Maths NCERT Exemplar Solutions Class 12th Chapter Twelve: Overview, Questions, Preparation

If the feasible region for a LPP is __________, then the optimal value of the objective function $Z=ax+by$ may or may not exist.

In a LPP, if the objective function $Z=ax+by$ has the same maximum value on two corner points of the feasible region, then every point on the line segment joining these two points gives the same __________ value.

A feasible region of a system of linear inequalities is said to be __________ if it can be enclosed within a circle.

A corner point of a feasible region is a point in the region which is the __________ of two boundary lines.

If the feasible region for a LPP is unbounded, maximum or minimum of the objective function $Z=ax+by$  may or may not exist.

In a LPP, the minimum value of the objective function $Z=ax+by$ is always 0 if the origin is one of the corner points of the feasible region.

The maximum value of the objective function $Z=ax+by$ in a LPP always occurs at only one corner point of the feasible region.

If $a$ and $b$ are real numbers such that $(2+\alpha {)}^4=a+b\alpha$ , where $\alpha =\frac{-1+i\sqrt[]{3}}{2}$ , then $a+b$ is equal to:

Contrapositive of the statement:

If a function $f$ is differentiable at $a$ , then it is also continuous at $a$ ', is:

If for some positive integer $n$ , the coefficients of three consecutive terms in the binomial expansion of $(1+x{)}^{\mathrm{N}+5}$ are in the ratio $5:10:14$ , then the largest coefficient in the expansion is:

Let $\lambda \ne 0$ be in R. If $\alpha$ and $\beta$ are the roots of the equation, $x^2-x+2\lambda =0$ and $\alpha$ and $\gamma$ are the roots of the equation, $3x^2-10x+27\lambda =0$ , then $\frac{\beta \gamma }{\lambda }$ is equal to:

The circle passing through the intersection of the circles, $x^2+y^2-6x=0$ and $x^2+y^2-4y$ $=0$ , having its centre on the line, $2x-3y+12=0$ , also passes through the point:

The area (in sq. units) of the largest rectangle $ABCD$ whose vertices $A$ and $B$ lie on the $x$ -axis and vertices $C$ and $D$ lie on the parabola, $y=x^2-1$ below the $x$ -axis, is:

Suppose the vectors $x_1,x_2$ and $x_3$ are the solutions of the system of linear equations, $Ax$ $=\mathrm{b}$ when the vector $\mathrm{b}$ on the right side is equal to ${\mathrm{b}}_1,{\mathrm{b}}_2$ and ${\mathrm{b}}_3$ respectively. If $x_1=\left[\begin{array}{l}1\\ 1\\ 1\end{array}\right],x_2=\left[\begin{array}{l}0\\ 2\\ 1\end{array}\right],x_3=\left[\begin{array}{l}0\\ 0\\ 1\end{array}\right],b_1=\left[\begin{array}{l}1\\ 0\\ 0\end{array}\right],b_2=\left[\begin{array}{l}0\\ 2\\ 0\end{array}\right]$ and $b_3=\left[\begin{array}{l}0\\ 0\\ 2\end{array}\right]$ , then the determinant of $A$ is equal of $A$ is equal to :

The solution of differential equation $\frac{dy}{dx}-\frac{y+3x}{{\mathrm{l}\mathrm{o}\mathrm{g}}_e?(y+3x)}+3=0$ is:

(where $\mathrm{C}$ is a constant of integration.)

Let $f:(0,\mathrm{\infty })\to (0,\mathrm{\infty })$ be a differentiable function such that $f\left(1\right)=e$ and ${\mathrm{l}\mathrm{i}\mathrm{m}}_{t\to x} \frac{t^2{f}^2\left(x\right)-x^2{f}^2\left(t\right)}{t-x}=0$ . If $f\left(x\right)=1$ , then $x$ is equal to:

The distance of the point $(1,-\mathrm{2,3})$ from the plane $x-y+z=5$ measured parallel to the line $\frac{x}{2}=\frac{y}{3}=\frac{z}{-6}$ is:

In a game two players $A$ and $B$ take turns in throwing a pair of fair dice starting with player $A$ and total of scores on the two dice, in each throw is noted. A wins the game if he throws a total of 6 before $B$ throws a total of 7 and $B$ wins the game if he throws a total of 7 before $A$ throws a total of six. The game stops as soon as either of the players wins. The probability of $A$ winning the game is:

The angle of elevation of a cloud $\mathrm{C}$ from a point $\mathrm{P},200\mathrm{m}$ above a still take is ${30}^{\circ }$ . If the angle of depression of the image of $C$ in the lake from the point $P$ is ${60}^{\circ }$ , then $PC$ (in $\mathrm{m}$ ) is equal to

Let $x=4$ be a directrix to an ellipse whose centre is at the origin and its eccentricity is $\frac{1}{2}$ . If $P(1,\beta ),\beta >0$ is a point on this ellipse, then the equation of the normal to it at $P$ is:

Let ${\bigcup }_{i=1}^{50} X_i={\bigcup }_{i=1}^n Y_i=T$ , where each $X_i$ contains 10 elements and each $Yi$ contains 5 elements. If each element of the set $T$ is an element of exactly 20 of sets $X_i$ 's and exactly 6 of sets $Y_i$ 's then $n$ is equal to:

If the perpendicular bisector of the line segment joining the points $P\left(\mathrm{1,4}\right)$ and $Q(k,3)$ has $y$ -intercept equal to -4 , then a value of $k$ is:

Let $a_1,a_2,\dots a_n$ be a given A.P. whose common difference is an integer and $S_n=a_1+a_2$ $+\dots +a_n$ . If $a_1=1,a_1=300$ and $15\le n\le 50$ , then the ordered pair $\left(S_{n-4},a_{n-4}\right)$ is equal to:

A test consists of 6 multiple choice questions, each having 4 alternative answers of which only one is correct. The number of ways, in which a candidate answers all six questions such that exactly four of the answers are correct, is

Let $PQ$ be a diameter of the circle $x^2+y^2=9$ . If $\alpha$ and $\beta$ are the lengths of the perpendiculars from $P$ and $Q$ on the straight line, $x+y=2$ respectively, then the maximum value of $\alpha \beta$ is......

Let $\left\{x\right\}$ and $\left[x\right]$ denote the fractional part of $x$ and the greatest integer $\le x$ respectively of $a$ real number $x$ . if ${\int }_0^n \left\{x\right\}dx,{\int }_0^n \left[x\right]dx$ and $10\left(n^2-n\right),(n\in N,n>1)$ are three consecutive terms of a G.P. then $\mathrm{n}$ is equal to(Integration)

Let $\alpha$ and $\beta$ be roots of $x^2-3x+p=0$ and $\gamma$ and $\delta$ be the roots of $x^2-6x+q=0$ . If $\alpha ,\beta$ , $\gamma ,\delta$ , form a geometric progression. Then ratio $(2q+p):(2q-p)$ is:

Let $f$ be a twice differentiable function on $\left(\mathrm{1,6}\right)$ . If $f\left(2\right)=8,f^{\mathrm{\text{'}}}\left(2\right)=5,f^{\mathrm{\text{'}}}\left(x\right)\ge 1$ and $f$ " $\left(x\right)\ge 4$ , for all $x\in \left(\mathrm{1,6}\right)$ , then: