MCQ
Match the integrals in Column $I$ with the values in Column $II$ and indicate your answer by darkening the appropriate bubbles in the $4 \times 4$ matrix given in the $ORS$.
Column $I$ Column $II$
$(A)$ $\int_{-1}^1 \frac{\mathrm{dx}}{1+\mathrm{x}^2}$ $(p)$ $\frac{1}{2} \log \left(\frac{2}{3}\right)$
$(B)$ $\int_0^1 \frac{\mathrm{dx}}{\sqrt{1-\mathrm{x}^2}}$ $(q)$ $2 \log \left(\frac{2}{3}\right)$
$(C)$ $\int_2^3 \frac{\mathrm{dx}}{1-\mathrm{x}^2}$ $(r)$ $\frac{\pi}{3}$
$(D)$ $\int_1^2 \frac{d x}{x \sqrt{x^2-1}}$ $(s)$ $\frac{\pi}{2}$
  • A
    $A-p\ \ B-r\ \ C-p\ \ D-s$
  • B
    $A-r\ \ B-s\ \ C-p\ \ D-q$
  • $A-s\ \ B-s\ \ C-p\ \ D-r$
  • D
    $A-q\ \ B-r\ \ C-q\ \ D-s$

Answer

Correct option: C.
$A-s\ \ B-s\ \ C-p\ \ D-r$
c
$(A)$. $\quad \int_{-1}^1 \frac{\mathrm{dx}}{1+\mathrm{x}^2}=\frac{\pi}{2}$

$(B)$. $\quad \int_0^1 \frac{\mathrm{dx}}{\sqrt{1-\mathrm{x}^2}}=\frac{\pi}{2}$

$(C)$. $\quad \int_2^3 \frac{\mathrm{dx}}{1-\mathrm{x}^2}=\frac{1}{2} \ln \frac{2}{3}$

$(D)$. $\quad \int_1^2 \frac{d x}{x \sqrt{x^2-1}}=\frac{\pi}{3}$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Similar questions

If the points $(2k, k), (k, 2k)$ and $(k, k)$ with $k > 0$ enclose a triangle of area $18$ square unit then centroid of triangle is equal to
The equation $13[{(x - 1)^2} + {(y - 2)^2}] = 3{(2x + 3y - 2)^2}$ represents
The straight line $2 x-3 y=1$ divides the circular region $x^2+y^2 \leq 6$ into two parts. If 

$S=\left\{\left(2, \frac{3}{4}\right),\left(\frac{5}{2}, \frac{3}{4}\right),\left(\frac{1}{4},-\frac{1}{4}\right),\left(\frac{1}{8}, \frac{1}{4}\right)\right\},$ then the number of point(s) in $S$ lying inside the smaller part is

The middle term in the expansion of ${(1 + x)^{2n}}$ is
The number of $5 -$tuples $(a, b, c, d, e)$ of positive integers such that

$I.$ $a, b, c, d, e$ are the measures of angles of a convex pentagon in degrees

$II$. $a \leq b \leq c \leq d \leq e$

$III.$ $a, b, c, d, e$ are in arithmetic progression is

For any $3 \times 3$ matrix $M$, let $| M |$ denote the determinant of $M$. Let

$E=\left[\begin{array}{ccc}1 & 2 & 3 \\ 2 & 3 & 4 \\ 8 & 13 & 18\end{array}\right], P=\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0\end{array}\right]$ and $F=\left[\begin{array}{ccc}1 & 3 & 2 \\ 8 & 18 & 13 \\ 2 & 4 & 3\end{array}\right]$

If $Q$ is a nonsingular matrix of order $3 \times 3$, then which of the following statements is (are) $TRUE$?

$(A)$F $=P E P$ and $P^2=\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right]$

$(B)$ $\left| EQ + PFQ ^{-1}\right|=| EQ |+\left| PFQ ^{-1}\right|$

$(C)$ $\left|( EF )^3\right|>| EF |^2$

$(D)$ Sum of the diagonal entries of $P ^{-1} EP + F$ is equal to the sum of diagonal entries of $E + P ^{-1} FP$

$1 - 2{\sin ^2}\left( {\frac{\pi }{4} + \theta } \right) = $
$S=\{(x, y, z): x, y, z \in Z, x+2 y+3 z=42$ $\mathrm{x}, \mathrm{y}, \mathrm{z} \geq 0\}$ ...........
If all the words (with or without meaning) having five letters, formed using the letters of the word $SMALL$ and arranged as in a dictionary; then the position of the word $SMALL$ is :
${\cos ^{ - 1}}\frac{1}{2} + 2{\sin ^{ - 1}}\frac{1}{2}$ is equal to