IIT JEE 1982 Mathematics: The actual Subjective Questions

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  • Q1 Let ff 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)=1f(x)=1, f(y)1f(y)≠1, f(z)2f(z)≠2. Then determine f1(1)f^{-1}(1).

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

  • Q3 Show that the equation esinxesinx4=0e^{\sin x}-e^{-\sin x}-4=0 has no solution.

  • Q4 Find the co-ordinates of the points of intersection of the curves y=cosxy=\cos x and y=sin3xy=\sin3x, if π2xπ2-\frac \pi 2 ≤x ≤ \frac \pi 2.

  • Q5 A is a point on the parabola y2=4axy^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 lima0ax1x=lna\displaystyle \lim_{a\to 0} \frac{a^x-1}{x}=\ln a to find limx02x1(1+x)1/21 \displaystyle \lim_{x\to 0}\frac {2^x-1}{(1+x)^{1/2}-1} .

  • Q7 Determine the values of a,b,ca, b, c for which the function f(x)f(x) is continuous at x=0x=0. f(x)={sin[a(x+1)]+sinxx,x<0c,x=0(x+bx2)1/2x1/2bx3/2,x>0f(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 ff be a twice differentiable function, such that f(x)=f(x)f''(x)=-f(x), f(x)=g(x)f'(x)=g(x), h(x)=[f(x)]2+[g(x)]2h(x)=[f(x)]^2+[g(x)]^2, h(5)=11h(5)=11. Find h(10)h(10) if h(5)=11h(5)=11.

  • Q9 mnmn squares of equal size are arranged to form a rectangle of dimension mm by nn, where m,nNm, 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 a1a_1 a2a_2, ......, ana_n are in arithmetic progression, where ai>0a_i>0 for all ii, show that 1a1+a2+1a2+a3+1a3+a4+...+1an1+an=n1a1+an\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 x2+xx+1x22x2+3x13x3x3x2+2x+32x12x1=xA+B\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+Bwhere A and B are determinants of order 3, not involving xx.

  • Q12 For any real tt, x=12(et+et), y=12(etet)x=\frac12(e^t+e^{-t}), \ y=\frac12(e^t-e^{-t}) is a point on the hyperbola x2y2=1x^2-y^2=1. Show that the area bounded by the hyperbola and the lines joining the centre to the points corresponding to t1t_1 and t1-t_1 is t1t_1.

  • Q13 If f(x) and g(x) are differentiable functions for 0x10≤x≤1 such that f(0)=2f(0)=2, g(0)=0g(0)=0, f(1)=6f(1)=6, g(1)=2g(1)=2, then show that there exists cc satisfying 0<c<10<c<1 and f(c)=2g(c)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)(0, c) from the parabola y=x2y=x^2 , where 0c10≤c≤1.

  • Q17 If ax2+bxc xR+\large ax^2+\frac{b}{x}≥c \ \forall x \in R^+ where a>0a>0 and b>0b>0, then show that 27ab24c327ab^2≥4c^3

  • Q18 Show that 0πxf(sinx) dx=π20πf(sinx) dx\large \int_0^{\pi}xf(\sin x) \ dx=\frac \pi 2 \int_0^{\pi}f(\sin x)\ dx.

  • Q19 Find the value of 132xsin(πx) dx\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.

Note by Shubhamkar Ayare
2 years, 8 months ago

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