A particle moves along the curve
Question: A particle moves along the curve $y=(2 / 3) x^{3}+1$. Find the points on the curve at which the $y$-coordinate is changing twice as fast as the $x$-coordinate. Solution: Here, $y=\frac{2}{3} x^{3}+1$ $\Rightarrow \frac{d y}{d t}=2 x^{2} \frac{d x}{d t}$ $\Rightarrow 2 \frac{d x}{d t}=2 x^{2} \frac{d x}{d t}$ $\left[\because \frac{d y}{d t}=2 \frac{d x}{d t}\right]$ $\Rightarrow x=\pm 1$ Substituting the value of $x=1$ and $x=-1$ in $y=\frac{2}{3} x^{3}+1$, we get $\Rightarrow y=\frac{5...
Read More →If AM and GM of the roots of a quadratic equation are 10 and 8 respectively then obtain the quadratic equation.
Question: If AM and GM of the roots of a quadratic equation are 10 and 8 respectively then obtain the quadratic equation. Solution: To find: The quadratic equation. Given: (i) AM of roots of quadratic equation is 10 (ii) GM of roots of quadratic equation is 8 Formula used: (i) Arithmetic mean between a and $b=\frac{a+b}{2}$ (ii) Geometric mean between $a$ and $b=\sqrt{a b}$ Let the roots be p and q Arithmetic mean of roots $p$ and $q=\frac{p+q}{2}=10$ $\Rightarrow \frac{p+q}{2}=10$ ⇒ p + q = 20 ...
Read More →A kite is 120 m high and 130 m of string is out.
Question: A kite is 120 m high and 130 m of string is out. If the kite is moving away horizontally at the rate of 52 m/sec, find the rate at which the string is being paid out. Solution: In the right triangle $A B C$, Here, $A B^{2}+B C^{2}=A C^{2}$ $\Rightarrow x^{2}+(120)^{2}=y^{2}$ $\Rightarrow 2 x \frac{d x}{d t}=2 y \frac{d y}{d t}$ $\Rightarrow \frac{d y}{d t}=\frac{x}{y} \frac{d x}{d t}$ $\Rightarrow \frac{d y}{d t}=\frac{50}{130} \times 52$ $\left[\because x=\sqrt{(130)^{2}-(120)^{2}}=50...
Read More →Sand is being poured onto a conical pile
Question: Sand is being poured onto a conical pile at the constant rate of 50 cm3/ minute such that the height of the cone is always one half of the radius of its base. How fast is the height of the pile increasing when the sand is 5 cm deep. Solution: Let $r$ be the radius, $h$ be the height and $V$ be the volume of the conical pile at any time $t$. Then, $V=\frac{1}{3} \pi r^{2} h$ $\Rightarrow V=\frac{1}{3} \pi(2 h)^{2} h$ $\left[\because h=\frac{r}{2}\right]$ $\Rightarrow V=\frac{4}{3} \pi \...
Read More →The volume of metal in a hollow sphere is constant.
Question: The volume of metal in a hollow sphere is constant. If the inner radius is increasing at the rate of 1 cm/sec, find the rate of increase of the outer radius when the radii are 4 cm and 8 cm respectively. Solution: Let $r_{1}$ be the inner radius and $r_{2}$ be the outer radius and $V$ be the volume of the hollow sphere at any time $t .$ Then, $V=\frac{4}{3} \pi\left(r_{1}^{3}-r_{2}^{3}\right)$ $\Rightarrow \frac{d V}{d t}=4 \pi\left(r_{1}^{2} \frac{d r_{1}}{d t}-r_{2}^{2} \frac{d r_{2}...
Read More →The radius of a cylinder is increasing at the rate 2 cm/sec.
Question: The radius of a cylinder is increasing at the rate 2 cm/sec. and its altitude is decreasing at the rate of 3 cm/sec. Find the rate of change of volume when radius is 3 cm and altitude 5 cm. Solution: Let $r$ be the radius, $h$ be the height and $V$ be the volume of the cylinder at any time $t .$ Then, $V=\pi r^{2} h$ $\Rightarrow \frac{d V}{d t}=2 \pi r h \frac{d r}{d t}+\pi r^{2} \frac{d h}{d t}$ $\Rightarrow \frac{d V}{d t}=\pi r\left(2 h \frac{d r}{d t}+r \frac{d h}{d t}\right)$ $\R...
Read More →The surface area of a spherical bubble
Question: The surface area of a spherical bubble is increasing at the rate of 2 cm2/s. When the radius of the bubble is 6 cm, at what rate is the volume of the bubble increasing? Solution: Let $r$ be the radius, $S$ be the surface area and $V$ be the volume of the sphere at any time $t$. Then, $S=4 \pi r^{2}$ $\Rightarrow \frac{d S}{d t}=8 \pi r \frac{d r}{d t}$ $\Rightarrow \frac{d r}{d t}=\frac{1}{8 \pi r} \frac{d S}{d t}$ $\Rightarrow \frac{d r}{d t}=\frac{2}{8 \pi \times 6}$ $\Rightarrow \fr...
Read More →A man 2 metres high walks at a uniform speed of 6 km/h
Question: A man 2 metres high walks at a uniform speed of 6 km/h away from a lamp-post 6 metres high. Find the rate at which the length of his shadow increases. Solution: LetABbe the lamp post. Let at any timet,the manCDbe at a distance ofxkm from the lamp post andym be the length of his shadow CE. Since triangles $A B E$ and $C D E$ are similar, $\frac{A B}{C D}=\frac{A E}{C E}$ $\Rightarrow \frac{6}{2}=\frac{x+y}{y}$ $\Rightarrow \frac{x}{y}=\frac{6}{2}-1=2$ $\Rightarrow \frac{d y}{d t}=\frac{...
Read More →If 56 x 32y is divisible by 18,
Question: If 56 x 32y is divisible by 18, find the least value of y. Solution: It is given that, the number 56 x 32y is divisible by 18. Then, it is also divisible by each factor of 18. Thus, it is divisible by 2 as well as 3. Now, the number is divisible by-2, its units digit must be an even number that is 0, 2,4, 6, Therefore, the least value of y is 0. Again, the number is divisible by 3 also, sum of its digits is a multiple of 3. i.e. 5 + 6 + x + 3 + 2 + y is a multiple of 3 = 16 + x + y = 0...
Read More →A man 2 metres high walks at a uniform speed of 6 km/h
Question: A man 2 metres high walks at a uniform speed of 6 km/h away from a lamp-post 6 metres high. Find the rate at which the length of his shadow increases. Solution: LetABbe the lamp post. Let at any timet,the manCDbe at a distance ofxkm from the lamp post andym be the length of his shadow CE. Since triangles $A B E$ and $C D E$ are similar, $\frac{A B}{C D}=\frac{A E}{C E}$ $\Rightarrow \frac{6}{2}=\frac{x+y}{y}$ $\Rightarrow \frac{x}{y}=\frac{6}{2}-1=2$ $\Rightarrow \frac{d y}{d t}=\frac{...
Read More →If 123123A4 is divisible by 11,
Question: If 123123A4 is divisible by 11, find the value of A. Solution: Given, 12312344 is divisible by 11, then we have (1 + 3 + 2 + 4) (2 + 1 + 3 + 4) is a multiple of 11. i.e. (6+ 4)-10 =0,11,22,,.. = A-4 = 0,11,22, = A-4 = 0 [A is a digit of the given number] = A = 4...
Read More →Show that the product of n geometric means between a and b is equal
Question: Show that the product of n geometric means between a and b is equal to the nth power of the single GM between a and b. Solution: To prove: Product of n geometric means between a and b is equal to the nth power of the single GM between a and b. Formula used:(i) Geometric mean between $a$ and $b=\sqrt{a b}$ (ii) Sum of $n$ terms of A.P. $=\frac{(n)(n+1)}{2}$ Let the $\mathrm{n}$ geometric means between and $\mathrm{b}$ be $\mathrm{G}_{1}, \mathrm{G}_{2}, \mathrm{G}_{3}, \ldots \mathrm{G}...
Read More →If 148101B095 is divisible by 33,
Question: If 148101B095 is divisible by 33, find the value of B. Solution: Given that the number 148101S095 is divisible by 33, therefore it is also divisible by 3 and 11 both. Now, the number is divisible by 3, its sum of digits is a multiple of 3. i.e. 1 + 4+ 8+1 + 0+1 + B+ 0+ 9+ 5 is a multiple of 3. 29 + B = 0, 3, 6,9, = B=1,4,7 (i) Also, given number is divisible by 11, therefore (1+ 8 + 0+ B + 9)-(4 + 1+ 1+ 0 + 5)=0, 11,22, = (18 + B) -11 = 0,11,22 B+ 7 = 0,11,22 = B+7 = 11 = B = 4 (ii) Fr...
Read More →If from a two-digit number, we subtract the number
Question: If from a two-digit number, we subtract the number formed by reversing its digits then the result so obtained is a perfect cube. How many such numbers are possible? Write all of them. Solution: Let ab be any two-digit number. Then, the digit formed by reversing it digits is ba. Now, ab-ba = (10a+b)-(10b +a) =(10a-a)+(b-10b) = 9a 9b = 9(a b) Further, since ab-ba is a perfect cube and is a multiple of 9. .-.The possible value of a b is 3. i.e. a = b + 3 Here, b can take value from 0 to 6...
Read More →Water is running into an inverted cone at the rate of π cubic metres per minute.
Question: Water is running into an inverted cone at the rate of cubic metres per minute. The height of the cone is 10 metres, and the radius of its base is 5 m. How fast the water level is rising when the water stands 7.5 m below the base. Solution: Let $r$ be the radius, $h$ be the height and $\mathrm{V}$ be the volume of the cone at any time $t$. Then, $V=\frac{1}{3} \pi r^{2} h$ $\Rightarrow \frac{d V}{d t}=\frac{1}{3} \pi r^{2} \frac{d h}{d t}+\frac{2}{3} \pi r h \frac{d r}{d t}$ Now, $\frac...
Read More →756x is a multiple of 11,
Question: 756x is a multiple of 11, find the value of x. Solution: We are given that, 756x is a multiple of 11. Then, we have to find the value of x. Since, 756x is divisible by 11, then (7 + 6) (5 + x) is a multiple of 11, i.e. 8-x = 0,11,22, = 8- x = 0 = x = 8...
Read More →1y3y6is divisible by 11.
Question: 1y3y6is divisible by 11. Find the value of y. Solution: It is given that, 1y3y6 is divisible by 11. Then, we have (1 + 3 + 6) (y + y) = 0,11,22, = 10-2y= 0,11,22, = 10-2y = 0 [other values give a negative number] = 2y = 10 = y= 5...
Read More →A balloon in the form of a right circular cone surmounted by a hemisphere,
Question: A balloon in the form of a right circular cone surmounted by a hemisphere, having a diametre equal to the height of the cone, is being inflated. How fast is its volume changing with respect to its total heighth, whenh= 9 cm. Solution: Let $r$ be the radius of the hemisphere, $h$ be the height and $V$ be the volume of the cone. Then, $H=h+r$ $\Rightarrow H=3 r$ $[\because h=2 r]$ $\Rightarrow \frac{d H}{d t}=3 \frac{d r}{d t}$ When $H=9 \mathrm{~cm}, r=3 \mathrm{~cm}$ Volume $=\frac{1}{...
Read More →The top of a ladder 6 metres long is resting against a vertical wall on a level pavement,
Question: The top of a ladder 6 metres long is resting against a vertical wall on a level pavement, when the ladder begins to slide outwards. At the moment when the foot of the ladder is 4 metres from the wall, it is sliding away from the wall at the rate of 0.5 m/sec. How fast is the top-sliding downwards at this instance?How far is the foot from the wall when it and the top are moving at the same rate? Solution: Let the bottom of the ladder be at a distance ofxm from the wall and its top be at...
Read More →The top of a ladder 6 metres long is resting against a vertical wall on a level pavement,
Question: The top of a ladder 6 metres long is resting against a vertical wall on a level pavement, when the ladder begins to slide outwards. At the moment when the foot of the ladder is 4 metres from the wall, it is sliding away from the wall at the rate of 0.5 m/sec. How fast is the top-sliding downwards at this instance?How far is the foot from the wall when it and the top are moving at the same rate? Solution: Let the bottom of the ladder be at a distance ofxm from the wall and its top be at...
Read More →If a, b, c are in AP, x is the GM between a and b; y is the GM between
Question: If a, b, c are in AP, x is the GM between a and b; y is the GM between b and c; then show that $b^{2}$ is the AM between $x^{2}$ and $y^{2}$. Solution: To prove: $b^{2}$ is the AM between $x^{2}$ and $y^{2}$. Given: (i) a, b, c are in AP (ii) $x$ is the GM between $a$ and $b$ (iii) $y$ is the GM between $b$ and $c$ Formula used: (i) Arithmetic mean between a and $b=\frac{a+b}{2}$ (ii) Geometric mean between $a$ and $b=\sqrt{a b}$ As a, b, c are in A.P. ⇒ 2b = a + c (i) As x is the GM b...
Read More →Find the value of k,
Question: Find the value of k, where 31K2 is divisible by 6. Solution: Given, 31k2 is divisible by 6. Then, it is also divisible by 2 and 3 both. Now, 31K2 is divisible by 3, sum of its digits is a multiple of 3. i.e. 3+ 1 + k + 2 = 0, 3, 6, 9,12, = k+ 6 = 0, 3,6, 9,12 = k = 0 or 3, 6, 9...
Read More →Find an angle θ
Question: Find an angle $\theta$ (i) which increases twice as fast as its cosine. (ii) whose rate of increase twice is twice the rate of decrease of its cosine. Solution: (i) Let $x=\cos \theta$ Differentiating both sides with respect to $t$, we get $\frac{\mathrm{d} x}{\mathrm{~d} t}=\frac{\mathrm{d}(\cos \theta)}{\mathrm{d} t}$ $=-\sin \theta \frac{\mathrm{d} \theta}{\mathrm{d} t}$ But it is given that $\frac{\mathrm{d} \theta}{\mathrm{d} t}=2 \frac{\mathrm{d} x}{\mathrm{~d} t}$ $\Rightarrow \...
Read More →212 x 5 is a multiple of 3 and 11.
Question: 212 x 5 is a multiple of 3 and 11. Find the value of x. Solution: Since, 212 x 5 is a multiple of 3, 2 +1 + 2 +x+5 = 0, 3, 6,9,12,15,18, = 10 + x = 0, 3, 6,.. = x =2, 5, 8 (i) Again, 2125 is a multiple of 11, (2 + 2 + 5) (1 + x) = 0,11,22, 33 = 8 x = 0,11,22, = x = 8 (ii) From Eqs. (i) and (ii), we have x = 8...
Read More →Prove the following
Question: If $27 \div A=33$, then find the value of $\mathrm{A}$ Solution: We observe that, 4 x 3 can never be a single digit number 2, so 4 x 3 must be a two-digit number, whose tens digit is 2 and units digit is the number less than or equal to 4. Therefore, the value of 4 can be 9, as the values of 4 from 1 to 8 do not fit....
Read More →