where O is the origin.A circle with center C inside the squire is drawn to touch the line pair xy =0 and another circle with center c₂ and radius twice that of c₂ and the radius twice the c₁ is drawn to touch the circle c₁ and the other line pair .The radius of the circle with center c₁ is

a)√2/V3(√2+1)           b) 2√2/√2(√2+1)         c) √2/3(√2+1)         d) √2+1/3√2

Ques28:- Suppose the functioni_ng(x)=x²ⁿ⁺¹ +a_nx+b_n; (n ϵ  N)satisfies the equation ∫_–¹₁(px+q)g_a(x)dx=0

For all linear functions (px+q) then

a)a_n=b_n=0            b) b_n=0;a_n=-3/2n+3       c) a_n =0; b_n=-3/2n+3          d) a_n=3/2n+3;b_n=-3/2n+3

Ques29:- A function y = f(x) has a second order derivatives f”(x)=6(x-1) if its graph passes through the point(2,1) and at that point the tangent to the graph is y=3x-5, then the function is

a)(x-1)³           b) (x+1)³        c) (x+1)²      d) (x-1)²

Ques30:-If a and d are two complex numbers , then the sum to (n+1) terms of the following series aC₀-(a+d)C₁+(a+2d)C₂-…… is

a)a/2ⁿ          b) na          c) 0      d) none of these

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