How many 2 - digit numbers are divisible by 7?
Question: How many 2 - digit numbers are divisible by $7 ?$ Solution: The first 2 digit number divisible by 7 is 14, and the last 2 digit number divisible by 7 is 98, so it forms AP with common difference 7 $14, \ldots, 98$ $98=14+(n-1) \times 7$ $n=22$...
Read More →If we subtract
Question: If we subtract 3x2y2from x2y2, then we get (a) 4x2y2 (b) 2x2y2 (c) 2x2y2 (d) 4x2y2 Solution: (d) 4x2y2 We have, The given two monomials are like terms. Subtract 3x2y2from x2y2= x2y2 (- 3x2y2) = x2y2+ 3x2y2 = x2y2(1 + 3) = 4x2y2...
Read More →The sum of –7pq and 2pq is
Question: The sum of 7pq and 2pq is (a) 9pq (b) 9pq (c) 5pq (d) 5pq Solution: (d) 5pq The given two monomials are like terms. Then sum of -7pq and 2pg = 7pq + 2pq = (-7 + 2) pq = -5pq...
Read More →Find the 8th term from the end of the AP
Question: Find the $8^{\text {th }}$ term from the end of the AP $7,9,11, \ldots, 201 .$ Solution: To find: $8^{\text {th }}$ term from the end $d=9-7$ $d=2$ Also $201=7+n \times 2-2$ $n=98$ So $8^{\text {th }}$ term from end will be $7+90 \times 2$ $\Rightarrow 187$...
Read More →Which of the following is correct?
Question: Which of the following is correct? (a) (a b)2= a2+ 2ab b2 (b) (a b)2= a2 2ab + b2 (c) (a b)2= a2 b2 (d) (a + b)2= a2+ 2ab b2 Solution: (b) (a b)2= a2 2ab + b2 We have, = (a b) (a b) = a (a b) b (a b) = a2 ab ba + b2 = a2 2ab + b2...
Read More →In a polynomial,
Question: In a polynomial, the exponents of the variables are always (a) integers (b) positive integers (c) non-negative integers (d) non-positive integers Solution: (c) In a polynomial, the exponents of the variables are either positive integers or 0. Constant term C can be written as C x. We do not consider the expressions as a polynomial which consist of the variables having negative/fractional exponent....
Read More →How many terms are there in the AP
Question: How many terms are there in the AP 13, 16, 19, ., 43? Solution: To find: number of terms in AP Also $\mathrm{d}=16-13$ $\mathrm{~d}=3$ Also $43=13+n \times 3-3$ So n = 11...
Read More →The product of a monomial
Question: The product of a monomial and a binomial is a (a) monomial (b) binomial (c) trinomial (d) None of these Solution: (b) Monomial consists of only single term and binomial contains two terms. So, the multiplication of a binomial by a monomial will always produce a binomial, whose first term is the product of monomial and the binomials first term and second term is the product of monomial and the binomials second term....
Read More →The 9th term of an AP is 0. Prove that its 29th term is double the 19th term.
Question: The $9^{\text {th }}$ term of an AP is 0 . Prove that its $29^{\text {th }}$ term is double the $19^{\text {th }}$ term. Solution: Given : $9^{\text {th }}$ term is 0 To prove: $29^{\text {th }}$ term is double the $19^{\text {th }}$ term $a+8 d=0$ $a=-8 d$ $29^{\text {th }}$ term is $a+28 d$ $\Rightarrow 20 d$ $19^{\text {th }}$ term is $a+18 d$ $\Rightarrow 10 d$ Hence proved $29^{\text {th }}$ term is double the $19^{\text {th }}$ term...
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: Solution: $\log (\sin x)$ $\sqrt{B a s i c}$ 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 $f$ is any real-valued function which is the composition of two functions $u$ and $v$, i.e. $f=v(u(x))$. For the sak...
Read More →Find three arithmetic means between 6 and - 6.
Question: Find three arithmetic means between 6 and - 6. Solution: let the three AM be $\mathrm{X}_{1}, \mathrm{X}_{2}, \mathrm{X}_{3}$. So new AP will be $6, x_{1}, x_{2}, x_{3},-6$ Also - 6 = 6 + 4d $d=-3$ $x_{1}=3$ $x_{2}=0$ $x_{3}=-3$...
Read More →In an AP it is given that
Question: In an AP it is given that $S_{n}=q n^{2}$ and $S_{m}=q m^{2}$. Prove that $S_{q}=q^{3}$. Solution: Given: $S_{n}=q n^{2}, S_{m}=q m^{2}$ To prove: $\mathrm{S}_{\mathrm{q}}=\mathrm{q}^{3}$ Put $n=1$ we get $a=q \ldots \ldots$ equation 1 Put n = 2 2a + d = 4q equation 2 Using equation 1 and 2 we get d = 2q So $\mathrm{S}_{\mathrm{q}}=\frac{\mathrm{q}}{2}(2 \mathrm{q}+(\mathrm{q}-1) \times 2 \mathrm{q})$ $\mathrm{S}_{\mathrm{q}}=\mathrm{q}^{3}$ Hence proved....
Read More →How many faces,
Question: How many faces, edges and vertices does a pyramid have with n sided polygon as its base? Solution: In a pyramid, the number of vertices is 1 more than the number of sides of the polygon base, i.e. vertices = n + 1 Also, the number of faces is 1 more than the number of sides of the polygonal base, i.e. faces = n+ 1 But the number of edges is 2 times the number of sides of the polygonal base, i.e. edges = 2 n ....
Read More →If 9 times the 9th term of an AP is equal to 13 times the 13th term,
Question: If 9 times the $9^{\text {th }}$ term of an AP is equal to 13 times the $13^{\text {th }}$ term, show that its $22^{\text {nd }}$ term is 0 . Solution: Given : $9 \times\left(9^{\text {th }}\right.$ term $)=13 \times\left(13^{\text {th }}\right.$ term $)$ To prove: $22^{\text {nd }}$ term is 0 $9 \times(a+8 d)=13 \times(a+12 d)$ $9 a+72 d=13 a+156 d$ $-4 a=84 d$ $a=-21 d \ldots . .$ Equation 1 Also $22^{\text {nd }}$ term is given by a + 21d Using equation 1 we get $-21 d+21 d=0$ Hence...
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: $\sin (\log x)$ 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. $f=v(u(x))$. Fo...
Read More →If the sum of n terms of an AP is given by
Question: If the sum of $n$ terms of an $A P$ is given by $S_{n}=\left(2 n^{2}+3 n\right)$, then find its common difference. Solution: Given: $S_{n}=\left(2 n^{2}+3 n\right)$ To find: find common difference To find: find common difference Put $n=1$ we get $S_{1}=5$ OR we can write $a=5 \ldots$ equation 1 Similarly put $\mathrm{n}=2$ we get $\mathrm{S}_{2}=14 \mathrm{OR}$ we can write $2 a+d=14$ Using equation 1 we get d = 4...
Read More →How many edges does each
Question: How many edges does each of following solids have? (a) Cone (b)Cylinder (c) Sphere (d)Octagonal Pyramid (e) Hexagonal Prism (f)Kaleidoscope Solution: (a) Cone has one edge. (b) Cylinder has two edges. (c) Sphere has no edge. (d) Octagonal pyramid has 16 edges. (e) Hexagonal prism has 18 edges. (f) Kaleidoscope has 9 edges. Note See edges in previous questions solution figures....
Read More →If a, b, c are in AP, show that
Question: If a, b, c are in AP, show that $\frac{a(b+c)}{b c}, \frac{b(c+a)}{c a}, \frac{c(a+b)}{a b}$ are also in AP. Solution: To prove $\frac{a(b+c)}{b c}, \frac{b(c+a)}{c a}, \frac{c(a+b)}{a b}$ are in A.P. Given: a, b, c are in A.P. Proof: a, b, c are in A.P. If each term of an A.P. is multiplied by a constant, then the resulting sequence is also an A.P. Multiplying the A.P. with (ab + bc + ac) $\Rightarrow(a)(a b+b c+a c)$, (b) $(a b+b c+a c)$, (c) $(a b+b c+a c)$, are in A.P. Multiplying ...
Read More →If a length of 100 m i% represented on a map by 1 cm,
Question: If a length of 100 m i% represented on a map by 1 cm, then the actual distance corresponding to 2 cm is 200 m. Solution: True When a length 100 m is respresented on a map by 1 cm. Then, actual distance corresponding to 2 cm = 2 x 100 = 200 m...
Read More →A cylinder is a 3-D shape having
Question: A cylinder is a 3-D shape having two circular faces of different radii. Solution: False In a cylinder, the radii of the two circular faces are same. If the radii of two circular faces are different, then it will become frustum....
Read More →All cubes are prism.
Question: All cubes are prism. Solution: True A cube is a prism because it has a square base, a congruent square top and the lateral sides are parallelograms....
Read More →A cuboid has atleast 4 diagonals.
Question: A cuboid has atleast 4 diagonals. Solution: True In a cuboid, the number of diagonals is not least then 4....
Read More →Solve this
Question: If $a\left(\frac{1}{b}+\frac{1}{c}\right), b\left(\frac{1}{c}+\frac{1}{a}\right), c\left(\frac{1}{a}+\frac{1}{b}\right)$ are in AP, prove that $a^{2}(b+c), b^{2}(c+a)$ $\mathbf{c}^{2}(\mathbf{a}+\mathbf{b})$ are in AP. Solution: To prove: $a^{2}(b+c), b^{2}(c+a), c^{2}(a+b)$ are in A.P. Given: $a\left(\frac{1}{b}+\frac{1}{c}\right), b\left(\frac{1}{c}+\frac{1}{a}\right), c\left(\frac{1}{a}+\frac{1}{b}\right)$ are in A.P. Proof: $a\left(\frac{1}{b}+\frac{1}{c}\right), b\left(\frac{1}{c}...
Read More →Pyramids do not have a diagonal.
Question: Pyramids do not have a diagonal. Solution: True , Pyramids are polyhedron with a polygon as its base and other faces as triangles meeting at a common vertex and diagonal is a line joining the two opposite vertex. So, in pyramids, two opposite vertex cannot be formed. So, we can say pyramids has no diagonal....
Read More →Every, solid shape has a unique net.
Question: Every, solid shape has a unique net. Solution: False A net is a flat figure that can be folded to form a closed, three-dimensional object. So, for an object, more than one net is possible but it is not true for the objects of all shapes....
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