# Board Paper of Class 10 2015 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
*f*: A → B is a bijective function and if*n*(A) = 5, then*n*(B) is equal to:

(a) 10

(b) 4

(c) 5

(d) 25 VIEW SOLUTION

- Question 2
The next term of $\frac{1}{20}$ in the sequence $\frac{1}{2},\frac{1}{6},\frac{1}{12},\frac{1}{20}$, ...... is:

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

(b) $\frac{1}{22}$

(c) $\frac{1}{30}$

(d) $\frac{1}{18}$ VIEW SOLUTION

- Question 3
The common ratio of the G.P. ${a}^{m-n},{a}^{m},{a}^{m+n}$ is :

(a)*a*^{m}

(b)*a*^{–m}

(c)*a*^{n}

(d)*a*VIEW SOLUTION^{–n}

- Question 4

- Question 5
*b = a + c*and the equation*ax*^{2}+*bx + c*= 0 has equal roots then :

(a)*a = c*

(b)*a = – c*

(c)*a*= 2*c*

(d)*a*= – 2*c*VIEW SOLUTION

- Question 6
If $\left(\begin{array}{cc}3x+7& 5\\ y+1& 2-3x\end{array}\right)=\left(\begin{array}{cc}1& y-2\\ 8& 8\end{array}\right)$, then the value of
*x*and*y*respectively are:

(a) –2, 7

(b) $-\frac{1}{3},7$

(c) $-\frac{1}{3},-\frac{2}{3}$

(d) 2, –7 VIEW SOLUTION

- Question 7
The centroid of the triangle with vertices at (–2, –5), (–2, 12) and (10, –1) is :

(a) (6, 6)

(b) 4, 4)

(c) (3, 3)

(d) (2, 2) VIEW SOLUTION

- Question 8
The value of
*k*if the straight lines 3*x*+ 6*y*+ 7 = 0 and 2*x*+*ky*= 5 are perpendicular is:

(a) 1

(b) – 1

(c) 2

(d) $\frac{1}{2}$ VIEW SOLUTION

- Question 9
If the sides of two similar triangles are in the ratio 2 : 3, then their areas are in the ratio:

(a) 9 : 4

(b) 4 : 9

(c) 2 : 3

(d) 3 : 2 VIEW SOLUTION

- Question 10
If the tangents PA and PB from an external point P to a circle with centre O are inclined to each other at an angle of 40° then ∠POA =

(a) 70°

(b) 80°

(c) 50°

(d) 60° VIEW SOLUTION

- Question 11
If
*x = a*sec θ,*y*=*b*tan θ, then the value of $\frac{{x}^{2}}{{a}^{2}}-\frac{{y}^{2}}{{b}^{2}}=:$

(a) 1

(b) –1

(c) tan^{2}θ

(d) cosec^{2}θ VIEW SOLUTION

- Question 12

- Question 13
The ratios of the respective heights and the respective radii to two cylinders are 1 : 2 and 2 : 1 respectively. Then their respective volumes are in the ratio:

(a) 4 : 1

(b) 1 : 4

(c) 2 : 1

(d) 1 : 2 VIEW SOLUTION

- Question 14

- Question 15
Probability of getting 3 heads and 3 tails in tossing a coin 3 times is :

(a) $\frac{1}{8}$

(b) $\frac{1}{4}$

(c) $\frac{3}{8}$

(d) $\frac{1}{2}$ VIEW SOLUTION

- Question 16
Draw Venn diagram A ⋃ (B ⋂ C). VIEW SOLUTION

- Question 17
Let A = {1, 2, 3, 4, 5}, B = N and
*f*: A → B be defined by*f*(*x*) =*x*^{2}. Find the range of*f*. Identify the type of function. VIEW SOLUTION

- Question 18
Find the first term and common difference of the A.P. $\frac{1}{2},\frac{5}{6},\frac{7}{6},\frac{3}{2},...........,\frac{17}{6}$ VIEW SOLUTION

- Question 19
Simplify $\frac{5x+20}{7x+28}$. VIEW SOLUTION

- Question 20
If α and β are the roots of 3
*x*^{2}– 5*x*+ 2 = 0, then find the value of $\frac{\mathrm{\alpha}}{\mathrm{\beta}}+\frac{\mathrm{\beta}}{\mathrm{\alpha}}$. VIEW SOLUTION

- Question 21
$\mathrm{A}=\left(\begin{array}{ccc}8& 5& 2\\ 1& -3& 4\end{array}\right)$ then find A
^{T}and (A^{T})^{T}. VIEW SOLUTION

- Question 22
If $\mathrm{A}=\left(\begin{array}{cc}2& 3\\ -9& 5\end{array}\right)-\left(\begin{array}{cc}1& 5\\ 7& -1\end{array}\right)$ then find the additive inverse of A. VIEW SOLUTION

- Question 23
The centre of a circle is at (–6, 4). If one end of a diameter of the circle is at the origin, then find the other end. VIEW SOLUTION

- Question 24

- Question 25

- Question 26
A ramp for unloading a moving truck has an angle of elevation of 30°. If the top of the ramp is 0.9 m. above the ground level, then find the length of the ramp. VIEW SOLUTION

- Question 27
If the circumference of the base of a solid right circular cylinder is 154 cm and its height is 16 cm, find its curved surface area. VIEW SOLUTION

- Question 28
Find the range and the coefficient of range of 43, 24, 38, 56, 22, 39, 45. VIEW SOLUTION

- Question 29
A bag contains 6 white balls numbered from 1 to 6 and 4 red balls numbered from 7 to 10.

A ball is drawn at random. Find the probability of getting:

(i) an even numbered ball.

(ii) a white ball. VIEW SOLUTION

- Question 30
(a) Prove the following identity. $\sqrt{{\mathrm{sec}}^{2}\mathrm{\theta}+{\mathrm{cosec}}^{2}\mathrm{\theta}}=\mathrm{tan}\mathrm{\theta}+\mathrm{cot}\mathrm{\theta}$

OR(b) The thickness of a hemispherical bowl is 0.25 cm. The inner radius of the bowl is 5 cm. Find the outer curved surface area of the bowl $\left(\mathrm{Take}\mathrm{\pi}=\frac{22}{7}\right)$. VIEW SOLUTION

- Question 31
Use Venn diagrams to verify De Morgan's law of complementation (A ⋃ B)' = A' ⋂ B'. VIEW SOLUTION

- Question 32
Let A = {4, 6, 8, 10} and B = {3, 4, 5, 6, 7} and
*f*: A → B be defined by $f\left(x\right)=\frac{1}{2}x+1$, then represent*f*by

(i) an arrow diagram

(ii) a set of ordered pairs and

(iii) a table VIEW SOLUTION

- Question 33
Find the sum of first
*n*terms of the series 6 + 66 + 666 + .............. VIEW SOLUTION

- Question 34
The sum of four consecutive terms of an A.P is 20 and the sum of their squares is 120. Find those numbers. VIEW SOLUTION

- Question 35

- Question 36
Find the square root of : $4+25{x}^{2}-12x-24{x}^{3}+16{x}^{4}$ VIEW SOLUTION

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

- Question 38
Prove that (0, 5) (–2, –2), (5, 0) and (7, 7) are the vertices of a rhombus. VIEW SOLUTION

- Question 39
If all sides of a parallelogram touch a circle, show that the parallelogram is a rhombus. VIEW SOLUTION

- Question 40
If tan θ + sin θ =
*m*, tan θ – sin θ =*n*and*m ≠ n*, then show that ${m}^{2}-{n}^{2}=4\sqrt{mn}$. VIEW SOLUTION

- Question 41
A sector containing an angle of 120° is cut off from a circle of radius 21 cm. and folded into a cone by joining the radii. Find the curved surface area of the cone. $\left(\mathrm{\pi}=\frac{22}{7}\right)$ VIEW SOLUTION

- Question 42
An iron right circular cone of diameter 8 cm and height 12 cm is melted and recast into spherical lead shots each of radius 4 mm. How many lead shots can be made? VIEW SOLUTION

- Question 43
The following table shows the marks obtained by 48 students in a quiz competition in mathematics. Calculate the standard deviation.

Data *x*6 7 8 9 10 11 12 Frequency *f*3 6 9 13 8 5 4

- Question 44
The probability that a new car will get an award for its design is 0.25, the probability that it will get an award for efficient use of fuel is 0.35 and the probability that it will get both the awards is 0.15. Find the probability that :

(i) it will get at least one of the two awards.

(ii) it will get only one of the awards. VIEW SOLUTION

- Question 45
Find the L.C.M. [Least Common Multiple] of the following:

${x}^{3}+{y}^{3},{x}^{3}-{y}^{3},{x}^{4}+{x}^{2}{y}^{2}+{y}^{4}$ORFind the equation of the line whose gradient is $\frac{3}{2}$ and which passes through P, where P divides the line segment joining A(–2, 6) and B(3, –4) in the ratio 2 : 3 internally. VIEW SOLUTION

- Question 46
(a) Draw a circle of diameter 10 cm. From a point P, 13 cm. away from its centre, draw the two tangents PA and PB to the circle and measure their lengths.OR(b) Construct a ΔABC in which the base BC = 5 cm, ∠BAC = 40° and the median from A to BC is 6 cm. Also measure the length of the altitude from A. VIEW SOLUTION

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

OR

(b) A bank gives 10% S.I (Simple Interest) on deposits for senior citizens.Draw the graph for the relation between the sum deposited and the interest earned for one year.VIEW SOLUTION

Hence find

(i) The interest on the deposit of Rs 650

(ii) The amount to be deposited to earn an interest of Rs 45.