Board Paper of Class 10 2023 Maths (Basic) Delhi(Set 3) - Solutions

General Instructions
Read the following instructions carefully and follow them:
1. This question paper contains 38 questions. All questions are compulsory.
2. Question paper is divided into FIVE sections Section A, B, C, D and E.
3. In section A, question number 1 to 18 are multiple choice questions (MCQs) and question number 19 and 20 are Assertion - Reason based questions of 1 mark each.
4. In section B, question number 21 to 25 are very short answer (VSA) type questions of 2 marks each.
5 in section C, question number 26 to 31 are short answer (SA) type questions carrying 3 marks each.
6. In section D, question number 32 to 35 are long answer (LA) type questions carrying 5 marks each.
7. In section E, question number 36 to 38 are case based integrated units of assessment questions carrying 4 marks each. Internal choice is provided in 2 marks question in each case study.
8. There is no overall choice. However, an internal choice has been provided in 2 questions in Section B, 2 questions in Section C, 2 questions in Section D and 3 questions in Section E.
9. Draw neat figures wherever required. Take π = 22/7 wherever required if not stated.
10. Use of calculators is not allowed.

Question 1
The value of k for which the equations 3x – y + 8 = 0 and 6x – ky + 16 = 0 represent coincident lines is :
$(a) \frac{1}{2} (b) - \frac{1}{2}$
(c) 2
(d) –2 VIEW SOLUTION

Question 2
A circle f radius 5.2 cm has two tangents AB and CD parallel to each other. What is the distance between the two tangents?
(a) 5.2 cm
(b) 10.4 cm
(c) 20.8 cm
(d) can't find VIEW SOLUTION

Question 3
The number of polynomials having zeroes –3 and 4 is :
(a) 1
(b) 2
(c) 3
(d) more than 3 VIEW SOLUTION

Question 4
If the perimeter and the area of a circle are numerically equal, then the radius of the circle is :
(a) 2 units
(b) π units
(c) 4 units
(d) 2 π units VIEW SOLUTION

Question 5
A quadratic equation whose one root is 2 and the sum of whose roots is zero, is:
(a) $x^{2} + 4 = 0$
(b) $x^{2} - 2 = 0$
(c) $4 x^{2} - 1 = 0$
(d) $x^{2} - 4 = 0$ VIEW SOLUTION

Question 6
Which of the following is not a quadratic equation?
(a) $2 {(x - 1)}^{2} = 4 x^{2} - 2 x + 1$
(b) $2 x - x^{2} = x^{2} + 5$
(c) ${(\sqrt{2} x + \sqrt{3})}^{2} + x^{2} = 3 x^{2} - 5 x$
(d) ${(x^{2} + 2 x)}^{2} = x^{4} + 3 + 4 x^{3}$ VIEW SOLUTION

Question 7
How many tangents can be drawn to a circle from a point on it?
(a) One
(b) Two
(c) Infinite
(d) Zero VIEW SOLUTION

Question 8
Which of the following cannot be the probability of an event?

(a) 0.1
(b) $\frac{5}{3}$
(c) 3%
(d) $\frac{1}{3}$ VIEW SOLUTION

Question 9
The angle of elevation of the top of a 30 m high tower at a point 30 m away from the base of the tower is:
(a) 30^∘
(b) 45^∘
(c) 60^∘
(d) 90^∘ VIEW SOLUTION

Question 10
The mode of the numbers 2, 3, 3, 4, 5, 4, 4, 5, 3, 4, 2, 6, 7 is:
(a) 2
(b) 3
(c) 4
(d) 5 VIEW SOLUTION

Question 11
If the pair of linear equations $x - y = 1, x + k y = 5$ has a unique solution x = 2, y = 1, then the value of k is:
(a) −2
(b) −3
(c) 3
(d) 4 VIEW SOLUTION

Question 12
(HCF × LCM) for the numbers 70 and 40 is :
(a) 10
(b) 280
(c) 2800
(d) 70 VIEW SOLUTION

Question 13
The number $(5 - 3 \sqrt{5} + \sqrt{5})$ is :
(a) an integer
(b) a rational number
(c) an irrational number
(d) a whole number VIEW SOLUTION

Question 14
If p(x) = x² + 5x + 6, then p(−2) is :
(a) 20
(b) 0
(c) −8
(d) 8 VIEW SOLUTION

Question 15
If ∆ABC ∼ ∆DEF and $∠ A = 47 °, ∠ E = 83 °$ then $∠ C$ is equal :
(a) 47^∘
(b) 50^∘
(c) 83^∘
(d) 130^∘ VIEW SOLUTION

Question 16
From a well -shuffled deck of 52 playing cards, a card is drawn at random. What is the probability of getting a red queen?
(a) $\frac{1}{52}$
(b) $\frac{1}{26}$
(c) $\frac{1}{13}$
(d) $\frac{12}{13}$ VIEW SOLUTION

Question 17
What is the length of arc of a circle of radius 7 cm which subtends an angle of 90° at the centre of the circle?

(a) 22 cm

(b) 11 cm

$(c) \frac{77}{2} cm (d) \frac{11}{2} cm$ VIEW SOLUTION

Question 18
(3 sin² 30° – 4 cos² 60°) is equal to :
$(a) \frac{5}{4} (b) - \frac{3}{4} (c) - \frac{1}{4} (d) - \frac{9}{4}$ VIEW SOLUTION

Question 19
Assertion (A): A tangent to a circle is perpendicular to the radius through the point of contact.
Reason (R): The lengths of tangents drawn from an external point to a circle are equal.
(a) Both Assertion (A) and Reason (R) are true and Reason (R) gives the correct explanation of Assertion (A).
(b) Both Assertion (A) and Reason (R) are true but Reason (R) does not give the correct explanation of Assertion (A).
(c) Assertion (A) is true but Reason (R) is false.
(d) Assertion (A) is false but Reason (R) is true. VIEW SOLUTION

Question 20
Assertion (A): If one root of the quadratic equation 4x² − 10x + (k − 4) = 0 is reciprocal of the other, then value of k is 8.
Reason (R): Roots of the quadratic equation x² − x + 1 = 0 are real.
(a) Both Assertion (A) and Reason (R) are true and Reason (R) gives the correct explanation of Assertion (A).
(b) Both Assertion (A) and Reason (R) are true but Reason (R) does not give the correct explanation of Assertion (A).
(c) Assertion (A) is true but Reason (R) is false.
(d) Assertion (A) is false but Reason (R) is true. VIEW SOLUTION

Question 21
Find the discriminant of the quadratic equation 3x²−2x + $\frac{1}{3}$ = 0 and hence find the nature of its roots.
OR

Find the roots of the quadratic equation x² − x − 2 = 0. VIEW SOLUTION

Question 22
In the adjoining figure, A, B and C are points on OP, OQ and OR respectively such that AB||PQ and AC||PR. Show that BC||QR.
VIEW SOLUTION

Question 23
In the adjoining figure, PT is a tangent at T to the circle with center O. If $∠ TPO$ = 30º, find the value of x.
VIEW SOLUTION

Question 24
Find the coordinates of the point which divides the join of A(−1, 7) and B(4, −3) in the ratio 2 : 3.
OR

If the points A (2, 3), B (−5, 6), C (6, 7) and D (p, 4) are the vertices of a parallelogram ABCD, find the value of p. VIEW SOLUTION

Question 25
In a right triangle PQR, right angled at Q. If tanP = $\sqrt{3}$ , then evaluate 2 sinP cosP. VIEW SOLUTION

Question 26
Prove that $\frac{1 + secθ}{secθ} = \frac{\sin^{2} θ}{1 - cosθ}$ VIEW SOLUTION

Question 27
An unbiased coin is tossed twice. Find the probability of getting:
(a) at least one head.
(b) exactly one tail.
(c) at most one head. VIEW SOLUTION

Question 28
Find the zeroes of the quadratic polynomial x² + 6x + 8 and verify the relationship between the zeroes and the coefficients. VIEW SOLUTION

Question 29
Find the area of the sector of a circle of radius 7 cm and of central angle 90°. Also, find the area of corresponding major sector. VIEW SOLUTION

Question 30
Prove that the lengths of tangents drawn from an external point to a circle are equal.
OR

Two concentric circles with centre O are of radii 3 cm and 5 cm. Find the length of chord AB of the larger circle which touches the smaller circle at P.
VIEW SOLUTION

Question 31
A lending library has a fixed charge for first three days and an additional charge for each day thereafter. Rittik paid ₹27 for a book kept for 7 days and Manmohan paid ₹21 for a book kept for 5 days. Find the fixed charges and the charge for each extra day.
OR

Find the values of 'a' and 'b' for which the system of linear equations 3x + 4y = 12, (a + b) x + 2 (a – b) y = 24 has infinite number of solutions. VIEW SOLUTION

Question 32
The boilers are used in thermal power plants to store water and then used to produce steam. One such boiler consists of a cylindrical part in middle and two hemispherical parts at its both ends. Length of the cylindrical part is 7m and radius of cylindrical part is $\frac{7}{2} m .$
Find the total surface area and the volume of the boiler. Also, find the ratio of the volume of cylindrical part to the volume of one hemispherical part.
VIEW SOLUTION

Question 33
The following table gives the monthly consumption of electricity of 100 families:

Monthly Consumption (in units) 130-140 140-150 150-160 160-170 170-180 180-190 190-200

Number of families 5 9 17 28 24 10 7

Find the median of the above data. VIEW SOLUTION

Question 34
The first term of an A.P. is –5 and the last term is 45. If the sum of all the terms of the A.P. is 120, find the number of terms and the common difference.
OR

If the sum of first 7 terms of an A.P. is 49 and that of first 17 terms is 289, find the sum of first n terms. VIEW SOLUTION

Question 35
As observed from the top of a 75 m high light house from the sea-level, the angles of depression of two ships are 30° and 45°. If one ship is exactly behind the other on the same side of the light house, find the distance between the two ships. $(use \sqrt{3} = 1.73)$
OR

From a point P on the ground, the angle of elevation of the top of a 10 m tall building is 30°. A flag is hoisted at the top of the building and the angle of elevation of the top of the flagstaff from P is 45°. Find the length of the flagstaff and the distance of the building from the point P.
$(use \sqrt{3} = 1.73)$ VIEW SOLUTION

Question 36
Khushi wants to organize her birthday party. Being health conscious, she decided to serve only fruits in her birthday party. She bought 36 apples and 60 bananas and decided to distribute fruits equally among all.

Based on the above information, answer the following questions:

(i) How many guests Khushi can invite at the most?

(ii) How many apples and bananas will each guest get ?

(iii) (a) If Khushi decides to add 42 mangoes, how many guests Khushi can invite at the most ?
OR

(b) If the cost of l dozen of bananas is ₹60, the cost of 1 apple is ₹15 and cost of 1 mango is ₹20, find the total amount spent on 60 bananas, 36 apples and 42 mangoes.
VIEW SOLUTION

Question 37
Observe the figures given below carefully and answer the questions:
Figure A

Figure B

Figure C

(i) Name the figure(s) wherein two figures are similar.

(ii) Name the figure(s) wherein the figures are congruent.

(iii) (a) Prove that congruent triangles are also similar but not the converse.
OR

(b) What more is least needed for two similar triangles to be congruent?
VIEW SOLUTION

Question 38
Use of mobile screen for long hours makes your eye sight weak and give you headaches. Children who are addicted to play "PUBG" can get easily stressed out. To raise social awareness about ill effects of playing PUBG, a school decided to start 'BAN PUBG' campaign, in which students are asked to prepare campaign board in the shape of a rectangle. One such campaign board made by class X student of the school is shown in the figure.

Based on the above information, answer the following questions:

(i) Find the coordinates of the point of intersection of diagonals AC and BD.

(ii) Find the length of the diagonal AC.

(iii) (a) Find the area of the campaign Board ABCD.
OR

(b) Find the ratio of the length of side AB to the length of the diagonal AC.
VIEW SOLUTION

More Board Paper Solutions for Class 10 Math

Board Paper Solutions for Other Subjects

Monthly Consumption (in units)	130-140	140-150	150-160	160-170	170-180	180-190	190-200
Number of families	5	9	17	28	24	10	7