Integral Calculus & Area · JEE Main previous year questions

2012 11 questions

1. Q12012 · 26 May 2012 · Shift 1

   If $a,b,c\in \mathrm{R}$ and 1 is a root of equation $a{x}^2+bx+c=0$, then the curve $y=4a{x}^2+3bx+2c$, $a\ne 0$ intersect $x$-axis at

   1. A.two distinct points whose coordinates are always rational numbers
   2. B.no point
   3. C.exactly two distinct points
   4. D.exactly one point
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   Answer: C

2. Q172012 · 26 May 2012 · Shift 1

   If $[x]$ is the greatest integer $\le x$, then the value of the integral ${\int }_{-0.9}^{0.9}\left([x^2]+\log \left(\frac{2-x}{2+x}\right)\right)dx$ is

   1. A.$0.486$
   2. B.$0.243$
   3. C.$1.8$
   4. D.0
   Show answer

   Answer: D

3. Q12012 · 12 May 2012 · Shift 1

   If $a,b,c,d$ and $p$ are distinct real numbers such that $(a^2+b^2+c^2)p^2-2p(ab+bc+cd)+(b^2+c^2+d^2)\le 0$, then

   1. A.$a,b,c,d$ are in A.P.
   2. B.$ab=cd$
   3. C.$ac=bd$
   4. D.$a,b,c,d$ are in G.P.
   Show answer

   Answer: D

4. Q162012 · 12 May 2012 · Shift 1

   The integral of $\frac{x^2-x}{x^3-x^2+x-1}$ w.r.t. $x$ is

   1. A.$\frac{1}{2}\log (x^2+1)+c$
   2. B.$\frac{1}{2}\log |x^2-1|+c$
   3. C.$\log (x^2+1)+c$
   4. D.$\log |x^2-1|+c$
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   Answer: A

5. Q172012 · 12 May 2012 · Shift 1

   If $\frac{d}{dx}G(x)=\frac{e^{\tan x}}{x},x\in (0,\pi /2)$, then ${\int }_{1/4}^{1/2}\frac{2}{x}\cdot e^{\tan (\pi x^2)}dx$ is equal to

   1. A.$G(\pi /4)-G(\pi /16)$
   2. B.$2[G(\pi /4)-G(\pi /16)]$
   3. C.$\pi [G(1/2)-G(1/4)]$
   4. D.$G(1/\sqrt{2})-G(1/2)$
   Show answer

   Answer: A

6. Q182012 · 12 May 2012 · Shift 1

   The area enclosed by the curves $y=x^2,y=x^3,x=0$ and $x=p$, where $p>1$, is $1/6$. Then $p$ equals

   1. A.$8/3$
   2. B.$16/3$
   3. C.2
   4. D.$4/3$
   Show answer

   Answer: D

7. Q192012 · 12 May 2012 · Shift 1

   If a straight line $y-x=2$ divides the region $x^2+y^2\le 4$ into two parts, then the ratio of the area of the smaller part to the area of the greater part is

   1. A.$3\pi -8:\pi +8$
   2. B.$\pi -3:3\pi +3$
   3. C.$3\pi -4:\pi +4$
   4. D.$\pi -2:3\pi +2$
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   Answer: D

8. Q202012 · 12 May 2012 · Shift 1

   Statement 1: The degrees of the differential equations $\frac{dy}{dx}+y^2=x$ and $\frac{d^{2}y}{d{x}^2}+y=\sin x$ are equal. Statement 2: The degree of a differential equation, when it is a polynomial equation in derivatives, is the highest positive integral power of the highest order derivative involved in the differential equation, otherwise degree is not defined.

   1. A.Statement 1 is true, Statement 2 is true; Statement 2 is not a correct explanation of Statement 1.
   2. B.Statement 1 is false, Statement 2 is true.
   3. C.Statement 1 is true, Statement 2 is false.
   4. D.Statement 1 is true, Statement 2 is true; Statement 2 is a correct explanation of Statement 1.
   Show answer

   Answer: D

9. Q272012 · 12 May 2012 · Shift 1

   If $\left|\begin{array}{ccc}-2a& a+b& a+c\\ b+a& -2b& b+c\\ c+a& b+c& -2c\end{array}\right|=\alpha (a+b)(b+c)(c+a)\ne 0$ then $\alpha$ is equal to

   1. A.$a+b+c$
   2. B.$abc$
   3. C.4
   4. D.1
   Show answer

   Answer: C

10. Q172012 · 7 May 2012 · Shift 1

    Let $f(x)$ be an indefinite integral of ${\cos }^{3}x$. Statement 1: $f(x)$ is a periodic function of period $\pi$. Statement 2: ${\cos }^{3}x$ is a periodic function.

    1. A.Statement 1 is true, Statement 2 is false.
    2. B.Both the Statements are true, but Statement 2 is not the correct explanation of Statement 1.
    3. C.Both the Statements are true, and Statement 2 is correct explanation of Statement 1.
    4. D.Statement 1 is false, Statement 2 is true.
    Show answer

    Answer: D

11. Q182012 · 7 May 2012 · Shift 1

    If ${\int }_e^{x}tf(t)dt=\sin x-x\cos x-\frac{x^2}{2}$, for all $x\in R-\{0\}$, then the value of $f\left(\frac{\pi }{6}\right)$ is

    1. A.1/2
    2. B.1
    3. C.0
    4. D.$-1/2$
    Show answer

    Answer: D