If n objects are distributed at random among n persons, the probability that at least one of them will not get anything is

\[1-\frac{(n-1)!}{{{n}^{n-1}}}\]

If \[\vec{a},\text{ }\vec{b},\text{ }\vec{c},\text{ }\vec{d}\] are the position vectors of points A, B, C and D respectively such that \[(\vec{a}-\vec{d}).(\vec{b}-\vec{c})=(\vec{b}-\vec{d}).(\vec{c}-\vec{a})=0\] then D is the

Centroid of \[\Delta \text{ }ABC\]

Circumcentre of \[\Delta \,ABC\]

Orthocentre of \[\Delta \,ABC\]

If \[A=\left( \begin{matrix} i & 1 \\ 0 & i \\ \end{matrix} \right)\], then \[{{A}^{4}}\]equals [AMU 1999]

\[\left( \begin{matrix} 1 & 4 \\ 0 & 1 \\ \end{matrix} \right)\]

\[\left( \begin{matrix} 1 & -4i \\ 0 & 1 \\ \end{matrix} \right)\]

\[\left( \begin{matrix} -1 & -4i \\ 0 & -1 \\ \end{matrix} \right)\]

\[\left( \begin{matrix} -i & 4 \\ 0 & i \\ \end{matrix} \right)\]

\[\left( \begin{matrix} 1 & 4 \\ 0 & 1 \\ \end{matrix} \right)\]

The equation \[{{\sin }^{4}}x+{{\cos }^{4}}x+\sin 2x+\alpha =0\]is solvable for

\[-\frac{1}{2}\le \alpha \le \frac{1}{2}\]

\[-\frac{3}{2}\le \alpha \le \frac{1}{2}\]

If \[A=\left[ \begin{matrix} 1 & 0 \\ 1 & 1 \\ \end{matrix} \right]\] and \[I=\left[ \begin{matrix} 1 & 0 \\ 0 & 1 \\ \end{matrix} \right]\], then which one of the following holds for all \[n\ge 1\], (by the principal of mathematical induction) [AIEEE 2005]

\[{{A}^{n}}={{2}^{n-1}}A-(n-1)I\]

\[{{A}^{n}}={{2}^{n-1}}A+(n-1)I\]

\[{{A}^{n}}={{2}^{n-1}}A-(n-1)I\]

If \[\sin {{18}^{0}}=\frac{\sqrt{5}-1}{4},\] then what is the value of \[\sin 18{}^\circ \]?

\[\frac{\sqrt{3+\sqrt{5}}+\sqrt{5+\sqrt{5}}}{4}\]

\[\frac{\sqrt{3+\sqrt{5}}+\sqrt{5-\sqrt{5}}}{4}\]

\[\frac{\sqrt{3+\sqrt{5}}+\sqrt{5+\sqrt{5}}}{4}\]

\[\frac{\sqrt{3-\sqrt{5}}+\sqrt{5-\sqrt{5}}}{4}\]

\[\frac{\sqrt{3+\sqrt{5}}-\sqrt{5-\sqrt{5}}}{4}\]

If \[\mathbf{u}=\mathbf{i}\times (\mathbf{a}\times \mathbf{i})+\mathbf{j}\times (\mathbf{a}\times \mathbf{j})+\mathbf{k}\times (\mathbf{a}\times \mathbf{k}),\]then [RPET 1989, 97; MNR 1986, 93; MP PET 1987, 98, 99, 2004; UPSEAT 2000, 02; Kerala (Engg.) 2002]

\[\mathbf{u}=\mathbf{i}+\mathbf{j}+\mathbf{k}\]

For the curve \[b{{y}^{2}}={{(x+a)}^{3}}\]the square of subtangent is proportional to [RPET 1999]

\[{{\text{(Subnormal)}}^{\text{3/2}}}\]

\[{{\text{(Subnormal)}}^{1/2}}\]

\[{{\text{(Subnormal)}}^{\text{3/2}}}\]

If \[f:R\to R\,\,\And \,\,g:R\to R\] be two given functions, then 2 min \[\{f(x)-g(x),0\}\] equals

\[f(x)+g(x)-\left| g(x)-f(x) \right|\]

\[f(x)+g(x)+\left| g(x)-f(x) \right|\]

\[f(x)-g(x)+\left| g(x)-f(x) \right|\]

\[f(x)-g(x)-\left| g(x)-f(x) \right|\]

Students of two schools appeared for a common test carrying 100 marks. The arithmetic means of their marks of school I and II are 82 and 86 respectively. If the number of students of school II is 1.5 times the number of students of school I, what is the arithmetic mean of the marks of all the students of both are schools?

This cannot be calculated with the given data

A unit vector perpendicular to the vector \[4\mathbf{i}-\mathbf{j}+3\mathbf{k}\] and \[-2\mathbf{i}+\mathbf{j}-2\mathbf{k}\] is [MNR 1995]

\[\frac{1}{3}\,(2\mathbf{i}+\mathbf{j}+2\mathbf{k})\]

\[\frac{1}{3}\,(\mathbf{i}-2\mathbf{j}+2\mathbf{k})\]

\[\frac{1}{3}\,(-\mathbf{i}+2\mathbf{j}+2\mathbf{k})\]

\[\frac{1}{3}\,(2\mathbf{i}+\mathbf{j}+2\mathbf{k})\]

\[\frac{1}{3}\,(2\mathbf{i}-2\mathbf{j}+2\mathbf{k})\]

Let \[f(x)\] be define on \[[-2,2]\] and is given by \[f(x)=\left\{ \begin{matrix} -1,\,-2\le x\le 0 \\ x-1,\,0\le x\le 2 \\ \end{matrix} \right.\], then \[f(\left| x \right|)\] is defined as

\[f(\left| x \right|)=\left\{ \begin{matrix} 1-2\le x\le 0 \\ 1-x,0

\[f(\left| x \right|)=x-1\forall x\in R\]

\[f(\left| x \right|)=\left\{ \begin{matrix} -x-1,-2\le x\le 0 \\ x-1,0

For any two vectors a and b, \[{{(\mathbf{a}\times \mathbf{b})}^{2}}\] is equal to [Roorkee 1975, 79, 81, 85]

\[{{a}^{2}}{{b}^{2}}-{{(\mathbf{a}\,.\,\mathbf{b})}^{2}}\]

The range of value of \[\alpha \] such that \[(0,\alpha )\] lies on or inside the triangle formed by the lines \[y+3x+2=0,\] \[3y-2x-5=0,\] \[4y+x-14=0\]is

\[\frac{1}{2}\le \alpha \le 1\]

\[\frac{5}{3}\le \alpha \le \frac{7}{2}\]

The solution of the differential equation \[3{{e}^{x}}\tan ydx+(1-{{e}^{x}}){{\sec }^{2}}ydy=0\] is [MP PET 1993; AISSE 1985]

\[{{(1-{{e}^{x}})}^{3}}\tan y=c\]

\[\tan y=c{{(1-{{e}^{x}})}^{3}}\]

\[{{(1-{{e}^{x}})}^{3}}\tan y=c\]

The equation to the straight line passing through the point \[(a{{\cos }^{3}}\theta ,\ a{{\sin }^{3}}\theta )\] and perpendicular to the line \[x\sec \theta +y\,\text{cosec}\,\theta =a,\] is [AMU 1975]

\[x\cos \theta +y\sin \theta =a\cos \ 2\theta \]

\[x\cos \theta -y\sin \theta =a\cos \ 2\theta \]

\[x\cos \theta +y\sin \theta =a\cos \ 2\theta \]

\[x\sin \theta +y\cos \theta =a\cos \ 2\theta \]

Let S be any set and P (S) be its power set, We define a relation R on P(S) by ARB to mean \[A\subseteq B;\forall A,B\in P(S).\] Then R is

Both equivalence and partial order relation

Not an equivalence but partial order relation

Both equivalence and partial order relation

For any vector \[\overset{\to }{\mathop{p}}\,\], the value of\[\frac{3}{2}\left\{ |\overset{\to }{\mathop{p}}\,\times \hat{i}{{|}^{2}}+|\overset{\to }{\mathop{p}}\,\times \hat{j}{{|}^{2}}+|\overset{\to }{\mathop{p}}\,\times \hat{k}{{|}^{2}} \right\}\] is

\[4\overset{\to 2}{\mathop{p}}\,\]

\[\overset{\to 2}{\mathop{p}}\,\]

\[2\overset{\to 2}{\mathop{p}}\,\]

\[3\overset{\to 2}{\mathop{p}}\,\]

\[4\overset{\to 2}{\mathop{p}}\,\]

The equation of director circle of the circle \[{{x}^{2}}+{{y}^{2}}={{a}^{2}},\]is [BIT Ranchi 1990]

\[{{x}^{2}}+{{y}^{2}}=4{{a}^{2}}\]

\[{{x}^{2}}+{{y}^{2}}=\sqrt{2}{{a}^{2}}\]

\[{{x}^{2}}+{{y}^{2}}-2{{a}^{2}}=0\]

Under what condition are the two lines\[y=\frac{m}{\ell }x+\alpha ,z=\frac{n}{\ell }x+\beta ;\] and \[y=\frac{m'}{\ell '}x+\alpha ',z=\frac{n'}{\ell '}x+\beta '\] Orthogonal?

\[\alpha \alpha '+\beta \beta '+1=0\]

\[(\alpha +\alpha ')+(\beta +\beta ')=0\]

8666 + Mcqs in Mathematics in Joint Entrance Exam - Main (JEE Main) Page 2 McqOptions

51.	If \[A=\{1,2,3,4,5\}\], then the number of proper subsets of A is
A.	31
B.	38
C.	48
D.	54
Answer» B. 38

Discussion

52.	If n objects are distributed at random among n persons, the probability that at least one of them will not get anything is
A.	\[1-\frac{(n-1)!}{{{n}^{n-1}}}\]
B.	\[\frac{(n-1)!}{{{n}^{n}}}\]
C.	\[1-\frac{(n-1)!}{{{n}^{2}}}\]
D.	None of these
Answer» B. \[\frac{(n-1)!}{{{n}^{n}}}\]

Discussion

53.	A vertex of square is (3, 4) and diagonal \[x+2y=1,\] then the second diagonal which passes through given vertex will be
A.	\[2x-y+2=0\]
B.	\[x+2y=11\]
C.	\[2x-y=2\]
D.	None of these
Answer» D. None of these

Discussion

54.	If the coordinates of the points A, B, C, D, be \[(a,\ b),\] \[({a}',\ {b}'),\] \[(-a,\ b)\] and \[({a}',\ -{b}')\] respectively, then the equation of the line bisecting the line segments AB and CD is
A.	\[2{a}'y-2bx=ab-{a}'{b}'\]
B.	\[2ay-2{b}'\ x=ab-{a}'{b}'\]
C.	\[2ay-2{b}'x={a}'b-a{b}'\]
D.	None of these
Answer» C. \[2ay-2{b}'x={a}'b-a{b}'\]

Discussion

55.	The line \[x+y=2\]is tangent to the curve \[{{x}^{2}}=3-2y\] at its point [MP PET 1998]
A.	(1, 1)
B.	(?1, 1)
C.	(\[\sqrt{3}\], 0)
D.	(3, ?3)
Answer» B. (?1, 1)

Discussion

56.	If \[{{t}_{n}}\] denotes the nth term of a G.P. whose common ratio is r, then the progression whose nth term is \[\frac{1}{t_{n}^{2}+t_{n+1}^{2}}\] is
A.	A.P.
B.	G.P.
C.	H.P.
D.	None of these
Answer» C. H.P.

Discussion

57.	If \[\vec{a},\text{ }\vec{b},\text{ }\vec{c},\text{ }\vec{d}\] are the position vectors of points A, B, C and D respectively such that \[(\vec{a}-\vec{d}).(\vec{b}-\vec{c})=(\vec{b}-\vec{d}).(\vec{c}-\vec{a})=0\] then D is the
A.	Centroid of \[\Delta \text{ }ABC\]
B.	Circumcentre of \[\Delta \,ABC\]
C.	Orthocentre of \[\Delta \,ABC\]
D.	None of these
Answer» D. None of these

Discussion

58.	If A is any set, then
A.	\[A\cup {A}'=\varphi \]
B.	\[A\cup {A}'=U\]
C.	\[A\cap {A}'=U\]
D.	None of these
Answer» C. \[A\cap {A}'=U\]

Discussion

59.	In how many ways vertices of a square can be coloured with 4 distinct colour if rotations are considered to be equivalent, but reflections are distinct?
A.	65
B.	70
C.	71
D.	None of these
Answer» C. 71

Discussion

60.	If \[A=\left( \begin{matrix} i & 1 \\ 0 & i \\ \end{matrix} \right)\], then \[{{A}^{4}}\]equals [AMU 1999]
A.	\[\left( \begin{matrix} 1 & -4i \\ 0 & 1 \\ \end{matrix} \right)\]
B.	\[\left( \begin{matrix} -1 & -4i \\ 0 & -1 \\ \end{matrix} \right)\]
C.	\[\left( \begin{matrix} -i & 4 \\ 0 & i \\ \end{matrix} \right)\]
D.	\[\left( \begin{matrix} 1 & 4 \\ 0 & 1 \\ \end{matrix} \right)\]
Answer» D. \[\left( \begin{matrix} 1 & 4 \\ 0 & 1 \\ \end{matrix} \right)\]

Discussion

61.	The equation \[{{\sin }^{4}}x+{{\cos }^{4}}x+\sin 2x+\alpha =0\]is solvable for
A.	\[-\frac{1}{2}\le \alpha \le \frac{1}{2}\]
B.	\[-3\le \alpha \le 1\]
C.	\[-\frac{3}{2}\le \alpha \le \frac{1}{2}\]
D.	\[-1\le \alpha \le 1\]
Answer» D. \[-1\le \alpha \le 1\]

Discussion

62.	If \[A=\left[ \begin{matrix} 1 & 0 \\ 1 & 1 \\ \end{matrix} \right]\] and \[I=\left[ \begin{matrix} 1 & 0 \\ 0 & 1 \\ \end{matrix} \right]\], then which one of the following holds for all \[n\ge 1\], (by the principal of mathematical induction) [AIEEE 2005]
A.	\[{{A}^{n}}=nA+(n-1)I\]
B.	\[{{A}^{n}}={{2}^{n-1}}A+(n-1)I\]
C.	\[{{A}^{n}}=nA-(n-1)I\]
D.	\[{{A}^{n}}={{2}^{n-1}}A-(n-1)I\]
Answer» D. \[{{A}^{n}}={{2}^{n-1}}A-(n-1)I\]

Discussion

63.	If \[f:R\to R\] is given by \[f(x)=\frac{{{x}^{2}}-4}{{{x}^{2}}+1},\] then the function f is
A.	many-one onto
B.	many-one into
C.	one-one into
D.	one-one onto
Answer» B. many-one into

Discussion

64.	Three cards are drawn at random from a pack of 52 cards. What is the chance of drawing three aces
A.	\[\frac{3}{5525}\]
B.	\[\frac{2}{5525}\]
C.	\[\frac{1}{5525}\]
D.	None of these
Answer» D. None of these

Discussion

65.	If \[\sin {{18}^{0}}=\frac{\sqrt{5}-1}{4},\] then what is the value of \[\sin 18{}^\circ \]?
A.	\[\frac{\sqrt{3+\sqrt{5}}+\sqrt{5-\sqrt{5}}}{4}\]
B.	\[\frac{\sqrt{3+\sqrt{5}}+\sqrt{5+\sqrt{5}}}{4}\]
C.	\[\frac{\sqrt{3-\sqrt{5}}+\sqrt{5-\sqrt{5}}}{4}\]
D.	\[\frac{\sqrt{3+\sqrt{5}}-\sqrt{5-\sqrt{5}}}{4}\]
Answer» B. \[\frac{\sqrt{3+\sqrt{5}}+\sqrt{5+\sqrt{5}}}{4}\]

Discussion

66.	The value of c, for which the line \[y=2x+c\]is a tangent to the circle \[{{x}^{2}}+{{y}^{2}}=16\], is [MP PET 2004; Karnataka CET 2005]
A.	\[-16\sqrt{5}\]
B.	20
C.	\[4\sqrt{5}\]
D.	\[16\sqrt{5}\]
Answer» D. \[16\sqrt{5}\]

Discussion

67.	\[{{\cos }^{2}}{{76}^{o}}+{{\cos }^{2}}{{16}^{o}}-\cos {{76}^{o}}\cos {{16}^{o}}=\] [EAMCET 2002]
A.	-0.25
B.	44228
C.	0
D.	3/4
Answer» E.

Discussion

68.	If the line \[3x-4y=\lambda \]touches the circle \[{{x}^{2}}+{{y}^{2}}-4x-8y-5=0\], then \[\lambda \]is equal to [Roorkee 1972; Kurukshetra CEE 1996]
A.	- 35, -15
B.	- 35, 15
C.	35, 15
D.	35, -15
Answer» C. 35, 15

Discussion

69.	The point at which the tangent to the curve \[y=2{{x}^{2}}-x+1\] is parallel to \[y\text{ }=\text{ 3}x+\text{9 }\]will be [Karnataka CET 2001]
A.	(2, 1)
B.	(1, 2)
C.	(3, 9)
D.	(?2, 1)
Answer» C. (3, 9)

Discussion

70.	If \[\mathbf{u}=\mathbf{i}\times (\mathbf{a}\times \mathbf{i})+\mathbf{j}\times (\mathbf{a}\times \mathbf{j})+\mathbf{k}\times (\mathbf{a}\times \mathbf{k}),\]then [RPET 1989, 97; MNR 1986, 93; MP PET 1987, 98, 99, 2004; UPSEAT 2000, 02; Kerala (Engg.) 2002]
A.	\[\mathbf{u}=0\]
B.	\[\mathbf{u}=\mathbf{i}+\mathbf{j}+\mathbf{k}\]
C.	\[\mathbf{u}=2\mathbf{a}\]
D.	\[\mathbf{u}=\mathbf{a}\]
Answer» D. \[\mathbf{u}=\mathbf{a}\]

Discussion

71.	Ravish writes letters to his five friends and addresses the corresponding envelopes. In how many ways can the letters be placed in the envelopes so that at least two of them are in the wrong envelopes?
A.	109
B.	118
C.	119
D.	None of these
Answer» D. None of these

Discussion

72.	A cricket team has 15 members, of whom only 5 can bowl. If the names of the 15 members are put into a hat and 11 drawn at random, then the chance of obtaining an eleven containing at least 3 bowlers is
A.	\[\frac{7}{13}\]
B.	\[\frac{11}{15}\]
C.	\[\frac{12}{13}\]
D.	None of these
Answer» D. None of these

Discussion

73.	The length of normal to the curve \[x=a\,(\theta +\sin \theta ),\] \[y=a(1-\cos \theta )\] at the point \[\theta =\pi /2\]is [RPET 1999]
A.	\[2a\]
B.	\[a/2\]
C.	\[\sqrt{2}\,a\]
D.	\[a/\sqrt{2}\]
Answer» D. \[a/\sqrt{2}\]

Discussion

74.	If \[\theta \] is the angle between the vectors a and b, then \[\frac{\|a\times b\|}{\|a\,.\,b\|}\] equal to [Karnataka CET 1999]
A.	\[\tan \theta \]
B.	\[-\tan \theta \]
C.	\[\cot \theta \]
D.	\[-\cot \theta \]
Answer» B. \[-\tan \theta \]

Discussion

75.	For the curve \[b{{y}^{2}}={{(x+a)}^{3}}\]the square of subtangent is proportional to [RPET 1999]
A.	\[{{\text{(Subnormal)}}^{1/2}}\]
B.	Subnormal
C.	\[{{\text{(Subnormal)}}^{\text{3/2}}}\]
D.	None of these
Answer» C. \[{{\text{(Subnormal)}}^{\text{3/2}}}\]

Discussion

76.	Two friends A and B have equal number of daughters. There are three cinema tickets which are to be distributed among the daughters of A and B. The probability that all the tickets go to daughters of A is 1/20. The number of daughters each of them have is
A.	4
B.	5
C.	6
D.	3
Answer» E.

Discussion

77.	Consider the following statements: For non-empty sets. A, B and C 1. \[A-(B-C)=(A-B)\cup C\] 2. \[A-(B\cup C)=(A-B)-C\] Which of the statements given above is/are correct?
A.	1 only
B.	2 only
C.	Both 1 and 2
D.	neither 1 nor 2
Answer» C. Both 1 and 2

Discussion

78.	If \[f:R\to R\,\,\And \,\,g:R\to R\] be two given functions, then 2 min \[\{f(x)-g(x),0\}\] equals
A.	\[f(x)+g(x)-\left\| g(x)-f(x) \right\|\]
B.	\[f(x)+g(x)+\left\| g(x)-f(x) \right\|\]
C.	\[f(x)-g(x)+\left\| g(x)-f(x) \right\|\]
D.	\[f(x)-g(x)-\left\| g(x)-f(x) \right\|\]
Answer» E.

Discussion

79.	Students of two schools appeared for a common test carrying 100 marks. The arithmetic means of their marks of school I and II are 82 and 86 respectively. If the number of students of school II is 1.5 times the number of students of school I, what is the arithmetic mean of the marks of all the students of both are schools?
A.	84
B.	84.2
C.	84.4
D.	This cannot be calculated with the given data
Answer» D. This cannot be calculated with the given data

Discussion

80.	A unit vector perpendicular to the vector \[4\mathbf{i}-\mathbf{j}+3\mathbf{k}\] and \[-2\mathbf{i}+\mathbf{j}-2\mathbf{k}\] is [MNR 1995]
A.	\[\frac{1}{3}\,(\mathbf{i}-2\mathbf{j}+2\mathbf{k})\]
B.	\[\frac{1}{3}\,(-\mathbf{i}+2\mathbf{j}+2\mathbf{k})\]
C.	\[\frac{1}{3}\,(2\mathbf{i}+\mathbf{j}+2\mathbf{k})\]
D.	\[\frac{1}{3}\,(2\mathbf{i}-2\mathbf{j}+2\mathbf{k})\]
Answer» C. \[\frac{1}{3}\,(2\mathbf{i}+\mathbf{j}+2\mathbf{k})\]

Discussion

81.	Let \[f(x)\] be define on \[[-2,2]\] and is given by \[f(x)=\left\{ \begin{matrix} -1,\,-2\le x\le 0 \\ x-1,\,0\le x\le 2 \\ \end{matrix} \right.\], then \[f(\left\| x \right\|)\] is defined as
A.	\[f(\left\| x \right\|)=\left\{ \begin{matrix} 1-2\le x\le 0 \\ 1-x,0<x\le 2 \\ \end{matrix} \right.\]
B.	\[f(\left\| x \right\|)=x-1\forall x\in R\]
C.	\[f(\left\| x \right\|)=\left\{ \begin{matrix} -x-1,-2\le x\le 0 \\ x-1,0<x\le 2 \\ \end{matrix} \right.\]
D.	None of these
Answer» D. None of these

Discussion

82.	Using mathematical induction, the numbers \[{{a}_{n}}'s\]are defined by \[{{a}_{0}}=1,\,\,{{a}_{n+1}}=3{{n}^{2}}+n+{{a}_{n'}}\] \[(n\ge 0).\]Then, \[{{a}_{n}}\] is equal to
A.	\[{{n}^{3}}+{{n}^{2}}+1\]
B.	\[{{n}^{3}}-{{n}^{2}}+1\]
C.	\[{{n}^{3}}-{{n}^{2}}\]
D.	\[{{n}^{3}}+{{n}^{2}}\]
Answer» C. \[{{n}^{3}}-{{n}^{2}}\]

Discussion

83.	For any two vectors a and b, \[{{(\mathbf{a}\times \mathbf{b})}^{2}}\] is equal to [Roorkee 1975, 79, 81, 85]
A.	\[{{a}^{2}}-{{b}^{2}}\]
B.	\[{{a}^{2}}+{{b}^{2}}\]
C.	\[{{a}^{2}}{{b}^{2}}-{{(\mathbf{a}\,.\,\mathbf{b})}^{2}}\]
D.	None of these
Answer» D. None of these

Discussion

84.	If \[{{x}^{2}}+2x+2xy+my-3\] has two rational factors, then the value of m will be [RPET 1990]
A.	\[-6,-2\]
B.	\[-6,2\]
C.	\[6,-2\]
D.	6, 2
Answer» D. 6, 2

Discussion

85.	The range of value of \[\alpha \] such that \[(0,\alpha )\] lies on or inside the triangle formed by the lines \[y+3x+2=0,\] \[3y-2x-5=0,\] \[4y+x-14=0\]is
A.	\[5<\alpha \le 7\]
B.	\[\frac{1}{2}\le \alpha \le 1\]
C.	\[\frac{5}{3}\le \alpha \le \frac{7}{2}\]
D.	None of these
Answer» D. None of these

Discussion

86.	The solution of the differential equation \[3{{e}^{x}}\tan ydx+(1-{{e}^{x}}){{\sec }^{2}}ydy=0\] is [MP PET 1993; AISSE 1985]
A.	\[\tan y=c{{(1-{{e}^{x}})}^{3}}\]
B.	\[{{(1-{{e}^{x}})}^{3}}\tan y=c\]
C.	\[\tan y=c(1-{{e}^{x}})\]
D.	\[(1-{{e}^{x}})\tan y=c\]
Answer» B. \[{{(1-{{e}^{x}})}^{3}}\tan y=c\]

Discussion

87.	\[AB=0\], if and only if [MNR 1981; Karnataka CET 1993]
A.	\[A\ne O,B=O\]
B.	\[A=O,B\ne O\]
C.	\[A=O\]or \[B=O\]
D.	None of these
Answer» E.

Discussion

88.	If \[A=\left[ \begin{matrix} i & 0 \\ 0 & i \\ \end{matrix} \right]\] and \[B=\left[ \begin{matrix} 0 & -i \\ -i & 0 \\ \end{matrix} \right]\], then \[(A+B)(A-B)\] is equal to [RPET 1994]
A.	\[{{A}^{2}}-{{B}^{2}}\]
B.	\[{{A}^{2}}+{{B}^{2}}\]
C.	\[{{A}^{2}}-{{B}^{2}}+BA+AB\]
D.	None of these
Answer» B. \[{{A}^{2}}+{{B}^{2}}\]

Discussion

89.	Three vertices of a parallelogram taken in order are \[(-1,\,-6)\], \[(2,\,-5)\] and \[(7,\,2)\]. The fourth vertex is [Kerala (Engg.) 2002]
A.	(1, 4)
B.	(4, 1)
C.	(1, 1)
D.	(4, 4)
Answer» C. (1, 1)

Discussion

90.	The equation to the straight line passing through the point \[(a{{\cos }^{3}}\theta ,\ a{{\sin }^{3}}\theta )\] and perpendicular to the line \[x\sec \theta +y\,\text{cosec}\,\theta =a,\] is [AMU 1975]
A.	\[x\cos \theta -y\sin \theta =a\cos \ 2\theta \]
B.	\[x\cos \theta +y\sin \theta =a\cos \ 2\theta \]
C.	\[x\sin \theta +y\cos \theta =a\cos \ 2\theta \]
D.	None of these
Answer» B. \[x\cos \theta +y\sin \theta =a\cos \ 2\theta \]

Discussion

91.	A three digit number is formed by using numbers 1, 2, 3 and 4. The probability that the number is divisible by 3, is
A.	\[\frac{2}{3}\]
B.	\[\frac{2}{7}\]
C.	\[\frac{1}{2}\]
D.	\[\frac{3}{4}\]
Answer» D. \[\frac{3}{4}\]

Discussion

92.	The equation of the tangent to curve \[y=b{{e}^{-x/a}}\] at the point where it crosses y-axis is [Karnataka CET 2002]
A.	\[ax+by=1\]
B.	\[ax-by=1\]
C.	\[\frac{x}{a}-\frac{y}{b}=1\]
D.	\[\frac{x}{a}+\frac{y}{b}=1\]
Answer» E.

Discussion

93.	If four vertices of a regular octagon are chosen at random, then the probability that the quadrilateral formed by them is a rectangle is [AMU 1999]
A.	\[\frac{1}{8}\]
B.	\[\frac{2}{21}\]
C.	\[\frac{1}{32}\]
D.	\[\frac{1}{35}\]
Answer» E.

Discussion

94.	Let S be any set and P (S) be its power set, We define a relation R on P(S) by ARB to mean \[A\subseteq B;\forall A,B\in P(S).\] Then R is
A.	Equivalence relation
B.	Not an equivalence but partial order relation
C.	Both equivalence and partial order relation
D.	None of these
Answer» C. Both equivalence and partial order relation

Discussion

95.	ABCDEF is a regular hexagon where centre O is the origin. If the position vectors of A and B are \[\hat{i}-\hat{j}+2\hat{k}\] and \[2\hat{i}+\hat{j}-\hat{k}\] respectively then \[\overrightarrow{BC}\] is equal to
A.	\[\hat{i}+\hat{j}-2\hat{k}\]
B.	\[-\hat{i}+\hat{j}-2\hat{k}\]
C.	\[3\hat{i}+3\hat{j}-4\hat{k}\]
D.	None of these
Answer» C. \[3\hat{i}+3\hat{j}-4\hat{k}\]

Discussion

96.	If \[\overset{\to }{\mathop{a}}\,,\,\overset{\to }{\mathop{b}}\,,\,\overset{\to }{\mathop{c}}\,\] are three non-coplanar vectors, then the value of \[\frac{\overset{\to }{\mathop{a}}\,.(\overset{\to }{\mathop{b}}\,\times \overset{\to }{\mathop{c}}\,)}{(\overset{\to }{\mathop{c}}\,\times \overset{\to }{\mathop{a}}\,).\overset{\to }{\mathop{b}}\,}+\frac{\overset{\to }{\mathop{b}}\,.(\overset{\to }{\mathop{a}}\,\times \overset{\to }{\mathop{c}}\,)}{\overset{\to }{\mathop{c}}\,.(\overset{\to }{\mathop{a}}\,\times \overset{\to }{\mathop{b}}\,)}\] is:
A.	0
B.	2
C.	1
D.	None of these
Answer» B. 2

Discussion

97.	For any vector \[\overset{\to }{\mathop{p}}\,\], the value of\[\frac{3}{2}\left\{ \|\overset{\to }{\mathop{p}}\,\times \hat{i}{{\|}^{2}}+\|\overset{\to }{\mathop{p}}\,\times \hat{j}{{\|}^{2}}+\|\overset{\to }{\mathop{p}}\,\times \hat{k}{{\|}^{2}} \right\}\] is
A.	\[\overset{\to 2}{\mathop{p}}\,\]
B.	\[2\overset{\to 2}{\mathop{p}}\,\]
C.	\[3\overset{\to 2}{\mathop{p}}\,\]
D.	\[4\overset{\to 2}{\mathop{p}}\,\]
Answer» D. \[4\overset{\to 2}{\mathop{p}}\,\]

Discussion

98.	Radical axis of the circles \[3{{x}^{2}}+3{{y}^{2}}-7x+8y+11=0\] and \[{{x}^{2}}+{{y}^{2}}-3x-4y+5=0\] is [RPET 2001]
A.	\[x+10y+2=0\]
B.	\[x+10y-2=0\]
C.	\[x+10y+8=0\]
D.	\[x+10y-8=0\]
Answer» C. \[x+10y+8=0\]

Discussion

99.	The equation of director circle of the circle \[{{x}^{2}}+{{y}^{2}}={{a}^{2}},\]is [BIT Ranchi 1990]
A.	\[{{x}^{2}}+{{y}^{2}}=4{{a}^{2}}\]
B.	\[{{x}^{2}}+{{y}^{2}}=\sqrt{2}{{a}^{2}}\]
C.	\[{{x}^{2}}+{{y}^{2}}-2{{a}^{2}}=0\]
D.	None of these
Answer» D. None of these

Discussion

100.	Under what condition are the two lines\[y=\frac{m}{\ell }x+\alpha ,z=\frac{n}{\ell }x+\beta ;\] and \[y=\frac{m'}{\ell '}x+\alpha ',z=\frac{n'}{\ell '}x+\beta '\] Orthogonal?
A.	\[\alpha \alpha '+\beta \beta '+1=0\]
B.	\[(\alpha +\alpha ')+(\beta +\beta ')=0\]
C.	\[\ell \ell '+mm'+nn'=1\]
D.	\[\ell \ell '+mm'+nn'=0\]
Answer» E.

Discussion

Explore topic-wise MCQs in Joint Entrance Exam - Main (JEE Main).

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