Conditions on the Parameters of Logarithm Function

Q. The most general values of x for which $\sin x+\cos x=min\{1,a^2-4a+6\}$ are given by

1. 
2. 
3. None of these
4. 

Q. Number of positive integral solutions of xyz=210 is (1) 27 (2) 81 (3) 72 (4) 69

Q.

The number of ordered triplets of non-negative integers which are solutions of the equation x + y + z = 100 is

1. 4851

2. 5081

3. 6005

4. 5151

Q.

If the set of factors of a whole number 'n' including 'n' itself but not '1' is denoted by $F\left(n\right),F\left(16\right)\cap F\left(40\right)$ = $F\left(x\right)$ then 'x' is

1. 10

2. 4

3. 8

4. 6

Q. The domain of the function $\sqrt{\left(lo{g}_{0.5}x\right)}$ is

1. $(0,\infty )$
2. $(0,1]$
3. $(0.5,1)$
4. $(1,\infty )$

Q.

For logarithm ${\log }_{0.2}$ $\frac{x+2}{x}$ to be defined. Which of the following is/are true?

1. x ∈ (0, ∞)

2. x ∈ (-∞, -2)

3. x ∈ R

4. x ∈ (-2, 0)

Q.

Find the real values of x satisfying ${\log }_{0.3}$ (10x + 3) < ${\log }_{0.3}$ (7x - 4).

1. x ∈

2. 

3. x ∈

4. 

Q. 10. lxcos(x)l=lx-1l find number of solutions where x belongs to [-3.14, 3.14].

Q. If log 2, log (2^x − 1) and log (2^x + 3) are in A.P., write the value of x.

Q. For $y=lo{g}_a$ x to be defined a must be

1. Any + ive real number
2. Any number
3. $\ge e$
4. any + ive real number $\ne$ 1

Q. For $y=lo{g}_{a}x$ to be defined 'a' must be

1. Any positive real number
2. Any number
3. Any positive real number $\ne 1$
4. $\ge e$

Q.

Find the set of real values of x for which ${\log }_{12}$ ($x^2$ - x) < 1.

1. x ∈ (-∞, 0) ∪ (1, ∞)

2. (-∞, 0) ∪ (1, ∞)

3. x ∈ (-3, 0)

4. x ∈ (-3, 0) ∪ (1, 4)

Q.

Equation $12x^2-10xy+2y^2+11x-5y+2=0$ represent a pair of straight lines. Which of the following is/are true about these lines?

1. Slope of line 1 is -2

2. Slope of line 1 is 2

3. Slope of line 2 is 3

4. Slope of line 2 is -3

Q. Find the derivative of the following functions (it is to be understood that $a,b,c,d,p,q,r$ and $s$ are fixed non-zero constant and $m$ and $n$ are integers) : $(ax+b){}^n$

Q.

Solve the inequality $\frac{3-x}{x+2}$ > 1

1. - 2 < x < 0

2. -2 < x < 3

3. 

4. -1 < x < 3

Q.

Find the solution of the inequation 0 < |x-3| $\le$ 1 contains the interval.

1. x ∈ [-2, 4]

2. x ∈ [2, 4]

3. x ∈ (2, 4)

4. x ∈ (-2, 4)

Q. Assume that $P\left(A\right)=P\left(B\right)$. Show that $A=B$

Q. The sum of values of $x$ satisfying ${\log }_{1/2}(x-1)+{\log }_{1/2}(x+1)-{\log }_{1/\sqrt{2}}(7-x)=1$ is

Q. For $y=lo{g}_a$ x to be defined a must be

1. Any + ive real number
2. Any number
3. $\ge e$
4. any + ive real number $\ne$ 1

Q.

The number of values of $x\in [0,n\pi ],n\in I$ that satisfy $lo{g}_{|sinx|}(1+cosx)=2$ is

1. 0

2. n

3. 2n

4. None of these

Q. If ${\log }_3(2x+6)$ is defined, then the complete set of values of $x$ is

1. $(-3,\infty )$
2. $(6,\infty )$
3. $(3,\infty )$
4. $(-6,\infty )$

Q. The number of real values of the parameter k for which $(lo{g}_{16}x{)}^2-lo{g}_{16}x+lo{g}_{16}k=0$ with real coefficients will have exactly one solution is

1. 2
2. 1
3. 4
4. None of these