SolveIt is under development
SolveItNCERT · CBSE · NEET

NCERT Exemplar · Class 11 Mathematics Mathematical Reasoning

37 questions · 37 still being checked

This chapter is from the older syllabus and is not in the current NCERT textbook, so you may skip it.

EXERCISE 14.3 1–10 (part 1 of 4)

  1. Exercise 1

    Which of the following sentences are statements? Justify
    (i)
    A triangle has three sides.
    (ii)
    0\displaystyle 0 is a complex number.
    (iii)
    Sky is red.
    (iv)
    Every set is an infinite set.
    (v)
    15+8>23\displaystyle 15+8>23.
    (vi)
    y+9=7\displaystyle 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\displaystyle 180°.
    (x)
    sin⁡2x+cos⁡2x=0\displaystyle \sin ^2 x+\cos ^2 x=0

    Matches the book, not yet reviewed

    This working reaches the answer NCERT prints. It has not yet been read through by hand.

    NCERT’s answer
    (i) to (v) and (viii) to (x) are statements.
    (i)
    Statement (true).
    (ii)
    Statement (true):
    \[0 = 0 + 0i \]
    (iii)
    Statement (false).
    (iv)
    Statement (false): \(\displaystyle \{1,2\}\) is a finite set.
    (v)
    Statement (false):
    \[15+8 = 23 \not> 23 \]
    (vi)
    Not a statement: true for \(\displaystyle y=-2\), false for \(\displaystyle y=0\).
    (vii)
    Not a statement: it is a question.
    (viii)
    Statement (true).
    (ix)
    Statement (true):
    \[\angle A+\angle C = 180^\circ, \qquad \angle B+\angle D = 180^\circ \]
    (x)
    Statement (false):
    \[\sin^2 x+\cos^2 x = 1 \neq 0 \]
    Answer: (i), (ii), (iii), (iv), (v), (viii), (ix), (x) are statements; (vi) and (vii) are not.

  2. Exercise 2

    Find the component statements of the following compound statements.
    (i)
    Number 7\displaystyle 7 is prime and odd.
    (ii)
    Chennai is in India and is the capital of Tamil Nadu.
    (iii)
    The number 100\displaystyle 100 is divisible by 3\displaystyle 3, 11\displaystyle 11 and 5.
    (iv)
    Chandigarh is the capital of Haryana and U.P.
    (v)
    7\displaystyle \sqrt{7} is a rational number or an irrational number.
    (vi)
    0\displaystyle 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\displaystyle 5 - sided polygon.

    Matches the book, not yet reviewed

    This working reaches the answer NCERT prints. It has not yet been read through by hand.

    NCERT’s answer
    (i)
    p : Number $\displaystyle 7$ is prime
    q : Number $\displaystyle 7$ is odd
    (ii)
    p : Chennai is in India
    q : Chennai is capital of Tamil Nadu
    (iii)
    p : $\displaystyle 100$ is divisble by $\displaystyle 3$
    q : $\displaystyle 100$ is divisible by $\displaystyle 11$
    r : $\displaystyle 100$ is divisible by $\displaystyle 5$
    (iv)
    p : Chandigarh is capital of Haryana
    q : Chandigarh is the capital of U.P
    (v)
    p : \(\displaystyle \sqrt{7}\) is a rational number
    q : \(\displaystyle \sqrt{7}\) is an irrational number
    (vi)
    p : $\displaystyle 0$ is less than every positive integer
    q : $\displaystyle 0$ is less than every negative integer
    (vii)
    p : plants use sunlight for photosynthesis
    q : plants use water for photosynthesis
    r : plants use carbondioxide for photosynthesis
    (viii)
    p : two lines in a plane intersect at one point
    q : two lines in a plane are parallel
    (ix)
    p : a rectangle is a quadrilateral
    q : a rectangle is a $\displaystyle 5$- sided polygons.
    (i)
    \(\displaystyle p\): $\displaystyle 7$ is prime; \(\displaystyle q\): $\displaystyle 7$ is odd.
    (ii)
    \(\displaystyle p\): Chennai is in India; \(\displaystyle q\): Chennai is the capital of Tamil Nadu.
    (iii)
    \(\displaystyle p\): $\displaystyle 100$ is divisible by $\displaystyle 3$; \(\displaystyle q\): $\displaystyle 100$ is divisible by $\displaystyle 11$; \(\displaystyle r\): $\displaystyle 100$ is divisible by 5.
    (iv)
    \(\displaystyle p\): Chandigarh is the capital of Haryana; \(\displaystyle q\): Chandigarh is the capital of U.P.
    (v)
    \(\displaystyle p\): \(\displaystyle \sqrt{7}\) is a rational number; \(\displaystyle q\): \(\displaystyle \sqrt{7}\) is an irrational number.
    (vi)
    \(\displaystyle p\): $\displaystyle 0$ is less than every positive integer; \(\displaystyle q\): $\displaystyle 0$ is less than every negative integer.
    (vii)
    \(\displaystyle p\): Plants use sunlight for photosynthesis; \(\displaystyle q\): Plants use water for photosynthesis; \(\displaystyle r\): Plants use carbon dioxide for photosynthesis.
    (viii)
    \(\displaystyle p\): Two lines in a plane intersect at one point; \(\displaystyle q\): Two lines in a plane are parallel.
    (ix)
    \(\displaystyle p\): A rectangle is a quadrilateral; \(\displaystyle q\): A rectangle is a $\displaystyle 5$-sided polygon.
    Answer: two components in (i), (ii), (iv), (v), (vi), (viii), (ix); three in (iii) and (vii).

  3. Exercise 3

    Write the component statements of the following compound statements and check whether the compound statement is true or false.
    (i)
    57\displaystyle 57 is divisible by 2\displaystyle 2 or 3.
    (ii)
    24\displaystyle 24 is a multiple of 4\displaystyle 4 and 6.
    (iii)
    All living things have two eyes and two legs.
    (iv)
    2\displaystyle 2 is an even number and a prime number.

    Matches the book, not yet reviewed

    This working reaches the answer NCERT prints. It has not yet been read through by hand.

    NCERT’s answer
    (i)
    Compound statement is true and its component statements are :
    p : $\displaystyle 57$ is divisible by $\displaystyle 2$ and q : $\displaystyle 57$ is divisble by $\displaystyle 3$
    (ii)
    component statement is true and its component statements are :
    p : $\displaystyle 24$ is multiple of $\displaystyle 4$ and q : $\displaystyle 24$ is multiple of $\displaystyle 6$
    (iii)
    component statement is false and is component statements are
    p : All living things have two eyes
    q : All living things have two legs
    (iv)
    component statement is true and its component statements are :
    p : $\displaystyle 2$ is an number ; q : $\displaystyle 2$ is a prime number
    (i)
    \(\displaystyle p\): $\displaystyle 57$ is divisible by $\displaystyle 2$; \(\displaystyle q\): $\displaystyle 57$ is divisible by 3.
    \[57 = 2\times 28 + 1, \qquad 57 = 3\times 19 \]
    \(\displaystyle p\) false, \(\displaystyle q\) true: \(\displaystyle p\vee q\) true.
    (ii)
    \(\displaystyle p\): $\displaystyle 24$ is a multiple of $\displaystyle 4$; \(\displaystyle q\): $\displaystyle 24$ is a multiple of 6.
    \[24 = 4\times 6 = 6\times 4 \]
    Both true: \(\displaystyle p\wedge q\) true.
    (iii)
    \(\displaystyle p\): All living things have two eyes; \(\displaystyle q\): All living things have two legs.
    \(\displaystyle q\) false (a snake has no legs): \(\displaystyle p\wedge q\) false.
    (iv)
    \(\displaystyle p\): $\displaystyle 2$ is an even number; \(\displaystyle q\): $\displaystyle 2$ is a prime number.
    Both true: \(\displaystyle p\wedge q\) true.
    Answer: (i) true, (ii) true, (iii) false, (iv) true.

  4. Exercise 4

    Write the negation of the following simple statements
    (i)
    The number 17\displaystyle 17 is prime.
    (ii)
    2+7=6\displaystyle 2+7=6.
    (iii)
    Violets are blue.
    (iv)
    5\displaystyle \sqrt{5} is a rational number.
    (v)
    2\displaystyle 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\displaystyle 366 days.
    (ix)
    All similar triangles are congruent.
    (x)
    Area of a circle is same as the perimeter of the circle.

    Matches the book, not yet reviewed

    This working reaches the answer NCERT prints. It has not yet been read through by hand.

    NCERT’s answer
    (i)
    The number $\displaystyle 17$ is not prime
    (ii)
    \(\displaystyle 2 + 7 \neq 6\)
    (iii)
    Violet are not blue
    (iv)
    \(\displaystyle \sqrt{5}\) is not a rational number
    (v)
    $\displaystyle 2$ is a prime number
    (vi)
    There exists a real number which is not an irrational number
    (vii)
    Cow has not four legs
    (viii)
    A leap year has not $\displaystyle 366$ days
    (ix)
    There exist similar triangles which are not congruent
    (x)
    Area of a circle is not same as the perimeter of the circle
    (i)
    The number $\displaystyle 17$ is not prime.
    (ii)
    \(\displaystyle 2+7\neq 6\).
    (iii)
    Violets are not blue.
    (iv)
    \(\displaystyle \sqrt{5}\) is not a rational number.
    (v)
    $\displaystyle 2$ is a prime number.
    (vi)
    Some real number is not irrational.
    (vii)
    Cow does not have four legs.
    (viii)
    A leap year does not have $\displaystyle 366$ days.
    (ix)
    Some similar triangles are not congruent.
    (x)
    Area of a circle is not the same as the perimeter of the circle.
    Answer: the negations (i)-(x) above.

  5. Exercise 5

    Translate the following statements into symbolic form
    (i)
    Rahul passed in Hindi and English.
    (ii)
    x\displaystyle x and y\displaystyle y are even integers.
    (iii)
    2\displaystyle 2, 3\displaystyle 3 and 6\displaystyle 6 are factors of 12.
    (iv)
    Either x\displaystyle x or x+1\displaystyle x+1 is an odd integer.
    (v)
    A number is either divisible by 2\displaystyle 2 or 3.
    (vi)
    Either x=2\displaystyle x=2 or x=3\displaystyle x=3 is a root of 3x2−x−10=0\displaystyle 3 x^2-x-10=0
    (vii)
    Students can take Hindi or English as an optional paper.

    Matches the book, not yet reviewed

    This working reaches the answer NCERT prints. It has not yet been read through by hand.

    NCERT’s answer
    (i)
    \(\displaystyle p \wedge q\) where p : Rahul passed in Hndi; q : Rahul passed in English
    (ii)
    \(\displaystyle p \wedge q\) where p : x is even integer ; q : y is even integer
    (iii)
    \(\displaystyle p \wedge q \wedge r\) where p : $\displaystyle 2$ is factor of $\displaystyle 12$; q : $\displaystyle 3$ is factor of $\displaystyle 12$; r : $\displaystyle 6$ is factor of $\displaystyle 12$
    (iv)
    \(\displaystyle p \vee q\) where p : x is an odd integer ; q : \(\displaystyle x + 1\) is an odd integer
    (v)
    \(\displaystyle p \vee q\) where p : a number is divisible by $\displaystyle 2$, q : it is divisibe by $\displaystyle 3$
    (vi)
    \(\displaystyle p \vee q\) where p : \(\displaystyle x = 2\) is a root of \(\displaystyle 3x^{2} - x - 10 = 0\), q : \(\displaystyle x = 3\) is a root of \(\displaystyle 3x^{2} - x - 10 = 0\)
    (vii)
    \(\displaystyle p \vee q\) where p : student can take Hindi as an optional paper and q : student can take English as an optional paper.
    (i)
    \(\displaystyle p\): Rahul passed in Hindi; \(\displaystyle q\): Rahul passed in English.
    \[p\wedge q \]
    (ii)
    \(\displaystyle p\): \(\displaystyle x\) is an even integer; \(\displaystyle q\): \(\displaystyle y\) is an even integer.
    \[p\wedge q \]
    (iii)
    \(\displaystyle p\): $\displaystyle 2$ is a factor of $\displaystyle 12$; \(\displaystyle q\): $\displaystyle 3$ is a factor of $\displaystyle 12$; \(\displaystyle r\): $\displaystyle 6$ is a factor of 12.
    \[p\wedge q\wedge r \]
    (iv)
    \(\displaystyle p\): \(\displaystyle x\) is an odd integer; \(\displaystyle q\): \(\displaystyle x+1\) is an odd integer.
    \[p\vee q \]
    (v)
    \(\displaystyle p\): A number is divisible by $\displaystyle 2$; \(\displaystyle q\): A number is divisible by 3.
    \[p\vee q \]
    (vi)
    \(\displaystyle p\): \(\displaystyle x=2\) is a root of \(\displaystyle 3x^2-x-10=0\); \(\displaystyle q\): \(\displaystyle x=3\) is a root of \(\displaystyle 3x^2-x-10=0\).
    \[p\vee q \]
    (vii)
    \(\displaystyle p\): Students can take Hindi as an optional paper; \(\displaystyle q\): Students can take English as an optional paper.
    \[p\vee q \]
    Answer: (i) \(\displaystyle p\wedge q\); (ii) \(\displaystyle p\wedge q\); (iii) \(\displaystyle p\wedge q\wedge r\); (iv), (v), (vi), (vii) \(\displaystyle p\vee q\).

  6. Exercise 6

    Write down the negation of following compound statements
    (i)
    All rational numbers are real and complex.
    (ii)
    All real numbers are rationals or irrationals.
    (iii)
    x=2\displaystyle x=2 and x=3\displaystyle x=3 are roots of the Quadratic equation x2−5x+6=0\displaystyle x^2-5 x+6=0.
    (iv)
    A triangle has either 3\displaystyle 3-sides or 4\displaystyle 4-sides.
    (v)
    35\displaystyle 35 is a prime number or a composite number.
    (vi)
    All prime integers are either even or odd.
    (vii)
    ∣x∣\displaystyle |x| is equal to either x\displaystyle x or −x\displaystyle -x.
    (viii)
    6\displaystyle 6 is divisible by 2\displaystyle 2 and 3.

    Matches the book, not yet reviewed

    This working reaches the answer NCERT prints. It has not yet been read through by hand.

    NCERT’s answer
    (i)
    It is false that all rational numbers are real and complex
    (ii)
    It is false that all real numbers are rational or irrational
    (iii)
    \(\displaystyle x = 2\) is not a root of the quadratic equation \(\displaystyle x^{2} - 5x + 6 = 0\) or \(\displaystyle x = 3\) is not a root of the quadratic equation \(\displaystyle x^{2} - 5x + 6 = 0\)
    (iv)
    A triangle has neither $\displaystyle 3$-sides nor $\displaystyle 4$-sides
    (v)
    $\displaystyle 35$ is not a prime number and it is not a complex number
    (vi)
    It is false that all prime integers are either even or odd
    (vii)
    \(\displaystyle |x|\) is not equal to x and it not eqaul to \(\displaystyle -x\)
    (viii)
    $\displaystyle 6$ is not divisible by $\displaystyle 2$ or it is not divisible by 3.
    \[\sim(p\wedge q) = \sim p\vee \sim q, \qquad \sim(p\vee q) = \sim p\wedge \sim q \]
    (i)
    Some rational number is not real or not complex.
    (ii)
    Some real number is neither rational nor irrational.
    (iii)
    \(\displaystyle x=2\) is not a root, or \(\displaystyle x=3\) is not a root, of \(\displaystyle x^2-5x+6=0\).
    (iv)
    A triangle has neither $\displaystyle 3$ sides nor $\displaystyle 4$ sides.
    (v)
    $\displaystyle 35$ is neither prime nor composite.
    (vi)
    Some prime integer is neither even nor odd.
    (vii)
    \(\displaystyle |x|\neq x\) and \(\displaystyle |x|\neq -x\).
    (viii)
    $\displaystyle 6$ is not divisible by $\displaystyle 2$ or not divisible by 3.
    Answer: the negations (i)-(viii) above.

  7. Exercise 7

    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 will not study.
    (iv)
    The unit digit of an integer is 0\displaystyle 0 or 5\displaystyle 5 if it is divisible by 5.
    (v)
    The square of a prime number is not prime.
    (vi)
    2b=a+c\displaystyle 2 b=a+c, if a,b\displaystyle a, b and c\displaystyle c are in A.P.

    Matches the book, not yet reviewed

    This working reaches the answer NCERT prints. It has not yet been read through by hand.

    NCERT’s answer
    (i)
    If the number is odd number then its square is odd number
    (ii)
    If you take the dinner then you will get sweet dish
    (iii)
    If you will not study then you will fail
    (iv)
    If an integer is divisible by $\displaystyle 5$ then its unit digits are $\displaystyle 0$ or $\displaystyle 5$
    (v)
    If the number is prime then its square is not prime
    (vi)
    If a,b and c are in A.P then \(\displaystyle 2b = a + c\).
    (i)
    If a number is odd, then its square is odd.
    (ii)
    If dinner is over, then you will get a sweet dish.
    (iii)
    If you do not study, then you will fail.
    (iv)
    If an integer is divisible by $\displaystyle 5$, then its unit digit is $\displaystyle 0$ or 5.
    (v)
    If a number is prime, then its square is not prime.
    (vi)
    If \(\displaystyle a, b, c\) are in A.P., then \(\displaystyle 2b=a+c\).
    Answer: the conditionals (i)-(vi) above.

  8. Exercise 8

    Form the biconditional statement p↔q\displaystyle p \leftrightarrow q, where
    (i)
    p\displaystyle p : The unit digit of an integer is zero.
    q\displaystyle q : It is divisible by 5.
    (ii)
    p:\displaystyle p: A natural number n\displaystyle n is odd.
    q\displaystyle q : Natural number n\displaystyle n is not divisible by 2.
    (iii)
    p:\displaystyle p: A triangle is an equilateral triangle.
    q\displaystyle q : All three sides of a triangle are equal.

    Matches the book, not yet reviewed

    This working reaches the answer NCERT prints. It has not yet been read through by hand.

    NCERT’s answer
    (i)
    The unit digit of an integer is zero if and only if it is divisible by 5.
    (ii)
    A natural number n is odd if and only if it is not divisible by 2.
    (iii)
    A triangle is an equilateral triangle if and only if all three sides of triangle are equal.
    \[p\leftrightarrow q \;:\; p \text{ if and only if } q \]
    (i)
    The unit digit of an integer is zero if and only if it is divisible by 5.
    (ii)
    A natural number \(\displaystyle n\) is odd if and only if it is not divisible by 2.
    (iii)
    A triangle is an equilateral triangle if and only if all three of its sides are equal.
    Answer: the biconditionals (i)-(iii) above.

  9. Exercise 9

    Write down the contrapositive of the following statements:
    (i)
    If x=y\displaystyle x=y and y=3\displaystyle y=3, then x=3\displaystyle x=3.
    (ii)
    If n\displaystyle n is a natural number, then n\displaystyle n is an integer.
    (iii)
    If all three sides of a triangle are equal, then the triangle is equilateral.
    (iv)
    If x\displaystyle x and y\displaystyle y are negative integers, then xy\displaystyle x y is positive.
    (v)
    If natural number n\displaystyle n is divisible by 6\displaystyle 6, then n\displaystyle n is divisible by 2\displaystyle 2 and 3.
    (vi)
    If it snows, then the weather will be cold.
    (vii)
    If x\displaystyle x is a real number such that 0<x<1\displaystyle 0<x<1, then x2<1\displaystyle x^2<1.

    Check this one against your book

    NCERT’s printed answer for this exercise does not match its own question. This working follows the question as printed.

    \[p \Rightarrow q \;\equiv\; {\sim}q \Rightarrow {\sim}p, \qquad {\sim}(A \wedge B) \equiv {\sim}A \vee {\sim}B \]
    (i)
    If \(\displaystyle x \ne 3\), then \(\displaystyle x \ne y\) or \(\displaystyle y \ne 3\).
    (ii)
    If \(\displaystyle n\) is not an integer, then \(\displaystyle n\) is not a natural number.
    (iii)
    If a triangle is not equilateral, then its three sides are not all equal.
    (iv)
    If \(\displaystyle xy\) is not positive, then \(\displaystyle x\) and \(\displaystyle y\) are not both negative integers.
    (v)
    If \(\displaystyle n\) is not divisible by $\displaystyle 2$ or not divisible by $\displaystyle 3$, then \(\displaystyle n\) is not divisible by 6.
    (vi)
    If the weather is not cold, then it does not snow.
    (vii)
    If \(\displaystyle x\) is real and \(\displaystyle x^2 \ge 1\), then \(\displaystyle x \le 0\) or \(\displaystyle x \ge 1\).
    Answer: the contrapositives (i)-(vii) as written above.

  10. Exercise 10

    Write down the converse of 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 Taj Mahal.
    (iv)
    If the sum of squares of two sides of a triangle is equal to the square of third side of a 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\displaystyle x: y=3: 2, then 2x=3y\displaystyle 2 x=3 y.
    (vii)
    If S is a cyclic quadrilateral, then the opposite angles of S are supplementary.
    (viii)
    If x\displaystyle x is zero, then x\displaystyle x is neither positive nor negative.
    (ix)
    If two triangles are similar, then the ratio of their corresponding sides are equal.

    Matches the book, not yet reviewed

    This working reaches the answer NCERT prints. It has not yet been read through by hand.

    NCERT’s answer
    (i)
    If the rectangle R is rhombus then it is square.
    (ii)
    If tomorrow is Tuesday then today is Monday.
    (iii)
    If you must visit Taj Mahal you go to Agra.
    (iv)
    If the triangle is right angle then sum of squares of two sides of a triangle is equal to the square of third side.
    (v)
    If the triangle is equilateral then all three anlges of triangle are equal.
    (vi)
    If \(\displaystyle 2x = 3y\) then \(\displaystyle x:y = 3:2\)
    (vii)
    If the opposite angles of a quadrilaterals are supplementary then S is cyclic.
    (viii)
    If x is neither positive nor negative than x is 0.
    (ix)
    If the ratio of corresponding sides of two triangles are equal then trianges are similar.
    \[p \Rightarrow q \quad\longrightarrow\quad q \Rightarrow p \]
    (i)
    If \(\displaystyle R\) is a rhombus, then \(\displaystyle R\) is a square.
    (ii)
    If tomorrow is Tuesday, then today is Monday.
    (iii)
    If you must visit the Taj Mahal, then you go to Agra.
    (iv)
    If a triangle is right angled, then the sum of the squares of two sides equals the square of the third side.
    (v)
    If a triangle is equilateral, then all three angles are equal.
    (vi)
    If \(\displaystyle 2x = 3y\), then \(\displaystyle x : y = 3 : 2\).
    (vii)
    If the opposite angles of \(\displaystyle S\) are supplementary, then \(\displaystyle S\) is a cyclic quadrilateral.
    (viii)
    If \(\displaystyle x\) is neither positive nor negative, then \(\displaystyle x\) is zero.
    (ix)
    If the ratios of the corresponding sides of two triangles are equal, then the triangles are similar.
    Answer: the converses (i)-(ix) as written above.