RBSE Class 10 E-Mathematics Self Evaluation Test Papers in English

Students must start practicing the questions from RBSE 10th Maths Model Papers E-Mathematics Self Evaluation Test Papers in English Medium provided here.

Time: 2:45 Hours
Maximum Marks: 80

General Instructions:

• All the questions are compulsory.
• Write the answer of each question in the given answer book only.
• For questions having more than one part the answers to their parts are to be written together in continuity.
• Candidate must write first his/her Roll No. on the question paper compulsorily.
• Question numbers 17 to 23 have internal choices.
• The marks weightage of the questions are as follows :

Section	Number of Questions	Total Weightage	Marks for each question
Section A	1 (i to xii), 2(i to vi), 3(i to xii) = 30	30	1
Section B	4 to 16 =13	26	2
Section C	17 to 20 = 4	12	3
Section D	21 to 23 = 3	12	4

RBSE Class 10 E-Mathematics Self Evaluation Test Paper 1 in English

Section -A

Question 1.
Multiple Choice Questions :
(i) If a is irrational, then a² is : (1)
(A) rational
(B) irrational
(C) perkct square
(D) none of these

(ii) If p(x) = x² + 5x + 2. then p(3) + p(2) + p(0) is : (1)
(A) 40
(B) 44
(C) 8
(D) 42

(iii) If a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 are pairs of linear equations in two variables and they have intinite number of solutions, then : (1)
(A) \(\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}} \neq \frac{c_{3}}{c_{2}}\)
(B) \(\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}\)
(C) \(\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}\)
(D) \(\frac{a_{1}}{a_{2}} \neq \frac{b_{1}}{b_{2}}\)

(iv) If the equation x² + 9x+ p = 0 has real roots, then : (1)
(A )P ≤ \(\frac{81}{4}\)
(B)P < \(\frac{81}{4}\)
(C) P > \(\frac{-81}{4}\)
(D )P ≥ \(\frac{9}{2}\)

(v) The first and last terms of an AP are 2 and 26 respectively. If the sum of its terms is 126, then number of terms will be: (1)
(A) 7
(B) 6
(C) 9
(D) 8

(vi) To divide a line segment, AB in the ratio 3 : 4, draw a ray AX such that ∠BAX is an acute angle, then draw a ray BY ∥ AX such that ∠ABY = ∠BAX. Locate points A₁, A₂, A₃, on AX, B₁, B₂, B₃ on BY at equal distances (each = AA₁ = BB₁) and then points joined are : (1)
(A) A₁ and B₃
(B) A₄ and B₄
(C) A₃ and B₅
(D) A₃ and B₄

(vii) The vertices of the diagonal of a parallelogram are (3, -4) and (-6, 5). If third vertex is (-2, 1), them the co¬ordinates of its fourth vertex is : (1)
(A) (1, 0)
(B) (-1, 0)
(C) (-1, 1)
(D) (1,-1)

(viii) If √3 tan 2θ – 3 = 0. , then θ is equal to : (1)
(A) 60°
(B) 20°
(C) 45°
(D) 30°

(ix) The value of 1 – sin²30° is equal to : (1)
(A) \(\frac{\sqrt{3}}{2}\)
(B) \(\frac{1}{4}\)
(C) \(\frac{3}{4}\)
(D) \(\frac{4}{3}\)

(x) The median of data : 18. 14, 3, 19, 12, 15, 8, 10, 13, 21, 22 will be : (1)
(A) 12
(B) 13
(C) 14
(D) 15

(xi) If the mode of a data is 18 and mean is 24, then median is : (1)
(A) 18
(B) 24
(C) 22
(D) 21

(xii) If there is an event (E) such that P (E) = \(\frac{x}{15}\) and P (not E) = \(\frac{3}{5}\), then x = ? (1)
(A) 5
(B) 6
(C) 9
(D) 15

Question 2.
Fill in the blanks :
(i) The pair of equations. y = 0 and y = -7 has _________ solution (s). (1)

(ii) If four numbers are in AP whose sum is 24, then first term will be _________ (1)

(iii) To draw a pair of tangents to a circle which are inclined to each other at an angle of 45°, it is required to draw tangents at the end points of those two radii of the circle, the angle between which is _________ (1)

(iv) Mid point of the line joining points \(\left(\frac{3}{2}, \frac{1}{4}\right)\) and \(\left(\frac{5}{2}, \frac{1}{3}\right)\) is _________ (1)

(v) If tan θ = \(\frac{3}{4}\), then sin²θ – cos²θ is equal to _________

(vi) The data having only one mode is called _________ data. (1)

Question 3.
Very Short Answer Type Questions :
(i) Find HCF of408 and 1032. (1)

(ii) If one zero of polynomial 3x² + 12x = k is reciprocal of the other, then find the value of k. (1)

(iii) If the graph of polynomial intersects the x-axis at exactly three points, then find the number of zeroes for this. (1)

(iv) Find the equation of a line coincident to the line -5x + 7y = 2. (1)

(v) Find the value of k, for which x = -5 is a solution of the quadratic equation x² + kx + 5 = 0. (1)

(vi) If one root of equation ax² + bx + c = 0 is three times the other, then find \(\frac{a c}{b^{2}}\) (1)

(vii) Divide a line segment of length 7 cm internally in the ratio 2 : 3. (1)

(viii) Find the values of k, if the point P (2, 4) is equidistant from the points A (5, k) and B (k, 7) (1)

(ix) If in A45C, ZR= 90°, .4,5= V3 cm and ,4C= 2 cm, then find ZC. (1)

(x) If isosceles A ABC is right-angled at C, then the value of cos (A + B) and tan (A – B). (1)

(xi) Following are the lines in hours of 15 pieces of the components of aircraft engine. Find the median. (1)
715, 724, 725, 710, 729, 745, 694, 699, 696, 712, 734, 728, 716, 705, 719. (1)

(xii) Three coins are tossed once. Find the probability of getting at least two heads. (1)

Section – B

Question 4.
Use Euclid’s division Algorithm to find the HCF of 155 and 1385. (2)

Question 5.
Divide 3x + x² + 2x + 5 by 1 + 2x + x². Find quotient and remainder. (2)

Question 6.
If two numbers are such that the sum of four-time the first and thrice the second is 51 and five times the first exceeds seven times the second by 10. Find the numbers. (2)

Question 7.
Show that the equation 8x² – 2x – 3 = 0 has real and unequal roots. (2)

Question 8.
Find the common difference of the AP whose 11th term is 38 and the 16th term is 73. (2)

Question 9.
Find the sum of 10th terms of AP: 2,7,12, ……….. (2)

Question 10.
Construct a triangle similar to a given triangle ABC with its sides equal to \(\frac{2}{3}\) of the corresponding sides of the triangle ABC.

Question 11.
Draw a circle of radius 3.5 cm. Take two points A aid B on one of its extended diameters each at a distance of 8 cm. from its centre. Draw tangents to the circle from these two points A and B, (2)

Question 12.
Prove that; \(\frac{\cos ^{2} \theta}{1-\tan \theta}+\frac{-\sin ^{3} \theta}{\sin \theta-\cos \theta}\) = 1 + sin θ cos θ

Question 13.
Prove that : \(\sqrt{\frac{1+\cos \theta}{1-\cos \theta}}\) = cosec θ + cot θ. (2)

Question 14.
Find the mode of the following data: (2)

Question 15.
Calculate the mean of the scores of 20 students in a mathematics test (2)

Question 16.
A card is drawn at random from a well-shuffled deck of 52 cards. Find the probability of getting (j) a red king. (ii) a queen. (2)

Section – C

Question 17.
The sum of n, 2 n, 3n terms of an A.P. are S₁, S₂, S₃ respectively. Prove that: S₃ = 3(S₂ – S₁) (3)
OR
Find the sum of all three-digits numbers which leave the same remainder 2, when divided by 5, (3)

Question 18.
Find the area of the quadrilateral ABCD whose vertices arc A (-3, -1), B (-2, -4), C (4, -1) and D (3, 4). (3)
OR
Prove that the points (a, b + c), (b, c + a) and (c, a + b) are collinear. (3)

Question 19.
Wthout using trigonornetric tables, evaluate: \(\frac{\cos ^{2} 20^{\circ}+\cos ^{2} 70^{\circ}}{\sin ^{2} 20^{\circ}+\sin ^{2} 70^{\circ}}\) + sin² 64° cos 64° sin 26°. (3)
OR
Prove that : cos²θ + cos⁴θ = 1, if sinθ + sin²θ = 1. (3)

Question 20.
The number of studernts absent in a school was recorded everyday kir 147 days and the raw data presented in the form of the following liequency table of students absent

Obtain the median. (3)
The following distribution gives the ages of workers of a factory. Calculate the modal age. (3)
OR

Section – D

Question 21.
The perimeter of a rectangular plot is 22 cm where length is 5 cm more than the width of the plot. Represent this situation algebraically and graphically. (4)
OR
Cost of 4 tables and 3 chairs is ₹ 2750 and cost of 8 tables and 5 chairs is ₹ 7200. Represent: this situation algebraically and graphically. (4)

Question 22.
Draw a triangle ABC in which AB = 4 cm, BC = 6 cm and AC = 9 cm. Construct a triangle similar to ∆ABC with scale factor \(\frac{3}{2}\).
Justify the construction. Are the two triangles congruent? (4)
OR
Let PQR be a right-angled triangle in which PQ = 9 cm and QR = 12 cm, and ∠PQR = 90°. Q is perpendicular from Q on PR. The circle through Q, R. M is drawa. Construct the pair of tangents from the point P to the circle. (4)

Question 23.
The following table gives the heights of trees: (4)

Draw more than ogive. (4)

RBSE Class 10 E-Mathematics Self Evaluation Test Paper 2 in English

Section – A

Question 1.
Multiple Choice Questions :
(i) If LCM (21,35) = 105, then HCF (21,35) is: (1)
(A) 21
(B) 15
(C) 7
(D) 5

(ii) Sum of the zeroes of the polynomial ax³ + bx² + cx + d is : (1)
(A) \(\frac{c}{a}\)
(B) \(\frac{-d}{a}\)
(C) \(\frac{-b}{a}\)
(D) \(\frac{b}{a}\)

(iii) If x = a, y = b are the solutions of equations x – y = 6 and x+ y = 4, then the values of a and b are : (1)
(A) 5 and -1
(B) -5 and -1
(C) 5 and 1
(D) 4 and 5

(vi) The value of c for which the equation ax² + 3bx + c = 0 has equal roots is: (1)
(A) \(\frac{4 a^{2}}{9 b^{2}}\)
(B) \(\frac{9 b^{2}}{4 a}\)
(C) \(\frac{9 b^{2}}{4 a^{2}}\)
(D) \(\frac{4 a}{9 b}\)

(v) If (x + 2), (2x + 3) and (4x + 1) are three consecutive terms of an AP, then x is equal to : (1)
(A) 2
(B) 3
(C) 1
(D) 4

(vi) To construct a triangle similar to a given ABC with its sides of the corresponding sides of ∆ABC, draw a ray BX such that ∠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
(B) 8
(C) 13
(D) 3

(vii) The vertices of a triangle are (1, a), (2, b) and (c², -3). The condition for which the centroid lie on the x-axis is: (1)
(A) a + b = 0
(B) a + b = 3
(C) a – b = 3
(D) a – b = 0

(viii) (1± tan²θ) cos²θ is equal to: (1)
(A) sin²θ – cos²θ
(B) 1
(C) sec²θ
(D) sin²θ

(ix) \(\frac{\tan ^{2} \theta}{\sec \theta+1}\) – secθ is equal to: (1)
(A) -1
(B) 1
(C) 0
(D) none of these

(x) The median of a frequency distribution is found graphically with, the help of: (1)
(A) ogives
(B) a frequency curve
(C) a histogram
(D) a frequency polygon

(xi) If the mode of the data: 12, 15, 10, 10, 12, 14,9,x 17, 15, 14 is 10,then x is: (1)
(A) 12
(B) 15
(C) 10
(D) 14

(xii) One card drawn at random from a well shuffled deck of 52 cards. The probability that the card drawn is not an ace card: (1)
(A) \(\frac{3}{13}\)
(B) \(\frac{9}{13}\)
(C) \(\frac{10}{13}\)
(D) \(\frac{3}{4}\)

Question 2.
Fill in the blanks :
(i) If the system of linear equations in two variables is consistent, then the lines represented by two equations are _______ (1)

(ii) If the nth term of an AP is 2n + 1. then the sum of its first three terms is _______ (1)

(iii) If the scale factor is \(\frac{3}{5}\), then the new’ triangle constructed is _______ (1)

(iv) If distance between (3, 2) and (-1, x) is 5, thenx is equal to _______ (1)

(v) (sec θ + tan θ) (1 – sin θ) is equal to _______ (1)

(vi) The width of a class is also known as the _______ of the class. (1)

Question 3.
Very Short Answer Type Questions:
(i) Find the number, if HCF of twu numbers is 23, LCM of two numbers is 1449 and one number is 161. (1)

(ii) Find the sum of the reciprocal s of the roots of the equation x² + px + q = 0. (1)

(iii) If α and β are the zeroes of x² – 5x + 6 = 0, then find α + β. (1)

(iv) If 5x + 7y = 2 and (a + 1) x + (a – 1)y = 3a – 1 represent parallel lines, then find the value of a. (1)

(v) Represent the following situation mathematically. “The sum of a number and its reciprocal is \(\frac{1}{2}\), we need to find the numbers. (1)

(vi) Find the roots of quadratic equation x² – 5x +6 = 0. (1)

(vii) Divide a line segment of length 10 cm internally in the ratio 4 : 3. (1)

(viii) Find the distance between P(√a, √b) and Q(-√a, -√b). (1)

(ix) If 4 tan θ = 3, then find \(\frac{4 \sin \theta-\cos \theta}{4 \sin \theta+\cos \theta}\) (1)

(x) If sin θ – cos θ = 0, then find sin⁴θ + cos⁴θ. (1)

(xi) The points scored by a basket ball team in a series of 14 matches are given below:
18, 14, 3, 19, 12, 15, 8, 10, 13, 21, 22, 5, 27, 11. Find the median score. (1)

(xii) In a single throw of a die, find the probability of getting a multiple of 3. (1)

Section – B

Question 4.
Find x, if the HCF of 420 and 110 is expressible in the form of 420 × 5 + 110x. (2)

Question 5.
If the product of two zeroes of the polynomial f(y) = 3y³ – 5y² – 11y – 3 is -3, then find its third zero. (2)

Question 6.
Solve the following pair of the linear equations by cross multiplication method: 5x + y- 13 =0 and 4x + 2y – 8 = 0. (2)

Question 7.
Find the roots of equation x² – 3y – 10 = 0 by factorisation method. (2)

Question 8.
How many multiples of 4 lies between 10 and 250? (2)

Question 9.
If the numbers 3P + 14, 7P+ 1 and 12P – 5 are in AP, then find the value of P. (2)

Question 10.
Construct a right triangle in which the sides (other than hypotenuse) are of lengths 6 cm and 8 cm, Then construct another triangle similar to it whose sides are \(\frac{3}{5}\) times the corresponding sides of the first triangle. (2)

Question 11.
Draw a circle of radius 4.5 cm. From a point 10 cm away from its centre construct the pair of tangents to the circle. (2)

Question 12.
In ∆ABC, right-angled at B. AB = 24 cm and BC = 7 cm determine sin A and cos A. (2)

Question 13.
Find the value of tan 1c tan 2° tan 3° …………….. tan 89°. (2)

Question 14.
Calculate the median of the following distribution : (2)

Question 15.
The following distribution shows the marks scored by 140 students in an examination. Calculate the mode of this distribution: (2)

Question 16.
Out of400 bulbs in a box, 15 bulbs are defective, one bulb is taken out at random from the box. Find the probability that the drawn bulb is not defective. (2)

Section – C

Question 17.
The sum of p, q, r terms of anA.P. are a, b, c respectively. Show that: \(\frac{a}{p}\)(q – r) + \(\frac{b}{q}\) (r – p) + \(\frac{c}{r}\)(p – q) = 0. (3)
OR
Yasmeent saves ₹ 32 during the first month, ₹ 36 in the second month and ₹ 40 in the third month. If she continues to save in this manner, in how many months will she save ₹ 2000. (3)

Question 18.
If the mid-point of line joining A (5,-1) andB (k, -2) is (,v,y) and 2x + 2y + 1 = 0, find the value of k: (3)
OR
If the points A (0, a), B (6,0) and C (1, 1) are collinear, then prove that \(\frac{1}{a}+\frac{1}{b}\) = 1. (3)

Question 19.
If a right ∆ABC, right-angled at C, ∠ABC = 9, AC = 1 cm and BC = 2 cm, then find the value of sin²θ + tan²θ.
OR
Without using trigonometric tables, evaluate :
\(\frac{\sec ^{2} 54^{\circ}-\cot ^{2}+36^{\circ}}{{cosec}^{2} 5^{\circ}-\tan ^{2} 33^{\circ}}\) + 2sin²38° sec² 52° – sin² 45°

Question 20.
Daily wages of 110 workers, obtained in a survey are tabulated below:

Compute the mean daily wages of these workers.

OR
Weights of 50 eggs were recorded as given below:

Calculate the median weight to the nearest gram.

Section – D

Question 21.
Amit bought two pencils and three chocolates for ₹ 11 and Sumit bought one pencil and two chocolates for ₹ 7. Represent this situation in the form of a pair of linear equations. Find the price of one pencil and that of one chocolate graphically. (4)
OR
Check graphically whether the pair of equations 2x – y = 4; x + y = -1 is consistent. If so. solve them graphically. (4)

Question 22.
Construct an isosceles triangle ABC with the base BC = 6 cm, AB = AC and ∠A= 90°. Draw another similar triangle whose sides are \(\frac{4}{5}\) of the sides of ∆ABC. (4)
OR
Draw a line segment AB = 9 cm. Taking A as centre draw a circle of radius 4 cm and taking B as centre draw another circle of radius 3 cm. Construct tangents to each circle from the centre of the other circle. (4)

Question 23.
Draw a cumulative frequency curve and cumulative frequency polygon for the following frequency distribution by less than method: (4)

OR
A survey regarding the heights (in cm) of 50 girls of class X of a school was conducted and the following data was obtained: (4)

RBSE Class 10 E-Mathematics Self Evaluation Test Paper 3 in English

Section-A

Question 1.
Multiple Choice Questions
(i) If a = bq + r in Euclid’s Division Algorithm, then r satisfies : (1)
(A)0 < r ≤ b
(B) 0 < r < b
(C) 0 ≤ r < b
(D) 0 ≤ r ≤ b

Question 2.
If graph of quadratic polynomial ax² + bx + c is a downward parabola, then a is : (1)
(A) a < 0
(B) a > 0
(C) a < 0
(D) a > 0

(iii) If a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 are pair of linear equations in two variables and they have no solutions, if: (1)
(A) \(\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}\)
(B) \(\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}\)
(C) \(\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}}\)
(D) \(\frac{a_{1}}{a_{2}} \neq \frac{b_{1}}{b_{2}}\)

(iv) If the sum of a number and its reciprocal is 2\(\frac{1}{6}\), then number is: (1)
(A) 6\(\frac{1}{2}\)
(B) 3\(\frac{1}{6}\)
(C) \(\frac{3}{2}\)
(D) 2\(\frac{1}{6}\)

(v) The sum of m terms of an AP i n and the sum of n terms is in, then the sum of (m + n) terms will be: (1)
(A) m – n
(B) m + n
(C) \(\frac{1}{2}\)(m + n)
(D) – (m + n)

(vi) To divide a line segment AB in the ratio 5 : 7, first a ray AX is drawn so that ∠BAX is an acute angle and then at equal distances points arc marked on the ray AX such that the minimum number of these points is: (1)
(A) 8
(B) 10
(C) 11
(D) 12

(vii) Points (7, 10), (-2, 5) and (3, -4) are the vertices of a: (1)
(A) a cute angled triangle
(B) obtüsed angled triangle
(C) right-angled triangle
(D) equilateral triangle

(viii) Given that sin α = \(\frac{1}{2}\) and cos β = \(\frac{1}{2}\), then the value of (α + β) is: (1)
(A) 0°
(B) 30°
(C) 60°
(D) 90°

(ix) If tan A = √3 and cot B = √3, then (A – B)is equal to: (1)
(A) 30°
(B) 90°
(C) 60°
(D) 45°

(x) The mean of certain number of observations is ?. If each observation is added by a, then mean of new observationis: (1)
(A) ax̄
(B) \(\frac{\bar{x}}{a}\)
(C) x̄ + a
(D) x̄ – a

(xii) In a single throw of a die the probability of getting a number divisible by 3 is: (1)
(A) \(\frac{1}{3}\)
(B) \(\frac{2}{3}\)
(C) \(\frac{1}{2}\)
(D) \(\frac{1}{6}\)

Question 2.
(i) If the system of linear equations in two variables is inconsistent, then the lines represented by two lines are __________ (1)

(ii) The first term of an AP is ,n and its common difference is n then 7th term will be __________ (1)

(iii) A pair of tangents can be constructed from a point P to a circle of radius 3.5 cm situated at a distance of __________ from the centre. (1)

(iv) The point (-4, -5) lie in __________ quadrant. (1)

(v) \(\frac{9 \sec ^{2} \theta-9 \tan ^{2} \theta}{2}\) is equal to __________ (1)

(vi) __________ mark of a class is the average of its upper and lower limits. (1)

Question 3.
(i) Find the simplest form of \(\frac{148}{185}\) (1)

(ii) Find the zeroes ofpolynoniial 3x² – 3x – 6 (1)

(iii) 1f 2 + √3 is one root of x2 + p.r + q. then find the value of p and q. (1)

(iv) Find the point of intersection of line y = 3x and x = 3. (1)

(y) Find the roots of quadratic equation x² – 9x + 18 = 0 (1)

(vi) If one root 0f x² – x – k = 0 be the square of the other, then find the value of k. (1)

(vii) Construct a triagle with sides 5 cm, 7 cm and 8 cm and then construct another triangle similar to it whose sides are \(\frac{3}{5}\)times the corresponding sìdes of the given triangle. (1)

(viii) Find the distance between points (-5, 2) and (-3, 2). (1)

(ix) If cot θ = \(\frac{b}{a}\), Find the value of \(\frac{\sin \theta-\cos \theta}{\sin \theta+\cos \theta}\). (1)

(x) If tan (α + β)= √3 and tan(α – β) = \(\frac{1}{\sqrt{3}}\), where α ≥ β and α + β ≥ 900. then find α and β. (1)

(xi) Find the range of the data 25, 18, 20, 22, 16, 6, 17, 12, 30, 32, 10, 19, 8, 11 and 20. (1)

(xii) A box contains 20 balls bearing numbers 1, 2, 3, 4, ……….. 20. A bail is drawn at random from the box. Find the probability that the number on the ball is divisible by 2 or 3. (1)

Section – B

Question 4.
Prove that √3 + √5 is irrational. (2)

Question 5.
If α and β arc the zeroes of the polynomial kx² + 3x + 2, such that α² + β²+ αβ = find the value of k. (2)

Question 6.
If the system of equations ax + by c = O and lx + my – n = 0 has a unique solution, then prove that am ≠ lb. (2)

Question 7.
If x = 4 and x = 5 are roots of equation x² – 3px + 5q = O. then find the value of p. (2)

Question 8.
Which term of the AP; 7, 13, 19 is 205? (2)

Question 9.
If 18, a, b, -3 are in AP, then find a + b. (2)

Question 10.
Construct a triangle similar to a given triangle ABC with its sides equal to \(\frac{4}{3}\) of the corresponding sides of the triangle ABC (i.e.. of scale factor \(\frac{4}{3}\)). (2)

Question 11.
Draw a circle of radius 5 cm. Draw a tangent to this circle making an angle of 45° with a line passing through the centre. (2)

Question 12.
Prove that \(\frac{1}{1+\sin \theta}+\frac{1}{1-\sin \theta}\) = 2sec²θ. (2)

Question 13.
Evaluate: \(\frac{\sin 30^{\circ}}{\cos 30^{\circ}}+\frac{\sec 30^{\circ}}{{cosec} 30^{\circ}}+\frac{\tan 30^{\circ}}{\cot 60^{\circ}}\) (2)

Question 14.
The maximum bowling speeds in km per hour of 33 players of cricket coaching centre are given as follows:

Calculate the median bowling speed. (2)

Question 15.
Calculate the mean of the following data: (2)

Question 16.
Find the probability of getting 53 fridays in a leap year. (2)

Section – C

Question 17.
If S₁, S₂ and S₃ are the sum of n terms of three series in A P, the first term of each being I and the respective common diffeence being 1, 2, 3, prove that: S₁ + S₃ = 2S₂ (3)
OR
The sum of the first 15 terms of an AP is 750 and its first term is 15. Find its 20th term. (3)

Question 18.
If the points p,q),(m, n) and (p – m, q – n) are collinear, show that pn = qm.
OR
Find the area of the triangle formed by the joining the mid-points of the sides of the triangle whose vertices are (0, -1), (2, 1) and (0, 3). (3)

Question 19.
If sinθ + cosθ = √2 sin (90° – θ), find the value of cot θ. (3)
OR ‘
If acos θ – bsinθ = c,prove that asin θ + bcos θ =± \(\sqrt{a^{2}+b^{2}-c^{2}}\) (3)

Question 20.
A book contained 300 pages. In the first proofreading the following distribution of misprints was obtained:

Find the mean of the distribution. (3)
OR
50 persons were examined through X-ray and observations were noted as under:

Find the median of the distribution. (3)

Section – D

Question 21.
Mohan purchased 3 pencils and 1 pen for ₹ 10 and Sumit purchased 2 pencils and 3 pens for ₹ 16. Find the price of 1 pencil and 1 pen graphically. (4)
OR
Solve graphically the system of equations 4x – 5y – 20 = 0, 3x + 5y – 15 = 0 Find the coordinates of the vertices of the triangle formed by these two lines and the y-axis. (4)

Question 22.
Draw a right triangle in which the sides (other than hypotenuse) arc of lengths 4 cm and 3 cm. Then construct another triangle whose sides are \(\frac{5}{3}\) times the corresponding sides of the given triangle. (4)
OR
Draw a circle of radius 5 cm construct a pair of tangents to its so that they are inclined at 60°. Measure the lengths of the two tangents. (4)

Question 23.
The monthly profits (in ₹) of 120 shops are distributed as follows : (4)

Convert the above distribution to less than type cumulative frequency distribution and draw its ogive.
OR
Compute the median from the following data: (4)

RBSE Class 10 E-Mathematics Self Evaluation Test Paper 4 in English

Section-A

Question 1.
Multiple Choice Questions:
(i) π is: (1)
(A) a rational number
(B) an irrational number
(C) not real
(D) prime number

(ii) If graph of y = ax² + bx + c intersects the x-axis at most in two points, then quadratic polynomial can have real zeroes : (1)
(A) 3
(B) 2
(C) 1
(D) 0

(iii) The point of intersection of lines y = 3x and x = 3y is: (1)
(A) (0,0)
(B) (3, 3)
(C) (0, 3)
(D) (3, 0)

(iv) The positive value of p for which the equation x² +4x + p = 0 and x² +2px +16 = 0 will both have real and equal roots is : (1)
(A) 4
(B) 8
(C) 2
(D) 6

(v) What is next term of the AP : √2, √8, √18, ………… ? (1)
(A) 4√2
(B) 3√2
(C) 2√3
(D) 3√3

(vi) To divide a line segment AB in the ratio 2:5, first a ray AX is drawn so that ∠BAX is an acute angle and then at equal distances points are marked on the ray such that the minimum number of these points is : (1)
(A) 2
(B) 5
(C) 4
(D) 7

(vii) The perimeter of the triangle formed by the points (-3, 0), (2, 0) and (0, -5) is : (1)
(A) \(\sqrt{7}+\sqrt{11}+\sqrt{31}\)
(B) \(\sqrt{11}+\sqrt{23}+\sqrt{37}\)
(C) \(\sqrt{25}+\sqrt{29}+\sqrt{34}\)
(D) \(\sqrt{12}+\sqrt{29}+\sqrt{35}\)

(viii) Given that sin θ = \(\frac{a}{b}\), then cos θ is equal to : (1)
(A) \(\frac{b}{\sqrt{b^{2}-a^{2}}}\)
(B) \(\frac{b}{a}\)
(C) \(\frac{\sqrt{b^{2}-a^{2}}}{b}\)
(D) \(\frac{a}{\sqrt{b^{2}-a^{2}}}\)

(ix) If 2 cos 2x = 1, then x is equal to : (1)
(A) 30°
(B) 90°
(C) 0°
(D) 60°

(x) For the following distribution : (1)

The modal class is:
(A) 10 – 20
(B) 20 – 30
(C) 30 – 40
(D) none of these

(xi) In an arranged discrete series in which total number of observations n is even, median is: (1)
(A) \(\left(\frac{n}{2}\right)^{t h}\) terms
(B) \(\left(\frac{n}{2}+1\right)^{t h}\) terms
(C) The mean of \(\left(\frac{n}{2}\right)^{t h}\) terms and \(\left(\frac{n}{2}+1\right)^{t h}\) terms
(D) none of these

(xii) The probability of getting a defective pen in a lot of 144 pens is \(\frac{1}{12}\) . The number of defective pens in the lot is: (1)
(A) 14
(B) 16
(C) 10
(D) 12

Question 2.
Fill in the blanks :
(i) If the pair of equations cx – y = 2 and 6x – 2y = 3 will have infinitely many solutions, then the value of c will be ________ (1)

(ii) 15th term of the AP x – 7, x – 2, x + 3, ________ is. (1)

(iii) In construction, the scale factor is used to construct ________ triangles. (1)

(iv) The point (2,-3) lies in ________ quadrant. (1)

(v) If sin θ = cos θ, then θ is equal to ________. (1)

(vi) ________ is that value among the observations which occurs most often. (1)

Question 3.
Very Short Answer Type Questions:
(i) Find the HCF of 504 and 1188. (1)

(ii) How many zeroes the polynomial ax⁴ + bx³ + cx² + dx+ c has at most ? (1)

(iii) If sum and product of zeroes are 2 and 4 respectively then find the quadratic polynomial. (1)

(iv) If 4x – 3y = 9 and 2x + ky = 11 has unique solution, then find the value of k. (1)

(v) Find the value of k, for which x = 3 is a solution of the quadratic equation x² – kx +6 = 0. (1)

(vi) If 2 is a root of equation x² +px + 12 = 0 and quadratic equation x² + px + q = 0 has equal roots, then find q. (1)

(vii) Draw a line segment of length 7.6 cm and divide ¡tin the ratio 5 : 8 (1)

(viii) Find the value of k, where distance (3, k) and (4, 1) is \(\sqrt{10}\) unit. (1)

(ix) Find the value of sin θ, cos (90° – θ) + cos θ sin (90° – θ). (1)

(x) Find the value of( A + B) if tan A cot B. (1)

(xi) Write the relation between mean, mode and median. (1)

(xii) A card is drawn from a packet of 100 cards numbered 1 to 100. Find the probability of drawing a number which is a perfect square. (1)

Section – B

Question 4.
Explain why 7 × 5 × 3 × 2 + 3 is a composite number. (2)

Question 5.
Find a quadratic polynomial, the sum and product of whose zeroes arc -3. 2 respectively. (2)

Question 6.
Solve the following pair of equations by substitution method: 7x – 5y = 2 and x + 2y = 3. (2)

Question 7.
The sum of two numbers is 9. 1f thc sum of their reciprocals is find the numbers. (2)

Question 8.
Find the 8th term of the AP: 95,91. 87, 83 ………………… (2)

Question 9.
Find the sum of first 100 even natural numbers. (2)

Question 10.
Draw a triangle ABC in which AB = 5 cm, BC = 6 cm and ∠ABC = 60°. Construct a triangle similar to ∆ABC with scale factor \(\frac{5}{7}\) Justify the construction. (2)

Question 11.
Draw a pair of tangents to a circle of radius 4.2 cm which are inclined to each other at an angle of 30. (2)

Question 12.
If3 tan 2θ = √3, find the value of θ. (2)

Question 13.
Prove that : (cosec θ – cot θ)² = \(\frac{1-\cos \theta}{1+\cos \theta}\) (2)

Question 14.
Calculate the median of the ftllowing data : (2)

Question 15.
The mean of the following distribution is 53. Find the missing frequency p: (2)

Question 16.
A child has a die whose six faces shows letters as A, B. C, D, F. A. The die is thro once. Find the probability of getting
(i)A. (ii.) D? (2)

Section – C

Question 17.
If the mth term of an A.P is \(\frac{1}{n}\) and the nth term is \(\frac{1}{m}\), show that the sum of mn terms \(\frac{1}{2}\)(mn + 1). (3)
OR
In an AP, the first team is -4. the last term is 29 and the sum of all its terms is 150, Find the common difference of the AP. (3)

Question 18.
If the points A (x, y). B (3, 6) and C (-3, 4) are collinear, show that x – 3y + 15 = 0. (3)
OR
Find the area of ∆ABC whose vertices are A ( 5, 7), B (-4, -5) and C (4, 5). (3)

Question 19.
Prove that:
\(\frac{\sin A}{{cosec} A+\cot A}+\frac{\sin A}{{cosec} A-\cot A}\) = 2 (3)
OR
Prove that:
sec² θ – \(\frac{\sin ^{2} \theta-2 \sin ^{4} \theta}{2 \cos ^{4} \theta-\cos ^{2} \theta}\) = 1 (3)

Question 20.
Weekly incomes of 600 families is tabulated below: (3)

Calculate the median weekly wages of the workers. (3)
OR
The mean of the following distribution is 22. Find the missing frequency: (3)

Section – D

Question 21.
In a parking place, the parking charge of 5 cars and 4 scooters is ₹ 37and parking charge of 6 cars and 3 scooters is ₹ 39. Represent this situation algebraically and graphically. (4)
OR
Solve each of the following system of linear equations graphically. Also tind the coordinates of the points where the lines meets x-axis in each systems x + 2y – 5 = 0 and 2x – 3y + 4 = 0. (4)

Question 22.
Draw a triangle ABC with BC =6 cm, ∠C = 30° and ∠A = 105°. Then construct another triangle whose sides are \(\frac{2}{3}\) times the corrsponding sides of ∆ABC. (4)
OR
Draw a circle of radius 4 cm. Construct a pair of tangents to it, the angle between which is 60°. Also justify the construction. Measure the distance between the centre of the circle and point of intersection of tangents. (4)

Question 23.
Draw an ogive and the cumulative frequency polygon for the following frequency distribution by less than method: (4)

OR
A student noted the number of cars passing through a spot on a road for 47 periods each of 2 minutes and summarised it in the table given below. Find the mode of the data. (4)