Write the negation of the following statements:
Question: Write the negation of the following statements: (i)p: For every positive real numberx, the numberx 1 is also positive. (ii)q: All cats scratch. (iii)r: For every real numberx, eitherx 1 orx 1. (iv)s: There exists a numberxsuch that 0 x 1. Solution: (i) The negation of statementpis as follows. There exists a positive real numberx, such thatx 1 is not positive. (ii) The negation of statementqis as follows. There exists a cat that does not scratch. (iii) The negation of statementris as fo...
Read More →Find the equation of the normal at the point $left(a m^{2}, a m^{3}ight)$ for the curve $a y^{2}=x^{3}$.
Question: Find the equation of the normal at the point Solution: The equation of the given curve is $a y^{2}=x^{3}$. On differentiating with respect tox, we have: $2 a y \frac{d y}{d x}=3 x^{2}$ $\Rightarrow \frac{d y}{d x}=\frac{3 x^{2}}{2 a y}$ The slope of a tangent to the curve at $\left(x_{0}, y_{0}\right)$ is $\left.\frac{d y}{d x}\right]_{\left(x_{0}, y_{0}\right)}$. $\Rightarrow$ The slope of the tangent to the given curve at $\left(a m^{2}, a m^{3}\right)$ is $\left.\frac{d y}{d x}\righ...
Read More →Which of the following statements are true and which are false? In each case give a valid reason for saying so.
Question: Which of the following statements are true and which are false? In each case give a valid reason for saying so. (i)p: Each radius of a circle is a chord of the circle. (ii)q: The centre of a circle bisects each chord of the circle. (iii)r: Circle is a particular case of an ellipse. (iv)s: Ifxandyare integers such thatxy, then x y. (v)t: $\sqrt{11}$ is a rational number. Solution: (i) The given statementpis false. According to the definition of chord, it should intersect the circle at t...
Read More →By giving a counter example, show that the following statements are not true.
Question: By giving a counter example, show that the following statements are not true. (i)p: If all the angles of a triangle are equal, then the triangle is an obtuse angled triangle. (ii) $q$ : The equation $x^{2}-1=0$ does not have a root lying between 0 and 2 . Solution: (i) The given statement is of the form ifqthenr. q: All the angles of a triangle are equal. r: The triangle is an obtuse-angled triangle. The given statementphas to be proved false. For this purpose, it has to be proved that...
Read More →If 27x =93x, find x.
Question: If $27^{x}=\frac{9}{3^{x}}$, find $\mathrm{x}$ Solution: We are given $27^{x}=\frac{9}{3^{x}}$. We have to find the value of $x$ Since $\left(3^{3}\right)^{x}=\frac{3^{2}}{3^{x}}$ By using the law of exponents $\frac{a^{m}}{a^{n}}=a^{m-n}$ we get, $3^{3 x}=3^{2-x}$ On equating the exponents we get, $\begin{aligned} 3 x =2-x \\ 3 x+x =2 \\ 4 x =2 \\ x =\frac{2}{4} \end{aligned}$ $x=\frac{1}{2}$...
Read More →Show that the following statement is true by the method of contrapositive.
Question: Show that the following statement is true by the method of contrapositive. $p:$ If $x$ is an integer and $x^{2}$ is even, then $x$ is also even. Solution: $p$ : If $x$ is an integer and $x^{2}$ is even, then $x$ is also even. Let $q: x$ is an integer and $x^{2}$ is even. $r . x$ is even. To prove thatpis true by contrapositive method, we assume thatris false, and prove thatqis also false. Letxis not even. To prove that $q$ is false, it has to be proved that $x$ is not an integer or $x^...
Read More →Find the points on the curve
Question: Find the points on the curve $x^{2}+y^{2}-2 x-3=0$ at which the tangents are parallel to the $x$-axis. Solution: The equation of the given curve is $x^{2}+y^{2}-2 x-3=0$. On differentiating with respect tox, we have: $2 x+2 y \frac{d y}{d x}-2=0$ $\Rightarrow y \frac{d y}{d x}=1-x$ $\Rightarrow \frac{d y}{d x}=\frac{1-x}{y}$ Now, the tangents are parallel to thex-axis if the slope of the tangent is 0. $\therefore \frac{1-x}{y}=0 \Rightarrow 1-x=0 \Rightarrow x=1$ But, $x^{2}+y^{2}-2 x-...
Read More →Show that the statement “For any real numbers a and b, a2 = b2
Question: Show that the statement "For any real numbers $a$ and $b, a^{2}=b^{2}$ implies that $a=b^{\prime}$ is not true by giving a counter-example. Solution: The given statement can be written in the form of if-then as follows. If $a$ and $b$ are real numbers such that $a^{2}=b^{2}$, then $a=b$. Let $p: a$ and $b$ are real numbers such that $a^{2}=b^{2}$. q:a=b The given statement has to be proved false. For this purpose, it has to be proved that if $p$, then $\sim q$. To show this, two real n...
Read More →For the curve
Question: For the curve $y=4 x^{3}-2 x^{5}$, find all the points at which the tangents passes through the origin. Solution: The equation of the given curve is $y=4 x^{3}-2 x^{5}$. $\therefore \frac{d y}{d x}=12 x^{2}-10 x^{4}$ Therefore, the slope of the tangent at a point $(x, y)$ is $12 x^{2}-10 x^{4}$. The equation of the tangent at (x,y) is given by, $Y-y=\left(12 x^{2}-10 x^{4}\right)(X-x)$ ....(1) When the tangent passes through the origin (0, 0), thenX=Y= 0. Therefore, equation (1) reduce...
Read More →Show that the statement “For any real numbers a and b, a2 = b2
Question: Show that the statement "For any real numbers $a$ and $b, a^{2}=b^{2}$ implies that $a=b$ " is not true by giving a counter-example. Solution: The given statement can be written in the form of if-then as follows. If $a$ and $b$ are real numbers such that $a^{2}=b^{2}$, then $a=b$. Let $p: a$ and $b$ are real numbers such that $a^{2}=b^{2}$. $q: a=b$ The given statement has to be proved false. For this purpose, it has to be proved that if $p$, then $\sim q$. To show this, two real numbe...
Read More →Show that:
Question: Show that: (i) $\frac{1}{1+x^{a-b}}+\frac{1}{1+x^{b-a}}=1$ (ii) $\left[\left\{\frac{x^{a(a-b)}}{x^{a(a+b)}}\right\} \div\left\{\frac{x^{b(b-a)}}{x^{b(b+a)}}\right\}\right]^{a+b}=1$ (iii) $\left(x^{\frac{1}{a-b}}\right)^{\frac{1}{a-c}}\left(x^{\frac{1}{b-c}}\right)^{\frac{1}{b-a}}\left(x^{\frac{1}{c-a}}\right)^{\frac{1}{c-b}}=1$ (iv) $\left(\frac{x^{a^{2}+b^{2}}}{x^{a b}}\right)^{a+b}\left(\frac{x^{b^{2}+c^{2}}}{x^{b c}}\right)^{b+c}\left(\frac{x^{c^{2}+a^{2}}}{x^{a c}}\right)^{a+c}=x^{...
Read More →Show that the statement
Question: Show that the statement $p$ : "If $x$ is a real number such that $x^{3}+4 x=0$, then $x$ is 0 " is true by (i) direct method (ii) method of contradiction (iii) method of contrapositive Solution: $p:$ "If $x$ is a real number such that $x^{3}+4 x=0$, then $x$ is $0 "$. Let $q$ : $x$ is a real number such that $x^{3}+4 x=0$ r:xis 0. (i) To show that statementpis true, we assume thatqis true and then show thatris true. Therefore, let statementqbe true. $\therefore x^{3}+4 x=0$ $x\left(x^{...
Read More →Find the points on the curve
Question: Find the points on the curve $y=x^{3}$ at which the slope of the tangent is equal to the $y$-coordinate of the point. Solution: The equation of the given curve is $y=x^{3}$. $\therefore \frac{d y}{d x}=3 x^{2}$ The slope of the tangent at the point (x,y) is given by, $\left.\frac{d y}{d x}\right]_{(x, y)}=3 x^{2}$ When the slope of the tangent is equal to the $y$-coordinate of the point, then $y=3 x^{2}$. Also, we have $y=x^{3}$. $\therefore 3 x^{2}=x^{3}$ $\Rightarrow x^{2}(x-3)=0$ $\...
Read More →Given statements in (a) and (b). Identify the statements given below as contrapositive or converse of each other.
Question: Given statements in (a) and (b). Identify the statements given below as contrapositive or converse of each other. (a) If you live in Delhi, then you have winter clothes. (i) If you do not have winter clothes, then you do not live in Delhi. (ii) If you have winter clothes, then you live in Delhi. (b) If a quadrilateral is a parallelogram, then its diagonals bisect each other. (i) If the diagonals of a quadrilateral do not bisect each other, then the quadrilateral is not a parallelogram....
Read More →Write each of the following statement in the form “if-then”.
Question: Write each of the following statement in the form if-then. (i) You get a job implies that your credentials are good. (ii) The Banana trees will bloom if it stays warm for a month. (iii) A quadrilateral is a parallelogram if its diagonals bisect each other. (iv) To get A+in the class, it is necessary that you do the exercises of the book. Solution: (i) If you get a job, then your credentials are good. (ii) If the Banana tree stays warm for a month, then it will bloom. (iii) If the diago...
Read More →Show that the tangents to the curve y
Question: Show that the tangents to the curve $y=7 x^{3}+11$ at the points where $x=2$ and $x=-2$ are parallel. Solution: The equation of the given curve is $y=7 x^{3}+11$. $\therefore \frac{d y}{d x}=21 x^{2}$ The slope of the tangent to a curve at $\left(x_{0}, y_{0}\right)$ is $\left.\frac{d y}{d x}\right]_{\left(x_{0}, y_{0}\right)}$ Therefore, the slope of the tangent at the point wherex= 2 is given by, $\left.\frac{d y}{d x}\right]_{x=-2}=21(2)^{2}=84$ It is observed that the slopes of the...
Read More →Write the contrapositive and converse of the following statements.
Question: Write the contrapositive and converse of the following statements. (i) Ifxis a prime number, thenxis odd. (ii) It the two lines are parallel, then they do not intersect in the same plane. (iii) Something is cold implies that it has low temperature. (iv) You cannot comprehend geometry if you do not know how to reason deductively. (v)xis an even number implies thatxis divisible by 4 Solution: (i) The contrapositive is as follows. If a numberxis not odd, thenxis not a prime number. The co...
Read More →Rewrite the following statement with “if-then” in five different ways conveying the same meaning.
Question: Rewrite the following statement with if-then in five different ways conveying the same meaning. If a natural number is odd, then its square is also odd. Solution: The given statement can be written in five different ways as follows. (i) A natural number is odd implies that its square is odd. (ii) A natural number is odd only if its square is odd. (iii) For a natural number to be odd, it is necessary that its square is odd. (iv) For the square of a natural number to be odd, it is suffic...
Read More →State whether the “Or” used in the following statements is “exclusive “or” inclusive. Give reasons for your answer.
Question: State whether the Or used in the following statements is exclusive or inclusive. Give reasons for your answer. (i) Sun rises or Moon sets. (ii) To apply for a driving licence, you should have a ration card or a passport. (iii) All integers are positive or negative. Solution: (i) Here, or is exclusive because it is not possible for the Sun to rise and the moon to set together. (ii) Here, or is inclusive since a person can have both a ration card and a passport to apply for a driving lic...
Read More →Check whether the following pair of statements is negation of each other. Give reasons for the answer.
Question: Check whether the following pair of statements is negation of each other. Give reasons for the answer. (i)x+y=y+xis true for every real numbersxandy. (ii) There exists real numberxandyfor whichx+y=y+x. Solution: The negation of statement (i) is as follows. There exists real numberxandyfor whichx+yy+x. This is not the same as statement (ii). Thus, the given statements are not the negation of each other....
Read More →Identify the quantifier in the following statements and write the negation of the statements.
Question: Identify the quantifier in the following statements and write the negation of the statements. (i) There exists a number which is equal to its square. (ii) For every real numberx,xis less thanx+ 1. (iii) There exists a capital for every state in India. Solution: (i) The quantifier is There exists. The negation of this statement is as follows. There does not exist a number which is equal to its square. (ii) The quantifier is For every. The negation of this statement is as follows. There ...
Read More →For each of the following compound statements first identify the connecting words and then break it into component statements.
Question: For each of the following compound statements first identify the connecting words and then break it into component statements. (i) All rational numbers are real and all real numbers are not complex. (ii) Square of an integer is positive or negative. (iii) The sand heats up quickly in the Sun and does not cool down fast at night. (iv) $x=2$ and $x=3$ are the roots of the equation $3 x^{2}-x-10=0$. Solution: (i) Here, the connecting word is and. The component statements are as follows. p...
Read More →Find the component statements of the following compound statements and check whether they are true or false.
Question: Find the component statements of the following compound statements and check whether they are true or false. (i) Number 3 is prime or it is odd. (ii) All integers are positive or negative. (iii) 100 is divisible by 3, 11 and 5. Solution: (i) The component statements are as follows. p: Number 3 is prime. q: Number 3 is odd. Both the statements are true. (ii) The component statements are as follows. p: All integers are positive. q: All integers are negative. Both the statements are false...
Read More →Find the equation of the tangent line to the curve
Question: Find the equation of the tangent line to the curve $y=x^{2}-2 x+7$ which is (a) parallel to the line 2xy+ 9 = 0 (b) perpendicular to the line 5y 15x= 13. Solution: The equation of the given curve is $y=x^{2}-2 x+7$. On differentiating with respect tox, we get: $\frac{d y}{d x}=2 x-2$ (a) The equation of the line is 2xy+ 9 = 0. 2xy+ 9 = 0 ⇒y= 2x+ 9 This is of the formy=mx+c. Slope of the line = 2 If a tangent is parallel to the line 2xy+ 9 = 0, then the slope of the tangent is equal to ...
Read More →Are the following pairs of statements negations of each other?
Question: Are the following pairs of statements negations of each other? (i) The numberxis not a rational number. The numberxis not an irrational number. (ii) The numberxis a rational number. The numberxis an irrational number. Solution: (i) The negation of the first statement is the numberxis a rational number. This is same as the second statement. This is because if a number is not an irrational number, then it is a rational number. Therefore, the given statements are negations of each other. ...
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