IITJEE Mains Mock test -4

IITJEE Mock test

SECTION – I   Straight Objective Type

Question 1:-   If $2^{\cfrac{2\pi}{Sin^{-1}x}}-2(a+2)2^{\cfrac{\pi}{Sin^{-1}x}}+8a\lt 0$ for at least one real x, then

(1)   $a\in \left[ -\infty,\cfrac{1}{8}\right)\cup(2,\infty)$

(2)   $a\in R – \{2\}$

(3)   $a\lt 2$

(4)   $\cfrac{1}{8}\le a\lt 2$

Question 2:-   A and B throw a pair of dice alternatly till one of them wins. If A wins by throwing 7 while B wins by throwing 5 as a sum respectively, then the probability that one of the dice shows 3 on the last throw if A starts the game is

(1)   $\cfrac{11}{36}$

(2)   $\cfrac{11}{28}$

(3)   $\cfrac{7}{22}$

(4)   None of these.

Question 3:-  Let $f(x)=Sinx(Cosx+\sqrt{Cos^2x+Cos^2\theta})$ where ‘$\theta$’ is a given constant, then maximum valume of $f(x)$ is

(1)   $\sqrt{1+Cos^2\theta}$

(2)   $\vert Cos\theta \vert$

(3)  $\vert Sin\theta \vert$

(4)  $\sqrt{1+Sin^2\theta}$

Question 4:- Let a relation R in the set N of natural numbers be defined as $$(x,y)\iff x^2-8xy+7y^2=0\;\forall\,x,y \in N.$$ The relation R is

(1)   reflexive

(2)   symmetric

(3)   transitive

(4)   None of these.

Question 5:-  Suppose $f(x)=x^3+ax^2+2bx+c,$ where a, b, c are chosen respectively by throwing a dice three times. Then the probability that f(x) is decreasing function for some x is 

 (1)   $\cfrac{1}{3}$

(2)   $\cfrac{13}{36}$

(3)   $\cfrac{2}{9}$

(4)   None of these.

Question 6:-  If $p_i=\cfrac{\vert x-a \vert}{x-a},$  ( where i=1, 2, 3, …..,p ) and $a_1 > a_2 > a_3 >……>a_p,$ then $$\lim_{x \to a_q} (\prod_{r=1}^p p_r ),\;1\le q\le p$$ is equal to

(1)  $(-1)^{q+1}$

(2)  $(-1)^{q-1}$

(3)  $(-1)^{q}$

(4)   Limit does not exist.

Question 7:-  If $x\in \Bigl(-\cfrac{\pi}{2},\cfrac{\pi}{2} \Bigr),$ then the value of $tan^{-1}\Bigl(\cfrac{tanx}{4}\Bigr) + tan^{-1}\Bigl(\cfrac{3Sin2x}{5+3Cos2x}\Bigr)$ is

(1)   $\cfrac{x}{2}$

(2)   $x$

(3)   $2x$

(4)   $3x$

Question 8:-  A line intersects the ellipse $\cfrac{x^2}{4a^2}+\cfrac{y^2}{a^2}=1$ at A and B , and parabola $y^2=4a(x+2a)$ at C and D. The line segment AB subtends a right angle at the centre of the ellipse. Then, the locus of the point of intersection of tangents to the parabola of C and D is

(1)   $y^2-a^2=\cfrac{5}{4}(x-4a)^2$

(2)   $y^2-2a^2=10(x-4a)^2$

(1)   $y^2+a^2=\cfrac{5}{2}(x-4a)^2$

(1)   $y^2+4a^2=5(x+4a)^2$

Question 9:-  P is any point on the circum-circle of $\Delta ABC$ other than the vertices. H is the orthocentre of $\Delta ABC$, M is the mid-point of PH and D is the mid-point of BC. Then,

(1)   DM is parallel to AP.

(2)   DM is perpendicular to AP.

(3)   AP is opposite side of DM.

(4)   None of these.

Question 10:-  Let $f(x)=(x-3)^3x^n,\;n \in N$. Then $f(x)$ has a

(1)   minimum at x=0, if n is even.

(2)   maximum at x=0, if n is even.

(3)   maximum at x=3 $\forall n \in N$

(4)   None of these.

Question 11:-  Two persons who are 500 m apart, observe the direction and the angle of elevation of a ballon at the same instant. One finds the elevation to be $\cfrac{\pi}{3}$ and direction being South-West, while the other finds the elevation to be $\cfrac{\pi}{4}$ and direction being west. Height of the balloon is

(1)   $500\sqrt{\cfrac{3}{4-\sqrt{6}}}$ m

(2)   $250\sqrt{\cfrac{3}{4+\sqrt{6}}}$ m

(3)   $250\sqrt{\cfrac{3}{4-\sqrt{6}}}$ m

(1)   $500\sqrt{\cfrac{3}{4+\sqrt{6}}}$ m

Question 12:-  Let  $p$ and $q$ be two statement, then $\sim(\sim p\land q)\land (p\lor q)$ is logically equivalent to

(1)   p

(2)   q

(3)   $p\land q$

(4)   $p\lor\sim q$

Question 13:-  The area of the region bounded by the curves $y=x^2,\;y=\vert 2-x^2 \vert$ and $y=2$ which lies to the right of the line $x=1$, is

(1)   $\Bigl(\cfrac{12-20\sqrt{3}}{2}\Bigr)$ sq. units

(2)   $\Bigl(\cfrac{20-\sqrt{2}}{3}\Bigr)$ sq. units

(3)   $\Bigl(\cfrac{12-20\sqrt{2}}{3}\Bigr)$ sq. units

(4)   $\Bigl(\cfrac{12-20\sqrt{2}}{3}\Bigr)$ sq. units

Question 14:-  The sum of squares of the perpendicular drawn from the points (0,1) and (0,-1) to any tangent to a curve is 2. Then, the equation of the curve is

 (1)   $2y=c(x+2)$

(2)    $y=c(x\pm 1)$

(3)    $y=c(x+2)$

(4)    $y=c(x\pm 2)$

Question 15:-  The coordinates of four vertices of a quadrilateral are (-2,4), (-1,2), (1,2), and (2,4) taken in order. The equation of the line passing through the vertex (-1,2) and dividing the quadrilateral in two equal areas is 

(1)   x+1=0

(2)   x+y=1

(3)   x-y+3=0

(4)   None of these.

Question 16.  PSQ is a focal chord of a parabola whose focus is S and vertex is A. PA, QA and produced to meet the directrix in R and T. Then angle $\angle RST$ is equal to

(1)   $90^0$

(2)   $60^0$

(3)   $45^0$

(4)    $30^0$

Question 17:-  If a, b, c are distinct positive number such that (b+c-a), (c+a-b), and (a+b-c) are positive, then the expression (b+c-a)(c+a-b)(a+b-c)-abc is

(1)   positive

(2)   negative

(3)   non-positive

(4)   non-negative

Question 18:-  $\sum_{r=0}^n {n \choose r} \cfrac{1+rln 10}{(1+ln 10^n)^r}$ equals

(1)   1

(2)   -1

(3)    n

(4)    None of these.

Question 19:-  $\int (1+x-x^2)e^{x+x^{-1}} dx$ is equal to

(1)    $x.e^{x+x^{-1}}+C$

(2)    $-x.e^{x+x^{-1}}+C$

(1)    $(x+1)e^{x+x^{-1}}+C$

(1)    $(x-1)e^{x+x^{-1}}+C$

Question 20:-  If a function $y=f(x)$ is defined as $y=\cfrac{1}{t^2-t-6}$ and $t=\cfrac{1}{x-2},\; t\in R$. Then $f(x)$ is discontinuous at 

(1)   $2, \cfrac{2}{3}, \cfrac{7}{3}$

(2)   $2, \cfrac{3}{2}, \cfrac{7}{3}$

(3)   $2, \cfrac{3}{2}, \cfrac{5}{3}$

(4)   None of these.

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

Question 21.  The sum of rational term(s) in $(\sqrt{3}+\sqrt[3]{2}+\sqrt[4]{5})^8$ is equal to

Question 22.  The number of integer values of K for which the equation

7Cosx + 5Sinx = 2K + 1 has solution

Question 23.  If $Sin^{-1}x+Sin^{-1}y+Sin^{-1}z=\pi$ then $x^4+y^4+z^4+4x^2y^2z^2=k(x^2y^2+y^2z^2+z^2x^2)$. where k is equal to

Question 24.  If $f(x)=\cfrac{9^2}{9^2+9}$, then $\sum_{r=1}^4029 f(\cfrac{r}{2015})$ is equal to

Question 25.  If $Z_1$ and $Z_2$ are lying on $\vert Z_1-1 \vert +\vert Z_2+1 \vert = 3$ respectively. If $A_1=min{\vert Z_1-Z_2 \vert}$ and $A_2=max{\vert Z_1 – Z_2 \vert}$

End of test – 4.

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