MCQ

MCQ 11 Mark

If $4a + 2b + c = 0$, then the equation $3ax^2+ 2bx + c = 0$ has atleast one real root lying in the interval:

A
$(0, 1)$
B
$(1, 2)$
✓
$(0, 2)$
D
None of these.

Answer

Correct option: C.

$(0, 2)$

Let, $f(x) = ax^3+ bx^2 + cx + d$
$f(0) = d$
$f(2) = 8a + 4b + 2c + d$
$= 2(4a + 2b + c) + d$
$= 2 \times 0 + d$
$= 0$
$f $ is continuous and differentiable on $(0, 2)$
$f(0) = f(2)$
Using Rolle's theorem,
$f'(x) = 0$ for $(0, 2)$
$3ax^2+ 2bx + c = 0$
$f(x)$ has atleast one root in the interval $(0, 2).$
Hence $f'(x)$ must have root in the interval $(0, 2).$

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MCQ 21 Mark

For the function $\text{f}(\text{x})=\text{x}+1\text{x},\text{x}\in[1,3],$ the value of $c$ for the Lagrange's mean value theorem is:

A
$1$
✓
$\sqrt3$
C
$2$
D
none of these

Answer

Correct option: B.

$\sqrt3$

We have
$\text{f}(\text{x})=\text{x}+\frac{1}{\text{x}}=\frac{\text{x}^2+1}{\text{x}}$
Clearly, $f(x)$ is continuous on $[1, 3]$ and derivable on $(1, 3).$
Thus, both the conditions of Lagrange's theorem is satisfied.
Concequently there exists $\text{c}\in(1,3)$ such that
$\text{f}'(\text{c})=\frac{\text{f}(3)-\text{f}(1)}{3-1}=\frac{\text{f}(3)-\text{f}(1)}{2}$
Now, $\text{f}(\text{x})=\frac{\text{x}^2+1}{\text{x}}$
$\text{f}'(\text{x})=\frac{\text{x}^2-1}{\text{x}^2},\text{f}(1)=2,\text{f}(3)=\frac{10}{3}$
$\therefore\ \text{f}'(\text{x})=\frac{\text{f}(3)-\text{f}(1)}{2}$
$\Rightarrow\frac{\text{x}^2-1}{\text{x}^2}=\frac{4}{6}$
$\Rightarrow\frac{\text{x}^2-1}{\text{x}^2}=\frac{2}{3}$
$\Rightarrow3\text{x}^2-3=2\text{x}^2$
$\Rightarrow\text{x}=\pm\sqrt3$
Thus, $\text{c}=\sqrt3\in(1,3)$ such that $\text{f}'(\text{c})=\frac{\text{f}(3)-\text{f}(1)}{3-1}.$

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MCQ 31 Mark

The value of $c$ in Rolle's theorem for the function $\text{f}(\text{x})=\frac{\text{x}(\text{x}+1)}{\text{e}^{\text{x}}}$ defined on $[-1, 0]$ is:

A
$0.5$
B
$\frac{1+\sqrt5}{2}$
✓
$\frac{1-\sqrt5}{2}$
D
$-0.5$

Answer

Correct option: C.

$\frac{1-\sqrt5}{2}$

$\frac{1-\sqrt5}{2}$
$\text{f}(\text{x})=\frac{\text{x}(\text{x}+1)}{\text{e}^{\text{x}}}$ defined on $[-1, 0]$
$\Rightarrow f(-1) = 0$ also $f(0) = 0$
Now, $f(x) = e^{-x}(x^2 + x)$
$\Rightarrow f'(x) = e^{-x}(2x + 1) - (x^2 + x)e^{-x}$
$\Rightarrow f'(x) = e^{-x}(2x + 1 - x^2 + x)$
$\Rightarrow f'(x) = e^{-x}(-x^2 + x - 1)$
$\Rightarrow f'(x) = 0$
$\Rightarrow e^{-x}(-x^2 + x - 1) = 0$
$\Rightarrow -x^2 + x - 1 = 0$
$\Rightarrow x^2 - x + 1 = 0$
$\Rightarrow\text{x}=\frac{1\pm\sqrt5}{2}$
As, $\text{x}\in[-1,0]$
$\text{x}=\frac{1-\sqrt5}{2}$

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MCQ 41 Mark

When the tangent to the curve $\text{y}=\text{x}\log\text{x}$ is parallel to the chord joining the points $(1, 0)$ and $(e, e),$ the value of $x$ is:

✓
$\text{e}^{\frac{1}{1}-\text{e}}$
B
$\text{e}^{(\text{e}-1)(2\text{e}-1)}$
C
$\text{e}^{\frac{2\text{e}-1}{\text{e}-1}}$
D
$\frac{\text{e}-1}{\text{e}}$

Answer

Correct option: A.

$\text{e}^{\frac{1}{1}-\text{e}}$

Given:
$\text{y}=\text{f}(\text{x})=\text{x}\log\text{x}$
Differentiating the given function with respect to $x,$ we get
$\text{f}'(\text{x})=1+\log\text{x}$
$\Rightarrow$ Slope of the tangent to the curve $=1+\log\text{x}$
Also,
Slope of the chord joining the points $(1, 0)$ and $(e, e), (\text{m})=\frac{\text{e}}{\text{e}-1}$
The tangent to the curve is parallel to chord joining the points $(1, 0)$ and $(e, e).$

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MCQ 51 Mark

The value of $c$ in Rolle's theorem when $f(x) = 2x^3 - 5x^2 - 4x + 3$, $\text{x}\in\Big[\frac{1}{3},3\Big]$ is$:$

✓
$2$
B
$-\frac{1}{3}$
C
$-2$
D
$\frac{2}{3}$

Answer

Correct option: A.

$2$

$f(x) = 2x^3 - 5x^2 - 4x + 3$
Differentiating the given function with respect to $x,$ we get
$f'(x) = 6x^2 - 10x - 4$
$\Rightarrow f'(c) = 6c^2 - 10c - 4$
$\therefore$ $f'(c) =0$
$\Rightarrow 3c^2 - 5c - 2 = 0$
$\Rightarrow 3c^2 - 6c + c - 2 = 0$
$\Rightarrow 3c(c - 2) + c - 2 = 0$
$\Rightarrow (3c + 1)(c - 2) = 0$
$\Rightarrow\text{c}=2, \frac{-1}{3}$
$\therefore\ \text{c}=2\in\Big(\frac{1}{3},3\Big)$
Thus, $\text{c}=2\in\Big(\frac{1}{3},3\Big)$ for which Rolle's theorem holds.
Hence, the required value of $c$ is $2$.

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MCQ 61 Mark

Function $\text{f}(\text{x})=\cos\text{x}-2\lambda\text{x}$ is monotonic decreasing when:

✓
$\lambda>\frac{1}{2}$
B
$\lambda<\frac{1}{2}$
C
$\lambda<2$
D
$\lambda>2$

Answer

Correct option: A.

$\lambda>\frac{1}{2}$

$\text{f}(\text{x})=\cos\text{x}-2\lambda\text{x}$
$\text{f}'(\text{x})=-\sin\text{x}-2\lambda$
For $f(x)$ to be decreasing, we must have
$\text{f}'(\text{x})<0$
$\Rightarrow-\sin\text{x}-2\lambda<0$
$\Rightarrow\sin\text{x}+2\lambda>0$
$\Rightarrow2\lambda>-\sin\text{x}$
We know that the maximum value of $-\sin\text{x}$ is $1.$
$\Rightarrow2\lambda>1$
$\Rightarrow\lambda>\frac{1}{2}$

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MCQ 71 Mark

If from Lagrange's mean value theorem, we have
$\text{f}'(\text{x}_1)=\frac{\text{f}(\text{b})-\text{f}(\text{a})}{\text{b}-\text{a}},$ then:

A
$\text{a}<\text{x}_1\leq\text{b}$
B
$\text{a}\leq\text{x}_1<\text{b}$
✓
$\text{a}<\text{x}_1<\text{b}$
D
$\text{a}\leq\text{x}_1\leq\text{b}$

Answer

Correct option: C.

$\text{a}<\text{x}_1<\text{b}$

We have
$\text{f}(\text{x})=\text{x}+\frac{1}{\text{x}}=\frac{\text{x}^2+1}{\text{x}}$
In the Lagrange's mean value theorem, $\text{c}\in(\text{a},\text{b})$ such that $\text{f}'(\text{c})=\frac{\text{f}(\text{b})-\text{f}(\text{a})}{\text{b}-\text{a}}$
So, if there is $x_1$ such that $\text{f}'(\text{x}_1)=\frac{\text{f}(\text{b})-\text{f}(\text{a})}{\text{b}-\text{a}},$ then $\text{x}_1\in(\text{a},\text{b})$
$\Rightarrow\text{a}<\text{x}_1<\text{b}$

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MCQ 81 Mark

If the polynomial equation $\text{a}_0\text{x}^{\text{n}}+\text{a}_{\text{n}-1}\text{x}^{\text{n}-1}+\text{a}_{\text{n}-2}\text{x}^{\text{n}-2}+...\text{a}_2\text{x}^2+\text{a}_1\text{x}+\text{a}_0=0$ n positive integer,has two different real roots $\alpha$ and $\beta,$ then between $\alpha$ and $\beta,$ the equation $\text{n}\text{a}_{\text{n}}\text{x}^{\text{n}-1}+(\text{n}-1)\text{a}_{\text{n}-1}\text{x}^{\text{n}-2}+...+\text{a}_1=0$ has:

A
Exactly one root.
B
Almost one root.
✓
At least one root.
D
No root.

Answer

Correct option: C.

At least one root.

We observe that, $\text{n}\text{a}_{\text{n}}\text{x}^{\text{n}-1}+(\text{n}-1)\text{a}_{\text{n}-1}\text{x}^{\text{n}-2}+...+\text{a}_1=0$ is the derivative of the polynomial $\text{a}_{\text{n}}\text{x}^{\text{n}}+\text{a}_{\text{n}-1}\text{x}^{\text{n}-1}+\text{a}_{\text{n}-2}\text{x}^{\text{n}-2}+...\text{a}_2\text{x}^2+\text{a}_1\text{x}+\text{a}_0=0$Polynomial function is continuous everywhere in $R$ and concequently derivative in $R.$
Therefore, $\text{a}_{\text{n}}\text{x}^{\text{n}}+\text{a}_{\text{n}-1}\text{x}^{\text{n}-1}+\text{a}_{\text{n}-2}\text{x}^{\text{n}-2}+...\text{a}_2\text{x}^2+\text{a}_1\text{x}+\text{a}_0$ is continuous on $\alpha,\beta$ and derivative on $\alpha,\beta.$
Hence, it is satisfies the both the conditions of Rolle's theorem.
By algebric interpretation of Roll's theorem, we know that between any two roots of a function $f(x),$
there exists atleast one root of its derivative.
Hence, the equation $\text{n}\text{a}_{\text{n}}\text{x}^{\text{n}-1}+(\text{n}-1)\text{a}_{\text{n}-1}\text{x}^{\text{n}-2}+...+\text{a}_1=0$ will have atleast one root between $\alpha$ and $\beta.$

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MCQ 91 Mark

The value of c in Lagrange's mean value theorem for the function $f(x) = x(x - 2)$ when $\text{x}\in[1,2]$ is:

A
$1$
B
$\frac{1}{2}$
C
$\frac{2}{3}$
✓
$\frac{3}{2}$

Answer

Correct option: D.

$\frac{3}{2}$

We have
$f(x) = x(x - 2)$
It can be rewritten as $f(x) = x^2 - 2x$
We know that a polynomial function is everywhere continuous and differentiable.
Since, $f(x)$ is polynomial, it is continuous on $[1, 2]$ and differentiable on $[1, 2]$.
So, there must exist at least one real number $\text{c}\in(1,2)$ such that
$\text{f}'(\text{c})=\frac{\text{f}(2)-\text{f}(1)}{2-1}$
$=\frac{\text{f}(2)-\text{f}(1)}{1}$
Now, $f(x) = x^2 - 2x$
$\Rightarrow f'(x) = 2x - 2$
and $f(1) = -1, f(2) = 0$
$\therefore\ \text{f}'(\text{x})=\frac{\text{f}(2)-\text{f}(1)}{2-1}$
$\Rightarrow\text{f}(\text{x})=\frac{0+1}{1}$
$\Rightarrow2\text{x}-2=1$
$\Rightarrow\text{x}=\frac{3}{2}$
$\therefore\ \text{c}=\frac{3}{2}\in(1,2)$

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MCQ 101 Mark

The value of $c$ in Rolle's theorem for the function $f(x) = x^3 - 3x$ in the interval $\big[0,\sqrt3\big]$ is:

✓
$1$
B
$-1$
C
$\frac{3}{2}$
D
$\frac{1}{3}$

Answer

Correct option: A.

$1$

$f(x) = x^3 - 3x$ in the interval $\big[0,\sqrt3\big]$
$\text{f}(0)=0$ and $\text{f}\big(\sqrt3\big)=0$
$f'(x) = 3x^2 - 3$
$f'(c) = 3c^2 - 3$
$f'(c) = 0$
$3c^2 - 3 = 0$
$3c^2= 3$
$c^2= 1$
$\text{c}=\pm1$
$\text{x}\in\big[0,\sqrt3\big]$
Hence, $x = 1$

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MCQ 111 Mark

If $\text{f}(\text{x})=\text{e}^{\text{x}}\sin\text{x}$ in $[0,\pi],$ then $c$ in Rolle's theorem is:

A
$\frac{\pi}{6}$
B
$\frac{\pi}{4}$
C
$\frac{\pi}{2}$
✓
$\frac{3\pi}{4}$

Answer

Correct option: D.

$\frac{3\pi}{4}$

$\text{f}(\text{x})=\text{e}^{\text{x}}\sin\text{x}$
$\text{f}'(\text{x})=\text{e}^{\text{x}}\cos\text{x}+\text{e}^{\text{x}}\sin\text{x}$
$\text{f}'(\text{c})=0$
$\text{e}^\text{c}(\cos\text{c}+\sin\text{c})=0$
$\cos\text{c}+\sin\text{c}=0$
$\cos\text{c}=-\sin\text{c}$
$\tan\text{c}=-1$
$\text{c}=\frac{3\pi}{4}\in(0,\pi)$

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MCQ 121 Mark

Rolle's theorem is applicable in case of $\phi(\text{x})=\text{a}^{\sin\text{x}},\text{a}>\text{a}$ in:

A
Any interval.
✓
Any interval $[0,\pi]$
C
Any interval $\Big[0,\frac{\pi}{2}\Big]$
D
None of these.

Answer

Correct option: B.

Any interval $[0,\pi]$

$\phi(\text{x})$ is continuous and differentiable function then using statement of Roll's theorem $f(a) = f(b).$
Hence, here $\sin0=0$ also $\sin\pi=0.$

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