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IITJEE Mains Mock test -2 - Best Maths Practice Material

IITJEE Mains Mock test -2

SECTION – IStraight Objective Type

Question 1:-  For the system of equations   $$log_{10}(x^3-x^2)=log_5y^2$$ $$log_{10}(y^3-y^2)=log_5z^2$$ $$log_{10}(z^3-z^2)=log_5x^2$$

Which of following is/are true ?

(1)   there are infinite number of solution.

(2)   there is unique solution with $x, y, z\in Q$

(3)   there are exactly two solution with $x, y, z\in Q$

(4)   The system equation has no solution.

Question 2:-  Three points A, B, and C are considered on a parabola. The tangent to the parabola at these points from MNP ( PN being tangent at A, PM at B and MN at C ). If the line through B and parallel to axis of parabola intersect AC at L then quadrilateral LMNP

(1)   is always a parallelogram.

(2)   Can never be parallelogram.

(3)    is parallelogram only when ordinates of A, B, C are in A.P.

(4)    has exactly two sides parallel only.

Question 3:-  Let  $f(x)=\cfrac{e^{tanx}-e^x+ln(Secx+tanx)-x}{tanx-x}$  be a continous function at x=0.  The value of f(0) equals

(1)    $\cfrac{1}{2}$

(2)    $\cfrac{2}{3}$

(3)    $\cfrac{3}{2}$

(4)     2

Question 4:-  Let  $\alpha , \beta , \gamma$ are the roots of the equation $8x^3+1001x+2008=0$. The value of  $(\alpha + \beta)^3+(\beta +\gamma)^3+(\gamma+\alpha)^3$ is

(1)   251

(2)   751

(3)   735

(4)   753

Question 5:-  Number of four digit numbers of the form N=abcd which satisfy following three condition  $(i)\;4000<N<6000$  (ii) N is a multiple of 5  (iii)  $3\le b<c\le 6$ is equal to

(1)   12

(2)   18

(3)   24

(4)   48

Question 6:-  A coin that comes up head with probability  p>0 and tails with probability 1-p > 0 independently on each flip, is flipped eight times. Suppose the probability of three heads and five tails is equal to $\cfrac{1}{25}$ of the probability of five heads and three tails. The value of p is

(1)   $\cfrac{5}{6}$

(2)   $\cfrac{2}{3}$

(3)   $\cfrac{1}{6}$

(4)   $\cfrac{3}{4}$

Question 7:-  The first two terms of a G.P, add up to 12. The sum of the $3^{rd}$ and $4^{th}$ term is 48. If the term of the G.P are alternatively positive and negative, then the first term is

(1)   – 4

(2)   – 12

(3)   12

(4)   14

Question 8. If a curve is represented parametrically by the equation $x=4t^3+3$ and $y=4+3t^4$ and $\cfrac{(\cfrac{d^2x}{dy^2})}{(\cfrac{dx}{dy})^n}$ is a constant then the value of n, is

(1)   3

(2)   4

(3)   5

(4)   6

Question 9:-  The minimum value of the function $f(x)=x^{\frac{3}{2}}+x^{-\frac{3}{2}} – 4\Bigl(x+\cfrac{1}{x}\Bigr)$ for all permissible real x, is

(1)   -10

(2)   -6

(3)   -7

(4)   -8

Question 10:- The complete set of ‘a’ for which there exist one line that is tangent to the graph of the curve $y=x^3-ax$ at one point and normal to the graph at another point is given by

(1)   $a\in \left(-\infty,-\cfrac{4}{3} \right]$

(2)   $a\in \left[-\cfrac{4}{3},\infty \right)$

(3)   $a\in \left[\cfrac{4}{3},\infty \right)$

(4)   $a\in \left(-\infty,\cfrac{4}{3} \right]$

Question 11:- Let a relation R in the set N of natural number be defined as $(x,y)\in R$ if and only if $x^2-4xy+3y^2=0\,\forall\;x,y \in N$. The relation R is

(1)   reflexive relation.

(2)   symmetric relation.

(3)   transitive relation.

(4)   equivalaence relation.

Question 12:- The top of a hill observed from the top and bottom of a building of height h is at angle of $\theta$ and $\phi$ respectively. The height of the hill is

(1)   $\cfrac{hCot\phi}{Cot\phi – Cot\theta}$

(2)   $\cfrac{hCot\theta}{Cot\theta – Cot\phi}$

(3)   $\cfrac{htan\theta}{tan\theta – tan\phi}$

(4)   None of these.

Question 13:-  Mean deviation about mean from the following data is

 xi : 3 9 17 23 27 fi : 8 10 12 9 5

(1)    7.15

(2)    7.09

(3)    8.05

(4)    None of these.

Question 14:-  The distance between the line  $x=2+t,\;y=1+t,\;z=-\cfrac{1}{2}-\cfrac{t}{2}$ and the plane $\vec r.(\vec i +2\vec j +6\vec k)=10,$ is

(1)   $\cfrac{1}{6}$

(2)   $\cfrac{1}{\sqrt{41}}$

(3)   $\cfrac{1}{7}$

(4)   $\cfrac{9}{\sqrt{41}}$

Question 15:-  If $\vec a,\vec b, \vec c$ are non-zero vector then value of the scalar $((\vec a \times \vec b)\times \vec a)((\vec b \times \vec a)\times \vec b)$ equals

(1)   $-(\vec a.\vec b)\vert \vec a \times \vec b \vert ^2$

(2)   ${\vec a}^2\vert \vec a \times \vec b \vert^2$

(3)   $\vert \vec b \vert^2\vert \vec a \times \vec b \vert^2$

(4)   $(\vec a.\vec b)\vert \vec a \times \vec b \vert ^2$

Question 16.  If $\cfrac{dy}{dx}=(e^y-x)^{-1}$ where $y(0)=0$, then y is expressed explicitly as

(1)   $\cfrac{1}{2}ln(1+x^2)$

(2)   $ln(1+x^2)$

(3)   $ln(x+\sqrt{1+x^2})$

(4)   $ln(x+\sqrt{1-x^2})$

Question 17:-  The area of the region bounded by the curve $C:y=\cfrac{x+1}{x^2+1}$ and the line $L:y=1$, is

(1)   $1-\cfrac{1}{2}ln2+\cfrac{\pi}{4}$

(2)   $1+ln2+\cfrac{\pi}{4}$

(3)   $\cfrac{1}{2}ln2+\cfrac{\pi}{4}-1$

(4)   $1+ln2-\cfrac{\pi}{2}$

Question 18:-  If  $\lim_{n \to \infty} (\cfrac{n.3^n}{n(x-2)^n+n.3^{n+1}-3^n})= \cfrac{1}{3}$  where  $n \in N$, then the number of integer(s) in the range $’x’$ is

(1)   3

(2)   4

(3)   5

(4)   infinite.

Question 19:- If Z is a complex number satisfying the equation $\vert Z-(1+i)\vert^2=2$ and $w=\cfrac{2}{Z}$, then the locus traced by $’w’$ in arggand plane is

(1)   x-y-1=0

(2)   x+y-1=0

(3)   x-y+1=0

(4)   x+y+1=0

Question 20:- The angle between pair of tangents drawn to the curve $7x^2-12y^2=84$ from $M(1,2)$ is

(1)   $2tan^{-1}\cfrac{1}{2}$

(2)   $2tan^{-1}2$

(3)   $2(tan^{-1}\cfrac{1}{3}+tan^{-1}\cfrac{1}{2})$

(4)   $2tan^{-1}3$

SECTION – II Numerical  Value Type.  ( xxxxxxxxx.xxx form )

Question 21.  If in a triangle ABC, a=5, b=4, and $Cos(A-B)=\cfrac{31}{32}$, then the third side c is equal to

Question 22.  The number or real values of  $x$ that satisfy the equation   $Sin(2Cos^{-1}(Cot(2tan^{-1}x)))=0$, is/are

Question 23.  Let $a_1,a_2,…a_n$ be the possible values of ‘a’ such that the slope of one of the line represented by $ax^2-6xy+y^2=0$ is suare of the other , then the value of $$\sum_{i=1}^n (-1)^ia_i$$

Question 24.  A circle of radius ‘r’ is inscribed in a trapezium of area $900\sqrt{2}$ ; two of whose sides are along the co-ordinate axis and the two parallel sides have slope of ‘-1’. then ‘r’ is

Question 25.  If $A=[a_{ij}]$, where $a_{ij}=i^{100}+j^{100}$, then $101(\lim_{n \to \infty} \Bigl(\cfrac{\sum_{i=1}^n a_{ij}}{n^{101}}\Bigr))$ equals

End of test – 2

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Mock test 1