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.
If you have missed previous test click : –