Find the ratio in which the point
Question: Find the ratio in which the point R(5, 4, -6) divides the join of P(3, 2, -4) and Q(9, 8, -10). Solution: Let the ratio be k:1 in which point R divides point P and point Q. Using $\left(\frac{m x_{2}+n x_{1}}{m+n}, \frac{m y_{2}+n y_{1}}{m+n}, \frac{m z_{2}+n z_{1}}{m+n}\right)$, we get, Here $m$ and $n$ are $k$ and 1 . The point which this formula gives is already given, i.e. $\mathrm{R}(5,4,-6)$ and the joining points are $\mathrm{P}(3,2,-4)$ and $\mathrm{Q}(9,8,-10)$. $\left.(5,4,-6...
Read More →Write down the contrapositive
Question: 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 xy is positive. (v) If 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. Solution: (i) If ...
Read More →Form the bi conditional statement p ↔ q,
Question: Form the bi conditional statement p q, where (i) p: The unit digit of an integer is zero. q: 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. Solution: (i) p: The unit digit of an integer is zero. q: It is divisible by 5. In the bi conditional statement, we use if and only if. p: The unit digit of an integer is zero. q: It is divisible by 5. The...
Read More →Find the coordinates of the point that divides the join of
Question: Find the coordinates of the point that divides the join of A(-2, 4, 7) and B(3, - 5, 8) extremally in the ratio 2 : 1. Solution: The coordinates of point $R$ that divides the line segment joining points $P\left(x_{1}\right.$, $\left.\mathrm{y}_{1}, \mathrm{z} 1\right)$ and $Q\left(x_{2}, y_{2}, z_{2}\right)$ externally in the ratio $m: n$ are $\left(\frac{m x_{2}-n x_{1}}{m-n}, \frac{m y_{2}-n y_{1}}{m-n}, \frac{m z_{2}-n z_{1}}{m-n}\right)$ Point $A(-2,4,7)$ and $B(3,-5,8), m$ and $n$...
Read More →Rewrite each of the following statements
Question: 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 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. Solution: (i) The square of an odd number is odd. In the conditional statement, expression is If p, then q Now, The giv...
Read More →If f '(x) changes its sign from positive to negative
Question: If $f^{\prime}(x)$ changes its sign from positive to negative as $x$ increases through $c$ in the interval $(c-h, c+h)$, then $x=c$ is a point of____________ Solution: First derivative test states that iff'(x) changes sign from positive to negative asxincreases throughc, thencis a point of local maxima, andf(c) is local maximum value. Thus, if $f^{\prime}(x)$ changes its sign from positive to negative as $x$ increases through $c$ in the interval $(c-h, c+h)$, then $x=c$ is a point of l...
Read More →Find the value
Question: Let A(2, 1, -3) and B(5, -8, 3) be two given points. Find the coordinates of the point of trisection of the segment AB. Solution: The coordinates of point $R$ that divides the line segment joining points $P\left(x_{1}\right.$,$\left.\mathrm{y}_{1}, \mathrm{z}_{1}\right)$ and $Q\left(x_{2}, y_{2}, z_{2}\right)$ in the ratio $m: n$ are $\left(\frac{m x_{2}+n x_{1}}{m+n}, \frac{m y_{2}+n y_{1}}{m+n}, \frac{m z_{2}+n z_{1}}{m+n}\right)$ Point $A(2,1,-3)$ and $B(5,-8,3), m$ and $n$ are 2 an...
Read More →If f(x) has the second order derivative
Question: If $f(x)$ has the second order derivative at $x=c$ such that $f^{\prime}(c)=0$ and $f^{\prime \prime}(c)0$, then $c$ is a point of____________ Solution: Second derivative test: Let $f(x)$ be a function defined on an interval I and $c \in I .$ Suppose $f(x)$ be twice differentiable at $x=c .$ Then, $x=c$ is a point of local minima if $f^{\prime}(c)=0$ and $f^{\prime \prime}(c)0$. In this case, $f(c)$ is then the local minimum value of $f(x)$. So, if $f(x)$ has the second order derivativ...
Read More →Write down the negation of following compound statements
Question: 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 and x = 3 are roots of the Quadratic equation x 2 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. Solution: (i) All rational numbers are real an...
Read More →The maximum value
Question: The maximum value of $f(x)=\sin x+\cos x$ is_______________. Solution: The given function is $f(x)=\sin x+\cos x$. $f(x)=\sin x+\cos x$ Differentiating both sides with respect tox, we get $f^{\prime}(x)=\cos x-\sin x$ For maxima or minima, $f^{\prime}(x)=0$ $\Rightarrow \cos x-\sin x=0$ $\Rightarrow \cos x=\sin x$ $\Rightarrow \tan x=1$ $\Rightarrow x=\frac{\pi}{4}, \frac{5 \pi}{4}$ (Let us only consider the values of $x \in[0,2 \pi]$ ) Now, $f^{\prime \prime}(x)=-\sin x-\cos x$ At $x=...
Read More →Find the coordinates of the point which divides the join
Question: Find the coordinates of the point which divides the join of A(3, 2, 5) and B(-4, 2, -2) in the ratio 4 : 3. Solution: The coordinates of point R that divides the line segment joining points P $(\mathrm{X} 1 $\left.\mathrm{y}_{1}, \mathrm{z}_{1}\right)$ and $Q\left(x_{2}, y_{2}, z_{2}\right)$ in the ratio $m: n$ are $\left(\frac{m x_{2}+n x_{1}}{m+n}, \frac{m y_{2}+n y_{1}}{m+n}, \frac{m z_{2}+n z_{1}}{m+n}\right)$ Point A( 3, 2, 5 ) and B( -4, 2, -2 ), m and n are 4 and 3 respectively....
Read More →The maximum value
Question: The maximum value of $f(x)=\sin x+\cos x$ is_______________. Solution: The given function is $f(x)=\sin x+\cos x$. $f(x)=\sin x+\cos x$ Differentiating both sides with respect tox, we get $f^{\prime}(x)=\cos x-\sin x$ For maxima or minima, $f^{\prime}(x)=0$ $\Rightarrow \cos x-\sin x=0$ $\Rightarrow \cos x=\sin x$ $\Rightarrow \tan x=1$ $\Rightarrow x=\frac{\pi}{4}, \frac{5 \pi}{4}$ (Let us only consider the values of $x \in[0,2 \pi]$ ) Now, $f^{\prime \prime}(x)=-\sin x-\cos x$ At $x=...
Read More →Translate the following statements into symbolic form
Question: 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 3x 2 x 10 = 0 (vii) Students can take Hindi or English as an optional paper. Solution: (i) Rahul passed in Hindi and English. The given sentence is a compound statement in which components are p: Rahul p...
Read More →Find the point in xy-plane which is equidistant from the points
Question: Find the point in xy-plane which is equidistant from the points A(2, 0, 3), B(0, 3, 2) and C(0, 0, 1) Solution: The general point on xy plane is D(x, y, 0). Consider this point is equidistant to the points A(2, 0, 3), B(0, 3, 2) and C(0, 0, 1). AD = BD $\sqrt{(x-2)^{2}+(y-0)^{2}+(0-3)^{2}}=\sqrt{(x-0)^{2}+(y-3)^{2}+(0-2)^{2}}$ Squaring both sides $(x-2)^{2}+(y-0)^{2}+(0-3)^{2}=(x-0)^{2}+(y-3)^{2}+(0-2)^{2}$ $x^{2}-4 x+4+y^{2}+9=x^{2}+y^{2}-6 y+9+4$ $-4 x=-6 y \ldots(1)$ Also, AD = CD $...
Read More →The minimum value
Question: The minimum value of $f(x)=\sin x$ in $\left[\frac{-\pi}{2}, \frac{\pi}{2}\right]$ isThe minimum value of $f(x)=\sin x$ in $\left[\frac{-\pi}{2}, \frac{\pi}{2}\right]$ is_________ Solution: The given function is $f(x)=\sin x, x \in\left[\frac{-\pi}{2}, \frac{\pi}{2}\right]$. $f(x)=\sin x$ Differentiating both sides with respect tox, we get $f^{\prime}(x)=\cos x$ For maxima or minima, $f^{\prime}(x)=0$ $\Rightarrow \cos x=0$ $\Rightarrow x=-\frac{\pi}{2}$ or $x=\frac{\pi}{2}$, for all $...
Read More →The minimum value
Question: The minimum value of $f(x)=\sin x$ in $\left[\frac{-\pi}{2}, \frac{\pi}{2}\right]$ isThe minimum value of $f(x)=\sin x$ in $\left[\frac{-\pi}{2}, \frac{\pi}{2}\right]$ is_________ Solution: The given function is $f(x)=\sin x, x \in\left[\frac{-\pi}{2}, \frac{\pi}{2}\right]$. $f(x)=\sin x$ Differentiating both sides with respect tox, we get $f^{\prime}(x)=\cos x$ For maxima or minima, $f^{\prime}(x)=0$ $\Rightarrow \cos x=0$ $\Rightarrow x=-\frac{\pi}{2}$ or $x=\frac{\pi}{2}$, for all $...
Read More →Write the negation of the following simple statements
Question: Write the negation of the following simple statements (i) The number 17 is prime. (ii) 2 + 7 = 6. (iii) Violets are blue. (iv) 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 same as the perimeter of the circle. Solution: (i) The number 17 is prime. Negation ofstatement p is not p. The negation of p is symbolize...
Read More →Find the point in yz-plane which is equidistant from the points
Question: Find the point in yz-plane which is equidistant from the points A(3, 2, -1), B(1, -1, 0) and C(2, 1, 2). Solution: The general point on yz plane is D(0, y, z). Consider this point is equidistant to the points A(3, 2, -1), B(1, -1, 0) and C(2, 1, 2). $\therefore \mathrm{AD}=\mathrm{BD}$ $\sqrt{(0-3)^{2}+(y-2)^{2}+(z+1)^{2}}=\sqrt{(0-1)^{2}+(y+1)^{2}+(z-0)^{2}}$ Squaring both sides, $(0-3)^{2}+(y-2)^{2}+(z+1)^{2}=(0-1)^{2}+(y+1)^{2}+(z-0)^{2}$ $9+y^{2}-4 y+4+z^{2}+2 z+1=1+y^{2}+2 y+1+z^{...
Read More →Write the component statements of the following compound
Question: 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. Solution: (i) 57 is divisible by 2 or 3. A compound statement is a combination of two statements (Components). So, the components of the given statement 57 is divisible by 2 or 3 are p: 57 is divisible...
Read More →Find the coordinates of the point which is equidistant from the points
Question: Find the coordinates of the point which is equidistant from the points A(a, 0, 0), B(0, b, 0), C(0, 0, c) and O(0, 0, 0). Solution: Consider, D(x,y,z) point equidistant from points A(a, 0, 0), B(0, b, 0), C(0, 0, c) and O(0, 0, 0). AD = 0D $\sqrt{(x-a)^{2}+(y-0)^{2}+(z-0)^{2}}=\sqrt{(x-0)^{2}+(y-0)^{2}+(z-0)^{2}}$ Squaring both sides, $(x-a)^{2}+(y-0)^{2}+(z-0)^{2}=(x-0)^{2}+(y-0)^{2}+(z-0)^{2}$ $x^{2}+2 a x+a^{2}+y^{2}+z^{2}=x^{2}+y^{2}+z^{2}$ $a(2 x-a)=0$ as $a \neq 0$ $x=a / 2$ BD =...
Read More →If the solve the problem
Question: If $f(x)=\frac{1}{4 x^{2}+2 x+1}$, then its maximum value is____________ Solution: The given function is $f(x)=\frac{1}{4 x^{2}+2 x+1}$. The function $f(x)$ would attain its maximum value, when the value of $4 x^{2}+2 x+1$ is minimum. Let $g(x)=4 x^{2}+2 x+1$ $\therefore g^{\prime}(x)=8 x+2$ For maxima or minima, $g^{\prime}(x)=0$ $\Rightarrow 8 x+2=0$ $\Rightarrow x=-\frac{1}{4}$ Now, $g^{\prime \prime}(x)=80$ So, $x=-\frac{1}{4}$ is the point of local minimum of $g(x)$ Minimum value ...
Read More →If the solve the problem
Question: If $f(x)=\frac{1}{4 x^{2}+2 x+1}$, then its maximum value is____________ Solution: The given function is $f(x)=\frac{1}{4 x^{2}+2 x+1}$. The function $f(x)$ would attain its maximum value, when the value of $4 x^{2}+2 x+1$ is minimum. Let $g(x)=4 x^{2}+2 x+1$ $\therefore g^{\prime}(x)=8 x+2$ For maxima or minima, $g^{\prime}(x)=0$ $\Rightarrow 8 x+2=0$ $\Rightarrow x=-\frac{1}{4}$ Now, $g^{\prime \prime}(x)=80$ So, $x=-\frac{1}{4}$ is the point of local minimum of $g(x)$ Minimum value ...
Read More →Find the component statements of the following compound statements.
Question: 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. (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 r...
Read More →Find the point on the z-axis which is equidistant from the points
Question: Find the point on the z-axis which is equidistant from the points A(1, 5, 7) and B(5, 1, -4). Solution: Consider, C(0,0,z) point which lies on z axis and is equidistant from points A(1, 5, 7) and B(5, 1, -4) AC = BC $\sqrt{(0-1)^{2}+(0-5)^{2}+(z-7)^{2}}=\sqrt{(0-5)^{2}+(0-1)^{2}+(z+4)^{2}}$ Squaring both sides, $(0-1)^{2}+(0-5)^{2}+(z-7)^{2}=(0-5)^{2}+(0-1)^{2}+(z+4)^{2}$ $1+25+z^{2}-14 z+49=25+1+z^{2}+8 z+16$ $-22 z=-33$ Z = 1.5 The point C is (0,0,1.5)....
Read More →The function
Question: The function $f(x)=2 x^{3}-3 x^{2}-12 x+4$, has (a) two points of local maximum (b) two points of local minimum (c) one maximum and one minimum (d) no maximum no minimum Solution: The given function is $f(x)=2 x^{3}-3 x^{2}-12 x+4$ $f(x)=2 x^{3}-3 x^{2}-12 x+4$ Differentiating both sides with respect tox, we get $f^{\prime}(x)=6 x^{2}-6 x-12$ $\Rightarrow f^{\prime}(x)=6\left(x^{2}-x-2\right)$ $\Rightarrow f^{\prime}(x)=6(x+1)(x-2)$ For maxima or minima, $f^{\prime}(x)=0$ $\Rightarrow ...
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