SECTION – I Straight 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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