MP Board Notes
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89. Choose and write the correct options : Let A = {1, 2, 3), then number of relations containing (1, 2) and (1, 3) which are reflexive and
90. Fill in the blanks : (i) Every differentiable function is ………..... (ii) Area lying in the first quadrant and bounded by the circle x2 + y2 = 4
91. Write true / false in the following statements : (i) If A and B are two events such that P(A) ≠ 0 and P(BIA)=1, then A ⊂ B.
92. Match the correct pairs : (i) ∫ cot x dx = (a) 1/a tan-1 x/a + c
93. Give answers in one word / sentence each : (i) What is the rate of change of the area of a circle with respect to its radius / at / = 6 cm ?
94. Find X and Y, if X+Y= [(7 0,2 5)] and X+Y= [(7 0,2 5)]
95. Compute the indicated product :
96. Examine whether the function f given by f(x) = x2 is continuous at x = 0.
97. Find dy / dx, if x–y = Π
98. Evaluate :
99. Evaluate : ∫(1-x) √x dx
100. Find the angle between two vectors a ⃗ and b ⃗ with magnitudes 1 and 2 respectively and when a ⃗.b ⃗=1.
101. Find a vector in the direction of vector a ⃗=i ̂-2j ̂ that has magnitude 7 units.
102. Find the direction cosines of x, y and z-axis.
103. Find the equation of the line which passes through the point (1, 2, 3) and is parallel to the vector
104. A stone is dropped into a quiet lake and waves move in circles at a speed of 4 cm per second. At the instant, when the radius of the
105. Use differential to approximate √36.6.
106. Find two positive numbers whose sum, is 15 and the sum of whose squares is minimum.
107. If the radius of a sphere is measured as 9 cm with an error 0.03 cm, then find the approximate error in calculating its volume.
108. Find a unit vector perpendicular to each of the vectors (a ⃗+b ⃗) and (a ⃗–b ⃗), where
109. Show that the point A(-2i ̂+3j ̂+5k ̂),B(i ̂+2j ̂+3k ̂ )and C(7i ̂-k ̂)are collinears.
110. Find the angle between the line x+1 / 2 = y/3 = z–3/6 and the plane 10x + 2y–11z = 3.
111. Show that the lines x + 3 / –3 = y – 1 / 1 = z – 5 / 5 and x + 1 / - 1 = y – 2 / 2 = z – 5 / 5 are coplanar.
112. Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} as ' R={(a, b): b = a +1} is reflexive, symmetric or transitive.
113. Show that * : R×RR defined by a*b = a + 2b is neither commutative nor associative.
114. Express tan^(-1)[(cosx/(1-sinx)),-3π/2
115. Show that sin^(-1)[3/5-sin^1 [8/17=cos^(-1) [84/85]]]
116. Show that A=[(2 3,1 -4)]and B=[ (1 -2,-1 3)] then verify that (AB)–1= B–1 A–1
117. Prove that :
118. Find the distance of a point (2, 5, –3) from the plane
119. Find the vector and Cartesian equations of the plane which passes through the point (5, 2, 4) and perpendicular to the line with direction
120. Solve the following linear programming problem graphically :
121. A cooperative society of farmers has 50 hectares of land to grow two crops X and Y. The profit from crops X and X per hectare are
122. Of the students in a college, it is known that 60% reside in hostel and 40% are day scholars (not residing in hostel). Previous year results
123. An unbiased die is thrown twice. Let the event A be "Odd number on the first throw" and B the event "Odd number on the second throw".
124. From a lot of 30 bulbs which include 6 defective, a sample of 4 bulbs is drawn at random with replacement. Find the probability
125. Ten cards numbered 1 to 10 are placed in a box, mixed up thoroughly and then one card is drawn randomly. If it is known that the
126. Express the matrix B=[(2 -2 -4,-1 3 4,1 -2 -3)]as the sum of a symmetric and a skew symmetric matrix.
127. By using elementary operations, find the inverse of the matrix A=[(2 1,1 1)].
128. Verily Rolle's theorem for the function f(x) = x2 + 2x – 8, x = ∈ [-4, 2]
129. Differentiate ax w.r.t. x, where a is a positive constant.
130. Evaluate ∫^(π/2) 0 [sin^4 x/sin^4[x+cos^4 x] dx]
131. Evaluate ∫〖((x^2+1) e^x)/(x+1)^2 dx〗
132. Using integration find the area of region bounded by the triangle whose vertices are (1, 0), (2, 2) and (3,1).
133. Find the area of the parabola y2 = 4 ax bounded by its latus rectum.
134. Solve the differential equation dy/dx + y/x = x2
135. Find the particular solution of the differential equation log(dy/dx) = 3x+4y given that y=0 when x=0.