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SECTION – I Straight Objective Type
Question 1. The range of function $ f(x)=(1+Sec^{-1}x)(1+Cos^{-1}x) $ is
(1) R
(2) $(- \infty , 0]\bigcup [4,\infty)$
(3) $\{1,(1+{\pi})^2\}$
(4) $\{0,(1+{\pi}^2)\}$
Question 2. $ \int e^{Sin{-1}x} \Bigl(\cfrac{ln x}{\sqrt{1-x^2}}+\cfrac{1}{x}\Bigr) dx $ is
(1) $(lnx) e^{Sin^{-1}x}+C$
(2) $\cfrac{e^{Sin{-1}x}}{x}+C$
(3) $\cfrac{e^{Sin{-1}x}}{ln x}+C$
(4) $e^{Sin^{-1}}\Bigl(ln x + \cfrac{1}{x}\Bigr)+ C$
Question 3. If $ A=\begin{bmatrix} \alpha & 0 \\ 1&1 \end{bmatrix}$ and $B=\begin{bmatrix} 1& 0 \\ 3 & 1 \end{bmatrix}$ , then value of $\alpha$ for which $A^2=B$ is
(1) 1
(2) -1
(3) i
(4) no real value of alpha.
Question 4. If $\begin{vmatrix} 2a+b+c & a+2b+c & a+b+2c \\ a+2b+c &a+b+2c & 2a+b+c \\ a+b+2c & 2a+b+c & a+2b+c \end{vmatrix}\, =\lambda \begin{vmatrix} a & b & c \\ b & c & a \\ c & a & b \end{vmatrix} $ then the value of $\lambda$ is
(1) 4
(2) -4
(3) 0
(4) 4 (a+b+c)
Question 5. Consider the following equation in x and y
$ (x-2y-1)^2+(4x+3y-4)^2+(x-2y-1)(4x+3y-4)=0 $ $$ $$ How many solutions to (x,y) with real , does the equation have.
(1) 0
(2) 1
(3) 2
(4) more than 2.
Question 6. If the line x-2y-1=0 intersect parabola $ y^2=4x $ at P and Q . Normal at P and Q intersect each other at R. Then co-ordinate of R is .
(1) (-1,4)
(2) (17,4)
(3) (18,4)
(4) (19,4)
Question 7. $\int_{-1}^{1} \cfrac{e^{-\cfrac{1}{x}}}{x^2(1+e^{-\cfrac{2}{x}})} dx $ is equal to.
(1) $\cfrac{\pi}{2}-2tan^{-1}e$
(2) $\cfrac{\pi}{2}-2Cot^{-1}e$
(3) $2tan^{-1}e$
(4) $\pi – 2tan^{-1}e$
Question 8. $\int_0^{\pi} x(Sin^2(Sinx)+Cos^2(Cosx))dx\,=$
(1) $\pi^2$
(2) $\cfrac{\pi^2}{2}$
(3) $\cfrac{\pi^2}{4}$
(4) $\cfrac{\pi^2}{8}$
Question 9. Tangent drawn at any point ‘P’ of the curve $y=\sqrt{4x-x^2}$ meet $y-axis$ at point A . If O is origin then
(1) OP = AP
(2) OP = OA
(3) OA = AP
(4) OP = 2.OA
Question 10. The number of arrangements which can be made out of the letter word ALGEBRA , without changing the relative order ( positions ) of vowels and consonants , is
(1) 72
(2) 54
(3) 36
(4) 64
Question 11. Out of 4 children , 2 women and 4 men , four persons are selected at random . The probability that there are exactly 2 children among the selected is
(1) $\cfrac{11}{21}$
(2) $\cfrac{2}{7}$
(3) $\cfrac{10}{21}$
(4) $\cfrac{3}{7}$
Question 12. The extremities of the diagonal of a rectangle are $(-4,4)$ and $(6,-1)$. A circle circumscribes the rectangle and cuts intercept of length AB on y-axis. The length of AB is
(1) 11
(2) 12
(3) 13
(4) 14
Question 13. The median of a set of 9 distinct observation observations is 20.5 . If each of the largest 4 observations of the set is increased by 2 , then the median of the new set
(1) is increased by 2.
(2) is decreased by 2.
(3) is two times the original median.
(4) remains the same as that of the original set.
Question 14. A relation R is defined on the set of circles such that ” $C_1$ R $C_2\,\Rightarrow$ circle $C_1$ and $C_2$ touch each other externally ” then relation R is
(1) Reflexive and symmetric but not transitive.
(2) Symmetric only.
(3) Symmetric and transitive but not reflexive.
(4) Equivalence.
Question 15. P is a variable point on ellipse $\cfrac{x^2}{a}+\cfrac{y^2}{b}\,=\,2$ whose foci are $F_1\;,\;F_2$ then maximum area of $\Delta PF_1F_2$
(1) $2a\sqrt{a^2-b^2}$
(2) $2b\sqrt{a^2-b^2}$
(3) $a\sqrt{2a^2-2b^2}$
(4) $b\sqrt{2a^2-2b^2}$
Question 16. A function from integers to integers is defined as $f(n)=\begin{cases} n+3, & \text{if n is odd} \\ \cfrac{n}{2} , & \text{ if n is even} \end{cases} $.
If k is an odd integer and f(f(f(x))) = 27 then the sum of digits of k is
(1) 3
(2) 6
(3) 9
(4) 12
Question 17 . $lim_{x \to 0} \Bigl(\cfrac{Sinx}{tan(5x)}\cfrac{(1-Cos4x)(5^x-4^x)}{x^3}\Bigr)$ equals to
(1) $\cfrac{8}{3} ln\Bigl(\cfrac{5}{2}\Bigr)$
(2) $\cfrac{8}{5} ln\Bigl(\cfrac{5}{2}\Bigr)$
(3) $\cfrac{8}{5} ln\Bigl(\cfrac{5}{4}\Bigr)$
(4) $\cfrac{8}{3} ln\Bigl(\cfrac{5}{4}\Bigr)$
Question 18. General solution of equation $\vert Sinx \vert\,=\,Cosx $ is
(1) $2n{\pi}\,+\,\frac{\pi}{4}\,,n\in I$
(2) $n{\pi}\,+\,\frac{\pi}{4}\,,n\in I$
(3) $2n{\pi}\,\pm \,\frac{\pi}{4}\,,n\in I$
(4) $n{\pi}\,+\,\frac{\pi}{4}\,,n\in I$
Question 19. If $a_1,a_2,a_3,…..,\,a_{20}$ are in A.M’s between 13 and 67 , then the maximum value of $a_1a_2a_3….a_{20}$ is
(1) ${20}^{20}$
(2) ${40}^{20}$
(3) ${60}^{20}$
(4) ${80}^{20}$
Question 20. If $f(x)= \begin{cases} x^2+2, & \text{$x$<0} \\ 3,& \text{$x$=0} \\ x+2,& \text{$x$>0} \end{cases} $ then which of the following is FALSE .
(1) f(x) has a local maximum at x = 0.
(2) f(x) is strictly decreasing on the left of x=0
(3) f'(x) is strictly increasing on the left of x=0
(4) f(x) is strictly increasing on the right of x=0
SECTION – II Numerical Value Type. ( xxxxxxxxx.xxx form )
Question 21. Area bounded by the region $R = {(x,y):y^2\le x\le \vert y \vert }$ is.
Question 22. The maximum value of $f(x)=2bx^2-x^4-3b$ is $g(b)$ , where b>0 . If b varies then the minimum value of g(b) is
Question 23. For a certain curve $y=f(x)$ satisfying $\cfrac{d^2y}{dx^2}=6x-4\;,f(x) $ has local minimum value 5 when x=1 . The global maximum value of $f(x)\,if\,0\le x \le 2$ , is
Question 24. If $f(x)=min\{\vert x \vert^2 – 5\vert x \vert , 1\} $ then $f(x)$ is non differentiable at $\lambda$ points , then $\vert \lambda \vert$ equals.
Question 25. If $a,b,c$ are distinct odd integers and $\omega$ is non real cube root of unity , then minimum value of $\vert a\omega^2 + b +c\omega \vert $ , is
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