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Q1 Let \(f\) be a one-one function with domain {x, y, z} and range {1, 2, 3}. It is given that exactly one of the following statements is true and the remaining two are false: \(f(x)=1\), \(f(y)≠1\), \(f(z)≠2\). Then determine \(f^{-1}(1)\).

Q2 Find the equation of the circle passing through the point \((-4, 3)\) and touching the lines \(x+y=2\) and \(x-y=2\).

Q3 Show that the equation \(e^{\sin x}-e^{-\sin x}-4=0\) has no solution.

Q4 Find the co-ordinates of the points of intersection of the curves \(y=\cos x\) and \(y=\sin3x\), if \(-\frac \pi 2 ≤x ≤ \frac \pi 2\).

Q5 A is a point on the parabola \(y^2=4ax\). The Normal at A cuts the parabola again at point B. If AB subtends a right angle at the vertex of the parabola, then find the slope of AB.

Q6 Use the formula \(\displaystyle \lim_{a\to 0} \frac{a^x-1}{x}=\ln a\) to find \( \displaystyle \lim_{x\to 0}\frac {2^x-1}{(1+x)^{1/2}-1}\) .

Q7 Determine the values of \(a, b, c\) for which the function \(f(x)\) is continuous at \(x=0\). \[f(x)=\begin{cases} \Large \frac{\sin[a(x+1)]+\sin x}{x}, \normalsize x<0 \\ c, x=0\\ \Large \frac{(x+bx^2)^{1/2}-x^{1/2}}{bx^{3/2}}, \normalsize x>0 \end{cases}\]

Q8 Let \(f\) be a twice differentiable function, such that \(f''(x)=-f(x)\), \(f'(x)=g(x)\), \(h(x)=[f(x)]^2+[g(x)]^2\), \(h(5)=11\). Find \(h(10)\) if \(h(5)=11\).

Q9 \(mn\) squares of equal size are arranged to form a rectangle of dimension \(m\) by \(n\), where \(m, n \in\mathbb N\). Two squares will be called "neighbours" if they have exactly one common side. A number is written in each square such that the number in any square is the arithmetic mean of the numbers written in neighbouring squares. Show that this is possible only if all the numbers used are equal.

Q10 If \(a_1\) \(a_2\), \(...\), \(a_n\) are in arithmetic progression, where \(a_i>0\) for all \(i\), show that \[\frac{1}{a_1+\sqrt{a_2}}+\frac{1}{\sqrt{a_2}+\sqrt{a_3}}+\frac{1}{\sqrt{a_3}+\sqrt{a_4}}+...+\frac{1}{\sqrt{a_{n-1}}+\sqrt{a_n}}=\frac{n-1}{\sqrt{a_1}+\sqrt{a_n}}\]

Q11 Without expanding a determinant at any stage, show that \[\begin{vmatrix} x^2+x & x+1 & x-2\\ 2x^2+3x-1 & 3x & 3x-3\\ x^2+2x+3 & 2x-1 & 2x-1\\ \end{vmatrix} = xA+B\]where A and B are determinants of order 3, not involving \(x\).

Q12 For any real \(t\), \(x=\frac12(e^t+e^{-t}), \ y=\frac12(e^t-e^{-t})\) is a point on the hyperbola \(x^2-y^2=1\). Show that the area bounded by the hyperbola and the lines joining the centre to the points corresponding to \(t_1\) and \(-t_1\) is \(t_1\).

Q13 If f(x) and g(x) are differentiable functions for \(0≤x≤1\) such that \(f(0)=2\), \(g(0)=0\), \(f(1)=6\), \(g(1)=2\), then show that there exists \(c\) satisfying \(0<c<1\) and \(f'(c)=2g'(c)\).

Q14 Does there exist a geometric progression containing 27, 8 and 12 as three of its terms? If it exists, how many such progressions are possible?

Q15 A and B are two candidates seeking admission in IIT. The probability that A is selected is 0.5 and the probability that A and B are selected is at most 0.3. Then is it possible that the probability of B getting selected is 0.9?

Q16 Find the shortest distance between the point \((0, c)\) from the parabola \(y=x^2\) , where \(0≤c≤1\).

Q17 If \(\large ax^2+\frac{b}{x}≥c \ \forall x \in R^+\) where \(a>0\) and \(b>0\), then show that \(27ab^2≥4c^3\)

Q18 Show that \(\large \int_0^{\pi}xf(\sin x) \ dx=\frac \pi 2 \int_0^{\pi}f(\sin x)\ dx\).

Q19 Find the value of \(\large \int_{-1}^\frac 32 |x \sin (\pi x)| \ dx\).

**Note** Do not post the solution to these problems in the comments. Instead consider, if you want to, posting them under the respective problem. If you can tell the marks of any of these questions, do inform.

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