Testing Theme: Creative Worksheets

Standard 10 Mathematics - Common Quarterly Examination - September 2024

PART-I

Marks: 14x1=14

Answer all the questions.

1) If \(n(A \times B) = 6\) and \(A = \{1, 3\}\) then \(n(B)\) is

a) 1

b) 2

c) 3

d) 6

2) Let \(A = \{1, 2, 3, 4\}\) and \(B = \{4, 8, 9, 10\}\). A function \(f: A \rightarrow B\) given by \(f = \{(1, 4), (2, 8), (3, 9), (4, 10)\}\) is a

a) Many one function

b) Identity function

c) One to one function

d) Into function

3) Given \(F_1 = 1, F_2 = 3\) and \(F_n = F_{n-1} + F_{n-2}\) then \(F_5\) is

a) 3

b) 5

c) 8

d) 11

4) The next term of the sequence \(\frac{3}{16}, \frac{1}{8}, \frac{1}{12}, \frac{1}{18}, ...\) is

a) \(\frac{1}{24}\)

b) \(\frac{1}{27}\)

c) \(\frac{2}{3}\)

d) \(\frac{1}{81}\)

5) The square root of \(\frac{256x^8y^4z^{10}}{25x^6y^6z^6}\) is equal to

a) \(\frac{16}{5} \frac{x^2z^4}{y^2}\)

b) \(\frac{16}{5} \frac{y^2}{x^2z^4}\)

c) \(\frac{16}{5} \frac{y}{xz^2}\)

d) \(\frac{16}{5} \frac{xz^2}{y}\)

6) \(\sqrt{a^2x^2 + 2abx + b^2}\) square root

a) \(|ax-b|\)

b) \(|ax+b|\)

c) \(|ab+x|\)

d) \(|ab-x|\)

7) If \(\triangle ABC\) is an isosceles triangle with \(\angle C = 90^\circ\) and \(AC = 5 \text{ cm}\), then AB is

a) 2.5 cm

b) 5 cm

c) 10 cm

d) \(5\sqrt{2}\) cm

8) The slope of the line which is perpendicular to a line joining the pts (0, 0) and (-8, 8) is

a) -1

b) 1

c) \(\frac{1}{3}\)

d) -8

9) If \((\sin\alpha + \csc\alpha)^2 + (\cos\alpha + \sec\alpha)^2 = K + \tan^2\alpha + \cot^2\alpha\), then the value of K is equal to

a) 9

b) 7

c) 5

d) 3

10) If \(x = a \tan\theta\) and \(y = b \sec\theta\) then

a) \(\frac{y^2}{b^2} - \frac{x^2}{a^2} = 1\)

b) \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\)

c) \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\)

d) \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 0\)

11) In \(\triangle LMN, \angle L = 60^\circ, \angle M = 50^\circ\). If \(\triangle LMN \sim \triangle PQR\) then the value of \(\angle R\) is

a) 40°

b) 70°

c) 30°

d) 110°

12) If in triangles ABC and EDF, \(\frac{AB}{DE} = \frac{BC}{FD}\) then they will be similar when

a) \(\angle B = \angle E\)

b) \(\angle A = \angle D\)

c) \(\angle B = \angle D\)

d) \(\angle A = \angle F\)

13) If \(3\sqrt{x} = 9\) find the value of 'x'

a) 3

b) 9

c) 27

d) \(\sqrt{3}\)

14) \(\sqrt{3} \sin\theta - \cos\theta = 0\) then find the value of 'θ'

a) 0°

b) 30°

c) 45°

d) 60°

PART-II

Marks: 10x2=20

Answer any 10 questions. Question No. 28 is compulsory.

15) Define function.

16) \(A = \{0, 1\}, B = \{0, 1\}, C = \{0, 1\}\) then find \((A \times B) \times C\)

17) Find f.g. and g.f when \(f(x) = 2x+1\) and \(g(x) = x^2-2\)

18) Find the sum \(3+1+\frac{1}{3}+.........\infty\)

19) Solve: \(x+y = 1, x-y = 3\).

20) The perimeters of two similar triangles ABC and PQR are respectively 36 cm and 24 cm. If PQ = 10 cm, find AB.

21) Find the distance between from the points (3, 4) (5, 5)

22) Find the equation of a line which passes through (5, 7) and makes intercepts on the axes equal in magnitude but opposite in sign.

23) Prove that \(\frac{\sec\theta}{\sin\theta} - \frac{\sin\theta}{\cos\theta} = \cot\theta\)

24) If the Highest Common Factor of 210 and 55 is expressible in the form \(55x-325\) find x.

25) If the first term of an infinite G.P. is 8 and its sum to infinity is \(\frac{32}{3}\) then find the common ratio.

26) IF \(\triangle ABC \sim \triangle DEF\) such that area of \(\triangle ABC\) is 9 cm² and the area of \(\triangle DEF\) is 16 cm² and BC = 2.1 cms. Find the length of EF.

27) Find the square root \(361x^4y^2\)

28) Area of a rectangle \(\frac{(x-4)(x+3)}{3x-12}\) km², length \(\frac{x-3}{3}\) km then find breath.

PART-III

Marks: 10x5=50

Answer any 10 questions. Q.No. 42 is compulsory.

29) A function 'f' is defined by \(f(x) = 2x-3\)

(i) find \(\frac{f(0) + f(1)}{2}\)

(ii) find x such that \(f(x) = 0\)

(iii) find x such that \(f(x) = x\)

(iv) find x such that \(f(x) = f(1-x)\)

30) \(f(x) = x^2, g(x) = 2x\) and \(h(x) = x+4\) prove that fo(goh) = (fog)oh

31) If the function \(f:R \rightarrow R\) is defined by \(f(x) = \begin{cases} 2x+7 & x < -2 \\ x^2-2 & -2 \le x < 3 \\ 3x-2 & x \ge 3 \end{cases}\) then find the value of

(i) \(f(4)\)

(ii) \(f(-2)\)

(iii) \(f(4)+2f(1)\)

(iv) \(\frac{f(1)-3f(4)}{f(-3)}\)

32) The sum of three consecutive terms that are in A.P. is 27 and their product is 288. Find the three terms.

33) The product of three consecutive terms of a Geometric Progression is 343 and their sum is \(\frac{91}{3}\). Find the three terms.

34) Find the sum of n terms of the series 5+55+555+...

35) Solve: \(3x-2y+z=2, 2x+3y-z=5, x+y+z=6\).

36) Find the square root of \(64x^4-16x^3+17x^2-2x+1\).

37) Simplify: \(\frac{b^2+3b-28}{b^2+4b+4} \div \frac{b^2-49}{b^2-5b-14}\)

38) Two poles of height 'a' metres and 'b' metres are 'p' metres apart. Prove that the height of the point of intersection of the lines joining the top of each pole to the foot of the opposite pole is given by \(\frac{ab}{a+b}\) metres.

39) Find the area of the quadrilateral (-9, 0) (-8, 6) (-1, -2) and (-6, -3)

40) Find the slope and y intercept of \(\sqrt{3}x+(1-\sqrt{3})y = 3\).

41) If \(cosec\theta + \cot\theta = P\) then Prove that \(\cos\theta = \frac{P^2-1}{P^2+1}\)

42) If \(x^4+3x^3+5x^2+26x+56\) and \(x^4+2x^3-4x^2-x+28\) HCF is \(x^2+5x+7\) find it LCM.

PART-IV

Marks: 2x8=16

Answer the following.

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

(OR)

Construct a \(\triangle PQR\) such that QR = 6.5 cm, \(\angle P = 60^\circ\) and the altitude from P to QR is of length 4.5 cm.

44) Draw the graph of \(xy = 24. x, y > 0\) using the graph find

(i) y when x = 3 and (ii) x when y = 6

(OR)

Varshika drew 6 circles with different sizes. Draw a graph for the relationship between the diameter and circumference of each circle as shown in the table and use it to find the circumference of a circle when its diameter is 6 cm.

Diameter x cm	1	2	3	4	5
Circumference y cm	3.1	6.2	9.3	12.4	15.5