Mathematics Questions

Browse 839 Mathematics questions with expert-verified, step-by-step solutions — NCERT, exemplar and previous-year questions, organised by chapter.

1. Q21
   Solve the following pair ​of linear equations by ​the ​substitution method:

   1. (i)$x+y=14$ and $x-y=4$
   2. (ii)$s-t=3$ and $\frac{s}{3}+\frac{t}{2}=6$ ​
   3. (iii)$3x-y=3$ and $9x-3y=9$
   4. (iv)$0.2x+0.3y=1.3$ ​and $0.4x+0.5y=2.3$ ​
   5. (v)$\sqrt{2}x+\sqrt{3}y=0$ and $\sqrt{3}x-\sqrt{8}y=0$
   6. (vi)$\frac{3x}{2}-\frac{5y}{3}=-2$ and $\frac{x}{3}+\frac{y}{2}=\frac{13}{6}$
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2. Q22
   Solve $2x+3y=11$ and $2x-4y=-24$ ​and ​hence ​find the ​value of '$m$' for ​which $y=mx+3$.
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3. Q23
   Form the pair of linear equations for the following ​problems and find their solution by the substitution ​method:

   1. (i)The difference between two ​numbers is $26$ and one number is three times the other. Find them.
   2. (ii)The larger of two supplementary angles exceeds the smaller by $18$ degrees. Find them.
   3. (iii)The coach of a cricket team buys $7$ bats and $6$ balls for Rs $3800$. Later, she buys $3$ bats ​and $5$ balls for Rs $1750$. Find the cost of each bat and each ball.
   4. (iv)The taxi ​charges in a city consist of a fixed charge together with the charge for the distance covered. For a distance of $10$ km, the charge paid is Rs $105$ and for ​a journey of $15$ km, the charge paid is Rs $155$. What are the fixed charges and the charge per km? How much does a person have to pay for travelling a distance of $25$ km?
   5. (v)A fraction becomes $\frac{9}{11}$ if $2$ is added to ​both the numerator and the denominator. If $3$ is ​added to both the numerator and the denominator it becomes $\frac{5}{6}$. Find the fraction.
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4. Q24
   Check ​whether ​the ​following are quadratic equations:

   1. (i)$(x+1{)}^2=2(x-3)$ ​
   2. (ii)$x^2-2x=(-2)(3-x)$
   3. (iii)$(x-2)(x+1)=(x-1)(x+3)$
   4. (iv)$(x-3)(2x+1)=x(x+5)$
   5. (v)$(2x-1)(x-3)=(x+5)(x-1)$
   6. (vi)$x^2+3x+1=(x-2{)}^2$ ​
   7. (vii)$(x+2{)}^3=2x(x^2-1)$ ​
   8. (viii)$x^3-4x^2-x+1=(x-2{)}^3$
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5. Q25
   Represent ​the following situation in the form ​of ​a quadratic equation: ​(i) ​The area of a rectangular plot is 528 m${}^2$. The length of the ​plot ​(in metres) is one more than twice its breadth. We need to find the length and breadth ​of the plot.
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6. Q26
   Find ​the roots of the following ​quadratic equations by ​factorisation: ​

   1. (i)$x^2-3x-10=0$ ​
   2. (ii)$2x^2+x-6=0$
   3. (iii)$\sqrt{2}{x}^2+7x+5\sqrt{2}=0$
   4. (iv)$2x^2-x+\frac{1}{8}=0$
   5. (v)$100x^2-20x+1=0$
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7. Q27
   Solve the problems given in Example 1. [Example 1:

   1. (i)John ​and Jivanti together have 45 marbles. Both of them lost 5 marbles each, and ​the product of the number of marbles ​they now have is 124. Find how many marbles they had to start with. ​
   2. (ii)A cottage industry produces ​a certain number of toys in a day. The cost of production of each toy (in rupees) was found to be 55 minus the number of toys produced in a day. ​On a particular day, the total cost of production ​was ₹ 750. Find the number of toys ​produced on that day.]
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8. Q28
   Find two ​numbers whose ​sum ​is ​27 and product ​is 182.
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9. Q29
   Find ​two ​consecutive positive ​integers, sum ​of ​whose ​squares is 365.
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10. Q30
    The altitude ​of a right ​triangle ​is $7$ cm less than its base. ​If the ​hypotenuse is $13$ ​cm, find the other two sides.
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11. Q31
    A cottage industry produces a certain number of pottery ​articles ​in a day. It was observed on a particular day that ​the cost of production of each article (in rupees) ​was $3$ more than twice the number of articles produced on that ​day. If the total cost of production ​on that day was Rs. $90$, find the ​number ​of articles produced and the cost of each article.
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12. Q32
    In ​which of the following situations does the list of numbers involved make an arithmetic progression, and ​why?

    1. (i)The taxi fare after each km when the fare is Rs $15$ for the first km and Rs $8$ for each ​additional ​km.
    2. (ii)The amount of air present ​in a cylinder when a vacuum ​pump removes $\frac{1}{4}$ of the air remaining in the cylinder at a time.
    3. (iii)The cost of digging a well after every metre of digging, when it costs Rs $150$ for the ​first metre and rises by Rs $50$ for each subsequent metre.
    4. (iv)The amount of money in the account every year, when Rs $10000$ is deposited at compound interest at $8\%$ per annum.
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13. Q33
    Write the first four ​terms of ​the AP when the first term $a$ and ​the ​common difference $d$ are given as ​follows: ​

    1. (i)$a=10$, $d=10$ ​
    2. (ii)$a=-2$, $d=0$
    3. (iii)$a=4$, $d=-3$
    4. (iv)$a=-1$, $d=\frac{1}{2}$
    5. (v)$a=-1.25$, $d=-0.25$
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14. Q34
    For the following ​APs, write the first term and ​the common ​difference: ​

    1. (i)$3,1,-1,-3,\dots$ ​
    2. (ii)$-5,-1,3,7,\dots$ ​
    3. (iii)$\frac{1}{3},\frac{5}{3},\frac{9}{3},\frac{13}{3},\dots$
    4. (iv)$0.6,1.7,2.8,3.9,\dots$
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15. Q35
    Which ​of ​the following are APs? ​If they form ​an AP, find the common difference $d$ and ​write three ​more terms.

    1. (i)$2,4,8,16,\dots$
    2. (ii)$2,\frac{5}{2},3,\frac{7}{2},\dots$
    3. (iii)$-1.2,-3.2,-5.2,-7.2,\dots$
    4. (iv)$-10,-6,-2,2,\dots$
    5. (v)$3,3+\sqrt{2},3+2\sqrt{2},3+3\sqrt{2},\dots$ ​
    6. (vi)$0.2,0.22,0.222,0.2222,\dots$
    7. (vii)$0,-4,-8,-12,\dots$
    8. (viii)$-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\dots$
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16. Q36
    Fill in the ​blanks in the following table, given ​that $a$ is ​the first ​term, $d$ the common difference and $a_n$ ​the $n$-th term of the AP.

    1. (i)$a=7$, $d=3$, $n=8$, $a_n=?$
    2. (ii)$a=-18$, $d=?$, $n=10$, $a_n=0$
    3. (iii)$a=?$, $d=-3$, $n=18$, $a_n=-5$ ​
    4. (iv)$a=-18.9$, $d=2.5$, $n=?$, $a_n=3.6$
    5. (v)$a=3.5$, $d=0$, $n=105$, $a_n=?$
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17. Q37
    Choose ​the ​correct choice in ​the following and justify:

    1. (i)30th term ​of the ​AP $10,7,4,\dots$ (A) 97 ​(B) 77 (C) $-77$ (D) $-87$
    2. (ii)​11th term of the AP $-3,-\frac{1}{2},2,\dots$ (A) ​28 (B) 22 (C) $-38$ (D) $-48\frac{1}{2}$
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18. Q38
    In the ​following ​APs, ​find the missing terms in ​the ​boxes:

    1. (i)$2,\square ,26$
    2. (ii)$\square ,13,\square ,3$
    3. (iii)$5,\square ,\square ,9\frac{1}{2}$
    4. (iv)$-4,\square ,\square ,\square ,\square ,6$ ​
    5. (v)$\square ,38,\square ,\square ,\square ,-22$
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19. Q39
    Which term ​of the ​AP: $3,8,13,18,\dots ,$ ​is 78?
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20. Q40
    Find ​the ​number of terms ​in each of ​the following APs:

    1. (i)$7,13,19,\dots ,205$
    2. (ii)$18,15\frac{1}{2},13,\dots ,-47$
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