Ch 1Real Numbers
2-Mark Questions
Express each number as a product of its prime factors: (i) 140 (ii) 156 (iii) 3825 (iv) 5005 (v) 7429
(i) $140 = 2^2 \times 5 \times 7$ (ii) $156 = 2^2 \times 3 \times 13$ (iii) $3825 = 3^2 \times 5^2 \times 17$ (iv) $5005 = 5 \times 7 \times 11 \times 13$ (v) $7429 = 17 \times 19 \times 23$
Find the LCM and HCF of the following pairs of integers and verify that LCM × HCF = product of the two numbers: (i) 26 and 91 (ii) 510 and 92 (iii) 336 and 54
(i) HCF $= 13$ , LCM $= 182$ ; check $13 \times 182 = 2366 = 26 \times 91$ . (ii) HCF $= 2$ , LCM $= 23460$ ; check $2 \times 23460 = 46920 = 510 \times 92$ . (iii) HCF $= 6$ , LCM $= 3024$ ; check $6 \times 3024 = 18144 = 336 \times 54$ .
Find the LCM and HCF of the following integers by applying the prime factorisation method: (i) 12, 15 and 21 (ii) 17, 23 and 29 (iii) 8, 9 and 25
(i) HCF $= 3$ , LCM $= 420$ . (ii) HCF $= 1$ , LCM $= 11339$ . (iii) HCF $= 1$ , LCM $= 1800$ .
Ch 2Polynomials
2-Mark Questions
The graphs of $y = p(x)$
(i) 0 (ii) 1 (iii) 3 (iv) 2 (v) 4 (vi) 3
Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients. (i) $x^2 - 2x - 8$
(i) Zeroes: $4, -2$ ; sum $= 2 = -\dfrac{-2}{1}$ , product $= -8 = \dfrac{-8}{1}$ . (ii) Zeroes: $\dfrac{1}{2}, \dfrac{1}{2}$ ; sum $= 1 = -\dfrac{-4}{4}$ , product $= \dfrac{1}{4} = \dfrac{1}{4}$ . (iii) Zeroes: $\dfrac{3}{2}, -\dfrac{1}{3}$ ; sum $= \dfrac{7}{6} = -\dfrac{-7}{6}$ , product $= -\dfrac{1}{2} = \dfrac{-3}{6}$ . (iv) Zeroes: $0, -2$ ; sum $= -2 = -\dfrac{8}{4}$ , product $= 0 = \dfrac{0}{4}$ . (v) Zeroes: $\sqrt{15}, -\sqrt{15}$ ; sum $= 0 = -\dfrac{0}{1}$ , product $= -15 = \dfrac{-15}{1}$ . …
Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively. (i) $\dfrac{1}{4}, -1$
(i) $4x^2 - x - 4$ (ii) $3x^2 - 3\sqrt{2}x + 1$ (iii) $x^2 + \sqrt{5}$ (iv) $x^2 - x + 1$ (v) $4x^2 + x + 1$ (vi) $x^2 - 4x + 1$
Ch 3Pair of Linear Equations in Two Variables
2-Mark Questions
Form the pair of linear equations in the following problems, and find their solutions graphically. (i) 10 students of Class X took part in a Mathematics quiz. If the number of girls is 4 more than the number of boys, find the number of boys and girls who took part in the quiz. (ii) 5 pencils and 7 pens together cost ₹50, whereas 7 pencils and 5 pens together cost ₹46. Find the cost of one pencil and that of one pen.
(i) Boys $= 3$ , girls $= 7$ . (ii) Cost of one pencil $= ₹3$ , cost of one pen $= ₹5$ .
On comparing the ratios $\dfrac{a_1}{a_2}$
(i) Intersect at a point. (ii) Coincident lines. (iii) Parallel lines.
On comparing the ratios $\dfrac{a_1}{a_2}$
(i) Consistent. (ii) Inconsistent. (iii) Consistent. (iv) Consistent. (v) Consistent.
Ch 4Quadratic Equations
2-Mark Questions
Check whether the following are quadratic equations: (i) $(x + 1)^2 = 2(x - 3)$
(i) Yes (ii) Yes (iii) No (iv) Yes (v) Yes (vi) No (vii) No (viii) Yes
Represent the following situations in the form of quadratic equations: (i) The area of a rectangular plot is $528\ \text{m}^2$
(i) $2x^2 + x - 528 = 0$ , where $x$ is the breadth. (ii) $x^2 + x - 306 = 0$ , where $x$ is the smaller integer. (iii) $x^2 + 32x - 273 = 0$ , where $x$ is Rohan’s present age. (iv) $x^2 - 8x - 1280 = 0$ , where $x$ is the train’s speed in km/h.
Find the roots of the following quadratic equations by factorisation: (i) $x^2 - 3x - 10 = 0$
(i) $x = 5, -2$ (ii) $x = \dfrac{3}{2}, -2$ (iii) $x = -\sqrt{2}, -\dfrac{5}{\sqrt{2}}$ (or $-\dfrac{5\sqrt{2}}{2}$ ) (iv) $x = \dfrac{1}{4}, \dfrac{1}{4}$ (v) $x = \dfrac{1}{10}, \dfrac{1}{10}$
Ch 5Arithmetic Progressions
2-Mark Questions
In which of the following situations, does the list of numbers involved make an arithmetic progression, and why? (i) The taxi fare after each km when the fare is ` 15 for the first km and ` 8 for each additional km. (ii) The amount of air present in a cylinder when a vacuum pump removes $\dfrac{1}{4}$
(i) Yes, the fares are $15, 23, 31, 39, \ldots$ with common difference $8$ . (ii) No, the remaining air is multiplied by $\dfrac{3}{4}$ each time, so the differences are not equal. (iii) Yes, the costs are $150, 200, 250, 300, \ldots$ with common difference $50$ . (iv) No, compound interest multiplies the amount each year, so the yearly increases are not equal.
Write first four terms of the AP, when the first term a and the common difference d are given as follows: (i) a = 10, d = 10 (ii) a = -2, d = 0 (iii) a = 4, d = - 3 (iv) a = - 1, d = $\dfrac{1}{2}$
(i) $10, 20, 30, 40$ (ii) $-2, -2, -2, -2$ (iii) $4, 1, -2, -5$ (iv) $-1, -\dfrac{1}{2}, 0, \dfrac{1}{2}$ (v) $-1.25, -1.50, -1.75, -2.00$
For the following APs, write the first term and the common difference: (i) 3, 1, - 1, - 3, . . . (ii) - 5, - 1, 3, 7, . . . (iii) $\dfrac{1}{3}, \dfrac{5}{3}, \dfrac{9}{3}, \dfrac{13}{3}, . . .$
(i) $a = 3$ , $d = -2$ . (ii) $a = -5$ , $d = 4$ . (iii) $a = \dfrac{1}{3}$ , $d = \dfrac{4}{3}$ . (iv) $a = 0.6$ , $d = 1.1$ .
1-Mark Questions (MCQ)
Choose the correct choice in the following and justify : (i) 30th term of the AP: 10, 7, 4, . . . , is (A) 97 (B) 77 (C) -77 (D) - 87 (ii) 11th term of the AP: $-3, -\dfrac{1}{2}, 2, . . .$
(i) Choice (C), $-77$ . (ii) Choice (B), $22$ .
Ch 6Triangles
2-Mark Questions
Fill in the blanks using the correct word given in brackets : (i) All circles are ________. (congruent, similar) (ii) All squares are ________. (similar, congruent) (iii) All ________ triangles are similar. (isosceles, equilateral) (iv) Two polygons of the same number of sides are similar, if (a) their corresponding angles are ________ and (b) their corresponding sides are ________. (equal, proportional)
(i) similar (ii) similar (iii) equilateral (iv) (a) equal, (b) proportional
Give two different examples of pair of (i) similar figures. (ii) non-similar figures.
(i) Any two circles; any two squares. (ii) A circle and a square; a triangle and a square.
State whether the following quadrilaterals are similar or not: Fig. 6.8 shows quadrilateral PQRS with each side 1.5 cm and quadrilateral ABCD with each side 3 cm, with right angles marked in ABCD.
The quadrilaterals are not similar.
Ch 7Coordinate Geometry
2-Mark Questions
Find the distance between the following pairs of points : (i) (2, 3), (4, 1) (ii) (- 5, 7), (- 1, 3) (iii) (a, b), (- a, - b)
(i) $2\sqrt{2}$ (ii) $4\sqrt{2}$ (iii) $2\sqrt{a^2+b^2}$
Find the distance between the points (0, 0) and (36, 15). Can you now find the distance between the two towns A and B discussed in Section 7.2.
The distance is 39 units, so the distance between the towns is 39 km.
Determine if the points (1, 5), (2, 3) and (- 2, - 11) are collinear.
The points are not collinear.
Ch 8Introduction to Trigonometry
2-Mark Questions
In $\triangle ABC$
(i) $\sin A=\dfrac{7}{25}$ , $\cos A=\dfrac{24}{25}$ . (ii) $\sin C=\dfrac{24}{25}$ , $\cos C=\dfrac{7}{25}$ .
In Fig. 8.13, find tan P - cot R.
$\tan P-\cot R=0$ .
If $\sin A = \dfrac{3}{4}$
$\cos A=\dfrac{\sqrt7}{4}$ and $\tan A=\dfrac{3}{\sqrt7}=\dfrac{3\sqrt7}{7}$ .
1-Mark Questions (MCQ)
Choose the correct option and justify your choice : (i) $\dfrac{2\tan30^\circ}{1+\tan^230^\circ}$
(i) A, sin 60° (ii) D, 0 (iii) A, 0° (iv) C, tan 60°
Ch 9Some Applications of Trigonometry
2-Mark Questions
A circus artist is climbing a 20 m long rope, which is tightly stretched and tied from the top of a vertical pole to the ground. Find the height of the pole, if the angle made by the rope with the ground level is 30° (see Fig. 9.11).
The height of the pole is 10 m.
A tree breaks due to storm and the broken part bends so that the top of the tree touches the ground making an angle 30° with it. The distance between the foot of the tree to the point where the top touches the ground is 8 m. Find the height of the tree.
The height of the tree is $8\sqrt3$ m.
A contractor plans to install two slides for the children to play in a park. For the children below the age of 5 years, she prefers to have a slide whose top is at a height of 1.5 m, and is inclined at an angle of 30° to the ground, whereas for elder children, she wants to have a steep slide at a height of 3m, and inclined at an angle of 60° to the ground. What should be the length of the slide in each case?
The slide lengths should be 3 m and $2\sqrt3$ m, respectively.
Ch 10Circles
2-Mark Questions
How many tangents can a circle have?
A circle can have infinitely many tangents.
Fill in the blanks : (i) A tangent to a circle intersects it in ____ point (s). (ii) A line intersecting a circle in two points is called a ____. (iii) A circle can have ____ parallel tangents at the most. (iv) The common point of a tangent to a circle and the circle is called ____.
(i) one (ii) secant (iii) two (iv) point of contact
Draw a circle and two lines parallel to a given line such that one is a tangent and the other, a secant to the circle.
One valid construction is: draw a circle, draw any line that cuts it at two points, and then draw a line parallel to it just touching the circle. The first line is a secant and the parallel touching line is a tangent.
1-Mark Questions (MCQ)
A tangent PQ at a point P of a circle of radius 5 cm meets a line through the centre O at a point Q so that OQ = 12 cm. Length PQ is :
Correct option: (D) $\sqrt{119}$ cm.
Ch 11Areas Related to Circles
2-Mark Questions
Find the area of a sector of a circle with radius 6 cm if angle of the sector is $60^\circ$
The area of the sector is $\dfrac{132}{7}$ cm $^2$ (about $18.86$ cm $^2$ ).
Find the area of a quadrant of a circle whose circumference is 22 cm.
The area of the quadrant is $\dfrac{77}{8}$ cm $^2$ or $9.625$ cm $^2$ .
The length of the minute hand of a clock is 14 cm. Find the area swept by the minute hand in 5 minutes.
The area swept is $\dfrac{154}{3}$ cm $^2$ (about $51.33$ cm $^2$ ).
1-Mark Questions (MCQ)
Tick the correct answer in the following : Area of a sector of angle $p$
Correct option: (C) $\dfrac{p}{360}\times\pi R^2$ .
Ch 12Surface Areas and Volumes
2-Mark Questions
2 cubes each of volume 64 cm $^3$
The surface area of the resulting cuboid is $160$ cm $^2$ .
A vessel is in the form of a hollow hemisphere mounted by a hollow cylinder. The diameter of the hemisphere is 14 cm and the total height of the vessel is 13 cm. Find the inner surface area of the vessel.
The inner surface area of the vessel is $572$ cm $^2$ .
A toy is in the form of a cone of radius 3.5 cm mounted on a hemisphere of same radius. The total height of the toy is 15.5 cm. Find the total surface area of the toy.
The total surface area of the toy is $214.5$ cm $^2$ .
Ch 13Statistics
2-Mark Questions
A survey was conducted by a group of students as a part of their environment awareness programme, in which they collected the following data regarding the number of plants in 20 houses in a locality. Find the mean number of plants per house. Number of plants: 0 - 2, 2 - 4, 4 - 6, 6 - 8, 8 - 10, 10 - 12, 12 - 14 Number of houses: 1, 2, 1, 5, 6, 2, 3 Which method did you use for finding the mean, and why?
The mean number of plants per house is $8.1$ . The direct method is convenient because the class marks and frequencies are small.
Consider the following distribution of daily wages of 50 workers of a factory. Daily wages (in ₹): 500 - 520, 520 - 540, 540 - 560, 560 - 580, 580 - 600 Number of workers: 12, 14, 8, 6, 10 Find the mean daily wages of the workers of the factory by using an appropriate method.
The mean daily wage is $₹545.20$ .
The following distribution shows the daily pocket allowance of children of a locality. The mean pocket allowance is Rs 18. Find the missing frequency $f$
The missing frequency is $f=20$ .
Ch 14Probability
2-Mark Questions
Complete the following statements: (i) Probability of an event E + Probability of the event 'not E' = ____. (ii) The probability of an event that cannot happen is ____. Such an event is called ____. (iii) The probability of an event that is certain to happen is ____. Such an event is called ____. (iv) The sum of the probabilities of all the elementary events of an experiment is ____. (v) The probability of an event is greater than or equal to ____ and less than or equal to ____.
(i) $1$ (ii) $0$ ; an impossible event (iii) $1$ ; a sure or certain event (iv) $1$ (v) $0$ and $1$
Which of the following experiments have equally likely outcomes? Explain. (i) A driver attempts to start a car. The car starts or does not start. (ii) A player attempts to shoot a basketball. She/he shoots or misses the shot. (iii) A trial is made to answer a true-false question. The answer is right or wrong. (iv) A baby is born. It is a boy or a girl.
Only (iii), when the answer is chosen at random, has equally likely outcomes. The others need not have equally likely outcomes.
Why is tossing a coin considered to be a fair way of deciding which team should get the ball at the beginning of a football game?
Because a fair coin has two equally likely outcomes: head and tail.
1-Mark Questions (MCQ)
Which of the following cannot be the probability of an event?
Correct option: (B) $-1.5$ .
Frequently asked questions
- Express each number as a product of its prime factors: (i) 140 (ii) 156 (iii) 3825 (iv) 5005 (v) 7429
- (i) $140 = 2^2 \times 5 \times 7$ (ii) $156 = 2^2 \times 3 \times 13$ (iii) $3825 = 3^2 \times 5^2 \times 17$ (iv) $5005 = 5 \times 7 \times 11 \times 13$ (v) $7429 = 17 \times 19 \times 23$
- Find the LCM and HCF of the following pairs of integers and verify that LCM × HCF = product of the two numbers: (i) 26 and 91 (ii) 510 and 92 (iii) 336 and 54
- (i) HCF $= 13$ , LCM $= 182$ ; check $13 \times 182 = 2366 = 26 \times 91$ . (ii) HCF $= 2$ , LCM $= 23460$ ; check $2 \times 23460 = 46920 = 510 \times 92$ . (iii) HCF $= 6$ , LCM $= 3024$ ; check $6 \times 3024 = 18144 = 336 \times 54$ .
- Find the LCM and HCF of the following integers by applying the prime factorisation method: (i) 12, 15 and 21 (ii) 17, 23 and 29 (iii) 8, 9 and 25
- (i) HCF $= 3$ , LCM $= 420$ . (ii) HCF $= 1$ , LCM $= 11339$ . (iii) HCF $= 1$ , LCM $= 1800$ .
- The graphs of $y = p(x)$
- (i) 0 (ii) 1 (iii) 3 (iv) 2 (v) 4 (vi) 3
These important questions are selected from the NCERT Class 10 Maths textbook book-back exercises to help you revise the most useful questions. Mark weightage (5/2/1) follows the usual exam pattern and may vary by exam — always check your latest syllabus and question pattern. Open each chapter for the complete set of questions and answers.