Download App

STD 12 - 7.2 definite integral — Mathematics STD 12 Science — Question

CBSE BoardEnglish MediumSTD 12 ScienceMathematicsSTD 12 - 7.2 definite integral1 Mark
MCQ
If $\int\limits_{a}^{x}{ty(t)dt={{x}^{2}}+y(x)}$ then $y$ as a function of $x$ is
  • ✓
    $y = 2 -(2 + a^2) {e^{\frac{{{x^2} - {a^2}}}{2}}}$
  • B
    $y = 1 -(2 + a^2) {e^{\frac{{{x^2} - {a^2}}}{2}}}$
  • C
    $y = 2 -(1 + a^2) {e^{\frac{{{x^2} - {a^2}}}{2}}}$
  • D
    none
✓

Answer

Correct option: A.
$y = 2 -(2 + a^2) {e^{\frac{{{x^2} - {a^2}}}{2}}}$
a
Given $\int_{a}^{x} t y(t) d t=x^{2}+y(x)$

$\Rightarrow x y=2 x+\frac{d y}{d x}$

$\Rightarrow x(y-2)=\frac{d y}{d x}$

$\Rightarrow \int x d x=\int \frac{d y}{y-2}$

$\Rightarrow \frac{x^{2}}{2}=\ln |y-2|+\ln c$

$\Rightarrow e^{\frac{x^{2}}{2}}=c(y-2) \quad-(2)$

Now, put $x=a$ in eqn ( 1$)$

$\Rightarrow y=-a^{2}$

So, by eqn ( 2$)$ $\therefore e^{\frac{a^{2}}{2}}=c\left(-a^{2}-2\right)$

$\Rightarrow c=-\frac{e^{\frac{a^{2}}{2}}}{\left(a^{2}+2\right)}$

Put this value in (2) $\therefore e^{\frac{x^{2}}{2}}=-\frac{e^{\frac{a^{2}}{2}}}{\left(a^{2}+2\right)}(y-2)$

$\Rightarrow-y+2=\left(a^{2}+2\right) e^{\frac{x^{2}-a^{2}}{2}}$

$\Rightarrow y=2-\left(2+a^{2}\right) e^{\frac{x^{2}-a^{2}}{2}}$

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

Explore more

More "M.C.Q (1 Marks)" from STD 12 - 7.2 definite integral→STD 12 - 7.2 definite integral — all question types→Mathematics STD 12 Science question papers→CBSE Board STD 12 Science question papers→CBSE Board question paper generator→

Similar questions

The value of $\int\limits_0^{{{\sin }^2}x} {{{\sin }^{ - 1}}\sqrt t \,dt}  + \int\limits_0^{{{\cos }^2}x} {{{\cos }^{ - 1}}\sqrt t \,dt} $ in $(0, \pi/2)$ is-
→
The value of $ 'a' $ so that the volume of parallelopiped formed by $i + aj + k,j + a\,k$ and $a\,i + k$ becomes minimum is
→
If $A$ is a square matrix such that $A ^2=1,$ then $A ^{-1}$ is equal to:
→
Let $\Omega$ be the set of all $3 \times 3$ symmetric matrices all of whose entries are either $0$ or $1$ . Five of these entries are $1$ and four of them are $0$ .

$1.$ The number of matrices in $\Omega$ is

$(A)$ $12$ $(B)$ $6$ $(C)$ $9$ $(D)$ $3$

$2.$ The number of matrices $A$ in $\Omega$ for which the system of linear equations

$A\left[\begin{array}{l}x \\ y \\ z\end{array}\right]=\left[\begin{array}{l}1 \\ 0 \\ 0\end{array}\right]$ has a unique solution, is

$(A)$ less than $4$

$(B)$ at least $4$ but less than $7$

$(C)$ at least $7$ but less than $10$

$(D)$ at least $10$

$3.$ The number of matrices $A$ in $\Omega$ for which the system of linear equations

$A\left[\begin{array}{l}x \\ y \\ z\end{array}\right]=\left[\begin{array}{l}1 \\ 0 \\ 0\end{array}\right]$ is inconsistent, is

$(A)$ $0$ $(B)$ more than $2$ $(C)$ $2$ $(D)$ $1$

→
Choose the correct answer from the given four options: If $y = x^4 - 10$ and if $x$ changes from $2$ to $1.99,$ what is the change in $y:$
→
The function $f(x) = x + \sin x$ has
→
The probability distribution of a discrete random variable $X$ is given below :
$X$2345
$P(X)$$\frac{5}{k}$$\frac{7}{k}$$\frac{9}{k}$$\frac{11}{k}$
The value of $k$ is
→
The derivative of the function $\cot^{-1}\Big|(\cos2\text{x})^{\frac{1}{2}}\Big|\text{ at }\text{x}=\frac{\pi}{6}$ is:
→
The function, $f(x)=(3 x-7) x^{2 / 3}, x \in R,$ is increasing for all $x$ lying in
→
A parallelopiped is formed by planes drawn through the point (2, 3, 5) and (5, 9, 7) parallel to the coordinate planes. The length of a diagonal of the parallelopiped is:
→