JEE Main Mathematics - Logarithms and Their Properties
Exam Duration: 60 Mins Total Questions : 30
If p, q, r are in A.P. the pth, qth and rth terms of any G.P.
- (a)
are in A.P.
- (b)
are in G.P.
- (c)
have their reciprocals in A.P.
- (d)
NONE OF THESE
The equation sin x=\(\pi\over2\) +x=0 has a real root in the interval
- (a)
\(({0,{\pi\over2}})\)
- (b)
\(({\pi\over2},\pi)\)
- (c)
\(({\pi,{3\pi\over2}})\)
- (d)
None of these
The lines represented by ax2+2hxy-ax2=0 are
- (a)
coicident
- (b)
perpendicular
- (c)
parallel
- (d)
NONE OF THESE
Which of the following expressions are meaningful?
- (a)
\(\vec { u } .(\vec { v } \times \vec { w } )\)
- (b)
\((\vec { u } .\vec { v } ).\vec { w } \)
- (c)
\((\vec { u } .\vec { v } )\times \vec { w } \)
- (d)
\(\vec { u } \times (\vec { v } .\vec { w } )\)
The integral
\(\int { \frac { d\theta }{ 5+4cos\theta } , } \) equals
- (a)
\(\frac { 1 }{ 3 } { tan }^{ -1 }\left( \frac { tan(\theta /2) }{ 3 } \right) +c\)
- (b)
\(\frac { }{ } { tan }^{ -1 }\left( \frac { tan(\theta /2) }{ 3 } \right) +c\)
- (c)
\(\frac { 1 }{ 6 } { tan }^{ -1 }\left( \frac { tan(\theta /2) }{ 3 } \right) +c\)
- (d)
\(\frac { 2 }{ 3 } { tan }^{ -1 }\left( \frac { tan(\theta /2) }{ 3 } \right) +c\)
If \(X=\left\{ { 8 }^{ n }-7n-1:n\epsilon N \right\} \) and \(Y=\left\{ 49\left( n-1 \right) :n\epsilon N \right\} \) , then
- (a)
X C Y
- (b)
Y C X
- (c)
X = Y
- (d)
None of these
If \(3(a+2c)=4(b+3d),\) then the equation \(ax^3+bx^2+cx+d=0\) will have
- (a)
no real solution
- (b)
atleast one real root in (-1, 0)
- (c)
atleast one real root in (0,1)
- (d)
None of the above
Which of the following is correct?
- (a)
If A is a square matrix, (A + A' ) is a symmetric matrix
- (b)
If A is a square matrix, (A - A') is a skew symmetric matrix
- (c)
Every square matrix can be expressed as the sum of a symmetric and skew symmetric matrix
- (d)
Some elements of the skew symmetric matrix must be zero
If the sum of the binomial coefficients in the expansion of \(\left( x+\frac { 1 }{ x } \right) ^{ n }\) is 64, then the term independent of x is equal to
- (a)
10
- (b)
20
- (c)
40
- (d)
60
If ai > 0,i=1, 2, 3, ..., 50 and a1 + a2 + a3 + ...+a50 = 50, then the minimum value of \(\frac { 1 }{ { a }_{ 1 } } +\frac { 1 }{ { a }_{ 2 } } +\frac { 1 }{ { a }_{ 3 } } +...+\frac { 1 }{ { a }_{ 50 } } \)is equal to
- (a)
150
- (b)
100
- (c)
50
- (d)
(50)2
The value of \(\cos { \frac { \pi }{ 15 } } \cos { \frac { 2\pi }{ 15 } } \cos { \frac { 3\pi }{ 15 } } \cos { \frac { 4\pi }{ 15 } } \cos { \frac { 5\pi }{ 15 } } \cos { \frac { 6\pi }{ 15 } } \cos { \frac { 7\pi }{ 15 } } \) is equal to
- (a)
1/26
- (b)
1/27
- (c)
1/28
- (d)
none of these
The number of vectors of unit length perpendicular to the vectors \(\overrightarrow { a } \) = (1, 1, 0) and \(\overrightarrow { b } \)= (0, 1, 1) is
- (a)
1
- (b)
2
- (c)
3
- (d)
infinite
If A, B are eccentric angles of the extremities of a focal chord of an ellipse, then eccentricity of the ellipse is
- (a)
\(\frac { \cos { \alpha } +\cos { \beta } }{ cos(\alpha -\beta ) } \)
- (b)
\(\frac { \sin { \alpha } -\sin { \beta } }{ sin(\alpha -\beta ) } \)
- (c)
\(sec\alpha +sec\beta \)
- (d)
\(\frac { \sin { \alpha } -\sin { \beta } }{ sin(\alpha +\beta ) } \)
In a triangle ABC, 2a2 + 4b2 + c2 = 4ab + 2ac, then the numerical value of cas B is equal to
- (a)
0
- (b)
\(\frac { 3 }{ 8 } \)
- (c)
\(\frac { 5 }{ 8 } \)
- (d)
\(\frac { 7 }{ 8 } \)
In any ∆ABC, \(\Pi \left( \frac { { sin }^{ 2 }A+sinA+1 }{ sin\quad A } \right) \)is always greater than
- (a)
9
- (b)
3
- (c)
27
- (d)
none of these
P(n):2.7n+3.5n-5 is divisible by
- (a)
24, ∀n∈N
- (b)
21, ∀n∈N
- (c)
35, ∀n∈N
- (d)
50, ∀n∈N
\(\left( x+\frac { 1 }{ x } \right) ^{ 6 }\) =
- (a)
x6 + 6x4 + 15x2 -20 - \(\frac { 15 }{ { x }^{ 2 } } +\frac { 6 }{ { x }^{ 4 } } -\frac { 1 }{ { x }^{ 6 } } \)
- (b)
x6 + 6x4 + 15x2 + 20+\(\frac { 15 }{ { x }^{ 2 } } +\frac { 6 }{ { x }^{ 4 } } +\frac { 1 }{ { x }^{ 6 } } \)
- (c)
x6 + 2x4 + 5x2 + 20 - \(\frac { 15 }{ { x }^{ 2 } } -\frac { 6 }{ { x }^{ 4 } } +\frac { 1 }{ { x }^{ 6 } } \)
- (d)
2x6 + 6x4 + 15x2 - 20+\(\frac { 15 }{ { x }^{ 2 } } -\frac { 6 }{ { x }^{ 4 } } +\frac { 1 }{ { x }^{ 6 } } \)
Find the equation of the parabola which is symmetric about the y-axis, and passes through the point (2, -3).
- (a)
x2=4y
- (b)
4y=3x2
- (c)
3x2=-4x
- (d)
3x=-4x2
The solution curve of \(\frac { dy }{ dx } =\frac { { y }^{ 2 }-2xy-{ x }^{ 2 } }{ { y }^{ 2 }+2xy-{ x }^{ 2 } } \) y(-1) = 1 is
- (a)
a straight line
- (b)
parabola
- (c)
circle
- (d)
ellipse
Find x from the equation
cosec(900 +θ)+xcosθcot(900+θ)=sin(900+θ)
- (a)
cotθ
- (b)
tanθ
- (c)
-tanθ
- (d)
-cotθ
The value of b for which the function f(x) =sinx-bx+c is decreasing for x∈R is given by
- (a)
b < 1
- (b)
b > 1
- (c)
b > 1
- (d)
b < 1
Match the following
Column I | Column II |
---|---|
(i) \(tan^{-1}(\frac{\sqrt{1+x^{2}}+\sqrt{1-x^2}}{\sqrt{1+x^2}-{\sqrt{1-x^2}}})=\) | (p) \(tan^{-1}\frac{4}{3}-x\) |
(ii) \(cos^{-1}(\frac{3}{5}cosx+\frac{4}{5}sinx),\)where \(x∈(-\frac{3\pi}{4},\frac{\pi}{4})\)is equal to | (q) \(\sqrt{\frac{1+x^2}{2+x^2}}\) |
(iii) \(cot^{-1}(\frac{\sqrt{1+sinx}+\sqrt{1-sinx}}{\sqrt{1+sinx}-\sqrt{1-sinx}}),\) where \(x∈(0,\frac{\pi}{4})\) is equal to | (r) \(\frac{\pi}{4}+\frac{1}{2}cos^{-1}x^2\) |
(iv) cos(tan-1(sin(cot-1x)))= | (s) \(\frac{x}{2}\) |
- (a)
(i) ⟶ (s), (ii) ⟶ (q), (iii) ⟶ (r), (iv) ⟶ (p)
- (b)
(i) ⟶ (r), (ii) ⟶ (s), (iii) ⟶ (r), (iv) ⟶ (p)
- (c)
(i) ⟶ (r), (ii) ⟶ (p), (iii) ⟶ (s), (iv) ⟶ (q)
- (d)
(i) ⟶ (p), (ii) ⟶ (r), (iii) ⟶ (q), (iv) ⟶ (s)
Fmily y = Ax + A3 of curves is represented y the differential equation of degree
- (a)
1
- (b)
2
- (c)
3
- (d)
4
If X={8n-7n-1|n∈N} and Y={49n-49|n∈N}. Then
- (a)
X⊂Y
- (b)
Y⊂X
- (c)
X=Y
- (d)
X∩Y=ф
The value of c in Rolle's theorem for the function f(x) = x3 - 3x in the interval [0, \(\sqrt { 3 } \)] is
- (a)
1
- (b)
-1
- (c)
\(\frac { 3 }{ 2 } \)
- (d)
\(\frac { 1 }{ 3 } \)