Online IIT JEE-Tests - Mathematics Practice # 1 | IIT JEE 2011 | IIT Papers | IIT Solutions | IIT JEE Cutoff | IIT Key

Question

The function $f : N \rightarrow 1$ defined by $f(x)= \frac {1}{2}(x-1)$ , when x is odd $f(x)=\frac {1}{2}x$ , when x is even is

Possible Answers

Selected	Possible Answer
	one – one onto
	one – one but not onto
	onto but not one – one
	neighter one – one nor onto

Question

For all $\emptyset,(x-a) \cos \emptyset +(y-b) \sin \emptyset = 2$ is target to the circle having radius

Possible Answers

Selected	Possible Answer

	1
	$\sqrt{2}$
	none of these

Question

If the slope of a focal chord of a parabola is $\tan$ then the ratio of the length of focal chord to the latus rectum is:

Possible Answers

Selected	Possible Answer
	$\cos^2 \emptyset$
	cosec $^2 \emptyset$
	$\sec^2 \emptyset$
	$\cot^2 \emptyset$

Question

If $f(x)=\cos [\pi^2]x+\cos[-\pi^2]x$ where [x] denotes greatest integer function

Possible Answers

Selected	Possible Answer
	$f(\pi/4)=2$
	$f(\pi/2)=-1$
	$f(-\pi)=3$
	$f(\pi)=1$

Question

x is the product of three natural number. The probability that x is divisible by 10 is

Possible Answers

Selected	Possible Answer
	0.427
	0.451
	0.27
	none of these

Question

If n is a natural number and a multiple of 3 such that $(1+x+x^2)^n=\overset {2n}{\underset {r=0}{\sum}}a_rx^r$ , then $\quad a_{\circ}.^nC_{\circ}-a_1.^nC_1+a_2.^nC_2- \cdots+(-1)^{2n}a_n.^nC_n=$

Possible Answers

Selected	Possible Answer
	$^nC_{n/2}$
	$(-1)^{n/3} \quad ^nC_{n/3}$
	$^nC_{n/3}$
	0

Question

Period of x-[x] is

Possible Answers

Selected	Possible Answer
	1
	0
	2
	3

Question

Read the passage given below and answer the questions (8-10)

Consider a sequence of $terms : t_1, t_2, t_3\cdots \cdots t_n.$ The sequence represents a $G.P. \quad if \quad t_{n+1}=Kt_n$ for all $n \in N$ where

is a constant. Further the sequence represents an $A.P. \quad if \quad t _{n+1}=t_n+K$ for all $n\in N$ where

is a constant. If $t_n-2t_{n+1}+t_{n+2}=0$ , then again the sequence is an A.P.

Consider a sequence $a_1, a_2, a_3\cdots \cdots a_n, \quad where \quad a_n=\int_{0}^{\pi}\frac {\sin (2n-1)x}{\sin x}dx,$ then the sequence is

Possible Answers

Selected	Possible Answer
	A. P. and G.P. both
	G. P.
	A. P.
	neighter A. P. nor G. P.

Question

The sum to n terms of the sequence is

Possible Answers

Selected	Possible Answer
	$\frac {\pi}{2}(n-1+\pi)$
	$\pi(2^n-1)$
	$2^{n-1}\pi$
	$n\pi$

Question

If A is any matrix, then the product AA i.e. A^2 is defined only when A is matrix of order

Possible Answers

Selected	Possible Answer
	m = n
	$m \underline > n$
	m > n
	m < n

Question

Read the passage given below and answer the questions (11-13)

If $f:A\rightarrow B$ , then the map $f: B \rightarrow A$ is said to be inverse of

, if $y=f(x) \Leftrightarrow x=f^{-1}(y)$ . To obtain $f^{-1}(x)$ we put y=f(x)

and express

in terms of

and then we replace

in the expression of

in terms of y. Also the graph of y=f(x)

and $y=f^{-1}(y)$ are symmetric about the line y=x

and point of intersection of any two of $y=f^{-1}(x).y=f(x)$ and y=x

lies on the third.

(Q11)

Let a function $f:R \rightarrow (- \infty, 1)$ is given by $f(x)=1-2^{-x}$ then its inverse is the function

Possible Answers

Selected	Possible Answer
	$-\log_2(1-x)$
	$1-\log_2(-x)$
	$1+ \log_2(-x)$
	$\log_x(1-x)$

Question

The number of root of the equation $1+\log_2(1-x)=$

Possible Answers

Selected	Possible Answer
	1
	Many
	0
	2

Question

The number of root of the equation $x^2-x+1=\frac {1}{2}+\sqrt{x-\frac {3}{4}}, x\underline > \frac {3}{2}$

Possible Answers

Selected	Possible Answer
	$x=\frac {3}{4}.\frac {\sqrt {3}}{2}$

	it is an identity

Question

The number of different metrics which can be formed using 12 different real numbers is

Possible Answers

Selected	Possible Answer

Question

If f'(x) exists for all $x \in [a,b] \quad and \quad \frac {f(b)-f(c)}{b-c}=\frac {f(c)f(a)}{c-a}$ where $a<c<b, \quad then \quad f'(\alpha)=0$ for

Possible Answers

Selected	Possible Answer
	at least two $\alpha \in (a,b)$
	at least one $\alpha \in (a,b)$
	infinity many $\alpha \in (a,b)$
	no $\alpha \in (a,b)$

Question

If f(x) is continuous in $[1,3] \quad and \quad f(1)=f(3)$ , then f(c)=f(2+c) for

Possible Answers

Selected	Possible Answer
	odd number of value of c in [1, 2]
	exaclty 1 value of c in [1, 2]
	no value of c in [1, 2]
	even number of value of c in [1, 2]

Question

A tangent to the ellipse x^2+4y^2=4 meets the ellipse x^2+2y^2=6 at and . Then the angle between the tangent to the ellipse at and is

Possible Answers

Selected	Possible Answer
	none of these
	$30^{\circ}$
	$45^{\circ}$
	$60^{\circ}$

Question

If $a=\cos \alpha +\cos \beta +\cos \gamma, \quad b=\sin \alpha+\sin \beta+\sin \gamma$ and

$\cos (\alpha-\beta)+\cos (\beta-\gamma)+\cos (\gamma-\alpha)=-\frac {3}{2}$ then a^3+b^3=

Possible Answers

Selected	Possible Answer
	-1
	none of these
	0
	1

Question

If f(x)=0 be a cubic equation with real co-efficient and $\alpha, \beta$ be the distinct real roots of equation f'(x)=0 and $f(\alpha)f(\beta)<0$ , then the number of real roots of equation f(x)=0 is

Possible Answers

Selected	Possible Answer
	1
	3
	0
	2

Question

Statement – I $\overset {lt}{x\rightarrow 0}\frac {2^{\tan^2x-2^{\sin^2x}}}{\tan^2x.\sin^2x}=\log_e2$

Statement – II $\overset {lt}{x \rightarrow 0}\frac {a^x-1}{x}=\log_ea$

Possible Answers

Selected	Possible Answer
	Both Statement I and II are true and Statement II is the correct explaination of Statement I.
	Statement I is false but Statement II is true.
	Both Statement I and II are true but Statement II is not the correct explaination of Statement I.
	Statement I is true but Statement II is false.

Question

Statement – I If $In=\int_{}^{}\tan^n\alpha \beta \gamma \quad then \quad 5 (I_4+I_6)=\tan^5 \alpha$

Statement – II If $In =\int_{}^{}\tan^n \alpha \beta \gamma \quad then \quad \frac {\tan^{n-1}\alpha}{n}-I_{n-2}$ where $n \in N$

Possible Answers

Selected	Possible Answer
	Both Statement I and II are true but Statement II is not the correct explaination of Statement I.
	Statement I is false but Statement II is true.
	Both Statement I and II are true and Statement II is the correct explaination of Statement I.
	Statement I is true but Statement II is false.

Question

Statement – I $arg Z_1-arg Z_2=\pi$

$\Rightarrow origin.Z_1, Z_2$ are collinear and Z_1,Z_2 lie on the same side of the origin

Statement – II $|Z_1-Z_2|=|Z_1|+|Z_2| \Rightarrow origin Z_1, Z_2$ are collinear and lie on opposite side of the origin

Possible Answers

Selected	Possible Answer
	Both Statement I and II are true and Statement II is the correct explaination of Statement I.
	Statement I is true but Statement II is false.
	Statement I is false but Statement II is true.
	Both Statement I and II are true but Statement II is not the correct explaination of Statement I.

Question

The area of the triangle with vertices at the point (a,b+c), (b,c+a), (c,a+b) is

Possible Answers

Selected	Possible Answer
	ab+bc+ca
	0
	None of these
	a+b+c

Question

The equation of the straight line which is perpendicular to y=x and passes through (3,2) will be given by

Possible Answers

Selected	Possible Answer
	x-y=5
	x+y=1
	x+y=5
	x-y=1

Question

The area of the triangle with vertices at (-4,1), (1,2), (4,-3) is

Possible Answers

Selected	Possible Answer
	14
	16
	15
	None of these

Question

The triangle joining the points A(2,7), B(4,-1), C(-2,6) is

Possible Answers

Selected	Possible Answer
	Isosceles
	None of these
	Equilateral
	Right angled

Question

The equation of the straight line which passes through the point (1,-2) and cuts off equal intercepts from the axis will be

Possible Answers

Selected	Possible Answer
	x+y=1
	x+y+1=0
	x-y-2=0
	x-y=1

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Online IIT JEE-Tests - Mathematics Practice # 1

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