Maths

${\displaystyle {\int }_0^{2/3}\frac{dx}{4+9x^2}}equals={\displaystyle {\int }_0^{2/3}\frac{dx}{2^2+{\left(3x\right)}^2}}$

 $=\frac{1}{2}*\frac{{\left[ta{n}^{-1}\frac{3x}{2}\right]}_0^{2/3}}{3}\\ =\frac{1}{6}\left[ta{n}^{-1}\frac{3}{2}*\frac{2}{3}-ta{n}^{-1}\frac{3}{2}*0\right]\\ =\frac{1}{6}\left[ta{n}^{-1}\left(1\right)-tan\left(0\right)\right]\\ =\frac{1}{6}[\pi /4-0]\\ =\pi /24$

? Option (c) is correct.

This is a  Mathematical Reasoning Solutions Type Questions as classified in NCERT Exemplar

The given statement can be written in five different ways: –

(i) A triangle is equiangular implies that it is an obtuse angled triangle.

(ii) A triangle is equiangular only if the triangle is an obtuse angled triangle.

(iii) For a triangle to be equiangular, it is necessary that the triangle is an obtuse angled triangle.

(iv) For a triangle to be an obtuse angled triangle, it is sufficient that the triangle is equiangular.

(v) If a triangle is an obtuse angled triangle, then the triangle is not equiangular.

This is a  Mathematical Reasoning Solutions Type Questions as classified in NCERT Exemplar

The compound statement with 'And' is as follows 25 is a multiple of 5 and 8.

This is false statement because 25 is not a multiple of 8.

The compound statement with 'Or' is as follows 25 is a multiple of 5 or 8.

This is true statement because 25 is not a multiple of 8 but it is a multiple of 5.

This is a  Mathematical Reasoning Solutions Type Questions as classified in NCERT Exemplar

(i) You watch television if and only if your mind is free.

(ii) You get an A grade if and only if you do all the homework regularly.

(iii) A quadrilateral is equiangular if and only if it is a rectangle.

This is a  Mathematical Reasoning Solutions Type Questions as classified in NCERT Exemplar

(i) The statement p in the form 'if then' is as follows.

If you log on to the server, then you have password.

(ii) The statement q in the form 'if then' is as follows.

If it rains, then there is a traffic jam.

(iii) The statement r in the form 'if them' is as follows.

If you can access the website, then you pay a subscription fee.

This is a  Mathematical Reasoning Solutions Type Questions as classified in NCERT Exemplar

(i) The negation of statement p is as follows.

There exists a positive real number x, such that x – 1 is not positive.

(ii) The negation of statement q is

There exists a CAT that does not scratch.

(iii) The negation of statement r is

There exist a real number x, such that neither x>1 nor x<1.

(iv) The negation of statement s is as follow.

There does not exist a number x, such that 0

This is a  Mathematical Reasoning Solutions Type Questions as classified in NCERT Exemplar

(i) False: As per the definition of a chord, it should intersect the circle at two distinct points.

(ii) False: The centre of a circle only bisects the diameter, which is the chord of the circle.

(iii) True: The equation of an ellipse is $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$

If we put a = b = 1, then

x² + y² = 1, which is an equation of a circle

(iv) True: Given x > y. Multiplying by –1, –x < y.

(v) False: Since | | is a prime number, therefore √11$$ is irrational.

This is a  Mathematical Reasoning Solutions Type Questions as classified in NCERT Exemplar

(ii) q: The equation x² – 1 = 0 does not have a root lying between 0 and 2.

(i) Let us consider an equilateral triangle ABC where $\angle A=\angle B=\angle C=60°,$ then triangle ABC is not obtuse angled since all its angles are equal

$\therefore$ The statement p is not true.

(ii) 1 lies between 0 and 2. Also, 1 satisfies the equation x² – 1 = 0

$\left[?1^2-1=1-1=0\right]$

$\therefore$ The statement q is not true.