Write the sum of first n odd natural numbers.
Question: Write the sum of first n odd natural numbers. Solution: n odd natural numbers are given by 3,5,7,9,. $S=\frac{n}{2} \times(6+2 \times n-2)$ $S=\frac{n}{2} \times(4+2 \times n)$ $S=n^{2}+2 n$...
Read More →On dividing
Question: On dividing 57p2qr by 114pq, we get (a) pr (b) pr (c) pr (d) 2pr Solution: (c) pr On dividing 57p2qr by 114pq, It can be expanded as = (57 p p q r)/(114 p q) = 57pr/114 [divide both numerator and denominator by 57] = pr...
Read More →Write the sum of first n even natural numbers.
Question: Write the sum of first n even natural numbers. Solution: Even natural numbers are $2,4,6,8 \ldots \ldots$ $S=\frac{n}{2} \times(4+2 \times n-2)$ $S=n^{2}+2 n$...
Read More →Factorised form of
Question: Factorised form of p2 17p 38 is (a) (p 19) (p + 2) (b) (p 19) (p 2) (c) (p + 19) (p + 2) (d) (p + 19) (p 2) Solution: (a) (p 19) (p + 2) Factorised form of p2 17p 38 is = p2 19p + 2p 38 Take out the common factors, = p (p 19) + 2 (p 19) Again take out the common factor, = (p 19) (p + 2)...
Read More →Solve this
Question: If $\frac{3+5+7+9+\ldots \text { up to } 35 \text { terms }}{5+8+11+\ldots . \text { up ton terms }}=7$ find the value of n. Solution: To find: the value of n. We can write it as $\frac{\frac{35}{2}(6+34(5-3))}{\frac{\mathrm{n}}{2}(10+3(\mathrm{n}-1))}=7$ $3 n^{2}+7 \times n-370=0$ Therefore $n=37 / 3,10$ Rejecting $37 / 3$ we get $n=10$...
Read More →Factorised form of
Question: Factorised form of r2 10r + 21 is (a) (r 1) (r 4) (b) (r 7) (r 3) (c) (r 7) (r + 3) (d) (r + 7) (r + 3) Solution: (b) (r 7) (r 3) Factorised form of r2 10r + 21 is = r2 7r 3r + 21 Take out the common factors, = r (r 7) 3 (r 7) Again take out the common factor, = (r 7) (r 3)...
Read More →Factorised form of 23xy – 46x + 54y – 108 is
Question: Factorised form of 23xy 46x + 54y 108 is (a) (23x + 54) (y 2) (b) (23x + 54y) (y 2) (c) (23xy + 54y) ( 46x 108) (d) (23x + 54) (y + 2) Solution: (a) (23x + 54) (y 2) Factorised form of 23xy 46x + 54y 108 is = 23xy (2 23x) + 54y (2 54) Take out the common factors, = 23x (y 2) + 54 (y 2) Again take out the common factor, = (y 2) (23x + 54)...
Read More →Square of 9x – 7xy is
Question: Square of 9x 7xy is (a) 81x2+ 49x2y2 (b) 81x2 49x2y2 (c) 81x2+ 49x2y2126x2y (d) 81x2+ 49x2y2 63x2y Solution: (c) 81x2+ 49x2y2126x2y As per the condition in the question, (9x 7xy)2 The standard identity = (a b)2= a2 2ab + b2 Where, a = 9x, b = 7xy Then, (9x 7xy)2= (9x)2 (2 9x 7xy) + (7xy)2 = 81x2 126x2y + 49x2y2...
Read More →In an AP, the pth term is q and (p + q)th term is 0.
Question: In an AP, the $p^{\text {th }}$ term is $q$ and $(p+q)^{\text {th }}$ term is 0 . Show that its $q^{\text {th }}$ term is $p$. Solution: Given: $p^{\text {th }}$ term is $q$ and $(p+q)^{\text {th }}$ term is 0 . To prove: $q^{\text {th }}$ term is $p$. $p^{\text {th }}$ term is given by $q=a+(p-1) \times d \ldots \ldots$ equation 1 $(p+q)^{\text {th }}$ term is given by $0=a+(p+q-1) \times d$ $0=a+(p-1) \times d+q \times d$ Using equation1 $0=q+q \times d$ $d=-1$ Put in equation1 we ge...
Read More →Common factor of 17abc,
Question: Common factor of 17abc, 34ab2, 51a2b is (a) 17abc (b) 17ab (c) 17ac (d) 17a2b2c Solution: (b) 17ab The given factors can be written in expanded form as, 17abc = 17 a b c 34ab2= 2 17 a b b 51a2b = 3 17 a a b So, common factors in the above is 17 a b = 17ab...
Read More →Prove the following
Question: a2 b2is equal to (a) (a b)2 (b) (a b) (a b) (c) (a + b) (a b) (d) (a + b) (a + b) Solution: (c) (a + b) (a b) (a2 b2) = (a + b) (a b) is one of the standard identity....
Read More →Coefficient of y in the
Question: Coefficient of y in the term y/3 is (a) 1 (b) 3 (c) -1/3 (d) 1/3 Solution: (c) -1/3 -y/3 can also be written as y (-1/3) So, Coefficient of y is -1/3...
Read More →Which of the following are like terms?
Question: Which of the following are like terms? (a) 5xyz2, 3xy2z (b) 5xyz2, 7xyz2 (c) 5xyz2, 5x2yz (d) 5xyz2, x2y2z2 Solution: (b) 5xyz2, 7xyz2 Like terms are formed from the same variables and the powers of these variables are also the same. But coefficients of like terms need not be the same....
Read More →Square of 3x – 4y is
Question: Square of 3x 4y is (a) 9x2 16y2 (b) 6x2 8y2 (c) 9x2+ 16y2+ 24xy (d) 9x2+ 16y2 24xy Solution: (d) 9x2+ 16y2 24xy As per the condition in the question, (3x 4y)2 The standard identity = (a b)2= a2 2ab + b2 Where, a = 3x, b = 4y Then, (3x 4y)2= (3x)2 (2 3x 4y) + (4y)2 = 9x2 24xy + 16y2...
Read More →The first and last terms of an AP are 1 and 11 respectively
Question: The first and last terms of an AP are 1 and 11 respectively. If the sum of its terms is 36, find the number of terms. Solution: Given: the sum of its terms is 36, the first and last terms of an AP are 1 and 11. To find: the number of terms Sum of AP using first and last terms is given by $\mathrm{S}_{\mathrm{n}}=\frac{\mathrm{n}}{2}(\mathrm{a}+\mathrm{l})$ $36 \times 2=\mathrm{n}(1+11)$ $\mathrm{n}=6$...
Read More →Product of
Question: Product of 6a2 7b + 5ab and 2ab is (a) 12a3b 14ab2+ 10ab (b) 12a3b 14ab2+ 10a2b2 (c) 6a2 7b + 7ab (d) 12a2b 7ab2+ 10ab Solution: (b) 12a3b 14ab2+ 10a2b2 Now we have find product of trinomial and monomial, = (6a2 7b + 5ab) 2ab = (2ab 6a2) (2ab 7b) + (2ab 5ab) = 12a3b 14ab2+ 10a2b2...
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: $e^{x} \sin 5 x$ Solution: $\sqrt{B a s i c}$ Idea: Second order derivative is nothing but derivative of derivative i.e. $\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}=\frac{\mathrm{d}}{\mathrm{dx}}\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)$ $\sqrt{T h e}$ 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. $f=v(u(x))$. F...
Read More →Volume of a rectangular box (cuboid)
Question: Volume of a rectangular box (cuboid) with length = 2ab, breadth = 3ac and height = 2ac is (a) 12a3bc2 (b) 12a3bc (c) 12a2bc (d) 2ab +3ac + 2ac Solution: (a) 12a3bc2 We know that, volume of cuboid = length breadth height Given, length = 2ab, breadth = 3ac, height = 2ac = 2ab 3ac 2ac = (2 3 2) ab ac ac = 12a3bc2...
Read More →What is the 10th common term between the AP
Question: What is the $10^{\text {th }}$ common term between the APs $3,7,11,15,19, \ldots$ and 1,6 , 11, 16, ? Solution: To find: $10^{\text {th }}$ common term between the APs Common difference of $1^{\text {st }}$ series $=4$ Common difference of $2^{\text {nd }}$ series $=5$ LCM of common difference will give us a common difference of new series $\Rightarrow 5 \times 4$ $\Rightarrow 20$ The first term of new AP will be 11 , so the $10^{\text {th }}=$ term of this series is $\Rightarrow 11+20...
Read More →Area of a rectangle with length
Question: Area of a rectangle with length 4ab and breadth 6b2is (a) 24a2b2 (b) 24ab3 (c) 24ab2 (d) 24ab Solution: (b) 24ab3 We know that, area of rectangle = length breadth Given, length = 4ab, breadth = 6b2 = 4ab 6b2 = 24ab3...
Read More →Product of the following monomials
Question: Product of the following monomials 4p, 7q3, 7pq is (a) 196 p2q4 (b) 196 pq4 (c) 196 p2q4 (d) 196 p2q3 Solution: (a) 196 p2q4 = 4p ( 7q3) (7pq) = (4 (-7) (-7)) p q3 pq = 196p2q4...
Read More →Sum of a – b + ab, b + c – bc
Question: Sum of a b + ab, b + c bc and c a ac is (a) 2c + ab ac bc (b) 2c ab ac bc (c) 2c + ab + ac + bc (d) 2c ab + ac + bc Solution: (a) 2c + ab ac bc We have, = (a b + ab) + (b + c bc) + (c a ac) = a b + ab + b + c bc + c a ac Now, grouping like terms = (a a) + (-b + b) + (c + c) + ab bc ac = 2c + ab bc ac...
Read More →Solve this
Question: If $7^{\text {th }}$ and $13^{\text {th }}$ terms of an AP be 34 and 64 respectively then find its $18^{\text {th }}$ term. Solution: Given: $7^{\text {th }}$ term is 34 and $8^{\text {th }}$ term is 64 To find: find its $18^{\text {th }}$ term 34 = a + 6d .equation1 64 = a + 12d equation2 Subtract equation1 from equation2 we get d = 5 Put in equation1 we get a = 4 So $18^{\text {th }}$ term is 4 + 17 5 = 89...
Read More →Which of the following is a binomial?
Question: Which of the following is a binomial? (a) 7 a + a (b) 6a2+ 7b + 2c (c) 4a 3b 2c (d) 6 (a2+ b) Solution: (d) 6 (a2+ b) Expressions that contain exactly two terms are called binomials. = 6 (a2+ b) = 6a2+ b...
Read More →Like term as
Question: Like term as 4m3n2is (a) 4m2n2 (b) 6m3n2 (c) 6pm3n2 (d) 4m3n Solution: (b) 6m3n2 Like terms are formed from the same variables and the powers of these variables are also the same. But coefficients of like terms need not be the same....
Read More →