# Mathematics Sample Paper for IIT Mains

Mathematics Sample Paper for IIT Mains available for those who are going to apper in the IIT Mains Entrance Examination.Mathematics Sample Paper of IIT Mains Entrance.We are proud to offer JEE Main Model Papers for JEE Main 2014 aspirants so that they can excel in their endeavor to do well and achieve the rank they predict for.

**Questions:**

**This is the Sample Paper for IIT Mains-Mathematics**

**Ques1:- The value of lim _{h->0} (a+h)^{2}sin (a+h)-a^{2}sina/h is**

a)2a sin a +a^{2} cos a b) 2a sin a – a^{2} cos a

c) 2a cos a + a^{2} sin a d) none of these

**Ques2****:- if∫√cos 2x/sin x dx = -log │cot x +√cot ^{2} x -1│ +A+C then A is equal to**

a)1/√2 log│√2 +√1-tan^{2 } x/√2-+√1-tan^{2 } x│

b) 1/√2 log│√2 +√1-cos^{2 } x/√2-+√1-cos^{2 } x│

c) 1/√2 log│√2 +√1-sin^{2 } x/√2-+√1-sin^{2 } x│

d) none of these

**Ques 3:- The value of ∫ ^{e2}_{e-1}│log_{e} x/x│ dx is**

a)3/2 b) 5/2 c) 3 d)5

**Ques 4****:- The area of the figure bounded by the straight lines x=0,x=2 and the curves y=2 ^{x},y=2x-x^{2}, is**

a)(4/log 2 +8/3) sq unit b) (4/log 2 -8/3)sq unit c) (8/log 3 -4/3)sq unit

d) (3/log 2 – 4/3)sq unit

**Ques 5****:- If the vector –I +j –k bisects the angle between the vector c and the vector 3i+4j, then the unit vector in the direction of c is**

a)–1/15(11-10+2) b) 1/15(11+10+2) c)-1/15(11+10-2) d) )-1/15(11+10+2)

**Ques 6****:-A circle is drawn to pass through the extremities of the latusectum of the parabola y ^{2} =8x . it is given that, this circle also touches the directrix of the parabola . Radius of this circle is equal to**

a)4 b)√21 c)√26 d) 3

**Ques7****:- the equation of the ellipse whose axes are coincident with the coordinate axes and which touches the straight lines 3x -2y-20=0 and x+ 6y – 20=0 is**

a)X^{2}/5 +y^{2}/8 =1 b) x^{2}/40 +y^{2}/10 = 10 c) x^{2}/40 +y2/10 =1 d) x^{2}/10+y^{2}/40=1

**Ques8****:- Each of these questions contains two statement I (Assertion) and staementII (reason).Thus Statement I -The number of real solution of the equation sin x = 2 ^{x} +2^{-x} is zero, Statement II │sin x │ <= 1. 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

**Ques 9****:- Each of these questions contains two statement I (Assertion) and staementII (reason).Thus Statement I if r ^{->}.a^{->}=0, r^{->}.c^{-> }=0 for some non zero vector r^{->}, then a^{->}, b^{->} and c^{->} are coplanar vectors, Statement II if a^{->} b^{->} and c^{->} are coplanar , then a^{->}+ b^{->}+ c^{->}=0. 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

**Ques10****:- If two distinct chords, drawn from the point (p,q) on the circle X ^{2} + y^{2} = px + qy ,(where pq≠0) are bisected by the x- axis , then**

a)p^{2}= q^{2} b) p^{2}=8 q^{2 } c) p^{2}<8 q^{2 }d) p^{2}> 8 q^{2}

**Ques11****:- A five digit number is formed by writing the digit s 1,2,3,4,5 in a random order without repetition. Then the probability that the number is divisible by 4 is**

a)3/5 b18/5 c)1/5 d)6/5

**Ques12****:- If A and B are different matrices satisfying A ^{3}= B^{3} andA^{2}B=B^{2}A then **

a)Det (A^{2}+B^{2}) must be zero b) det (A-B )must be zero c)det (A^{2}+B ^{2}) as well as det (A-B) must be zero d) At least one of the det (A^{2}+B^{2}) or det(A-B) must be zero

**Ques13**:**– By the Roles’ theorem between any two real zeros of a polynomial p(x) lies a zero of p’(x).If a,b,c ϵ R and 3b ^{2}-8ac<0, then the equation f(x)= ax^{4}+bx^{3}+cx^{2}+2006x+2007 has**

a)All real roots b) all imaginary roots c) cannot have all real roots d)two real and two imaginary root

**Ques 14****:- By the Roles’ theorem between any two real zeros of a polynomial p(x) lies a zero of p’(x).If a+b+c=0 , then equation 3ax ^{2}+2bx+c=0 has**

a)At least one root in [0,1] b) one positive and one negative root c) no real roots d) none of these above

**Ques15****:- If (1+x) ^{n} = E_{r}^{n}_{=0 } ^{n}C_{r} x^{t}, then (1+c_{1/}c_{0})( 1+c_{2/}c_{1})……( 1+c_{n/}c_{n-1}) is equal to**

a)(n+1)^{n-1}/(n-1)! b) n^{n-1}/(n-1)! c)(n+1)^{ n}/n! d)(n+1)^{n+1}/n!

**Ques16**:**– if f’(x) =sin (log x) and y= f(2x+3/3-2x), then dy/dx at x=1 is equal to**

a)6sin lag(5) b)5sin log(6) c) 12sin log (5) d) 5 sin log(12)

**Ques17****:- The distance between the line r ^{->}=2i^{^}-2j^{^}+3k^{^} +α(i^{^}-j^{^}+4K^{^}) and the plane r^{->}.(i^{^}+5j^{^}+K^{^})=5 is**

a)10/3 b)3/10 c)10/3√3 d) 10/9

**Ques18****: The equation of the straight line passing through the point (4,3) and making intercepts on the coordinate axes whose sum is -1 , is**

a)x/2+y/3=-1 and x/-2+y/1=-1

b) x/2 –y/3 =-1 and x/-2 + y/1 =-1

c)x/2 +y/3 =1 and x/-2+y/1=1

d) x/2 –y/3 =1 and x/-2 +y/1 =1

**Ques19**:**-Negation of the preposition “if we control population growth , we prosper”,is**

a) If we do not control population growth , we do not prosper

b) If we do not control population growth, we prosper

c) We control population growth but we do not prosper

d) We do not control population growth but we prosper

**Ques20**:**– Let x, y be real satisfying sin x+cos y +1 and ****sin y +cos x=-1 . Then which of the following must be correct ?**

a)Sin(x+yion , 0 b)cos(x-y)=0 c) cos(x-y)=0 d) cos2x=cosy

** Ques21****:-Number of solution of the equation sinx=]x], where[.] denotes the largest integer function is**

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

**Ques22****:-A bag contains a white and b black balls. Two players A and B alternatively draw a ball from the bag, replacing the ball each time after the draw. A begins the game .If the probability of A winning (that is drawing a white ball) is twice the probability of B winning, then the ratio a:b is equal to**

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

**Ques23****:-If f(x) =(x ^{3}) the**

a)f is continuous but not derivable at x=0 b) f”(0^{*})=2 c) f”(0^{*})=1 d) f is not derivable at x=0

**Ques24****:- If the value of the determinant of matrix a 1 1**

**1 b 1**

**1 1 c ****Is positive then**

a)abc>1 b) abc>-6 c)abc <-8 d) abc>-2

**Ques 25****:- If f(x) is a function satisfying f(x+y)=f(x)f(y) for all x,y,ϵ N such that ****f(1)=3 and _{ }E^{n}_{x=1} f(x) =120, then the value of n is**

a)4 b)5 c)6 d)none of these

**Ques 26****:- The value of tan{ cos ^{-1}(-2/7)-∏/2} is**

a)None of these b) 2/3 c) 1/√5 d) 4/√5

**Ques27****:- A boat is being rowed away from a cliff of 150 m height .A t the top of the cliff the angle of depression of boat changes from 60degree to 45 degree in 4 min, Then the speed of the boat in (m/h) is**

a)None of these b) 4500/√3(√3-1) c) 45000/√3 d) 4500/√3(√3+1)

**Ques 28**:**– Let f:R -. R be a function defines f(x) = x ^{2}+2x +5/x^{2}+x+1 is**

a)One – one and into b) one-one and onto

c)many-one and onto d)many one and into

**Ques 29**:**-The equation of a line of intersection of plane s 4x + 4y – 5z =12 and 8x +12y -13z=32 can be written as**

a)X-1/2=y+2/-3=z/4 b) x-i/2=y-2/3=z/4 c)x/2=y+1/3= z-2/4 d) x/2=y/3=z-2/4

**Ques 30****:- Statement I if e ^{xy} + log(xy)+cos(xy)+5=0 then dy/dx=-y/x , Statement II d/dx(xy)=0 **

**ð Dy/dx=-y/x**

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 true

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