Inequalities Involving Mathematical Means

Q.

Three numbers are in GP, whose sum is 13 and the sum of whose squares is 91. Find the numbers.

Q.

If both the roots of $k(6x^2+3)+rx+2x^2-1=0$ and $2(6k+2)x^2+px+2(3k-1)=0$ are common, then 2r-p is equal to

1. -1

2. 1

3. 2

4. 0

Q.

The roots of the equation $x^{\frac{2}{3}}+x^{\frac{1}{3}}-2$ = 0 are

1. 1, -4

2. 1, -8

3. 1, 8

4. 1, 4

Q.

If a, b and c are distinct positive real numbers and $a^2+b^2+c^2$ = 1, then ab + bc + ca is

1. less than 1

2. equal to 1

3. greater than 1

4. any real number

Q.

If two roots of the equation $x^3-3x+2=0$ are same, then the roots will be

1. 1, 1, -2

2. - 2, 3, 3

3. -2, -2, 1

4. 2, 2, 3

Q. For three finite sets, n(A) = 34, n(B) = 41, n(C)52, n(An B) = 15, n(Bn C) = 19, n(Cn A) = 12A possible value of n(AU BuC) is(1) 97(2) 93(3) 98(4) 100

Q.

The incentre of the triangle formed by (0, 0), (5, 12), (16, 12) is

1. (7, 9)

2. (9, 7)

3. (-9, 7)

4. (-7, 9)

Q.

Every even power of an odd number greater than 1 when divided by 8 leaves 1 as the remainder.The inductive step for the above statement is

1. P(k) = (2k + 1)²ⁿ = 8a + 1 implies P(k + 1) : (2k + 3)^{2n + 2} = 8b + 1 for some natural number a, b

2. P(k) = (2k - 1)² = 8a + 1 implies P(k + 1); (2k + 1)² = 8b + 1 for some natural numbers a, b

3. P(k) = (2k + 1)² = 8a + 1 implies P(k + 1):(2k + 3)² = 8b + 1 for some natural numbers a, b

4. P(1) = 3² = 8.1 + 1

Q. If $x^2+y^2+z^2=1$, then $yz+zx+xy$ lies in the interval

1. $[\frac{1}{2},2]$
2. $[-1,2]$
3. $[-\frac{1}{2},1]$
4. $[-1,\frac{1}{2}]$

Q.

If a, b and c are three positive real numbers, which one of the following are true?

1. $a^2+b^2+c^2\ge bc+ca+ab$
   

2. $a^3+b^3+c^3\ge 3abc$
   

3. $(b+c)(c+a)(a+b)\ge 8abc$
   

4. $\frac{bc}{a}+\frac{ca}{b}+\frac{ab}{c}\ge a+b+c$

Q.

Suppose there are four roads from station A to station B and 3 roads between station B and station C. Find the number of ways in which one can drive from station A to station C via station B.

1. 16

2. 12

3. 4

4. 7

Q.

If $x_1$ and $x_2$ are the values of x satisfying the equation |x| = 6, then $x_1$ + $x_2$ = , then

__

Q. explain ques 2 and ques 5 of module pf 5 try yourself

Q.

Insert appropriate symbol (<, > or = ) between the given measurements:

$5.250g\_\_\_\_\_\_\_2.5kg$

Q. If the difference of the roots of $x^2-px+q=0$ is unity, then
(a) $p^2+4q=1$
(b) $p^2-4q=1$
(c) $p^2+4q^2=(1+2q{)}^2$
(d) $4p^2+q^2=(1+2p{)}^2$

Q.

Insert appropriate symbol (<, > or = ) between the given measurements:

$2800g\_\_\_\_\_\_\_28mg$

Q.

Insert appropriate symbol (<, > or = ) between the given measurements:

$9.9000mL\_\_\_\_\_\_\_9L$

Q. If a, b are the roots of the equation $x^2+x+1=0,\mathrm{then}{a}^2+b^2=$
(a) 1
(b) 2
(c) −1
(d) 3

Q. $\underset{x\to \infty }{\lim }{\left\{\frac{3x^2+1}{4x^2-1}\right\}}^{\frac{x^3}{1+x}}$

Q. If α and β are the roots of $4x^2+3x+7=0$, then the value of $\frac{1}{\alpha }+\frac{1}{\beta }$ is
(a) $\frac{4}{7}$
(b) $-\frac{3}{7}$
(c) $\frac{3}{7}$
(d) $-\frac{3}{4}$

Q. The least value of the expression $2{\log }_{10}x-{\log }_x\left(0.01\right)$, for $x>1$, is

1. $10$
2. $2$
3. $-0.01$
4. $4$

Q.

Find the sum of the roots of the equation |$x^2$| - 36|x|

1. 36

2. -36

3. -1

4. 0

Q. If a, b, c and d are four positive real numbers such that abcd=1 what is the minimum value of (1+a), (1+b), (1+c) and (1+d)

Q. Evaluate the following:
(i) ¹⁴C₃
(ii) ¹²C₁₀
(iii) ³⁵C₃₅
(iv) ^{n + 1}C_n
(v) $\sum _{r=1}^5{}^5{C}_r$.

Q. If $X=\left\{8^n-7n-1:n\in N\right\}\mathrm{and}Y=\left\{49\left(n-1\right):n\in N\right\}$, then prove that $X\subseteq Y.$

Q. If z is a complex number such that |z-1|=1, z not equal to 2, then the real part of z-2/z is

Q. If α, β are roots of the equation $4x^2+3x+7=0,\mathrm{then}1/\alpha +1/\beta$ is equal to
(a) 7/3
(b) −7/3
(c) 3/7
(d) −3/7

Q.

Sum of common roots of the equations

$z^3$ + 2$z^2$ + 2z + 1 = 0 and $z^{100}$ + $z^{32}$ + 1 = 0 is equal to :

1. 0

2. -1

3. 1

4. None

Q. 22. To receive Grade 24, in a course, one must obtain an average of 90 marks ormore in five examinations (each of 100 marks). If Sunita's marks in first fourexaminations are 87, 92, 94 and 95, find minimum marks that Sunita must obtairnin fifth examination to get grade A' in the course.

Q. 28. exists and is equal to ø (where ø is a finite real number), then A) b=-1 B) ø=5 C) c=-8 D) b+c=9 (One or more than one options correct type)