For any two non-zero rational numbers
Question: For any two non-zero rational numbers x and y, x4 y4is equal to (a) (x y)0 (b) (x y)1 (c) (x y)4 (d) (x y)8 Solution: (c) (x y)4 (By law of exponent: (a)m(b)m= (ab)m)...
Read More →Prove the following
Question: (3/4)5(5/3)5is equal to (a) (3/45/3)5 (b) (3/4 5/3)1 (c) (3/4 5/3)0 (d) (3/4 5/3)10 Solution: (a) (3/45/3)5 (By law of exponent: (a)m(b)m= (ab)m...
Read More →Find the geometric series whose
Question: Find the geometric series whose $5^{\text {th }}$ and $8^{\text {th }}$ terms are 80 and 640 respectively. Solution: The $n^{\text {th }}$ term of $a$ GP is $a_{n}=a r^{n-1}$ It's given in the question that $5^{\text {th }}$ term of the GP is 80 and $8^{\text {th }}$ term of GP is 640 . So, $a_{5}=a r^{4}=80 \rightarrow(1)$ $a_{8}=a r^{7}=640 \rightarrow(2)$ $\frac{(2)}{(1)} \rightarrow \frac{\mathrm{ar}^{7}}{\mathrm{ar}^{4}}=\mathrm{r}^{3}=\frac{640}{80}=8$ Common ratio, r = 2, $a r^{...
Read More →Prove the following
Question: (1/10)0is equal to (a) 0 (b) 1/10 (c) 1 (d) 10 Solution: (c) 1 Since, a0= 1 (by law of exponent)...
Read More →The usual form for
Question: The usual form for 2.03 10-5 (a) 0.203 (b) 0.00203 (c) 203000 (d) 0.0000203 Solution: (d) 0.0000203...
Read More →The standard form for
Question: The standard form for 234000000 is (a) 2.34 108 (b) 0.234 109 (c) 2.34 10-8 (d) 0.23410-9 Solution: (a) 2.34 108 Explanation: 234000000 = 234 106= 2.34 102 106= 2.34 108...
Read More →The standard form for
Question: The standard form for 0.000064 is (a) 64 104 (b) 64 10-4 (c) 6.4 105 (d) 6.4 10-5 Solution: (d) 6.4 10-5...
Read More →The value of
Question: The value of (7-1 8-1)-1 (3-1 4-1)-1is: (a) 44 (b) 56 (c) 68 (d) 12 Solution: (a) 44 Explanation: (7-1 8-1)-1 (3-1 4-1)-1 = (1/7-1/8)-1 (1/3-1/4)-1 = (1/56)-1 (1/12)-1 = 56 12 = 44...
Read More →Which term of the GP
Question: Which term of the GP $\sqrt{3}, 3,3 \sqrt{3} \ldots$ is $729 ?$ Solution: Given GP is $\sqrt{3}, 3,3 \sqrt{3} \ldots$ The given GP is of the form, $a, a r, a r^{2}, a r^{3} \ldots .$ Where $r$ is the common ratio. First term in the given GP, $a_{1}=a=\sqrt{3}$ Second term in GP, $a_{2}=3$ Now, the common ratio, $r=\frac{a_{2}}{a_{1}}$ $r=\frac{3}{\sqrt{3}}=\sqrt{3}$ Let us consider 729 as the $n^{\text {th }}$ term of the GP. Now, $\mathrm{n}^{\text {th }}$ term of GP is, $\mathrm{a}_{...
Read More →For a non-zero integer x,
Question: For a non-zero integer x, (x4)-3is equal to: (a) x12 (b) x-12 (c) x64 (d) x-64 Solution: (b) x-12 Explanation: (x4)-3= x4(-3)= x-12 (By the law of exponents: (am)n=amn)...
Read More →For a non-zero integer x,
Question: For a non-zero integer x, x7 x12is equal to: (a) x5 (b) x19 (c) x-5 (d) x-19 Solution: (c) x-5 Explanation: x7 x12= x7-12= x-5 (By the law of exponents: am an=am-n)...
Read More →Find the second order derivatives of each of the following functions:
Question: Find the second order derivatives of each of the following functions: $\log (\log x)$ Solution: $\sqrt{B a s i c}$ Idea: Second order derivative is nothing but derivative of derivative i.e. $\frac{d^{2} y}{d x^{2}}=\frac{d}{d x}\left(\frac{d y}{d x}\right)$ $\sqrt{T h e}$ idea of chain rule of differentiation: If $\mathrm{f}$ is any real-valued function which is the composition of two functions $u$ and $v$, i.e. $f=v(u(x))$. For the sake of simplicity just assume $t=u(x)$ Then $f=v(t) ...
Read More →Solve the following
Question: (9)3 (9)8is equal to: (a) (9)5 (b) (9)-5 (c) ( 9)5 (d) ( 9)-5 Solution: (d) ( 9)-5 Explanation: (9)3 (9)8= (-9)3-8= (-9)-5 (By the law of exponents: am an=am-n)...
Read More →Which term of the GP
Question: Which term of the GP $\frac{1}{4}, \frac{-1}{2}, 1 \ldots . .$ is $-128 ?$ Solution: Given GP is $\frac{1}{4}, \frac{-1}{2}, 1 \ldots$ The given GP is of the form, $a, a r, a r^{2}, a r^{3} \ldots .$ Where r is the common ratio The first term in the given GP, $\mathrm{a}=\mathrm{a}_{1}=\frac{1}{4}$ The second term in GP, $\mathrm{a}_{2}=-\frac{1}{2}$ Now, the common ratio, $r=\frac{a_{2}}{a_{1}}$ $r=-\frac{4}{2}=-2$ Let us consider $-128$ as the $\mathrm{n}^{\text {th }}$ term of the G...
Read More →Prove the following
Question: (-7/5)-1is equal to: (a) 5/7 (b) 5/7 (c) 7/5 (d) -7/5 Solution: (b) 5/7 Explanation: (-7/5)-1= 1/(-7/5) = -5/7...
Read More →Prove the following
Question: (-5/7)-5is equal to: (a) (5/7)-5 (b) (5/7)5 (c) (7/5)5 (d) (-7/5)5 Solution: (d) (-7/5)5 Explanation: (-5/7)-5=1/(-5/7)5=(-7/5)5 (By the law of exponents: a-n= 1/an)...
Read More →Which of the following is equal to
Question: Which of the following is equal to (-3/4)-3? (a) (3/4)-3 (b) (3/4)-3 (c) (4/3)3 (d) (-4/3)3 Solution: (d) (-4/3)3 Explanation: (-3/4)-3= 1/(-3/4)3= (-4/3)3 (By the law of exponents: a-n= 1/an)...
Read More →If x be any integer different from zero and m,
Question: If x be any integer different from zero and m, n be any integers, then (xm)nis equal to: (a) xm+n (b) xmn (c) xm/n (d) xm-n Solution: (b) xmn(By the law of exponents)...
Read More →If x be any integer different from zero
Question: If x be any integer different from zero and m be any positive integer, then x-mis equal to: (a) xm (b) xm (c) 1/xm (d) -1/xm Solution: (c) 1/xm(By the law of exponents)...
Read More →Which term of the GP 3, 6, 12, 24…. Is 3072?
Question: Which term of the GP 3, 6, 12, 24. Is 3072? Solution: Given GP is 3, 6, 12, 24. The given GP is of the form, $a, a r, a r^{2}, a r^{3} \ldots .$ Where r is the common ratio. First term in the given GP, a1 = a = 3 Second term in GP, a2 = 6 Now, the common ratio, $r=\frac{a_{2}}{a_{1}}$ $r=\frac{6}{3}=2$ Let us consider 3072 as the $\mathrm{n}^{\text {th }}$ term of the GP. Now, $n^{\text {th }}$ term of GP is, $a_{n}=a r^{n-1}$ $3072=3.2^{n-1}$ $\frac{3072 \times 2}{3}=2^{n}$ $2^{n}=2^{...
Read More →If x be any non-zero integer,
Question: If x be any non-zero integer, then x-1is equal to (a) x (b) 1/x (c) x (c) -1/x Solution: (b) 1/x (By the law of exponents)...
Read More →If y be any non-zero integer,
Question: If y be any non-zero integer, then y0is equal to: (a) 1 (b) 0 (c) 1 (c) Not defined Solution: (a) 1 (By the law of exponent)...
Read More →If x be any non-zero integer and m,
Question: If x be any non-zero integer and m, n be negative integers, then xm xnis equal to: (a) xm (b) xm+n (c) xn (d) xm-n Solution: (b) xm+n(By the law of exponents)...
Read More →The multiplicative inverse of
Question: The multiplicative inverse of (-5/9)-99is: (a) (-5/9)99 (b) (5/9)99 (c) (9/-5)99 (d) (9/5)99 Solution: (-5/9)99 Explanation: Take the reference of Q.8 mentioned above....
Read More →Find the second order derivatives of each of the following functions:
Question: Find the second order derivatives of each of the following functions: $x \cos x$ Solution: $\sqrt{B a s i c}$ Idea: Second order derivative is nothing but derivative of derivative i.e. $\frac{d^{2} y}{d x^{2}}=\frac{d}{d x}\left(\frac{d y}{d x}\right)$ $\sqrt{T h e}$ idea of chain rule of differentiation: If $f$ is any real-valued function which is the composition of two functions $u$ and $v$, i.e. $f=v(u(x))$. For the sake of simplicity just assume $t=u(x)$ Then $f=v(t) .$ By chain ru...
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