S - 100 1lendarray Which of the following is only CORRECT combination-A- I - P Q B- II - Q C- III - R D- IV - P R T

The correct option is D (IV)→(P)(R)(T)(I)→x+y+z=1 . x+4=x_1≥0 y+4=y_1≥0 z+4=z_1≥0} x_1+y_1+z_1=13 So number of solutions =^13+3 1C_3 1=^15C_2=105 (II)(1+1/ ...

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Byju's Answer

Standard XII

Mathematics

Domain

S & 100 1lend...

Question

$\begin{matrix} List- I & List-II (I) & Number of integral solutions of & (P) & 132 x + y + z = 1, x \geq - 4, y \geq - 4, z \geq - 4 is less than (Q) & 99 (I I) & Greatest term in the expression of \frac{4}{3 \sqrt{2}} {(1 + \frac{1}{\sqrt{2}})}^{12} is (R) & 120 (I I I) & If a_{1}, a_{2}, a_{3} . . . a_{100} are in H.P. then value of \sum_{i = 1}^{99} \frac{a_{i} a_{i + 1}}{a_{1} a_{100}} is - (S) & 100 (I V) & If 8 points out of 11 are in smae straight line then number of triangles formed is less then & (T) & 125 \end{matrix}$

Which of the following is only CORRECT combination?

(I) \to (P) (Q)

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(I I) \to (Q)

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(I I I) \to (R)

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(I V) \to (P) (R) (T)

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Solution

The correct option is D $(I V) \to (P) (R) (T)$
$(I) \to$ $x + y + z = 1 \begin{matrix} x + 4 = x_{1} \geq 0 y + 4 = y_{1} \geq 0 z + 4 = z_{1} \geq 0 \end{matrix} ⎫ ⎪ ⎬ ⎪ ⎭ x_{1} + y_{1} + z_{1} = 13$
So number of solutions $=^{13 + 3 - 1} C_{3 - 1} =^{15} C_{2} = 105$

$(I I) {(1 + \frac{1}{\sqrt{2}})}^{12} \to | T_{r + 1} | \geq | T_{r} |^{12} C_{r} {(\frac{1}{\sqrt{2}})}^{r} \geq^{12} C_{r - 1} {(\frac{1}{\sqrt{2}})}^{r - 1} \frac{12 - r + 1}{r} \geq \sqrt{2} r \leq \frac{13}{\sqrt{2} + 1} r \leq 5.385$
So $r = 5$
$T_{6}$ is numerically greatest term.
$T_{6} =^{12} C_{5} {(\frac{1}{\sqrt{2}})}^{5} ∴ \frac{4}{3 \sqrt{2}} \times^{12} C_{5} \times {(\frac{1}{\sqrt{2}})}^{5} = 132$

$(I I I) \frac{1}{a_{1}}, \frac{1}{a_{2}}, \frac{1}{a_{3}}, . . . \frac{1}{a_{100}}$ in A.P.
$\frac{1}{a_{2}} - \frac{1}{a_{1}} = \frac{1}{a_{3}} - \frac{1}{a_{2}} = . . . = \frac{1}{a_{100}} - \frac{1}{a_{99}} \frac{a_{1} - a_{2}}{a_{1} a_{2}} = d \frac{1}{a_{1} a_{100}} \sum_{i = 1}^{99} a_{i} a_{i + 1} = \frac{1}{a_{1} a_{100}} (a_{1} a_{2} + a_{2} a_{3} + . . . + a_{99} a_{100}) = \frac{1}{a_{1} a_{100}} (\frac{a_{1} - a_{2}}{d} + \frac{a_{2} - a_{3}}{d} + . . . + \frac{a_{99} - a_{100}}{d}) = \frac{a_{1} - a_{100}}{a_{1} a_{100} \times d}$
Now $\frac{1}{a_{100}} = \frac{1}{a_{1}} + 99 d \Rightarrow d = \frac{a_{1} - a_{100}}{a_{1} a_{100}} \times \frac{1}{99} \Rightarrow \frac{a_{1} - a_{100}}{a_{1} a_{100} \times d} = 99$

$(I V)$ Number of triangles $=^{11} C_{3} -^{8} C_{3} = 109$

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Similar questions

\begin{matrix} List- I & List-II (I) & Number of integral solutions of & (P) & 132 x + y + z = 1, x \geq - 4, y \geq - 4, z \geq - 4 is less than (Q) & 99 (I I) & Greatest term in the expression of \frac{4}{3 \sqrt{2}} {(1 + \frac{1}{\sqrt{2}})}^{12} is (R) & 120 (I I I) & If a_{1}, a_{2}, a_{3} . . . a_{100} are in H.P. then value of \sum_{i = 1}^{99} \frac{a_{i} a_{i + 1}}{a_{1} a_{100}} is - (S) & 100 (I V) & If 8 points out of 11 are in same straight line then number of triangles formed is less then & (T) & 125 \end{matrix}

Which of the following is only INCORRECT combination?

Q. Consider the curve

sin x + sin y = 1

, lying in the first quadrant , then

\begin{matrix} List- I & List-II (I) & lim x \to π / 2 \frac{d^{2} y}{d x^{2}} = & (P) & 0 (II) & lim x \to 0^{+} x^{3 / 2} \frac{d^{2} y}{d x^{2}} = & (Q) & 1 (III) & lim x \to 0^{+} x^{2} \frac{d^{2} y}{d x^{2}} = & (R) & \frac{1}{\sqrt{2}} (IV) & lim x \to π / 2 \frac{d y}{d x} = & (S) & \frac{1}{2 \sqrt{2}} (T) & \sqrt{2} (U) & 3 \end{matrix}

Which of the following is the only CORRECT combination?

\begin{matrix} List I & List II (I) \int_{0}^{2} \frac{x^{4} + 1}{(x^{2} + 2)^{\frac{3}{4}}} d x = & (P) \sqrt{3 \sqrt{2}} (II) \int_{1}^{3} (\sqrt{1 + (x - 1)^{3}} + \sqrt[3]{x^{2} - 1}) d x = & (Q) \sqrt{2 \sqrt{3}} (III) \int_{0}^{2} \sqrt[3]{x^{3} - 3 x^{2} + 8 x - 6} d x = & (R) 6 (IV) \int_{e}^{π} \frac{2^{\frac{x}{e}} - 2^{\frac{π}{x}}}{x} d x = & (S) 0 (T) \frac{π - e}{2} \end{matrix}

Which of the following is only CORRECT combination?

\begin{matrix} List- I & List-II (I) & If {log}_{| cos x |} (1 + sin x) = 2, x \in [- 2 π, 2 π], & (P) & - 2 then the value(s) of x is/are (II) & If max | | x - 4 | - | x - 3 | | = a and & (Q) & - 1 min | x - 4 | + | x - 3 | = b, then a + b = (III) & If f (x^{3} - 6 x^{2} + 11 x - 3) = x - 1, then & (R) & 0 f (3) = (IV) & If x cos θ - y sin θ + z = 1 + cos ϕ & (S) & 1 x sin θ + y cos θ + z = 1 - sin ϕ x cos (θ + ϕ) - y sin (θ + ϕ) = z, then x^{2} + y^{2} + z^{2} = (T) & 2 (U) & 3 \end{matrix}

Which of the following is the only INCORRECT combination?

Q. A bag contains some white and black balls, all combinations being equally likely. The total number of balls in the bag is

12

. Four balls are drawn at random from the bag without replacement.

\begin{matrix} List - I & List -II (I) & probability that all the four balls are black & (P) & \frac{14}{33} (II) & If the bag contains 10 black and 2 white balls, & (Q) & \frac{1}{5} then the probability that all four balls are black (III) & If all the 4 balls are black then the probability, & (R) & \frac{70}{429} that the bag contains 10 black balls (IV) & Probability that two balls are black and two are & (S) & \frac{13}{165} white is (T) & \frac{1}{3} (U) & \frac{2}{9} \end{matrix}

Which of the following is the only CORRECT combination?

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