# 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= 2x^{2 }.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+x^{2 }/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 }.x^{2 }+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 S^{2} so where thus a^{2} 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 S^{2} 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 y^{2}=16x is tangent to (x-6)^{2} +y^{2 }=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 x^{2}=p(x+1)-q=0 then the value of (α^{2}+2 α+1/ α^{2}+2 α+q)+( (β^{2}+2V+1/ β^{2}+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+x^{2})+…… will be

a)1-x^{n}/1-x b)x(1-x^{n})1-x

c)n(1-x)-x(1-x^{n})/(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 x^{2}+ y^{2 }-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 a^{2 } b^{2 } c^{2 }

(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)36x^{2}+9y^{2}=l ^{2} b)36x^{2}+9y^{2}=4l ^{2} c)9x^{2}+36y^{2}=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,b^{2}=2/3 b)a=2/3, b^{2}=1 c)a=2, b^{2}=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 x^{2}/a^{2}+y^{2}/b^{2}=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 c_{2} and radius twice that of c_{2} and the radius twice the c_{1} is drawn to touch the circle c_{1 }and the other line pair .The radius of the circle with center c_{1} 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)=x^{2n+1 }+a_{n}x+b_{n}; (n ϵ N)satisfies the equation ∫_{–}^{1}_{1}(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 aC_{0}-(a+d)C_{1}+(a+2d)C_{2}-…… is

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

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