If y = y(x), y ∈ [0, π/2] is the solution of the differential equation sec y (dy/dx) - sin(x+y) - sin(x-y) = 0, with y(0) = 0, then 5y'(π/2) is equal to ________.

sec y dy/dx = 2sinxcosy. sec²y dy = 2sinx dx. tan y = -2cosx + C. y (0)=0 ⇒ 0=-2+C ⇒ C=2. tan y = 2-2cosx. y' = (-2sinx)/sec²y. 5y' (π/2) = 5 (2sin (π/2)/sec² (π/2) sec²y dy/dx = 2sinx. y' (π/2)? At x=π/2, tan y = 2. sec²y = 1+tan²y = 5. 5 (2sin (π/2) = 5 (2)=10.

For real numbers α and β, consider the following system of linear equations: x + y - z = 2, x + 2y + αz = 1, 2x - y + z = β. If the system has infinite solutions, then α + β is equal to ________.

Δ = |1,1, -1; 1,2, α 2, -1,1| = 1 (2+α)-1 (1-2α)-1 (-1-4) = 2+α+2α-1+5 = 3α+6=0 ⇒ α=-2. Δ? = |2,1, -1; 1,2, α β, -1,1| = 2 (2+α)-1 (1-αβ)-1 (-1-2β) = 4+2α-1+αβ+1+2β = 4+2α+αβ+2β=0. 4-4-2β+2β=0. This holds. Δ? = |1,2, -1; 1,1, α 2, β,1| = 1 (1-αβ)-2 (1-2α)-1 (β-2) = 1-αβ-2+4α-β+2 = 1+4α-αβ-β=0. 1-8+2β-β=0 ⇒ -7+β=0 ⇒ β=7. α+β = -2+7 = 5.

Let a plane P pass through the point (3, 7, -7) and contain the line, (x-2)/(-3) = (y-3)/2 = (z+2)/1. If distance of the plane P from the origin is d, then d² is equal to ________.

Vector on plane: (3-2, 7-3, -7- (-2) = (1,4, -5). Line direction vector (-3,2,1). Normal to plane n = (1,4, -5)× (-3,2,1) = (14,14,14) or (1,1,1). Plane: 1 (x-3)+1 (y-7)+1 (z+7)=0 ⇒ x+y+z-3=0. d = |-3|/√3 = √3. d²=3.

Let F: → R be a twice differentiable function on (3, 5) such that F(x) = e⁻ˣ ∫₃ˣ (3t² + 2t + 4F'(t))dt. If F'(4) = (αeβ - 224)/(eβ - 4)², then α + β is equal to ________.

e? F (x) = ∫ (3t²+2t+4F' (t)dt. e? F (x)+e? F' (x) = 3x²+2x+4F' (x). (e? -4)F' (x) = 3x²+2x-e? F (x). F' (4) = (48+8-e? F (4)/ (e? -4). Also F (3)=0, F (x)= (x³+x²-36)/ (e? -4) from solution. F (4)= (64+16-36)/ (e? -4) = 44/ (e? -4). F' (4) = (56-e? (44/ (e? -4)/ (e? -4) = (56 (e? -4)-44e? )/ (e? -4)² = (12e? -224)/ (e? -4)². α=12, β=4. α+β=16.

Let f(x) be a determinant. Then the maximum value of f(x) is equal to ________.

f (x) = |sin²x, -2+cos²x, cos2x; 2+sin²x, cos²x, cos2x; sin²x, cos²x, 1+cos2x|. R? →R? -R? , R? →R? -R? f (x) = |sin²x, -2+cos²x, cos2x; 2, 2-2cos²x, 0; 0, 2-2cos²x, 1|. f (x) = sin²x (2-2cos²x) - (-2+cos²x) (2) + cos2x (2 (2-2cos²x). This seems tedious. From the solution, f (x)=4+2cos2x. Max value when cos2x=1, f (x)=6.

The connective in the statement “2 + 7 > 9 or 2 + 7 (a) and (b) Or (c) > (d) <

This is an Objective Type Questions as classified in NCERT Exemplar I n ' 2 + 7 > 9 o r 2 + 7 < 9 ' t h e c o n n e c t i v e i s ' o r ' . H e n c e , t h e c o r r e c t o p t i o n i s ( b ) .

Let a = i + j + 2k and b = -i + 2j + 3k. Then the vector product (a + b) × ((a × ((a - b) × b)) × b) is equal to:

5(34 i - 5 j + 3 k )

5(30 i - 5 j + 7 k )

7(30 i – 5 j + 7 k )

The value of lim(n→∞) [ Σ(k=1 to n) (2k-1) + 8n ] / [ Σ(k=1 to n) (2k-1) + 4n ] is equal to:

1 + 2logₑ(3/2)

2 - logₑ(2/3)

5 + logₑ(2/3)

A ray of light through (2, 1) is reflected at a point P on the y-axis and then passes through the point (5, 3). If this reflected ray is the directrix of any ellipse with eccentricity 1/3 and the distance of the nearer focus from this directrix is 8/√53, then the equation of the other directrix can be:

11x + 7y + 8 = 0 or 11x + 7y - 15 = 0

2x - 7y - 39 = 0 or 2x - 7y - 7 = 0

2x - 7y + 29 = 0 or 2x - 7y - 7 = 0

11x - 7y - 8 = 0 or 11x + 7y + 15 = 0

The compound statement (P ∨ Q) ∧ (¬P) ⇒ Q is equivalent to:

¬(P ⇒ Q) ⇔ P ∧ ¬Q

P ∧ ¬Q

A ray of light through (2, 1) is reflected at a point P on the y-axis and then passes through the point (5, 3). If this reflected ray is the directrix of any ellipse with eccentricity 1/3 and the distance of the nearer focus from this directrix is 8/√53, then the equation of the other directrix can be:

2x - 7y + 29 = 0 or 2x - 7y - 7 = 0

2x - 7y - 39 = 0 or 2x - 7y - 7 = 0

11x + 7y + 8 = 0 or 11x + 7y - 15 = 0

11x - 7y - 8 = 0 or 11x + 7y + 15 = 0

Let P and Q be two distinct points on a circle which has center at C(2, 3) and which passes through origin O. If OC is perpendicular to both the line segments CP and CQ, then the set {P, Q} is equal to:

{(2+2√2, 3-√5), (2-2√2, 3+√5)}

{(-1, 5), (5, 1)}

{(2+2√2, 3+√5), (2-2√2, 3-√5)}

Two tangents are drawn from the point P(-1, 1) to the circle x² + y² - 2x – 6y + 6 = 0. If these tangents touch the circle at points A and B, and if D is a point on the circle such that length of the segments AB and AD are equal, then the area of the triangle ABD is equal to:

(3√2 + 2)

3(√2 - 1)

Maths NCERT Exemplar Solutions Class 11th Chapter Fourteen: Overview, Questions, Preparation

Updated on Sep 8, 2025 11:39 IST

By Payal Gupta, Retainer

Table of content

Mathematical Reasoning Short Answer Type Questions
Mathematical Reasoning Objective Type Questions
JEE Mains 2021
JEE Mains 2021

Mathematical Reasoning Short Answer Type Questions

1. Which of the following sentences are statements? Justify.

i. A triangle has three sides.

ii. 0 is a complex number.

iii. Sky is red.

iv. Every set is an infinite set.

v. 15 + 8 > 23.

vi. y + 9 = 7.

vii. Where is your bag?

viii. Every square is a rectangle.

ix. Sum of opposite angles of a cyclic quadrilateral is 180°.

x. sin²x + cos² x = 0.

Sol.

\begin{array}{l} A s w e k n o w, a s t a t e m e n t i s a s e n t e n c e w h i c h i s e i t h e r t r u e o r f a l s e b u t n o t b o t h simultaneously . \\ (i) I t i s t r u e s t a t e m e n t . A s t r i a n g l e i s a c l o s e d p o l y g o n o f t h r e e s i d e s . \\ (i i) I t i s t r u e s t a t e m e n t . A s 0 + 0 i \\ (i i i) I t i s f a l s e s t a t e m e n t . A s s k y i s b l u e . \\ (i v) I t i s f a l s e s t a t e m e n t . N o, e v e r y s e t i s n o t a n infinite s e t . \\ (v) I t i s f a l s e s t a t e m e n t . 15 + 8 = 23 > 23 \\ (v i) I t i s n o t a s t a t e m e n t . B e c a u s e y + 9 = 7, t h e v a l u e o f y i s n o t g i v e n . \\ (v i i) I t i s n o t a s t a t e m e n t . I t i s a q u e s t i o n . \\ (v i i i) I t i s t r u e s t a t e m e n t . A s s q u a r e i s a rectangle w h o s e a l l s i d e s a r e e q u a l . \\ (i x) I t i s t r u e s t a t e m e n t . \\ (x) I t i s f a l s e s t a t e m e n t . A s s i n^{2} x + c o s^{2} x = 1 \end{array}

2. 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.

Sol.

\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}

Commonly asked questions

Prove the following statement by the contradiction method:

$p$ : The sum of an irrational number and a rational number is irrational.

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

$\begin{array}{l} L e t p i s f a l s e i . e ., t h e s u m o f a n i r r a t i o n a l n u m b e r a n d a r a t i o n a l n u m b e r i s r a t i o n a l . \\ L e t \sqrt[]{λ} i s i r r a t i o n a l a n d n i s r a t i o n a l n u m b e r \\ \Rightarrow \sqrt[]{λ} + n = r (r a t i o n a l) \\ \Rightarrow \sqrt[]{λ} = r - n \\ W e o b s e r v e t h a t \sqrt[]{λ} i s i r r a t i o n a l w h e r e a s (r - n) i s r a t i o n a l w h i c h i s a c o n t r a d i c t i o n . \\ S o, o u r supposition i s w r o n g . \\ H e n c e, p i s t r u e . \end{array}$

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.

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

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

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.

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

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

Prove by the direct method that for any integer $' n'$ , $n^{3} - n$ is always even.
Hint: Two cases (i) $n$ is even, (ii) $n$ is odd.

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

$\begin{array}{l} W e h a v e t w o c a s e s : \\ C a s e I : I f n i s e v e n \\ L e t n = 2 k w h e r e k \in N \\ ∴ n^{3} - n = {(2 k)}^{3} - (2 k) \\ = 2 k (4 k^{2} - 1) = 2 m, \\ w h e r e m = k (4 k^{2} - 1) \\ T h e r e f o r e, (n^{3} - n) i s e v e n . \\ C a s e I I : I f n i s o d d \\ L e t n = (2 k + 1), k \in N \\ ∴ n^{3} - n = {(2 k + 1)}^{3} - (2 k + 1) \\ = (2 k + 1) [{(2 k + 1)}^{2} - 1] \\ = (2 k + 1) [4 k^{2} + 4 k + 1 - 1] \\ = (2 k + 1) (4 k^{2} + 4 k) = 4 k (2 k + 1) (k + 1) \\ = 2.2 k (2 k + 1) (k + 1) = 2 λ w h e r e λ = 2 k (2 k + 1) (k + 1) \\ T h e r e f o r e, (n^{3} - n) i s e v e n . \\ H e n c e, (n^{3} - n) i s a l w a y s e v e n . \end{array}$

Check the validity of the following statements:

i. $p$ : 125 is divisible by 5 and 7.

ii. $q$ : 131 is a multiple of 3 or 11.

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

$\begin{array}{l} (i) G i v e n t h a t : \\ p : 125 i s d i v i s i b l e b y 5 a n d 7 . \\ L e t q : 125 i s d i v i s i b l e b y 5 . \\ a n d r : 125 i s d i v i s i b l e b y 7 . \\ H e r e q i s t r u e a n d r i s f a l s e . \\ \Rightarrow q \land r i s f a l s e . \\ H e n c e, p i s n o t v a l i d . \\ (i i) G i v e n t h a t : \\ q : 131 i s a m u l t i p l e o f 3 o r 11 . \\ L e t p : 131 i s a m u l t i p l e o f 3 . \\ a n d r : 131 i s a m u l t i p l e o f 11 . \\ H e r e p i s n o t t r u e a n d r i s a l s o n o t t r u e i . e ., f a l s e . \\ \Rightarrow p \lor r i s f a l s e . \\ H e n c e, q i s n o t v a l i d . \end{array}$

Prove by the direct method that for any real numbers $x, y$ , if $x = y$ , then $x^{2} = y^{2}$ .

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

$\begin{array}{l} L e t p : x = y, x, y \in R \\ O n s q u a r i n g b o t h s i d e s w e h a v e \\ x^{2} = y^{2} : q (s a y) \\ \Rightarrow p = q H e n c e, p r o v e d . \end{array}$

Using the contrapositive method, prove that if $n^{2}$ is an even integer, then $n$ is also an even integer.

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

$\begin{array}{l} L e t p : n^{2} i s a n e v e ninteger . \\ q : n i s a l s o a n e v e n integer . \\ A l s o l e t \sim q i s t r u e i . e ., n i s n o t a n e v e n integer . \\ \Rightarrow n^{2} i s n o t a n e v e n integer . \\ \Rightarrow \sim p i s t r u e . [? S q u a r e o f a n o d d integer i s o d d] \\ H e n c e, \sim q i s t r u e \Rightarrow \sim p i s t r u e . \end{array}$

Which of the following sentences are statements? Justify.

i. A triangle has three sides.

ii. 0 is a complex number.

iii. Sky is red.

iv. Every set is an infinite set.

v. 15 + 8 > 23.

vi. y + 9 = 7.

vii. Where is your bag?

viii. Every square is a rectangle.

ix. Sum of opposite angles of a cyclic quadrilateral is 180°.

x. sin²x + cos² x = 0.

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

$\begin{array}{l} A s w e k n o w, a s t a t e m e n t i s a s e n t e n c e w h i c h i s e i t h e r t r u e o r f a l s e b u t n o t b o t h simultaneously . \\ (i) I t i s t r u e s t a t e m e n t . A s t r i a n g l e i s a c l o s e d p o l y g o n o f t h r e e s i d e s . \\ (i i) I t i s t r u e s t a t e m e n t . A s 0 + 0 i \\ (i i i) I t i s f a l s e s t a t e m e n t . A s s k y i s b l u e . \\ (i v) I t i s f a l s e s t a t e m e n t . N o, e v e r y s e t i s n o t a n infinite s e t . \\ (v) I t i s f a l s e s t a t e m e n t . 15 + 8 = 23 > 23 \\ (v i) I t i s n o t a s t a t e m e n t . B e c a u s e y + 9 = 7, t h e v a l u e o f y i s n o t g i v e n . \\ (v i i) I t i s n o t a s t a t e m e n t . I t i s a q u e s t i o n . \\ (v i i i) I t i s t r u e s t a t e m e n t . A s s q u a r e i s a rectangle w h o s e a l l s i d e s a r e e q u a l . \\ (i x) I t i s t r u e s t a t e m e n t . \\ (x) I t i s f a l s e s t a t e m e n t . A s s i n^{2} x + c o s^{2} x = 1 \end{array}$

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.

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

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

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.

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

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

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.

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

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

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.

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

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

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.

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

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

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.

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

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

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 $x^{2} < 1$ .

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

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

Identify the quantifiers in the following statements:

i. There exists a triangle that is not equilateral.

ii. For all real numbers $x$ and $y$ , $x y = y x$ .

iii. There exists a real number that is not a rational number.

iv. For every natural number $x$ , $x + 1$ is also a natural number.

v. For all real numbers $x$ with $x > 3$ , $x^{2} > 9$ .

vi. There exists a triangle that is not an isosceles triangle.

vii. For all negative integers $x$ , $x^{3}$ is also a negative integer.

viii. There exists a statement in the above statements that is not true.

ix. There exists an even prime number other than 2.

x. There exists a real number $x$ such that $x^{2} + 1 = 0$ .

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

$\begin{array}{l} Q u a n t i f i e r m e a n s a p h r a s e l i k e' t h e r e e x i s t s',' f o r a l l' a n d' f o r e v e r y' e t c . \\ (i) T h e r e e x i s t s (i i) F o r a l l \\ (i i i) T h e r e e x i s t s (i v) F o r e v e r y \\ (v) F o r a l l (v i) T h e r e e x i s t s \\ (v i i) F o r a l l (v i i i) T h e r e e x i s t s \\ (i x) T h e r e e x i s t s (x) T h e r e e x i s t s \end{array}$

Mathematical Reasoning Objective Type Questions

1. Which of the following is a statement.

(a) x is a real number.

(b) Switch off the fan.

(c) 6 is a natural number.

(d) Let me go.

Sol:

$\begin{array}{l} Since s t a t e m e n t i s a s e n t e n c e w h i c h i s e i t h e r t r u e o r f a l s e . \\ S o, 6 i s a n a t u r a l n u m b e r w h i c h i s t r u e . \\ H e n c e, t h e c o r r e c t o p t i o n i s (c) . \end{array}$

2. Which of the following is not a statement

(a) Smoking is injurious to health.

(b) 2 + 2 = 4

(c) 2 is the only even prime number.

(d) Come here.

Sol.

\begin{array}{l} T o g i v e o r d e r c a n n o t b e a s t a t e m e n t . \\ S o,' C o m e h e r e' i s n o t a s t a t e m e n t . \\ H e n c e, t h e c o r r e c t o p t i o n i s (d) . \end{array}

Commonly asked questions

Which of the following is a statement.

(a) x is a real number.

(b) Switch off the fan.

(c) 6 is a natural number.

(d) Let me go.

This is an Objective Type Questions as classified in NCERT Exemplar

Which of the following is not a statement

(a) Smoking is injurious to health.

(b) 2 + 2 = 4

(c) 2 is the only even prime number.

(d) Come here.

This is an Objective Type Questions as classified in NCERT Exemplar

$\begin{array}{l} T o g i v e o r d e r c a n n o t b e a s t a t e m e n t . \\ S o,' C o m e h e r e' i s n o t a s t a t e m e n t . \\ H e n c e, t h e c o r r e c t o p t i o n i s (d) . \end{array}$

The connective in the statement

“2 + 7 > 9 or 2 + 7 < 9” is

(a) and

(b) Or

(c) >

(d) <

This is an Objective Type Questions as classified in NCERT Exemplar

$\begin{array}{l} I n' 2 + 7 > 9 o r 2 + 7 < 9' t h e c o n n e c t i v e i s' o r' . \\ H e n c e, t h e c o r r e c t o p t i o n i s (b) . \end{array}$

The connective in the statement “Earth revolves round the Sun and Moon is a satellite of earth” is

(a) or

(b) Earth

(c) Sun

(d) and

This is an Objective Type Questions as classified in NCERT Exemplar

$\begin{array}{l} C o n n e c t i v e w o r d i s " a n d " . \\ H e n c e, t h e c o r r e c t o p t i o n i s (d) . \end{array}$

The negation of the statement “A circle is an ellipse” is

(a) An ellipse is a circle.

(b) An ellipse is not a circle.

(c) A circle is not an ellipse.

(d) A circle is an ellipse.

This is an Objective Type Questions as classified in NCERT Exemplar

$\begin{array}{l} L e t p : A c i r c l e i s a n e l l i p s e . \\ \sim p : A c i r c l e i s n o t a n e l l i p s e . \\ H e n c e, t h e c o r r e c t o p t i o n i s (c) . \end{array}$

The negation of the statement “7 is greater than 8” is

(a) 7 is equal to 8.

(b) 7 is not greater than 8.

(c) 8 is less than 7.

(d) none of these

This is an Objective Type Questions as classified in NCERT Exemplar

$\begin{array}{l} L e t p : 7 i s g r e a t e r t h a n 8 . \\ \sim p : 7 i s n o t g r e a t e r t h a n 8 . \\ H e n c e, t h e c o r r e c t o p t i o n i s (b) . \end{array}$

The negation of the statement “72 is divisible by 2 and 3” is

(a) 72 is not divisible by 2 or 72 is not divisible by 3.

(b) 72 is not divisible by 2 and 72 is not divisible by 3.

(c) 72 is divisible by 2 and 72 is not divisible by 3.

(d) 72 is not divisible by 2 and 72 is divisible by 3.

This is an Objective Type Questions as classified in NCERT Exemplar

$\begin{array}{l} L e t p : 72 i s d i v i s i b l e b y 2 a n d 3 . \\ a n d q : 72 i s d i v i s i b l e b y 2 . \\ r : 72 i s d i v i s i b l e b y 3 . \\ ∴ \sim q : 72 i s n o t d i v i s i b l e b y 2 . \\ \sim r : 72 i s n o t d i v i s i b l e b y 3 . \\ S o, \sim (q \land r) : \sim q \lor \sim r \\ \Rightarrow 72 i s n o t d i v i s i b l e b y 2 o r 72 i s n o t d i v i s i b l e b y 3 . \\ H e n c e, t h e c o r r e c t o p t i o n i s (b) . \end{array}$

The negation of the statement “Plants take in CO₂ and give out O₂” is

(a) Plants do not take in CO₂ and do not give out O₂.

(b) Plants do not take in CO₂ or do not give out O₂.

(c) Plants take in CO₂ and do not give out O₂.

(d) Plants take in CO₂ or do not give out O₂.

This is an Objective Type Questions as classified in NCERT Exemplar

$\begin{array}{l} L e t p : P l a n t s t a k e i n C O_{2} a n d g i v e o u t O_{2} . \\ q : P l a n t s t a k e i n C O_{2} . \\ a n d r : P l a n t s g i v e o u t O_{2} . \\ ∴ \sim q : P l a n t s d o n o t t a k e i n C O_{2} . \\ \sim r : P l a n t s d o n o t g i v e o u t O_{2} . \\ S o, \sim (q \land r) = (\sim q \lor \sim r) : P l a n t s d o n o t t a k e i n C O_{2} o r d o n o t g i v e o u t O_{2} . \\ H e n c e, t h e c o r r e c t o p t i o n i s (b) . \end{array}$

The negation of the statement “Rajesh or Rajni lived in Bangalore” is

(a) Rajesh did not live in Bangalore or Rajni lives in Bangalore.

(b) Rajesh lives in Bangalore and Rajni did not live in Bangalore.

(c) Rajesh did not live in Bangalore and Rajni did not live in Bangalore.

(d) Rajesh did not live in Bangalore or Rajni did not live in Bangalore.

This is an Objective Type Questions as classified in NCERT Exemplar

$\begin{array}{l} L e t p : R a j e s h a n d R a j n i l i v e s i n B a n g a l o r e . \\ q : R a j e s h l i v e d i n B a n g a l o r e . \\ a n d r : R a j n i l i v e d i n B a n g a l o r e . \\ ∴ \sim q : R a j e s h d i d n o t l i v e i n B a n g a l o r e . \\ \sim r : R a j n i d i d n o t l i v e i n B a n g a l o r e . \\ S o, \sim (q \lor r) = (\sim q \land \sim r) : R a j e s h d i d n o t l i v e i n B a n g a l o r e a n d R a j n i d i d n o t \\ l i v e i n B a n g a l o r e . \\ H e n c e, t h e c o r r e c t o p t i o n i s (c) . \end{array}$

The negation of the statement “101 is not a multiple of 3” is

(a) 101 is a multiple of 3.

(b) 101 is a multiple of 2.

(c) 101 is an odd number.

(d) 101 is an even number.

This is an Objective Type Questions as classified in NCERT Exemplar

$\begin{array}{l} L e t p : 101 i s n o t a m u l t i p l e o f 3 . \\ \sim p : 101 i s a m u l t i p l e o f 3 . \\ H e n c e, t h e c o r r e c t o p t i o n i s (a) . \end{array}$

The contrapositive of the statement “If 7 is greater than 5, then 8 is greater than 6” is

(a) If 8 is greater than 6, then 7 is greater than 5.

(b) If 8 is not greater than 6, then 7 is greater than 5.

(c) If 8 is not greater than 6, then 7 is not greater than 5.

(d) If 8 is greater than 6, then 7 is not greater than 5.

This is an Objective Type Questions as classified in NCERT Exemplar

$\begin{array}{l} L e t p : 7 i s g r e a t e r t h a n 5 . \\ q : 8 i s g r e a t e r t h a n 6 . \\ ∴ p \to q i . e ., \sim p : 7 i s n o t g r e a t e r t h a n 5 . \\ a n d \sim q : 8 i s n o t g r e a t e r t h a n 6 . \\ S o, \sim p \to \sim q : I f 8 i s n o t g r e a t e r t h a n 6, t h e n 7 i s n o t g r e a t e r t h a n 5 . \\ H e n c e, t h e c o r r e c t o p t i o n i s (c) . \end{array}$

The converse of the statement “If x > y, then x + a > y + a” is

(a) If x < y, then x + a < y + a.

(b) If x + a > y + a, then x > y.

(c) If x < y, then x + a > y + a.

(d) If x > y, then x + a < y + a.

This is an Objective Type Questions as classified in NCERT Exemplar

$\begin{array}{l} L e t p : x > y \\ a n d q : x + a > y + a \\ p \to q \\ C o n v e r s e o f t h e a b o v e s t a t e m e n t i s q \to p . \\ T h e r e f o r e, i f x + a > y + a t h e n x > y . \\ H e n c e, t h e c o r r e c t o p t i o n i s (b) . \end{array}$

The converse of the statement “If sun is not shining, then sky is filled with clouds” is

(a) If sky is filled with clouds, then the sun is not shining.

(b) If sun is shining, then sky is filled with clouds.

(c) If sky is clear, then sun is shining.

(d) If sun is not shining, then sky is not filled with clouds.

This is an Objective Type Questions as classified in NCERT Exemplar

$\begin{array}{l} L e t p : S u n i s n o t s h i n i n g . \\ q : S k y i s f i l l e d w i t h c l o u d s . \\ S o, t h e c o n v e r s e o f t h e s t a t e m e n t p \to q i s q \to p . \\ i . e ., I f S k y i s f i l l e d w i t h c l o u d s, t h e n t h e S u n i s n o t s h i n i n g . \\ H e n c e, t h e c o r r e c t o p t i o n i s (a) . \end{array}$

The contrapositive of the statement “If p, then q”, is

(a) If q, then p.

(b) If p, then ~ q.

(c) If ~ q, then ~ p.

(d) If ~ p, then ~ q.

This is an Objective Type Questions as classified in NCERT Exemplar

$\begin{array}{l} H e r e t h e s t a t e m e n t i s " I f p, t h e n q " \\ i . e . p \to q \\ C o n t r a p o s i t i v e o f t h e s t a t e m e n t p \to q i s (\sim q) \to (\sim p) \\ i . e . I f \sim q, t h e n \sim p . \\ H e n c e, t h e c o r r e c t o p t i o n i s (c) . \end{array}$

The statement “If x² is not even, then x is not even” is converse of the statement

(a) If x² is odd, then x is even.

(b) If x is not even, then x² is not even.

(c) If x is even, then x² is even.

(d) If x is odd, then x² is even.

This is an Objective Type Questions as classified in NCERT Exemplar

$\begin{array}{l} L e t p : x^{2} i s n o t e v e n . \\ a n d q : x i s n o t e v e n . \\ S o, t h e c o n v e r s e o f t h e s t a t e m e n t p \to q i s q \to p \\ i . e . I f x i s n o t e v e n, t h e n x^{2} i s n o t e v e n . \\ H e n c e, t h e c o r r e c t o p t i o n i s (b) . \end{array}$

The contrapositive of statement ‘If Chandigarh is capital of Punjab, then Chandigarh is in India’ is

(a) If Chandigarh is not in India, then Chandigarh is not the capital of Punjab.

(b) If Chandigarh is in India, then Chandigarh is capital of Punjab.

(c) If Chandigarh is not capital of Punjab, then Chandigarh is not in India.

(d) If Chandigarh is capital of Punjab, then Chandigarh is not in India.

This is an Objective Type Questions as classified in NCERT Exemplar

$\begin{array}{l} L e t p : C h a n d i g a r h i s c a p i t a l o f P u n j a b . \\ a n d q : C h a n d i g a r h i s i n I n d i a . \\ \sim p : C h a n d i g a r h i s n o t t h e c a p i t a l o f P u n j a b . \\ \sim q : C h a n d i g a r h i s n o t i n I n d i a . \\ I f (\sim q), t h e n (\sim p) \\ i . e . I f C h a n d i g a r h i s n o t i n I n d i a, t h e n C h a n d i g a r h i s n o t t h e c a p i t a l o f P u n j a b . \\ H e n c e, t h e c o r r e c t o p t i o n i s (a) . \end{array}$

Which of the following is the conditional p → q?

(a) q is sufficient for p.

(b) p is necessary for q.

(c) p only if q.

(d) if q, then p.

This is an Objective Type Questions as classified in NCERT Exemplar

$\begin{array}{l} W e k n o w t h a t p \to q i s s a m e a s p o n l y i f q . \\ H e n c e, t h e c o r r e c t o p t i o n i s (c) . \end{array}$

The negation of the statement “The product of 3 and 4 is 9” is

(a) It is false that the product of 3 and 4 is 9.

(b) The product of 3 and 4 is 12.

(c) The product of 3 and 4 is not 12.

(d) It is false that the product of 3 and 4 is not 9.

This is an Objective Type Questions as classified in NCERT Exemplar

$\begin{array}{l} T h e n e g a t i o n o f t h e g i v e n s t a t e m e n t i s f a l s e . i . e ., I t i s f a l s e t h a t t h e p r o d u c t o f 3 a n d 4 i s 9 . \\ H e n c e, t h e c o r r e c t o p t i o n i s (a) . \end{array}$

Which of the following is not a negation of “A natural number is greater than zero”

(a) A natural number is not greater than zero.

(b) It is false that a natural number is greater than zero.

(c) It is false that a natural number is not greater than zero.

(d) None of the above

This is an Objective Type Questions as classified in NCERT Exemplar

$\begin{array}{l} T h e n e g a t i o n o f t h e g i v e n s t a t e m e n t i s f a l s e . i . e ., I t i s f a l s e t h a t a n a t u r a l n u m b e r i s n o t \\ g r e a t e r t h a n z e r o . \\ H e n c e, t h e c o r r e c t o p t i o n i s (c) . \end{array}$

Which of the following statement is a conjunction?

(a) Ram and Shyam are friends.

(b) Both Ram and Shyam are tall.

(c) Both Ram and Shyam are enemies.

(d) None of the above.

This is an Objective Type Questions as classified in NCERT Exemplar

$\begin{array}{l} L e t t h e t w o s t a t e m e n t s p a n d q b e s i m p l e s t a t e m e n t s . I f t h e y a r e c o n n e c t e d w i t h a n d . \\ T h e n, t h e r e s u l t i n g c o m p o u n d s t a t e m e n t p a n d q i s c a l l e d a c o n j u c t i o n o f p a n d q . \\ H e n c e, t h e c o r r e c t o p t i o n i s (d) . \end{array}$

State whether the following sentences are statements or not:

(a) The angles opposite to equal sides of a triangle are equal.

(b) The moon is a satellite of earth.

(c) May God bless you!

(d) Asia is a continent.

(e) How are you?

This is an Objective Type Questions as classified in NCERT Exemplar

$\begin{array}{l} (a) I t i s a s t a t e m e n t . \\ (b) I t i s a s t a t e m e n t . \\ (c) I t i s n o t a s t a t e m e n t . \\ (d) I t i s a s t a t e m e n t . \\ (e) I t i s n o t a s t a t e m e n t . \end{array}$

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