If x = 1 and y = 6 is a solution of the equation
Question: If $x=1$ and $y=6$ is a solution of the equation $8 x-a y+a^{2}=0$, find the values of $a$. Solution: We are given, $8 x$-ay $+a^{2}=0(1,6)$ is a solution of equation $8 x-a y+a^{2}=0$ Substituting $x=1$ and $y=6$ in $8 x-a y+a^{2}=0$, we get $8 \times 1-a \times 6+a^{2}=0$ $\Rightarrow a^{2}-6 a+8=0$ Using quadratic factorization $a^{2}-4 a-2 a+8=0 a(a-4)-2(a-4)=0(a-2)(a-4)=0$ a = 2, 4...
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Question: $\int_{0}^{\frac{2}{3}} \frac{d x}{4+9 x^{2}}$ equals A. $\frac{\pi}{6}$ B. $\frac{\pi}{12}$ C. $\frac{\pi}{24}$ D. $\frac{\pi}{4}$ Solution: $\int \frac{d x}{4+9 x^{2}}=\int \frac{d x}{(2)^{2}+(3 x)^{2}}$ Put $3 x=t \Rightarrow 3 d x=d t$ $\therefore \int \frac{d x}{(2)^{2}+(3 x)^{2}}=\frac{1}{3} \int \frac{d t}{(2)^{2}+t^{2}}$ $=\frac{1}{3}\left[\frac{1}{2} \tan ^{-1} \frac{t}{2}\right]$ $=\frac{1}{6} \tan ^{-1}\left(\frac{3 x}{2}\right)$ $=\mathrm{F}(x)$ By second fundamental theore...
Read More →If x = 2a + 1 and y = a -1 is a solution of the equation
Question: If x = 2a + 1 and y = a -1 is a solution of the equation 2x - 3y + 5 = 0, find the value of a. Solution: We are given, 2x - 3y + 5 = 0 (2a + 1, a - 1) is the solution of equation 2x - 3y + 5 = 0. Substituting x = 2a + 1 and y = a - 1 in 2x - 3y + 5 = 0, We get 2 2a + (1- 3) x a - 1 + 5 = 0 ⟹ 4a + 2 - 3a + 3 + 5 = 0 ⟹ a + 10 = 0 ⟹ a = - 10...
Read More →If f(x) = sin
Question: Iff(x) = sin [2]x+ sin []2x, where [x] denotes the greatest integer less than or equal tox, then (a)f(/2) = 1 (b)f() = 2 (c)f(/4) = 1 (d) None of these Solution: (a)f(/2) = 1 $f(x)=\sin \left[\pi^{2}\right] x+\sin \left[-\pi^{2}\right] x$ $\Rightarrow f(x)=\sin [9.8] x+\sin [-9.8] x$ $\Rightarrow f(x)=\sin 9 x-\sin 10 x$ $f\left(\frac{\pi}{2}\right)=\sin 9 \times \frac{\pi}{2}-\sin 10 \times \frac{\pi}{2}$ $\Rightarrow f\left(\frac{\pi}{2}\right)=1-0=1$...
Read More →If f(x) = sin
Question: Iff(x) = sin [2]x+ sin []2x, where [x] denotes the greatest integer less than or equal tox, then (a)f(/2) = 1 (b)f() = 2 (c)f(/4) = 1 (d) None of these Solution: (a)f(/2) = 1 $f(x)=\sin \left[\pi^{2}\right] x+\sin \left[-\pi^{2}\right] x$ $\Rightarrow f(x)=\sin [9.8] x+\sin [-9.8] x$ $\Rightarrow f(x)=\sin 9 x-\sin 10 x$ $f\left(\frac{\pi}{2}\right)=\sin 9 \times \frac{\pi}{2}-\sin 10 \times \frac{\pi}{2}$ $\Rightarrow f\left(\frac{\pi}{2}\right)=1-0=1$...
Read More →The sum of digits of a two digit number is 13.
Question: The sum of digits of a two digit number is 13. If the number is subtracted from the one obtained by interchanging the digits, the result is 45. What is the number? Solution: Let the digits at units and tens place of the given number bexandyrespectively. Thus, the number is. The sum of the digits of the number is 13. Thus, we have After interchanging the digits, the number becomes. The difference between the number obtained by interchanging the digits and the original number is 45. Thus...
Read More →Find the value of λ, if x = -λ and y = 5/2
Question: Find the value of , if x = - and y = 5/2 is a solution of the equation x + 4y 7 = 0 Solution: We are given, x + 4y - 7 = 0 (- , - 5) is a solution of equation 3x + 4y = k Substituting x = -and y =5/2in x + 4y - 7 = 0, We get;-+ 4 (5/2) - 7 = 0 -+ 4 5/2- 7 = 0 = 10 - 7 = 3...
Read More →If x = – 1, y = 2 is a solution of the equation
Question: If x = 1, y = 2 is a solution of the equation 3x + 4y = k, find the value of k. Solution: We are given, 3x + 4y = k Given that, (-1, 2) is the solution of equation 3x + 4y = k. Substituting x = -1 and y = 2 in 3x + 4y = k, We get; 3x - 1 + 4 2 = k K = - 3 + 8 k = 5...
Read More →If f : R
Question: If $f: \mathrm{R} \rightarrow \mathrm{R}$ be given by for all $f(x)=\frac{4^{x}}{4^{x}+2} x \in \mathrm{R}$, then (a)f(x) =f(1 x) (b)f(x) +f(1 x) = 0 (c)f(x) +f(1 x) = 1 (d)f(x) +f(x 1) = 1 Solution: (c)f(x) +f(1 x) = 1 $f(x)=\frac{4^{x}}{4^{x}+2} ; x \in \mathrm{R}$ $f(1-x)=\frac{4^{1-x}}{4^{1-x}+2}$ $=\frac{4}{2 \times 4^{x}+4}$ $=\frac{2}{4^{x}+2}$ $f(x)+f(1-x)=\frac{4^{x}}{4^{x}+2}+\frac{2}{4^{x}+2}$ $=\frac{4^{x}+2}{4^{x}+2}=1$...
Read More →If f : R
Question: If $f: \mathrm{R} \rightarrow \mathrm{R}$ be given by for all $f(x)=\frac{4^{x}}{4^{x}+2} x \in \mathrm{R}$, then (a)f(x) =f(1 x) (b)f(x) +f(1 x) = 0 (c)f(x) +f(1 x) = 1 (d)f(x) +f(x 1) = 1 Solution: (c)f(x) +f(1 x) = 1 $f(x)=\frac{4^{x}}{4^{x}+2} ; x \in \mathrm{R}$ $f(1-x)=\frac{4^{1-x}}{4^{1-x}+2}$ $=\frac{4}{2 \times 4^{x}+4}$ $=\frac{2}{4^{x}+2}$ $f(x)+f(1-x)=\frac{4^{x}}{4^{x}+2}+\frac{2}{4^{x}+2}$ $=\frac{4^{x}+2}{4^{x}+2}=1$...
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Question: A. $\frac{\pi}{3}$ B. $\frac{2 \pi}{3}$ C. $\frac{\pi}{6}$ D. $\frac{\pi}{12}$ Solution: $\int \frac{d x}{1+x^{2}}=\tan ^{-1} x=\mathrm{F}(x)$ By second fundamental theorem of calculus, we obtain $=\tan ^{-1} \sqrt{3}-\tan ^{-1} 1$ $=\frac{\pi}{3}-\frac{\pi}{4}$ $=\frac{\pi}{12}$ Hence, the correct answer is D....
Read More →Check which of the following are solutions of the equation
Question: Check which of the following are solutions of the equation 2x - y = 6 and Which are not: (i) (3, 0) (ii) (0, 6) (iii) (2, - 2) (iv) (3, 0) (v) (1/2 , - 5) Solution: We are given, 2x - y = 6 (i) In the equation 2x - y = 6, We have L.H.S = 2x - y and R.H.S = 6 Substituting x = 3 and y = 0 in 2x - y, We get L.H.S = 2 3 - 0 = 6 ⟹ L.H.S = R.H.S ⟹ (3, 0) is a solution of 2x - y = 6. (ii) In the equation 2x - y = 6, We have L.H.S= 2x - y and R.H.S = 6 Substituting x = 0 and y = 6 in 2x - y We...
Read More →The sum of two numbers is 8. If their sum is four times their difference, find the numbers.
Question: The sum of two numbers is 8. If their sum is four times their difference, find the numbers. Solution: Let the numbers arexandy. One of them must be greater than or equal to the other. Let us assume thatxis greater than or equal toy. The sum of the two numbers is 8. Thus, we have The sum of the two numbers is four times their difference. Thus, we have $x+y=4(x-y)$ $\Rightarrow x+y=4 x-4 y$ $\Rightarrow 4 x-4 y-x-y=0$ $\Rightarrow 3 x-5 y=0$ So, we have two equations $x+y=8$ $3 x-5 y=0$ ...
Read More →Write two solutions of the form
Question: Write two solutions of the form x = 0, y = a and x = b, y = 0 for each of the following equations: (i) 5x - 2y =10 (ii) - 4x + 3y =12 (iii) 2x + 3y = 24 Solution: (i) We are given, 5x - 2y = 10 Substituting x = 0 in the given equation, We get; 5 0 - 2y = 10 - 2y = 10 - y = 10/2 y = - 5 Thus x = 0 and y = - 5 is the solution of 5x - 2y = 10 Substituting y = 0 in the given equation, we get 5x 2 0 = 10 5x = 10 x = 10/5 x = 2 Thus x = 2 and y = 0 is a solution of 5x - 2y = 10 (ii) We are g...
Read More →If 3f
Question: If $3 f(x)+5 f\left(\frac{1}{x}\right)=\frac{1}{x}-3$ for all non-zero $x$, then $f(x)=$ (a) $\frac{1}{14}\left(\frac{3}{x}+5 x-6\right)$ (b) $\frac{1}{14}\left(-\frac{3}{x}+5 x-6\right)$ (c) $\frac{1}{14}\left(-\frac{3}{x}+5 x+6\right)$ (d) None of these Solution: (d) None of these $3 f(x)+5 f\left(\frac{1}{x}\right)=\frac{1}{x}-3$ ...91) Multiplying (1) by 3 : 15 f\left(\frac{1}{x}\right)+9 f(x)=\frac{3}{x}-9 \ldots \ldots(2) Replacing $x$ by $\frac{1}{x}$ in $(1)$ : $3 f\left(\frac{...
Read More →If 3f
Question: If $3 f(x)+5 f\left(\frac{1}{x}\right)=\frac{1}{x}-3$ for all non-zero $x$, then $f(x)=$ (a) $\frac{1}{14}\left(\frac{3}{x}+5 x-6\right)$ (b) $\frac{1}{14}\left(-\frac{3}{x}+5 x-6\right)$ (c) $\frac{1}{14}\left(-\frac{3}{x}+5 x+6\right)$ (d) None of these Solution: (d) None of these $3 f(x)+5 f\left(\frac{1}{x}\right)=\frac{1}{x}-3$ ...91) Multiplying (1) by 3 : 15 f\left(\frac{1}{x}\right)+9 f(x)=\frac{3}{x}-9 \ldots \ldots(2) Replacing $x$ by $\frac{1}{x}$ in $(1)$ : $3 f\left(\frac{...
Read More →Write two solutions for each of the following equations:
Question: Write two solutions for each of the following equations: (i) 5x - 2y = 7 (ii) x = 6y (iii) x + y = 4 (iv) (2/3)x - y = 4. Solution: (i) We are given, 3x + 4y = 7 Substituting x = 1 In the given equation, We get 3 1 + 4y = 7 4y = 7- 3 -4 = 4Y Y = 1 Thus x = 1 and y = 1 is the solution of 3x + 4y = 7 Substituting x = 2 in the given equation, we get 3 2 + 4y = 7 4y = 7 - 6 y = 1/4 Thus x = 2 and y = 1/4 is the solution of 3x + 4y = 7(ii) We are given, x = 6y Substituting x = 0 in the give...
Read More →A lending library has a fixed charge for the first three days and additional charge for each day thereafter.
Question: A lending library has a fixed charge for the first three days and additional charge for each day thereafter. Saritha paid Rs 27 for a book kept for seven days, while Susy paid Rs 21 for the book she kept for five days. Find the fixed charge and the charge for each extra day. Solution: To find: (1) the fixed charge (2) The charge for each day Let the fixed charge be Rsx And the extra charge per day be Rsy. According to the given conditions, $x+4 y=27$ $x+4 y-27=0$....(1) $x+2 y=21$ $x+2...
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Question: $\int_{0}^{1}\left(x e^{x}+\sin \frac{\pi x}{4}\right) d x$ Solution: Let $I=\int_{0}^{1}\left(x e^{x}+\sin \frac{\pi x}{4}\right) d x$ $\int\left(x e^{x}+\sin \frac{\pi x}{4}\right) d x=x \int e^{x} d x-\int\left\{\left(\frac{d}{d x} x\right) \int e^{x} d x\right\} d x+\left\{\frac{-\cos \frac{\pi x}{4}}{\frac{\pi}{4}}\right\}$ $=x e^{x}-\int e^{x} d x-\frac{4 \pi}{\pi} \cos \frac{x}{4}$ $=x e^{x}-e^{x}-\frac{4 \pi}{\pi} \cos \frac{x}{4}$ $=\mathrm{F}(x)$ By second fundamental theorem...
Read More →The cost of ball pen is Rs 5 less than half of the cost of fountain pen.
Question: The cost of ball pen is Rs 5 less than half of the cost of fountain pen. Write this statement as a linear equation in two variables. Solution: Let the cost of fountain pen be y and cost of ball pen be x. According to the given equation, we have x = y/2 5 ⟹ 2x = y - 10 ⟹ 2x - y + 10 = 0 Here y is the cost of one fountain pen and x is that of one ball pen....
Read More →Write each of the following as an equation in two variables:
Question: Write each of the following as an equation in two variables: (i) 2x = 3 (ii) y = 3 (iii) 5x = 7/ 2 (iv) y = 3/2x Solution: (i) We are given, 2x = - 3 Now, in two variable forms the given equation will be 2x + 0y + 3 = 0 (ii) We are given, y = 3 Now, in two variable forms the given equation will be 0 x + y - 3 = 0 (iii) We are given, 5x = 7/2 Now, in two variable forms the given equation will be 5x + 0y + 7/2 = 0 10x + 0y - 7 = 0 (iv) We are given, y = 32x (Taking L.C.M on both sides) N...
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Question: $\int_{0}^{2} \frac{6 x+3}{x^{2}+4} d x$ Solution: Let $I=\int_{0}^{2} \frac{6 x+3}{x^{2}+4} d x$ $\int \frac{6 x+3}{x^{2}+4} d x=3 \int \frac{2 x+1}{x^{2}+4} d x$ $=3 \int \frac{2 x}{x^{2}+4} d x+3 \int \frac{1}{x^{2}+4} d x$ $=3 \log \left(x^{2}+4\right)+\frac{3}{2} \tan ^{-1} \frac{x}{2}=\mathrm{F}(x)$ By second fundamental theorem of calculus, we obtain $I=\mathrm{F}(2)-\mathrm{F}(0)$ $=\left\{3 \log \left(2^{2}+4\right)+\frac{3}{2} \tan ^{-1}\left(\frac{2}{2}\right)\right\}-\left\...
Read More →Express the following linear equations in the form ax + by + c = 0
Question: Express the following linear equations in the form ax + by + c = 0 and indicate the values of a, b and c in each case: (i) - 2x + 3y = 12 (ii) x - y/2 - 5 = 0 (iii) 2x + 3y = 9.35 (iv) 3x = -7y (v) 2x + 3 = 0 (vi) y - 5 = 0 (vii) 4 = 3x (viii) y = x/2 ; Solution: (i) We are given - 2x + 3y = 12 - 2x + 3y - 12 = 0 Comparing the given equation with ax + by + c = O We get, a = - 2; b = 3; c = -12 (ii) We are given x - y/2 - 5= 0 Comparing the given equation with ax + by + c = 0, We get, a...
Read More →The coach of a cricket team buys 7 bats and 6 balls for Rs 3800.
Question: The coach of a cricket team buys 7 bats and 6 balls for Rs 3800. Later, he buys 3 bats and 5 balls for Rs 1750. Find the cost of each bat and each ball. Solution: Given: (i) 7 bats and 6balls cost is Rs3800 (ii) 3 bats and 5balls cost is Rs1750 To find: Cost of 1 bat and 1 ball Let (i) the cost of 1 bat = Rs.x. (ii) the cost of 1 ball = Rs.y. According to the given conditions, we have $7 x+6 y=3800$ $7 x+6 y-3800=0$.....(1) $3 x+5 y=1750$ $3 x+5 y-1750=0$....(2) Thus, we get the follow...
Read More →If f(x)=
Question: If $f(x)=64 x^{3}+\frac{1}{x^{3}}$ and $\alpha, \beta$ are the roots of $4 x+\frac{1}{x}=3$. Then, (a)f() =f() = 9 (b)f() =f() = 63 (c)f()f() (d) none of these Solution: (a) $f(\alpha)=f(\beta)=-9$ Given: $f(x)=64 x^{3}+\frac{1}{x^{3}}$ $\Rightarrow f(x)=\left(4 x+\frac{1}{x}\right)\left(16 x^{2}+\frac{1}{x^{2}}-4\right)$ $\Rightarrow f(x)=\left(4 x+\frac{1}{x}\right)\left(\left(4 x+\frac{1}{x}\right)^{2}-12\right)$ $\Rightarrow f(\alpha)=\left(4 \alpha+\frac{1}{\alpha}\right)\left(\le...
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