Let a, b, c be non-zero real numbers such that ; _0^1 (1 + cos^8x… - Vidyadip

MCQ

Let $a, b, c$ be non-zero real numbers such that ; $\int\limits_0^1 {} (1 + cos^8x) (ax^2 + bx + c) dx$ $= \int\limits_0^2 {} (1 + cos^8x) (ax^2 + bx + c) dx$ , then the quadratic equation $ax^2 + bx + c = 0$ has :

A
no root in $(0, 2)$
✓
atleast one root in $(0, 2)$
C
a double root in $(0, 2)$
D
none

✓

Answer

Correct option: B.

atleast one root in $(0, 2)$

b
$\int\limits_0^1 {(1 + {{\cos }^8}x)\,\,f(x)\,\,dx} $

$=\int\limits_0^2 {(1 + {{\cos }^8}x)\,\,f(x)\,\,dx} $

$=\int\limits_0^1 {(1 + {{\cos }^8}x)\,\,f(x)\,\,dx} \,\, + \,\,\int\limits_1^2 {(1 + {{\cos }^8}x)\,\,f(x)\,\,dx} $
Hence $\,\int\limits_1^2 {(1 + {{\cos }^8}x)\,\,f(x)\,\,dx} = 0$
$\Rightarrow (1+cos^8x) f(x) = 0$ at least once in $(1,2)$
but $1 + \cos^8x \neq 0$
$\Rightarrow f(x) = ax^2 + bx + c$ vanishes at least once in $(1,2)$

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 7.2 definite integral→7.2 definite integral — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

If ${2^x} + {2^y} = {2^{x + y}}$, then ${{dy} \over {dx}} = $

The function $(x-\sin x)$ decreases for

The binary operation × defined on N by a × b = a + b + ab for all a, b ∈ N is:

Commutative only.
Associative only.
Both commutative and associative.
None of these.

If $x =1$ is a critical point of the function $f(x)=\left(3 x^{2}+a x-2-a\right) e^{x},$ then

The value of the integral $\int_0^\pi(1-|\sin 8 x|) d x$ is

The area (in sq. units) of the region $A=\{(x, y):(x-1)[x] \leq y \leq 2 \sqrt{x}, 0 \leq x \leq 2\}$ where $[t]$ denotes the greatest integer function, is

If the random variable X has the following distribution:

X:	0	1	2	3	4	5	6	7	8
P(X):	a	3a	5a	7a	9a	11a	13a	15a	17a

then the value of a is:

$\frac{7}{81}$
$\frac{5}{81}$
$\frac{2}{81}$
$\frac{1}{81}$

The optimal value of the objective function is attained at the points:

On x - axis
On y - axis
Which are corner points of the feasible region
None of these

$\int_{-1}^1 \frac{|x-2|}{x-2} d x, x \neq 2$ is equal to

$\int_{}^{} {\frac{x}{{{x^4} + {x^2} + 1}}dx} $ equal to