# Board Paper of Class 10 2006 Maths - Solutions

(1) Check the question paper for fairness of printing. If there is any lack of fairness, inform the Hall Supervisor immediately.(2) Use Blue or Black ink to write and underline and pencil to draw diagrams.Note : This question paper contains four sections.

- Question 1
If A = {
*p, q, r, s*}, B = {*r, s, t, u*}, then A\B =

(a) {*p, q*}

(b) {t, u}

(c) {r, s}

(d) {*p, q, r, s*} VIEW SOLUTION

- Question 2
The 8
^{th}term of the sequence 1, 1, 2, 3, 5, 8, ... is :

(a) 25

(b) 24

(c) 23

(d) 21 VIEW SOLUTION

- Question 3
1 + 2 + 3 + ... +
*n*=*k*then 1^{3}+ 2^{3}+ 3^{3}+ ... +*n*^{3}is equal to:

(a)*k*^{2}

(b)*k*^{3}

(c) $\frac{k\left(k+1\right)}{2}$

(d) (*k*+ 1)^{3}VIEW SOLUTION

- Question 4
The system of equations
*x*– 4*y*= 8, 3*x*– 12*y*= 24 :

(a) has infinitely many solutions

(b) has no solution

(c) has a unique solution

(d) may or may not have a solution VIEW SOLUTION

- Question 5
If
*ax*^{2}+*bx + c*= 0 has equal roots, then '*c*' is equal to :

(a) $\frac{{b}^{2}}{2a}$

(b) $\frac{{b}^{2}}{4a}$

(c) $-\frac{{b}^{2}}{2a}$

(d) $-\frac{{b}^{2}}{4a}$ VIEW SOLUTION

- Question 6
If $\mathrm{A}=\left(\begin{array}{cc}4& -2\\ 6& -3\end{array}\right)$, then A
^{2}is:

(a) $\left(\begin{array}{cc}16& 4\\ 36& 9\end{array}\right)$

(b) $\left(\begin{array}{cc}8& -4\\ 12& -6\end{array}\right)$

(c) $\left(\begin{array}{cc}-4& 2\\ -6& 3\end{array}\right)$

(d) $\left(\begin{array}{cc}4& -2\\ 6& -3\end{array}\right)$ VIEW SOLUTION

- Question 7
The equation of a straight line parallel to
*y*-axis and passing through the point (–2, 5) is :

(a)*x*– 2 = 0

(b)*x*+ 2 = 0

(c)*y*+ 5 = 0

(d)*y*– 5 = 0 VIEW SOLUTION

- Question 8
The equation of a straight line having slope 3 and
*y*– intercept – 4 is:

(a) 3*x**– y*– 4 = 0

(b) 3*x**+ y*– 4 = 0

(c) 3*x**– y +*4 = 0

(d) 3*x**+ y*+ 4 = 0 VIEW SOLUTION

- Question 9
In ΔABC, DE is || to BC, meeting AB and AC at D and E. If AD = 3 cm, DB = 2 cm and AE = 2.7 cm, then AC is equal to :

VIEW SOLUTION

(a) 6.5 cm

(b) 4.5 cm

(c) 3.5 cm

(d) 5.5 cm

- Question 10
AB and CD are two chords of a circle which when produced to meet at a point P such that AB = 5 cm, AP = 8 cm and CD = 2 cm then PD =

(a) 12 cm

(b) 5 cm

(c) 6 cm

(d) 4 cm VIEW SOLUTION

- Question 11
$\left(1+{\mathrm{tan}}^{2}\mathrm{\theta}\right).{\mathrm{sin}}^{2}\mathrm{\theta}=$

(a) sin^{2}θ

(b) cos^{2}θ

(c) tan^{2}θ

(d) cot^{2}θ VIEW SOLUTION

- Question 12
In the adjoining figure, $\mathrm{sin}\mathrm{\theta}=\frac{15}{17}$. Then BC =

(a) 85 m

(b) 65 m

(c) 95 m

(d) 75 m VIEW SOLUTION

- Question 13
The total surface area of a solid hemisphere of diameter 2 cm is equal to:

(a) 12 cm^{2}

(b) 12π cm^{2}

(c) 4π cm^{2}

(d) 3π cm^{2}VIEW SOLUTION

- Question 14
If
*t*is the standard deviation of*x, y, z*, then the standard deviation of*x*+ 5,*y*+ 5,*z*+ 5 is :

(a) $\frac{t}{3}$

(b)*t*+ 5

(c)*t*

(d)*xyz*VIEW SOLUTION

- Question 15
A bag contains 5 black balls, 4 white balls and 3 red balls. If a ball is selected at random, the probability that it is not red is:

(a) $\frac{5}{12}$

(b) $\frac{4}{12}$

(c) $\frac{3}{12}$

(d) $\frac{3}{4}$ VIEW SOLUTION

- Question 16
Given A = {
*a*,*x*,*y*,*r*,*s*}, B = {1, 3, 5, 7, –10} verify the commutative property of set union. VIEW SOLUTION

- Question 17
A = {5, 6, 7, 8} ; B = {–11, 4, 7, –10, –7, –9, –13} and
*f*= {(*x, y*) :*y*= 3 – 2*x**, x*∈ A,*y*∈ B}

(i) Write down the elements of*f*.

(ii) What is the range? VIEW SOLUTION

- Question 18

- Question 19
Simplify: $\frac{{x}^{3}}{x-2}+\frac{8}{2-x}$ VIEW SOLUTION

- Question 20
If α and β are the roots of the equation 3
*x*^{2}– 6*x*+ 4 = 0, find the value of α^{2}+ β^{2}. VIEW SOLUTION

- Question 21
Find the product of the matrices, if exists.

$\left(\begin{array}{ccc}2& 9& -3\\ 4& -1& 0\end{array}\right)\left(\begin{array}{cc}4& 2\\ -6& 7\\ -2& 1\end{array}\right)$ VIEW SOLUTION

- Question 22
Find the values of
*x, y*and*z*if $\left[\begin{array}{c}x+y\\ y+z\\ z-5\end{array}\right]=\left[\begin{array}{c}7\\ 9\\ 0\end{array}\right]$ VIEW SOLUTION

- Question 23
If the centroid of a triangle is at (1, 3) and two of its vertices are (–7, 6) and (8, 5) then find the third vertex of the triangle. VIEW SOLUTION

- Question 24
Find the value of '
*a*', if the straight lines 5*x*– 2*y*– 9 = 0 and*ay*+ 2*x*– 11 = 0 are perpendicular to each other. VIEW SOLUTION

- Question 25
In the figure TP is a tangent to the circle. A and B are two points on the circle. If ∠BTP = 72° and ∠ATB = 43° find ∠ABT.

VIEW SOLUTION

- Question 26
If ΔABC is right angled at 'C', then find the values of cos (A + B) and sin (A + B). VIEW SOLUTION

- Question 27
Curved surface area and circumference at the base of a solid right circular cylinder are 4400 sq. cm and 110 cm respectively. Find its height and diameter. VIEW SOLUTION

- Question 28
If the coefficient of variation of a collection of data is 57 and its S.D. is 6.84, then find the mean. VIEW SOLUTION

- Question 29
20 cards are numbered from 1 to 20. One card is drawn at random. What is the Probability that the number on the card is:

(i) A multiple of 4

(ii) Not a multiple of 6 VIEW SOLUTION

- Question 30
(a) The central angle and radius of a sector of a circular disc are 180° and 21 cm respectively. If the edges of the sector are joined together to make a hollow cone, then find the radius of the cone.

OR

(b) Prove that: $\frac{1+\mathrm{sec}\mathrm{\theta}}{\mathrm{sec}\mathrm{\theta}}=\frac{{\displaystyle {\mathrm{sin}}^{2}}{\displaystyle}{\displaystyle \mathrm{\theta}}}{1-\mathrm{cos}\mathrm{\theta}}$ VIEW SOLUTION

- Question 31
A radio station surveyed 190 students to determine the types of music they liked. The survey revealed that 114 liked rock music, 50 liked folk music and 41 liked classical music, 14 liked rock music and folk music, 15 liked rock music and classical music, 11 liked classical music and folk music, 5 liked all the three of music.

Find:

(i) How many did not like any of the 3 types?

(ii) How many liked any two types only

(iii) How many liked folk music but not rock music? VIEW SOLUTION

- Question 32
A function
*f*: [–3, 7) →**R**is defined as follows: $f\left(x\right)=\left\{\begin{array}{ccc}4{x}^{2}-1& ;& -3\le x<2\\ 3x-2& ;& 2\le x\le 4\\ 2x-3& ;& 4x7\end{array}\right.$

Find :

(i)*f*(–2) –*f*(4)

(ii) $\frac{f\left(3\right)+f\left(-1\right)}{2f\left(6\right)-f\left(1\right)}$ VIEW SOLUTION

- Question 33
Find the sum to
*n*terms of the series 7 + 77 + 777 + ..... VIEW SOLUTION

- Question 34
If 7 times the 7
^{th}term of an Arithmetic Progression is equal to 11 times its 11^{th}term, show that its 18^{th}term is zero. Can you find the first term and the common difference? Justify your answer. VIEW SOLUTION

- Question 35
Factorize: ${x}^{3}-5{x}^{2}-2x+24$ VIEW SOLUTION

- Question 36
Simplify: $\frac{1}{{x}^{2}+3x+2}+\frac{1}{{x}^{2}+5x+6}-\frac{2}{{x}^{2}+4x+3}$ VIEW SOLUTION

- Question 37
If $\mathrm{A}=\left(\begin{array}{cc}3& 2\\ -1& 4\end{array}\right);\mathrm{B}=\left(\begin{array}{cc}-2& 5\\ 6& 7\end{array}\right)\mathrm{and}\mathrm{C}=\left(\begin{array}{cc}1& 1\\ -5& 3\end{array}\right)$ verify that A(B + C) = AB + AC. VIEW SOLUTION

- Question 38
Find the area of the quadrilateral whose vertices are (–4, 5), (0, 7), (5, –5) and (–4, –2). VIEW SOLUTION

- Question 39
State and prove Thales theorem. VIEW SOLUTION

- Question 40
A person in a helicopter flying at a height of 700 m, observes two objects lying opposite to each other on either banks of a river. The angles of depression of the objects are 30° and 45°; find the width of the river. $\left(\sqrt{3}=1.732\right)$ VIEW SOLUTION

- Question 41
The radii of two circular ends of a frustum shaped bucket are 15 cm and 8 cm. If its depth is 63 cm, find the capacity of the bucket in litres. $\left(\mathrm{Take}\mathrm{\pi}=\frac{22}{7}\right)$ VIEW SOLUTION

- Question 42
A circus tent is to be erected in the form of a cone surmounted on a cylinder. The total height of the tent is 49 m. Diameter of the base is 42 m and height of the cylinder is 21 m. Find the cost of canvas needed to make the tent, if the cost of canvas is Rs 12.50/m
^{2}. $\left(\mathrm{Take}\mathrm{\pi}=\frac{22}{7}\right)$ VIEW SOLUTION

- Question 43

- Question 44
A box contains 4 white balls, 6 red balls, 7 black balls and 3 blue balls. One ball is drawn at random from the bag. Find the probability that the ball drawn is:

(i) neither white nor black

(ii) red or white, and

(iii) either white or red or black or blue VIEW SOLUTION

- Question 45
(a) If the equation $\left(1+{m}^{2}\right){x}^{2}+2mcx+{c}^{2}-{a}^{2}=0$ has equal roots, then prove that ${c}^{2}={a}^{2}\left(1+{m}^{2}\right)$.

OR(b) Find the equation of the straight line segment whose end points are the points of intersection of the straight lines

2VIEW SOLUTION*x*– 3*y*+ 4 = 0,*x*– 2*y*+ 3 = 0 and the midpoint of the line joining the points (3, –2) and (–5, 8).

- Question 46
(a) Draw the two tangents from a point which is 9 cm away from the centre of a circle of radius 3 cm. Also, measure the lengths of the tangents.

OR

(b) Construct a cyclic quadrilateral PQRS given PQ = 5 cm, QR = 4 cm, ∠QPR = 35° and ∠PRS = 70°. VIEW SOLUTION

- Question 47
(a) Draw the graph of
*y*=*x*^{2}+ 2*x*– 3 and hence find the roots of*x*^{2}–*x*– 6 = 0.

OR(b)No. of workers *x*3 4 6 8 9 16 No. of days *y*96 72 48 36 32 18

Draw graph for the data given in the table. Hence find the number of days taken by 12 workers to complete the work. VIEW SOLUTION