The value of
Question: The value of (-2/3)4is equal to: (a) 16/81 (b) 81/16 (c) -16/81 (d) 81/ 16 Solution: (a) 16/81 Explanation: (-2/3)4= (-2/3)(-2/3)(-2/3)(-2/3) = 16/81...
Read More →The value of
Question: The value of (2)23-1is (a) 32 (b) 64 (c) 32 (d) 64 Solution: (c) 32 Explanation: (2)23-1=(-2)6-1=(-2)5=-32...
Read More →Find the 10th and nth terms of the GP
Question: Find the 10th and nth terms of the GP $-\frac{3}{4}, \frac{1}{2},-\frac{1}{3}, \frac{2}{9} \ldots$ Solution: Given GP is $-\frac{3}{4}, \frac{1}{2},-\frac{1}{3}, \frac{2}{9} \ldots .$ 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, $\mathrm{a}=\mathrm{a}_{1}=-\frac{3}{4}$ The second term in $\mathrm{GP}, \mathrm{a}_{2}=\frac{1}{2}$ Now, the common ratio, $\mathrm{r}=\frac{\mathrm{a}_{2}}{\mathrm{a}_{1}}$ $r=-...
Read More →The multiplicative inverse
Question: The multiplicative inverse of 10-100is (a) 10 (b) 100 (c) 10100 (d) 10-100 Solution: (c) 10100 Explanation: By the law of exponent we know: a-n= 1/an. So, 10-100= 1/10100 The multiplicative inverse for any integer a is 1/a, such that; a x 1/a = 1 Hence, the multiplicative inverse for 1/10100is 10100 as, 1/10100x 10100= 1...
Read More →The reciprocal of
Question: The reciprocal of (2/5)-1is: (a) 2/5 (b) 5/2 (c) 5/2 (d) 2/5 Solution: (b) 5/2 Explanation: By the law of exponent we know: a-n= 1/an. Hence, (2/5)-1=1/(2/5)=5/2...
Read More →The value of
Question: The value of (2/5)-2is: (a) 4/5 (b) 4/25 (c) 25/4 (d) 5/2 Solution: (c) 25/4 Explanation: By the law of exponent we know: a-n= 1/an. Hence, (2/5)-2= 1/(2/5)2= 1/(4/25) = 25/4...
Read More →The value of
Question: The value of 35 3-6is: (a) 35 (b) 3-6 (c) 311 (d) 3-11 Solution: (c) 311 Explanation: By the law of exponents, we know, am/an=am-n Hence, 35 3-6= 35-(-6)= 311...
Read More →Find the 7th and nth terms of the GP
Question: Find the $7^{\text {th }}$ and $n$th terms of the GP $0.4,0.8,1.6 \ldots$ Solution: Given GP is 0.4, 0.8, 1.6. 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, $a_{1}=a=0.4$ Second term in GP, $a_{2}=0.8$ Now, the common ratio, $\mathrm{r}=\frac{\mathrm{a}_{2}}{\mathrm{a}_{1}}$ $r=\frac{0.8}{0.4}=2$ Now, $\mathrm{n}^{\text {th }}$ term of GP is, $\mathrm{a}_{\mathrm{n}}=\mathrm{ar}^{\mathrm{n}-1}$ So, the $7^{\tex...
Read More →The value of
Question: The value of 1/(4)-2is: (a) 16 (b) 8 (c) 1/16 (d) 1/8 Solution: (a) 16 Explanation: 1/(4)-2= 1/(1/42) = 42= 16...
Read More →Prove the following
Question: 3-2can be written as: (a) 32 (b) 1/32 (c) 1/3-2 (d) -2/3 Solution: (b) 1/32 Explanation: By the law of exponent we know: a-n= 1/an. Hence, 3-2=1/32...
Read More →For a fixed base,
Question: For a fixed base, if the exponent decreases by 1, the number becomes (a) one-tenth of the previous number (b) ten times of the previous number (c) hundredth of the previous number (d) hundred times of the previous number Solution: (a) One-tenth of the previous number Explanation: Suppose for 106, when the exponent is decreased by 1, it becomes 105. Hence, 105/106= 1/10....
Read More →Find the 17th and nth terms of the
Question: Find the $17^{\text {th }}$ and nth terms of the GP $2,2 \sqrt{2}, 4,8 \sqrt{2} \ldots .$ Solution: Given GP is $2,2 \sqrt{2}, 4,8 \sqrt{2} \ldots \ldots$ 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, $a_{1}=a=2$ Second term in GP, $a_{2}=2 \sqrt{2}$ Now, the common ratio, $\mathrm{r}=\frac{\mathrm{a}_{2}}{\mathrm{a}_{1}}$ $r=\frac{2 \sqrt{2}}{2}=\sqrt{2}$ Now, $\mathrm{n}^{\text {th }}$ term of GP is, $\math...
Read More →Prove the following
Question: In2n, n is known as (a) base (b) constant (c) exponent (d) variable Solution: (c) We know that an is called the nth power of a; and is also read as a raised to the power n. The rational number a is called the base and n is called the exponent (power or index). In the same way in 2n,n is known as exponent....
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: $\tan ^{-1} x$ Solution: Basic idea: $\sqrt{T h e}$ 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 $f=v(t)$. By chain rule, we can write the derivative of $f$ w.r.t to $x$ as: $\frac{\mathrm{df}}{\mathrm{dx}}=\frac{\mathrm{dv}}{\mathrm{dt}} \times \frac{\mat...
Read More →Solve this
Question: Find the $6^{\text {th }}$ and $n$th terms of the GP $2,6,18,54 \ldots$ Solution: Given: GP is 2, 6, 18, 54. 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, $a_{1}=a=2$ Second term in GP, $a_{2}=6$ Now, the common ratio, $r=\frac{a_{2}}{a_{1}}$ $r=\frac{6}{2}=3$ Now, $\mathrm{n}^{\text {th }}$ term of GP is, $a_{n}=a r^{n-1}$ So, the $6^{\text {th }}$ term in the GP, $\mathrm{a}_{6}=\mathrm{ar}^{5}$ $=2 \times 3^{5...
Read More →abc + bca + cab is a monomial.
Question: abc + bca + cab is a monomial. Solution: True The given expression seems to be a trinomial but it is not as it contains three like terms which can be added to form a monomial, i.e. abc + abc + abc = 3abc...
Read More →An equation is true
Question: An equation is true for all values of its variables. Solution: False As equation is true only for some values of its variables, e.g. 2x 4= 0 is true, only for x =2....
Read More →The product of one negative
Question: The product of one negative and one positive term is a negative term. Solution: True When we multiply a negative term by a positive term, the resultant will be a negative term, i-e. (-) x (+) = (-)....
Read More →The product of two negative
Question: The product of two negative terms is a negative term. Solution: False Since, the product of two negative terms is always a positive term, i.e. (-) x (-) = (+)....
Read More →The factorisation of
Question: The factorisation of 2x + 4y is-. Solution: 2 (x + 2y) We have, 2x + 4y = 2x + 2 x 2y = 2 (x + 2y)...
Read More →The common factor method
Question: The common factor method of factorisation for a polynomial is based on-property. Solution: Distributive In this method, we regroup the terms in such a way, so that each term in the group contains a common literal or number or both....
Read More →The numerical coefficient
Question: The numerical coefficient in -37abc is. Solution: -37 The constant term (with their sign) involved in term of an algebraic expression is called the numerical coefficient of that term....
Read More →Volume of a rectangular box
Question: Volume of a rectangular box with length 2x, breadth 3y and height 4z is . Solution: 24 xyz We know that, the volume of a rectangular box, V = Length x Breadth x Height = 2x x 3y x 4z = (2 x 3 x 4) xyz = 24 xyz...
Read More →Factorised form of
Question: Factorised form of 18mn + 10mnp is ________. Solution: Factorised form of 18mn + 10mnp is 2mn (9 + 5p) = (2 9 m n) + (2 5 m n p) = 2mn (9 + 5p)...
Read More →Common factor of
Question: Common factor of ax2+ bx is __________. Solution: Common factor of ax2+ bx is x (ax + b)...
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