JEE Advance - Mathematics (1981)

1
Show that : $$\mathop {\lim }\limits_{n \to \infty } \left( {{1 \over {n + 1}} + {1 \over {n + 2}} + .... + {1 \over {6n}}} \right) = \log 6$$
उत्तर
(A)
The limit can be evaluated by recognizing it as a Riemann sum approximation of the integral of 1/x from 1 to 6.
2
For a biased die the probabilities for the different faces to turn up are given below :
This die tossed and you are told that either face $$1$$ or face $$2$$ has turned up. Then the probability that it is face $$1$$ is ...............
उत्तर
(A)
5/21
3
An anti-aircraft gun can take a maximum of four shots at an enemy plane moving away from it. The probabilities of hitting the plane at the first, second, third and fourth shot are $$0.4, 0.3, 0.2$$ and $$0.1$$ respectively. What is the probability that the gun hits the plane?
उत्तर
(C)
0.69
4
Let $$\overrightarrow A ,\overrightarrow B ,\overrightarrow C $$ be vectors of length $$3, 4, 5$$ respectively. Let $$\overrightarrow A $$ be perpendicular to $$\overrightarrow B + \overrightarrow C ,\overrightarrow B $$ to $$\overrightarrow C + \overrightarrow A $$ to $$\overrightarrow A + \overrightarrow B .$$ Then the length of vector $$\overrightarrow A + \overrightarrow B + \overrightarrow C $$ is ..........
उत्तर
(B)
5√2
5
Let $$\overrightarrow A ,\overrightarrow B $$ and $${\overrightarrow C }$$ be unit vectors suppose that $$\overrightarrow A .\overrightarrow B = \overrightarrow A .\overrightarrow C = 0,$$ and thatthe angle between $${\overrightarrow B }$$ and $${\overrightarrow C }$$ is $$\pi /6.$$ Then $$\overrightarrow A = \pm 2\left( {\overrightarrow B \times \overrightarrow C } \right).$$
उत्तर
(B)
FALSE
6
The scalar $$\overrightarrow A .\left( {\overrightarrow B + \overrightarrow C } \right) \times \left( {\overrightarrow A + \overrightarrow B + \overrightarrow C } \right)$$ equals :
उत्तर
(A)
$$0$$
7
Find the area bounded by the curve $${x^2} = 4y$$ and the straight
उत्तर
(C)
9/8 sq. units
8
Suppose that the normals drawn at three different points on the parabola $${y^2} = 4x$$ pass through the point $$(h, k)$$. Show that $$h>2$$.
उत्तर
(D)
The statement is false for h < 2.
9
The general solution of the trigonometric equation sin x+cos x=1 is given by:
उत्तर
(C)
$$x = n\pi + {\left( { - 1} \right)^n}\,\,\,\,\,\,\,{\pi \over 4} - {\pi \over 4}$$ ; $$n = 0,\, \pm 1,\, \pm 2..$$
10
For complex number $${z_1} = {x_1} + i{y_1}$$ and $${z_2} = {x_2} + i{y_2},$$ we write $${z_1} \cap {z_2},\,\,if\,\,{x_1} \le {x_2}\,\,and\,\,{y_1} \le {y_2}.$$
Then for all complex numbers $$z\,\,with\,\,1 \cap z,$$ we have $${{1 - z} \over {1 + z}} \cap 0.$$
उत्तर
(B)
FALSE
11
The complex numbers $$z = x + iy$$ which satisfy the equation $$\,\left| {{{z - 5i} \over {z + 5i}}} \right| = 1$$ lie on
उत्तर
(A)
the x-axis
12
Let the complex number $${{z_1}}$$, $${{z_2}}$$ and $${{z_3}}$$ be the vertices of an equilateral triangle. Let $${{z_0}}$$ be the circumcentre of the triangle. Then prove that $$z_1^2 + z_2^2 + z_3^2 = 3z_0^2$$.
उत्तर
(A)
The given equation holds true if and only if the triangle is equilateral and $$z_0$$ is its circumcenter.
13
For every integer n > 1, the inequality $${(n!)^{1/n}} < {{n + 1} \over 2}$$ holds.
उत्तर
(B)
FALSE
14
Five balls of different colours are to be placed in there boxes of different size. Each box can hold all five. In how many different ways can be place the balls so that no box remains emply?
उत्तर
(B)
300
15
The area enclosed within the curve $$\left| x \right| + \left| y \right| = 1$$ is .................
उत्तर
(B)
2 sq. units
16
Let A be the centre of the circle $${x^2}\, + \,{y^2}\, - \,2x\,\, - 4y\, - 20 = 0\,$$. Suppose that the tangents at the points B (1, 7) and D (4. - 2) on the circle meet at the point C. Find the area of the quadrilateral ABCD.
उत्तर
(B)
72 sq units
17
Find the equations of the circle passing through (- 4, 3) and touching the lines x + y = 2 and x - y = 2.
उत्तर
A
B
18
The equation $${{{x^2}} \over {1 - r}} - {{{y^2}} \over {1 + r}} = 1,\,\,\,\,r > 1$$ represents
उत्तर
(D)
none of these
19
Each of the four inequalties given below defines a region in the $$xy$$ plane. One of these four regions does not have the following property. For any two points $$\left( {{x_1},{y_1}} \right)$$ and $$\left( {{x_2},{y_2}} \right)$$ in the region, the point $$\left( {{{{x_1} + {x_2}} \over 2},{{{y_1} + {y_2}} \over 2}} \right)$$ is also in the region. The inequality defining this region is
उत्तर
(C)
$${x^2} - {y^2} \le 1$$
20
Suppose $${\sin ^3}\,x\sin 3x = \sum\limits_{m = 0}^n {{C_m}\cos \,mx} $$ is an identity in x, where C₀, C₁ ,....C_n are constants, and $${C_n} \ne 0$$ , then the value of n is _____.
उत्तर
(C)
6
21
Let $$y = {e^{x\,\sin \,{x^3}}} + {\left( {\tan x} \right)^x}$$. Find $${{dy} \over {dx}}$$
उत्तर
(C)
${e^{x\,\sin {x^3}}}\left[ {\sin {x^3} + 3{x^3}\cos {x^3}} \right] + {\left( {\tan x} \right)^x}\left[ {{{x\sec^2 x} \over {\tan x}} + \log \,\tan x} \right]$
22
Let the angles $$A, B, C$$ of a triangle $$ABC$$ be in A.P. and let $$b:c = \sqrt 3 :\sqrt 2 $$. Find the angle $$A$$.
उत्तर
(D)
75°
23
Let $$a, b, c$$ be positive real numbers Let
$$\theta = {\tan ^{ - 1}}\sqrt {{{a\left( {a + b + c} \right)} \over {bc}}} + {\tan ^{ - 1}}\sqrt {{{b\left( {a + b + c} \right)} \over {ca}}} $$ $$ + {\,\,\tan ^{ - 1}}\sqrt {{{c\left( {a + b + c} \right)} \over {ab}}} $$
Then $$\tan \theta = $$ ____________
उत्तर
(A)
0
24
Find the value of : $$\cos \left( {2{{\cos }^{ - 1}}x + {{\sin }^{ - 1}}x} \right)$$ at $$x = {1 \over 5}$$, where
$$0 \le {\cos ^{ - 1}}x \le \pi $$ and $$ - \pi /2 \le {\sin ^{ - 1}}x \le \pi /2$$.
उत्तर
(A)
$$\frac{-2\sqrt{6}}{5}$$
25
For all $$x$$ in $$\left[ {0,1} \right]$$, let the second derivative $$f''(x)$$ of a function $$f(x)$$ exist and satisfy $$\left| {f''\left( x \right)} \right| < 1.$$ If $$f(0)=f(1)$$, then show that $$\left| {f\left( x \right)} \right| < 1$$ for all $$x$$ in $$\left[ {0,1} \right]$$.
उत्तर
(B)
The problem statement is true, and the result follows from Taylor's theorem with remainder term.
26
Let $$x$$ and $$y$$ be two real variables such that $$x>0$$ and $$xy=1$$. Find the minimum value of $$x+y$$.
उत्तर
(C)
2
27
Use the function $$f\left( x \right) = {x^{1/x}},x > 0$$. to determine the bigger of the two numbers $${e^\pi }$$ and $${\pi ^e}$$
उत्तर
(A)
$$e^\pi$$ is bigger
28
Evaluate $$\int {\left( {{e^{\log x}} + \sin x} \right)\cos x\,\,dx.} $$
उत्तर
(C)
x \sin x + \cos x - \frac{1}{4}\cos 2x + C
29
The value of the definite integral $$\int\limits_0^1 {\left( {1 + {e^{ - {x^2}}}} \right)} \,\,dx$$
उत्तर
(D)
none of these
30
Let $$a, b, c$$ be non-zero real numbers such that
$$\int\limits_0^1 {\left( {1 + {{\cos }^8}x} \right)\left( {a{x^2} + bx + c} \right)dx = \int\limits_0^2 {\left( {1 + {{\cos }^8}x} \right)\left( {a{x^2} + bx + c} \right)dx.} } $$
Then the quadratic equation $$a{x^2} + bx + c = 0$$ has
उत्तर
(B)
at least one root in $$(0, 2)$$