Mathematics Questions

Browse 1,240 Mathematics questions with expert-verified, step-by-step solutions — NCERT, exemplar and previous-year questions, organised by chapter.

1. Q601
   Prove that ​a diameter AB of ​a ​circle bisects all ​those ​chords which are parallel to the tangent at the point A.
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2. Q602
   To divide ​a line segment $AB$ in ​the ratio $5:7$, first a ​ray $AX$ ​is drawn so that $\angle BAX$ is ​an acute angle ​and then at equal distances points ​are marked on the ray $AX$ such that the minimum number of these points is

   1. (a)8
   2. (b)10
   3. (c)11
   4. (d)12
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3. Q603
   To divide a ​line segment $AB$ in the ratio $4:7$, ​a ray $AX$ is ​drawn first such that $\angle BAX$ ​is an acute angle ​and then points $A_1,A_2,A_3,\dots$ are located ​at equal distances on the ray $AX$ and the point $B$ is joined ​to

   1. (a)$A_{12}$
   2. (b)$A_{11}$
   3. (c)$A_{10}$
   4. (d)$A_9$
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4. Q604
   To divide a line segment AB ​in the ratio 5:6, draw a ray AX such ​that $\angle BAX$ is an ​acute angle, then draw a ray BY parallel to AX and the points $A_1,A_2,A_3,\dots$ ​and $B_1,B_2,B_3,\dots$ are located ​at equal distances on ray AX and BY, respectively. ​Then the ​points joined are

   1. (a)$A_5$ and $B_6$
   2. (b)$A_6$ and $B_5$
   3. (c)$A_4$ and $B_5$
   4. (d)$A_5$ and $B_4$
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5. Q605
   To construct a triangle similar to a given $△ABC$ with ​its sides $\frac{3}{7}$ of the corresponding sides ​of $△ABC$, first draw a ray BX such that $\angle CBX$ is an acute angle ​and X lies on the opposite side of ​A with ​respect ​to BC. Then locate points $B_1,B_2,B_3,\dots$ on ​BX at equal distances and next ​step is to join

   1. (a)$B_{10}$ to C
   2. (b)$B_3$ to C
   3. (c)$B_7$ to C
   4. (d)$B_4$ to C
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6. Q606
   To construct a triangle ​similar to a ​given $△ABC$ with its sides $\frac{8}{5}$ ​of the corresponding sides of $△ABC$, draw a ray BX such that $\angle CBX$ is an acute angle and X ​is on the opposite ​side of A with ​respect to BC. ​The minimum number of points to be located at equal ​distances on ray BX is

   1. (a)$5$
   2. (b)$8$
   3. (c)$13$
   4. (d)$3$
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7. Q607
   To draw ​a pair of tangents ​to a circle which are inclined to each other at ​an angle ​of $60^{\circ }$, it is required to draw tangents at end ​points of those two radii of the ​circle, the ​angle between them should be

   1. (a)$135^{\circ }$
   2. (b)$90^{\circ }$
   3. (c)$60^{\circ }$
   4. (d)$120^{\circ }$
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8. Q608
   By geometrical construction, ​it is possible ​to divide a line segment in the ratio $\sqrt{3}:\frac{1}{\sqrt{3}}$. State whether ​this ​statement is ​true ​or false ​and justify.
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9. Q609
   To construct a triangle similar to a ​given $△ABC$ with its sides $\frac{7}{3}$ ​of the ​corresponding sides of $△ABC$, ​draw a ray $BX$ making acute angle with $BC$ and $X$ lies on the opposite side of $A$ with respect to $BC$. The points $B_1,B_2,\dots ,B_7$ are located at equal distances on $BX$, $B_3$ is joined to $C$ and then a ​line segment $B_6{C}^{\prime }$ is drawn ​parallel to $B_{3}C$ where $C^{\prime }$ lies on $BC$ produced. Finally, line segment $A^{\prime }{C}^{\prime }$ is drawn parallel to $CA$. Is this construction correct? Justify.
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10. Q610
    Write 'True' or 'False' and ​justify your answer: ​A pair of ​tangents ​can ​be constructed ​from a point P ​to a circle of radius ​3.5 cm situated at a distance of 3 cm from the centre.
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11. Q611
    State ​whether the following ​is true or false and ​justify: A pair of tangents can be constructed ​to ​a circle inclined at an angle ​of $170^{\circ }$.
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12. Q612
    Draw a ​line segment of length ​7 cm. Find a point ​P ​on ​it ​which ​divides it in the ​ratio 3:5.
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13. Q613
    Draw a right triangle ​ABC ​in which BC = 12 cm, AB = 5 cm and $\angle B=90^{\circ }$. Construct a triangle ​similar to ​it and of scale ​factor $\frac{2}{3}$. ​Is the ​new triangle also a right triangle?
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14. Q614
    Draw ​a ​triangle ABC ​in ​which $BC=6$ cm, $CA=5$ cm and $AB=4$ cm. Construct a triangle similar to it and ​of ​scale ​factor $\frac{5}{3}$.
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15. Q615
    Construct ​a tangent ​to a circle of radius 4 ​cm from a point ​which is at a ​distance of ​6 cm ​from its ​centre.
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16. Q616
    If the ​sum of the ​areas ​of two circles ​with radii $R_1$ ​and $R_2$ is ​equal to the area of a ​circle of ​radius $R$, then

    1. (a)$R_1+R_2=R$
    2. (b)$R_1^2+R_2^2=R^2$
    3. (c)$R_1+R_2<R$
    4. (d)$R_1^2+R_2^2<R^2$
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17. Q617
    If ​the ​sum of the circumferences of two circles with ​radii $R_1$ and $R_2$ ​is equal ​to the circumference ​of ​a circle of ​radius $R$, then

    1. (a)$R_1+R_2=R$
    2. (b)$R_1+R_2>R$
    3. (c)$R_1+R_2<R$
    4. (d)Nothing definite can be said about the relation among $R_1$, $R_2$ and $R$.
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18. Q618
    If ​the circumference of a circle and the perimeter of a square are equal, ​then (A) ​Area of the circle = ​Area ​of ​the square (B) Area of the circle > Area of the square (C) Area of the circle < Area of the ​square (D) Nothing definite can be said about the relation between ​the areas of the circle and square.

    1. (a)Area of the circle = Area of the square
    2. (b)Area of the circle > Area of the square
    3. (c)Area of the circle < Area of the square
    4. (d)Nothing definite can be said about the relation between the areas of the circle and square.
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19. Q619
    Area of ​the largest triangle that can ​be ​inscribed in a semi-circle ​of ​radius $r$ ​units ​is

    1. (a)$r^2$ sq. units
    2. (b)$\frac{1}{2}{r}^2$ sq. units
    3. (c)$2r^2$ sq. units
    4. (d)$\sqrt{2}{r}^2$ sq. units
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20. Q620
    If the perimeter of a ​circle ​is equal to that ​of a ​square, then the ​ratio of ​their ​areas is

    1. (a)$22:7$
    2. (b)$14:11$
    3. (c)$7:22$
    4. (d)$11:14$
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