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New answer posted

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Mathematical Reasoning Maths NCERT Exemplar Solutions Class 11th Chapter Fourteen

Write down the converse of the following statements:

i. If a rectangle $R$ is a square, then $R$ is a rhombus.

ii. If today is Monday, then tomorrow is Tuesday.

iii. If you go to Agra, then you must visit the Taj Mahal.

iv. If the sum of squares of two sides of a triangle is equal to the square of the third side of the triangle, then the triangle is right-angled.

v. If all three angles of a triangle are equal, then the triangle is equilateral.

vi. If $x : y = 3 : 2$ , then $2 x = 3 y$ .

vii. If $S$ is a cyclic quadrilateral, then the opposite angles of $S$ are supplementary.

viii. If $x$ is zero, then

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Write down the converse of the following statements:

i. If a rectangle $R$ is a square, then $R$ is a rhombus.

ii. If today is Monday, then tomorrow is Tuesday.

iii. If you go to Agra, then you must visit the Taj Mahal.

iv. If the sum of squares of two sides of a triangle is equal to the square of the third side of the triangle, then the triangle is right-angled.

v. If all three angles of a triangle are equal, then the triangle is equilateral.

vi. If $x : y = 3 : 2$ , then $2 x = 3 y$ .

vii. If $S$ is a cyclic quadrilateral, then the opposite angles of $S$ are supplementary.

viii. If $x$ is zero, then $x$ is neither positive nor negative.

ix. If two triangles are similar, then the ratio of their corresponding sides is equal.

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0 Follower 2 Views

Payal Gupta

Contributor-Level 10

This is a short answer type question as classified in NCERT Exemplar

$\begin{array}{l} (i) I f t h e r e c t a n g l e R i s r h o m b u s, t h e n i t i s a s q u a r e . \\ (i i) I f t o m o r r o w i s T u e s d a y, t h e n t o d a y i s M o n d a y . \\ (i i i) I f y o u m u s t v i s i t t h e T a j M a h a l, t h e n y o u g o t o A g r a . \\ (i v) I f t h e t r i a n g l e i s r i g h t t r i a n g l e, t h e n t h e s u m o f t h e s q u a r e s o f t w o s i d e s o f a t r i a n g l e \\ i s e q u a l t o t h e s q u a r e o f t h e t h i r d s i d e o f t h e t r i a n g l e . \\ (v) I f t h e t r i a n g l e i s e q u i l a t e r a l, t h e n a l l t h r e e a n g l e s o f a t r i a n g l e a r e e q u a l . \\ (v i) I f 2 x = 3 y t h e n x : y = 3 : 2 . \\ (v i i) I f t h e o p p o s i t e a n g l e s o f a q u a d r i l a t e r a l a r e s u p p l e m e n t a r y, t h e n S i s c y c l i c . \\ (v i i i) I f x i s n e i t h e r p o s i t i v e n o r n e g a t i v e t h e n x = 0 . \\ (i x) I f t h e r a t i o o f c o r r e s p o n d i n g s i d e s o f t w o t r i a n g l e s a r e e q u a l, t h e n t r i a n g l e s a r e s i m i l a r . \end{array}$

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New answer posted

a year ago

Mathematical Reasoning Maths NCERT Exemplar Solutions Class 11th Chapter Fourteen

Write down the contrapositive of the following statements:

i. If $x = y$ and $y = 3$ , then $x = 3$ .

ii. If $n$ is a natural number, then $n$ is an integer.

iii. If all three sides of a triangle are equal, then the triangle is equilateral.

iv. If $x$ and $y$ are negative integers, then $x y$ is positive.

v. If a natural number $n$ is divisible by 6, then $n$ is divisible by 2 and 3.

vi. If it snows, then the weather will be cold.

vii. If $x$ is a real number such that $0 < x < 1$ , then $x^{2} < 1$ .

0 Follower 4 Views

Payal Gupta

Contributor-Level 10

This is a short answer type question as classified in NCERT Exemplar

$\begin{array}{l} W e k n o w t h a t t h e c o n t r a p o s i t i v e o f p \to q i s (\sim q) \to (\sim p) \\ (i) I f x \neq 3, t h e n x \neq y o r y \neq 3 . \\ (i i) I f n i s n o t a n i n t e g e r, t h e n i t i s n o t a n a t u r a l n u m b e r . \\ (i i i) I f t h e t r i a n g l e i s n o t e q u i l a t e r a l, t h e n a l l t h r e e s i d e s o f t h e t r i a n g l e a r e n o t e q u a l . \\ (i v) I f x y i s n o t p o s i t i v e integer, t h e n x o r y i s n o t n e g a t i v e integer. \\ (v) I f n a t u r a l n u m b e r' n' i s n o t d i v i s i b l e b y 2 o r 3, t h e n n i s n o t d i v i s i b l e b y 6 . \\ (v i) T h e w e a t h e r w i l l n o t b e c o l d, i f i t d o e s n o t s n o w . \\ (v i i) I f x^{2} > 1 t h e n x i s n o t r e a l n u m b e r s u c h t h a t 0 < x < 1 . \end{array}$

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New answer posted

a year ago

Mathematical Reasoning Maths NCERT Exemplar Solutions Class 11th Chapter Fourteen

Form the biconditional statement $p \leftrightarrow q$ , where:

$i. p$ : The unit digit of an integer is zero.

It is divisible by 5.

$ii. p$ : A natural number $n$ is odd.

$q$ : Natural number $n$ is not divisible by 2.

$iii. p$ : A triangle is an equilateral triangle.

$q$ : All three sides of a triangle are equal.

0 Follower 10 Views

Payal Gupta

Contributor-Level 10

This is a short answer type question as classified in NCERT Exemplar

$\begin{array}{l} (i) p \leftrightarrow q : U n i t d i g i t o f a n integer i s z e r o i f a n d o n l y i f i t i s d i v i s i b l e b y 5 . \\ (i i) p \leftrightarrow q : A n a t u r a l n u m b e r i s o d d i f a n d o n l y i f i t i s n o t d i v i s i b l e b y 2 . \\ (i i i) p \leftrightarrow q : A t r i a n g l e i s a n e q u i l a t e r a l t r i a n g l e i f a n d o n l y i f a l l t h r e e s i d e s o f t r i a n g l e \\ a r e e q u a l . \end{array}$

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New answer posted

a year ago

Mathematical Reasoning Maths NCERT Exemplar Solutions Class 11th Chapter Fourteen

Rewrite each of the following statements in the form of conditional statements:

i. The square of an odd number is odd.

ii. You will get a sweet dish after the dinner.

iii. You will fail if you do not study.

iv. The unit digit of an integer is 0 or 5 if it is divisible by 5.

v. The square of a prime number is not prime.

vi. 2b = a + c if a, b, and c are in A.P.

0 Follower 2 Views

Payal Gupta

Contributor-Level 10

This is a short answer type question as classified in NCERT Exemplar

$\begin{array}{l} (i) I f p, t h e n q \Rightarrow I f t h e n u m b e r i s o d d, t h e n i t s s q u a r e i s o d d n u m b e r . \\ (i i) q i f p \Rightarrow I f t a k e t h e d i n n e r, t h e n y o u w i l l g e t s w e e t d i s h . \\ (i i i) p o n l y i f q \Rightarrow I f y o u d o n o t s t u d y, t h e n y o u w i l l f a i l . \\ (i v) p i s s u f f i c i e n t f o r q \Rightarrow I f a n integer i s d i v i s i b l e b y 5, t h e n i t s u n i t d i g i t s a r e 0 a n d 5 . \\ (v) q i s n e c e s s a r y f o r p \Rightarrow I f a n y n u m b e r i s p r i m e, t h e n i t s s q u a r e i s n o t p r i m e . \\ (v i) q i m p l i e s p \Rightarrow I f a, b, c a r e i n A . P t h e n 2 b = a + c . \end{array}$

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New answer posted

a year ago

Mathematical Reasoning Maths NCERT Exemplar Solutions Class 11th Chapter Fourteen

Write down the negation of the following compound statements:

i. All rational numbers are real and complex.

ii. All real numbers are rational or irrationals.

iii. x = 2 and x = 3 are roots of the quadratic equation x² – 5x + 6 = 0.

iv. A triangle has either 3 sides or 4 sides.

v. 35 is a prime number or a composite number.

vi. All prime integers are either even or odd.

vii. x is equal to either x or –x.

viii. 6 is divisible by 2 and 3.

0 Follower 3 Views

Payal Gupta

Contributor-Level 10

This is a short answer type question as classified in NCERT Exemplar

$\begin{array}{l} (i) p : A l l r a t i o n a l n u m b e r s a r e r e a l . \\ \sim p : A l l r a t i o n a l n u m b e r s a r e n o t r e a l . \\ q : A l l r a t i o n a l n u m b e r s a r e c o m p l e x . \\ \sim q : A l l r a t i o n a l n u m b e r s a r e n o t c o m p l e x . \\ \sim (p \land q) = (\sim p \lor \sim q) : A l l r a t i o n a l n u m b e r s a r e n e i t h e r r e a l n o r c o m p l e x . \\ (i i) p : A l l r e a l n u m b e r s a r e r a t i o n a l s . \\ q : A l l r e a l n u m b e r s a r e i r r a t i o n a l s . \\ T h e n e g a t i o n o f t h e a b o v e s t a t e m e n t i s \\ \sim (p \lor q) = (\sim p \land \sim q) : A l l r e a l n u m b e r s a r e n o t r a t i o n a l s a n d a l l r e a l n u m b e r s a r e \\ n o t i r r a t i o n a l s . \\ (i i i) p : x = 2 i s r o o t o f t h e e q u a t i o n x^{2} - 5 x + 6 = 0 . \\ q : x = 3 i s r o o t o f t h e e q u a t i o n x^{2} - 5 x + 6 = 0 . \\ T h e n e g a t i o n o f t h e a b o v e s t a t e m e n t i s \\ \sim (p \land q) = (\sim p \lor \sim q) : x = 2 i s n o t t h e r o o t o f t h e e q u a t i o n x^{2} - 5 x + 6 = 0 o r x = 3 i s \\ n o t t h e r o o t o f t h e e q u a t i o n x^{2} - 5 x + 6 = 0 . \\ (i v) p : A t r i a n g l e h a s 3 - s i d e s . \\ q : A t r i a n g l e h a s 4 - s i d e s . \\ T h e n e g a t i o n o f t h e a b o v e s t a t e m e n t i s \\ \sim (p \lor q) = (\sim p \land \sim q) : A t r i a n g l e h a s n e i t h e r 3 - s i d e s n o r 4 - s i d e s . \\ (v) p : 35 i s a p r i m e n u m b e r . \\ q : 35 i s a c o m p o s i t e n u m b e r . \\ T h e n e g a t i o n o f t h e a b o v e s t a t e m e n t i s \\ \sim (p \lor q) = (\sim p \land \sim q) : 35 i s n o t a p r i m e n u m b e r a n d i t &thins \end{array}$

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New question posted

a year ago

Mathematical Reasoning Maths NCERT Exemplar Solutions Class 11th Chapter Fourteen

Write down the negation of the following compound statements:

i. All rational numbers are real and complex.

ii. All real numbers are rational or irrationals.

iii. x = 2 and x = 3 are roots of the quadratic equation x² – 5x + 6 = 0.

iv. A triangle has either 3 sides or 4 sides.

v. 35 is a prime number or a composite number.

vi. All prime integers are either even or odd.

vii. x is equal to either x or –x.

viii. 6 is divisible by 2 and 3.

0 Follower 3 Views

New answer posted

a year ago

Mathematical Reasoning Maths NCERT Exemplar Solutions Class 11th Chapter Fourteen

Translate the following statements into symbolic form:

i. Rahul passed in Hindi and English.

ii. x and y are even integers.

iii. 2, 3, and 6 are factors of 12.

iv. Either x or x + 1 is an odd integer.

v. A number is either divisible by 2 or 3.

vi. Either x = 2 or x = 3 is a root of 3x² – x – 10 = 0.

vii. Students can take Hindi or English as an optional paper.

0 Follower 3 Views

Payal Gupta

Contributor-Level 10

This is a short answer type question as classified in NCERT Exemplar

$\begin{array}{l} (i) p : R a h u l p a s s e d i n H i n d i . \\ q : R a h u l p a s s e d i n E n g l i s h . \\ p \land q : R a h u l p a s s e d i n H i n d i a n d E n g l i s h . \\ (i i) p : x i s a n e v e n i n t e g e r . \\ q : y i s a n e v e n i n t e g e r . \\ p \land q : x a n d y a r e e v e n i n t e g e r . \\ (i i i) p : 2 i s a f a c t o r o f 12 . \\ q : 3 i s a f a c t o r o f 12 . \\ r : 6 i s a f a c t o r o f 12 . \\ p \land q \land r : 2, 3 a n d 6 a r e f a c t o r s o f 12 . \\ (i v) p : x i s a n o d d i n t e g e r . \\ q : x + 1 i s a n o d d i n t e g e r . \\ p \lor q : E i t h e r x o r x + 1 i s a n o d d i n t e g e r . \\ (v) p : A n u m b e r i s d i v i s i b l e b y 2 . \\ q : A n u m b e r i s d i v i s i b l e b y 3 . \\ p \lor q : A n u m b e r i s d i v i s i b l e b y 2 o r 3 . \\ (v i) p : x = 2 i s a r o o t o f t h e e q u a t i o n 3 x^{2} - x - 10 = 0 . \\ q : x = 3 i s a r o o t o f t h e e q u a t i o n 3 x^{2} - x - 10 = 0 . \\ p \lor q : E i t h e r x = 2 o r x = 3 i s t h e r o o t o f t h e e q u a t i o n 3 x^{2} - x - 10 = 0 . \\ (v i i) p : H i n d i i s t h e o p t i o n a l p a p e r . \\ q : E n g l i s h i s t h e o p t i o n a l p a p e r . \\ p \lor q : E i t h e r H i n d i o r E n g l i s h i s o p t i o n a l p a p e r . \end{array}$

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New answer posted

a year ago

Mathematical Reasoning Maths NCERT Exemplar Solutions Class 11th Chapter Fourteen

Write the negation of the following simple statements:

i. The number 17 is prime.

ii. 2 + 7 = 6.

iii. Violets are blue.

iv. $\sqrt$ 5 is a rational number.

v. 2 is not a prime number.

vi. Every real number is an irrational number.

vii. Cow has four legs.

viii. A leap year has 366 days.

ix. All similar triangles are congruent.

x. Area of a circle is the same as the perimeter of the circle.

0 Follower 6 Views

Payal Gupta

Contributor-Level 10

This is a short answer type question as classified in NCERT Exemplar

$\begin{array}{l} (i) T h e n u m b e r 17 i s n o t p r i m e . \\ (i i) 2 + 7 \neq 6 . \\ (i i i) V i o l e t s a r e n o t b l u e . \\ (i v) \sqrt[]{5} i s n o t a r a t i o n a l n u m b e r . \\ (v) 2 i s a p r i m e n u m b e r . \\ (v i) E v e r y r e a l n u m b e r i s n o t a n i r r a t i o n a l n u m b e r . \\ (v i i) C o w d o e s n o t h a v e f o u r l e g s . \\ (v i i i) A l e a p y e a r d o e s n o t h a v e 366 d a y s . \\ (i x) T h e r e e x i s t s i m i l a r t r i a n g l e s w h i c h a r e n o t c o n g r u e n t . \\ (x) A r e a o f a c i r c l e i s n o t s a m e a s t h e p e r i m e t e r o f t h e c i r c l e . \end{array}$

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New answer posted

a year ago

Mathematical Reasoning Maths NCERT Exemplar Solutions Class 11th Chapter Fourteen

Write the component statements of the following compound statements and check whether the compound statement is true or false:

i. 57 is divisible by 2 or 3.

ii. 24 is a multiple of 4 and 6.

iii. All living things have two eyes and two legs.

iv. 2 is an even number and a prime number.

0 Follower 2 Views

Payal Gupta

Contributor-Level 10

This is a short answer type question as classified in NCERT Exemplar

$\begin{array}{l} (i) H e r e t h e g i v e n s t a t e m e n t i s t h e f o r m p \lor q w h i c h h a s t h e t r u t h v a l u e T w h e n e v e r e i t h e r \\ p o r q b o t h h a v e t h e t r u t h v a l u e T . \\ H e n c e, I t i s a t r u e s t a t e m e n t a n d i t s c o m p o n e n t s t a t e m e n t s a r e \\ p : 57 i s d i v i s i b l e b y 2 . (F a l s e) \\ q : 57 i s d i v i s i b l e b y 3 . (T r u e) \\ (i i) H e r e t h e g i v e n s t a t e m e n t i s t h e f o r m p \land q w h i c h h a s t h e t r u t h v a l u e T w h e n e v e r b o t h \\ p o r q h a v e t h e t r u t h v a l u e T . \\ H e n c e, I t i s a t r u e s t a t e m e n t a n d i t s c o m p o n e n t s t a t e m e n t s a r e \\ p : 24 i s a m u l t i p l e o f 4 . (T r u e) \\ q : 24 i s a m u l t i p l e o f 6 . (T r u e) \\ (i i i) I t i s a f a l s e s t a t e m e n t . S ince p \land q w h i c h h a s t h e t r u t h v a l u e F w h e n e v e r e i t h e r \\ p o r q o r b o t h h a v e t h e t r u t h v a l u e F . I t s c o m p o n e n t s t a t e m e n t s a r e \\ p : A l l l i v i n g t h i n g s h a v e t w o e y e s . (F a l s e) \\ q : A l l l i v i n g t h i n g s &thins \end{array}$

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New answer posted

a year ago

Mathematical Reasoning Maths NCERT Exemplar Solutions Class 11th Chapter Fourteen

Find the component statements of the following compound statements:

i. Number 7 is prime and odd.

ii. Chennai is in India and is the capital of Tamil Nadu.

iii. The number 100 is divisible by 3, 11, and 5.

iv. Chandigarh is the capital of Haryana and U.P.

v. $\sqrt$ 7 is a rational number or an irrational number.

vi. 0 is less than every positive integer and every negative integer.

vii. Plants use sunlight, water, and carbon dioxide for photosynthesis.

viii. Two lines in a plane either intersect at one point or they are parallel.

ix. A rectangle is a quadrilateral or a 5-sided polygon.

0 Follower 4 Views

Payal Gupta

Contributor-Level 10

This is a short answer type question as classified in NCERT Exemplar

$\begin{array}{l} (i) p : N u m b e r 7 i s p r i m e . \\ q : N u m b e r 7 i s o d d . \\ (i i) p : C h e n n a i i s i n I n d i a . \\ q : C h e n n a i i s t h e c a p i t a l o f T a m i l N a d u . \\ (i i i) p : 100 i s d i v i s i b l e b y 3 . \\ q : 100 i s d i v i s i b l e b y 11. \\ r : 100 i s d i v i s i b l e b y 5. \\ (i v) p : C h a n d i g a r h i s t h e c a p i t a l o f H a r y a n a . \\ q : C h a n d i g a r h i s t h e c a p i t a l o f U . P . \\ (v) p : \sqrt[]{7} i s a r a t i o n a l n u m b e r . \\ q : \sqrt[]{7} i s a n i r r a t i o n a l n u m b e r . \\ (v i) p : 0 i s l e s s t h a n e v e r y p o s i t i v e i n t e g e r . \\ q : 0 i s l e s s t h a n e v e r y n e g a t i v e i n t e g e r . \\ (v i i) p : P l a n t s u s e s u n l i g h t f o r p h o t o s y n t h e s i s . \\ q : P l a n t s u s e w a t e r f o r p h o t o s y n t h e s i s . \\ r : P l a n t s u s e c a r b o n - d i o x i d e f o r p h o t o s y n t h e s i s . \\ (v i i i) p : T w o l i n e s i n a p l a n e i n t e r s e c t a t o n e p o i n t . \\ q : T w o l i n e s i n a p l a n e a r e p a r a l l e l . \\ (i x) p : A r e c t a n g l e i s a q u a d r i l a t e r a l . \\ q : A r e c t a n g l e i s a 5 - s i d e s p o l y g o n . \end{array}$

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