The set of values of $m$ for which $mx^2 - 6mx + 5m + 1 > 0$ for all real $x$ is
(A) $m < \frac{1}{4}$
(B) $m \geq 0$
(C) $0 \leq m \leq \frac{1}{4}$
(D) $0 \leq m < \frac{1}{4}$

The set of values of $m$ for which $m x ^ { 2 } - 6 m x + 5 m + 1 > 0$ for all real $x$ is
(a) $m < \frac { 1 } { 4 }$
(b) $m \geq 0$
(c) $0 \leq m \leq \frac { 1 } { 4 }$
(d) $0 \leq m < \frac { 1 } { 4 }$.

The set of values of $m$ for which $m x ^ { 2 } - 6 m x + 5 m + 1 > 0$ for all real $x$ is
(a) $m < \frac { 1 } { 4 }$
(b) $m \geq 0$
(c) $0 \leq m \leq \frac { 1 } { 4 }$
(d) $0 \leq m < \frac { 1 } { 4 }$.

The set of values of $m$ for which $mx^2 - 6mx + 5m + 1 > 0$ for all real $x$ is
(A) $m < \frac{1}{4}$
(B) $m \geq 0$
(C) $0 \leq m \leq \frac{1}{4}$
(D) $0 \leq m < \frac{1}{4}$

The set of values of $m$ for which $m x ^ { 2 } - 6 m x + 5 m + 1 > 0$ for all real $x$ is
(A) $m < \frac { 1 } { 4 }$
(B) $m \geq 0$
(C) $0 \leq m \leq \frac { 1 } { 4 }$
(D) $0 \leq m < \frac { 1 } { 4 }$

Let $\mathbb { R } ^ { 2 }$ denote $\mathbb { R } \times \mathbb { R }$. Let
$$S = \left\{ ( a , b , c ) : a , b , c \in \mathbb { R } \text { and } a x ^ { 2 } + 2 b x y + c y ^ { 2 } > 0 \text { for all } ( x , y ) \in \mathbb { R } ^ { 2 } - \{ ( 0,0 ) \} \right\}$$
Then which of the following statements is (are) TRUE?
(A) $\left( 2 , \frac { 7 } { 2 } , 6 \right) \in S$
(B) If $\left( 3 , b , \frac { 1 } { 12 } \right) \in S$, then $| 2 b | < 1$.
(C) For any given $( a , b , c ) \in S$, the system of linear equations
$$\begin{aligned} & a x + b y = 1 \\ & b x + c y = - 1 \end{aligned}$$
has a unique solution.
(D) For any given $( a , b , c ) \in S$, the system of linear equations
$$\begin{aligned} & ( a + 1 ) x + b y = 0 \\ & b x + ( c + 1 ) y = 0 \end{aligned}$$
has a unique solution.

The number of integral values of $m$ for which the quadratic expression $( 1 + 2 m ) x ^ { 2 } - 2 ( 1 + 3 m ) x + 4 ( 1 + m ) , x \in R$ is always positive, is
(1) 7
(2) 3
(3) 6
(4) 8

The number of integral values of $m$ for which the equation, $1 + m ^ { 2 } x ^ { 2 } - 21 + 3 m x + 1 + 8 m = 0$ has no real root, is
(1) 2
(2) 3
(3) Infinitely many
(4) 1

The integer $k$, for which the inequality $x ^ { 2 } - 2 ( 3 k - 1 ) x + 8 k ^ { 2 } - 7 > 0$ is valid for every $x$ in $R$ is:
(1) 4
(2) 2
(3) 3
(4) 0

If the set of all $\mathrm { a } \in \mathbf { R }$, for which the equation $2 x ^ { 2 } + ( a - 5 ) x + 15 = 3 \mathrm { a }$ has no real root, is the interval $( \alpha , \beta )$, and $X = \{ x \in Z : \alpha < x < \beta \}$, then $\sum _ { x \in X } x ^ { 2 }$ is equal to :
(1) 2109
(2) 2129
(3) 2119
(4) 2139

Consider the two functions
$$\begin{aligned} & f ( x ) = x ^ { 2 } + 2 a x + 4 a - 3 \\ & g ( x ) = 2 x + 1 \end{aligned}$$
We are to find the condition on $a$ for which $f ( x ) \geqq g ( x )$ for all $x$ and also find the range of values of the minimum of $f ( x )$ under this condition.
We must find the condition under which
$$x ^ { 2 } + \mathbf { A } ( a - \mathbf { A } ) x + \mathbf { A C } a - \mathbf { A D } \geq 0$$
for all $x$.
For each of $\mathbf { E } \sim \mathbf{ H }$ in the following questions, choose the correct answer from among (0) $\sim$ (7) below each question.
(1) The required condition is that $a$ satisfy the quadratic inequality $\mathbf{E}$. Hence $a$ is in the range $\mathbf { F }$. (0) $a ^ { 2 } - 5 a + 4 \geqq 0$
(1) $a ^ { 2 } - 6 a + 5 \geqq 0$
(2) $a ^ { 2 } - 5 a + 4 \leqq 0$
(3) $a ^ { 2 } - 6 a + 5 \leqq 0$
(4) $a \leqq 1$ or $5 \leqq a$
(5) $1 \leqq a \leqq 5$ (6) $1 \leqq a \leqq 4$ (7) $a \leqq 1$ or $4 \leqq a$
(2) Let $m$ be the minimum value of $f ( x )$. Then, since $m = \mathbf { G }$, the range of values which $m$ can take under the condition in (1) is $\mathbf { H }$. (0) $a ^ { 2 } + 4 a - 3$
(1) $4 a ^ { 2 } + 4 a - 3$
(2) $- a ^ { 2 } + 4 a - 3$
(3) $2 a ^ { 2 } - 4 a + 3$
(4) $- 5 \leqq m \leqq 1$
(5) $- 8 \leqq m \leqq 1$ (6) $- 8 \leqq m \leqq - 1$ (7) $- 5 \leqq m \leqq - 1$